G.S. Smith &; R.L. Snyder, “A Criterion for
`Rating Powder Diffraction Patterns and
`Evaluating the Reliability of Powder Pattern
`Indexing”, ]. Appl. Cryst, V. 12 (1979) pp. 60-65
`
`RS 1035 - 000001
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`
`60
`
`J’. Appf. Cryst. (1979). I2, 60-65
`
`F”: A Criterion for Rating Powder Diffraction Patterns and Evaluating the Reliability
`of Powder-Pattern Indexing*
`
`Bv GORDON S. Sum: AND Ronsnr L. SNYDER‘|'
`
`Lawrence Livermore Laboratory, University of Caii_'forriia, Livermore, California 94550, USA
`
`[Received 1'? April 1978; accepted9 August 1978}
`
`Abstract
`
`At present, over 30000 powder dil'fraction patterns
`are available as references. It is proposed that the pat-
`terns on file as well as new patterns submitted for pub-
`lication be assigned quantitative quality factors. A
`simple-to-use figure of merit. FN, covering both accur-
`acy in the measurement of the positions of the diffrac-
`tion lines and com leteness of the pattern, has been
`derived: F” =(l/IFEFI) {N/N,,,,,,), where NM,
`is
`the
`number of independent diffraction lines possible up to
`the Nth observed line and HE is the average absolute
`discrepancy between observed and calculated 29 val-
`ues. This figure of merit provides a rapid evaluation of
`powder patterns, in much the same way as the R factor
`provides a rating for single-crystal structure determina-
`tions. This figure of merit also provides a means to
`assess the reliability of a unit derived solely from pow-
`der data. The present FN ranking scheme is shown to
`be superior to de Wolffs M 2., for ranking patterns. It
`is recommended that use of the latter be discontinued
`for that purpose. Guidelines are given on the use and
`implementation of the F, rating of powder diffraction
`patterns.
`
`Introduction
`
`At present, approximately 30000 powder diffraction
`patterns are available for use as references in the
`Powder Diffraction File published by the Joint Com-
`mittee on Powder Diffraction Standards {JCPDS}.
`Concern about the quality of any given experimental
`powder pattern usually centers around three questions:
`(1) How accurate are the positions of the diffraction
`lines? (2) How complete is the pattern? (3) How accur-
`ate are the intensities‘? At present, such assessments
`typically involve intuitive estimates, e.g. the agreement
`between observed and calculated at spacings seems
`reasonable, and the experimentation appears to have
`been carefully done.
`We propose that powder patterns be given quanti-
`tative figures of merit. For new patterns presented for
`publication, such ratings would be most helpful
`to
`
`" Work performed under the auspices of the US Department of
`Energy at Lawrence Livermore Laboratory under contract No.
`W-“M05-Eng-48.
`T Sabbatical guest at Lawrence Livermore Laboratory during
`1977: permanent address is NYS College of Ceramics. Alfred Uni-
`versity. Alfred, New York 14802. USA.
`
`referees in deciding whether these data should be pub-
`lished as submitted or should be returned to the
`
`authors for upgrading. For patterns that have been
`indexed by computer methods, a quantitative figure of
`merit would enable the user to assess the reliability of
`the indexing so obtained. For older patterns already
`in the JCPDS Powder Diffraction File, such ratings
`would provide quality factors for the user. Thus, in
`phase identification. a quality factor on each reference
`pattern would enable the user to search with a test
`window on d spacings related to both the quality of the
`experiment and the quality of the reference pattern.
`For frequently encountered phases, rankings would
`promote reexamination of those patterns having listed
`figures of merit below current standards. Finally, and
`perhaps most importantly, we believe that a quantita-
`tive rating scheme would in time generally improve the
`quality of published powder diffraction data.
`
`Quantitative figure of merit, FN
`
`The figure of merit recommended here is
`F” = ..__1.._ .1
`|A26| NW,‘
`
`(1)
`
`where N W, is the number of possible diffraction lines
`up to the Nth observed line and HE is the average
`absolute discrepancy between the observed and cal-
`culated 29 values.
`
`Our F” combines the concepts of accuracy in the
`positions of the diffracted lines [expressed in the first
`term of the equation) as well as completeness of the
`measured pattern (expressed in the second term of the
`equation). Note that FN is defined in terms of merit
`rather than as a residual: the higher the accuracy or the
`more complete the pattern, the larger is F”.
`The units for FN are reciprocal degrees. This permits
`easy interpretation of a particular F,» value. For ex-
`ample, an F” of 100 means the average discrepancy in
`29 was 5001'’; an F, of 50, an average discrepancy of
`50-02“, and so on. (The 5 symbol arises because the
`ratio NJNM, can never exceed unity.) The values of
`N;‘N,,,,, expected for high-quality data are discussed
`later.
`Note also that N in {I} is considered a variable. Thus,
`FN is a cumulative function that can be terminated at
`any line number in the powder pattern or continued
`
`0021-8898,i'?9,-’0l0060-06501.00
`
`© [979 International Union of Crystallography
`
`RS 1035 - 000002
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`RS 1035 - 000002
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`

`
`GORDON 3. SMITH AND ROBERT L SNYDER
`
`61
`
`to the last observed line. In other words, a partial or
`total appraisal ofa powder pattern is possible. It is also
`clear that FN can be used as a criterion for assessing the
`reliability ofa unit cellderived exclusively from powder
`diffraction data. In such cases. the quality factor for the
`pattern would be synonymous with the reliability fac-
`tor for the proposed indexing
`The above relationship for a pattern is similar to de
`Wollfs (1963) M20 figure of merit, which is often used
`in estimating the reliability of a particular indexing of
`a powder pattern. We had anticipated that the de Wolff
`index could also be employed as the quality factor for
`patterns in the JCPDS file. However, we found that
`M20, in addition to being defined only in terms of the
`20th observed line, is strongly dependent on the crystal
`class and the space group and is less strongly dependent
`on unit-cell volume. In other words, rankings by the
`M 29 criterion do not properly reflect the accuracy with
`which the pattern was measured. Because of these in-
`adequacies (detailed below), we recommend that F"
`be used rather than M30.
`The FN figure of merit does not, of course, say any-
`thing about the reliability of the measured intensities.
`At present, because measured intensities are used in-
`frequently to derive a structural model, assessment of
`the accuracy of intensity measurements by an R-factor
`calculation is not possible for most patterns. However,
`if structures are increasingly determined by the powder
`diffraction technique or if calculated powder patterns
`become sufficiently plentiful,
`then F, could be ex-
`panded to include a term relating to the reliability of
`the intensity measurements. For the present, we be-
`lieve that increased attention on the part of experi-
`menters toward improved accuracy in the measure-
`ments of the positions of diffracted lines and pattern
`completeness will lead to improvements in their inten-
`_sity measurement techniques.
`
`Reexamination of de Wolff's M,3 figure of merit
`
`The M 10 figure of merit was defined by de Wolff{l968)
`as
`
`M2o=Q1ofl2m_lNzol,
`
`(2)
`
`Of critical importance is the finding that K in (3)
`depends strongly on the crystal system, being related
`to its overall multiplicity. In other words, .'(,,,,,.,,,,,,<
`Kmonmlinic < Konhorhumbie ‘*1 Ktetragnnals Khexasnnal '< Kcubie'
`{The relative standings of tetragonal and hexagonal
`materials have not yet been established.) Thus, it fol-
`lows that materials possessing high crystallographic
`symmetry will have large M 29 values, other factors in
`(2) being equal.
`Furthermore, K is also space-group dependent,
`being especially affected by systematic absences as a
`result of glide planes. For example, in the orthorhombic
`system, for a given accuracy in 29 and for a given unit-
`oell volume,
`the order of M ,0 is: Pmmm<Pbmm<
`Pbcm 4. Porn. This is because. for cells of equal volume,
`dm in Pmmm is generally larger than dm for Pbmm%
`An additional difficulty with Mm concerns the IAQI
`term in the denominator of (2). A stra_igmforward dif-
`ferentiation of Bragg's law shows that IAQI depends on
`"2. Fig.
`I shows diagrammaticall
`the relation of
`L4
`| to unit-cell volume. Note that |_zll—Q_| is not a good
`measure of accuracy, being inherently related to the
`size of the unit cell. This effect on M 30 partially offsets
`the V ' 3” dependency put in by the use of Qm. The net
`effect, however, is an approximate V‘ "3 dependency
`built into M20.
`Note also that M H, is undefined for a pattern having
`less than 20 diffraction lines, a not uncommon occur-
`rence for high-symmetry materials. We have found no
`straightforward way of generalizing M 10 so that values
`of M N are even approximately independent of the line
`number, N, for all crystal systems. One such attempt
`is shown in the last column of Table 1; for this calcula-
`tion MN was defined as
`
`QN
`Nobs
`M"'=2|A |N,,,,,,‘
`
`where Q20 is the Q value { = lid’) for the 20th observed
`line, N10 is the number of different diffraction lines
`possible up to the 20th observed line, and ]Zl_Q'I is the
`average absolute discrepancy between the observed
`and calculated Q values for the 20 observed lines.
`This definition of accuracy in terms of Q and AQ,
`rather than 26,
`leads to a number of difficulties.
`Previous work (Smith,
`l9':’7a, b) has shown that for
`high-quality powder data the at value of the 20th line
`is related to the volume, V, of the unit cell by
`V = Kd’
`.
`10
`ZI3
`xtzmwmm.
`
`Hence,
`
`M20 =
`
`3
`
`l
`J
`14)
`
`Fig. 1. Schematic dependence of E on volume {V} of unit cell.
`
`RS 1035 - 000003
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`

`
`62
`
`F”: A CRITERION FOR RATING POWDER DIFFRACTION PATTERNS
`
`For the National Bureau of Standards {NBS} cubic
`patterns, values for Mitt were generally found to in-
`crease markedly with line number, as the present ex-
`ample shows. On the other hand. for the high-quality
`triclinic patterns this increase is usually less steep be-
`cause of the effect of the pattern-completeness factor,
`N/NW5. in a few of these patterns, the compensation
`is almost perfect, and M3,. appears to remain nearly
`constant. However, this will not be true for patterns
`from high-symmetry materials.
`
`Examples and comparisons of ratings
`
`Table 1 shows an example (K;SiF,, from NBS, 1955)
`from a computer printout of a line—by-line evaluation
`of all the cubic powder patterns published by NBS.
`Overall, of 36 possible lines (second column}. 33 were
`observed (first column); the average discrepancy in 28
`
`Table 1. Example of line-by—t‘t‘ne figure—of-merit
`calculation for the NBS cubic pattern for KzSiF6
`
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`
`was 001S6°; and F33 =59. For all the NBS cubic pat-
`terns [326 in number), the average value of F for the
`30th or last line is 74-8. This indicates that 26 values
`
`were measured on an average to an accuracy of at least
`0013”. [A full evaluation of all the cubic and triclinic
`patterns in the Powder Diffraction File has been com-
`pleted (Snyder, Smith, Kahara, Johnson & Nichols,
`
`1977}.Talille 2 compares F20 and M39 for three groups of
`
`compounds: triclinic, orthorhombic, and cubic. The
`diffraction data are either from NBS or from de Wolff
`
`(Powder Diffraction File). Some interesting inconsis-
`tencies are seen in the Mm values. Powder patterns
`for compounds F and R have the same average dis-
`crepancy in 26‘, yet R, a cubic material, has an M 2,;
`value about six times larger. Similar considerations
`apply to the patterns for compounds I (orthorhombic)
`and 0 (cubic). Comparison of compounds N and 0
`(both cubic}, shows that the pattern for 0 is less accur-
`ate and less complete, yet 0 has an almost three times
`larger M 2,, value. Thus, in the M 1., ranking scheme, the
`quality factor for 0 would be higher and the reliability
`of the unit cell derived from the powder data would be
`considered higher. Such discrepancies do not exist in
`our ranking scheme.
`
`Recommendations for image of F,
`1. Value ofN in F,
`Ideally, N should be the last line in the pattern. In this
`way, the entire pattern would be evaluated. However,
`for low-symmetry materials, especially those having
`large unit-cell volumes, the high- (and even mid-)29
`regions become very crowded with calculated a‘ values.
`A particular observed at value is then ‘easily matched’
`with a calculated value. Hence, the average discrepancy
`in 29 tends to decrease artificially.
`As a compromise, it
`is recommended that N be
`taken as 30, or as the last line if there are fewer than 30
`
`Compound
`B-Mg{Cl-I ,CO0)1
`A
`Kzznzv I 001 s
`B
`Nt{NO,);.6I-I30
`C
`CuSeO. . SH ,0
`D
`Nal . 2H ;O
`E
`Pb{COO),
`F
`G M82320:
`H
`Cd{O HINO,
`I
`NaHgCI 3
`J
`KCaCl,
`K
`Bl2S3
`L
`NaMn F,
`M
`N H30;
`0
`ZnTe
`P
`Cr,Rh
`Q
`C501 F_-,
`R
`KNiI-'3
`S
`L1BaF;
`
`.
`
`O
`
`Orthorhombic
`
`CubIC
`
`V0“)
`946
`392
`473
`383
`249
`l35
`l 72
`844
`708
`572
`502
`255
`I92 7
`70]
`223
`102
`93
`65
`64
`
`Table 2. Comparison between F,0 and M 20
`System
`Triclinic
`
`ll‘-T2—9l'l°l
`OCQ9
`00 I 7
`{H} I 2
`0'0 I 5
`0'00?
`0'0 I 4
`0010
`0007
`01113
`0&3
`001 3
`0016
`0'01 5
`0%?
`0003
`0'0”
`0' 01 5
`0 014
`0'0} I
`
`li5'ldDl
`4' 1
`7'4
`6'3
`3'7
`4-6
`9' 5
`7' I
`4-4
`4' 5
`5'6
`9'4
`f4'0
`9' 7
`7'0
`9'?
`l2'4
`174
`169
`I 3‘3
`
`Nae
`34
`34
`36
`27
`23
`24
`36
`25
`23
`24
`30
`26
`20
`20
`2 I
`22
`20
`20
`20
`
`M20
`24
`14
`26
`24
`57
`34
`41
`65
`56
`64
`35
`42
`43
`1 35
`39 l
`235
`l 54
`202
`259
`
`P20
`63
`34
`45
`50
`I20
`53
`54
`I l T
`I 14
`I01
`5 2
`50
`63
`14 I
`120
`34
`69
`73
`9 I
`
`
`
`RS 1035 - 000004
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`
`GORDON S. SMITH AND ROBERT L. SNYDER
`
`63
`
`lines in the entire pattern (N -< 30). A study of F1., versus
`N shows that FN increases slowly beyond N =30 be-
`cause of the easy-matching phenomenon described
`above.
`
`Reflections that are systematically absent because
`of lattice centering or space-group symmetry elements
`should, of course, be excluded in the counting of N,,,,,,.
`
`2. Computation of F”
`The actual computation of [7170] should be done by
`computer, either as part of a program for least-squares
`refinement of lattice constants or as part ofan indexing
`program that includes refinement of lattice constants.
`We have sometimes found, especially for high-quality
`data, that a least-squares refinement of the original lat-
`tice constants lowered |_.d_2_0l by a factor of two or more.
`A computer calculation also eliminates problems as-
`sociated with the accumulation of round-oil‘ errors.
`(We are preparing a computer code for distribution
`that will calculate FN, given the unit-cell constants and
`space group. In the developmental state is another pro-
`gram that will determine the space group — more ac-
`curately, the dilfraction aspect {Crystal Data, Deter-
`minative Tables, 1972) — for any crystal class from the
`hkl‘ values of an indexed pattern.)
`
`3. Format for overall F,
`It is recommended that both accuracy and complete-
`ness be reported because both are equally important.
`(We believe standards of high quality will become
`established for both factors in the future.) The pro-
`posed format for the reporting of FM is:
`
`F”-—-overall value (|AIO|, NW).
`
`As an example, consider the pattern for Cr3Rh
`(NBS, 1968). Twenty lines were reported ofa possible
`
`Fig. 2. Uncertainties in 20 for Cu Ks, radiation as a function of the
`number of decimal places reported for d.
`
`d {A}
`
`22; L761 is calculated to be 0009. Thus, F 202101
`(0009, 22].
`
`4. Reporting of powder diffraction data
`The primary data are the observed 29 values. We
`strongly recommend that these be reported along with
`A26. defined as 29,,.,,-—2ti,,., [ in degrees). for each line.
`Reporting only a derived result, such as d values, in-
`troduces unnecessary errors as a result of round off
`and uncertainty in the exact value of 2. used. For
`Cu Kat, radiation, the uncertainty in 26 that is caused
`by reporting a‘ values to two, three, and four decimal
`places is shown in Fig. 2. There are numerous examples
`in the literature of patterns with d values as low as 0-80
`given to only two decimal places, thereby causing a 26
`uncertainty of over 5°! These artificial errors would
`have been eliminated if the observed 26' values had
`been reported.
`At the very least, the overall F, for a pattern should
`be reported. It would be preferable to report a line-by-
`line evaluation. The proposed format for presenting
`such data is shown in Table 3. (The 26.5, and A20
`values were reconstructed, because these were not
`originally reported.)
`
`Table 3. Evaluation ofpowder difli-action patternfor
`K2SiF;,
`
`d,.,, {A}
`4699
`2-377
`2-453
`2349
`
`I
`100
`65
`5
`1'1
`
`Md
`l l 1
`220
`31 l
`222
`
`2611135 (0)
`18-369
`3] -058
`36-601
`33'283
`
`A2B(‘)'
`—0{|l4
`-001?
`-0013
`-0-020
`
`F.-
`72 [0-014, I]
`4310016, 3]
`52 (0015, 4}
`50 (0016, 5)
`
`‘ A29 = 29..., -— 29...“
`
`5. Implementation of FN ratings
`Only a few changes in the usual procedures of data
`transmission are needed to enable the user (and others
`in the data chain) to assess immediately the reliability
`of the line measurements and the completeness of a
`given powder pattern. A [low chart to implement such
`evaluations is shown in Fig. 3. Before submitting new
`powder data to the referee, authors would: (1) put their
`powder data through a computer indexing program
`or (2) obtain symmetry and unit-cell information from
`a data base (such as Crystal Data, 1972) (or, alterna-
`tively, study single crystals to establish the unit cell)
`and attach the F” figure of merit. Thereafter,
`this
`powder pattern has the FN value attached to it. For
`existing
`data,
`professional
`data-base managers
`(JCPDS} would use these same methods to attach the
`F” figure of merit to patterns in their files. These values
`would be included on a future update of their data base
`(presently on magnetic tape). If a pattern of a ‘Fre-
`quently Encountered Phase‘ [a term used by JCPDS
`[19773] is incapable of being indexed, then a retake is
`commissioned.
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`
`64
`
`.F‘_...: A CRITERION FOR RATING POWDER DIFFRACTION PATTERNS
`
`Such an implementation is realizable, in our opinion,
`for several reasons:
`(1) Computer indexing of NBS or de Wolll quality
`data is successful probably 80 to 90"/,, of the time and
`should be equally successful for other high-quality
`data.
`[A review paper covering recent advances in
`computer indexing is being prepared (Shirley, 1979).]
`(2) The technology of achieving NBS or de Wolll
`data quality has been well documented for at least 20
`years.
`(3) A new update of Crystal Data (1972) is expected
`in the near future.
`technology is well
`(4) The necessary computer
`within almost everyone‘s pocketbook. A number of
`fast computer indexers are currently available. Further-
`more, we estimate that the cost in computing time for
`evaluating the entire JCPDS Powder Diffraction File
`would be about S 1000.
`
`Discussion
`
`The basis for the FN rating scheme rests primarily on
`the premise that the error in measuring 20 is generally
`independent of the magnitude of 26. Moreover, there
`is no reason to believe that the error in measuring 26
`would depend on crystal system, space group, and the
`
`like. That 29 is a good measure of accuracy throughout
`the 29 scale has been empirically established (Snyder,
`Smith, Kahara, Johnson 49:. Nichols, 1977) through an
`examination of all the NBS cubic patterns. (‘Easy
`matching’ is not a problem with these cubic patterns.)
`For example, the weak lines seem to be measured as
`accurately as the moderate-to-strong lines; further-
`more, the known aberrations of difiractometer geom-
`etry apparently are ameliorated through the use of
`internal 29 standards and careful experimentation, at
`least at the present levels of excellent: [0-00 to 004°
`errors in 29). The root-mean-square (r.m.s.) value of
`A26 could have been used in FN instead of F23]. The
`behavior of r.m.s. A26 essentially parallels that of
`1329]. In the interest of simplicity,
`the latter was
`adopted for use in F”.
`Almost nothing is known about standards of excel-
`lence for judging completeness of a powder pattern.
`Only recently Smith (1976) established for triclinic
`compounds that
`the relationship between N,
`the
`number of observed diffraction lines, and N,,,,, the
`number of independent diilraction lines possible up
`to N, is
`
`=Kl(1—K2N)-
`
`(5)
`
`Existing data:
`
`Unknown
`unit cell
`
`Space group
`tinder
`
`Space group
`tinder
`
`Unit cell
`data base
`
`Unit cell
`data base
`
`F N I
`calcula tron
`
`N .
`calcula ttnn
`
`publication
`
`?rolessionaIlv managed
`data base
`
`Magnetic
`tape
`
`Fig. 3. Flow chart for implementation of numerical ratings of powder difiraction patterns
`
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`GORDON 5. SMITH AND ROBERT L. SNYDER
`
`65
`
`Completeness of a pattern thus decreases linearly with
`increasing line number, N. For 40 high-quality patterns
`of triclinic compounds, the constants in (5) are found
`to be: K, =0-833 and K1 =0-0065. Thus, the 30th ob-
`served line is typically the 45th possible line. However,
`excursions of as much as :7 from 45 are not uncom-
`mon. For the 132 cubic patterns from NBS that had at
`least 30 observed lines the most frequent value of
`N,,,,,,, corresponding to an N of 30, is 32; a majority
`of these patterns had NM,
`in the range, 30 to 34.
`Completeness is generally expected to be greatest for
`cubic materials, other crystal systems being interme-
`diate between cubic and triclinic. These standards, of
`course, apply to patterns from reasonably well behaved
`materials and not to patterns from materials showing
`pseudosymmetry, subcell formation, and the like. For
`patterns from materials having unusual diffracted in-
`tensity distributions, e.g. for systematically weak re-
`flections for odd values of one index, the completeness
`factors will likewise be low.
`
`Conclusions
`
`1. A simple to use, quantitative figure of merit, cover-
`ing both accuracy of the positions of diffracted lines
`and completeness of a powder diffraction pattern, has
`been derived. This now provides the author, referee,
`editor, data-base manager, and user with a rapid,
`quantitative rating in much the same way as the R
`
`factor provides such a rating for single-crystal structure
`determinations.
`2. The present FN ranking scheme is shown to be
`superior to the M 2,, ranking scheme. It is recommended
`that use of the latter be discontinued.
`3. Guidelines are given on the use and implementa-
`tion of the FN rating of powder patterns.
`
`We wish to thank Q. C. Johnson. M. C. Nichols, and
`E. W. Kahara for helpful discussions.
`
`References
`
`Crystal Dara, Dererminarive Tables (I972). 3rd ed., edited by
`J. D. H. DONNAY & H. M. ONDIK. National Bureau of
`Standards (US) and Joint Committee on Powder Diffrac-
`tion Standards, Washington, D.C.
`JCPDS (l9‘??). Powder Diffraction Search Manual for Fre-
`quently Encountered Phases. Publication FEP-27, Inter-
`national Centre for Diffraction Data. Swarthmore, Pa.
`NBS (1955). Natl Bur. Stand. (US) CI‘:-c. 539, 5, 50.
`NBS |(51968). Nan‘ Bur. Stand. (US), Monogr. 25, Section 6,
`p.
`.
`SHIRLEY, R. (1979). J. Appl. Crysr. To be published.
`SMITH, G. S. (1976). J. Appl. Crysr. 9. 424-428.
`SMITH, G. S. (l9‘??a). J. Appf. Cryst. 10, 252-255.
`SMITH, G. S. (l9‘7‘?b). Unpublished.
`Suvosn, R. L., SMITH, G. 8., Kansas, E., Jormson, Q. C. &
`NICHOLS, M. C. (1977). Lawrence Livermore Laboratory.
`In preparation.
`WDLFF, P. M. DE (1968). J. Appl. Crysr. 1, 108-113.
`
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