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Eberl Equation Native F&4 _952155587tive F[W[WF` Ole53510 F' PIC  (L MTimes MPolish!ߣ##f#ff_!zMUDMASTER: A PROGRAM FOR CALCULATING CRYSTALLITE SIZE DISTRIBUTIONS AND STRAIN FROM THE SHAPES OF X-RAY DIFFRACTION PEAKS D. D. Eberl D. D. Eberlࡱ; vtrpnljhf | z [ _..... 37 Figure 1: Illite....................................................................................... 37 Figure 2: 2-glycol smectite......................................................................... 38 Figure 3: 2-Water Smectite ........................................................................ 38 Figure 4: Na-1-Water Smectite.................................................................... 39 Figure 5: Mg-Tri-Tri-Chlorite .................................................................... 39 Figure 6: Pyrophyllite.............................................................................. 40 Figure 7: Talc........................................................................................ 40 Figure 8: Kaolinite.................................................................................. 41 Figure 9: Serpentine................................................................................ 41 Figure 10: PVP-smectite + illite................................................................... 42 Appendix 4: Order of Calculation and Key Equations................................................ 43 1. Choosing the XRD peak....................................................................... 43 2. Correction of intensities for LpG2............................................................ 44 3. Removal of background........................................................................ 44 4. Removal of K-alpha 2 component of radiation.............................................. 45 5. Flipping the peak................................................................................ 47 6. Fourier analysis of interference function maximum........................................ 47 7. Removal of instrumental broadening......................................................... 48 8. Correction of Fourier coefficients for strain................................................. 49 9. Calculation of mean crystallite size and distribution........................................ 50 10. Calculation of lognormal parameters........................................................ 51 11. Calculation of volume......................................................................... 52 Appendix 5: How to get the latest version of MudMaster by FTP................................... 53 MUD MASTER The muddy rivers of spring Are snarling Under muddy skies. The mind is muddy. As yet, for the mind, new banks Of bulging green Are not; Sky-sides of gold Are not. The mind snarls. Blackest of pickanines, There is a master of mud. The shaft of light Falling, far off, from sky to land, That is he-- The peach-bud maker, The mud master, The master of the mind. Wallace Stevens MUDMASTER: A PROGRAM FOR CALCULATING CRYSTALLITE SIZE DISTRIBUTIONS AND STRAIN FROM THE SHAPES OF X-RAY DIFFRACTION PEAKS _________________________________________ D. D. Eberl, V. Drits, J. rodo, and R. Nesch ______________________________________________________________ INTRODUCTION Particle size may strongly influence the physical and chemical properties of a substance (e.g. its rheology, surface area, cation exchange capacity, solubility, etc.), and its measurement in rocks may yield geological information about ancient environments (sediment provenance, degree of metamorphism, degree of weathering, current directions, distance to shore, etc.). Therefore mineralogists, geologists, chemists, soil scientists, and others who deal with clay-size material would like to have a convenient method for measuring particle size distributions. Nano-size crystals generally are too fine to be measured by light microscopy. Laser scattering methods give only average particle sizes; therefore particle size can not be measured in a particular crystallographic direction. Also, the particles measured by laser techniques may be composed of several different minerals, and may be agglomerations of individual crystals. Measurement by electron and atomic force microscopy is tedious, expensive, and time consuming. It is difficult to measure more than a few hundred particles per sample by these methods. This many measurements, often taking several days of intensive effort, may yield an accurate mean size for a sample, but may be too few to determine an accurate distribution of sizes. Measurement of size distributions by X-ray diffraction (XRD) solves these shortcomings. An X-ray scan of a sample occurs automatically, taking a few minutes to a few hours. The resulting XRD peaks average diffraction effects from billions of individual nano-size crystals. The size that is measured by XRD may be related to the size of the individual crystals of the mineral in the sample, rather than to the size of particles formed from the agglomeration of these crystals. Therefore one can determine the size of a particular mineral in a mixture of minerals, and the sizes in a particular crystallographic direction of that mineral. A shortcoming of the XRD technique is that the size that is measured is the X-ray scattering domain size (crystallite size), which may or may not be equal to crystal size. Therefore, it is important to check the XRD measurements by another method. Comparisons between crystallite thicknesses measured by XRD and crystal thicknesses measured by other methods indicate that these values are the same for fundamental illite crystals (Eberl et al., 1998). The XRD method is based on the observation that XRD peaks are broadened regularly as a function of decreasing crystallite size. This effect permits accurate measurement of mean crystallite sizes for periodic crystals that range from about 2 nm to about 100 nm. The upper limit for size determination depends significantly on the accuracy of instrumental standards. Reflections for non-periodic systems (for example, basal reflections of irregular mixed-layer structures as well as hkl reflections of minerals containing stacking faults) can not be analyzed by this approach. The crystallite size is equal to  EMBED Equation.2 , where N is the number of hkl planes responsible for a reflection. This expression takes into account the fact that layers and planes are not the same, because layers have a thickness. For example, the analysis of kaolinite  EMBED Equation.2  peaks will give crystallite dimensions perpendicular to the ab plane of the unit cell. If the mean number of diffracting planes is 10, then the number of layer spacings is 9, and the mean thickness is 9 x 7.2 = 64.8 . The term strain refers to small fluctuations in the d-spacing of the substance. These d-spacings may be distributed symmetrically or asymmetrically about the substances mean d-spacing, which corresponds to symmetrical and asymmetrical strain. This instruction manual covers how to use the program. The theory for its operation is described by Drits et al. (1998), and is based on the previous work of Bertaut (1950) and Warren and Averbach (1950) that was applied to diffraction by metals. The program is available by FTP (see Appendix 5), from ddeberl@usgs.gov, or by writing D. D. Eberl, U.S. Geological Survey, 3215 Marine Street, Boulder, Colorado, USA, 80303-1066. SYSTEM REQUIREMENTS AND DISCLAIMER MudMaster will run under either IBM or Macintosh systems. Use of MudMaster requires Microsoft Excel, version 5.0 or greater (the program also is available in version 4.0), and an elementary knowledge of how to use the Excel program (pasting data, changing the axis on a chart, etc.). MudMaster works best on a computer having 16 megabytes or more of RAM. Ten or more megabytes should be assigned to run the Excel program if your system offers an option to assign memory to a program. The program occupies about 4.6 Mb of disk space. Although this program has been used by the USGS, no warranty, expressed or implied, is made by the USGS or the United States Government as to the accuracy and functioning of the program and related program material nor shall the fact of distribution constitute any such warranty, and no responsibility is assumed by the USGS in connection herewith. STRUCTURE OF MUDMASTER The workbook MudMaster (MudMastr.xlw) is composed of one macrosheet (1-mm.xls) linked to one worksheet (2-mm.xls). In addition, the macrosheet PeakPicker (pp.xls) also is linked to 1-mm.xls. These sheets perform the following functions: PeakPicker. This macrosheet accepts pasted XRD intensity data as a function of two-theta angle. It is used to inspect the X-ray pattern, to choose the peak to be analyzed, and to calculate an approximate mean thickness for the crystallites by an integral peak width method. The XRD peak may be chosen over a two-theta range that is either symmetrical with respect to the peak maximum, or over any two-theta range that is specified. After the XRD peak has been chosen, PeakPicker automatically can transfer the chosen XRD intensity data into MudMaster and start the MudMaster program. 1-mm.xls (hereafter sheet 1). This macrosheet processes the XRD intensities of the peak to be analyzed. These intensities can be pasted directly into this sheet, or can be transferred automatically from PeakPicker. Sheet 1 corrects the XRD peak intensity distribution for the Lorentz-polarization factor (Lp) and layer scattering intensity (G2) and removes the background, leaving the interference function, which contains information concerning crystallite size distributions and strain. Because experimental data will contain the Ka1-Ka2 doublet, the program also can remove the Ka2 component to the peak if this option is chosen. The interference function  EMBED Equation.2  maximum, plotted as a function of  EMBED Equation.2 , then is decomposed into a Fourier series. The Fourier coefficients [A(n) and B(n)] then can be corrected for instrumental broadening (if standards have been introduced into the program - see below) and analyzed for strain, if these options are chosen. Then the corrected Fourier coefficients are pasted into sheet 2, where they are analyzed further for crystallite size and strain. 2-mm.xls (hereafter sheet 2). This worksheet uses Fourier coefficients calculated in sheet 1 to perform a Bertaut-Warren-Averbach-type analysis to calculate the mean crystallite size, the crystallite size distribution, the root mean square of the strain, and the strain distributions for the sample. The final results of this analysis are displayed on this sheet, together with other useful data. The crystallite size and strain distributions may be smoothed and truncated to eliminate noise, and the mean size and the size distribution may be corrected for instrumental broadening and for symmetrical strain (if this option was chosen on sheet one). INSTALLATION OF MUDMASTER To install the program, copy MudMastr.xlw and samp.xls onto your hard disk. Make backup copies, and then open the originals. A copy must have the name of the original to run properly. If the Mac version of the program is stuffed (.sea extension), the program will decompress when opened. If the IBM version of the program is compressed (.zip extension), one should run pkunzip.exe to decompress the files. Do not change the name of the program. The correct LpG2s for the mineral and, if necessary, for the mineral edges to be analyzed must be pasted into columns K and L in sheet 1 prior to analysis. These LpG2s must have the same two-theta step size as the XRD peak to be analyzed, and should start at two degrees two-theta. The recommended LpG2s for the interior and edges of clay crystallites are listed in Table 4, Appendix 2. LpG2s calculated in steps of 0.02o two-theta are located in a file named LpG2.xls that accompanies the program. If another step size is required, then LpG2s having any step size can be calculated using the spreadsheet CALCLPG2.xls which also accompanies the program. Eventually one may want to analyze an instrumental standard, and enter the Fourier coefficients for the standard's XRD peaks into sheet 1, as is discussed below. The current version of the program contains instrumental standards that can be used with our experimental setup (Siemens D500 diffractometer, diffracted beam Sller slits, 1 divergence and 0.15 receiving slits, graphite monochromator, 40 kv and 30 ma tube current). These standards are left in the program as an example, but may not be correct for your experimental setup, and really do not work very well even using our setup. Luckily, with our experimental system, machine broadening does not appear to be a problem, for example, for illites that are 30 nm or less in mean thickness. DATA REQUIRED The XRD data should be of high quality (use long count times, a monochromator, a stable generator, excellent equipment adjustment, properly prepared specimen, etc.) and, generally, should be collected in steps of 0.02 two-theta, although any step size can be accommodated by entering the correct LpG2s into columns K and L, sheet 1, as is discussed above. The program picks the XRD peak position from the peaks maximum intensity; therefore, a peak with a noisy top may be picked in the wrong position. Data should be entered as increasing angles of two-theta. The program can analyze an XRD peak that has 750 analytical points (e.g. a peak that is 15 two-theta broad, having 0.02 steps), and can accept an XRD pattern that contains 2900 analytical points (e.g. 2 to 60 two-theta with 0.02 steps). EXAMPLES Example 1: A Simple Analysis Sample inputs are summarized in Appendix 1, Tables 1-3. Example 1 is the simplest example of how to analyze a sample. The intensities to be analyzed were calculated using the NEWMOD program (Reynolds, 1985) for an 003 reflection for an illite having a lognormal crystallite thickness distribution and a mean crystallite thickness of 10 nm. The peak was calculated for Cu K-alpha radiation only (i.e. no Ka1- Ka2 doublet), and does not contain instrumental broadening. The steps for analysis are: 1. Copy the data for sample LOGNOR, 10 from the samp.xls sheet. This is done by clicking the heading for this column (column B) and by choosing the Copy command under the Edit menu. These intensities have been calculated for the 003 illite reflection for a two-theta range of 23.5 to 30.1, for illite crystallites having a lognormal distribution. 2. Paste this data into column C on sheet 1 by clicking the column heading C on sheet 1, and then by choosing the Paste command under the Edit menu. The actual data should begin in cell C2. Pasting the entire column automatically clears old data in column C. One can move from PeakPicker to sheet 1 to sheet 2 by clicking the tabs in the lower left corner of the screen. 3. Open the file LpG2.xls, copy the columns containing the LpG2s for illite (0.89K) and for prophyllite and paste them into columns K and L in sheet 1, respectively. The illite LpG2 is for the crystal interiors, and the pyrophyllite LpG2 is for the crystal edges. The use of crystal edges is an approximation, and it would be better to calculate an LpG2 for an illite having a reduced K-content according to the proportion of edges (equivalents K =  EMBED Equation.2 , where  EMBED Equation.2  is the mean thickness, in this case 10 nm; see Drits et al., 1998), and use these values in column K. 4. Fill out the INPUT PARAMETERS in columns A and B as follows: Sample name: Lognormal 10 Input this name in cell A3. Starting two-theta: 23.5 Input this number in cell A5. It is the starting two-theta angle for the peak to be analyzed which in this case is the same as the starting two-theta for the XRD pattern. Ending two-theta: 30.1 Input this number in cell A7. This number is the ending two-theta angle for the peak to be analyzed. If the ending two-theta is chosen beyond the end of the intensity data set, the program will not run. The intensity data should contain no zeros. Step: 0.02 Input this number in cell A9. It must match the step size of the LpG2s that have been entered into columns K and L. Increment in n for Fourier analysis: 1 Input this number in cell A11. Generally use 1 for clays, although it is sometimes necessary to use a larger value for non-clays that have a small d and a large crystallite size. Calculate what maximum thickness for the crystallites? 50 Input this number in cell A13. Normally a thickness at least 5 times the mean thickness should be entered here. This value is used to calculate the maximum n to be used in the Fourier analysis for the crystallite size distribution. Using a small thickness saves calculation time, generally does not affect the calculated mean thickness, but may truncate the thickness distribution if the entered value is too small. Approximate Mean Thickness (nm): 10 Input this number in cell A15. This program can be iterated to find this number (see below), or the approximate mean found in PeakPicker (see below) can be entered. This mean is used to calculate the proportion of edges for the LpG2 calculation. Reflection order: 3 Input this number in cell A17. The XRD reflection to be analyzed is the illite 003. Mineral: 1 Input this number in cell A22. For example, 1 is the index number for illite. The program uses this input to decide which way to flip the peak when the Autoflip option is employed (see below). Expandability (%): 0 Input this number in cell A24. This is the expandability of illite/smectite measured from XRD peak positions of a glycol-solvated sample prior to K-saturation and dehydration. The input is used to calculate the proportion of crystal edges. Because the calculated pattern is simply illite, a zero is entered. Wavelength detector sees (): 1.5418 Input this number in cell A26 for Cu Ka radiation. If the experimental setup includes a monchromator that removes Ka2 radiation, then input the value for Ka1 here. Wavelength K-alpha 1 (): 1.54051 Input this number in cell A28 for Cu Ka1 radiation. Wavelength K-alpha 2 (): 1.54433 Input this number in cell A28 for Cu Ka2 radiation. K-alpha 2/K-alpha 1 intensity ratio?: 0.5 This ratio, normally 0.5, is input in cell A30. It is used in Ka2 removal, and will be inactive in the present calculation. Correct intensities for LpG2? 2 Input this number in cell B6. If zero is entered, no LpG2 correction is made; 1 = correction made only for the interiors of the crystals (the LpG2s for which should have been pasted into column K); 2 = correction made for the interiors and edges of the crystals (the LpG2 for which should have been pasted into column L): 3 = correction is made only for the edges (this option generally is not used). Calculate symmetrical strain? 0 Input this number in cell B8. It is possible to calculate symmetrical strain only if two or more reflections are analyzed. Use of this option will be discussed in a following section. Remove Ka2 radiation? 0 Input this number in cell B10. The NEWMOD patterns are calculated using the Ka wavelength; therefore, there is no need to remove a Ka2 component. Removing the Ka2 component slows the calculation. Normally, it is adequate to use Ka radiation, even for patterns of natural materials, if the mean crystallite size is less than about 20 nm, and if the two-theta for the peak is less than about 50. Do the flip? (0 = no; 1 = left to right; 2 = right to left; 3 = autoflip): 3 Input this number in cell B13. With this option one can restrict analysis to half of the interference function if the other half is disturbed. Theoretically, the interference function is symmetrical; therefore, the other half is generated by rotating half of the function around a vertical axis passing through the interference functions maximum intensity. The rotation can be either from left to right (enter "1") or from right to left (enter "2"). If a "3" is entered, the program automatically will choose the correct flip direction for the clay to be analyzed, according to Table 4 in Appendix 2. The flipping option may be used if the minerals layer scattering intensity is equal to zero (or extremely small) in the vicinity of the XRD peak, or if XRD intensities from other minerals obscure half of the peak. The flip option will be discussed further in Example 2 below. Iterate calculation? 0 Input this number in cell B15. Entering a 1 will cause the program to iterate until the approximate mean, previously entered into cell A15 and updated during the calculation, equals the calculated mean, thereby insuring that the program used the proper proportion of edges when calculating LpG2. Other information: NEWMOD calculated pattern Input this information in cell B17. Update screen? 0 Input this number in cell B19. Calculation is faster if the screen is not updated. The screen automatically will update at the end of the calculations. Input a number other than zero to watch the program make the calculations. 5. Click the button labeled, Start MudMaster. 6. The results will appear on sheet 2 in less than a minute when using a 8500/120 Power Macintosh computer. The results can be printed by choosing Print in the file menu. The results are as follows: Sample name: The sample name, Lognormal 10, appears in cell A2. The mineral chosen for analysis, ILLITE, is given in cell H1. Best mean (nm; extrapolated): 10.0 This area-weighted, mean crystallite size is calculated, according to the theory of Warren and Averbach (1950), by extrapolation of the steepest slope of a plot of the corrected Fourier coefficients [H(S)] versus S to an H(S) of zero (see plot labeled Extrapolated Mean in sheet 2). The extrapolated mean is said to be the best mean because it is unaffected by smoothing or by ripples that sometimes affect the mean calculated from the distribution, and also because it is unaffected by the maximum thickness entered in cell A13 on sheet 1. Mean (nm; distribution): 10.0 This mean is calculated from the distribution that is shown in the chart labeled, "Thickness (Area-weighted frequency)." It should equal the best mean. Its value may change if one changes the smoothing power or the distribution limit (see below). The solid curve given in the distribution figure is for a theoretical lognormal curve that has been calculated from the measured distribution. Alpha (from nm): 2.16 This parameter, which is mean of the natural logarithms of the crystallite sizes, is calculated from the distribution. Beta^2: 0.29 This parameter, which is the variance of the natural logarithms of the crystallite sizes, is calculated from the distribution. Volume-weighted mean (nm3): 13.3 This volume weighted mean is calculated from the distribution (see Appendix 4). Position interference function (two-theta): 26.80 This value corresponds to the position of the peak intensity of the interference function 003 maximum. d-spacing (): 3.327 This value corresponds to the 003 maximum of the interference function. Root mean square strain (): Not calculated The strain calculation requires at least two peaks for analysis, and therefore was not calculated. It will be discussed in more detail below. Approximate fundamental particle thickness (nm): 10.0 This value is calculated from the approximate mean thickness and expandability entered on sheet 1 according to an equation given by Drits et al. (1998). If the expandability entered into sheet 1 is greater than zero, then this value refers to the mean thickness of fundamental illite particles found in mixed-layer illite/smectite. Otherwise this value is the same as the approximate mean thickness entered into sheet 1. Calculation time (min; from PeakPicker) minutes This value is number of minutes the calculation took from the start of the program PeakPicker. This value has no meaning for the present calculation because PeakPicker was not used. Hidden knobs set at recommendation?: Yes A yes indicates that all of the recommended settings are used on sheet 1, cells E5 to E29. These hidden knobs and switches are used in program development, and normally should not be changed. Cells H25:H52: These cells repeat the values entered on sheet 1. 7. One can input parameters on sheet 2 to refine the results of the calculations for the crystallite size distribution and the strain: Dist. Limit? (nm): Input 50 in B2. This option can be used to truncate the distribution at a chosen crystallite size. It is used to eliminate ripples and noise at large sizes in the distribution. If the distribution is disturbed by such noise, the distribution limit can be adjusted so as to reduce the difference between the distribution mean given in cell H5 and the extrapolated mean given in cell H3. Smooth power: Input 1 in B4. This option smoothes the crystallite size, strain and volume distributions. The larger the number entered, the greater the smoothing. A small smoothing power (e.g. 0 to 1) is best, because large smoothing powers will distort the distributions. The distributions are smoothed by using a moving average that is centered on the number to be smoothed. A smoothing power of 1 includes one number on either side of the number to be smoothed, and therefore averages 3 numbers; a smoothing power of 2 includes two numbers on either side, and therefore averages 5 numbers to calculate the smoothed number; etc. Symmetrical strain correction?: Input 0 in D2. If a 1 is entered here, and if at least two XRD peaks related by n in Braggs law (e.g. the 001 and the 002; or the 114 and the 228) have been analyzed for a sample under the strain option (cell B8 in sheet 1), the means given in cells H3 and H5, and the distributions given in the charts are corrected for the effects of symmetrical strain. Prior to correction, symmetrical strain causes reflections to give progressively smaller mean crystallite sizes with increasing reflection order. The strain calculation is discussed in more detail in Example 3, below. Instrumental correction?: Input 0 in D4. If a 1 is entered here, the mean size and distributions are corrected for instrumental broadening, provided instrumental standards have been entered into columns CS to DB in sheet 1 (see below). A zero is entered here for this analysis because NEWMOD calculated patterns do not contain machine broadening. Normally, correction for instrumental broadening is not needed if the mean crystallite size is less than about 30 nm. Correction for instrumental broadening may distort distributions. This procedure has not been fully refined. 8. The results can now be printed by pushing the printer button on the Tools Bar, or by choosing Print... in the File menu. Example 2: Using PeakPicker and doing the flip The following steps indicate how to use PeakPicker in an analysis of the kaolinite 001 reflection. Clear column E in macro sheet pp.xls, and then copy the data for sample GA KAOLINITE from the samp.xls sheet, and paste it into column E in PeakPicker. The first data point should be entered into cell E2, and the sample name should be in cell E1. 2. Copy the kaolinite LpG2 from the spread sheet LpG2.xls into column K on sheet 1. 3. Fill in the INPUT in columns B and C in pp.xls as follows: Step size (deg. two-theta): 0.02 Input this number in cell B4. This value is the two-theta angle step size at which the data was collected. It normally is 0.02 two-theta. Update screen?: 0 Input this number in cell B6. Entering zero yields the faster calculation time. Enter 1 to watch the calculation take place. Plot pattern? (1 = yes): 1 Input this number in cell B8. With this option set to 1, the intensity data that was input in column E will be plotted as a function of two-theta angle in a chart to be labeled with the samples name. Pick symmetrical peak? (1 = yes): 1 Input this number in cell B10. If 1 is chosen, the program automatically will pick a peak that is symmetrical in two-theta with respect to the peak maximum. In other words, the number of degrees two-theta between the left side of the peak (to be entered as the exact starting angle in cell C8) and the peak maximum will be used for the right side of the peak. If this cell is not set equal to one, the two-theta range for the peak will be chosen between the angles entered into the exact starting angle (cell C8) and the ending angle (cell C10). For either option, if the ending two-theta is chosen beyond the end of the intensity data set, the program will not run. Calculate approximate mean? 2 Input this number in cell B13. A 1 is entered to calculate an approximate thickness by the integral peak width method of Drits et al. (1997; developed for illite); 2 and 3 assume an asymptotic or lognormal shape to the distribution, respectively, and use the equations relating integral peak width and thickness given in Figure 8 in Eberl et al. (1998). These equations, developed for illite fundamental particle thicknesses, frequently work for other clays. d 001 (): 7.2 Input this number in cell B15. This input is used to calculate the approximate thickness using an integral peak width method. Sample name: GA Kaolinite Do not enter anything in cell C4. The sample name automatically will be entered here from the top of column E when the program is started. Starting angle for all intensities: 2 Input this number in cell C6. The XRD pattern for this kaolinite starts at 2 two-theta. Exact starting angle for analysis: 6 Input this number in cell C8. This two-theta angle is the left side of the peak to be analyzed. See Table 4 in Appendix 2 for the recommended settings for kaolinite, or see the chart in pp.xls by scrolling down. End angle for analysis: 13 Input this number in cell C10. This two-theta angle approximately marks the right side of the peak to be analyzed. Reflection order: 1 Input this number in cell C12. We are analyzing the kaolinite 001 reflection. Calculate strain?: 0 Input this number in cell C14. The strain option will be discussed below. Autopaste?: 0 Input this number in cell C17. Set it to zero to do calculations in PeakPicker only. Enter 1 to paste results from the PeakPicker calculation into sheet 1. Enter 2 to paste results into sheet 1 and to start the MudMaster program. 4. Click the button Start PeakPicker. The XRD pattern will be plotted in the chart labeled with the samples name. The kaolinite 001 reflection can be examined more closely by enlarging the chart (pull out the corner of the chart) and changing its scale (double click on the values next to the x- and y-axes). An approximate mean of 13.6 appears in cell A13. This number is not needed because the kaolinite calculation does not need to know the proportion of edges. Now set cell B8 to 0 (because the pattern already has been plotted), B10 to 0 (to avoid the minimum in LpG2 when calculating the interference function), and cell C17 to 1, so that the data will be transferred to MudMaster. Start the program again. 5. The intensity data now has been processed, and pasted into MudMaster. Now fill in the input options on sheet 1 as follows: Sample name: GA Kaolinite (Filled in automatically from PeakPicker.) Starting two-theta: 6.00 (Filled in automatically from PeakPicker.) Ending two-theta: 13.00 (Filled in automatically from PeakPicker.) Step size: 0.02 (Filled in automatically from PeakPicker.) Increment in n for Fourier analysis: 1 Calculate what maximum thickness for the crystallites?: 100 Approximate mean thickness: 13.6 This number is from the approximate mean thickness found by using PeakPicker. Actually, for this calculation this number is not needed because crystal edges are not included in the calculation. Reflection order: 1 (Filled in automatically from PeakPicker.) Mineral: 9 Expandability: 0 Wavelength detector sees (): 1.5418 Wavelength K-alpha 1 (): 1.54051 Wavelength K-alpha 2 (): 1.54433 K-alpha 2/K-alpha 1 intensity ratio?: 0.5 Correct for LpG2?: 1 Only the interior LpG2 is used for kaolinite (see recommendations in Table 4). Calc. symm. strain?: 0 (Filled in automatically from PeakPicker.) Remove K-alpha 2 radiation?: 0 Inputting a number larger than 1 in cell B10 will cause the program to separate Ka radiation into its two components by a Fourier technique (Gangulee, 1970), and the Ka1 component would be used in subsequent analysis. The number that is input will determine the maximum n in the Fourier analysis used for this separation. Generally, 300 has given good results. Inputting a zero will cause the radiation that the detector sees (normally Ka radiation) to be used. Generally, for determining clay thicknesses it is sufficient to use Ka radiation. Do the flip? 3 If one decides to flip, then one chooses the left half of the kaolinite 001 interference function maximum for analysis, because the layer scattering intensity for the right half approaches zero (Figure 8 in Appendix 3). In Figure 8, the Lorentz-polarization function (Lp) times the layer scattering intensity (G2) is plotted on top of the kaolinite XRD pattern. The intensity of an XRD peak = (Lp)(G2)(j), where j is the interference function, a function which carries the crystallite size and strain information. Division of the XRD data by LpG2 to find the interference function may lead to an unrealistically large intensity for the right half of the  EMBED Equation.2  interference function maximum and to an asymmetrical form. Therefore only the left half of the peak can be used. Table 4 and the other figures in Appendix 3 show that the problem also exists for other clay XRD peaks, for example, the illite 001 and 002 illite reflections. Entering a 1 flips the peak from left to right; entering a 2 flips it from right to left; and entering a 3 automatically flips the peak according to the recommendations given in Table 4 in Appendix 2. Iterate the calculation? 0 There is no need to iterate, because edges are not used in the calculation. Other information: Well-crystallized CMS source clay; 5 sec/step Update screen? 0 5. Start MudMaster. Using a Power Macintosh 8500/120 computer, the results of the calculation appear in less than one minute on sheet 2. The two-theta scale needs to be changed on the appropriate chart (double click on the scale to change it) in order to display the interference function. One should always check the interference function plot to see if it looks realistic. The profile of the maximum should be symmetrical, and must have a background that reaches zero on either side of the peak. The calculation gives a best mean of 16.8 nm and a mean for the distribution of 16.8 nm using a distribution limit of 65, a smoothing power of 1, and no correction for instrumental broadening. The thickness distribution shown in the chart (diamond symbols) is very different from the theoretical lognormal distribution calculated from the same data (solid curve). One also should always check the chart for the extrapolated mean to be sure that the solid line finds the steepest slope of the plotted data. If the solid line lies above the data, one should extend the range for which the program searches for the steepest slope in cells L4 and L5 in sheet 2. Example 3: Symmetrical strain analysis The following is an example of how to correct the mean crystallite size for symmetrical strain: Copy the data for sample K-Zemp-H from the samp.xls sheet, and paste it into column E in PeakPicker. This pattern is of an illite/smectite sample (Zempleni illite/smectite) that has been K-saturated, heated at 300 C overnight to collapse all smectite layers, and then X-rayed in dry air. Prior to dehydration it was 15% expandable. Copy the LpG2 for illite (0.89K) from the file LpG2.xls into column K in sheet 1. Copy the LpG2 for illite (0.40K) into column L in sheet 1. Analyze the pattern in PeakPicker as was described above for kaolinite. Choose a two-theta range and other settings according to the recommendations for illite in Table 4. Enter the settings in sheet 1 as follows: Sample name: K-Zemp-H (entered automatically from PeakPicker) Starting two-theta: 4.4 (entered automatically from PeakPicker) Ending two-theta: 9.2 (entered automatically from PeakPicker) Step size: 0.02 (entered automatically from PeakPicker) Increment in n for Fourier analysis: 1 Calculate what max thickness?: 50 Approximate mean thickness: 10.4 (calculated in PeakPicker from option 3, cell B13) Reflection order: 1 Mineral: 1 Expandability (%): 15 (measured for a glycol-solvated sample) Wavelength detector sees: 1.5418 Wavelength K-alpha 1: 1.54051 Wavelength K-alpha 2: 1.54433 K-alpha2/K-alpha 1 intensity ratio? 0.5 Correct intensities for LpG2? 2 Calc. symm. strain? 1 The succession number of the peak in the analysis is entered here (also can be entered from PeakPicker). Entering a 0 or a 1 will erase all previously stored strain data. For the second peak to be analyzed, a 2 should be entered here; for the third a 3; etc. Remove K-alpha 2 radiation? 0 Do the flip? 3 Iterate calculation?: 1 Iterate the calculation only for the 001 reflection. Switch this option off for subsequent orders. Other information: K-saturated I/S heated to 300 C overnight Update screen during calculation? 0 5. Push the start button. The results in sheet 2 give an extrapolated mean of 10.7 nm and a distribution mean of 10.7 nm, using a smoothing power of 0 and a distribution limit of 35. 6. Choose a second peak for analysis that is related to the first peak by "n"' in Braggs law. For example, analyze the 002 peak in the same manner by running PeakPicker, and by entering the following parameters in sheet 1: Sample name: K-Zemp-H (entered automatically from PeakPicker) Starting two-theta: 13.20 (entered automatically from PeakPicker) Ending two-theta: 18.00 (entered automatically from PeakPicker) Step size: 0.02 (entered automatically from PeakPicker) Increment in n for Fourier analysis: 1 Calculate what max thickness? 50 Approximate mean thickness: 10.7 Reflection order: 2 Mineral: 1 Expandability (%): 15 Wavelength detector sees: 1.5418 Wavelength K-alpha 1: 1.54051 Wavelength K-alpha 2: 1.54433 K-alpha 2/K-alpha 1 intensity ratio?: 0.5 Correct intensities for LpG2? 2 Calc symm strain? 2 Remove K-alpha 2 radiation?: 0 Do the flip? 3 Iterate calculation? 0 Other information: K-saturated I/S heated to 300 C overnight Update screen during calculation? 0 7. Start the program. The means on sheet 2 now can be corrected for the presence of symmetrical strain by entering 1 in cell D2, but it is better to continue the analysis to include the 005 reflection, analyzed in a similar fashion, to receive a more accurate value for the strain. A quartz reflection is hidden beneath the illite 003 reflection, as can be deduced from the presence of another quartz reflection at 20.84 two-theta; therefore the 003 is not suitable for analysis. After the 005 has been analyzed, the extrapolated mean thickness, not corrected for strain, equals 5.9 nm; corrected for strain the mean equals 11.1 nm. The distribution, plotted using a distribution limit of 26 and a smoothing power of 1, is somewhat distorted because a small XRD peak lies in the tail of the 005 reflection. It would have been better to flip the 005 peak from right to left, rather than from left to right, as was done under the autoflip option (cell B13, sheet 1). A right to left flip is practical for the 005 illite reflection because LpG2 does not approach zero on the right side of this peak (see Figure 1). A plot of the root mean square of the symmetrical strain as a function of layer thickness is presented on sheet 2, as are plots of lnA(n) versus , the slopes of which are used to determine the strain. The root mean square of the strain equals 0.37 if the region that the program searches for the strain (set in cells L11 and L12, sheet 2) is set at 2 to 12, which includes a plateau in the root mean square strain values. If the root mean square of the strain is constant with thickness, as it is in this analysis because there is a plateau in the previously mentioned chart, then the strain has a Gaussian distribution, the positive half of which is graphed in a chart labeled Strain Distribution in sheet 2. If there is no such plateau in the root mean square strain, then the Strain Distribution chart is meaningless. Other charts on this page also are corrected for strain if a 1 is entered into cell D2. The data for this sample listed in Table 3 of Drits et al. (1998) were collected using an earlier version of the program, and have been superseded. Other examples: MOM, HUMPS, MUM, Mt. Washington The NEWMOD calculated illite XRD patterns labeled MOM and HUMPS in the samp.xls worksheet also can be pasted into 1-mm.xls and analyzed (using Cu Ka radiation) as illite 003 reflections between 23.5 and 30.1 two-theta. The distribution for MOM (use an approximate mean crystallite thickness of 30 nm) should spell out this word, and HUMPS (use an approximate mean crystallite size of 40 nm) should give a distribution containing 14 evenly spaced and equally sized modes between thicknesses of 2 to 200 nm. These two samples can be used to study the effects of smoothing, flipping, etc. on the shapes of the distributions. The data labeled MUM (mean = 48.5 nm) and Mt. Washington (mean = 6 nm) were calculated from 2 to 50 two-theta using NEWMOD by adding together separately calculated patterns for Ka1 and Ka2 radiation in the ratio 2:1. The former distribution should spell the word MUM, whereas the latter should yield a symmetrical distribution starting at 2 nm, ending at 10 nm, with a maximum at 6 nm. This latter distribution is better calculated using an LpG2 corrected for K-content, rather than using pyrophyllite edges, as was discussed above. These patterns can be used to study the effects of Ka2 removal, reflection order, flipping, etc. on the mean size and size distribution. STANDARDS In addition to crystallite size effects and strain, XRD peaks also will be broadened by XRD instrumental effects. These effects are removed by using an instrumental standard (which should be composed of many crystallites) containing effectively infinitely large crystallites that have no strain. Instrumental broadening does not lead to serious errors if the mean crystallite size is small. Thicker crystals require an instrumental standard, and for this purpose we use the greater than 20 micrometer size fraction of the National Bureau of Standards Standard Reference Material 675, a synthetic fluorophlogopite. This size fraction was prepared by repeated (seven times) settling in water. Then a thin, oriented film was prepared for X-ray diffraction analysis by drying a suspension of the material on a non-diffracting substrate (polished silicon wafer cut perpendicular to the 100 axis). A glass slide is inappropriate as a substrate because it gives too large of an X-ray background, which tends to broaden the XRD peaks. To develop a standard for your instrument, X-ray the standard using exactly the same slit system and step size that you plan to use for crystallite size analysis. Then enter the raw XRD intensities into PeakPicker and MudMaster using Ka1 radiation, as is described above. When the MudMaster analysis has finished, copy the values for the Fourier coefficients [A(n) and B(n)], found in columns BH and BI of sheet 1, and paste them into the appropriate columns for the angular range over which the standard peak is to operate (columns CS to BD in sheet 1). Ripples and noise in the distribution for the standard at large sizes can be eliminated by setting A(n) and B(n) for these sizes equal to constants, the constants being the last useful calculated values for A(n) and B(n). Then cell D4 in sheet 2 is set equal to 1 to remove the effect of instrumental broadening for a sample. The approach for using instrumental standards has yet to be resolved satisfactorily. We run most of our analyses without instrumental standards, an approximation which gives satisfactory results up to a mean size of at least 30 nm for illites. OTHER ANALYSIS HINTS One needs to include all of the tails of the XRD peaks to be analyzed. Analysis should extend from one half the distance between the peak to be analyzed and the adjacent reflections that are related to it by n in Braggs law. Failure to include all of the tails may lead to noisy distributions, incorrect means, and to rippled interference functions. The analyzed peak needs to be free from any other factors that might broaden it, i.e. non-periodicity in the analyzed direction of the crystallites, and from the presence of interfering XRD peaks. Analysis of illite and illite/smectite samples is particularly tricky, and one should refer to the papers by Drits et al. (1998) and Eberl et al. (1998). It may be possible to calculate a good mean size for many samples even if G2 is not known by using option 11 in cell A22, sheet 1. The LpG2 function for many clays can be calculated by using the accompanying program CALCLPG2.xls. The program yields area-weighted mean sizes for X-ray scattering domains. These domains may or may not equal true particle size; therefore, it is wise to verify the XRD measurements by another technique. As was discussed previously, this verification has been made for illite fundamental particle thickness measurements. For these minerals the X-ray scattering domains equal fundamental illite particle thicknesses (Eberl et al., 1998). ACKNOWLEDGMENTS We thank John Neil and Gene Whitney for reviewing this manuscript and the program. REFERENCES Bertaut, F., 1950, Raies de Debye-Scherrer et rpartition des dimensions des domaines de Bragg dans les poudres polycristallines: Acta Crystallographica, v. 3, p. 14-18. Drits, V. A., rodo, J., and Eberl, D. D., 1997, XRD measurement of the mean crystallite thickness of illite and illite/smectite: reappraisal of the Kubler index and the Scherrer equation: Clays & Clay Minerals, v. 45, p. 461-475. Drits, V. A., Eberl, D. D., and rodo, J., 1998, XRD measurement of the mean thickness, thickness distribution and strain for illite and illite/smectite crystallites by the Bertaut-Warren-Averbach technique: Clays & Clay Minerals, v. 46, p. 38-50. Eberl, D. D., Nesch, R., Sucha, V., and Tsipursky, S., 1998, Measurement of the thickness of fundamental illite particles by X-ray diffraction using PVP-10 intercalation: Clays & Clay Minerals, v. 46, p. 89-97. Eberl, D. D., rodo, J., Kralik, M., Taylor, B., and Peterman, Z. E., 1990, Ostwald ripening of clays and metamorphic minerals, Science, v. 248, p. 474-477. Gangulee, A., 1970, Separation of the a1- a 2 doublet in X-ray diffraction profiles: Journal of Applied Crystallography, v. 3, p. 272-277. Klug, H. P. and Alexander, L. E., 1974, X-ray Diffraction Procedures for Polycrystalline and Amorphous Materials, second edition, John P. Wiley and Sons, Inc., N.Y., 966 p. Reynolds, R.C., Jr., 1985, NEWMOD, a Computer Program for the Calculation of One-Dimensional Diffraction Patterns of Mixed-Layered Clays: R. C. Reynolds, Jr., 8 Brook Dr., Hanover, New Hampshire, 03755. Warren, B. E., and Averbach, B. L., 1953, The effect of cold-work distortion on X-ray patterns: Journal of Applied Physics, v. 21, p. 595-599. APPENDIX 1: SUMMARY OF PROGRAM INPUTS Table 1. Summary of PeakPicker (pp.xls) Inputs. InputCellValueCommentsStep size (degrees 2-theta)B42-theta valueAnalytical step size (usually 0.02 two-theta).Update screen?B60 or 1Input "0" for faster calculation, "1" to watch calculation.Plot pattern?B81 = yesInput "1" plots entire XRD pattern in chart labeled with the samples name.Pick symm. peak? B101 = yes"1" picks XRD peak symmetrically with respect to starting angle (C8) and peak maximum; otherwise the peak is picked between angles entered in cells C8 and C10.Calc approx mean?B130 = no; or 1, 2 or 3.Calculates an approximate thickness (see cell A13) by various integral peak width methods. 1= DSE method; 2 = assumes asymptotic shape to size distribution; 3 = assumes lognormal shape to size distribution.d 001 ()B1510 for illite; 7.2 for kaolinite; etc.This value is used in the calculation of the approximate crystallite thickness by an integral peak width method (see cell B13).Sample nameC4Name of sampleEntered automatically from cell E1.Start angle for all intensitiesC62-theta value2-theta value for start of XRD patternExact start angle for analysisC82-theta valueExact starting angle (left, low-angle side) of XRD peak to be analyzed. Be sure to include all of peak's tail (see Table 4). End angle for analysisC102-theta valueApproximate end angle of this peak. C10 must be greater than C8.Reflection orderC12Whole numberOrder of reflection analyzed.Calc strain?C140, 1, 2, 3, 4 or 50 = no; or give analysis number, up to a maximum of 5.Autopaste?C170, 1 or 2"0" stays in pp.xls program; "1' pastes results in 1-mm.xls; "2" pastes and starts 1-mm.xls. Table 2. Summary of MudMaster Sheet 1 (1-mm.xls) Inputs. InputCellValueCommentsSample nameA3sample nameFilled in automatically if PeakPicker is used.Starting two-thetaA52-theta value2-theta for start (left side) of XRD peak to be analyzed; filled in automatically if PeakPicker is usedEnding two-thetaA72-theta value2-theta for end (right side) of XRD peak to be analyzed; filled in automatically if PeakPicker is used.Step sizeA9degrees 2-thetaBe sure that step size of LpG2, entered into columns K and L, sheet 1, is the same as this entry. Filled in automatically if PeakPicker is used.Increment in n for FourierA11whole numberNormally use 1.Calculate what max thickness?A13whole numberEnter the largest thickness to be calculated for the crystallite size distribution.Approximate mean thicknessA15whole number, in nmApproximate mean crystallite size for the size distribution. See cell A13, in pp.xls.Reflection orderA17whole numberLargest common denominator for the indices of the analyzed peakMineralA221 through 12This entry is used to decide which way to flip the interference function if the Autoflip option is used in cell B13.Expandability (%)A240 to 100Enter expandability of mixed-layer illite/smectite prior to K-saturation and dehydration. Set at 0 for illite.Wavelength detector seesA26in -unitsUsually K-alpha (example, Cu Ka = 1.5418 ), unless Ka2 has been eliminated by using a monochromator. Wavelength K-alpha 1A28in -unitsWavelength to be used in MudMaster analysis. For example, Cu Ka1 = 1.54051 .Wavelength K-alpha 2A30in -unitsWavelength to be used in MudMaster analysis. For example, Cu Ka2 = 1.54433 .K-alpha 2/K-alpha 1 intensity ratioA27normally 0.5Use 0.5, unless you have found a different value using the method of Gangulee (1970).Correct intensities for LpG2?B60, 1 or 20 = do not correct XRD intensities for LpG2, 1 = correct only for interiors (column L) of crystals; 2 = correct for interiors and edges (columns L and K) of crystals; 3 = correct only for edges (column K). Be sure to input correct LpG2s into columns K and L.Calculate symm. strain?B80 through 50 turns this option off. Otherwise input the number of the peak in the strain analysis to a maximum of 5 peaks. A 0 or a 1 erases previously stored strain data.Remove K-alpha 2 radiation?B10>0 = yesGenerally, Ka radiation can be used for clays. Input, for example, 300 to set maximum n for Fourier analysis.Do the flip?B130, 1, 2 or 3"0" uses the analytical interference function for subsequent Fourier analysis; "1" takes half this peak and flips it left to right; "2" flips it right to left. "3" automatically flips peak in direction given in Table 4, Appendix 2. Iterate calculation?B151 = yesIterates calculation until approximate mean in cell A15 = calculated mean.Other informationB17 InformationUpdate screen?B191 = yesProgram runs faster if screen is not updated. It will update automatically at the end. Table 3. Summary of MudMaster Sheet 2 (2-mm.xls) Inputs. InputCellValueCommentsDistribution Limit?B2in nm Sets the limit to the distribution charts. Used to eliminate ripples and noise at large sizes.Smooth PowerB4whole number Increasing this number increases the degree of smoothing. Best to use a small number (0 to 1)Strain Correction?D20 = no; 1 = yesIf more than one peak has been analyzed for symmetrical strain (cell B8 in sheet 1), then entering a "1" will apply the strain correction to the means and the distributions. The peaks also will be corrected for instrumental broadening, if cell E7 in sheet 1 is set equal to 1.Instrument correction?D41 = yesInputting "1" will cause the mean and distribution to be corrected for instrumental broadening, if proper standards have been entered in 1-mm.xls. Strain corrected means will not be corrected for instrumental broadening unless a 1 was entered into cell E7, sheet 1, prior to calculation. APPENDIX 2: RECOMMENDED SETTINGS FOR CLAY ANALYSES Table 4. Recommended Settings for Analysis of Clay Basal Reflections Mineral:Reflection Order:Flip2-theta of max:Two-theta Range:Pick symm. peak? (1 = yes):Illite 001L to R8.8o4.4o to 9.2o0PVP-illite*, and K-sat, dehyd I/S** 002 003*** 004 005****L to R R to L L to R L to R17.7o 26.8o 36.0o 45.3o13.3o to 18o 25o to 31.52 31.5o to 36.4o 40.9o to 46o0 0 0 0Kaolinite and serpentine 001 002 003L to R None None12.3o 24.7o 37.5o6o to 13o 18o to 26o 31o to 44o0 0 12-glycol smectite 001 002 003 005L to R R to L R to L R to L5.2o 10.47o 15.7o 26.4o2o to 9o 10o to 13.1o 15o to 18.4o 26o to 29o0 0 0 02-H2O smectite 001 003 005L to R L to R L to R5.9o 17.7o 29.8o2.9o to 6o 14.7o to 18o 26.8o to 30o0 0 01-H2O smectite 001 002 004L to R R to L L to R7.1o 14.2o 28.8o3.5o to 8o 14o to 17.7o 25.2o to 29o0 0 0 Mg-tri-trichlorite 001 002 003 004 005 007L to R R to L R to L R to L L to R None6.2o 12.5o 19o 25.1o 31.5o 41.4o3o to 6.5o 12o to 15.6o 18o to 22.1o 24o to 28.2o 28.3o to 32o 42o to 47.8o0 0 0 0 0 1Pyrophyllite and talc 001 002 003L to R L to R L to R9.6o 19.3o 29.1o4.5o to 10o 14.3o to 20o 24o to 30o0 0 0* See Eberl et al. (1998) ** See Drits et al. (1998). *** Also can be flipped in opposite direction, or not flipped. APPENDIX 3: PLOTS OF LpG2 AND XRD PATTERNS FOR CLAY MINERAL BASAL REFLECTIONS These patterns are presented so that one can tell whether or not the Flip option needs to be used when analyzing a peak. Flipping is needed if LpG2 approaches zero in the vicinity of the peak to be analyzed. LpG2s are plotted using the thicker lines.           APPENDIX 4: ORDER OF CALCULATION AND KEY EQUATIONS The general order of calculation is as follows: 1. Choose the XRD peak to be studied. 2. Correct intensities for LpG2. 3. Remove background. 4. Remove K-alpha 2 component of radiation (optional). 5. Flip peak (optional). 6. Perform Fourier analysis of interference function maximum. 7. Remove instrumental broadening from Fourier coefficients (optional). 8. Correct Fourier coefficients for symmetrical (optional) and asymmetrical stain (latter option not installed). 9. Analyze Fourier coefficients for mean crystallite size and crystallite size distribution. 10. Calculate lognormal parameters. 11. Calculate volume distribution. The details of the calculations are as follows: 1. Choosing the XRD Peak: Choose an XRD peak to be analyzed for which the interference function maximum is broadened only by crystallite size and/or stain effects. PeakPicker automatically copies intensities for the peak in two-theta space with respect to the chosen left side of the peak, the peak maximum, and the right side of the peak. These intensities then are pasted into MudMaster. PeakPicker also can choose a peak between any two two-theta angles (asymmetrical option). The maximum intensity from the intensity file is chosen as the peak maximum, so it is important that the top of the peak be smooth. Some commercial programs insure that the XRD peak is smooth by using a mathematical function to model the peak, and this model is used in subsequent analysis. When using this approach it is difficult or impossible to determine crystallite size distributions accurately, because the shape of the XRD peak is determined by the shape of the model. Therefore the approach taken here is to collect good data, and, should it be necessary later in the program (in sheet 2), to smooth the first and second derivatives of the Fourier coefficients when calculating crystallite size distributions (see below). An approximate mean for the peak may be calculated by integral peak width methods (Drits et al., 1997; Eberl et al., 1998), and a line where the background was chosen is plotted on the XRD pattern. 2. Correction of Intensities for LpG2: The intensity I(2q), of an X-ray reflection is calculated as:  EMBED Equation.2  where Lp is the Lorentz-polarization function, G2 is the layer scattering intensity (square of the structure factor), f is the interference function, and bg is the background. LpG2 for several clay minerals have been calculated as a function of two-theta angle in 0.02 steps and stored in the file LpG2.xls. They were calculated using the program CALCLPG2.xls, which accompanies the MudMaster program. The correct (see Table 4) LpG2s for crystal interiors and crystal edges must be pasted I sheet 1, columns K and L, respectively, prior to analysis. MudMaster calculates the proportion of crystal edges from the approximate mean thickness, or, if K-saturated, dehydrated illite/smectite is being run, from both the approximate mean thickness and the expandability (see equation 31 in Drits et al., 1998), and weights the sum of the two structure factors accordingly. MudMaster then divides LpG2 into the raw intensities to find the interference function, which contains the crystallite size and strain information of interest. 3. Removal of Background: If the peak will not be flipped, the smallest intensity values on the left and right sides of the peak are chosen to calculate a curve having an exponential shape that defines the background for the peak. If the peak is to be flipped, then a constant background is chosen from the minimum intensity on the left or right side of the peak, depending on whether the peak will be flipped from left to right or from right to left, respectively. The original intensities at each two-theta angle are corrected for this background curve by subtraction. All intensities to the left and right sides of the left and right background minimums are set equal to zero. 5. Removal of K-alpha 2 Component of the Radiation (optional): The Fourier method of Gangulee (1970) is used to remove the Ka2 component of the radiation. This method requires that the Ka1- Ka2 angular separation be known, that the Ka1 and Ka2 components have approximately the same shape, and that the ratio R be known, where R = the maximum intensity of Ka2 divided by the maximum intensity of Ka1. Gangulee also gives a method for calculating R, because R may differ from the theoretical value of 0.5, depending on the experimental setup. Gangulees method for calculating R was used in an earlier version of the program, but R always was found to equal 0.5, and therefore the method was removed from MudMaster. However, the readers may want to try this method with their experimental setup to insure that R is 0.5. Ka2 removal is programmed as follows: (a) The number of data points (N) are found: N = (2q2 - 2 q 1)/s, where s = the two-theta step size, and 2 q 1 and 2 q 2 = the beginning and ending two-theta for the interference function maximum. (b) The period for the Fourier series is calculated by transforming the two-theta axis into an axis that is labeled with respect to x, where x ranges from [-(N/2) + 1] to [(N/2) + 1], with the maximum for the interference function intensity located at x = 0. (c) Delta (D), which is the number of two-theta steps between the Ka1 and Ka2 components, is calculated from:  EMBED Equation.2 , where s = the two-theta step size, 1/  EMBED Equation.2  is the position in -1 of the interference function  EMBED Equation.2  maximum. (d) The interference function maximum intensity distribution is normalized against the sum of the intensities:  EMBED Equation.2 . (e) A Fourier analysis of the normalized interference function is carried out according to:  EMBED Equation.2 , and  EMBED Equation.2  where n is the harmonic number. The maximum n used in this Fourier analysis is set in cell B10 on sheet 1. 300 seems to work in many cases. (f) Then p(n) and q(n) are calculated according to:  EMBED Equation.2 , and  EMBED Equation.2 , where R=0.5. (g) Then the Fourier coefficients for the Ka1 peak are calculated:  EMBED Equation.2 , and  EMBED Equation.2  (h) The Fourier coefficients for the Ka1 peak are used to construct a new intensity distribution for the interference function with Ka2 radiation removed:  EMBED Equation.2 . (i) The background then is removed in a manner similar to that described above, except that a linear trend is fitted to the minimum intensity values on the left and right sides of the peak. 5. Flipping the Peak: If the structure factor approaches zero in the vicinity of an XRD peak, then dividing the intensities by LpG2 may produce an extremely large intensity for the interference function that is related to the position of the minimum in the layer scattering intensity, rather than to the peak maximum. To avoid this difficulty, one can choose to analyze only half of the XRD peak, and the other half is generated by rotating the analyzed half peak 180 around an axis that passes through the peak maximum. 6. Fourier Analysis of the Interference Function: A Fourier analysis is performed on either the Ka or the Ka1 interference function intensities. This is accomplished as follows: (a) The previously calculated intensities, I(2q), for the interference function are normalized to the sum of the intensities:  EMBED Equation.2 . .(b) The two-theta abscissa is converted into s*, where:  EMBED Equation.2 , and  EMBED Equation.2 , where the variable q for the latter equation begins and ends halfway between previous and subsequent reflection orders of the peak, with the maximum of the interference function chosen at s* = 0.  EMBED Equation.2  is calculated in a similar fashion to the latter equation at the two-theta value for the maximum of the interference function, and  EMBED Equation.2 , where  EMBED Equation.2  is calculated at the minimum two-theta value for the start of the interference function peak. (c) The maximum n (nmax) for the Fourier analysis is calculated from the input (J) in cell A13:  EMBED Equation.2 , where d is the spacing in nm of the peak maximum. The program is limited to a maximum n of approximately 200/d. (d) A Fourier analysis is performed in which:  EMBED Equation.2 , and  EMBED Equation.2 , where Fourier coefficients are calculated for values of n that increase from zero in increments set in cell A11, sheet 1. 7. Removal of Instrumental Broadening (optional): Instrumental broadening is removed from the Fourier coefficients by use of the Stokes equation, found in Klug and Alexander (1974):  EMBED Equation.2 , and  EMBED Equation.2 , where cor, samp and std refer to the Stokes corrected values, the sample and the standard, respectively. Coefficients for the standard need to have been entered into the program previously, as is discussed in the "Standards" section of this document. It has been found for illite that this correction need not be made if the main thickness is less than approximately 30 nm. 9. Correction of Fourier Coefficients for Strain (optional): The Fourier coefficients (which, for strain analysis, can be corrected for instrumental broadening by setting cell E7 on sheet 1 to 1), A(n)cor and B(n)cor, now are corrected for strain broadening, if the strain option was chosen, as follows: (a) ln[A(n)cor] is calculated for the peak under investigation. These values are stored, according to the analysis number entered into the input in sheet 1, in separate columns for each peak. The square of the reflection order,  EMBED Equation.2 , also is calculated and stored. (b) Best-fit straight lines that relate ln[A(n)cor] values to the abscissa values, , are calculated for each n. Their intercepts on the Y-axis yield the strain corrected Fourier coefficients,  EMBED Equation.2 , and the slopes of the lines yield the mean strain,  EMBED Equation.2 , for each n. On the assumption that n (or crystallite size) is small:  EMBED Equation.2 . The root mean square of the strain (also known as the strain standard deviation) is the square root of this value. It is plotted in units as a function of crystallite size for small values of crystallite size in a chart labeled, Strain Standard Deviation, in sheet 2. The crystallite thickness values over which the root mean square strain is averaged can be changed in cells L11 to L12 in sheet 2. If its value is constant with changing crystallite thickness, then the strain distribution is Gaussian, and the positive half of the strain distribution is presented on a chart labeled, Strain Distribution. If the root mean square strain is not constant with crystallite thickness, then the Strain Distribution chart has no meaning. (c) Asymmetrical strain  EMBED Equation.2  was calculated in an earlier version of MudMaster, but was later removed because it did not appear to be useful, especially using flipped peaks. It was calculated as follows:  EMBED Equation.2 . It can be plotted for  EMBED Equation.2  as a function of crystallite size (S, in units). 9. Calculation of Mean Crystallite Size and Size Distribution: Mean crystallite size and size distributions are calculated as follows: (a) The Fourier coefficients determined previously, either corrected [ EMBED Equation.2  or  EMBED Equation.2 ] or not corrected [A(n)] for strain and instrumental broadening, all hereafter referred to simply as H(n), are used to calculate the mean n (here termed  EMBED Equation.2 ), and the distribution. The first and second derivative of H(n) are taken with respect to n:  EMBED Equation.2 , and  EMBED Equation.2 . (b) The hook correction is made by plotting H(n) on the y-axis versus n. size (x-axis), where size The straight line between adjacent points that has the steepest (most negative) slope is extrapolated to n = 0, where H(n) is set equal to one. Extrapolation of this same line in the opposite direction to H(n) = 0 yields the mean crystallite size in terms of n. Mean n is converted into mean size ( EMBED Equation.2 ) by the equation: (S) =  EMBED Equation.2 . The crystallite size distribution is determined by calculating the second derivative of a plot of H(n), not corrected for the hook effect, versus n. N is then converted into size This approach for making the hook correction and for determining the mean crystallite size and the crystallite size distribution was found to be the best method by far, based on trial and error analysis of NEWMOD calculated XRD patterns. (d) The first and then the second derivative of the plot of H(n) versus crystallite size are smoothed by using a moving average centered on the value to be smoothed. The amount of smoothing is entered in the "Smooth power" option in sheet 2. Both the first and the second derivatives are smoothed. A value of "0" entered here yields no smoothing; a "1" leads to a moving average of 3 centered on the value to be smoothed; a "2" yields a moving average of 5 (two on either side of the value and the value itself), etc. Ends of the data columns are smoothed as much as possible, given the entered smoothing power. (e) The distribution is truncated by entry of a distribution limit in sheet 2. Truncation eliminates noise in the size and strain distributions at large sizes. (f) The crystallite size distribution is normalized to the sum of the frequencies, and plotted in a chart on sheet 2. (g) A mean crystallite size is calculated from the distribution:  EMBED Equation.2 , where f(S) is the frequency of a given size. 10. Calculation of Lognormal Parameters: Many natural distributions of crystallite sizes can be approximated by lognormal distributions (Eberl et al., 1990). A lognormal distribution can be completely described by two parameters, a and b2:  EMBED Equation.2 , where f(S) is the frequency of a given size, as was calculated in the previous section, and S is the crystallite size; and  EMBED Equation.2 . Then the lognormal size distribution is calculated:  EMBED Equation.2 . This theoretical distribution is superimposed as a solid line on the measured crystallite size distribution in sheet 2. 11. Calculation of Volume: The volume weighted mean size,  EMBED Equation.2 , is calculated according to:  EMBED Equation.2 . APPENDIX 5: HOW TO GET THE LATEST VERSION OF MUDMASTER BY FTP Push return after each of these steps: Log into your server. Type ftp servcolkr.cr.usgs.gov. Do not type the quotation marks. For the requested name, type anonymous. For the password, type your e-mail address. To find the PC version of MudMaster, type: cd /pub/ddeberl/pc_version. To find the MAC version, type cd /pub/ddeberl/mac_version. There is a space between cd and /pub. Type ls to see the exact file names (the first character of ls is the lowercase letter L). Type bin. Type get mudmastr.zip or what ever file you choose to copy after the get command. The file name is case sensitive. Type exit. If you can not access the program by FTP, please send a message to ddeberl@usgs.gov . The program and other files have been compressed. IBM users can use the pkzip.exe program provided in the FTP account to decompress. The Macintosh .sea files are self-extracting.  The use of trade, product, industry, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. PAGE 4 PAGE 53 sm  |HH(FG(HH(d'hse:$F+' S&WordMicrosoft Word&Word   2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. ? 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P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""// V. VV++++/ VV++ /V+0VV+/VV+/V+,++++   $ϬJVVVV0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/2/V.KV+ +VKV+VV+V+VV+VV+ V+V +VMV+VVV+VVVVV+++V+VVV V>++VVVVVVV VVVV++V!VV VV +V++V VV [++VVV*+VV'++VV'+V'+V'+V'V0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/2K/' V+V'  V+VVVVVVVVVVVVVVVVVV+VVVVVVV0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/Kd/VVVVVVV+ ++V,+VVV+< + V>VVV+ VV@V+++<V+VV*VV+*++V(V V%VV++(VV++VV+0++5+++6V0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/d}/5+9V 8 +V&V&V&''+'+(1+4VV+V77+V9V:VV<VVV+3+V8+V; VVVV+6VV+; V+3V+.V++8V0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/}/8VV++8+V.V8V++:+V:+V+:V++:VV:VV+8+VVV+/VVVV7VVV;VV=V+VV+:V+VV+9VV7VVV7VV/VV+/VV+8VVV:VV+:V:VVV:+VVVV0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""//7VVV6VV+VVVV9VVVV;VVVV5V<VV+7+V1++V+V:+V+9+AV+V++@V++DV+E+JVVV ++@+VGVV VGVVV +F V> > > >V+=V>0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""//=V+>V>++=+PV++++++RV+S+VVV+TVVT+UU +++@V V+V@ V+VB V+V+VA +V+V@ VG ++G H I+ V+VI VGIVV+M+V0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""//MV+VO +Q V+R+ Q+ O  Q  VV+R  VT+ VVV+ V+VU VU VVVVV V VY +V+ +V[+  V[ V++ V+Z + VVV+X ++V+[V VV+VV+V+\+V VVV[ V+VVZ+V VVV[ ++++ VVVV+\ V +Vo VV +0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""//o ++VVVVVVo VV +Vn VV V+kVV++VjVV++V+gVV`+VV+VV+[+V+Y+++++XWVV++ZV+VZVVYVVVVVVbVV+V+VbVV++Vm++V+kV+f+VVk+V+n+Vp ++v++Vz ++ +++++V+ +V+ 0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""//+V+ + +rVV++++++VTVVVV[I VVVV+   Q VU+VV+ V + VV[+VV + ++ VVUV V +VV XVV V+ V XV VV V ++VTV + VHVV++V+V++V+V+0/ QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""/*/VC+V+ +@ +V GV+HV+GVVVBVV V+C++++A+V V++V V+j:!/(4 S&WordMicrosoft Word&Word &  2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. &)V HH)V  )V2)VVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VVVVV0VVV0/V0V+V0VV-+VVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""V,V + +++VVV)߬X VVVVVV+++VVVVVVVV++VV1V+ VV6V+MV+qDVV++qOV+V++s++:VVV+sV6VVV+r++VVVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""V,BVr++VVVV+V+VK+V+MV+V+VVJV+V3+5E+VD++8V8 +V8 8+ V+V7 VV6V +%V%+V%V%VV%+V%V$VVVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VBXV$VVV$V$VV$V%V%VV%V+V%V%V%V+V%V&VV&V+V&VV%V+++V%VV%VV%VV%VV%V+V%VV%VVVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VXnV%V+V%V+V$VV%V+V%V%VV+V$V+V$VVV$VV$VV$VV%VV$VV%VV$VVV&VV&V++V&VV%VV%VV%VV%VVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VnV%V&VV*VVV-VVV/VV0VV0VV2VV2+VV*+V0V1V-V0++V*++V%V0VV/V+V/+V%+V/+V/VVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VV/VV/VV/V/VVV.++V&+V-VV*VV-+V*VV-VV*+,++$+,+V00+1++0V+-/+VV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333                          ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { } ~  333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VV02,+8+2+078 +<++6:++6V:++1V+V+V6+V6;+V0+V43V3V+V2VVVVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VV3VV3V+3VGVV++H+V+KV++VKVVHVV+GV+JV JVVV5VVV4VV3VVV5VV7V+VV6V+; =V +?V+ VV>+ V< VVVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VV=V >V >V+ ++?V VVHVV+VVLV++FVVV+QV++NVV+V ++JVVJV VVKV VV++VHVVVFVVVDVVVVFVVVVDV+VV?VVV>V+V9VV9V +V+:V VVV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VV9V +V:V V:V +V;V V<V ?V ++BV ++DV +DVVCV++VCVVVEVVDV+V++DV+VVDVVVVFVVVFVVVGV+HVV+IVVVSV+V+_VVV+VVV++VV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VVZVV++_VV VVVVVjV VV++V^VVVVV+WV+V+[V+VV+UV+VVf+++V+++VViVV+VVt V+VV++VVVVVz++++ V++V+VVQV++VV?VVVVVVVa   <++!+!+ ;+V+VV+RVVV V + V+VV QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""VVZ+ ++ +V +W VV +[+V+ V VWVV ++ VV QVVVV +7+V++++V VVE+V"+VEV"+VVV V QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""" V)VHV#V++D+VH+ #V+V+C!VB!V7 +:i+{) S&WordMicrosoft Word&Word   2 @@$Use Word 6.0c or later to 2 @$view Macintosh picture. -; HH/=  /=2/=<; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";= 1++V+/V+.+VV/VV+V+.V+V-V$ <; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";2= ++'+V;VVV+++*+++VVVVVVVVV+V+  +  + +&+V<; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";2J=+J+++O +VNI+ +N++N+N++ V++B + V V + V V+ V+ +  V VVV<; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";Jb=VVV/+ 1V+1VV0V1V+1V2V+0VV+VVVVVVVVVVVV!VVV<; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";bz="V+V"VV"VV"VV"VV+V"VV++VV"VV+VV#VVV#VVVV#VV$VVV$V+V$VV#VV"VVV"V+V"V+V"VV#V+V#V$*++,+0++VV+<; QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD"""""";z=-+VV+2V2VV*+V/V/+*/)VV$V1 +VVV+V/VV/++$V,/V.+/VV.V.+V.++V$+'V)+V<; 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P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""DbzE2VV+VV2VVV+V++2VV2VV+2VVV++2VV,V+V V  V +V#V $V $V V%V &VV&VV%VVV&V&VV&VV'VVV+'VVV&V&DD QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""DzE&V&V&V&&V+.V13V3V4VVV5+V.++++V4+VV5V+V0VVV5 +VV-VV)+VV5V++V4V4V+VVV*V+V3+V+VDD QP _ 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P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""DE1VV2VV2VV2VV2VV8VVV8VVV;+VVV<VV<VVAV+AVVDVBVA+VFVVV++VGVVJV+VVVEVV +VVBVVVCVB+V+VVLVN++DD QP _ 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P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""D Eh+VfVVVgVVV++g+V++d+V VeVVV+RVTVVTVVUVUVVVT+VSTVVSVSV+WV +VVYVVVVbV+V++lVg+VVuVVV++w+++V++VV+++V++VDD QP _ hkl  ((())/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1984 div 608 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (N) 120 292 sh (d) 1038 292 sh 320 ns (hkl) 1365 414 sh 384 /Symbol f1 (-) 484 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() -20 302 sh (\)) 912 302 sh 320 ns (\() 1242 422 sh (\)) 1787 422 sh 384 /Times-Roman f1 (1) 743 292 sh end MTsave restore dMMATHA  N-1()d hkl()mࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; A N-1()d hkl()ࡱ; ࡱ; L  48{=leAYSIE;0I M T X \ _ bx s n f P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDDD""""""D "EnV+VVj V VV++ V+V  V+e++++VVV gVVVVVVV++++ ++++++++++VV8! ++P+VVVW+ V+V V V+VZ V++ +VUV+V + + VWVV + VY+V+ + V TV++V+ VV+ V O+++VDD QP _ P30ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ff33̙ff33̙̙̙̙ff̙33̙ffffffffffff33ff33333333ff333333ff33ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33ffffffffffff33ffffffff̙ffffff33ffffffffffffff33ffffffffffffffffffffffff33ffffff33ff33ff33ff33ffff3333ff33ffffffffffff33ff33333333ff333333333333̙33ff33333333333333ff33333333ff33ff33ff33ffff33ff3333ff3333333333333333ff333333333333333333ff333333ff33̙ff33ff33ffffffffffff33ff33333333ff333333ff33wwUUDD""wwUUDD""wwUUDD""wwwwwwUUUUUUDDDDObjInfoNative F Equation Native $F _952173549 _(FWW Ole                             ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ` ] ^ o a b c d e f g h i j r ep q t v du w | y cz {  ~ ܥhO Le%A!"D"D* * ---R3<1R3R3R3b330R3?J58(888888<:>:>:>:>:>:>:&@Xr@d:l-8K a 28888d:8--85Ole              PIC# $ % & ' ( ) * + , - . / 0  5 6 7 89 L @PICTD E F G H I J K L M N O P U V W XY ! `CompObje f g h i j k l m n o p u v w xy 0Y ࡱ;    dxpr  , MT Extra .  *  "  currentpoint ",Times( 00 ) l/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (00) -9 261 sh 384 /MT-Extra f1 (l) 386 261 sh end MTsave restore dMATH~ 00lmࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  00lࡱࡱ; Lࡱ; U dxpr  , MT Extra 8888-8-8<:-F.\*F\+*--8<:8`8MUDMASTER: A PROGRAM FOR CALCULATING CRYSTALLITE SIZE DISTRIBUTIONS AND STRAIN FROM THE SHAPES OF X-RAY DIFFRACTION PEAKS By D. D. Eberl1, V. A. Drits2, J. rodo3 and R. Nesch4 1U.S. Geological Survey, 3215 Marine St., Boulder, CO 80303-1066 USA; PIC             5L PICT$ % & ' ( ) * + , - . / 0 5 6 7 89 7U @CompObjE F G H I J K L M N O P U V W XY EY `ObjInfoe f g h i j k l m n o p u v w xy G .*  "  currentpoint ",Times( 00 ) l/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (00) -9 261 sh 384 /MT-Extra f1 (l) 386 261 sh end MTsave restore dMATH~ 00lmࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; 2Institute of Geology RAN. Pyzevskij 7, 109017 Moscow, Russia; 3Institute of Geological Sciences PAN, Senacka 1, 31002 Krakow, Poland; 4ETH, Zurich 8092, Switzerland. _________________________________________________________ U.S. Geological Survey Open-File Report 96-171 __________________________________________________________  Boulder, Colorado 1996 U.S. DEPARTMENT OF THE INTERIOR BRUCE BABBITT, Secretary U.S. GEOLOGICAL SURVEY Gordon Eaton, Director Last revised: March 23, 1998 Co 00lࡱࡱ; L\ࡱ; +dxpr  +"+ currentpoint ",Times .+ 2)Sin, Symbol)q) l" /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 448 3 PICT            L CompObj% & ' ( ) * + , - . / 0 5 6 7 89 _Y @ObjInfoE F G H I J K L M N O P U V W XY a `Equation Nativei j k l m n o p u v w xy b@ -1 roll exch div scale currentpoint translate 64 37 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -117 366 1003 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (2) -11 283 sh 384 /Times-Roman f1 (Sin) 184 283 sh 384 /Symbol f1 (q) 695 283 sh (l) 1039 283 sh end MTsave restore d0MATH$ )2Sinql۠ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; $ )2Sinqlࡱ; Ly ࡱ; Ldxpr  L"L currentpoint ",Times .+0).)8ver SEM photo of dickite crystals from the San Juan Mountains (Colorado) by Howard May. For additional information and a copy of the program, write or e-mail: Dennis D. Eberl U.S. Geological Survey 3215 Marine Street Boulder, Colorado 80303-1066 USA ddeberl@usgs.gov CONTENTS Page MudMaster by Wallace Stevens....................................................................... 5 Introduction................................................................................................ 6 System Requirements and Disclaimer................................................................... 7 Structure of MudMaster................................................................................. 8 Installation of MudMaster................................................................................ 9 Data Required.............................................................................................. 10 Examples.........................................................................9)T", Symbol) -)1 (())(@T"@"> ([)1]/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2432 div 704 3 -1 roll exch div scale currentpoint translate 64 45 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 910 70 moveto 222 0 rlineto stroke 1998 70 moveto 222 0 rlineto stroke /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -176 546 1963 31 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (0) -9 403 sh (89) 279 403 sh (1) 1483 403 sh 384 /Times-Roman f1 (.) 183 403 sh 384 /Times-Roman f1 (T) 904 403 sh (T) 1992 403 sh 384 /Symbol f1 (-) 1224 403 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1653 /Symbol f3 (\() 766 457 sh (\)) 1652 457 sh 384 1000 1921 /Symbol f3 ([) 643 495 sh (]) 2237 495 sh end MTsave restore d^MATHR  0.89)T-1()T[]mࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; R 0.89)T-1()T[]ࡱࡱ; La.......................... 11 Example 1: A Simple Analysis..................................................................... 11 Example 2: Using PeakPicker and Doing the Flip............................................... 17 Example 3: Symmetrical Strain Analysis.......................................................... 21 Other Examples: MOM, HUMPS, MUM, Mt. Washington.................................... 24 Standards....................................................................................Ole  FPICObj (+LPICTcument ObjInfol*,ࡱ;  dxpr  " currentpoint ",Times .+ T"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 416 3 -1 roll exch div scale currentpoint translate -1934 -19 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1998 70 moveto 222 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (T) 1992 403 sh end MTsave restore dMATH T۠ࡱ; ࡱ;  Tࡱࡱ; Lࡱ; U dxpr  , MT Extra .*  "  currentpoint ",Times( 00 ) l/MTsave save def 40 dict begin curr a ` ^ ] [ Z Y W V U S  R PPICTfo  U CompObjNative 03 Y ObjInfo0 F Equation Native 2 . entpoint 3 -1 roll sub neg 3 1 roll sub 640 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (00) -9 261 sh 384 /MT-Extra f1 (l) 386 261 sh end MTsave restore dMATH~ 00lmࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  00lࡱࡱ; ............... 25 Other Analytical Hints..................................................................................... 26 Acknowledgments......................................................................................... 27 References.................................................................................................. 28 Appendix 1: Summary of Program Inputs.............................................................. 29 Table 1. Summary of PeakPicker (pp.xls) Inputs...........L0 |ࡱ; dxpr  " currentpoint ",Times .+I) 2, Symbol)q (()))=) Lp)2)q (5()))G (R2 + 2)q (X()))f) 2)q (s()))+) bg/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5152 div 608 3 -1 roll exch div scale currentpoint translate 64 61 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (I) -6 387 sh (Lp) 1216 387 sh (G) 2296 387 sh (bg) 4666 387 sh 384 /Times-Roman f1 (2) 257 387 sh (2) 1772 387 sh (2) 2915 387 sh (2) 3769 387 sh 320 ns (2) 2589 217 sh 384 /Symbol f1 (q) 452 387 sh (q) 1967 387 sh (q) 3110 387 sh (f) 3439 387 sh (q) 3964 387 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1190 /Symbol f3 (\() 124 399 sh (\)) 656 399 sh 384 /Symbol f1 (=) 886 387 sh 384 1000 1190 /Symbol f3 (\() 1639 399 sh (\)) 2171 399 sh (\() 2782 399 sh (\)) 3314 399 sh (\() 3636 399 sh (\)) 4168 399 sh 384 /Symbol f1 (+) 4367 387 sh end MTsave restore dMATH} I2q()=Lp2q()G 2 2q()f2q()+bgmࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; .................................... 29 Table 2. Summary of MudMaster Sheet 1 (1-mm.xls) Inputs................................. 31 Table 3. Summary of MudMaster Sheet 2 (2-mm.xls) Inputs................................. 34 Appendix 2: Recommended settings for clay analyses................................................ 35 Table 4. Recommended Settings for Analysis of Clay Basal Reflections................... 35 Appendix 3: Plots of LpG2 and XRD patterns for clay mineral basal reflections............} I2q()=Lp2q()G 2 2q()f2q()+bgࡱ; ࡱ; L$x0ࡱ;  dxpr  , MT Extra .* " currentpoint ", Ole  FPICObj <?LPICTcument  CompObjl>AYSymbol(D) =,Times( 1*s" ("2)arcsin)!l +a+2 (e2)d + 00 ) l  (r())"d  (F()C)(-) 2)arcsin)!l +a+1 (2)d + 00 ) l  (())"  (()B) ([)]/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8384 div 896 3 -1 roll exch div scale currentpoint translate 64 36 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 689 409 moveto 178 0 rlineto stroke /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -155 483 3160 225 vb 16 th -155 483 6895 225 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (D) -2 508 sh 384 /Symbol f1 (=) 355 508 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 2000 /Symbol f3 (\() 2181 594 sh (\)) 4329 594 sh 384 /Symbol f1 (-) 4527 508 sh 384 1000 2000 /Symbol f3 (\() 5968 594 sh (\)) 8064 594 sh 384 1000 2289 /Symbol f3 ([) 898 637 sh (]) 8187 637 sh 320 1000 1165 /Symbol f3 (\() 3592 637 sh (\)) 4191 637 sh (\() 7327 637 sh (\)) 7926 637 sh 384 /Times-Roman f1 (1) 682 261 sh (2) 1030 508 sh (2) 3186 508 sh (2) 4817 508 sh (2) 6921 508 sh 320 ns (2) 2767 684 sh (00) 3710 629 sh (1) 6528 684 sh (00) 7445 629 sh 384 /Times-Roman f1 (s) 702 809 sh (d) 3388 508 sh (d) 7123 508 sh 384 /Times-Roman f1 (arcsin) 1262 508 sh (arcsin) 5049 508 sh 384 /Symbol f1 (l) 2316 508 sh (l) 6103 508 sh 320 ns (a) 2545 604 sh (a) 6332 604 sh 320 /MT-Extra f1 (l) 4044 629 sh (l) 7779 629 sh end MTsave restore dMATH ( D=1s2arcsin)l a 2  2d 00l() ()-2arcsin)l a 1  2d 00l() ()[]mFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; %%%%%%%%%%%%%%ԕٕܕD%`%a%c%e%f%g%h%j%k%l%n%p%r%s%t%v%x%y%{%|%~%%%]cuP uDP.     "#$%&'()*+,-.05789:;<=>?@ABCDEFGHIJKLMNOPQSVWZ\]^_`abcdefghijklmnopqrstuvwxyz{|}~ࡱ;   D=1s2arcsin)l a 2  2d 00l() ()-2arcsin)l a 1  2d 00l() ()[]ࡱ;ࡱ; LF{lhPIC CFLPICT  ]ObjInfoEGEquation Native Gࡱ; ]dxpr  , MT Extra .* " currentpoint ",Times( d + 00 ) l, Symbol  (())R/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 992 div 576 3 -1 roll exch div scale currentpoint translate -3333 -188 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (d) 3388 508 sh 320 /Times-Roman f1 (00) 3710 629 sh 320 /MT-Extra f1 (l) 4044 629 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 320 1000 1165 /Symbol f3 (\() 3592 637 sh (\)) 4191 637 sh end MTsave restore d7MATH+- d 00l()mࡱ; ࡱ; + d 00l()ࡱ; ࡱ; Ld|}?LMNOQRSej$$$$$$$$$$$$$$,$$$$$$$h$h$h$$$$$$$$$$$$$ !  !hh!%ࡱ; U dxpr  , MT Extra .*  "  currentpoint ",Times( 00 ) l/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 384 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (00) -9 261 sh 384 /MT-Extra f1 (l) 386 261 sh end MTsave restore dMATH~ 00lmࡱ; FMicrosoft Equation Editor 2.0XY +j@  K  ^ ($$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$  " "!! !&DNQE Equation.2ࡱ; ࡱ;  00lࡱࡱ; LoXTࡱ; pdxpr  p"p currentpoint ",Times .+ I) x, Oleion Native  3 PIC15632 PSF4L PICT  6 CompObj RU RY Symbol ( () ) +norm ( .=) I) x ( <() ))I)x ( `() ) ( P"O  /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3584 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 387 2481 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (I) -6 292 sh (x) 262 292 sh (I) 1753 292 sh (x) 2021 292 sh (I) 2889 292 sh (x) 3157 292 sh 320 ns (norm) 599 431 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 124 302 sh (\)) 463 302 sh 384 /Symbol f1 (=) 1425 292 sh 384 1000 1171 /Symbol f3 (\() 1883 302 sh (\)) 2222 302 sh (\() 3019 302 sh (\)) 3358 302 sh 448 /Symbol f1 (\345) 2520 337 sh end MTsave restore d~MATHrl Ix() norm =)Ix()Ix()  ۠ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; r Ix() norm =)Ix()Ix()  ࡱ; J tCvIK}$$$$$$$$$$$$$$$$$$$$ 5< "$`'0*-/2p5@8;=@CPF! 8< "$`'0*-/2p5@8;=@CPF #  "   " #  "  "ࡱ; L= ࡱ;  p&dxpr  &"& currentpoint ",Times .+A) ')', Symbol)=)I)x (2() ) +norm (Xcos)2)p)nx)N"  (h()+) (!-)N) 2",(N) PIC44291 WZFYL PICT  [p CompObj Y\ Y ObjInfo   2") (" /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4864 div 1216 3 -1 roll exch div scale currentpoint translate 64 63 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -110 344 4322 348 vb 8 th -91 286 1374 805 vb 8 th -91 286 1276 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 609 sh (I) 1415 609 sh (x) 1683 609 sh (nx) 3832 609 sh (N) 4363 609 sh 320 ns (norm) 2020 748 sh (N) 1053 1022 sh (N) 955 217 sh O/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 256 609 sh (\251) 373 609 sh (cos) 2777 609 sh 384 /Symbol f1 (=) 525 609 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1545 619 sh (\)) 1884 619 sh 384 1000 1291 /Symbol f3 (\() 3281 630 sh (\)) 4653 630 sh 320 /Symbol f1 (-) 856 1022 sh 448 ns (\345) 1046 654 sh 384 /MT_Times-Roman f1 (2) 3414 609 sh 320 ns (2) 1397 1022 sh (2) 1299 217 sh 384 /Symbol f1 (p) 3622 609 sh end MTsave restore dMATH A=Ix() norm cos)2pnxN() )-N2)N2 mࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  A=Ix() norm cos)2pnxN() )-N2)N2 ࡱࡱ; CuABDFHJLOPQRSTU`a|$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ "! 8< "$`'0*-/2p5@8;=@CPF! 8"$`'0*-/2p5@8;=@CPFL= ࡱ;  p&dxpr  &"& currentpoint ",Times .+B)')', Symbol)=)I) x (0() ) +norm (Wsin)2)p)nx)N"  (e()+) (!-)N) 2"*(N) 2"' (! /MTsave save def 40 dict begin currenPICTbj           p CompObj% & ' ( ) * + , - . / 0 `c5 6 7 89 Y @ObjInfoNativeI J K L M N O P U V W XY  `Equation Nativei j k l m n o p bF tpoint 3 -1 roll sub neg 3 1 roll sub 4768 div 1216 3 -1 roll exch div scale currentpoint translate 64 63 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -110 344 4215 348 vb 8 th -91 286 1329 805 vb 8 th -91 286 1231 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (B) -7 609 sh (I) 1370 609 sh (x) 1638 609 sh (nx) 3725 609 sh (N) 4256 609 sh 320 ns (norm) 1975 748 sh (N) 1008 1022 sh (N) 910 217 sh O/mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 211 609 sh (\251) 328 609 sh (sin) 2722 609 sh 384 /Symbol f1 (=) 480 609 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1500 619 sh (\)) 1839 619 sh 384 1000 1291 /Symbol f3 (\() 3174 630 sh (\)) 4546 630 sh 320 /Symbol f1 (-) 811 1022 sh 448 ns (\345) 1001 654 sh 384 /MT_Times-Roman f1 (2) 3307 609 sh 320 ns (2) 1352 1022 sh (2) 1254 217 sh 384 /Symbol f1 (p) 3515 609 sh end MTsave restore dMATH B=Ix() norm sin)2pnxN() )-N2)N2 mࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  B=Ix() norm sin)2pnxN() )-N2)N2 ࡱࡱ; L4 @/I\X0!#_%'*U,V,z,.// 00D37C:D:^:$$$$$$$$$$$$$$$$$$$$$$$$$$$T $$$$$$$$    &ࡱ; dxpr  " currentpoint ",Times .+ p) n, Symbol ( () ))=) 1)+) R) cos)2)p)n)D) N"n  ( M(),)G/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4064 div 512 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -111 347 3502 48 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (p) 0 312 sh (n) 330 312 sh (R) 1600 312 sh (n) 2961 312 sh (N) 3542 312 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 189 322 sh (\)) 536 322 sh 384 /Symbol f1 (=) 766 312 sh (+) 1302 312 sh 384 1000 1291 /Symbol f3 (\() 2410 333 sh (\)) 3832 333 sh 384 /Times-Roman f1 (1) 1058 312 sh (2) 2543 312 sh 384 /Times-Roman f1 (cos) 1906 312 sh 384 /Symbol f1 (p) 2751 312 sh 384 /Symbol f1 (D) 3153 312 sh end MTsave restore dbMATHV pn()=1+Rcos)2pnDN()۠ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; ^:$<>AAADDDEGhHI$$$$$ $$$$j$ E$E$$08p@ P !$`'0*-/2"8p@ P !$`'0*-/2!8p@ P !$`'0*-/2"8p@ P !$`'0*-/2! 5p@ P !$`'0*-/  V pn()=1+Rcos)2pnDN()ࡱࡱ; L4@ࡱ; mdxpr  m"m currentpoint ",Times .+ q) n, Symbol ( () ))=) R) sin)2OleObj3 FPICnfo loLPICTon Native CompObj1nqFY      !$%&)+./0146789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrtwxy|~)p)n)D) N"]  ( ;()-)(/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3488 div 512 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -111 347 2938 48 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (q) -9 312 sh (n) 332 312 sh (R) 1098 312 sh (n) 2397 312 sh (N) 2978 312 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 191 322 sh (\)) 538 322 sh 384 /Symbol f1 (=) 768 312 sh 384 1000 1291 /Symbol f3 (\() 1846 333 sh (\)) 3268 333 sh 384 /Times-Roman f1 (sin) 1394 312 sh 384 /Times-Roman f1 (2) 1979 312 sh 384 /Symbol f1 (p) 2187 312 sh 384 /Symbol f1 (D) 2589 312 sh end MTsave restore d\MATHPj qn()=Rsin)2pnDN()mࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; P qn()=Rsin)2pnDN()ࡱࡱ; L!+ࡱ; IILLLMN"OOQRhS=TU[VVWWoYKZ[_`PaIb$ $$$$$$$$$$$$$$$$$$ $$$$$ 8hp@ P !$`'0*-/$8hp@ P !$`'0*-/$08p@ P !$`'0*-/20PICion Native sv L PICT5632 F  CompObj ux Y ObjInfo  "  dxpr  " currentpoint ",Times .+A) ')n, Symbol (() ))=)A)')') n (>() ))p) n (S() ))+) B)')') n (|() ))q) n (() ) ()[)v](p) n (() ) ( 2 + +) q) n (() ) ( 2  ([)F]"-/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7808 div 736 3 -1 roll exch div scale currentpoint translate 64 60 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -206 643 5347 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 420 sh (n) 533 420 sh (A) 1408 420 sh (n) 2063 420 sh (p) 2404 420 sh (n) 2734 420 sh (B) 3434 420 sh (n) 4046 420 sh (q) 4378 420 sh (n) 4719 420 sh (p) 5502 420 sh (n) 5832 420 sh (q) 6737 420 sh (n) 7078 420 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 256 420 sh (\251) 1669 420 sh (\251) 1786 420 sh (\251) 3652 420 sh (\251) 3769 420 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 392 430 sh (\)) 739 430 sh 384 /Symbol f1 (=) 969 420 sh 384 1000 1171 /Symbol f3 (\() 1922 430 sh (\)) 2269 430 sh (\() 2593 430 sh (\)) 2940 430 sh 384 /Symbol f1 (+) 3139 420 sh 384 1000 1171 /Symbol f3 (\() 3905 430 sh (\)) 4252 430 sh (\() 4578 430 sh (\)) 4925 430 sh 384 1000 1424 /Symbol f3 ([) 1270 463 sh (]) 5048 463 sh 384 1000 1171 /Symbol f3 (\() 5691 430 sh (\)) 6038 430 sh 384 /Symbol f1 (+) 6444 420 sh 384 1000 1171 /Symbol f3 (\() 6937 430 sh (\)) 7284 430 sh 384 1000 2031 /Symbol f3 ([) 5359 524 sh (]) 7614 524 sh 320 /MT_Times-Roman f1 (2) 6167 248 sh (2) 7413 248 sh end MTsave restore dMATH&l An()=)An()pn()+Bn()qn()[]pn() 2 +qn() 2 []tiࡱ; vtrpnljhf ~ | z ~FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  An()=)An()pn()+Bn()qn()[]pn() 2 +qn() 2 []ࡱ; #5$B$P%Q%R%S%%%%%%%%%%%$$$$$$m h,P !$`'0*-/2`%$ >T< !$`'0*-/2p5@8;=@CPFC h 4.>hT< !$`'0*-/2p5@8;=@CPFLs"+ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  Bn()=)-An()qn()+Bn()pn()[]pnPICTon Native   CompObj2 |F*Y ObjInfo  , Equation Native ~ -  dxpr  " currentpoint ",Times .+B)') n, Symbol ( () ))=)-)A)')') n (C() ))q) n (X() ))+) B)')') n (() ))p) n (() ) (([)}](p) n (() ) ( 2 + +) q) n (() ) ( 2  ([)F]"-/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8000 div 736 3 -1 roll exch div scale currentpoint translate 64 60 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -206 643 5532 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (B) -7 420 sh (n) 488 420 sh (A) 1593 420 sh (n) 2248 420 sh (q) 2580 420 sh (n) 2921 420 sh (B) 3621 420 sh (n) 4233 420 sh (p) 4574 420 sh (n) 4904 420 sh (p) 5687 420 sh (n) 6017 420 sh (q) 6922 420 sh (n) 7263 420 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 211 420 sh (\251) 1854 420 sh (\251) 1971 420 sh (\251) 3839 420 sh (\251) 3956 420 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 347 430 sh (\)) 694 430 sh 384 /Symbol f1 (=) 924 420 sh (-) 1364 420 sh 384 1000 1171 /Symbol f3 (\() 2107 430 sh (\)) 2454 430 sh (\() 2780 430 sh (\)) 3127 430 sh 384 /Symbol f1 (+) 3326 420 sh 384 1000 1171 /Symbol f3 (\() 4092 430 sh (\)) 4439 430 sh (\() 4763 430 sh (\)) 5110 430 sh 384 1000 1424 /Symbol f3 ([) 1225 463 sh (]) 5233 463 sh 384 1000 1171 /Symbol f3 (\() 5876 430 sh (\)) 6223 430 sh 384 /Symbol f1 (+) 6629 420 sh 384 1000 1171 /Symbol f3 (\() 7122 430 sh (\)) 7469 430 sh 384 1000 2031 /Symbol f3 ([) 5544 524 sh (]) 7799 524 sh 320 /MT_Times-Roman f1 (2) 6352 248 sh (2) 7598 248 sh end MTsave restore dMATH'" Bn()=)-An()qn()+Bn()pn()[]pn() 2 +qn() 2 [] sࡱ; vtrpnljhf ~ | z ~() 2 +qn() 2 []ࡱ; ࡱ; L>$ࡱ; V"dxpr  """ currentpoint ",Times .+I, Symbol +a)1 (x (() )K&O&Normal a bc.A@.Default Paragraph Font ]a bcj@j Heading 2I 8hp@ P !$`'0*-/]^cd@d Heading 3C 8Pp@ P !$`'0*-/2]^cn@n Heading 4N0 ;< !$`'0*-/2p5@8;=@CPFU]^h@"h Heading 5E8p@ P )=)A)') n (L() ))cos)2)p)nx)N" ( -n)=)-(6 (4 (+)B)') n (() ))sin)2)p)nx)N" ( n)=)-( (/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8416 div 1088 3 -1 roll exch div scale currentpoint translate 64 52 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -124 387 4397 296 vb 16 th -124 387 7988 296 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (I) -6 588 sh (x) 632 588 sh (A) 1998 588 sh (n) 2536 588 sh (nx) 3869 588 sh (N) 4428 588 sh (B) 5694 588 sh (n) 6189 588 sh (nx) 7460 588 sh (N) 8019 588 sh 320 ns (n) 1395 1001 sh (n) 5093 1001 sh 320 /Symbol f1 (a) 134 684 sh 384 ns (p) 3659 588 sh (p) 7250 588 sh 320 /Times-Roman f1 (1) 327 684 sh 384 ns (2) 3451 588 sh (2) 7042 588 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 494 598 sh (\)) 833 598 sh 384 /Symbol f1 (=) 1063 588 sh 384 1000 1171 /Symbol f3 (\() 2395 598 sh (\)) 2742 598 sh 384 /Symbol f1 (+) 4793 588 sh 384 1000 1171 /Symbol f3 (\() 6048 598 sh (\)) 6395 598 sh 320 /Symbol f1 (=) 1580 1001 sh (-\245) 1780 1001 sh (\245) 1672 217 sh (=) 5278 1001 sh (-\245) 5478 1001 sh (\245) 5370 217 sh 448 ns (\345) 1628 633 sh (\345) 5326 633 sh /mt_vec StandardEncoding 256 array copy def /Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis/Udieresis/aacute /agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute/egrave /ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde/oacute /ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex/udieresis mt_vec 128 32 getinterval astore pop mt_vec dup 176 /brokenbar put dup 180 /twosuperior put dup 181 /threesuperior put dup 188 /onequarter put dup 190 /threequarters put dup 192 /Agrave put dup 201 /onehalf put dup 204 /Igrave put pop /Egrave/Ograve/Oacute/Ocircumflex/Otilde/.notdef/Ydieresis/ydieresis /Ugrave/Uacute/Ucircumflex/.notdef/Yacute/thorn mt_vec 209 14 getinterval astore pop mt_vec dup 228 /Atilde put dup 229 /Acircumflex put dup 230 /Ecircumflex put dup 231 /Aacute put dup 236 /Icircumflex put dup 237 /Iacute put dup 238 /Edieresis put dup 239 /Idieresis put dup 253 /yacute put dup 254 /Thorn put pop /re_dict 4 dict def /ref { re_dict begin /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup dup /FID ne exch /Encoding ne and { exch newfont 3 1 roll put } { pop pop } ifelse } forall newfont /FontName newfontname put newfont /Encoding mt_vec put newfontname newfont definefont pop end } def /Times-Roman /MT_Times-Roman ref 384 /MT_Times-Roman f1 (\251) 2259 588 sh (cos) 2903 588 sh (\251) 5912 588 sh (sin) 6546 588 sh end MTsave restore dMATH, I a1 x()=)An()cos2pnxN n=-  +)Bn()sin2pnxN n=- isࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  I a1 x()=)An()cos2pnx!$`'0*-/2 U]^c( @2(Footer!]c*B@B* Body Text  ]c$OQ$ Hyperlink ]^a bc*&@a*Footnote Reference ]a bc$@r$ Footnote Text],O, Body Text 2  ]c)@ Page Number activate this option and to 3433, . 5051ac. 5253,: 4647|HH(FG(HH(d'hN n=-  +)Bn()sin2pnxN n=- ࡱ; LX( Tࡱ; 2dxpr  " currentpoint ",Times .+ I) 2, Symbol)q ( ()) +norm OleObj3 FzPICnfo {LPICTon Native }2CompObj2FY ( 4=) I) 2)q ( B()))I)2)q ( l()) ( \"[ _/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 544 3 -1 roll exch div scale currentpoint translate 64 57 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -126 393 2869 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (I) -6 295 sh (I) 1946 295 sh (I) 3277 295 sh 320 ns (norm) 792 437 sh 384 /Times-Roman f1 (2) 257 295 sh (2) 2209 295 sh (2) 3540 295 sh 384 /Symbol f1 (q) 452 295 sh (q) 2404 295 sh (q) 3735 295 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1190 /Symbol f3 (\() 124 307 sh (\)) 656 307 sh 384 /Symbol f1 (=) 1618 295 sh 384 1000 1190 /Symbol f3 (\() 2076 307 sh (\)) 2608 307 sh (\() 3407 307 sh (\)) 3939 307 sh 448 /Symbol f1 (\345) 2908 340 sh end MTsave restore dMATH{^ I2q() norm =)I2q()I2q()  ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; { I2q() norm =)I2q()I2q()  ࡱ; Օ֕וٕؕݕޕߕa%g%l%t%h h hh hhh hh hhh $$ ;T< !$`'0*-/2p5@8;=@CPF P @ `',P !$`'0*-/2,P !$`'0*-/2ࡱ; LF( lࡱ; Wdxpr  " currentpoint ",Times .+s)*, Symbol) =) 0).)5( .s)* +2)q ( H-)s)* +2)q)max (JD)s)*"-L( ' * ( z *  /MTsave save def 40 dPICObj           L PICTfo% & ' ( ) * + , - . / 0 5 6 7 89 W @CompObjNativeI J K L M N O P U V W XY Y `ObjInfo9f g h i j k l m n o p F ict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 992 3 -1 roll exch div scale currentpoint translate 64 63 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1401 446 moveto 2455 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (s) -19 545 sh (s) 1414 294 sh (s) 2471 294 sh (s) 2559 846 sh 384 /Times-Roman f1 (*) 119 545 sh (.) 927 545 sh (*) 1619 294 sh (*) 2676 294 sh (*) 2764 846 sh 320 ns (max) 3249 393 sh 384 /Symbol f1 (=) 410 545 sh (-) 2256 294 sh (\346) 1205 355 sh (\350) 1205 780 sh (\366) 3865 355 sh (\370) 3865 780 sh 384 /Times-Roman f1 (0) 735 545 sh (5) 1023 545 sh 320 ns (2) 1809 393 sh (2) 2866 393 sh 320 /Symbol f1 (q) 1977 393 sh (q) 3034 393 sh 384 /Symbol f1 (D) 2324 846 sh end MTsave restore d~MATHr s*=0.5s* 2q -s* 2qmax Ds*()t ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; t%{%|%$" ;T< !$`'0*-/2p5@8;=@CPFr s*=0.5s* 2q -s* 2qmax Ds*()ࡱ; ࡱ; L 0ࡱ; Gdxpr  G"G currentpoint ",Times .+s)* +PICTon Native   CompObj0tive FY ObjInfo0 F Equation Native  Y )"6      !"#$%&'()*+,-./01234562, Symbol)q (=( '2)sin)q(2l" '/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2272 div 896 3 -1 roll exch div scale currentpoint translate 64 33 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1189 412 moveto 957 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (s) -19 511 sh 384 /Times-Roman f1 (*) 186 511 sh (sin) 1437 264 sh 320 /Times-Roman f1 (2) 376 610 sh 384 ns (2) 1210 264 sh 320 /Symbol f1 (q) 544 610 sh 384 ns (q) 1928 264 sh (l) 1558 812 sh 384 /Symbol f1 (=) 855 511 sh end MTsave restore dIMATH= 7 s* 2q =2sinqlsuࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; = s* 2q =2sinqlࡱ; ࡱ;   !"#$%&'()*+,-./012345"&%6!!!!!!!!! ! ! ! ! !!!!Lp,ࡱ; ,dxpr  ,", currentpoint ",Times .+ s)* +2, Symbol)q)max/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1408 div 480 3 -1 roll exch div scale currentpoint translate -2426 26 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (s) 2471 294 sh 384 /Times-Roman f1 (*) 2676 294 sh 320 ns (max) 3249 393 sh 320 /Times-Roman f1 (2) 2866 393 sh 320 /Symbol f1 (q) 3034 393 sh end MTsave restore d3MATH' s* 2qmaxroࡱ; ࡱ; ' s* 2qmaxࡱ; ࡱ; L] ,ࡱ; !!!!!!!!!!!!!!! !!!"!#!$!%!&!'!(!)!*!+!,!-!. / 0 1!2!3!4!5!6kCN W)E0$9AI=QZbibrzz HƶCXmnqrkXY\]oVjq`  "  ez#PICion Native  L PICT0269tive F CompObj0 F Y ObjInfo        !"#$'*,-./0123456789:;<=>?@ABDGJLMNOPQRSTUVWXYZ[\]^_`abcdefghijkmpqtvwxyz{|}~~dxpr  ~"~ currentpoint ", Symbol .+ D,Times)s)*) =) s)* +2)q)max ( M-)s)* +2)q)min/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4032 div 480 3 -1 roll exch div scale currentpoint translate 64 56 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (D) -2 264 sh 384 /Times-Roman f1 (s) 233 264 sh (s) 977 264 sh (s) 2637 264 sh 384 /Times-Roman f1 (*) 371 264 sh (*) 1182 264 sh (*) 2842 264 sh 320 ns (max) 1755 363 sh (min) 3415 363 sh 384 /Symbol f1 (=) 662 264 sh (-) 2422 264 sh 320 /Times-Roman f1 (2) 1372 363 sh (2) 3032 363 sh 320 /Symbol f1 (q) 1540 363 sh (q) 3200 363 sh end MTsave restore daMATHUJ Ds*=s* 2qmax -s* 2qminllࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; m P  }ix '   !3"#O$%&'()3*U+,-.9/0>1b234?5#5Z[\]_:mnopr]rv~$bn1:U Ds*=s* 2qmax -s* 2qminࡱࡱ; L\,ࡱ; +dxpr  +"+ currentpoint ",Times .+ s)* +2, Symbol)q)min/MTsave save def 40 dict be[!!" hh h hhh hh hh h h hh hh hhhh hh hh h$$ $$h$$$$$$$ $$$$$D$\$ $| $$$D$$$1KVhў z%%K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` ^:gin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 480 3 -1 roll exch div scale currentpoint translate -2592 56 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (s) 2637 264 sh 384 /Times-Roman f1 (*) 2842 264 sh 320 ns (min) 3415 363 sh 320 /Times-Roman f1 (2) 3032 363 sh 320 /Symbol f1 (q) 3200 363 sh end MTsave restore d3MATH' s* 2qminroࡱ; ࡱ; ' s* 2qminࡱ; OleObj           ( PICnfoNative) * + , - . / 0 5 6 7 89 )L @PICTon NativeI J K L M N O P F+ `CompObj4f g h i j k l m n o p FCY ࡱ; L* 4T@ࡱ; Qdxpr  Q"Q currentpoint ",Times .+ n +max, Symbol ( =)J) d"4  ( (()))+)1/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2592 div 512 3 -1 roll exch div scale currentpoint translate 64 40 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -110 344 1625 51 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (n) -3 312 sh (J) 1367 312 sh (d) 1661 312 sh 320 /Times-Roman f1 (max) 213 408 sh 384 /Symbol f1 (=) 912 312 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1291 /Symbol f3 (\() 1226 333 sh (\)) 1867 333 sh 384 /Symbol f1 (+) 2066 312 sh 384 /Times-Roman f1 (1) 2326 312 sh end MTsave restore dPMATHD  n max =)Jd()+12 ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; D n max =)Jd()+1ࡱ; IIbx}jHO!Wm1v2[|5BcN!#t%|%a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  $$$%%%2223-3/3HHHHHH ",.Wkm4HJ-ACJ^`ࡱ; L=l ࡱ; -&dxpr  &"& currentpoint ",Times .+A)n, Symbol ( () ))=)!I)s)* (A()) +norm (mcos)2)p)ns) * (!&s)*)=)-)1)2"A(PICnfoNative  IL PICTon Native FK- CompObj2 FlY ObjInfo  n 01)2"6 (2/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5088 div 1216 3 -1 roll exch div scale currentpoint translate 64 63 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -92 286 2046 805 vb 8 th -92 286 1708 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 609 sh (n) 416 609 sh (I) 1914 609 sh (s) 2169 609 sh (ns) 4418 609 sh 320 ns (norm) 2695 748 sh (s) 1170 1022 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 619 sh (\)) 622 619 sh 384 /Symbol f1 (=) 852 609 sh 384 1000 1177 /Symbol f3 (\() 2044 619 sh (\)) 2559 619 sh 320 /Symbol f1 (=) 1458 1022 sh (-) 1658 1022 sh 448 ns (\345) 1545 654 sh 384 /Times-Roman f1 (*) 2374 609 sh (cos) 3452 609 sh (*) 4815 609 sh 320 ns (*) 1290 1022 sh 384 /Times-Roman f1 (2) 4000 609 sh 320 ns (1) 1826 1022 sh (2) 2069 1022 sh (1) 1488 217 sh (2) 1731 217 sh 384 /Symbol f1 (p) 4208 609 sh end MTsave restore dMATH* An()=Is*() norm cos2pns* s*=-)12)12  ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  An()=/1ni}#7913:NP      . B D    m,.o<PRYmo":::::::::::::::::::::Is*() norm cos2pns* s*=-)12)12 ࡱࡱ; L=0 ࡱ; -&dxpr  &"& currentpoint ",Times .+B) n, Symbol ( () ))=)!I)s)*PICT4291 Fu- CompObj  Y ObjInfoNative   Equation Native F  (@()) +norm (lsin)2)p)ns) * (!%s)*)=)-)1)2"?(/1)2"5 (0/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4992 div 1216 3 -1 roll exch div scale currentpoint translate 64 63 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -92 286 2001 805 vb 8 th -92 286 1663 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (B) -7 609 sh (n) 371 609 sh (I) 1869 609 sh (s) 2124 609 sh (ns) 4311 609 sh 320 ns (norm) 2650 748 sh (s) 1125 1022 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 230 619 sh (\)) 577 619 sh 384 /Symbol f1 (=) 807 609 sh 384 1000 1177 /Symbol f3 (\() 1999 619 sh (\)) 2514 619 sh 320 /Symbol f1 (=) 1413 1022 sh (-) 1613 1022 sh 448 ns (\345) 1500 654 sh 384 /Times-Roman f1 (*) 2329 609 sh (sin) 3397 609 sh (*) 4708 609 sh 320 ns (*) 1245 1022 sh 384 /Times-Roman f1 (2) 3893 609 sh 320 ns (1) 1781 1022 sh (2) 2024 1022 sh (1) 1443 217 sh (2) 1686 217 sh 384 /Symbol f1 (p) 4101 609 sh end MTsave restore dMATH0 Bn()=Is*() norm sin2pns* s*=-)12)12 11ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  Bn()=Is*() norm sin2pns* s*=-)12)12 ࡱࡱ; L-Oࡱ;  Mdxpr  M"M:::::::::::::::::::::::::::::: !!` D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt D. D. Ebe currentpoint ",Times .+A)n, Symbol ( () ) +cor (+=) A)n (A() ) +std (]A) n (f() ) +samp (+) B) n (() ) +std (B) n (() ) +samp  (4[)](A) n (() ) +std(  2 ++) B) n (-() ) +std( ;2  ([)W]"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10656 div 768 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -218 677 7652 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 437 sh (n) 416 437 sh (A) 1756 437 sh (n) 2177 437 sh (A) 2930 437 sh (n) 3351 437 sh (B) 4763 437 sh (n) 5141 437 sh (B) 5892 437 sh (n) 6270 437 sh (A) 7796 437 sh (n) 8217 437 sh (B) 9334 437 sh (n) 9712 437 sh 320 ns (cor) 752 576 sh (std) 2505 576 sh (samp) 3679 576 sh (std) 5469 576 sh (samp) 6598 576 sh (std) 8545 576 sh (std) 10040 576 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 447 sh (\)) 622 447 sh 384 /Symbol f1 (=) 1317 437 sh 384 1000 1171 /Symbol f3 (\() 2036 447 sh (\)) 2383 447 sh (\() 3210 447 sh (\)) 3557 447 sh 384 /Symbol f1 (+) 4468 437 sh 384 1000 1171 /Symbol f3 (\() 5000 447 sh (\)) 5347 447 sh (\() 6129 447 sh (\)) 6476 447 sh 384 1000 2138 /Symbol f3 ([) 1618 551 sh (]) 7311 551 sh 384 1000 1171 /Symbol f3 (\() 8076 447 sh (\)) 8423 447 sh 384 /Symbol f1 (+) 9039 437 sh 384 1000 1171 /Symbol f3 (\() 9571 447 sh (\)) 9918 447 sh 384 1000 2031 /Symbol f3 ([) 7658 541 sh (]) 10458 541 sh 320 /Times-Roman f1 (2) 8552 265 sh (2) 10047 265 sh end MTsave restore dYMATHM8  An() cor =)An() std An() samp +Bn() std Bn() samp []An() std2 +Bn() std2 []ewࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; rlHD 2:Papers:MudMan.txt D. D. EberlHD 2:Papers:MudMan.txt@Ah DN8KuvDN8 D8KuvD8 D8KuvD8 D,8KuvD,8 D8KuvD8 D8KuvD8x W Dg8KuvDg8x W"#5XZ_;>mrXh$q$LU/01ࡱ; M An() cor =)An() std An() samp +Bn() std Bn() samp []An() std2 +Bn() std2 []ࡱࡱ; Ole              PIC# $ % & ' ( ) * + , - . / 0 5 6 7 89 L @PICTD E F G H I J K L M N O P U V W XY  nCompObje f g h i j k l m r p u w x y| Y L-Oࡱ;  Mdxpr  M"M currentpoint ",Times .+A)n, Symbol ( () ) +cor (+=) B) n (@() ) +samp (eA) n (n() ) +std (-) A) n (() ) +samp (   !"$)+,-./0123456789:;=@CEFGHIJKLMNOPQRSTUVWY\_abcdefghijklmnopqrstuvwxyz{|}~B) n (() ) +std  (4[)](A) n (() ) +std(  2 ++) B) n (-() ) +std( ;2  ([)W]"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 10656 div 768 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -218 677 7650 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 437 sh (n) 416 437 sh (B) 1754 437 sh (n) 2132 437 sh (A) 3180 437 sh (n) 3601 437 sh (A) 4718 437 sh (n) 5139 437 sh (B) 6185 437 sh (n) 6563 437 sh (A) 7794 437 sh (n) 8215 437 sh (B) 9332 437 sh (n) 9710 437 sh 320 ns (cor) 752 576 sh (samp) 2460 576 sh (std) 3929 576 sh (samp) 5467 576 sh (std) 6891 576 sh (std) 8543 576 sh (std) 10038 576 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 447 sh (\)) 622 447 sh 384 /Symbol f1 (=) 1317 437 sh 384 1000 1171 /Symbol f3 (\() 1991 447 sh (\)) 2338 447 sh (\() 3460 447 sh (\)) 3807 447 sh 384 /Symbol f1 (-) 4422 437 sh 384 1000 1171 /Symbol f3 (\() 4998 447 sh (\)) 5345 447 sh (\() 6422 447 sh (\)) 6769 447 sh 384 1000 2138 /Symbol f3 ([) 1618 551 sh (]) 7309 551 sh 384 1000 1171 /Symbol f3 (\() 8074 447 sh (\)) 8421 447 sh 384 /Symbol f1 (+) 9037 437 sh 384 1000 1171 /Symbol f3 (\() 9569 447 sh (\)) 9916 447 sh 384 1000 2031 /Symbol f3 ([) 7656 541 sh (]) 10456 541 sh 320 /Times-Roman f1 (2) 8550 265 sh (2) 10045 265 sh end MTsave restore dYMATHM8  An() cor =)Bn() samp An() std -An() samp Bn() std []An() std2 +Bn() std2 []ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; M An() cor =)Bn() samp An() std -An() samp Bn() std []}~789z{}  !!!!t"u"}""""""""""""UXԕZٕEܕtZD%Ce%f%-7x%An() std2 +Bn() std2 []ࡱࡱ; L4@ࡱ; \ dxpr   , MT Extra .* " currentpoint "(l,Times (2/MTsave save def 40 dict begin cuPICion Native        L PICT1457tive) * + , - . / 0 F\ @CompObjNativeI J K L M N O P F#Y `ObjInfo4f g h i j k l m n o p F% rrentpoint 3 -1 roll sub neg 3 1 roll sub 416 div 512 3 -1 roll exch div scale currentpoint translate 64 61 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /MT-Extra f1 (l) -8 387 sh 320 /Times-Roman f1 (2) 167 217 sh end MTsave restore d$MATH l 2ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  l 2ࡱࡱ; Ole   PIC `c L PICT   CompObj bd Y L ࡱ; E(dxpr  ("( currentpoint ",Times .+A)n, Symbol ( () ) +cor(D:/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1280 div 640 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add stPICT3960 F*E CompObj  <Y ObjInfo  > Equation Native  ?N ring dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 389 sh (n) 416 389 sh 320 ns (cor) 752 528 sh (D) 754 217 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 399 sh (\)) 622 399 sh end MTsave restore d>MATH2m An() corDàࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; 2 An() corDࡱ; ࡱ; Lr+ࡱ; dxpr  " currentpoint ", Symbol .+d,Times +n( 2" "  " "  /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roObjInfo:9e6Equation Native /  ll sub 800 div 736 3 -1 roll exch div scale currentpoint translate 64 61 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -102 320 102 0 vb 16 th 102 320 0 320 vb 16 th 102 320 563 0 vb 16 th -102 320 665 320 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (d) 132 419 sh 320 /Times-Roman f1 (n) 346 515 sh 320 /Times-Roman f1 (2) 339 249 sh end MTsave restore d6MATH* d n2 efࡱ currentpoint ",Times .+ S"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 448 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 8 moveto 172 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch catFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; * d n2 efࡱ; ࡱ; L"+ࡱ; Oleion Native  ] PIC53769tive F^L PICT1174 F` CompObj  Y  dxpr  , MT Extra .* " currentpoint ", Symbol(d,Times +n( 2" "  " "   +=) ln) A)n (<() ) +cor (\-) ln) A) n (y() ) +cor( D  ($[)r](d + 001  (())( 2 +"2)p ( 2 +l ( 2 + n ) l"  (()) ([):]"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 8096 div 736 3 -1 roll exch div scale currentpoint translate 64 60 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -102 320 102 1 vb 16 th 102 320 0 321 vb 16 th 102 320 563 1 vb 16 th -102 320 665 321 vb 8 th -110 344 7568 159 vb 16 th -207 643 6005 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (d) 132 420 sh (p) 6356 420 sh 320 /Times-Roman f1 (n) 346 516 sh (cor) 2352 559 sh (cor) 4289 559 sh (D) 4291 248 sh 384 ns (A) 1595 420 sh (n) 2016 420 sh (A) 3532 420 sh (n) 3953 420 sh (d) 4861 420 sh (n) 7267 420 sh 320 /Times-Roman f1 (2) 339 250 sh (001) 5183 541 sh (2) 5072 250 sh (2) 6582 250 sh (2) 6933 250 sh 384 ns (2) 6148 420 sh 384 /Symbol f1 (=) 805 420 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1875 430 sh (\)) 2222 430 sh 384 /Symbol f1 (-) 2885 420 sh 384 1000 1171 /Symbol f3 (\() 3812 430 sh (\)) 4159 430 sh 384 1000 2031 /Symbol f3 ([) 1106 524 sh (]) 4747 524 sh 384 1000 1291 /Symbol f3 (\() 7126 441 sh (\)) 7767 441 sh 384 1000 2018 /Symbol f3 ([) 6016 522 sh (]) 7890 522 sh 320 1000 1165 /Symbol f3 (\() 5065 549 sh (\)) 5649 549 sh 384 /Times-Roman f1 (ln) 1242 420 sh (ln) 3179 420 sh 384 /MT-Extra f1 (l) 6758 420 sh (l) 7597 420 sh end MTsave restore d!MATH*L d n2 ef=)lnAn() cor -lnAn() corD []d 001()2 2p 2 l 2 )nl()[]ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (S) -16 341 sh end MTsave restore dMATH Sn ࡱ; L>Wࡱ;  d n2 ef=)lnAn() cor -lnAn() corD []d 001()2 2p 2 l 2 )nl()[]ࡱࡱ; LXTࡱ; PIC53960 FL PICT   CompObj  Y ObjInfo   !dxpr  !"! currentpoint ",Times .+ AS +n""""/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1056 div 544 3 -1 roll exch div scale currentpoint translate 64 58 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -83 227 83 0 vb 16 th 83 227 0 227 vb 16 th 83 227 835 0 vb 16 th -83 227 918 227 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (AS) 123 326 sh 320 ns (n) 618 422 sh end MTsave restore d5MATH)  AS n efJࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; ) AS n efࡱ; ࡱ; ࡱ; vtrpnljhf | z [ _ ࡱ; L ࡱ; %dxpr  %, MT Extra .*% "% currentpoint ",Times(AS +n"""", Symbol (#=(.arcsin)!B) n (V() ) +cor (zA) n (() ) PICTbj            CompObj% & ' ( ) * + , - . / 0 5 6 7 89 Y @ObjInfoNativeI J K L M N O P U V W XY  `Equation Nativei j k l m n o p F +cor( D"x  (J()V)(#`2)p )l"-v-/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5312 div 1184 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -83 227 83 464 vb 16 th 83 227 0 691 vb 16 th 83 227 835 464 vb 16 th -83 227 918 691 vb 8 th -171 531 3821 31 vb 16 th 1392 691 moveto 3815 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (AS) 123 790 sh (B) 2466 420 sh (n) 2844 420 sh (A) 3853 420 sh (n) 4274 420 sh 320 ns (n) 618 886 sh (cor) 3180 559 sh (cor) 4610 559 sh (D) 4612 248 sh 384 /Symbol f1 (=) 1058 790 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 2703 430 sh (\)) 3050 430 sh (\() 4133 430 sh (\)) 4480 430 sh 384 1000 1945 /Symbol f3 (\() 2329 501 sh (\)) 5069 501 sh 384 /Times-Roman f1 (arcsin) 1410 420 sh 384 /Times-Roman f1 (2) 3009 1091 sh 384 /Symbol f1 (p) 3217 1091 sh 384 /MT-Extra f1 (l) 3427 1091 sh end MTsave restore dMATH> AS n ef=arcsin)Bn() cor An() corD ()2pl/Oࡱ;FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  AS n ef=arcsin)Bn() cor An() corD ()2plࡱ; ࡱ; LV ࡱ; ࡱ; L>Wࡱ;  dxpr   " currentpoint ",Times .+ S"/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 448 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewicKdxpr  K"K currentpoint ",Times .+ d + 001, Symbol  (()) ( "AS +n"!""9"; ( CS"C"B/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2400 div 640 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th /stb { newpath moveto 0 setlinewidth 2 copy rlineto } def /enb { rlineto neg exch neg exch rlineto closepath fill } def /hb { stb 0 thick enb } def /vb { stb thick 0 enb } def -83 227 1011 15 vb 16 th 83 227 928 242 vb 16 th 83 227 1763 15 vb 16 th -83 227 1846 242 vb 16 th 2110 8 moveto 172 0 rlineto stroke -174 541 2065 0 vb /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (d) -9 341 sh (AS) 1051 341 sh (S) 2094 341 sh 320 ns (n) 1546 437 sh 320 /Times-Roman f1 (001) 313 462 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 320 1000 1165 /Symbol f3 (\() 195 470 sh (\)) 779 470 sh end MTsave restore dkMATH_  )d 001() AS n efS4ࡱ;FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; _ )d 001() AS n efSࡱ; R F5ZW@CompObj\WordDocument^AObjectPoolkzWVWOleion Native F PIC55587tive FL PICT3510 FE CompObj   Y       !"#$%&'()*+,.146789:;<=>?@ABCEJLMNOPQRSTUVWXYZ[\]^_`abcdfilnopqrstuvwxyz{|}~L ࡱ; E(dxpr  ("( currentpoint ",Times .+A)n, Symbol ( () ) +cor(D:/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1280 div 640 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 389 sh (n) 416 389 sh 320 ns (cor) 752 528 sh (D) 754 217 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 399 sh (\)) 622 399 sh end MTsave restore d>MATH2m An() corDg ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; 2 An() c N   LK"J#$%&()*+,-$ =?DAFBCEFHQKDLMNOPRWTUVabcgg@|hikm~ npr>su>vwxz~k orDࡱ; ࡱ; LX Tࡱ; +(dxpr  ("( currentpoint ",Times .+ A)n, Symbol ( () ) +cor+/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 rollPIC55657       FL PICTfo% & ' ( ) * + , - . / 0 5 6 7 89 + @CompObjNativeI J K L M N O P  U V W XY -Y `ObjInfo9f g h i j k l m n o p F/ sub 1280 div 544 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (A) -5 292 sh (n) 416 292 sh 320 ns (cor) 752 431 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 275 302 sh (\)) 622 302 sh end MTsave restore d:MATH. An() corb ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; } \ @  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuwxyz{|}~. An() corࡱ; ࡱ; Lࡱ;  dxpr  "  currentpoint ",Times .+ M" /MTsave save def 40 dict begin currentpoint 3 -1 roll sPICTon Native F5 CompObj9tive FDY ObjInfo0 FF Equation Native  G/ ub neg 3 1 roll sub 448 div 416 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 8 moveto 327 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (M) -7 341 sh end MTsave restore dMATH Meࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  MࡱObjInfoNative   Equation Native \Fi SummaryInformation (FK _952339303 aFWW ࡱ; L ࡱ; S"Udxpr  "U""U currentpoint ", Symbol .+ ,Times)H) n ( () )( )n""! +n)) 0 (<=( J1(GM"H "G /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2720 div 1088 3 -1 roll exch div scale currentpoint translate 64 37 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 472 moveto 975 0 rlineto stroke 1012 0 moveto 0 944 rlineto stroke 2244 539 moveto 327 0 rlineto stroke 2212 472 moveto 391 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (\266) 23 324 sh (\266) 294 872 sh 384 /Times-Roman f1 (H) 212 324 sh (n) 631 324 sh (n) 483 872 sh (M) 2237 872 sh 320 ns (n) 1066 988 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 490 334 sh (\)) 837 334 sh 384 /Symbol f1 (=) 1878 571 sh 320 ns (\256) 1239 988 sh 320 /Times-Roman f1 (0) 1570 988 sh 384 ns (1) 2311 324 sh end MTsave restore dpMATHd  Hn()nb n0 =1Malࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; _G:976}5t d Hn()nb n0 =1Mࡱ; LU  ࡱ; +(hdxpr  (h"(h currentpoint ", Symbol .+,Times (2 +H) n (() )( Oleion Native Fj PIC56419tive  #FkL PICT3510 Fm+ CompObj "% Y  )n (2"#"'"+ n)) 0 (B=(Mf) M (R())(TM"T "M2/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3328 div 1280 3 -1 roll exch div scale currentpoint translate 64 38 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 567 moveto 1163 0 rlineto stroke 1200 0 moveto 0 1134 rlineto stroke 2641 634 moveto 327 0 rlineto stroke 2400 567 moveto 808 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (\266) 23 419 sh (\266) 285 1002 sh 320 /Times-Roman f1 (2) 223 249 sh (2) 685 832 sh (0) 1758 1178 sh 384 /Times-Roman f1 (H) 400 419 sh (n) 819 419 sh (n) 474 1002 sh (f) 2425 419 sh (M) 2718 419 sh (M) 2634 967 sh 320 ns (n) 1254 1178 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 678 429 sh (\)) 1025 429 sh 384 /Symbol f1 (=) 2066 666 sh 384 1000 1171 /Symbol f3 (\() 2581 429 sh (\)) 3070 429 sh 320 /Symbol f1 (\256) 1427 1178 sh end MTsave restore dMATHR  2 Hn()n 2 b n0 =fM()MࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;   2 Hn()n 2 b n0 =fM()MࡱD. D. Eberl'@?/T@LW@AW@Microsoft Word 6.0.1103ࡱ; d Sࡱࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ;  dxpr   " S 2 Sࡱ Oh+'0t     `h`dd\`hd``lddh`dh`\hd`QMacintosh HD:TEXT & FIG Applications:MS Office:Microsoft Word 6:Templates:Normal{MUDMASTER: A PROGRAM FOR CALCULATING CRYSTALLITE SIZE DISTRIBUTIONS AND STRAIN FROM THE SHAPES OF X-RAY DIFFRACTION PEAKS D. D. Eberl PICion Native '*FL PICT6951tive F CompObj3 ),FY ObjInfo   dth /thick exch def } def 16 th 0 8 moveto 172 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (S) -16 341 sh end MTsave restore dMATH Sn ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  Sࡱࡱ; LzܥhO \e%A!"D"D-* 7* ---3I1>3333403q@JO58(999999q:s:s:s:s:s:s:&@XA:-9K a 29999:9--9O5ࡱ; r/ /dxpr  /, MT Extra .//* "/ currentpoint ",Times( M" ) d + 00 ) l, Symbol// ((*)) //~( (l/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1504 div 640 3 -1 roll exch div scale curPICTbjNative  r CompObjNative 03FY ObjInfoNative F Equation Native 2FU rentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 8 moveto 327 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (M) -7 341 sh (d) 318 341 sh 320 /Times-Roman f1 (00) 640 462 sh 320 /MT-Extra f1 (l) 974 462 sh 384 ns (l) 1233 341 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 320 1000 1165 /Symbol f3 (\() 522 470 sh (\)) 1121 470 sh end MTsave restore dEMATH9' Md 00l() lolࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; 9 Md 00l() lࡱ; ࡱ; L!4@ࡱ; ;dxpr  ;";                          ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z ` ] ^ o a b c d e f g h i j ep q t du w y cz {  currentpoint ",Times .+ S", Symbol) =)f)S ( %() ))S ( /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1888 div 512 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 8 moveto 172 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (S) -16 341 sh (f) 992 341 sh (S) 1276 341 sh (S) 1601 341 sh 384 /Symbol f1 (=) 296 341 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1148 351 sh (\)) 1482 351 sh 448 /Symbol f1 (\345) 624 386 sh end MTsave restore dFMATH: S=fS()S  llࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; : S=fS9999-9-9q:-@.RC*@+ --9q:9`9MUDMASTER: A PROGRAM FOR CALCULATING CRYSTALLITE SIZE DISTRIBUTIONS AND STRAIN FROM THE SHAPES OF X-RAY DIFFRACTION PEAKS By D. D. Eberl1, V. A. Drits2, J. rodo3 and R. Nesch4 1U.S. Geological Survey, 3215 Marine St., Boulder, CO 80303-1066 USA; ()S  ࡱ; ࡱ; L @,ࡱ; Pdxpr  P"P currentpoint ", Symbol .+ a) =,Times)f) S ( '() ) (  ( 7ln) S ( @() )/MTsave save def 40 dOleObjNative   PICnfo9tive <?FL PICTon Native F CompObj5 >AFY ict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2560 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (a) -15 292 sh 384 /Symbol f1 (=) 347 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1199 302 sh (\)) 1533 302 sh (\() 1999 302 sh (\)) 2333 302 sh 448 /Symbol f1 (\345) 675 337 sh 384 /Times-Roman f1 (f) 1043 292 sh (S) 1327 292 sh (S) 2127 292 sh 384 /Times-Roman f1 (ln) 1696 292 sh end MTsave restore dSMATHG j a=fS()   lnS()ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; G a=fS()   lnS()ࡱ;ࡱ; L2Institute of Geology RAN. Pyzevskij 7, 109017 Moscow, Russia; 3Institute of Geological Sciences PAN, Senacka 1, 31002 Krakow, Poland; 4ETH, Zurich 8092, Switzerland. _________________________________________________________ U.S. Geological Survey Open-File Report 96-171 __________________________________________________________  Boulder, Colorado 1996 U.S. DEPARTMENT OF THE INTERIOR BRUCE BABBITT, Secretary U.S. GEOLOGICAL SURVEY Gordon Eaton, Director Last revised: March 23, 1998 CoPIC57169tive CFFL PICTon Native F, CompObj4 EHFY ObjInfo        "$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOQTUVY[\]^_`abcdefghjoqrstuvwxyz{|}~ࡱ; ,sdxpr  s"s currentpoint ", Symbol .+b,Times ( 2 + =)f) S (,() ) ( (>ln) S (G() ))-) a (:[)/] (l2g/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3680 div 640 3 -1 roll exch div scale currentpoint translate 64 59 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Symbol f1 (b) -23 421 sh (a) 3050 421 sh 320 /Times-Roman f1 (2) 197 251 sh (2) 3422 217 sh 384 /Symbol f1 (=) 505 421 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1357 431 sh (\)) 1691 431 sh (\() 2232 431 sh (\)) 2566 431 sh 384 /Symbol f1 (-) 2764 421 sh 384 1000 1424 /Symbol f3 ([) 1793 464 sh (]) 3305 464 sh 448 /Symbol f1 (\345) 833 466 sh 384 /Times-Roman f1 (f) 1201 421 sh (S) 1485 421 sh (S) 2360 421 sh 384 /Times-Roman f1 (ln) 1929 421 sh end MTsave restore dMATHs b 2 =fS()   lnS()-a[] 2ࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; s b 2 =fS()   lnS()-a[] 2ࡱ; ver SEM photo of dickite crystals from the San Juan Mountains (Colorado) by Howard May. For additional information and a copy of the program, write or e-mail: Dennis D. Eberl U.S. Geological Survey 3215 Marine Street Boulder, Colorado 80303-1066 USA ddeberl@usgs.gov CONTENTS Page MudMaster by Wallace Stevens....................................................................... 5 Introduction................................................................................................ ࡱ; Ll ࡱ;  (dxpr  ("( currentpoint ",Times .+g) S, Symbol (() ))=(>1(!)S)b) 2)p (!6()) (K0).)5"(0 ( # * (# ( Z * (Z PICT8419tive F# CompObj7tive LOFPY ObjInfo0 FR Equation Native N S (`exp)-(1(!2)b (2" (  * ( ( * ( + ln) S (() ))-) a ([)0] (2 ( r *  *  (  *  *  /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7136 div 1280 3 -1 roll exch div scale currentpoint translate 64 53 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1236 584 moveto 1572 0 rlineto stroke 4198 584 moveto 645 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (g) -10 683 sh (S) 307 683 sh (S) 1252 1024 sh (S) 5587 683 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 179 693 sh (\)) 513 693 sh 384 /Symbol f1 (=) 743 683 sh 384 1000 1171 /Symbol f3 (\() 1671 1034 sh (\)) 2233 1034 sh 384 /Symbol f1 (\351) 1077 372 sh (\353) 1077 1121 sh (\352) 1077 741 sh (\352) 1077 1055 sh (\371) 2830 372 sh (\373) 2830 1121 sh (\372) 2830 741 sh (\372) 2830 1055 sh (-) 3783 683 sh (\346) 4002 387 sh (\350) 4002 1024 sh (\347) 4002 805 sh (\366) 4852 387 sh (\370) 4852 1024 sh (\367) 4852 805 sh 384 1000 1171 /Symbol f3 (\() 5459 693 sh (\)) 5793 693 sh 384 /Symbol f1 (-) 5991 683 sh 384 1000 1424 /Symbol f3 ([) 5020 726 sh (]) 6532 726 sh 384 /Symbol f1 (\354) 3590 355 sh (\355) 3590 748 sh (\357) 3590 456 sh (\356) 3590 1142 sh (\357) 3590 1028 sh (\374) 6856 355 sh (\375) 6856 748 sh (\357) 6856 456 sh (\376) 6856 1142 sh (\357) 6856 1028 sh 384 /Times-Roman f1 (1) 1926 436 sh (2) 1804 1024 sh (1) 4424 436 sh (2) 4219 1019 sh 320 ns (0) 2364 852 sh (5) 2604 852 sh (2) 4627 849 sh (2) 6649 479 sh 384 /Symbol f1 (b) 1465 1024 sh (p) 2012 1024 sh (b) 4407 1019 sh (a) 6277 683 sh 320 /Times-Roman f1 (.) 2524 852 sh 384 ns (exp) 3024 683 sh (ln) 5156 683 sh end MTsave restore dMATH"2 gS()=1Sb2p() 0.5 []exp-12b 2 ()lnS()-a[] 2 {}ࡱ; FMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  gS()=1Sb2p() 0.5 []exp-12b 2 ()lnS()-a[] 2 {}ࡱ; 6 System Requirements and Disclaimer................................................................... 7 Structure of MudMaster................................................................................. 8 Installation of MudMaster................................................................................ 9 Data Required.............................................................................................. 10 Examples.........................................................................LX,Tࡱ; dxpr  " currentpoint ",Times .+ S +v" /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 480 div 544 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 8 moveto 382 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (S) -16 341 sh 320 ns (v) 201 437 sh end MTsave restore d+MATHI S vntࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ;  S vࡱ; Lpࡱ; .......................... 11 Example 1: A Simple Analysis..................................................................... 11 Example 2: Using PeakPicker and Doing the Flip............................................... 17 Example 3: Symmetrical Strain Analysis.......................................................... 21 Other Examples: MOM, HUMPS, MUM, Mt. Washington.................................... 24 Standards....................................................................................OleObj9tive Fm PICnfoNative X[FnL PICTon Native Fp CompObj3 Z]FY a ` ^ ] [ Z Y W V U S  R P#,dxpr  #,"#, currentpoint ",Times .+S +v"  , Symbol (=(S ( #2" (!S" " /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1408 div 1120 3 -1 roll exch div scale currentpoint translate 64 54 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 381 moveto 382 0 rlineto stroke 872 8 moveto 379 0 rlineto stroke 975 682 moveto 172 0 rlineto stroke 840 615 moveto 443 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (S) -16 714 sh (S) 856 467 sh (S) 959 1015 sh 320 ns (v) 201 810 sh 384 /Symbol f1 (=) 506 714 sh 320 /Times-Roman f1 (2) 1067 297 sh end MTsave restore dYMATHM S v =S 2 SࡱFMicrosoft Equation Editor 2.0DNQE Equation.2ࡱ; ࡱ; M S v =............... 25 Other Analytical Hints..................................................................................... 26 Acknowledgments......................................................................................... 27 References.................................................................................................. 28 Appendix 1: Summary of Program Inputs.............................................................. 29 Table 1. Summary of PeakPicker (pp.xls) Inputs............................................... 29 Table 2. Summary of MudMaster Sheet 1 (1-mm.xls) Inputs................................. 31 Table 3. Summary of MudMaster Sheet 2 (2-mm.xls) Inputs................................. 34 Appendix 2: Recommended settings for clay analyses................................................ 35 Table 4. 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