Peak Indexing and Lattice Parameter Refinement

8/10/2019 Peak Indexing and Lattice Parameter Refinement

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Peak indexing and Lattice Parameter refinement

Dr Nik Reeves-McLaren, Prof Neil C. Hyatt

Once we have identified what chemical phases are present within a sample, we may wish to

characterise these phases a little further. Can we assign Miller indices to each observed

Bragg reflection? What is the symmetry and size of the unit cell? This short primer is

intended to help people new to X-ray diffraction with such tricky subjects. Throughout this

guide, instructions for the user to choose in software tend to be given in bold.

Table of Contents

Introduction ................................................................................................................................ 2

Data Collection Strategies.......................................................................................................... 2

Indexing a powder pattern manually.......................................................................................... 3

Lattice Parameter RefinementGetting Started ....................................................................... 4

Lattice Parameter Refinement using WinXPOW ...................................................................... 7

Step 1: Creating a peak file for peak calibration.................................................................... 7

Step 2: Peak calibration ......................................................................................................... 8

Step 3: Creating a peak file for the phase of interest ........................................................... 10

Step 4: Pattern Fitting (Optional) ......................................................................................... 11

Step 5: Lattice Constant Refinement ................................................................................... 12


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Introduction

Every solid crystalline compound gives its own unique X-ray diffraction pattern, consisting

of a set of Bragg peaks. The diffractionpattern for a compound can be consideredanalogous to a fingerprint, or barcode, with the peak positions determined by the unit cell

symmetry and dimensions (or lattice parameters). When we collect XRD data, we can use

these fingerprints to identify not only what phases are present in our sample(s), but also

index the pattern to obtain information about the unit cell size and shape.

Data Collection Strategies

Lattice parameter refinement does not necessarily require data of very high quality. The only

important parameter is peak position, and the peaks will remain in the same place no matter

how long we collect data for. However, our ability to determine the peak position accurately

will depend to some degree on the peak shape; i.e.it will be difficult to determine the exact

position of a noisy peak, or one lost amongst the background!

For indexing and lattice parameter refinement, therefore, there are four main decisions thatshould be made before carrying out experimental work:

- Equipment: Bragg-Brentano (also known as reflection) geometry instruments (i.e. the

Philips PW1825, Siemens D500 and D5000) can be used, but specimen height

displacement errors can easily introduce significant error into refined lattice

parameters, and so use of an internal standard of known lattice parameter is essential.

For powder samples, the high resolution STOE PSD systems are excellent choices,

though an internal or external peak position standard must still be used; use of theSTOE imaging plate detectors is not recommended.

- Scan range: The resolution in d-spacing increases with 2, which means that for

lattice parameter refinement we want to collect data to high angles. Avoid collecting

unnecessary angular ranges where only background noise will occur.

- Scan rate: Try to collect at the fastest rate possible, and then see how good your data

are:

o For Bragg-Brentano geometry instruments, start with a scan rate of 1 per

minute. You should aim to record 2000-5000 counts on your largest peaks.


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o On the STOE PSD system, the data collection time necessary will depend on

the nature of your sample. For cubic samples, 1 or 2 hours will most likely be

more than sufficient. For lower symmetry, more time will be required.

- Step size: For lattice parameter refinement, peak resolution (i.e. being able to

distinguish two peaks that are very close together) and accurate peak position

determination are critical. Use step sizes of 0.02 2on Bragg-Brentano instruments,

or 2-4 points to be added on the STOE PSD systems.

Again: the most common pitfall with Bragg-Brentano experiments is specimen height

displacement. Loading the sample too high, with respect to the sample holder, will cause any

Bragg peaks to be shifted to higher angles; if the sample is too low, the peaks will be shifted

to lower angles. The effect on peak positions is almost, but not quite linear with 2. It is

always worth making sure your sample preparation is as good as possible!

Indexing a powder pattern manually

Indexing is the process of determining the size and shape of the unit cell for a given phase. Itinvolves assigning the correct Miller indices to each Bragg reflection. The complexity of the

task depends on the symmetry of the phaseit is easy for cubic, tetragonal and orthorhombic

systems, but gets significantly more complex when the angles are not 90 !

In these notes, I will deal with a cubic system:

We know that the Miller indices are integer numbers, and hence the (h2

+ k

2

+ l

2

) term mustalso be an integer value. Furthermore, the quotient / 4a2must be constant for any given


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diffraction pattern, since both and aare fixed values. This means that we can use ratios of

sin2values to help us determine the correct Miller indices for reflections.

Note that for each Bravais lattice type, only certain values are permitted for (h2+ k2+ l2):.

Primitive: All values of h, k, l are allowed

Body centred: Values of h + k + l = 2n only are allowed (where n is integral)

Face centred: Values of h, k, l all odd or all even are allowed (0 counts as even)

Also, there are certain prohibited values for (h2+ k2+ l2), e.g. 7, 15, 23, which cannot be the

result of the sum of three squares. Knowing all of this can be very useful in indexing, and

means we need at least seven reflections to distinguish between primitive and body centred

cells!

To index our pattern, we first collate a list of peak positions, which are then converted

manually to sin2values. We then divide these sin2values by allowed (h2+ k2+ l2) for

primitive, body centred or face centred unit cells. We then inspect the sin2/ (h2+ k2+ l2)

valuesare they all constant? If not, repeat for the next unit cell type. Once we find

constant values we can then assign Miller (hkl) indices for each reflection and determine the

cell constant, a.

Consider the following example, where a number of Bragg peaks were recorded from an

unknown material using Cu Kradiation.


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The resolution in d-spacing increase, while systematic errors decrease, with diffraction angle.

Therefore, to calculate the unit cell constant we should use high angle reflections in the

above example, we would choose the ninth reflection:


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From: West, A.R., Basic Solid State Chemistry, John Wiley & Sons Ltd, 1996.


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Lattice Parameter Refinement Getting Started

The WinXPOW

software can also be used to index and refine lattice parameters from yourdata, but this should be approached with caution. The software does not understand your

sample, and does not have any intelligence. It may give you an answer, but that does not

mean it is correctthat is for the user to judge! What you will need is a starting model:

- a space group, or at least the symmetry and centring (e.g.if you were looking at a

spinel, you should know that the space group for a spinel is likely Fd-3m, or at the

very least know that you are looking at something face-centred cubic), and also

- a good estimate of the lattice parameters.

This is made easier if the researcher has something to compare against. The most useful and

powerful source of assistance available in the Department of Materials Science &

Engineering is a database made by the International Center for Diffraction Data (ICDD)

called the Powder Diffraction File (PDF). The software we use to access this database is

made by STOE, and is called WinXPOW. The PDF contains a mixture of experimentally-

determined and calculated X-ray diffraction patterns. In the 2004 edition licensed by

Engineering Materials, there are a total of 134,000 patterns belonging to inorganic and

organic compounds. It is likely that one or more of these reference patterns belong either to

your material or to something analogous to iti.e. with the same crystal structure, and a

similar unit cell. This may give you the information you need.

Lattice Parameter Refinement using WinXPOW

Step 1: Creating a peak file for peak calibration

The first thing we need to do is to create a peak file a list of all the peak positions for a

given phase for our internal or external peak position standard. The Graphics (Raw

Data/Graphics) program can be used to view STOE *.raw files and create peak (*.pks) files

for indexing/refining unit cells.


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To open a STOE *.raw file, click File/Open. Then, we must find the standard peaks in your

diffraction pattern; clicking Options/Find Peaks will bring up a child window where the

Peak Search Parameters may be entered:

Estimated Halfwidthcharacteristic of both the sample and the instrument the data

were collected upon. Values of ~ 0.1 (for the Stoe PSD) and ~ 0.15 (for all our other

instruments) are reasonable.

Significance Level- sets a lower limit to the peak heights, defined as the ratio of the

maximum peak intensity to background.Default value of 2.5 is fine.

Intensity Limit- All peaks having an absolute height less than the input value will be

discarded. Entering low values significantly increases the chance that the software

will find artificial peaks in the random noise of the background!

Remove Alpha2 Peaks - Checking this box will scan the peak list for pairs of

Alpha1/Alpha2 peaks and discard the Alpha2 reflection.

Remove Spikes - Peaks of a halfwidth less than 25% of the estimated halfwidth

defined above will be discarded as spikes, or electrical noise.

Clicking OK will perform the peak search. If you are using an external standard, this is

straightforwardall the peaks will be from the reference material. Check the peaks have all

been selected appropriately, and save the peakfile.

If you are using an internal standard, the diffraction pattern will contain peaks from both the

standard material and from your specimen. At this stage, it is vital that we only include those

peaks from the reference material; we can manually add or edit peaks as well - double-click

the left mouse button to add a peak, right-click to remove a peak, or hold the left-mouse

button to drag a peak in to a chosen position. Check the peaks have all been selected

appropriately, and save the peakfile.

Step 2: Peak calibration

The Calibrate Peaks program allows the user to calculate correction terms for the 2 scale of

data collected with an internal (or external) standard reference material for the purposes of


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this guide it will be assumed that you are using silicon. The program calculates a correction

term which may be used to correct raw data files in the Raw Data Handlingprogram.

To perform a peak calibration on your data set, you will need a copy of the si-icdd.pksfile, a

peak file for the Si standard powder. You will also need a Si peak file, created from data

collected on the same day as your data (in the case of the Stoe PSD) or from a Si internal

standard (i.e. Si powder added to your sample). This latter course of action is recommended

if you wish to refine lattice parameters from data collected on any instrument other than the

Stoe PSD.

To perform the peak calibration:

1. Run Raw Data/Peak Calibration

2. Click File/Open. A child window will appear; enter the location of the si-icdd.pks

file in the upper box, and your Si file in the lower box.

3. From the menubar, select Calibrate/Match Peaks. In the child window that appears,

there are two options:

a. Intensity limit- a cutoff-value below which the program will ignore all peaks

from both the calibration and the observed peak files. Set to 0 %.

b.

2-Theta window: the 2theta range around a calibration peak which the

program will search for matching observed peaks. Set this to 0.05.

You will need at least three peaks, but more is better.

4. You may need to connect a measured peak to a calibration peak; to do this, left-click

the (blue) calibration peak, then attach the measured (grey) peak by left-clicking on it.

5. Once all your measured peaks are attached to calibration peaks, click

Calibrate/Calculate Polynomials. The results are shown in the lower part of the

main window. The 2theta offsets for all peak pairs are shown as black squares and the

calculated correction polynomials as curves of different colours.

6. Click Calibrate/Calibrate Peakfile. A child window will appear asking which

polynomial should be applied; choose the one with order > 0 with the lowest chi^2

value, and check that this fits the observed data well. Click OK and exit from the

Calibrate program withoutsaving changes.

7. Go to Raw Data/Raw Data Handlingand Openthe file you wish to calibrate. Click

Operations/Angle Correction. Make sure the use stored polynomial box is

ticked, and click OK.


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8. Save your corrected data.

The next thing we need to do is to create a peak file a list of all the peak positions for a

given phase. The Graphics (Raw Data/Graphics) program can be used to view STOE *.raw

files and create peak (*.pks) files for indexing/refining unit cells.

Step 3: Creating a peak file for the phase of interest

To open a *.raw file, click File/Open.

Then, we must find the peaks in your diffraction pattern; clicking Options/Find Peakswill

bring up a child window where the Peak Search Parameters may be entered:

Estimated Halfwidthcharacteristic of both the sample and the instrument the data

were collected upon. Values of ~ 0.1 (for the Stoe PSD) and ~ 0.15 (for all our other

instruments) are reasonable.

Significance Level- sets a lower limit to the peak heights, defined as the ratio of the

maximum peak intensity to background.Default value of 2.5 is fine.

Intensity Limit- All peaks having an absolute height less than the input value will be

discarded. Entering low values significantly increases the chance that the software

will find artificial peaks in the random noise of the background!

Remove Alpha2 Peaks - Checking this box will scan the peak list for pairs of

Alpha1/Alpha2 peaks and discard the Alpha2 reflection.

Remove Spikes - Peaks of a halfwidth less than 25% of the estimated halfwidth

defined above will be discarded as spikes, or electrical noise.

Clicking OK will perform the peak search. It is important to make sure that if we want to

refine the lattice parameters for one given phase, then at this point we should ensure that only

the peaks from that phase are selected. We can manually add or edit peaks as well - double-

click the left mouse button to add a peak, right-click to remove a peak, or hold the left-mouse

button to drag a peak in to a chosen position.


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At this stage, you may well choose to omit peaks if you are not confident about the position

that they have been placed in.

Step 4: Pattern Fitting (Optional)

The Pattern Fitting routine is used to determine profile parameters and integrated intensities

from observed powder patterns. A variety of constraints makes it possible to achieve

meaningful results even for complicated patterns. This routine does not require any

information on the structure of the phase.

1. Open up the *.raw and *.pks file for the pattern you wish to fit.

2. Click the Parametermenu option. You will notice three choices:

a. Overall background and asymmetry functions; unless the background is

strange, leave these to the program.

b. PeaksLists the 2values, absolute and integrated intensities, and halfwidths

of all the peaks.

c. GroupsParameters that may be fixed or varied during refinement.

3. Choose the Group option. Start by refining the halfwidths only. The initial

parameters should be W = 0.01, V = 0, U = 0. Tick Halfwidth as function, and

unfix Wmake sure that initially all the other parameters are fixed. Click Replace

to enforce any changes you have made, and then OK.

4. You are now ready to start the refinement: click Operations/Nonlinear Regression.

Once the refinement has converged, go back to Groupsand allow Uand Vto vary.

Remember to click Replace after youve made any changes in the Groups window!

5. Once convergence is achieved, allow the profile function to vary. There are two

Pearson and Pseudovoigt try both to see which fits your data best. The default

parameters for the profile function are 0.5 for Pseudovoigt and 2 for Pearson.

6. Inspect the pattern for missed peaks, extra peaks, poorly fitted peaks and edit

accordingly by clicking Operations/Edit Peaks.

7. Inspect the peak file for negative intensities, halfwidths that are very large (may be

two peaks?) or very small (may not be a peak!)

8. Some peaks at low 2 values may be asymmetric. This can be modelled using the

asymmetry function Click Parameter/Overall Parameter and click Use


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asymmetry and untick fixed. The starting value should be greater than 0.03.

Note: This step will not be necessary if you dont have any peak asymmetry!

9. Once you are happy with the fit, save the fitted peak file as .pft.

Once again, remember that more detailed help is available from the help files in the Pattern

Fitting program.

Step 5: Lattice Constant Refinement

WinXPOWsIndex & Refine module comprises three indexing routines and a lattice parameter

refinement option. The implemented indexing algorithms will give a list of possible unit

cells, and are based on:

Werner's TREOR program (see e.g. P.-E.Werner, L.Eriksson, M.Westdahl, J. Appl.

Cryst. 18 (1985) 367)

Visser's ITO routine (see e.g. J.W.Visser, J. Appl. Cryst. 2 (1969) 89)

Lour's DICVOL (see e.g. A.Boultif, D.Lour, J. Appl. Cryst. 24 (1991) 987).

The Index & Refine software can be accessed from the main WinX POW console window

under Cell. This program can be used to analyse peak files in *.pks or*.pth format.

The Index menu options

Under the Index heading, there are four entries. Before running any of the algorithms,

however, you first need to define the refinement parameters.

1. Click on Index/Parameters. A child window will appear, with three tabsone for

each of the available algorithms. For the algorithm you intend to use, fill in all the

boxes you cana detailed list of what each item means can be found in the help files

accessible from the Index program.

2. To run your algorithm of choice (try each one to see which provides the best possible

solutions), click Index/Run . This will start the calculation

processthis may take some time! You can then try refining your sample based on

these solutions.


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When Visser's and Lour's routines have finished, they open a child window displaying their

output. Werner's routine writes possible solutions into the main window and after

completing the search - the user may double-click on any of them to open a child window

showing the indexing of the data with the currently selected cell (highlighted in red in the

main window). Werner's algorithm may be interrupted anytime by clicking the toolbar button

with the red traffic light.

Lattice Constant Refinement

You may already have a possible solution, from literature, the Powder Diffraction File, or one

of the algorithms above. The software allows you to refine the lattice constants for your

observed data using the Refine menu option. To work through such a refinement:

1. Openyour peakfile (.pks) in the Index & Refineprogram.

2. Click Refine. A child window will appear, offering you various data entry options,

all of which should be self-explanatory. Fill these boxes in:

a. Symmetry- Enter the space group, if you know it, or the crystal system (e.g.

cubic) and Bravais lattice (e.g.Primitive) if you dont.

b.

Lattice Parameters Enter in the dimensions (in ) and angles of the cell

upon which you wish to refine your data.

c. Settings

i. 2Theta window- If the absolute difference between an observed peak

position and a calculated peak position is less than the 2Theta window

(in degrees), the observed peak will be considered indexed. The

program will try to reduce the window as much as is possible without

observed peaks becoming unindexed. A value of 0.1 is reasonable.

ii. 2Theta zero- allows the input of a constant 2theta-error introduced by

poor 2Theta-calibration of the diffractometer. The check box Refine

2Theta zero' is used to enable or disable refinement of this value

together with the cell parameters (not advised!).

iii. Use peak file indices- if checked, starting values for refinement are not

taken from the Lattice parameter' edit fields but are generated from all

peaks in the peak file which are indexed. Note: it is possible to enter


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hkl values into a peak file using Notepad just place the values

directly under the H, K and L headings.

3. Press OK to perform the refinement.

4. The result of the attempted refinement will now be displayed. There are basically two

possible outcomessuccess (to some level) or failure.

a. FailureYou will see a window much like the one below. Your solution is

not the answer, or the lattice parameters may be too far from the correct

values; try working them out from individual reflections first. Your starting

model must be very close to the final answer if you are to successfully refine

the lattice parameters!

b. Success? You will be prompted with a list of text, detailing the refined

values for your lattice constants. You should now plug these lattice constants

back into the Refine window, and try refining again. Keep doing this until

the values no longer change. You will also be told how many peaks are fitted

by the model; if all of them are fitted, then well done! If there are reflections

not accepted by the model, consider possible reasons for this is the model

correct? Could they be from secondary phases? Is the 2window too small?

If you know that any unidentified peaks should be there, try indexing these

manually in Notepadas described above.

For a good lattice parameter refinement, the Final 2window, i.e. the biggest value from

the obs-calc column, should be smaller than 0.03. This means that all of the observed

reflections fall within 0.03 of a peak whose position has been calculated using the refined

lattice parameters.

Once you have reached a satisfactory answer, you should save your refined peak file

(File/Save As). Saving the file also automatically saves a *.lrf file, with the same filename

as your *.pks file, with all the lattice refinement data. This *.lrf file can be viewed inNotepad, and contains all of the results of the lattice parameter refinement.

Peak Indexing and Lattice Parameter Refinement

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