Phase Analysis and Structure Refinement – A Tutorial Part 1: Phase

Dr. Frank Girgsdies, Nanostructures Group, Dept. of Inorganic Chemistry, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany 1

Powder X-ray Diffraction: Phase Analysis and Pattern Fitting

Phase Analysis and Structure Refinement – A Tutorial

Part 1: Phase Analysis Phase iden8fica8on by search/match methods

Part 2: Pa;ern Fi=ng Extrac8ng informa8on from powder pa;erns by various fit methods

Powder X-‐ray Diffrac8on: Phase Analysis and Pa;ern Fi=ng



Informa8on obtainable from powder diffrac8on:

• Qualita8ve analysis (phase iden8fica8on) → Part 1

• Quan8ta8ve analysis (phase mixtures) → Parts 1 & 2

• La=ce parameters → Part 2

• Crystallite sizes & defects (microstructure) → Part 2

• Crystal structure) → Part 2



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Informa8on content of an idealized diffrac8on pa;ern

peak area (integral intensity): real measure for peak intensity relative intensities... a) ...within one phase → crystal structure (contents of the unit cell, i.e. types and relative positions of atoms) b) ...between different phases (in a mixture) → phase quantification

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peak position: d-spacing → lattice parameters (size & shape of the unit cell)

peak height (maximum intensity): approximation for peak intensity

peak width: a) full width at half maximum (FWHM, a.k.a. “half width”) depends on peak profile b) integral breadth (= integral intensity / maximum intensity) less dependent on peak profile → crystallite size, defects (strain, disorder)

peak shape (peak profile): → crystallite size, defects (strain, disorder)

n λ = 2 d sin(θ)



The ideal crystal structure

la=ce → unit cell parameters (unit cell size & shape) → peak posi8ons

= crystal structure + basis la=ce

unit cell

basis → unit cell contents (atom types & posi8ons) → rela8ve peak intensi8es



The real (crystal) structure

Ideal crystals are infinite, but real crystals are not: •  crystallite/domain size effects → peak shape, peak broadening

In ideal crystals, all unit cells are iden8cal, but real crystals have defects: • missing/wrong/addi8onal atoms, thermal mo8on → rela8ve intensi8es

•  stacking faults, twin boundaries → peak shape, peak broadening, even peak shiYs

•  local la=ce distor8ons = varia8on of d-‐spacings = (micro)strain → peak shape, peak broadening



Phase Iden8fica8on (Search & Match)

In powder XRD, crystalline phases are iden8fied by the posi-ons and intensi-es of the diffrac8on peaks.

Note: The diffrac8on angle 2θ (θ = Bragg angle) is related to the d-‐spacing via the Bragg equa8on n λ = 2 d sin(θ). Thus, the peak posi8ons on the angular scale are wavelength dependent!

To iden8fy phases from XRD, you need: a) experimental diffrac8on data, b) one or more database(s) of reference pa;erns, c) a search/match soYware. Note: The soYware performing the search acts as an interface to the database, thus the soYware and database must be compa8ble with each other.




Depending on the search/match strategies supported by your soYware, your diffrac8on data may (or must) have one of the following formats:

a) raw (i.e. unprocessed) diffrac8on data in binary or ASCII format,

b) processed diffrac8on data (e.g. background subtrac8on, α2 stripping),

c) a peak list (posi8ons & intensi8es) derived from the measured data by peak search (crude) or peak fi=ng (more precise).




Some so3ware examples:

• WinXPow (STOE & CIE): automa8c search on peak list, or manual matching with raw data (or peak list)

•  DIFRAC EVA (Bruker AXS), older version: automa8c search on background subtracted raw data, automa8c search on peak list

•  DIFFRAC EVA (Bruker AXS), later version, and •  HighScore / HighScore Plus (Panaly8cal): automa8c search on raw data (background subtracted automa8cally), automa8c search on peak list



Search & Match with WinXPow



Search & Match with EVA (older version)



Search & Match with EVA (later version)




General strategies:

•  For a peak list search, the user has to define a 2θ tolerance interval within which an agreement between reference and data is considered to be a match.

•  Raw data search algorithms use the digital intensity informa8on, without any separate peak recogni8on. Thus, the background intensity has to be subtracted (by the user or the soYware) before the search is executed.

•  Note that α2-‐stripping is recommended in the older EVA manual, but it's prac8cal impact is usually unimportant or even nega8ve!




Analysis of phase mixtures:

•  If a posi8ve match is found, but some experimental peaks remain unexplained, another search has to be performed on the unexplained peaks (residual).

•  In the older DIFFRAC EVA version, a scissors tool can be used to manually exclude explained (or unwanted) regions of the experimental pa;ern before a residual search is conducted. This procedure can be tedious, but gives the user maximum control.

•  In later DIFFRAC EVA versions and HighScore, a residual scan can be prepared by omission of user controlled intervals around explained peaks. This method is easier, but especially in EVA the "one threshold for all peaks" approach does not work so well.



Phase Iden8fica8on (Search & Match) Databases:

We have to dis8nguish between reference pa;ern databases and crystal structure databases.

Pa;ern databases are used for search/match phase iden8fica8on. The entries contain informa8on about peak posi8ons and intensi8es, but tradi8onally no detailed structure informa8on (e.g. atomic coordinates). The crystal structures of the respec8ve phases may be known or unknown, the pa;erns may be indexed or not, measured or calculated.

Crystal structure databases contain structural informa8on, including space group, unit cell, and atomic coordinates, which is necessary for Rietveld based pa;ern fi=ng. They usually do not contain diffrac8on pa;ern informa8on.

Since diffrac8on pa;erns can be calculated from crystal structures, pa;ern databases may contain calculated pa;erns derived from crystal structure database entries.



Phase Iden8fica8on (Search & Match) Pa@ern databases:

The most commonly used commercial pa;ern databases are the various versions of the PDF (Powder Diffrac8on File) maintained by the ICDD (Interna8onal Centre for Diffrac8on Data), formerly known as JCPDS (Joint Commi;ee on Powder Diffrac8on Standards).

Note that the JCPDS was renamed to ICDD in 1978, so the s8ll ongoing cita8on of "JCPDS" in the literature is somewhat outdated.

The PDF-‐2 covers inorganic materials. Since 2003, calculated pa;erns based on ICSD and NIST crystal structures are included on a larger scale.

The PDF-‐4/Organics and PDF-‐4/Minerals cover organics/organometallics and minerals, respec8vely. PDF-‐4+ covers inorganic materials. All PDF-‐4 products include a high percentage of calculated pa;erns which are accompanied by corresponding crystal structure data.



Phase Iden8fica8on (Search & Match) Pa@ern databases (con-nued):

As a recent development, calculated pa;erns derived from the non-‐commercial COD (Crystallography Open Database) are supported by commercial search/match soYware (e.g. EVA 3 and HighScore) as alterna8ve to the commercial PDF products. Note that a simultaneous search over both databases is not supported due to license restric8ons from the ICDD.



Phase Iden8fica8on (Search & Match) PDF quality marks:

•  * (Star): high quality measured pa;ern; chemical composi8on well characterized, intensi8es measured objec8vely, no serious systema8c errors, average delta 2θ < 0.03°.

•  I (Indexed): pa;ern has been indexed (thus almost certainly single phase); reasonable range and even spread in intensi8es, no serious systema8c errors, average delta 2θ < 0.06°.

•  0 (Zero): diffrac8on data of poorly characterized material, or data known (or suspected) to be of low precision.

•  B (Blank): pa;erns which do not meet the "*", "I", or "0" criteria.

•  D (Deleted): pa;ern which was subsequently discredited, or was superseded by a later, be;er pa;ern (check comments of the entry for details). Remains available as part of the database for backward reference reasons.



Phase Iden8fica8on (Search & Match) PDF quality marks (con-nued):

•  R (Rietveld): pa;erns with d-‐values directly from Rietveld refinement; accepted only in unusual cases.

•  C (Calculated): pa;ern calculated from single crystal data; very precise d-‐values, but intensi8es may differ somewhat from experimental pa;ern. C pa;erns contain reference to crystal structure database entry used for calcula8on; useful for later Rietveld refinement.

•  A (Alterna8ve): used to mark alterna8ve C pa;erns if mul8ple entries for the same phase exist; useful to limit the number of choices.

Note: If you have the choice between several alterna8ve pa;erns for the same phase, you should chose a "*" or "C" pa;ern.



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: How many peaks must match between a reference and measured pa;ern? A: All peaks of the reference pa;ern should be present in the measured data, otherwise it is not a valid match.

obs

ref ý

obs

ref þ

obs

ref þ

Possible excep8ons: •  Small peaks are below detec8on limit (high noise level) •  Missing peaks may result from extreme preferred orienta8on effects or extreme anisotropic peak broadening → check if effect is hkl dependent

•  Bad sta8s8cs (few, large crystallites) → peaks would be very sharp; try grinding the sample thoroughly and measure again

valid match (single phase pa;ern)

valid match (mul8 phase pa;ern)

invalid or ques8onable match (some reference peaks missing in experimental pa;ern)



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: All reference peaks occur in the measured pa;ern, but the measurement

contains addi8onal peaks. What does this mean? A: The iden8fica8on is probably correct, but the measured pa;ern represents a

phase mixture. → Keep the reference pa;ern, then con8nue searching for references to explain the addi8onal peaks. Proceed un8l all peaks are explained.

obs

ref ?

obs

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Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: All measured peaks are explained with a reference, except for some very

small ones. I tried to iden8fy them as an impurity phase, but failed. What could this mean?

A: Possible reasons are: •  The peaks are ar8facts resul8ng from spectral impuri8es (other wavelengths,

e.g. Kβ, W L), i.e. weak "duplicates" of very strong peaks. •  The peaks are real and belong to the reference compound. Your experimental

pa;ern might be be;er (signal/noise ra8o) than the reference pa;ern. AYer all, diffractometer technology improves with 8me. → Try to check if the addi8onal peaks are compa8ble with the unit cell parameters of the reference compound.

obs

ref ?



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: There is a match between all measured and reference peak posi8ons,

but the rela8ve intensi8es disagree. What does this mean? A: Possible reasons are: •  The iden8fica8on is correct, but there are preferred orienta8on effects in the measured

data. → The intensity devia8ons should correlate systema8cally with hkl → Check with Rietveld fit including a preferred orienta8on model

•  The iden8fica8on is correct, but the reference intensi8es have a low level of precision. → Check if the reference intensi8es have only discrete, rounded values (e.g. 100%, 80%, 50% etc.). If so, the reference data originated from visual evalua8on of a photographic film (i.e. "very strong", "strong", "medium" etc. translated into percentage numbers).

•  The iden8fica8on is incorrect, the peak posi8ons coincide only by chance (not very likely, but possible).

obs

ref ?

obs

ref þ



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: The measured and reference pa;erns look very similar, but the peaks seem to

be shiYed. What does this mean? A: Check if the individual peak shiYs are

(a) constant in direc8on and magnitude, (b) constant in direc8on but with increasing magnitude at increasing angle, or (c) variable in their direc8on.

(a) A constant peaks shiY is most probably a measurement ar8fact.

If the defini8on of the 2θ scale of the diffractometer is off, then the shiY should be strictly constant (zero shi+, zero error). If the posi8on of the sample deviates from the center of the goniometer, then the shiY is not exactly, but close to constant (sample displacement, height error).



Phase Iden8fica8on (Search & Match) (b) Increasing peak shiYs with increasing angle are caused by differences in the

la=ce parameter(s). (c) The same is true for shiYs with opposite signs (some la=ce parameters

increase, some decrease). Differences in la=ce parameters may be caused by: •  Temperature differences (thermal expansion; could be even nega8ve in some

cases!) •  Chemical differences: instead of e.g. AO2, you may have

A1-‐xBxO2 (doping/solid solu8on), AO2-‐x (variable vacancies), BO2 (isostructural compound), ...

Note: ShiYs to lower angles mean expansion, shiYs to higher angles contrac8on of d-‐spacings/la=ce parameters.



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: I explained all diffrac8on peaks of my measured pa;ern with the reference

pa;ern of my target compound. Both peak posi8ons and intensi8es agree very well. May I say that my sample is phase pure?

A: No, not necessarily.

Strictly speaking, XRD will never be able to prove phase purity. There may be impuri8es which are invisible to XRD, either because their amount is below the detec8on limit, or because they are "XRD amorphous". However, if your measured pa;ern has a very low noise level and no unexplained features (e.g. broad bumps) in the background, then you have a good chance that your sample is indeed phase pure.



Phase Iden8fica8on (Search & Match) Pa@ern matching FAQ: Q: What do you mean by "XRD amorphous"? A: With "XRD amorphous" I mean absence of proper Bragg diffrac8on peaks.

The material might be truly amorphous, i.e. lack any long range ordering (crystal la=ce). However, the material could also be what I would call "nanocrystalline", i.e. crystalline with very small crystallite sizes. Remember that the intensity of diffrac8on peaks is given by their area, and that small crystallites lead to peak broadening. Hence, if the crystallite size is reduced, the peaks will not only become broader, but also lower. Furthermore, diffrac8on peaks have no abrupt end, thus they tend to merge smoothly with the background, becoming increasingly indis8nct. S8ll, such a "nanocrystalline" sample without recognizable Bragg peaks will show la=ce fringes in high resolu8on TEM.




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Peak broadening: no discrete border between crystalline and "nanocrystalline XRD amorphous" materials



Phase Iden8fica8on (Search & Match) Phase quan-fica-on by the RIR (Reference Intensity Ra-o) method In the PDF product family, many experimental and all calculated reference pa;erns are accompanied by I/Ic values. I/Ic is the ra8o between the intensi8es of the strongest peaks of the respec8ve phase and corundum in a binary 50:50 wt.% phase mixture. In ideal powder XRD, the intensi8es are approximately propor8onal to the weight frac8ons of the corresponding phases. Thus, if all components of a mul8phase mixture have been iden8fied with suitable reference pa;erns (with I/Ic values) and the reference pa;erns are scaled to match the intensi8es of the experimental pa;ern, then the ra8os of the I/Ic values weighted by the individual scaling factors yield the approximate weight frac8ons of the phases (Ic contribu8ons cancel out).




Phase quan-fica-on by the RIR (Reference Intensity Ra-o) method Note: •  This method requires that all phases are accounted for (no amorphous or uniden8fied phases, no missing I/Ic values).

•  Any distor8ons of rela8ve peak intensi8es, e.g. by preferred orienta8on effects or microabsorp8on, will lead to erroneous results.

•  In the RIR method, the peak height (instead of area) is used to scale the reference pa;erns to the measured data. Thus, stronger varia8ons in the peak width will distort the results (no propor8onality between peak height and area).



Pa;ern Fi=ng

Some general remarks on fiMng: •  A model (empirical or physically based) is used to calculate a theore8cal data set.

•  The variable parameters of the model are op8mized (usually by least squares algorithms) to obtain the best possible agreement (fit) between calculated and measured data.

•  The aim of fi=ng experimental data is oYen to extract physical parameters from the data.

•  Only with a physically based, appropriately chosen model, the refined parameters may be interpreted as physically meaningful quan88es.

•  However, even if a physical model is absent, fi=ng may s8ll be useful to parameterize the measurement data (data reduc8on).



Pa;ern Fi=ng

Some general remarks on fiMng: •  An increase in the number of refined parameters usually improves the fit (agreement between measured and simulated data) because the model becomes more flexible.

•  If the number of refined parameters is increased too much, they become less well defined (parameters start to correlate with each other).

•  In the worst case, you will have an excellent looking fit with a set of meaningless parameters ("overfi=ng"). → The number of refined parameters must stay small compared to the number of "observables".

•  The strategy for any fit should be to obtain the best possible fit with as few fit variables as possible (which is in most cases easier said than done). → The choice of a fit model is always a compromise and should be based on physical reasoning.



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: WPPM vs. WPPF

•  In powder diffrac8on, a dis8nc8on is usually made between Whole Powder Pa;ern Fi=ng (WPPF) and Whole Powder Pa;ern Modelling (WPPM).

•  In WPPF, the peak shape func8ons used are empirical, with the focus on the determina8on of crystal structure parameters (Rietveld method) and/or microstructural parameters (size/strain analysis).

•  In WPPM, the peak shapes are modelled based on physical assump8ons in order to analyze the microstructure and especially the defect structure (e.g. stacking faults) of the material.



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Rietveld method (crystal structure refinement)

•  The Rietveld method was originally invented to refine crystal structures from (neutron) powder diffrac8on data.

•  A Rietveld structure refinement primarily aims at the determina8on of the crystal structure geometry (atomic coordinates and la=ce parameters → bond distances).

•  Depending on the context and the data quality, site occupancies and thermal displacement parameters ("temperature factors") may also be refined.

•  The refinement of further parameters (peak profiles, background func8on, zero error and other correc8ons) only serves the purpose of determining the peak intensi8es as accurately as possible. → Preference for analy8cal peak shape func8ons.



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Rietveld method

•  Later, it was realized that the Rietveld scale factors obtained from fi=ng phase mixtures are related to the concentra8ons of the respec8ve phases → Rietveld based phase quan8fica8on.

•  For quan8fica8on and "advanced qualita8ve analysis", it is advisable to keep the atomic parameters of the reference structures fixed, thus reducing the number of refined parameters.

•  If a reference structure seems doub~ul or inappropriate for the samples inves8gated, one may a;empt to refine the structural parameters once on a reference measurement (data quality and phase purity as good as possible) to obtain a modified reference structure, then keep the structural parameters fixed again during further analysis.



Pa;ern Fi=ng

70686664626058565452504846444240383634323028262422201816141210

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t-ZrO2_refined 4.63 %m-ZrO2_ICSD-89426 95.37 %



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70686664626058565452504846444240383634323028262422201816141210

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t-ZrO2_refined 4.68 %m-ZrO2_refined 95.32 %



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Rietveld method

•  In a Rietveld fit, the intensi8es of the peaks are calculated from the crystal structure model (whether the model itself is refined or not).

•  Individual peak shapes of a phase are usually correlated via a peak shape func8on with refinable parameters. → May be used for microstructure (size/strain) analysis.

•  Individual peak posi8ons are correlated via the la=ce parameters. •  If the crystal structure model itself is not refined, the Rietveld method thus provides the lowest possible number of refined variables for WPPF.

Note: The correla8on of peak posi8ons and shape parameters reduces the number of variables and improves the extrac8on of intensi8es for overlapping peaks, which is of primary importance for a Rietveld structure refinement.



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Pawley and Le Bail methods

•  The Pawley and Le Bail methods have been developed to extract the reflec8on intensi8es from powder diffrac8on pa;erns for the purpose of structure solu8on.

•  Both methods may also be conceived as (or used as) structure-‐less methods in WPPF.

•  In the Pawley method, the individual peak intensi8es are refined fit variables. •  In the Le Bail method, the intensi8es are not refined directly. Instead, the par88oning scheme of overlapping reflec8ons is op8mized itera8vely in analogy to the Rietveld algorithm, but without the structure informa8on.



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Pawley and Le Bail vs. Rietveld

•  The Pawley and Le Bail methods share with the Rietveld method that the peak shapes and peak posi8ons of a phase are correlated.

•  The lack of a crystal structure model (which would determine the peak intensi8es) leads to a higher number of refinable parameters.

•  If a good Rietveld fit is possible, it is preferable over Pawley/Le Bail because of the lower number of fit parameters and the extra informa8on obtained (phase quan8fica8on).

•  If a Rietveld fit doesn't work well, a Pawley/Le Bail fit may be the be;er choice, as the increased flexibility usually leads to a be;er fit. → More accurate extrac8on of peak profiles (microstructure) and la=ce parameters.

•  However, you should ask yourself why the Rietveld approach has failed to produce the desired fit...



Pa;ern Fi=ng

Types of fiMng in powder diffrac-on: Single peak fiMng

•  Single peak fi=ng usually means that every individual peak has its own refined peak shape, posi8on and intensity parameters → very high number of variables in WPPF. Note: If you would do exclusively single peak fi=ng, this could barely be called WPPF at all. Never refine a peak shiY correc8on parameter in such a case!

•  The refinement of la=ce parameters (including peak shiY correc8on), microstructure etc. would have to be performed secondarily.

•  Single peak fi=ng is especially troublesome in case of peak overlap. •  Thus, single peak fi=ng should only be used for specific reasons or if there is no other alterna8ve.

•  Examples: extra peaks from uniden8fied impurity phases in a mixture, sample holder signals or other ar8facts, ...



Pa;ern Fi=ng

Powder pa@ern fiMng step by step: an overview •  Before you can fit a powder pa;ern, you need to iden8fy your phase(s) → see part 1 (phase iden8fica8on).

•  AYer iden8fica8on of the main protagonists, you should try to get corresponding crystal structure models for Rietveld based fi=ng. → Retrieve crystal structure data from structure databases or literature.

•  If the phase iden8fica8on can be based on a calculated pa;ern, it should give you a reference to the corresponding structure database entry (e.g. PDF-‐2 references to ICSD and NIST), or be accompanied by structural data (e.g. COD or PDF-‐4+).

•  If you cannot get crystal structure informa8on, you should at least try to get the la=ce parameters and space group (or at least crystal class) of that phase for use in Pawley/Le Bail fi=ng.

•  If you have none of the above informa8on, you are leY with single peak fi=ng for the corresponding phase.



Pa;ern Fi=ng

Powder pa@ern fiMng step by step: an overview •  Some8mes, you will realize during the pa;ern fi=ng that there are previously overlooked addi8onal phases in the samples (this is what I call "advanced qualita8ve analysis"). → Repeat the iden8fica8on and data retrieval procedure for the addi8onal phase(s).

•  If you are sa8sfied with the fit (i.e. with the agreement between calculated and measured data), you can proceed with extrac8ng physical data from the fit variables.

•  If all relevant phases were treated appropriately with Rietveld fi=ng, you will obtain the corresponding quan8ta8ve informa8on (weight percentages).

•  Refined la=ce parameters result for all phases represented by either Rietveld or Pawley/Le Bail fi=ng.

•  If a suitable peak shape func8on was chosen, Rietveld, Pawley and Le Bail phases will provide microstructural (size and/or strain) parameters.



Pa;ern Fi=ng

Powder pa@ern fiMng step by step: an overview •  In the case of single peak fi=ng, quan88es like la=ce and microstructure parameters can only be obtained by secondary evalua8on, usually with much lower precision than from correlated fi=ng.



Pa;ern Fi=ng

Crystal structure databases •  ICSD (Inorganic Crystal Structure Database) by FIZ Karlsruhe is the world's largest database for inorganic structures. Original defini8on: crystal structures of elements (including metals) and compounds containing at least one non-‐metal and no C-‐H bonds. As a recent development, the ICSD also incorporates some simple C-‐H containing compounds (especially formates, acetates etc.) and some intermetallic compounds.

•  CSD (Cambridge Structural Database) by CCDC (Cambridge Crystallographic Data Centre) covers small-‐molecule organic and metal-‐organic crystal structures (compounds with C-‐H bonds).

•  CRYSTMET by Toth Informa8on Systems covers metals, alloys, intermetallic compounds and minerals (cri8cally reviewed, physical metadata).



Pa;ern Fi=ng

Crystal structure databases •  NIST Crystal Data Iden8fica8on File contains inorganic and organic structures (hasn't been updated for several years).

•  Pearson's Crystal Data by ASM Interna8onal covers inorganic structures (cri8cally reviewed).

•  Furthermore, there are several databases for large organic molecules (mainly proteins).



Pa;ern Fi=ng

Crystal structure data •  Depending on the database source and the Rietveld soYware you are using, you may have to convert the crystal structure data format.

•  The most common format, however, is the CIF (Crystallographic Informa8on File) format, which is supported by most contemporary crystallographic soYware.

•  Note that the CIF is an "open format", i.e. it may always happen that some keywords are not understood by your soYware. Depending on the soYware, such keywords are simply ignored, or may lead to incorrect data import or even program crashes. In the worst case, you have to iden8fy the problem and edit the CIF file accordingly.



Pa;ern Fi=ng

Crystal structure data: possible piWalls •  Many space groups have alterna8ve se=ngs. In some cases, this involves different choices of crystallographic axes. In others, it is simply a difference in the choice of the coordinate origin.

•  Some8mes, the se=ng informa8on is coded in the space group symbol (different cell choices in monoclinic or orthorhombic space groups), but will be absent if the space group is iden8fied only via the space group number.

•  Alterna8ve origin choices can op8onally be coded by extending the Hermann-‐Mauguin symbol with the le;ers "Z" and "S".

•  Some space groups occur in different nota8ons. For example, space group No. 225 (Fm3m) is usually wri;en as Fm-‐3m, but some soYware or databases may use the nota8on Fm3-‐m or Fm3m instead.

•  The safest way to unambiguously describe the space group se=ng is to explicitly include the symmetry opera8ons in the CIF file, in addi8on to the space group symbol or number.

3



Pa;ern Fi=ng

Crystal structure data: possible piWalls •  Crystallographic soYware differs in how a space group is recognized from the data given in a CIF file.

•  Always check your crystal structures for plausibility aYer import. Erroneous interpreta8on of crystal structures by the soYware will lead to wrong intensity distribu8ons.

•  In the worst case, you may have to find out the correct se=ng by trial and error.

•  Some8mes, soYware may be offended by some forma=ng of the CIF file (e.g. non-‐integer oxida8on states). In this case, you may have to edit the CIF file accordingly.

•  Another possible source of error can be the end-‐of-‐line character problem between unix/linux and windows systems. → demonstra8on



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Some commonly used Rietveld so3ware (non-‐exhaus-ve list): •  Fullprof: freely available •  GSAS: freely available •  RIETAN: freely available •  TOPAS (Bruker AXS) and TOPAS Academic: commercial, differ in license fee and user interface

•  HighScore Plus (Panaly8cal): commercial, contains func8onality of HighScore with addi8onal Rietveld capabili8es

•  DDM: freely available, contains conven8onal Rietveld approach in addi8on to the Difference Deriva8ve Minimiza8on.



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Peak profiles •  The peak shapes of diffrac8on pa;ern result from a convolu8on of instrumental and sample contribu8ons.

•  For microstructural analysis, the instrument contribu8on must be separated from the sample contribu8on (not trivial, various approaches).

•  Instrumental profiles may be determined experimentally (using reference samples with negligible sample related peak broadening), or be calculated from known instrument geometries (e.g. Fundamental Parameters Approach, ray tracing methods).

•  Note that peak asymmetries from the instrumental profile will cause shiYs in the peak posi8ons. Thus, this effect needs to be accounted for to get reliable la=ce parameters.

•  For Rietveld refinement in the strict sense, empirical peak shape func8ons are commonly used.



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Peak profiles Commonly used peak profile func8ons are: •  Gaussian •  Lorentzian •  Voigt (a convolu8on of a Gaussian and a Lorentzian contribu8on) •  pseudo-‐Voigt (a linear combina8on of a Gaussian and a Lorentzian func8on; approaches the Voigt func8on very well)

•  Pearson VII (popular during the 1980s and 1990s, now superceded in popularity by the pseudo-‐Voigt)

Note: All above listed func8ons are symmetric. Some8mes, asymmetric versions (so-‐called "split" func8ons) are used to model peak asymmetry. Split func8ons have separate width an shape parameters for the parts leY and right of the maximum.



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Peak profiles Voigt func8ons with different Gaussian and Lorentzian contribu8ons: black: pure Gaussian red: pure Lorentzian The Gaussian has a narrower base and more rounded 8p, while the Lorentzian has a more pointed 8p and much more extending "feet".



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Correla-on of peak profiles •  In order to reduce the number of refined parameters and to be;er resolve peak overlap, the width/shape parameters of peaks belonging to a phase are usually correlated in WPPF via a func8on with refinable coefficients.

•  The most common way is to describe the peak shapes as func8ons of the diffrac8on angle θ.

•  Depending on the profile fi=ng approach implemented, the refined angular dependence will be either that of the total profile, or the sample contribu8on to the profile.

•  In X-‐ray diffrac8on, the peaks usually get broader with increasing angle. •  If the peak widths are more properly described as a func8on of the Miller indices hkl rather than the angle θ (so-‐called anisotropic peak broadening), then the microstructural parameters vary with the direc8on in the crystallites.



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Correla-on of peak profiles Commonly used correla8on func8ons are: •  The Caglio= formula ΓG2 = U tan2(θ) + V tan(θ) + W was introduced for fi=ng neutron diffrac8on data (pure Gaussian profiles).

•  The Thompson-‐Cox-‐Has8ngs formula is an extension of the Caglio= formula to account for the different profile behavior in synchrotron XRD with pseudo-‐Voigt profiles: ΓG2 = U tan2(θ) + V tan(θ) + W + P/cos(θ) ΓL = X/cos(θ) + Y tan(θ)

•  In the Double-‐Voigt approach, a Voigt func8on with 1/cos(θ) dependence (represen8ng crystallite size broadening) is convoluted with a tan(θ) dependent Voigt func8on (strain broadening) for the sample related peak broadening, while the instrumental contribu8on can be represented by the Fundamental Parameters Approach (convolu8onal fi=ng).



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Peak posi-on correc-ons Common correc8ons for peak posi8ons include •  Zero error or zero shi+: offset in the defini8on of the 2θ scale of the diffractometer; all peaks are shiYed by a constant value (i.e. θ independent).

•  Sample displacement or height error: sample is posi8oned off the diffractometer center; peak shiYs are a func8on of cos(θ).

•  Specimen transparency: incident beam penetrates into deeper layers of the sample (diffracted beam, on average, will come from a lower height); indis8nguishable from a real sample displacement, thus not handled separately.

•  Correc8ons connected with instrumental peak asymmetry (usually handled by the soYware).



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X-ray tube

detector

sample holder with sample

sample displacement

specimen transparency



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Peak posi-on correc-ons •  In WPPF, you usually need to refine a peak posi8on correc8on parameter. •  However, the effects of zero shi+ and sample displacement are so similar that you should never refine both parameters together!

•  You need to decide which of the two parameters you want to refine. •  In reality, probably both will affect your peak posi8ons. •  However, as both are very similar, the choice between one or the other will have very li;le impact on your results.

•  Personally, I chose displacement for reflec8on (Bragg-‐Brentano) and zero shi+ for transmission geometry.



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Intensity devia-ons Some8mes, the calculated rela8ve intensi8es of a phase do not match the observed intensi8es. Possible reasons can be: •  Incorrect crystal structure model (se=ng and origin choice, handling of special posi8ons) → calculated intensi8es are wrong.

•  Bad sampling sta8s8cs due to very large (thus comparably few) crystallites → peaks are extremely sharp.

•  Preferred orienta:on effects → crystallites have strongly anisotropic shape (platelets, needles); intensity devia8ons are func8on of hkl; direc8on of effect will be reversed for change between reflec8on and transmission geometry.



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2Theta20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.00

500

1000

1500

2000

2500

3000

3500

4000

4500

5000Ab

solu

te In

tens

ity

Example for very bad sta8s8cs (few, very large crystallites)



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Preferred orienta-on effect •  One of the basic assump8ons of powder diffrac8on is that you have a very large number of crystallites in random orienta8on.

•  Random orienta8on means that all possible orienta8ons of crystallites within the sample occur with the same probability.

•  Thus, for any possible orienta8on of the sample rela8ve to the primary beam and the detector, a sufficiently large and representa8ve frac8on of crystallites would be in the correct orienta8on to fulfill the Bragg equa8on. [Note that this implies that every observed reflec8on in a powder pa;ern (except higher orders) results from a different sub-‐set of crystallites in your sample!]

•  If anisotropic crystallite shapes are combined with a direc8ng mechanical force (sample prepara8on), then some crystallite orienta8ons will occur more oYen that others → assump8on of random orienta8on of crystallites in the powder is not valid anymore.



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Preferred orienta-on effect

Transmission geometry:

Reflec8on geometry:



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Preferred orienta-on effect •  In the case of a preferred orienta8on effect, some reflec8ons will be stronger, and some weaker than calculated.

•  OYen, you may no8ce that there is a trend with the Miller indices hkl: try to check if the effect is most pronounced for certain families of low-‐index reflec8ons, like h00, 0k0, 00l, hk0 etc.

•  If you suspect preferred orienta8on, you may want to check this hypothesis against independent evidence: -‐ Crystal morphology from electron microscopy -‐ Typical crystal habit or cleavage from literature, especially for minerals -‐ The crystal structure itself (maybe 2D structure, e.g. MoO3?) -‐ Second XRD measurement with alternate measurement geometry -‐ Second XRD measurement with different sample prepara8on



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Preferred orienta-on effect •  Most Rietveld programs contain one or more op8ons for the modeling of preferred orienta8on.

•  The most popular is the March-‐Dollase model. •  It requires that you explicitly chose the crystal direc8on of the preferred orienta8on and refines a parameter represen8ng the magnitude of this effect.

•  Chose a direc8on which either seems to be most enhanced or suppressed in your pa;ern, or base your choice on independent evidence as listed above.

•  Note that higher order reflec8ons (e.g. 001, 002, 003) represent the same direc8on in the crystal. Thus, if e.g. the 002 reflec8on is most pronounced and your crystal structure does not allow the presence of a 001 reflec8on due to the systema8c absences of its space group, the direc-on is s8ll called 001.



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Preferred orienta-on effect •  Another popular approach is the refinement of a spherical harmonics func8on. •  You may chose the "order" of the func8on, which is an even integer number. •  The higher the order, the more flexible the model (higher number of refined parameters).

•  Note that the actual number of variables also depends on the crystal symmetry (higher symmetry = fever variables).

•  Advantage and disadvantage in comparison to March-‐Dollase: -‐ higher number of refinable parameters -‐ model free (you do not need to chose the preferred direc8on, but you also will learn less about it).

•  Depending on how your Rietveld soYware handles it, you may run into physically unreasonable results (nega8ve intensi8es)!



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Preferred orienta-on effect •  The preferred orienta8on effect modulates the intensi-es (i.e. areas) of the peaks as a func8on of hkl.

•  Thus, you may some8mes be mislead by anisotropic peak broadening, which modulates the widths of the peaks as a func8on of hkl.

•  If the areas of the peaks stay constant but their widths vary, then their heights will vary, too.

•  Thus, you should check if the observed mismatch is really caused by the peak areas, or maybe by their widths. → Try switching from a Rietveld fit to a structure-‐less (Pawley or Le Bail) fit. If there is s8ll a profile mismatch, although you allowed the intensi8es to be refined, then you have a problem with the peak widths or profiles.



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Other effects influencing peak intensi-es •  There are several other effects which may distort the rela8ve intensi8es, usually as a func8on of 2θ: -‐ details of the instrument geometry -‐ beam spill (beam cross-‐sec8on at low angles is larger than sample area) -‐ surface roughness etc.

•  These effects are to numerous to be detailed here, please refer to you Rietveld soYware manual to see what op8onal correc8ons may be implemented.

•  The corresponding correc8ons are usually global, i.e. the intensi8es of all phases in a mixture will be affected the same way.

•  If you decide to use one of these intensity correc8on op8ons because it improves the fit, you should s8ll consider whether this choice is physically reasonable: For example, a surface roughness correc8on may be reasonable if your sample is coarse grained or highly absorbing, but not if your sample has a low absorp8on, very fine grain size and was carefully prepared.



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Peak broadening •  Peak broadening may be sample or instrument related, symmetric or asymmetric.



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Asymmetric peak broadening •  Peaks at lower angles are usually asymmetrically broadened towards the leY, which is an instrumental effect caused by axial beam divergence.

•  Axial divergence can be reduced by using axial sollers (on the tube and/or detector side).

•  Theore8cally, asymmetric broadening on the right flank would occur for peaks at very high angles (approaching 180° 2θ), while peaks around 90° 2θ should be symmetric.

•  If the measurement was done with Kα1+2 radia8on, the peaks may look increasingly asymmetric (shoulder on the right) for increasing angle (depends on the rela8on between peak width and angle).



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Asymmetric peak broadening •  Asymmetries which cannot be explained as above may be sample related. •  If the asymmetry affects all peaks of a phase and increases with the diffrac8on angle, then it may be caused by inhomogeneity of that phase.

•  For example, the d-‐spacings/la=ce parameters may have a non-‐symmetric distribu8on (in contrast to microstrain, which is symmetric) due to chemical varia8on between different crystallites, or even within crystallites (chemical gradient).

•  You should consider whether chemical varia8on is likely for your type of phase (solid solu8on, doping, variable stoichiometry).

•  A plausibility test may be made by inser8ng the same phase a second 8me into the (preferably Rietveld-‐) refinement with slightly different star8ng parameters. If the fit looks significantly be;er, this may be taken as evidence suppor8ng the inhomogeneity hypothesis.



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Asymmetric peak broadening •  If only some peaks are affected, maybe even in different direc8ons, the reason could be a symmetry reduc8on of your crystal structure. → Try to fit with structurally closely related, lower symmetry version of the crystal structure you are currently using (i.e. t-‐ZrO2 instead of c-‐ZrO2, MoO2 type instead of ru8le type).

•  If only few peaks are affected, it might be that an unrecognized impurity phase accidentally overlaps with the main phase peaks. → Try find a reference phase which has its main peaks in the posi8ons of the apparent "shoulders" of your main phase peaks.



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Symmetric peak broadening / microstructure •  Symmetric, sample related peak broadening is usually caused by the microstructural proper8es crystallite size and microstrain (short: size & strain).

•  Size broadening has been recognized and described by Paul Scherrer almost a century ago (1918). → Scherrer equa8on.

•  Smaller crystallites yield broader reflec8ons. •  XRD is a volume effect, hence crystallite sizes determined from XRD peak broadening ale volume weighted averages.

•  Size broadening can be best analyzed in the range of roughly 10 -‐ 1000 Å. For larger sizes, the effect becomes comparable to instrumental broadening. For smaller sizes, the peaks become so indis8nct that they merge with the background.



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Symmetric peak broadening / microstructure •  With XRD, you get either crystallite or domain sizes, but not par8cle sizes! •  If you do not know the exact microstructure of your material, you will not know whether it is crystallite or domain size. Hence, both terms are mostly used synonymously.

•  A very precise term would be "size of the coherently diffrac8ng domain".



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Symmetric peak broadening / microstructure

par8cle crystal crystallite domain ≥ ≥ ≥

[ illustra8on taken from TOPAS Users Manual, Bruker AXS ]



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Symmetric peak broadening / microstructure •  Many people s8ll use the Scherrer formula to extract size informa8on from the width of a single diffrac8on peak. This is not state of the art and not recommended!

My favorite quote from the Rietveld mailing list ([email protected]) Ma;eo Leoni (in reply to a newbie ques8on concerning how to chose a peak for Scherrer analysis): "I think you can use the same criterion you use to select numbers to play bingo. As (I hope!) you are going to use Scherrer formula on one peak just to get a number (and not to get any physically meaningful informa8on), just select the most isolated, most intense one. At least you play with a be;er sta8s8cs..."



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Symmetric peak broadening / microstructure •  Scherrer realized that the peak broadening from XRD yields only "apparent" crystallite sizes, which need to be converted by means of a shape factor if "true" crystallite sizes should be obtained: D = K λ / {FWHM cos(θ)} D: crystallite/domain size K: shape factor (close to unity, on average around 0.9)

•  Shape factors have been calculated for several regular geometries (sphere, cube, octahedron, tetrahedron).

•  The differ even for different miller indices (except for spheres).



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Symmetric peak broadening / microstructure •  Most people don't know (or care about?) their crystallite shapes, so they use something like an "overall average" shape factor.

•  In most cases, the details of the Scherrer analysis are not reported. → Different people would get different results from the same data set!

•  What is the point of using a shape factor to convert an apparent into a "true" size, if the true shape is not known???

•  Even if done correctly, the Scherrer approach will only result in reasonable values if no other sources of peak broadening (instrumental, strain etc.) are present.

•  You should be;er: 1) Use WPPF instead of single peak analysis 2) Use integral breadth instead of FWHM 3) Report apparent instead of "true" crystallite sizes (e.g. LVol-‐IB) 4) Consider instrumental broadening and the possibility of strain



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Symmetric peak broadening / microstructure •  LVol-‐IB is a measure for crystallize size which can be calculated directly from WPPF without using a shape factor: LVol-‐IB = λ / {IB cos(θ)} IB: Integral Breadth

•  "L" stands for a length (or height) of a column of stacked unit cells, while the index "Vol" points at the fact that the value represents a volume weighted mean.




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Symmetric peak broadening / microstructure •  Microstrain means that not all d-‐spacings in a crystallite are exactly equal, but show a certain distribu8on.

•  This can be envisioned as local, sta8c displacements of atoms out of their ideal average posi8on.

•  Both compression and expansion of la=ce planes from the op8mum distance consume energy. As both forces need to be in balance, the distribu8on is symmetric, i.e. the peak is broadened, but not shiYed.

•  Stokes and Wilson (1944) described strain broadening as follows: ε0 = IB / {4 tan(θ)}

•  The different angular dependence of the broadening can be used to separate size and strain effects (size broadening propor8onal to 1/cos(θ), strain broadening propor8onal to tan(θ))



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Symmetric peak broadening / microstructure •  In the Williamson-‐Hall analysis, the widths of all peaks of a phase (corrected for instrumental broadening) are extracted separately (uncorrelated, i.e. single peak fit).

•  The term widthcorr. cos(θ) (in radians) is plo;ed on the y-‐axis, while 4 sin(θ) is plo;ed on the x-‐axis for all reflec8ons.

•  If size and strain are isotropic (independent of the direc8on in the crystallite), then the data points should fall on a straight line.

•  The slope of the line represents the microstain, while the y-‐axis intercept at 4 sin(θ) = 0 represents the crystallite size.

•  As the Williamson-‐Hall approach requires extrac8on of single peak widths, it will work properly only in absence of significant peak overlap.



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Symmetric peak broadening / microstructure •  The Double-‐Voigt approach (Balzar, 1992) uses two convoluted Voigt func8ons to describe the sample related peak broadening of a phase over the whole angular range.

•  The width of one Voigt func8on is propor8onal to 1/cos(θ), while the other is propor8onal to tan(θ).

•  Thus, the peak width correla8on used in this WPPF approach can be interpreted as implying that a perfect isotropic Williamson-‐Hall type correla8on exists.

•  This way, the problems of separa8ng overlapping peaks is overcome. •  If the fit yielded by this method is good, we may assume a Williamson-‐Hall type correla8on and thus directly extract size and strain from the fit.



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Symmetric peak broadening / microstructure •  If the Double-‐Voigt fit is not good, then we have probably a case of anisotropic peak broadening.

•  Anisotropic peak broadening means that the distribu8on of peak widths follows not so much the diffrac8on angle but depends more on the direc8on in the crystallite.

•  As size and strain are separated based on their angular dependence, the anisotropic case leaves us with a phenomenological fit of the pa;ern (no extrac8on of size and strain).

•  In reality, the peak widths will probably depend on both θ and hkl with varying weight.

•  However, the simultaneous refinement of a θ and hkl dependence is usually not recommended because of the risk of overfi=ng. → In an intermediate case, you need to decide which model suits your case be;er.



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Symmetric peak broadening / microstructure •  If you fit anisotropic peak broadening with an isotropic model, the misfit will usually leave characteris8c symmetric signatures in the residual (difference between measured and calculated data).

•  Because the misfit is symmetric, you may s8ll extract reasonable la=ce parameters (peak posi8ons not affected, only error bars slightly increased).

•  The physical reason for anisotropic size broadening lies in anisotropic crystallite shapes with high aspect ra8os. Thus, it may occur together with preferred orienta8on.



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45444342414039383736353433323130292827262524232221201918171615

26,000

25,000

24,000

23,000

22,000

21,000

20,000

19,000

18,000

17,000

16,000

15,000

14,000

13,000

12,000

11,000

10,000

9,000

8,000

7,000

6,000

5,000

4,000

3,000

2,000

1,000

0

-1,000

m-TaON_ICSD-1032 100.00 %

monoclinic TaON with isotropic peak broadening



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45444342414039383736353433323130292827262524232221201918171615

13,50013,00012,50012,00011,50011,00010,50010,0009,5009,0008,5008,0007,5007,0006,5006,0005,5005,0004,5004,0003,5003,0002,5002,0001,5001,000

5000

-500-1,000-1,500-2,000-2,500-3,000-3,500-4,000-4,500

m-TaON_ICSD-1032 100.00 %

monoclinic TaON with anisotropic peak broadening, isotropic fit



Pa;ern Fi=ng

45444342414039383736353433323130292827262524232221201918171615

13,500

13,000

12,50012,000

11,500

11,000

10,500

10,0009,500

9,000

8,500

8,0007,500

7,000

6,500

6,0005,500

5,000

4,500

4,0003,500

3,000

2,500

2,0001,500

1,000

500

0

-500

m-TaON_ICSD-1032 100.00 %

monoclinic TaON with anisotropic peak broadening, anisotropic fit



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Symmetric peak broadening / microstructure •  Many people compare their XRD crystallite sizes with electron microscopy results.

•  It should be emphasized that this is by no means straight forward! •  XRD is an integral method, EM is local. •  Even if the EM operator is measuring a large number of crystallites to obtain a histogram (size distribu8on), you may see two different things (like e.g. crystallite vs. domain sizes).

•  A size distribu8on from EM can be used to produce various mean values, while XRD will always yield the volume weighted mean. → Request specifically the volume weighted mean size from your electron microscopist!



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Symmetric peak broadening / microstructure




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Structure refinement (Rietveld refinement in the original sense) This topic is too complex to be covered here in detail. Some brief general guidelines would be: •  Refinement of (individual) thermal displacement parameters ("temperature factors") or hydrogen posi8ons from powder XRD is very tricky and should be leY to the experts. Beware if you come across publica8ons in which this seems rou8ne.

•  Temperature factors are something like a waste bin in which many kinds of ar8facts may accumulate.

•  Thermal parameters and site occupancy factors correlate strongly with each other. → Don't refine them together without reasonable restrains.

•  Judge the quality of a fit not by R values, but by the look of the fit (e.g. the shape of the residual) and the plausibility of the resul8ng structure (reasonable bond distances, small e.s.d.s (es8mated standard devia8ons).



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R values in Rietveld refinement •  In single crystal structure refinement R values provide a rela8vely sound basis to judge the quality of a refined crystal structure model on an objec8ve scale.

•  In Rietveld refinement, this is not the case. •  Originally, R factors were defined for refinements in which the background had been subtracted previously.

•  Nowadays, the background is usually not subtracted, but refined with the other parameters (less bias). → Two kinds of R values, including or excluding the contribu8on of the background.

•  If you have a measurement with a high background and small diffrac8on peaks on top, than you can get excellent R factors with a lousy structure model, as long as you fit your background properly.

•  On the other hand, a perfect structure model may give bad R values if you e.g. cannot fit the peak profiles perfectly.



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R values in Rietveld refinement •  There are no good or bad R values in Rietveld refinement on an absolute scale. •  However, the R values have a meaning for a given measurement (be;er R value, be;er fit).

•  Many Rietveld programs will report an Rexp value. This is an es8mate by the soYware how good an ideal fit would be if only the experimental noise was leY as residual.

•  Another op8on is to switch from a Rietveld to a Pawley or Le Bail fit with otherwise the same set of parameters. The R value you obtain then is the op8mum you could theore8cally reach with a perfect Rietveld fit.

•  Suggested reading: Brian H. Toby: "R factors in Rietveld analysis: How good is good enough?" Powder Diffrac8on 21(1), 2006, 67-‐70.

Documents

Phase Analysis and Structure Refinement – A Tutorial Part 1: Phase