XRDTIME

A program for the collection of X-ray diffraction data

with variable step counting times

I.C. Madsen and R.J. Hill

CSIRO Division of Minerals

P.O. BOX 124 Port Melbourne 3207

Victoria

Australia

Phone +61 3 9647 0366

FAX +61 3 9646 3223

Email Ian.Madsen@minerals.csiro.au

Introduction

This document describes the background to, and the operation of, a method of data collection for X-ray diffraction data using variable step counting times. The program is distributed freely, but the work of the original authors should be referenced with any published results.

The Problem

For X-ray diffraction data collection, a number of factors act to progressively decrease the observed intensities as the Bragg angle increases.

The major factors causing this decrease are:-

_Lorentz-polarization factor

_Thermal vibration

_Atomic scattering factor

However the fall-off is partially compensated for by the effect of:-

_Reflection multiplicity

_Sample absorption (for cylindrical geometry)

The net result, however, is a fall in diffracted intensities that can be larger than two orders of magnitude across the pattern.

In spite of this intensity decline, diffraction data for Rietveld analysis is universally collected with the same counting time (or monitor count) for each step in the pattern. As a result, the peaks at high angles are collected with much lower precision than those at low angles. This is unfortunate since the mid to high angle regions of the pattern contain more information (peaks) than the low angle regions.

The Cause

The fall-off in the intensity of a diffraction pattern can be represented as follows

I ∝ _2 x Lp x TC x m x A	(1)

where	_2	is the scattering factor

Lp	is the Lorentz polarization factor

TC	is the thermal correction

m	is the reflection multiplicity

and	A	is the sample absorption factor

Lorentz-polarization factor

The Lp factor is the dominant effect in terms of variation as a function of 2 , exhibiting a range of about 400 across the pattern. The additional polarization correction introduced if a crystal monochromator is used only slightly reduces this variation.

Thermal correction

The thermal motion of atoms in the structure also serves to decrease the intensity of the high angle regions of the pattern, although the magnitude of the effect is less than the Lp factor. For B = 0.5Å2 ( a value found in typical oxide materials) the variation across the pattern is about 1.5. For larger values of B (3.0Å2), the correction factor can vary by as much as 10 between the two ends of the pattern.

Scattering factor

While the overall magnitude of the scattering factor is highly dependant on the average atomic number, the variation between the low and high ends of the pattern is much less dependant. A decrease in intensity of about 3 to 5 times is typical for a wide range of compositions. Note that XRDTIME does not take the absolute magnitude of scattering into account - only its variation as a function of diffraction angle.

Reflection multiplicity

The decrease in pattern intensity as a function of 2 exhibited by the preceding factors is partially compensated for by the effect of reflection multiplicity. On average, the reflections at high angles will have higher multiplicity than those at low angles. In addition this difference will be largest for the highest symmetry. An intensity increase of 6 to 8 times for cubic and 1 to 2 times for monoclinic materials can be expected. The correction applied has the form

m = a + b ln2	(2)

where m is the average reflection multiplicity and a and b are variables. See Madsen and Hill (1994) for details of the method of determination of these values.

Absorption factor

In conventional Bragg-Brentano diffraction, sample absorption decreases the intensities of all peaks equally, that is, there is no variation in peak intensity as a function of 2 . However, if cylindrical (capillary) geometry is used (Hill and Madsen, 1991), the affect of absorption of the beam in the sample is to increase the diffracted intensities at high 2 relative to low 2 .

The correction is dependant on the linear absorption coefficient ( ) and the radius of the tube (R). For example, for rutile (TiO2 ) packed in a 0.5mm diameter tube, and using CuK radiation, R is about 4.0. These values lead to an increase in the high angle intensities by a factor of 12 relative to the low angle peaks.

The solution

To compensate for the overall decrease in pattern intensity, an algorithm has been devised to calculate the optimum step-counting time, taking the afore-mentioned parameters into account . Equation (1) is used to calculate the expected intensity at each point in the pattern and the counting time is adjusted accordingly. The strategy uses shorter counting times at low 2 , where diffracted intensities are high, and longer times at high 2 to compensate for the lower intensities. Further details of this variable counting time (VCT) algorithm can be found in Madsen and Hill (1992) and Madsen and Hill (1994).

The Program

XRDTIME is a program, written in FORTRAN, for the calculation of variable counting time data collection conditions. The program is default driven , that is, the most commonly used conditions are suggested and will be accepted by simply pressing the enter key. The details of the program input are given below:-

Program Prompt

Meaning Note - the value displayed after the [D: will be accepted if no entry is made.

Enter START angle (deg) [D: 10 >

The lowest 2theta angle to be collected Enter STOP angle (deg) [D: 160 >

The highest 2theta angle to be collected Enter STEP WIDTH (deg) [D: 0.020 >

The increment between data points in 2theta Enter BLOCK WIDTH (deg) [D: 1 >

The width of the data blocks. The counting time will be changed after this many degrees. Enter TOTAL COUNTING TIME (sec) [D: 14400 >

The total data collection time for the data set in seconds (do not enter the number of seconds per step) Enter MINIMUM step count time [D: 0.20 >

By default, the program assigns 0.1 of the total counting time as a fixed time across the entire pattern. This ensures that adequate statistics are accumulated in the background regions of the low angle regions. The remaining available time (0.9 of the total) is allowed to vary in accordance with the VCT regime. Enter TUBE TARGET Fe Co Cu Mo Ag [D: Cu >

Used to define the wavelength for this data collection Is a monochromator fitted Y/N [D: Y >

Used to modify the Lp correction for the presence of a crystal monochromator in the diffractometer system. The use of a curved graphite, diffracted beam monochromator is assumed. Enter THERMAL VIBRATION (B) value [D: 0.5 >

The average thermal parameter for the material being examined. Typically 0.3 - 0.5 for oxide materials. Values of 2.0 - 3.0 might be acceptable for organic materials. Enter uR for DEBYE-SCHERRER

geometry 0.0 for flat plate [D: 0.0 >

The product of the linear absorption coefficient, the radius of the sample and the packing density if capillary geometry is used. typical values for packing density are 0.3 - 0.4. Enter 0.0 for flat plate geometry. Enter code for CRYSTAL SYSTEM

1 = Cubic 2 = Tetragonal 3 = Orthorhombic

4 = Hexagonal 5 = Monoclinic 6 = Triclinic [D:3 >

A code for the crystal system of the dominant phase in the sample. Enter CHEMICAL FORMULA ? Y/N [D: N >

The chemical formula is used to calculate the scattering factor curves for the sample. An N answer will assume a composition of Fe (an intermediate value of scattering for many materials). See the next box of the answer is Y

Enter SYMBOL for element # 1 (Return to end list) > Mg

Enter SYMBOL for element # 2 (Return to end list) > Al

Enter SYMBOL for element # 3 (Return to end list) > O

Enter SYMBOL for element # 4 (Return to end list) >

Enter number of MG atoms in formula [D:1> 1

Enter number of AL atoms in formula [D:1> 2

Enter number of O atoms in formula [D:1> 4

Atom type MG AL O

# of atoms 1.00 2.00 4.00

Enter the symbol for the each of the elements in the compound (15 maximum). Terminate the list by pressing enter with no entry. Then enter the number of each of the atoms in the formula. The entered composition will be displayed and used to calculate the average scattering factor curves. Do you wish to write these conditions to PW1710 compatible files Y/N [D:n>

Enter Y if you wish to write a file of commands to control the diffractometer. Note that the commands currently support the Philips PW1710 and PW3710 controllers. Modify subroutine WRITE to support your own diffractometer controller. The file is called COMMAND.LIS Do you want to collect FIXED TIME data Y/N [D:Y>

You can write the diffractometer commands to collect a fixed count time (FCT) data set after the variable count time (VCT) data set by entering Y . The FCT data will be collected with the same total count time as the VCT data.

Files supplied

The files supplied with XRDTIME are:-

XRDTIME.FOR	the source code for the main routine for the program

BLKABS.FOR	the source code for the capillary absorption correction routine

SCATFAC.FOR	the source code for the scattering factor routine

MAKEFILE	a makefile for compilation and linking of the three routines (setup for WATCOM FORTRAN77, but could be translated to other compilers.

XRDTIME.EXE the executable version of XRDTIME for PC

XRDTIME.OUT	a typical output list file

COMMAND.LIS	a typical list of Philips PW1710/PW3710 diffractometer controller commands

DOS4GW.EXE	a DOS extender required to run XRDTIME on the PC. Place this file anywhere on the path.

REFRMVAR.FOR	a program for reformatting the raw data returned by the (PW1710/3710) diffractometer controller. This will have to be modified if other controllers are used.

VCTDISTR.DOC	this document in Word for Windows 6.0 format

Typical output file

The following is a typical output file (XRDTIME.OUT) from the XRDTIME program

XRDTIME - calculation of variable count time data collection conditions

Calculation run at 16:04 on 15/03/1996

All published use should refer to

# Madsen and Hill, (1994), J.Appl.Cryst., 27, 385-392 #

Start = 10.000 Step = 0.020 Stop = 160.000 Block width (deg)= 5.

Total count time = 100000.0 sec Minimum step count time = 1.35 sec

Step count time for fixed time data collection = 13.35

CU target tube - wavelength = 1.54056

Monochromator correction INCLUDED in LP

Average temperature factor of 0.50 assumed.

mu-R used for absorption correction = 0.000

Crystal system used for multiplicity effect is Cubic

Formula used to calculate scattering factor curves =

Atom type MG AL O

# of atoms 1.00 2.00 4.00

Block Angle - Limits Avg. Avg. Avg. Avg. Avg. Average

# Low High LP Scatt Temp Trans Mult. Int./1000

1 10.000 14.980 78.08 4287.53 0.995 1.000 14.663 4884.23

2 15.000 19.980 38.62 3855.52 0.990 1.000 17.389 2563.81

3 20.000 24.980 22.85 3421.22 0.984 1.000 19.415 1493.85

4 25.000 29.980 14.99 3022.05 0.976 1.000 21.030 930.14

5 30.000 34.980 10.51 2669.72 0.968 1.000 22.373 607.34

6 35.000 39.980 7.73 2362.62 0.957 1.000 23.522 411.06

7 40.000 44.980 5.89 2095.23 0.946 1.000 24.527 286.19

8 45.000 49.980 4.61 1862.21 0.934 1.000 25.420 204.00

9 50.000 54.980 3.71 1659.24 0.921 1.000 26.223 148.50

10 55.000 59.980 3.04 1482.70 0.907 1.000 26.953 110.30

11 60.000 64.980 2.55 1329.38 0.893 1.000 27.622 83.59

12 65.000 69.980 2.18 1196.39 0.878 1.000 28.239 64.70

13 70.000 74.980 1.90 1081.09 0.863 1.000 28.813 51.21

14 75.000 79.980 1.70 981.15 0.848 1.000 29.347 41.51

15 80.000 84.980 1.55 894.50 0.833 1.000 29.849 34.52

16 85.000 89.980 1.45 819.34 0.818 1.000 30.321 29.48

17 90.000 94.980 1.39 754.11 0.803 1.000 30.766 25.87

18 95.000 99.980 1.36 697.47 0.788 1.000 31.189 23.34

19 100.000 104.980 1.36 648.26 0.774 1.000 31.590 21.62

20 105.000 109.980 1.40 605.50 0.760 1.000 31.971 20.54

21 110.000 114.980 1.46 568.34 0.747 1.000 32.336 19.99

22 115.000 119.980 1.54 536.07 0.735 1.000 32.685 19.89

23 120.000 124.980 1.67 508.08 0.723 1.000 33.019 20.22

24 125.000 129.980 1.82 483.86 0.713 1.000 33.340 20.96

25 130.000 134.980 2.02 462.97 0.703 1.000 33.648 22.16

26 135.000 139.980 2.28 445.05 0.694 1.000 33.945 23.90

27 140.000 144.980 2.61 429.80 0.685 1.000 34.231 26.34

28 145.000 149.980 3.05 416.94 0.678 1.000 34.508 29.74

29 150.000 154.980 3.64 406.29 0.672 1.000 34.775 34.58

30 155.000 160.000 4.50 397.63 0.667 1.000 35.034 41.83

Block # Angle - Limits Average Total Step Time

# pts Low High Int./1000 Time sec/step

1 250 10.000 14.980 4884.23 362.04 1.45

2 250 15.000 19.980 2563.81 384.24 1.55

3 250 20.000 24.980 1493.85 417.72 1.65

4 250 25.000 29.980 930.14 466.34 1.85

5 250 30.000 34.980 607.34 534.82 2.15

6 250 35.000 39.980 411.06 629.05 2.50

7 250 40.000 44.980 286.19 756.26 3.05

8 250 45.000 49.980 204.00 924.97 3.70

9 250 50.000 54.980 148.50 1144.49 4.60

10 250 55.000 59.980 110.30 1424.02 5.70

11 250 60.000 64.980 83.59 1771.14 7.10

12 250 65.000 69.980 64.70 2189.77 8.75

13 250 70.000 74.980 51.21 2677.71 10.70

14 250 75.000 79.980 41.51 3224.28 12.90

15 250 80.000 84.980 34.52 3809.10 15.25

16 250 85.000 89.980 29.48 4402.68 17.60

17 250 90.000 94.980 25.87 4969.51 19.90

18 250 95.000 99.980 23.34 5473.17 21.90

19 250 100.000 104.980 21.62 5881.78 23.55

20 250 105.000 109.980 20.54 6172.39 24.70

21 250 110.000 114.980 19.99 6332.92 25.35

22 250 115.000 119.980 19.89 6361.64 25.45

23 250 120.000 124.980 20.22 6264.97 25.05

24 250 125.000 129.980 20.96 6054.63 24.20

25 250 130.000 134.980 22.16 5745.04 23.00

26 250 135.000 139.980 23.90 5351.25 21.40

27 250 140.000 144.980 26.34 4887.76 19.55

28 250 145.000 149.980 29.74 4367.80 17.45

29 250 150.000 154.980 34.58 3803.11 15.20

30 251 155.000 160.000 41.83 3215.39 12.80

Total time = 100012.80

Data analysis

It is important that the data analysis software used to process VCT data be modified to handle the new format. with the step counting time and the step intensity being read in at each point. The step counting time is used to modify the calculated step intensities, not to simply scale the observed intensities. Since the weighting scheme is usually based on the accumulated count at each point, the correct weighting will be maintained.

References

Madsen I.C. and Hill R.J. (1991) "Rietveld Analysis Using Para-Focusing and Debye Scherrer Geometry Data Collected with a Bragg-Bentano Diffractometer", Zeitschrift fur Kristallographie, 196, 73-92

Madsen I.C. and Hill R.J. (1992), "Variable Step-Counting Times For Rietveld Analysis, or, Getting The Most Out of Your Experiment Time", Advances in X-ray Analysis, 35, 39-47

Madsen I.C. and Hill R.J. (1994), "Collection and Analysis of Powder Diffraction Dtaa with Near Constant Counting Statistics", J.Appl. Cryst, 27, 385-392