G.S. Pawley, Unit-Cell Refinementfrom
`
`Powder Dzfiraction Scans,
`
`J. Appl. Cryst, 14 (1981) 357-361
`
`RS 1039 - 000001
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`RS 1039 - 000001
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`

`
`0 J. App}. Cryst. (l98l). I4, 357-36]
`
`Unit-Cell Refinement From Powder Diffraction Scans
`
`BY G. S. Pawu=.v
`
`Department of Physics, Um’ver.n'ty of Edinburgh, Mayfiefd Road, Edinburgh EH9 3.12, Scotland
`
`{Received I6 February l98| ; accepted 20 March I981)
`
`Abstract
`
`A procedure for the refinement of the crystal unit cell
`from a powder diffraction scan is presented. in this
`procedure knowledge of the crystal structure is not
`required, and at the end of the refinement a list of
`indexed intensities is produced. This list may well be
`usable as the starting point for the application of direct
`methods.
`The
`problems
`of
`least-squares
`ill-
`conditioning due to overlapping reflections are over-
`come by constraints. An example, using decafluoro-
`cyclohexene, C(,F,o, shows the quality of fit obtained in
`a case which may even be a false minimum. The method
`should become more relevant as powder scans of
`improved resolution become available, through the use
`of pulsed neutron sources.
`
`Introduction
`
`In recent years, powder dilfraetion, both with neutrons
`and )(-rays. has gained considerable attention, very
`much as a result of the work of Rietveld{1969), and as a
`consequence ofthe increasing quality of the experimen-
`tal data. Future developments, such as the spallation
`neutron source at the Rutherford Laboratory, promise
`to give data with a resolution much superior to that at
`present available, and powder diffraction may well
`develop into a tool for the whole range of structure
`analysis, whereas at present it is mainly restricted to the
`realm of refinement of previously solved structures. The
`first stage ofsuch an analysis is the determination ofthe
`unit cell. This is
`followed by an examination of
`systematic absences to suggest a space group, only then
`leading on to the structure solutions and refinement.
`As all progress rests on the success of the first stage,
`this has to be regarded as the most important step. A
`number of procedures have been advocated for this
`step, such as those by Visscr {[969} and Taupin {I973}.
`The probability of success at
`this stage is a rapid
`function ofthe resolution of the diffraction peaks on the
`scan, and even the best neutron diffractometer of today
`—such as DIA at ILL, Grenoble — produces results with
`a fairly low success probability. However, ifa unit cell
`can be determined in this way,
`then the next stage
`customarily involves refinement including the refine-
`ment ofthe structure. As the structure may well be quite
`unknown it is advantageous to be able to refine the unit
`cell by itself. The procedure suggested in this paper
`
`achieves that aim, and also generates a set of indexed
`intensity values. This list ofintensity values may well in
`itself be usable as input data for a structure solution
`program.
`As a test of the method, the unsolved structure of
`decalluorocyclohcxene has been chosen from a number
`of similar materials because of the apparent success in
`determining the unit cell with Visser’s program. The
`data used is of measurements from D] A, and shows the
`complexity of a problem at this resolution, which is
`characterized by FWHM=0-6‘ at 20=36‘. The form
`of the intensity list is shown in Table 2. The limitation of
`the range of the scan may well prevent the use ofdirect
`methods in this case, but it should be remembered that
`the equivalent data set from a more resolved and more
`extensive data set as may be forthcoming from the SNS
`technique should be of sufficient quality for direct
`methods. The enhanced resolution in such measure-
`ments comes from the pulsed nature ofsuch techniques,
`always assuming that
`the pulse rate is accurately
`controlled so that
`frame overlap can be used to
`advantage. The greater equivalent scan width comes
`from the presence of a considerable short-wavelength
`neutron flux.
`
`Readers familiar with the problems of least-squares
`refinements will appreciate that because of the overlap-
`ping of intensity peaks there will be severe problems of
`ill-conditioning introduced by assuming all reflections
`to have independent variable intensities. This problem
`is surmounted by the use of slack constraints {Waser,
`1963), but
`in using this technique the intensities
`achieved (Table 1} become somewhat
`restricted as
`correlation cannot be removed. It is these restrictions
`which will place a burden on direct methods, and may
`well demand alterations of the algorithms thereof.
`
`The method
`
`It is assumed that the researcher is furnished with an
`experimental powder diffraction scan which has under-
`gone analysis, with apparent success, using a unit-cell
`analysis program, such as that of Visser U969}. He may
`then adopt
`the following procedure. For neutron
`diffraction results, such as those presented here,
`it
`is
`plausible to assume a Gaussian form for the diffraction
`peak shape, whereupon Gaussian curves can be fitted to
`well resolved indexed peaks, and from the resulting
`Bragg angles an accurate unit cell can be computed. The
`
`Mt“ I-I
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`I
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`0021-8898i"8HD6035?-US$01.00
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`-Q l9Bl International Union of Crystallography
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`RS 1039 - 000002
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`
`358
`
`UNIT-CELL REFINEMENT FROM POWDER DIFFRACTION SCANS
`
`which peak at the same point as one variable parameter.
`Thus at no time can the least-squares matrix become
`singular. Each cycle of refinement alters the unit cell and
`thus alters the grouping of rellections, and therefore the
`size of the least-squares matrix may change from one
`cycle to another. This is an example ofa strict constraint
`{see Pawley, 19?2} and can thus be written explicitly {for
`the above example)
`
`new parameter: p,,,,, = p,,, + p,, .
`
`(3)
`
`Such a procedure is all that is necessary if the spacing
`between the points in the scan is comparable to or
`greater than the width of a diffraction peak. However,
`such a situation is not productive, as the aim of the work
`is to locate peaks as accurately as possible. To do this,
`the scan spacing must be made considerably smaller
`than the peak width. As a result of this any two
`intensities which peak on adjacent scan points give rise
`to ill-conditioning. Such least-squares problems are
`usually a symptom of the close approach of the roles of
`two variable parameters, giving
`
`6y.-'
`6.0..
`
`__ Ey.-'
`_ Ba.
`
`(4)
`
`and leading to a pair of rows (and a pair of columns) in
`the matrix being almost
`the same. The degree of
`sameness lessens as the separation between the peaks
`concerned widens. There is no analytic way of
`expressing these facts, and therefore slack constraints
`must be used in this case {Wascr, 1963).
`A brief outline of the way in which slack constraints
`are used is beneficial here. Consider that the constraint
`to be applied is such that a function r should be allowed
`to vary from a predetermined value r° within a normal
`distribution with standard deviation 0'. The function is
`
`not a variable parameter of the model, but is a function
`of the model variables. Let a weight be defined as
`
`$:'rF°—F
`H‘: ——~—-—- ,
`n_Mp)a2
`
`(5)
`
`to M,
`where there are M, parameters in a fit
`observations. Ifp, is a paramelerin the fit. then we must
`add
`
`Fr‘ Fr‘
`W j— -T
`
`("P3 (‘Pt
`
`10 Mjk
`
`w(r"—r‘)
`
`fir‘
`
`Ep-
`
`J
`to V-
`
`(7)
`
`are components of the least-squares
`where M_,., and
`matrix and vector respectively. This in effect alters the
`residual which is to be minimized.
`
`In our particular example let p, and p,, represent two
`variable intensities which are calculated to be very close
`together in 23, and introduce r"=p,-—p_.,. We wish to
`restrict the difference in intensities r‘ by demanding that
`
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`0
`
`Table 1. Refinement values
`Entries with errors in parentheses were varied in the refinement.
`Unit cell
`
`ll-9IS(3)
`7-241(2)
`9-663 (2)
`113.34 tn
`T66 {ll
`0-I911)
`
`-1.34 [36]
`0-56 [2 5}
`0-34 {4}
`0-71 {5}
`2'30
`440
`(H
`3
`06
`T0
`1-83
`
`am
`b
`
`C .
`
`3 (°l
`Volume of unit cell {A3}
`Scan zero of 29 (°)
`Peak shape
`u m1
`in
`w
`Skewness
`Eat background unrefined
`Number ofpoints in scan, N
`Sean point separation, A (‘'1
`Slack constraint limit, in
`Constraint reduction parameter. y
`Value ch: in equation (11)
`Final slack constraint weight, w
`
`pitfall in this procedure is the difficulty in the choosing
`of well resolved peaks as any overlap confounds the
`process. Also a small error in the origin, 26(Bragg)=0,
`has a systematic effect on the result which can prove
`ruinous. A method which is not prone to these troubles
`is therefore desirable. These are some of the arguments
`which promoted Rietveld's development of the profile
`refinement technique, and point to such a technique for
`the refinement ofthe unit cell. Indeed Rietveld‘s method
`is itselfa refinement ofthe unit cell, but one in which the
`peak intensities are constrained by an assumed but
`relinable crystal structure.
`In the method here proposed, the profile refinement
`technique is advocated such that each reflection peak
`has a variable intensity. These intensities are not
`constrained by a crystal structure in the calculation, as
`the structure is as yet unknown, but in order to achieve
`a well-conditioned least-squares matrix some form of
`constraint is necessary. This can be understood from a
`simple argument as follows.
`Let y}’ and yf be the observed and calculated values of
`the intensity at the ith scan point, which occurs at 29,-. If
`pJ represents thejth variable parameter and determines
`the intensity for the jth reflection, the jth row of the
`least-squares matrix has terms
`
`__
`
`(632?)
`6y.-‘
`"
`;(5P_i) apt:
`
`‘
`
`1
`
`[
`
`)
`
`where lc varies over all the parameters, and N is the
`number of observed points in the scan. If two dilferent
`reflections peak at the same value of 26,, say the mth and
`nth reflections, then
`
`8y?
`__J_ =
`6p...
`
`61):‘
`
`69..
`
`(2)
`
`and the least-squares matrix is singular. The present
`procedure overcomes this by treating all reflections
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`RS 1039 - 000003
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`

`
`G. s. PAWLEY
`
`359
`
`O
`
`H
`
`r‘ be distributed about a zero mean with standard
`deviation 0”. The value of 0' must be chosen in ad vanoe
`
`arbitrarily, but with regard to the average peak
`intensity value. From the experience of the present
`example it would seem plausible to equate or with the
`average least-squares error found for
`the intensity
`parameters. in this case or would have been 23 in the
`final cycle, whereas a value of 70 was used throughout.
`As
`
`- =1, 3'
`UP;
`0P1:
`
`-1,
`
`w, W, —- w, —w
`we add
`to the components Mii, Mm Mix, Mii
`
`and
`to the components
`
`Vi, I/,,.
`
`The expression for w is similar to that of (5):
`N
`
`W:
`
`igllilyi ‘Bil/Gil 2
`“VT
`
`(8)
`
`9
`
`{
`
`l
`
`(11)
`
`for the case where pi and pi. peak at scan points which
`dilTer in 26 by A, where A is the separation between
`neighbouring points in the scan. In this expression, cri is
`the standard deviation assigned to the ilh point in the
`scan. For the present example all a.-=1.
`Ifthe separation between pi and p,, is Jim. the weight
`used in the program is *,t"' 1W, and a limit rt ,-, is placed on
`the value ofn which needs to he considered. For Ft) iii
`the separation between pi and p,, is deemed to be large
`enough for the two intensities to be unconstrained. For
`such values oh: the weight used is zero, thus preventing
`any constraints from being applied. The factor ~,-, the
`value ofn ,; and the value of the constant 0' in (I I} are all
`to be chosen by experience. In some respects A is also
`arbitrarily adjustable as in the present example. for
`which A=0-IO“ was chosen although the original
`measurements were made with a scan step of 005‘.
`Equation ( l l}shows that the constraining weight w is
`a function of the refinement, becoming smaller as the
`residual, the numerator of (1 1), becomes smaller. As it:
`gets smaller, the amount of constraint decreases as is
`reasonable in a successful refinement.
`Occasionally it is found that an intensity parameter is
`shifted too much by the least-squares process, giving a
`negative and therefore unphysical value. In such cases
`the value ofthe parameter is automatically reset to zero.
`In the standard profile refinement procedure the zero
`of2B is a variable, and this freedom is here retained. Any
`standard refinement with incorrect
`indexing will,
`through correlations, tend to have its zero of 23 forced
`to an incorrect value. This is undesirable in the present
`method, and it must therefore be arranged for peaks to
`become reindexed automatically in the course of
`refinement. In the argument which follows this is seen to
`be possible even for an isolated reflection, whereas this
`is not
`the case in the standard refinement simply
`Jhcl-1-I‘
`
`because the peak intensities are correlated through the
`structure constraint.
`
`The index of an isolated reflection can change as
`follows. If all the other peak positions suggest a certain
`change in the unit cell which would move the calculated
`index for an isolated rellection away from its intensity
`maximum,
`the intensity parameter would decrease
`accordingly, leaving a region of discrepancy between y?
`and yf for part of the observed peak. This discrepancy
`certainly hinders refinement, but, if the unit cell is nearly
`correct, the true index for the observed peak should be
`calculated nearby, and should be brought nearer to the
`observed peak by the change in unit cell. As soon as it is
`
`
`
`NEUTRONCOUNTS
`
`SCATTERING ANG LE (29)
`Fig. 1. The observed and calculated diffraction scan. Observations are
`points,
`the calculated scan is
`the continuous line, and the
`differences yf — y,-‘ are plotted beneath the main diagram.
`
`
`
`NEUTRONCOUNTS
`
`SCATTERING A NGLE (29)
`Fig. 2. Enlarged parts of the scan in two discrepant regions. In {b} there
`is a possibility of a reflection at 53".
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`
`UNIT-CELL REFINEMENT FROM POWDER DIF FRACTION SCANS
`360
`n within the width of the observed peak, the intensity
`parameter for the true index will grow, thereby helping
`in the move towards the best unit cell. The incorrect
`index moves away in search of its true peak!
`For peaks which are overlapping the scope of
`reindexing is quite obvious, and this is a flexibility
`which is not denied by the standard technique. However
`
`it must be remembered that automatic reindexing in the
`standard technique requires the concomitant change in
`the calculated crystal structure, and thus the flexibiiity
`is somewhat more restricted than in the unit-cell
`refinement.
`In the course of refinement, some parameters of the
`usual procedure are not relevant. Thus the overall scale
`
`Table 2. The indices h, k, 1'. the scan positions 26 in units ofd = 0-1°, the integrated irrrensities p{h,k,I) not correctedfor
`the Lorentz factor, and the least-squares errors etp)
`For reflections with the same scan position the intensities are equal, and the error is for the intensity sum.
`20
`P
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`
`G. S. PAWLEY
`
`361
`
`temperature factor are not
`factor and the overall
`needed. Such parameters as the peak shape parameters,
`at, u, W, the skewness parameter (see Rietveld, 1969;
`Pawley, 1930) and the background parameter are just as
`important as in the standard procedure.
`
`An example
`
`The alterations to a profile refinement program needed
`for the implementation of the unit-cell refinement are
`not too extensive, as the main task is the removal ofthe
`sections which generate reflection multiplicities and
`which govern intensities using the assumed structure.
`Alterations have been made to the program EDINP
`{Pawley, 1980), and the example here presented is the
`first extensive test.
`Decafluorocyclohexene is a liquid at room tempera-
`ture. A sample was frozen solid, ground into powder
`and cooled to 4-2 K, at which temperature the
`diffraction measurements were taken on the DIA
`instrument at the Institut Laue—Langevin in Grenoble.
`Indexing with Visser‘s program gave the following
`results. The ten best possible unit cells are listed along
`with various figures ofmerit, but there was only one cell
`which indexed all the reflections. for which a: I
`I -955,
`b='i-260.
`t'=9-682 A,
`if: ll3»5l'. This
`cell was
`favoured for further work because it was monoclinic
`rather than triclinic, as thelatter is much rarer in nature.
`Four triclinic cells were suggested, though none indexed
`more than I? of the 20 measured reflections given.
`Another reason for choosing this cell was that it gave a
`volume of TF1 A’, which is about equal
`to four
`molecular volumes.
`
`Refinement from this unit cell was stable and swift,
`giving the lit shown in Fig.
`l with the parameters of
`Table 1. This fit
`is indeed remarkably close to the
`experiment, the most serious disagreement being at 26
`=53”. Because this is so marginal the comparison is
`made more clear by enlargement into Fig. 2(b}. Another
`enlargement of a less serious region of discrepancy is
`shown in Fig. 2(a). Progress from this point requires first
`the deduction of the space group, followed by the
`structure determination.
`
`Space-group determination is perhaps made possible
`by introducing into the refinement
`the various sys-
`tematic absence conditions and finding the best
`fit
`under these conditions. When confidence in the initial
`indexing is greater than it is at present, this aspect might
`be programmed for the individual space groups. For the
`present it is probably wisest to attempt space group and
`structure solution together, using a knowledge of the
`most probable space groups. Scrutiny of Table 2 shows
`that 0&0 fork odd may all beabsent,and hot‘ for h+ lodd
`may also all be absent, indicating a space group P2,/n.
`
`The presence of 103 at 29:42-9° militates against the
`otherwise possible #101, it odd or alternatively h0l, I odd
`absences.
`
`Conclusion
`
`Although the structure of the test material is as yet
`unsolved, this aspect ofthe work is beyond the scope of
`the present paper. A refinement procedure has been
`presented and shown to be achieved in practice,
`obtaining the best possible unit cell from diffraction
`data, starting with a suggested unit cell. The amount of
`computational
`time required for this work is quite
`small, and it
`is therefore quite feasible to use the
`program automatically, starting from any unit cell
`given by an indexing routine. This should greatly
`improve the ability to assign figures of merit to the
`possible solutions.
`A suite of programs can be envisaged where the best
`results after unit-cell
`refinement undergo further
`refinements with possible systematic absences included,
`in an effort to predict the space group.
`When this is achieved it should be possible to put the
`intensities, such as given in Table 2 but more extensive,
`into a direct-methods program. Such a program will
`need to take special note of the correlations between
`intensities which effectively reduces the number of
`measured intensities. Although the ill-conditioning
`effect of these correlations has been overcome by the
`slack constraints, the correlation is in no way removed.
`It may seem that the loss in information may be too
`severe for direct methods to succeed, but the actual
`complexity of crystal structures of molecules the size of
`the present example, C¢,F.,,, is by no means high for
`standard direct methods. This may well mean that
`structure solution is possible in this way, and the author
`hopes that the experts in the field will not neglect the
`possibility.
`
`The author wishes to thank Drs A. W. Hewat and R.
`Shirley, with whom the experimental work was done
`and with whom attempts to solve structures using no
`more than powder diffraction data will continue.
`
`References
`
`PAWLEY, G. 5. {I972}. Advances in .')‘rrt.rcl't.tre' Re'.t'eart‘h by
`Diffraction Methods, Vol. 4, edited by W. HOPPE & R.
`Mason, pp. I-64. London: Pergamon Press.
`PAWLEY, G. S. (I980). J. Appl. Cryst. 13, 630-633.
`RIETVELD, H. M. (I969). J. Appi. Crysr. 2. 65-7].
`TAUPIN, D. (1973). J. Appl. C‘rys.'. 6, 380-385.
`VISSER. J. W. U969). J. Appl. C'ry.tt. 2, 89-95.
`WASER, J. (I963). Aria Cryst. I6,
`l09l—|094.
`
`RS 1039 - 000006
`
`RS 1039 - 000006