V. K. Pecharsky &: P.Y. Zavalij, ’’Fundamentals
`
`of Powder D1fj‘raction and Structural
`
`Characterization of Materials”,
`
`21"‘ Ed., Springer (2009) pp. 524-545
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`
`Vitaliri K, PL‘ChEJl'5k}'
`Anson Mtiiston Distingui.«;hcti Professor
`01'EItgiI1cCI'ir!g
`l)ep;1rtmciit of .\f1£IlL’[liIi{\ Science
`and Engimzeritig and Attics l.tiboi'atmI'y
`of LES Department 01' Iittci-gy
`imvat State University
`Atwss. Iowz1S(}Ol 1-3020
`USA
`Vilkp@illI]CSliIb.gC?\-'
`
`Peter Y. Ztivulij
`Director. X-R:t_v Caystallographic Center
`Dcpetrtmettt of C‘hcini.s.try and B‘m-.-hcmixiry
`Unix-crsity of" .\'1ill')-‘hind
`Cnilegc P11l'i\.
`.'\«l;tr;.‘l'.t|1d 'l()742—44.S4
`USA
`p':_;n'ali_i@Lttnd.cdti
`
`,
`_
`irctl bx-' S:3.!v.uJ0r D".-|i'x
`is Illfi
`it
`_ Inlltwvta the hook content
`C0»-or iullhlitllitltl, created hr Pctcr 7.il\-"(ill
`illllllltc t,»\u.:\'i}Sn-1. cnurtcxy
`painting "Tin: fttictzinmrphusi.-3 0!" .\l:trci»'su.~;“ uhcrc Nuiuisstis (pttlyul
`Lubov '/_;i\ -.tli_;‘) tulls in Inn: with his mum refiectiuit ttIiI'fi'.iution patttct n), tramfurnis mm an egg [il.‘.Ci[)l'0C{II
`latttictzi. and than mm :1 limi-ct fcr}'~I1I|
`.‘ill'uL'llll‘(‘ in Ll1)‘]}'SiC1II .\[}:1L‘L‘]. which hC&Il"}lIiHI1lI|l1C
`
`J
`
`ISBN: 97B—i}—38?—O‘)5T8—3
`DOI:
`I0.IU07f978-U—38'F—()9S?9—U
`
`::—ISBN; 9?8—U—387-t)95T9-(J
`
`Library of CI.)l1}_1I'CS.‘i Control [\'umht:t': 2(J08‘)30|2'2
`
`© Springer Sciencc+BLixi|1e.\s Mcdia. LLC 2009
`All rigltts t'r::servcd. Tliir. work rnuy tlni be ti'tIns|itit:d nr copied in whole or in part without the writteit
`|1{3f1Tl1\!aiUIl of the publishtsr ("Springer Science+Business Nlediu. L!.C. 2?-3 Spring Street. New ‘ftarlt.
`NY IUUI3. USA). Exttlpi fur hricl t.',>iCt!I‘pIS in cuimecttsan with rc\it::\!t«.'a tar
`.~'cholat'1} unuiy.-:i.-;i Line In
`connection with any form of inI'urm.mun storage amt! I-C“-it‘-‘a‘lt electronic ;ati:tpt;iIiun cnn1g1t|tct'~:nIt-.;11'c.
`or by Rilllllilf or dim-«imil;ii' Il‘:('l1'Il}LiE1lUg_\,‘ now known or Eicrcatltcr dL'V'C.ll)pt.T{l is Imbidtlen.
`The use in this pint:-licatliun U1-il';ILi{.‘.IHIIHCS.ITUUCIIIUIK‘-.M2lYiCl! :1aui'k.~', and .~an'ml;1r terms. evcn if they turn‘
`nnt idcntilicd as such. is nut in be taken as an nxprcusitmst 0!" opinion as tn whether (1! nut they are subject
`to pmprictm y tight.-.
`
`Printed. on atu.'1d—I'rcc paper
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`=.pri ngur cum
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`ii 7 The Rietveld Method
`
`sequence in which various groups of parameters are refined. Regardless of the rel-
`atively long history of the method, it is certainly true that almont everyone familiar
`with the technique has his/her own set of “unique" secrets about how to make the
`relinernent stable. complete, and triumphant. Therefore, for the remttinder of this
`chapter. we introduce the basic theory of Ric-tve1d's approach. A set‘it‘,s of hands-
`on exztmples dt.‘.'ITIt)nfill‘El[lf’tg the solution and refinement oi" cr'_vstatl structures with
`\'tll'it}'LIS tiegreex ol' complexity is {,L‘i.‘wCIT}bICd in the laxt ten cliztpters of this hook.
`Every example is strpplemented by actual experimental data found online. thu:-; al-
`lowing the reader many opportunities to follow our reasoning, as well as to create
`and test hisfher own strtrtegiex, leading to the sueeessfttl determination of the crysttri
`structure from powder diffraction data.
`Consider Fig. 15. IO. which shows both the observed and calculated powder dif-
`ftuctiori ptttterns of l_;iNi4_g5Stim_q for at rtttrrow range of Bragg tangles. AS.‘i1lt‘ll-
`itie that
`
`~ We have an udeqttute structural model, which makes both pltyhlctli and chemicztl
`sense
`
`.. 2::
`
`32‘C
`3OU
`G":
`10>-
`23'6':C
`
`o 5
`
`EE
`
`81.0
`
`Bragg angle, 26 (deg)
`Fig. 15.10 A frttgtnent of H. powder dillructitiri ptrttern of l_.El.Ni_13=.SIm .3 collected on -.1 Rigtlktl.
`TTRAX totutrng anode powder‘ diI'lr'actomr:tt:r using Cu Kt: radtzitrotr with in step 323 : 0.02‘.
`The tthxer wed scattered intensity [i’_"“‘J IS ‘l|l0\\'I‘l tl.‘.it'Ig the upw: cfrt-lt'.t': cttlcttlated intensity [}'f""l
`is shown Llhlllg the _fiHt'r.‘ t't'rt‘le.r connected with rlrm mhxl hrrc. The tli|Tert:ricu:s between the ob-
`nened and ccllcuiatted intensities are shown using the ripen trt{ftIl.;i'(!Y. The tlrirk .t'or'rtI hm‘ dr:rt\'n
`:rcro.~r:» the nperr :rr'mt_s,»lt».r corresponds to lr‘..“"‘ — lr'f""" -—' U. The dit'l'er'ent:e5 lzretween the observed
`and calcuinlcd lnrerrsities. are lI.‘;u.‘,|”)" plotted tcxing the scale identical to l'.“"-‘ and Yf""‘ with a
`conxtnnt displttcerrtent For elurily. The i-cu-riral trrk mm'k.§‘ {h(tJ‘.\') indicate calculated positions of
`Bragg petrks.
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`15.? The Rietvcltl Method
`
`537
`
`Slrtl(.‘ltI|‘t)N of complex organic and meIal~organic sti'ut':tui'es can be reliably
`determined using rigid body description
`Structures of small- and medium-size proteins, itnd other Intteromolcettles can be
`refined using stereo-chemical restraints;
`Charge-density distribution from powder dillraetion data can be studied beeattse
`of much improved £lCC1Il'E.tCy of powder diffraction experiment and profile simu-
`Iation-l4
`Structures of low crystallinity and nt1no—crystttllinc materials can be refinetl using
`it pair distribution function
`A wealth of noncrystallogrtiphic $LI'tIt;ttII‘étl information can be extracted from
`Rietveld refinement, which includes reliable phase eotnptisition, including amor-
`phous content. gratin size, Inicroslrain informallion. texture ttnalysis, and others
`
`15.7.1 Fundamentals of the Rietveld Method
`
`During refinement using the Rictvcld method, the following system of equations is
`solved by means; of Li nonlinear least sqttarcs minimizationijj
`
`Ylr.'ttlc' : kylrzlit
`Y-ruir :
`ni>.r
`2
`3
`
`'
`I
`fair" _ _
`Yr:
`i"YJi’7‘
`
`(£5.28)
`
`Here ij.""’“ is the observed and Yf""" is; the ealcuiatcd intensity of :1 point i of the
`powder diffraction pattern. it is the pattern scale factor. which is usually set at A = I
`because scattered intensity is measured on a relative scale and It is absorbed by the
`phase scale factor (e.g., see (15.30) and tl5.3|)), and it
`is the total number of the
`menstired tizttat points. Hence.
`it powder diffraction pattern in :1 digital Format,
`in
`which scattered intensity at every point is measured with high ktCCttt':1Cy, is indeed
`required for it successful implementation of the technique, In the Rietveld method
`the minimized function, <i>. is therefore. given by:
`
`on : git-,-(i',.""-" — i/_"""')3
`
`(15.29)
`
`5‘ V A. Streltmv. N. lshixawu, Synchrotron X-my ztnalysns of the electron density in HoFt-:3. Acta
`Cryst, B55, 32I (I999).
`‘5 In this section, we are concerned with an experiment. which consists of it single pattern. The
`Rietveld technique may also he used to contlttct refinement of the crystal structure employing
`multiple patternti collected from the same Inaterial. For cxaniple, conventional XAray data collected
`using diI't‘erent wtiveletigtlis, conventional and syncltrotrolt X-l'Zl}'.‘\. conventional or synchrotron
`X—ru‘_~':.' and neutron source may he used simtiltatiieouslv in :1 combined Rn.-t\'c|:J refinement The
`|'tintlainicI1ttI|.~ til" the comlunctl Rictreld rclincincnt are hrielly COl'|5l(lt:ICtl in Sect. ms.
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`I5 Solving (_‘i'y.~ttnI Stt'uctItt'e From Powder Diiitmctiott Data
`at function selected to represent a peak shape.
`In at multiple-phase diffraction
`pattern. these may be Ct)tlSIt'2tli1t.:(.i to be identical or refined independently for
`each identified phase (generally except for the asymrnetry, which is t:_;(J1'm‘n(]n
`for all of the phases since. for the most part. rtsytinnetty is L: Function of the
`axial and in—pl:1t1e divergence of the X-t‘;t_\‘ beam, see Fig. 8.l5l. it" U~'£1I’t‘£i!1lC(l
`both by [l1€.’qL1;llll)‘ of the data and eunsiderahle dilierenccs due to the pliysieal
`state of various; t:)l‘l£t‘ic.‘~' in the specimen.
`td) Unit cell dimen.a'ion.<;, usually front l
`to 6 independent pararneters for each crys-
`talline phase prexettt in the HpCt.‘iIttt‘.l’1.
`(Cl
`l’refet'red orientation. and if 11t:C{.‘.‘~'.‘l1.tI‘y. ab.«'orptioti. pormity, and extinction pa-
`rametetzx. which usually are independent for each phaxe.
`Scale l'aetot'.s, one for eacltpha.xe (1\’,-),'.ti1t:l in the case oi" multiple sets ofpowder
`diffiaction data. one per pnttem excluding the first. which is fixed at ll‘ -_ 1.
`Positional paranteters. of all independent atoms in the model of the crystal struc-
`ture ofeach crystalline phase. ttsttally front (J to 3 per atom.
`Population paratneters. if" certain site pos'ition.x' are occupied partially or by dif-
`let'ent types ol'atotns' sinittltatiteottsly. uxttally one per" site.”
`Atomic displacement paratnetet's_ which may he treated as an overall displace-
`ment paratnetet' (one for each phase or :1 group of zttonisj or individual atomic
`displacement paran1etcrs;. with the number of independent variables between
`one (isotropic approximation) and six (am.~;oti‘opicappi'oxirn;1tion)pcr site.
`The least Sqttztres p;n'aineter.s' listed in itenis ta) through (d) are the same in Paw
`ley and Le Bail full pattern decompo.~,ition. and in Rietveld relinentent. Other vari-
`ables. that is. those loutttl in iteim te] tltt'ott_:_;lt ii] in the list given here. are specific
`to the Rictveld ntethod. Although it
`is nearly ittipttssihle to pre_s'crihe it unixetsal
`and rigid 0t'dt:J' in which vztritztts grotips. of‘ pltysicall)-' dit'l'et'::nt p£i1‘£1I'I1C[L’:|'.H .‘&ll011l(l
`be included in :1 Rietveltl refinement. the most common parameter turn-on sequence
`based on their importance and potential for least .squat'es ittstahility, as suggested by
`Young.” is illttstrated in Table [52,
`The turn—on sequence described in Table l5.2 may be. and often is altered de-
`pending on many variables. which include data quality, accut'ae_v nfthe initial struc-
`tural model, and lmowledge ofthe ittsttumental L.‘DI1l.I‘tl'Jlttlt)n.\' to profile pZ1t'{lt11C[E:r.\'.
`It
`is important to realize that rarely. ilever, it is p:J5:;ible to rehne all t'L‘lC\‘:,tt1L vari-
`ables .~:imult21neous'ly from the beginning due to the eornplexity ofthe problem and
`many possiliilities for an out—ol‘—control least squares. AH noted hy Young. a sys-
`tematic. one—hy—ot1e tut't1~on sequence is nearly atlwatys the most effeetiwe tool to
`establish which pattutitetei‘ i.\ eattsing the trouble when the refinement does not go
`well. and it is not clear why‘.
`
`.?_¢.,.,_~.-..._,,-_.-_.—_»-r:'
`
`.-—.—.-_:_-H_.-
`
`. ntore than one \';£I'1iil’)!C per mite ntaj.
`re tmtall_\' L.‘l\[l'L‘.lllCl_\r' titflicult to tetinc .‘w'I.'I't}&ihl_\.
`5" RA. Young. introduction to the Rietvelti method.
`in. RA. Young. Eti.. The Rietveld method.
`O.~cI'o."d [_'rnver.~;lty Pi'e.s.x. Oxford. New York tl9‘J3).
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`|5.7' The Rietveld Method
`
`53.
`
`Table IS.2 A sttggestcd ptminteter turn-on sequettct: in :1 single-phttse. single-pattern Rietvcld
`refinement using constant wttxclcngtli X-my or neutron dultt (adopted from ‘r'otine_).5°
`
`Sequence
`Comrnerit
`Stable
`l.ine:tr in (l5.3Ul- (15.3-4)
`P:traIt1etet‘ot'gt'ou|'iol‘
`pttritmelet 5 ____‘p’_
`Phone smlt:
`Yes.
`Yer.
`,1
`Specimen
`No
`‘(as
`b
`displacement
`Linear bi.l.L‘kgI'0ul1d
`l.Lllll(.'l.2 p£1l'«'1I\1ClCI’\'
`More hLlCkgI'()lll'iCl
`W
`Coortlintttes of atoms
`Pi'cl'errt:d uricnteition
`Poptrlution and
`isotropic
`ilrsplztccme nts
`U. V . other profile
`prtrairteters
`Anisotropic
`displacctttcttts
`Zero shift
`
`Yes
`No
`Geuerztlly nu
`No
`No
`No
`N0
`
`No
`
`No
`
`No
`
`Yes;
`Yes
`Fairly
`Poorly
`Fairly
`Fairly
`Varies
`
`No
`
`Varies
`
`Yes
`
`as
`;|
`e
`e
`'
`I"
`Correlated
`
`'..Ii-Ii-'.vJ'.\J|\J|'\Jl\l
`
`Last
`
`Last
`
`It
`
`I. 5 or not
`
`“ When the scale l'ui:Ioi' I\' fin‘ off. or \.\'llt'1Il the model of the Liystnl structure is wrong or too far
`lrom reality. the refined scale factor m;i_v hecuntc incorrect.
`" ‘line specinicn l.llSplElCclTICTllp(tr:lIl.lt3[El'l.lS1.l.Ill)- varies trom suttiple to sttrnple. and it usually lttkt:.\
`up ~.0lt1t: oi" the effects of sample l.I'(tll.\p:ll‘Cl1t.‘}‘. Fora properly Llll1_1l1CLl goniometer; the zet'o-shill
`error should be negligible. Even if the gmtiornctcr is Ini-tzilignctl. the zcro—sItil't correction should
`remain sninploindepentlent.
`" When the background is large. ill least its constant component should be estimated and included
`an the very bcgiilning. In a polynomial approxinnattion, the background parameters tire linear. When
`more than the required background pill"dI'lll.’lC1’.*i are employed, .~'.evet'e correlations rnuy result.
`‘J When one or rnore lttttice patrarneters nre incorrect. one or more cttlcululcd Bragg punks can lock-
`on to it wrong observed [.)CEil(.ll1llE\ leading to it solid false minimum.
`‘U, V. W tend to be highly co|Tt:|oted and \-arious combinations of quite dil‘icrentv:tlt1es enn lend
`to essetttitiliy the sutne peak \t'tdth.~..
`‘ When coordinates of all atoms are retinctl. the plot of the observed and calculated dillraietion data
`should be used to determine whether preferred orientation should be included. Coordinates may
`strongly t.'0t'rcl2ttt: in the presence of pseudo-syminctry.
`
`15. 7.3 Restraints, Constraints, and Rigid-Bodies
`
`We already mentioned that all chemical and physical information must be cottsid—
`ered in order to succeed andlor speed up the process of solving the crystal structure
`from powder diflrrtction data. The same is true for Rietveld refinements. especially
`when diliraction data are of low resolution andfor the §U'uCElIl'eS are complex. Only
`relatively simple structures can be refined using good data without any or with mini-
`mul restrictions. instead ofemploying numerical tools in order to stabilize the refine-
`ment and obtain it sensible result {e.g.. using dumping. see Sect. 15.5) that are not
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`
`Rigid Bodies
`
`If» Solving Cr_\-xtal Structure from Powder Dithxiclion Data
`
`it whole (rigidly) using
`Rigid bodies are molecules or groups that are treated as:
`the Cartesian coordinate system.63 or sonietirnes using Euler angles.“ The most
`recent developments in the rigid body algorithms in the Rietveld refinement have
`been reviewed by Dinnebieizm and many applications of the technique have been
`ineorporatted in GSAS by Von DreeIe.7' Here. we only give a brief description in
`order to help the reader with a general ltl1LlCl'.‘~‘liiI1dli'1g of the rigid—body approach
`and its possible applications.
`Tran.~;i'oriiiatiotts of the cr'yst2illographic coordinates (it) into the Cartesian coor-
`dinates (X) and back are performed as.’
`
`X:H{x—t)andx_—.—H‘lX+t
`
`(I538)
`
`t'co:s;3
`bcors "/
`Lt
`l}
`bxin 0:“ sin }t
`0
`csinfi
`{J ——:’>cost1's;iny
`and t is the center of the rigid body consisting of n atoms; delined as
`
`H =2
`
`it {:1
`
`(15.40)
`
`The location oi" the center and OI'lC1't{E1IIOI't of the rigid body in the Cartesian co-
`ordinate esystein is fully defined by six parainetem: three coordinates of the center
`describing the displacement of the rigid body from the origin. and three angles de-
`scribing the orientation ofthe rigid body. Thus, instead of 32-: coordinates in general,
`the number of pzn'zirnete1's is reduced to no more than six and both the location and
`orientation of the rigid body can easily be atdjusted. If the rigid body is located on
`.1 finite symmetry element, the number of paranieters is reduced. For example, it
`molecule located on :1 mirror plane can only more in the plane (two positional pa-
`1':nneter.~'.). and it can only rotate around at normal to the plane (one orientational
`parameter); a molecule located on rt twofold axis; has one p0.\‘iti0r1:.Il (along the axis)
`and one rotational [around the axis) paranteter.
`Thermal motions ole rigid body are described using the so—called TLS matrices:
`
`.tindS—.
`
`(154!)
`
`T12 T13
`Tit
`1-13
`1'-12
`I’-it
`5n 5t2 Sn
`7]: T3;
`:3
`.I.= Li: L22
`53; S33 S3;
`7'13 T23
`721.‘;
`in in 1'»_s_i
`S31
`332 533
`"5 C:1t'tc.\i;tn coordinate ‘tyktcm uses there mutually perpendicular coordinate axes with equal scal-
`ing, c.g., l A or I run.
`6" The Euler angles were developed by Lconi'tutd Euler to (l{L\Cl'1l'.tC the orientzition of 3. rigid hotly.
`For more details" see http.f.-’en.wi1<ipedia.orgtwikifELulenaiigles.
`ll’ RE. Dinnebier, Rigid bodies in powder difliaction. A pt'.'tt.'1lL".li guide. Powder Diffiztction. 14.
`8:1 t I999}.
`
`T:
`
`7' Available at i1llp:.".t'\.=r'\i'w.L‘t,'p]='l :ac.ul-:f.»;olutitimlgaitxr.
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`
`15.? The Rietveld Method
`
`537
`
`where the T is translation, I, is lihration, and S is a screw matrix that describes
`
`thermal motions. Matrices T and L are symmetric (M,-j r M_,-,)_ and matrix S is not.
`The screw matrix defines mixing of the translational and librational motions and is
`Often 0. c.g.. for a centrosyntmetric rigid body. The maximum number of the TLS
`parameters is 20 (6+6+9—i)'. one pi1l'al‘I'It'.lt.’,I' is subtracted since only two diagonal
`elements ofS are independent. and they are usually refitted as Sim = 533 — Sn and
`5'93 .— S3]. — 323. When the center of a rigid body is located in a special position. the
`site symmetry imposes additional const1'ttints73 on the allowed rigid body motions,
`As a result. the number of independent parameters (which are the elements of the
`TLS matrices) is reduced. Hoinetiines signilicantly.
`The tensor U (see (9. ND) representing thermal motions of the ith atom {1 3 i 5 ii)
`is obtained from the TLS parameters as follows:
`
`="r—u—A,sTs"A,T+A,-LA?"
`
`(15.42)
`
`where matrix A, is constructed from Cartesian coordinates of an atom:
`
`A; =
`
`0
`--Z;
`Y;
`
`Z’.
`O
`—X,-
`
`_}/J.
`X;
`0
`
`[l5.f-“l-3)
`
`The anisotropic atomic displaccnient parameters U,-,- {see (9.9)) can be obtained
`from the tensor U by using the so-called g matrix, which is the same as that used
`to calculate the d-spacings,
`that is, (IN): = h‘ gh. also sec (8.10), (I444). and
`
`_ -4.’. where g =
`V g""‘qJ"*"
`
`a'2
`rr‘b' cos)"
`(t"c" cos ,8 “
`
`a"b‘t:os}"
`b‘?
`t';“<“ cot;0t'
`
`a‘c‘cos,8'
`b"c:'”cos0t‘
`c'3
`
`(IS.-44)
`
`The TLS approach is far from trivial, especially with respect to symmetry im-
`posed relationships between the elements of the matrices, and attention is needed
`in order to provide a stable and correct refinement.” Yet, the mechanism of rigid
`bodies results in many advantages. These are: elimination of meaningless changes
`in the geometry; at substantial reduction of the number of free variables. which can
`be determined with a higher accuracy; a better convergence to a correct structure
`when compared to the unconstrained refinement; inclusion of hydrogen atoms at
`early stages; and reliable anisotropic atomic displacement parameters. which is osui
`ally difficult to ex pcct from powder data. The following rules proposed by Dinnebicr
`should be followed:
`
`— If the center of the rigid body is also the center ofgravity. the components of the
`S matrix can normally be set to zero.
`
`73 V. Shomaker.
`B24. f13(l9§i6).
`
`|{.N. Trueblood, Lin the rigid body motion of molecules in crystal. Acta Ciyst.
`
`7-‘ “| E" one sticks too rigidly to onc‘.- pt’irIL'iple.\_ one would hardly see anybody." Agatha Christie.
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`I5 Solx ing Cry.~.ta| Structure from Powder Drllraetiorr Data
`533
`— Refining only the diagonal components of the T matrix corrstrained to be identi-
`cal. and keeping all elements of L and S eqtral to zero is the same as refintrrg an
`overall isotropic temper'atut'e factor for the rigid body.
`— Refining all elements of the T mntrirr. with I. : S :. U, is the statue as refitting an
`overall anisotropic temperature factor tor the rigid body.
`— For planar moleculest. it is often strllicient to refine only the components of the
`T matrix and the diagonal elements of the L matrix.
`~ If the rigid body is located on a finite symmetry element, some elenrents of the
`TLS matrices must be set to zero or constrained accordingly.
`An example on the rigid body refinement of :1 complex 16 atoms ligand in rt
`rnetaI—organic compound can be lorrnd in recent paper by Nielson at al|.7"' In addi-
`tion, at variety of tutorials and przrctical guides on how to use the rigid body refine—
`mcnt can be found on the GSAS web page.”
`
`iii».
`
`15. 7.4 Figures 0fMerz't and Quaiitfy ofRietve!d Refinement
`
`Similar to both Le Bail‘s and Paw|ey’s full pattern deeonrptrsititrm, the quality of
`the refinement using the Rietveld method is qtrztntilied by the corr'espontling ligureu
`of merit: profile residual. RP. weighted profile residual Rap. Bt'agg residuztl, RB.
`expected residutrl RL.,,p. and goodness of ht. 13 (see ('l5. |9)—(l5.23}). The Dllt’l')ll‘I/
`Watson d-statistic [(15.26) and (l5.27)) may be used to quantify :1 serial correlation
`between adjacent least squares residuals in at Rietveld refinement based on step-scan
`experimental data. As noted earlier, all but one (R5) residtr
`profile and strttctural p
`figure of merit, which is nearly ex-
`'
`'
`clusively dependent on structural pararneters and therefore. prirrtttrily characterizes
`the accuracy of the model of the crystal S[t‘L1CIltr{:_m
`Regardless of the importance
`sure the quality of the Rietveld re '
`
`by the difference. A)"; = l’,-"l" —
`
`tt.~ir.‘5 truc ealctrluted II'IICgt'(IlL'tIl
`' -tuully rtteasttred expcr'irrr::ntal|y.
`_
`y proratrng the obsert-ed exper'imentzt| profile pt'op:>t'ti0ntIl'ly to the
`eorttributiorn; i"ror1r multiple overlapped ealutrlatetl retleeriorr protiles zttrv r the ltaekgrottrtd l].I\l1CL‘tI
`strbtracted (see (t5.7)}, lollowed by the rttrnterical integration (sec (8.40)). In tltis. regard. Bragg
`residual also depends on the profile pararneterx although this dependence is far less errticetl when
`compared to R9. Rwp. and 13. In .~it)I11t-: refcrerrees_ R5, which is based on the .~.qtr:tre root-. oi" the
`integrated inrerisities. is used ;t~ an equivalent of R..-, The latter employs the absolute Willie» ntthe
`ob.<er'ved and calculated .‘rlI'l|ClUIL' t'nctot‘.\: Rt-.
`'—- :i:lr\‘i]:‘p|'t.a';
`, [l*'|"“"“l§J,»“E(ft',F:’l“;l
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`15 Solviiig Crystal Structure from Powder Diffraction Data
`
`y to do both in order to attain it sufficient approxima-
`tion of the background present in some powder patterns. Needless to say. measuring
`the background without a sample must be done using configuration (satnplc holder.
`optics, and power settings) that are identical to those employed dttting the actual
`measurement of the powder dilfiaetioii pattern.
`Complexity of the background may also be intrinsic to a sample, rellecting poor
`crystallinity. extreme disorder, or nzmocrystallinity. When deviations trom the con-
`ventiotial, weakly Bragg angle dependent behavior are weak. adding it few free vari-
`ables to fit the background usually helps. Otherwise, oneis best bet is to employ tools
`developed for total scattering analysis” rather than Rietvcld refinement.
`
`Poor Fit of Peak Shapes
`
`This usually occurs when one or more critical peak-shape parameters have been
`overlooked, or became unphysical during pre\-‘iotts refinement steps. The latter often
`happens when too many of the pealt-shape—related parameters have been set as free
`variables during early stages of the refinement. Common solutions include
`-— Checking the difference profile {see Figs.
`l5.5——l5.9) and identifying the cul-
`prit responsible for discrepancies in tltc majority of strong, welhresolved Bragg
`peaks. When the source ol‘ the problem has been identified, an incorrect patterne-
`ter must be reset and/or the forgotten parameter included into the refinement.
`It is possible that the selected peak-shape function does not describe this par~
`ticular experiment well. The best way to lind the right function is to exznnine ti
`well-charztcterized standard, such as LttB5 (SRNI-(>(3{}a. which is available from
`NISTTQ). Once the function that
`fits peak shapes of the standard is found.
`it
`should be used to fit profiles of all other samples measured rising the same dif-
`fractometer and the same goniornetcr optics.
`Asyrnrnetry correction is commonly overlooked. Nlztke sure that an appropriate
`model (see Sect. 8.5.2} is selected. and relevant parameters are included in the
`refinement.
`Sometimes, peak widths exhibit strong anisotropy due to anisotropic strain. cx~
`trctnc particle anisotropy. or stitching faults. If this is the case, it may be necessary
`to use appropriate corrections to the conventional. smoothly varying as :1 function
`of Bragg angle FWHM {e.g.. sec (8.34)).
`
`Mismatch Between Calculated and Observed Peak Positions
`
`Most corninonly, this problem occurs when expertmental data have been ttffected
`by errors due to zero shift, saniple displacement. or sample transparency. Setting
`73 T. ligttitii and 8.1 L. Billinge, lltitletm-.:tth the t3i-agg peaks. Pmgtzrnon Materials Series tl’ctga—
`rnon. Amsterdam, 2003].
`79 SRM is Standartl Retirreiiee Material. Consult
`littp.-fits.nist.govf:nct:sur'cmentsctvices}
`ref‘ei'et1ce-rnatcriatlsIindes.ct'nt for details.
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`I5 8 Concluding Rerrttarl-m
`
`543
`
`llttetuattions about certain minimum valttes. which ztre experiment- and structure-
`specific. In some euses, FOM.~ may begin to rise. More often than not, their steady or
`erratic rise indicates the undesired divergence of the nonlinear least sqtt-ares, which
`is usually associated with severe correlations between two or more vztriables. lf this
`condition is detected. the refinement should be stopped without 5;;tvitig the results.
`ttnd the ztrrny of free Variables !'c‘11tI£tiy.'r’.t.‘(.l to introdttce proper Cottstmittts.
`Liven ifthe relinement is stable. it should be tetmittated at :1 t:t:rtain point. Because
`of the finite accuracy of both the data and computations. it is unrealistic to wait until
`increments of all [cast sutnnes parameters becttmc zero. The latter usually never
`happens anyway, due to sitnpliliezttions introduced in the nonlincttr least squares
`algorithm {see (15.9)). In addition to the stabilixaticun of all figures of merit near
`their respective rninima. another important factor that should he t.-onsidered is the
`t'elutioti,~:hips between the ittcretttettts and c0t't'esptmding staltclitttl £it:\-'iEtl.lt)ll.\' tiller‘
`each least squares cycle. It is eomtnonly accepted that when all residuals converge
`and stabilize at their minimzt. ttnti when the absolute vttlttcs til‘ all incrctnents become
`
`.~;m;tller than the estiinttted stanclartl deviations of the eo'rrespt:nd1ng free vat'iables.m
`the least sqttztres refinement may be eon.<idet'e(l converged (indeed, the model must
`remain rational).
`
`15.8 Concluding Remarks
`
`In the retnaindcr of this book. we eottsit_let' multiple practical .H!t'tIt.‘1U1‘C solution ex-
`amples. For the most part. individual intensities and structure {actors are extracted
`by using Le Bail ’s algorithm of l'ul1—pattern decomposition. This technique is chosen
`instead of P:tw1ey’s approach because the former algorithm is usually more stable
`and it has been incorporated into several freely available softwzne pi'ogrutns. which
`are coupled with PLtll.C1'>i0I‘t and FoLtI‘iet' calcttlations. Tltesc are: Ll-lPM—Ri<:ticz1,3l
`GSAS,” and FOX.“ although other weIl—devel0peti and testeti t:0mpttteI' Codes are
`available.“ In mine of the more cttrnplex examples, however, we will employ matt-
`ual profile fitting. The latter 2.tppt'():.tt.‘l] is less “Use:'—t't‘iendly" in terms t)liaLttt>mutio1t.
`
`|fl0ll1} fratction olthc stttndurd t.ie\'i.ttion.
`ll" t,|su;tlly lowet than a small (at least
`*” L1-ll’?Vt—Rtetic;t
`tzttithnrs B A, Hunter and Cl, lloward) may be downloaded frmn ftpjlltp.
`autsto gov_attfpt|b!pltysiestnettti't)|ttrielve|cLI'Rieticz1,LHPl\«‘l95! or via :1 link at l1ttp:ffw'\tw. Ccp]-l.
`ttc.tIk
`
`l,;|i's0It and RB, Von D1'ec|t:) may be downluadeti 1mm l1ttp:."t‘www.
`'0‘! USAS tatlthors AC,
`cupl-Lac.ukfsolttttotitgsatsf. A convenient graphic user tntcrtttce tor GSAS {author Brian Toby} may
`be dam-nloudcd from the szttne site or from httpjfwww.ncnr.nist.goviprogtttnsferyslalltngrttphyfl
`3‘ R, (Terny, V Fuwe-i\'ir:olit1. J. Rohlféelvc. M, Htt§:3R_ Z, Mdtéj, R,
`l{u},ei, Expanding FOX:
`!\uto—itttlcxitt;:, grid cotnputittg. profile fitting. 1:. 16 in: CPD Newsletter “Real-Space and Hybrid
`Methods for Structure Solution from Powders.“ Issue 35. (200?); available at
`l1ttp..’!www.iucr—
`cpd‘org:‘Pl)FsICl"1')_?«5.total.pdt".
`5' One oi" these is EXPO — an integrtttcd palcknge for full pattern tlceotnpumtiott and for solv-
`ing ctf.-"stat structure by direct
`I!'Il.‘[l1t)d.‘«
`tatuthors A. Allomaire.
`I3. Cznrnzzini. G.Cttst:aI':1nn.
`C. Gi:tctwnz:r_o. A. Gtiagiiardi, A.Ci.(_i.
`.\=it)|iterni. R. Rlzzi. M.C. Butler. C. Polidort. and
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