D. Krawitz, "Introduction to Dijfmction in
`Materials Science and Engineering”,
`Wiley (2001), Ch. 8, pp 269-277
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`I. Cry.~ata|i:igr:aphy' 2. .\-Iutcrial.» science. 3. Diffrzlctinn. I. Title.
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`DIFFRACTJON FROM MORE COMPLEX STRUCTURES
`
`‘Go
`
`(1))
`
`FIGURE 9.4. Plots cf the (.3) Se and (by Be components 0! FM“) fora Gese structure with both
`Se and Ge on equipeint c. The values of x59 and Xge are 0.28? and 0046, respectively.
`
`had a major impact on structure analysis for new classes ofmale1‘ials such as high-
`remperature superconductors and high-strength rare earth anagracts. The meihod has
`been extended to X—rays and is also a means of pro\«‘iding quantitative phase analysis‘
`The basic elements 9!" the method. which apply In both neulmns and X—ray:.. will he
`presented.
`
`instrumental
`[it parameters can be grouped into {we categories.
`The principal
`and structural. The instrumental parameters include counter zero. an imensity scale
`factor. and the halfwidths of peaks. The struciural parameters include atomic coor-
`dinalcs. equipoint nceupation numbers. and Debye-Waller temperature faciors. The
`slructural parameters have already been discussed: they characterize ihe atoms and
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`FIGURE 9.5. Plots of the (.3) Se and (b) Ge componenls of F00, lor a Gese structure with both
`Se and Ge on eqmpoint c. The values of 259 and 239 are 0.098 and 0.210, respectively.
`
`VI
`__
`3.
`
`0Se
`
`FIGURE 9.6. Gese unit cell. Both elemenls occupy equipoinr c,
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`272
`
`DIFFFIACTION FROM MORE COMPLEX STRUCTURES
`
`the crystal structure of the diffraeting material. This brief treatment is based on the
`original paper (Rietveld, 1969).
`
`9.6
`
`instrumental Parameters
`
`The counter zero introduce.» a constant 29 shift to account for uncertainty in the
`ditfraetometer zero. It is determined through use at a standmd sample whose cell pa-
`rameters are well characterized, such as Al3O3. The intensity scale factor accounts;
`for the constants in front of intensity exprc.s.s1ons and involves quantities such as
`the physical constants associated with the scattering process and the power in the
`incident beam; see Section 8.4. The functional Form for the Scale factor is
`
`_v(c'rtic) = (‘
`
`- ylnbs)
`
`(9.| I)
`
`where _\-(caic) and _\'(ob.r) are the calculated and observed counts, respectively, at :1
`point on the pattern, and c is the scale Factor. The halfwidths of peaks are typically
`modeled using the following function, which has been found to describe the breadths
`of peaks across the angular range of the pattern, especially for neutrons:
`
`Hf = Utanjflt + v tan at + w
`
`(9.12;
`
`where H; is the halfwidth ol‘ the kth peak and 1}‘ V, and W are the HI parameters that
`are determined from experimental measurements of a standard sample. Analytical
`descriptions of Gaussian peaks are given by
`
`_\i',' = I';_- exp
`
`m4 tn2(2ti,- — Etitfi
`Hf
`
`L71)
`
`it the
`is the peak height, and fit
`is the ith position across the peak. I;
`where _t-,-
`position of the kth peak. Parameters for peak asymmetry and preferred orientation
`(texture) are also available.
`
`9.7 Structural Parameters
`
`'3
`The six unit cell parameters are input in n genet':tli7.t:d format based on the form ol
`l/(‘W for a trielinic cell:
`
`I
`I
`1, = Ftaitt + at-3 + C13 +1‘)H+ Em + Fhk).
`‘flit
`‘
`
`(9.14;
`
`where V : cell volume. A = !?2{‘2 s:in3or. B : UECE sin: 13. C = (13123 sin: y.
`D 2 2ri3bc:(co.<;fieos‘ 32 — cosa). E : 2ab3c.'t'co.~;ycostx - cos (5). and F =
`2abc3(eo.<oi cos ,8 — cos y); see Appendix A, I .
`The atomic coordinates are the functional atomic coordinates used to (lest-i-the
`the po.~;ition of the atoms in the unit cell. They are indicated by .1',_,-. _r,‘,.. and :,V,,
`the coordinates of" the jth atoin on the rth equipoint.
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`274
`
`DIFFRACTION FROM MOFIE COMPLEX STRUCTURES
`
`A1203 Standard
`
`L—-S cycle
`Lambda 1.4875 A.
`alrf" Wj '—l"
`~
`
`.
`
`56
`I
`
`Obsd. and Diff. Prcfili
`I
`r
`
`i
`
`l
`
`3.6
`2.0
`2—’I‘heta, deg
`
`FIGURE 9.1’. A Rietveld refinement fit for alumina (N203). The solid tine is the lit and the points
`are the data. The difference between the fit and data is shown under the plot. (Courtesy ol
`W. ‘r’elon.,\
`
`L :
`
`l
`
`W,- {_t',-(mlxr) —— ;\',(ml'r‘)
`
`7
`
`(9.20)
`
`which is ininimi7,ed with reszpccl to the fitting p£.li'l1t‘i]Bl€I'.\i. The weighting factor 1-L’, is
`often basal on the counting statistics For the ith point. G0(3dttc5s—of-lil values. giving
`;I measure of the quality of the lit. are returned. At this point in time a law hundred
`|‘}ill'£tlllCl6t'.\‘ can he relined. Certain patrumcters uro allowed to wiry and are fixed after
`they converge. enabling others to be refined.
`The fit of at pattern of alumina is shown in Figure 9.7. The solid line is the lit to
`the recorded cliffiaction p.'mci'n. which is plotted 213-; points (only visible at the tops of
`{I few peaks and in the backgmtiud regions because of the high quality of the lit). A
`residual line is shown below the diffraction pattern to help see errors.
`Two other tits. for Y(,{F€()_4l\/l|tn_(,)2ft
`and Nd5Fcn. are shown in Emrnplee 9.3
`zmcl 9.4,
`
`Example 9.3 Prrgfifc Re_{.5ric=.=.=.=1=n: Q} ‘{.9{Feu_4Mng_6)33’
`This is an extunple of the power ol" the Rietvcld method. The structure is cubic
`group Fm3m. !iirea':iat:‘a.-ta.’ Tublc=s number 225, There are four possilnle sites than
`the Fe and Mn atoms can ocuipy, given in Table E93, If the Fe and Mn atoms
`
`'E\ump|c t‘ntIi'Ic~._v (ll Dr. W. H Ycloii.
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`PROFILE REFINEMENT
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`Yr.(Feo.a-“no 5)::
`Random
`
`Ys[Feo.aM”n.s}n
`Fuliy Refined
`X’ = 1.33
`
`wflmmzhazw
`
`FIGURE E9.3. Hierveid profile refinements of Y5-(Fe0_4Mn0 5:23. Fn(a)1he Fe and Mn site mob-
`abiiity occupations are random and in (b) they have been reiineci to their correct uafues.
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`276
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`DIFFRACTION FROM MORE COMPLEX STRUCTURES
`
`Table E9.3. Structural Parameters for Y5(FEg_4Mflo_5)23
`
`Elc mcnt
`
`Y
`MLIKFC
`MnfFe
`Muff-‘e
`MnfFc
`
`Site
`
`I‘
`'
`
`Ran R
`
`3:1
`32
`32
`24
`4
`
`x
`
`U205
`{I [73
`(1379
`fixed
`fixed
`
`M :1 fraction
`
`—
`{H8
`0.(>?5
`0.326
`0,548
`
`Note: 'E‘I1r: .x'p:1¢c group is cubirc group number 225. I-'nz,5u:i For cq11ip:.)1nL.\ I.’ and f. unly the I tIU<)['t_ii|1.1ltJ
`is. \-Eli'iilMt?.
`
`lift: randomly assigned. that is, if the probability of any site being an Fe is 0.4. at
`clearly incowcct fit is obtained, as shown in P-‘ig_ut'c E‘).3{a). H(m't:\-C1‘. if the site
`occupation pamtnctcrs are refined by the Rictveld method. the pt'opc1'.'»tru:;1un: is
`obtained as indictttcd by the improved fit in Figure E9.3(b).
`
`Example 9.4 Rieri-Bid P:‘r2fife Rtffiitmmit rgfNt_i5Fe173
`
`The SIl‘L1C11_Ir{3 0|‘ Nd; Fen is considcrabiy more complicatuti than that for Y5(Feg4
`N1[‘1()_(,}3_'I.
`in Example 9.3.
`It shows that
`the Rietveld method can handle more
`
`chi 2= .5
`Nc£5Fef7
`I‘ !
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`no’300
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`ZSOL
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`200!-
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`‘<"$"*U1?“1"'J'-1!"!
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`“T
`
`:.:..._i_.___:_t::_:I
`60
`30
`
`2-THET.-1
`
`FIGURE E931. Fitetveid refinement of Nd5Fe1;.
`
`3 l;'x21mp[c cu1t1‘tcsyo!'D1'. W. 13. Yt.-Ion.
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