RIETVELD ANALYSIS AND MEM-BASED WHOLE-PATTERN FITTINGUNDER PARTIAL PROFILE RELAXATIONFUJIO IZUMINational Institute for Research in Inorganic Materials, 1-1 Namiki, Tsukuba, Ibaraki 3050044, JapanThe latest version of a Rietveld-analysis program RIETAN-2000 has a couple of advanced
features: partial profile relaxation and whole-pattern fitting based on a maximum-entropy
method (MEM). Partial profile relaxation means that primary profile parameters of (nearly)
isolated reflections can be individually refined independently of secondary profile parameters.
It was combined with split profile functions to achieve better fits between observed andcalculated patterns, particularly in samples showing anisotropic profile broadening andreflections with very small d spacings. RIETAN-2000 and an MEM program MEED were
integrated into a structure-refinement system named REMEDY. With REMEDY, the patterncalculated from structure factors obtained by MEM analysis is fit to the whole observed
pattern. Observed structure factors estimated at the end of the whole-pattern fitting are
analyzed again by the MEM. Whole-pattern fitting and MEM analyses are alternately repeateduntil R factors in the former no longer decrease. Some examples of applying this methodology
to structure refinements are introduced.
1.

Introduction

The Rietveld method [1] has been extensively used as a leading technique for refining
structure parameters of a variety of metal and inorganic compounds. The severe overlap ofreflections in the high-Q region (Q = |Q| = 2π/d; Q: scattering vector, d: lattice-plane spacing)in powder diffraction patterns limits the quality and amount of structural information
extractable from them in comparison with single-crystal data. However, advances insynchrotron X-ray powder diffraction [2] and time-of-flight (TOF) neutron powder diffraction
in recent years have enhanced the resolution, ∆d/d, pronouncedly, enabling us to determine
very complex structures whose analyses were formerly difficult with powder data.
Five years have passed since a Rietveld-analysis program RIETAN [3] was introduced in
the well-known book entitled “The Rietveld Method.” RIETAN has two versions for angledispersive diffraction [3, 4] and TOF neutron diffraction [5]. It has been widely utilized,
particularly in Japan, contributing to a large number of structural studies. For example, the1structures of the first High-Tc superconductors with fluorite and carbonate blocks were solved
with RIETAN for TOF neutron diffraction [6]. This article reviews recent advances in
RIETAN with a focus on our original technology: partial profile relaxation [7–9] and wholepattern fitting based on a maximum-entropy method (MEM) [8, 10].
2.

Partial Profile Relaxation

The profile function used in whole-pattern fitting (w.p.f.) generally contains two levels of
profile parameters: primary profile parameter (PPP) and, in its turn, secondary profile
parameter (SPP). The dependence of a PPP on θ (angle-dispersive diffraction) or d (TOF
neutron diffraction) is represented with a physical foundation or in an empirical way inequations including SPP’s. For example, the full-width-at-half-maximum (FWHM), H, in the
famous equation of Caglioti et al. [11]H = (U tan 2 θ k + V tan θ k + W )

1/ 2

,

(1)

is a PPP specific for reflection k with a Bragg angle of θk while U, V, and W are SPP’s
common to the whole 2θ (d) range. We refine not PPP’s but SPP’s in the least-squares fittingof whole powder patterns. Such an equation imposes a kind of an equality constraint on a PPP,
often failing to describe the relation between the PPP and θk (dk) satisfactorily. As the PPP
deviates from the equation relating it to θk (dk), the goodness-of-fit gets worse and worse. The
resulting misfit gives rise to serial correlation committing a prerequisite of Gaussian
distribution of residuals in least-squares methods.
We have devised an original technique called partial profile relaxation [7–9], where PPP’sof (nearly) isolated reflections specified by the user are locally refined independently of their
SPP’s. In Rietveld analysis under partial profile relaxation, the PPP’s of these reflections are
all or partially freed from equations relating the PPP’s to θk (dk) and diffraction indices, hkl.
On the other hand, peak positions and integrated intensities for the relaxed reflections arerespectively calculated from lattice and structure parameters in the same fashion as those forthe other reflections. Part of the PPP’s may be constrained by the equations applied toreflections not to be relaxed. Though the profiles of only low-Q reflections can besubstantially relaxed except for very simple structures, better fits in this region necessarily
lead to an improvement in the overall fit.
Partial profile relaxation stemmed from a simple idea but is well suited to samples showing
anisotropic profile broadening and reflections with very large d values. It is sound andpowerful in the point that neither assumption nor approximation is required in regard to thedependence of the PPP’s on θk (dk). The profile-relaxation technology allows us to fit more
flexible profile functions to observed profiles of relaxed reflections to improve the goodnessof-fit. It can be introduced not only into Rietveld analysis but into pattern decomposition such2

as Pawley [12] and Le Bail [13] refinements. Integrated intensities are refined in the Pawley
method and calculated after each cycle of least-squares fitting by Rietveld’s procedure (refer
to Subsection 4.2) [1] in the La Bail method.
The latest version RIETAN-2000 for angle-dispersive diffraction has grown into a versatile
program supporting whole-pattern fitting under partial profile relaxation by the Rietveld, Le
Bail, and maximum-entropy methods; Le Bail refinement under partial profile relaxation will
be published elsewhere. In each of these pattern-fitting methods, three methods of nonlinear
least squares can be used: Gauss-Newton, Fletcher’s modified Marquardt, and Powell’s
direction set methods [3, 4]. Powell’s method has proved to be most effective for escaping
from local minima in the three kinds of pattern fitting.
3.

Rietveld Analysis under Partial Profile Relaxation

3.1

Three Split Profile Functions

Partial profile relaxation was first implemented in the Rietveld-analysis program for TOF
neutron powder diffraction data measured on the Vega diffractometer at the KENS pulsed
neutron source [5, 7]. Later, we made it possible to analyze angle-dispersive diffraction databy whole-pattern fitting under partial profile relaxation [8, 9]. For this purpose, profile
relaxation was combined with two split profile functions of Toraya [14]: a pseudo-Voigt
functionf ( x) =(1 + A)[ηH + π ln 2 (1 − ηH )]×
ηL + π ln 2 (1 − ηL ) + A[ηH + π ln 2 (1 − ηH )]−1
ln 2  1 / 2 2
2   1 + A 2  x  2 
1 + A  2  x  2  
exp − ln 2(2)ηL
1 + 
 + (1 − ηL )  A   H  A   H 
π 
H

 
 πH and a Pearson VII function
2(1 + A)  AΓ ( mL − 1 / 2)
Γ ( mH − 1 / 2) f ( x) =+ 1/ m
π H  2 L − 1Γ ( mL )21 / mH − 1Γ ( mH ) 

1 + A 2  x  2 

1
/
m
L
− 1)
1 + (2 A   H 



−1

×

− mL

(3)

normalized in such a way that+∞∫−∞ f ( x )dx = 1.

(4)

In Eqs. (2) and (3), x = 2θi – 2θk, 2θi is the diffraction angle at the ith step, A is the asymmetry3parameter, η is the fraction of the Lorentzian component, H is the FWHM expressed as afunction of θk with Eq. (1), m is the decay parameter, and Γ is the gamma function. Subscripts
L and H are regions of 2θ lower and higher than the peak position, respectively. Bothequations are effective in the region x < 0. For x > 0, subscripts L and H should be replacedwith each other, and A with 1/A. These split functions were combined with partial profile
relaxation because of their high flexibility.
The dependence of PPP’s other than H in Eqs. (2) and (3) on θk is empirically expressedin
A = A0 + A1

( 2 − 1 /sin θ k ) + A2 (2 − 1 /sin2 θ k ) ,(5)

η = η0 + η1 (2θ k ),(6)m = −1.517 + 0.980[m0 + m1 (2θ k )]+ 1.578 /[m0 + m1 (2θ k )],(7)

where A0 , A1 , A2 , η0 , η1 , m0 , and m1 are SPP’s, namely, refinable parameters in whole-pattern
fitting. Two sets of PPP’s are assigned to the low- and high-angle sides of each diffraction
profile.
Extending Eq. (2) by assigning different FWHM’s, H1 and H2 , to the Lorentzian andGaussian components, respectively, we havef ( x) =(1 + A)[ηH + π ln 2 (1 − ηH )]×
ηL + π ln 2 (1 − ηL ) + A[ηH + π ln 2 (1 − ηH )]

2 −1
2 

1/ 2
2

2   1 + A 2  x  2 ln 2  1 + A   x    . (8)
+
−
η
1
(
1
η
)
exp
ln
2
+
−




 L
L
 π 
 A   H1  
 A   H2   
π
H
H
1
2




 



This modified split pseudo-Voigt function [8, 9] may be fit to observed profiles of only relaxedreflections. It is flexible enough to fit highly asymmetric profiles in low-2θ regions.
3.2

Examples of Profile-Relaxed Rietveld Refinements

Profile asymmetry mainly arising from axial divergence [15] is pronounced in the BraggBrentano geometry and constant-wavelength neutron diffraction. It may be inadequately
approximated with symmetric profile functions made asymmetric by various procedures. Flatspecimen and specimen-transparency errors in the parafocusing geometry [15] are particularly
difficult to represent analytically. With profile-relaxed Rietveld refinement, we have beensuccessfully analyzing the crystal structures of various zeolites exhibiting reflections in 2θ
regions lower than 20° [9, 16, 17].
Figure 1 exemplifies low-2θ parts of Rietveld-refinement patterns for a zeolite, hydrated

4

Na-LTA (Linde Type A) [9]. Its X-ray powder diffraction data were measured with two BraggBrentano-type goniometers: (a) a commercial goniometer with an aperture angle of Soller slits
5° and (b) a newly designed goniometer with a vertical θ :θ configuration and a pair of long
Soller slits with an aperture angle of 1°. Unless otherwise stated, we measured powder
diffraction data with CuKα radiation using the latter goniometer in all X-ray diffraction
studies that will be described from now on. On relaxation of PPP’s for five reflections whose
indices are noted in this figure, Eq. (8) is fit to extremely asymmetric profiles in (a) and fairly
symmetric ones in (b) comparably well. It will also be applicable to intensity data measured
by transmission-specimen methods. In what follows, profiles of relaxed reflections will be
all expressed in Eq. (8), and those of the other reflections in Eq. (2).
Figure 2 shows observed, calculated, and difference patterns for X-ray Rietveld refinementof hydrated Na-X (faujasite) with 23 sites at general positions [16]. The global minimum could
not be reached in the analysis of such a complex structure unless Powell’s direction set method[3, 4] were used. PPP’s of Eq. (8) were independently refined for 17 reflections to afford very
low R factors: Rwp = 7.60% (Re = 5.23%) and RB = 1.75%. Na+ ions are situated on framework
planes. A large amount of water molecules is incorporated in supercages consisting of 12membered rings. Each cage is distorted owing to the deviation of an Si:Al amount-ofsubstance ratio from unity.
Figure 3 demonstrates the result of neutron Rietveld refinement for AlPO4 -5 prepared by
microwave heating with triethylamine as template molecules [16]. Its intensity data were
measured on BT-1 at NIST. Marked incoherent scattering from hydrogen atoms intriethylamine is responsible for the high background. In Rietveld refinement adopting spacegroup P6cc, we relaxed the PPP’s of ten reflections including the highly asymmetric 100
reflection near 10°. Each template molecule with roughly the symmetry of point group C3v isoriented in the so-called head-to-tail manner along the [001] direction. R factors were Rwp =1.83% (Re = 1.90%) and RB = 4.25%. Equations representing the dependence of the PPP’s on
θk in Eq. (2), viz, Eqs. (1), (5), and (6), were applicable to the neutron diffraction data without
any modifications despite their original optimization for the analysis of X-ray diffraction data.
The performance of the conventional and profile-relaxed Rietveld methods in structurerefinements of five inorganic materials is embodied in the form of a bar graph, Fig. 4. In theconventional refinements, the pseudo-Voigt function of Thompson, Cox, and Hastings [18]was made asymmetric by a physically meaningful method of expressing the effect of axial
divergence on profile shape [19]. Figure 4 reveals that it is far from satisfactory to fit the
profile function adopting this pair to observed patterns in spite of its popularity; Rwp ’s werelowered more or less with Eq. (2) coupled with Eq. (8) for relaxed reflections. The latter
combination was particularly effective in zeolites with several reflections in low-2θ regions[9, 16, 17].

5

4.

Whole-Pattern Fitting on the Basis of the MEM

4.1

BackgroundIn single-crystal X-ray analysis, we often carry out difference (D) synthesis, that is, theinverse Fourier transform of differences, ∆Fk = Fok – Fck, between observed structure factors,
Fok, and structure factors, Fck, calculated from structure parameters. D synthesis is suitable fordetecting subtle differences between real and modeled structures.
Two serious problems hinder the satisfactory application of D synthesis to powder data.
One is the appearance of ‘ripples’ due to the termination effect even in D synthesis where
Fourier coefficients with higher orders are regarded as negligible. This effect makes it difficult
to extract physically meaningful residual distribution from D-synthesis maps. The other isdifficulty in obtaining accurate observed structure factors, |Fok|, owing to the collapse of thethree-dimensional reciprocal space onto the one-dimensional diffraction pattern.
Sakata et al. [20] have been applying the MEM to determination of electron-density
distribution with powder diffraction data. In MEM analysis, no model function is fit to theobserved pattern unlike Rietveld analysis using least-squares methods [1]; that is, we estimate
density distribution whose information entropy is maximized within errors in observed
diffraction data. Structural information contained in diffraction data is accordingly extracted
by the MEM and reflected on the density map if they have been appropriately measured.
Electron densities are calculated from Fok’s in Fourier synthesis. On the other hand, in the
MEM, we solve a kind of an inverse problem where electron densities are determined prior
to calculation of structure factors, FMk. The termination effect is, consequently, far less marked
in MEM analysis than in Fourier synthesis. Further, the MEM can estimate structure factorsof reflections in the high-Q region excluded in the analysis of intensity data; electron densities
can therefore be estimated from a limited number of integrated intensities.
The distribution of coherent scattering lengths, bc, is also determinable by the MEM
analysis of neutron diffraction data. For convenience, bc per unit volume will hereafter becalled nuclear density. The term “nuclear density” may be substituted for “electron density”
in subsequent general descriptions common to X-ray and neutron diffraction.
4.2

Derivation of Integrated Intensities from Powder Diffraction Data

In the MEM analysis of relatively simple structures with powder diffraction data, |Fok| isusually obtained by pattern decomposition such as individual profile fitting [21], the Pawley
method [12], and the Le Bail method [13]. However, most values of |Fok| for compounds
exhibiting heavily overlapped reflections can hardly be estimated by pattern decomposition
without introducing any structural models.
After Rietveld analysis, the ‘observed’ integrated intensity, Iok, for reflection k is6

approximately computed by apportioning the net diffraction intensity, yi – yb (2θi), at step i in
the ratio of contributions of overlapping reflections (K = 1, 2, 3, …..) [1]:Yik,
∑ K YiK

Iok = ∑[yi − yb (2θ i )]i(9)where yi is the observed intensity; yb (2θi) is the background function; Yik and YiK are thecontributions of reflections k and K to the net diffraction intensity at i, respectively. The
summations Σi and ΣK are carried out over all data points contributing to the profile ofreflection k and over all the reflections contributing to yi – yb (2θi), respectively. In the case ofisolated reflections, Eq. (9) reduces to
Iok = ∑[yi − yb (2θ i )].

(10)

i

Consequently, fairly accurate Iok is obtainable in a model-free fashion as long as the
background can be adequately approximated by yb (2θi). Equations (9) and (10) are also used
to derive Iok after each cycle in Le Bail refinement [13].
Yik is evaluated from scale factor s, multiplicity mk, preferred-orientation factor Pk, Lorentzpolarization factor Lk, structure factor Fck, and profile function f (x):Yik = smk Pk Lk Fck 2 f ( x ) .

(11)

Substituting K for k in this equation yields YiK. Fck is calculated from structure parameters(occupancies, fractional coordinates, and atomic displacement parameters) in Rietveld
analysis. The observed scattering amplitude, |Fok|, is determined from Iok Fok =  smk Pk Lk 

1/ 2

.

(12)

Takata et al. [22] applied the above procedure to the estimation of |Fok|’s analyzed by the
MEM. However, electron densities derived from the |Fok|’s are subject to the structural modelin Rietveld analysis because Eq. (11) contains Fck. This undesirable effect enlarges withincreasing degree of overlap of reflections, lowering the accuracy of final densities
necessarily. The MEM can extract structural details from the |Fok| data estimated in the above
way because they involve contributions neglected in the structural model. In addition, |Fok|for an isolated reflection can simply be determined without any structural model, as describedearlier. Nevertheless, the dependence of electron-density distribution on the structural model
casts a gloom over the combined use of the Rietveld and maximum-entropy methods.

4.3 REMEDY Cycles
To minimize the bias imposed on electron densities by the structural model, we devised

7

the following iterative procedure [8, 10]:
1. Intensity data are analyzed by the Rietveld method.
2. |Fok|’s are approximately evaluated with Eqs. (9)–(12).
3. The |Fok|’s with phases, ϕ, derived from the Rietveld analysis, are analyzed by the MEM
to afford three-dimensional electron densities, ρ (x, y, z). Then, FMk is computed by the
Fourier transform of ρ (x, y, z) in the unit cell with a volume of V :FMk = V ∫∫∫ ρ ( x, y, z )exp[2π i(hx + ky + lz )]dx dy dz .

(13)

4. Maps of ρ (x, y, z) are plotted.
5. After checking the electron-density maps, return to step 1 to modify the structural model
if necessary.
6. The model function is fit to the whole observed diffraction pattern by refining parameters
irrelevant to the crystal structure. Each structure factor in the model function is not
calculated from any structure parameters but fixed at FMk.
7. Terminate unless decreases in R factors in step 6 are significant compared with those in
the previous whole-pattern fitting.
8. Return to step 2.
Figure 5 illustrates the above sequence schematically. ‘Rietveld’, ‘MEM’, and ‘w.p.f.’,each in a pair of parentheses, denote analyses whereby structure factors are derived.
Fo (Rietveld) and Fo (w.p.f.) in this figure correspond to Fok, and Fc(MEM) is equivalent to FMk.
Contributions of anomalous dispersion to structure factors must be excluded in MEM analysis
in the same way as in Fourier synthesis whereas they should be included in whole-pattern
fitting. In other words, structure parameters are used only to add anomalous dispersion
corrections to structure factors in MEM-based whole-pattern fitting.
Structural details are changed in step 3 while the goodness-of-fit is improved by fixing eachstructure factor at FMk in step 6. The effect of the initial structural model on FMk diminisheswith increasing number of iterations. We can therefore approach the final structure reflecting
the observed intensity data more closely. Step 6 is not Rietveld analysis but MEM-based
whole-pattern fitting to obtain values of Fok containing additional information about structuraldetails. Takata et al. [22] utilized not all the processes shown in Fig. 5 but only the part ofiterative Rietveld/MEM analyses (inside the upper box consisting of broken lines) for
modifying structural models for metallofullerenes, proving its validity and capability.
We have recently developed a state-of-the-art system for MEM-based structure refinement,
REMEDY [8, 10], comprising RIETAN for angle-dispersive diffraction [3, 9] and an MEM
program MEED [23]. Thus, repetition of MEM analyses and whole-pattern fitting is called
REMEDY cycles, as noted in Fig. 5. We have revealed REMEDY to possess a great capacity
for determining real structures. Figure 6 is a flow diagram illustrating the normal sequence8

of structure refinement with REMEDY and two related programs: Fousyn and Mevius. Fousyn
is used for Fourier synthesis, and Mevius for plotting contour maps and calculating fractional
coordinates from electron densities. Integers in this figure correspond to step numbers
described above. RIETAN provides MEED with structural information, and vice versa,
through two files storing the Fok and FMk data.
4.4

Three Reflection Types

The algorithm for the MEM analysis of Fok’s is, per se, the same regardless of diffraction
procedures. Some special techniques are, however, required to analyze Fok’s derived frompowder data. The MEM deals with |Fok|’s of reflections whose phases have been estimatedby some means. A reflection whose |Fok| is computed from Eqs. (9)–(12) at the end of Rietveld
analysis is referred to as a type 1 reflection. Fok’s of overlapped reflections evaluatedaccording to Rietveld’s procedure [1] are influenced by the refined structure in a varying
degree. Then, the sum of |Fok|’s is assigned to a group of overlapped reflections belonging totype 2 to enhance the ability of estimation by the MEM. In the case of type 3 reflections, theirpeak positions are so near to 2θ (max) that parts of their profiles extend beyond 2θ (max).
Nevertheless, we should also calculate the FMk of a type 3 reflection because excluding themnecessarily brings about a bad profile fit near 2θ (max). The FMk of the type 3 reflection is thus
estimated from electron densities determined from only values of Fok for reflections of types
1 and 2. Type 2 reflections are decomposed to afford their individual Fok’s. Fok’s, sums of
Fok’s, and only hkl indices are output for reflections with types 1, 2, and 3, respectively.
4.5

Structural Data Input/Output by RIETAN and MEEDThe estimated standard deviation (e.s.d.), σ (|Fok|), of |Fok| for a type 1 reflection isdetermined from
1/2

2
Fok   σ ( Iok ) σ ( s)  2 σ ( Fok ) =+2   Iok  s  ,(14)

where σ (Iok) and σ (s) are the e.s.d.’s of Iok and s, respectively. The sum of |Fok|’s, Go , for agroup of type 2 reflections is given by ∑ k I ok 
Go =  s ∑ k mk Pk Lk 

1/ 2

(15)

with the summation Σk carried out over all the overlapped reflections. The e.s.d. of Go isestimated by

9

(

)

1/2

2


Go   σ ∑ k Iok 
σ (s)  2  +σ (Go ) = s  
2   ∑ k Iok 






(16)

analogous to Eq. (14).
At the end of the whole-pattern fitting, Iok is evaluated in a similar manner as Eq. (9):Yik′.
∑ K YiK′

Iok = ∑[yi − yb (2θ i )]i

(17)

Yik′ is the contribution of reflection k to the net diffraction intensity and given by
Yik′ = smk Pk Lk FMk 2 f ( x ) ,(18)

where Fck in Eq. (11) is replaced with FMk . On substitution of K for k in this equation, we
obtain YiK′ . |Fok| and σ (|Fok|) are respectively computed with Eqs. (12) and (14) in the samemanner as their calculations after the Rietveld analysis [1]. Because overlapped reflections
are decomposed during the first MEM analysis without any structural model, they are regarded
as type 1 reflections in subsequent processes.
When analyzing neutron diffraction data or X-ray diffraction data for centrosymmetricstructures, |Fok| for a profile-relaxed reflection can directly be refined in MEM-based wholepattern fitting with REMEDY. This unique feature is expected to accelerate the convergenceof structure refinement, provided that calculated profiles are in good agreement with observed
ones for the relaxed reflections.
4.6

Strategy for Stable Convergence

We should pay attention to electron-density maps obtained from FMk data in addition to theconvergence test in each iteration. There is particularly a fair chance for structural estimation
to be misdirected at the first iteration. Accordingly, we need to repeat steps 2–8 while
comparing density maps resulting from MEM analysis with those obtained from Fck’s. This
comparison is needed to judge whether the electron-density distribution is affected by the
structural model or estimated properly by the MEM. For reflections with unsatisfactory fits
between observed and calculated intensities, e.s.d.’s obtained with Eqs. (14) and (16) should
appropriately be increased to enhance the degree of freedom in estimation by the MEM; suchlarge residuals usually arise from imperfect representation of the real structure on condition
that the intensity data are collected appropriately. If this part were partitioned among
overlapped reflections in proportion to their values of YiK, the MEM would estimate thestructure in a way biased in favor of the structural model.
The efficiency of REMEDY cycles depends on the validity of the starting structural model.

10

If the structure refined in step 1 differs considerably from the real one, MEM analysis may
fail in structure refinement owing to the inaccurate partition of observed Bragg intensities
among overlapped reflections.
R factors in the whole-pattern fitting are used as the measure of the convergence in
REMEDY cycles. R factors in model-free MEM analysis are comparable regardless of the
progress of the REMEDY cycles because the convergence criterion is a constraint function
being equal to unity. Hence, R factors in the MEM should not be used to judge theconvergence of the iterations.
4.7

Examples of Structure Refinements with REMEDY

The visualization of diffraction data by the MEM is very useful for modifying structuralmodels imperfect with respect to positional disorder, defects, partially occupied sites, etc. We
investigated the real structure of Na-LTA dehydrated imperfectly with REMEDY [9, 17].
Figure 7 shows electron-density maps plotted for the (110) plane with the vertical axis parallelto the c axis. Figures 7(a) and 7(b) were obtained by Rietveld analysis followed by Fourier
synthesis and MEM analysis with the same number of Fok data, respectively. ‘Ripples’ andnegative densities dominate the Fourier map because of the termination effect, which makes
it virtually impossible to distinguish residual water molecules from ripples in (a). In contrast
with (a), such noises are suppressed in (b), so that residual water molecules can be
unambiguously detected at four positions indicated by arrows inside a β cage despite an
occupancy as low as 0.028. The MEM owes the predominance over Fourier synthesis to its
excellent capability to estimate structure factors of reflections whose peak positions exceed2θ (max).
With REMEDY, the order-disorder-type phase transition of KH2 PO4 was studied with its
neutron powder diffraction data measured on HRPD (λ = 1.16 Å) at JRR-3M [24]. For MEM
analysis, we used MEND [25] instead of MEED to deal with hydrogen with a negative bc of–3.7390 fm. The ferroelectric and paraelectric phases of KH2 PO4 are stable below and above
its Curie temperature of 123 K, respectively. The nuclear-density map plotted for the (001)plane in Fig. 8 resulted from repeating MEM analyses and whole-pattern fitting of theintensity data for the paraelectric phase at room temperature. Hydrogen atoms are statistically
distributed at a pair of positions with a distance between the twin peaks of ca. 0.36 Å, whichis in good agreement with values reported in the literature. On the other hands, a hydrogen
atom in the ferroelectric phase at 10 K tends to occupy a single position near to an oxygen
atom.
KxTi2–x/3 Lix/3 O4 (x = 0.8) contains lepidocrocite-related Ti2–x/3 Lix/3 O4 layers composed ofedge-sharing (Ti,Li)O6 octahedra and charge-balancing K+ ions in the interlayer domain [26].
Rietveld refinement of KxTi2–x/3 Lix/3 O4 with neutron powder diffraction data showed K+ ions11to be disordered between two Ti2–x/3 Lix/3 O4 layers. An unreasonably large isotropic atomic
displacement parameter, B, of ca. 10 Å2 for K evidently originated in its positional disorder.
Conventional Rietveld analysis adopting a split-atom model would fail to express the real
structure adequately in such a highly disordered system. Then, X-ray Rietveld refinement of
KxTi2–x/3 Lix/3 O4 was followed by three iterations of MEM analyses and whole-pattern fitting,which lowered Rwp a little from 11.56% to 10.63% (Re = 10.54%) and RB dramatically from
5.35% to 1.53% [27]. The striking decrease in RB must result from better representation of thestructural details, particularly the disordered distribution of K+ ions.
A final electron-density map determined for the (100) plane is plotted in Fig. 9. A winding
stream of K+ ions along the [001] direction was clearly visualized by virtue of the iterative
analyses by the MEM and whole-pattern fitting. Positional disordering of K+ ions is partly
ascribable to large differences in the oxidation states and ionic radii between Ti4+ and Li+ ionsin six-fold coordination. Thus, our first challenge to the determination of spatial distribution
for highly disordered species was quite successful.
We determined nuclear and electron densities in a high-temperature superconductor
HgBa2 CuO4+δ (Tc = 97 K) whose content of carbonate ions was minimized [10, 28]. Its
neutron powder diffraction data were collected on HRPD (λ = 1.823 Å) at JRR-3M. In bothneutron and X-ray diffraction, structure refinements with REMEDY gave lower R factors thanconventional Rietveld analysis. R factors were particularly reduced in X-ray diffraction. No
distribution of bonding electrons can be determined by the Rietveld method to refine structure
parameters contained in Fck . On the other hand, the MEM enables us to represent chemicalbonds by electron densities in a higher degree of freedom, which leads to the close approachof FMk’s toward Fok’s. In fact, densities of bonding electrons between metal and oxygen varied
appreciably during the REMEDY cycles.
Figure 10 displays (a) nuclear and (b) electron densities on the (100) plane of
HgBa2 CuO4+δ. Dotted lines denote a and c axes of the tetragonal unit cell. Interposition ofbonding electrons between Hg and O2 and between Cu and O1, and nearly electrostaticinteraction between Cu and O2 are clearly seen in this figure. Comparison of (a) with (b)shows that thermal vibrations of Cu and Hg predominate along the [001] direction
perpendicular to the CuO2 sheet and on the z = 0 plane perpendicular to an [O2–Hg–O2]2– ion,
respectively. These findings are reasonable considering the highly covalent character of theCu–O1 and Hg–O2 bonds. Although harmonic thermal motion was assumed in the Rietveld
analysis prior to the REMEDY cycles, isosurfaces of nuclear densities for O2 considerably
differed from ellipsoids. In fact, the analysis of anharmonic thermal vibration for O2 gavesignificant cubic and quartic anharmonic terms [25].
Interstitial oxygen defects, O3, at (1/2, 1/2, 0) are responsible for hole doping in
HgBa2 CuO4+δ . In conventional Rietveld analysis, B for a slightly occupied site like O3 is12

arbitrarily fixed at a typical value, e.g., 1 Å2 , to refine only its occupancy, g, on account ofvery high correlation between B and g. We took more elaborate approaches to obtain morereliable values of g(O3) and B(O3). That is, g(O3) was estimated at 0.14 by integrating nuclear
densities of O3, and B(O3) at 1.5 Å2 by evaluating mean square displacements. This technique
can generally be used to advantage in the determination of B and g for sites occupied partially.
5.

Closing Remarks

We have been routinely introducing partial profile relaxation into most of our Rietveld
refinements with angle-dispersive X-ray and neutron diffraction data [9, 16, 17, 24, 27] andTOF neutron diffraction data [7, 8, 28]. This technique more or less surpasses the conventional
Rietveld method in goodness-of-fit, improving accuracy in structure parameters with better
estimation of e.s.d.’s. It is also useful when combined with MEM-based whole-pattern fitting[10, 24, 27, 28]. To minimize the influence of the structural model in Rietveld analysis on
final densities determined with REMEDY, the goodness-of-fit must reach a very high level;
this can easily be accomplished with partial profile relaxation. At present, REMEDY cycles
would not function well in practice without this feature.
REMEDY is eminently suitable for1. modifying imperfect structural models for Rietveld analysis through the visualization of
intensity data [17, 22],
2. modeling chemical bonds, nonlocalized electrons, and anharmonic thermal motion [10, 28],
3. analyzing the crystal structure of a compound with a highly disordered atomic configuration[27].
The first application serves the construction of structural models [17, 22]; diffraction
intensities scattered in the reciprocal space are converted into electron densities filling the real
space. The second and third applications are attainable more satisfactorily with REMEDY thanwith programs for conventional Rietveld analysis because crystal structures are represented
not by structure parameters but by three-dimensional electron/nuclear densities on the useof REMEDY. Of course, X-ray and neutron diffraction should be utilized for (a) the more
adequate expression of chemical bonds and (b) the analysis of anharmonic thermal vibrationand disordered structures, respectively.
REMEDY is a newborn system that needs to be tested and polished up further. We will
apply it to a variety of compounds to improve its methodology and confirm its great
capability.

Acknowledgments
The encouragement and support of S. Kumazawa, T. Ikeda, and A. Yamamoto during the13present work are gratefully appreciated. Thanks are also due to T. Sasaki and W.-Z. Hu fortechnical assistance in the synthetic work, and Q. Huang and K. Oikawa for the neutron
diffraction experiments.
References[1] H. M. Rietveld, J. Appl. Crystallogr., 2, 65 (1969).
[2] D. E. Cox, “Synchrotron Radiation Crystallography,” ed by P. Coppens, Academic Press,
London (1992), Chap. 9.
[3] F. Izumi, “The Rietveld Method,” ed by R. A. Young, Oxford University Press, Oxford(1995), Chap. 13.
[4] F. Izumi, “Applications of Synchrotron Radiation to Materials Analysis,” ed by H.
Saisho and Y. Gohshi, Elsevier, Amsterdam (1996), Chap. 7.
[5] F. Izumi, H. Asano, H. Murata, and N. Watanabe, J. Appl. Crystallogr., 20, 411 (1987).
[6] F. Izumi, Trans. Am. Crystallogr. Assoc., 29, 11 (1993).
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1093 (1997).
[8] F. Izumi, S. Kumazawa, T. Ikeda, and T. Ida, “Powder Diffraction,” ed by S. P. Sen
Gupta, Allied Publishers, New Delhi (1998), pp. 24–36.
[9] F. Izumi and T. Ikeda, Mater. Sci. Forum, 321–324, 198 (2000).
[10] F. Izumi, S. Kumazawa, T. Ikeda, W.-Z. Hu, A. Yamamoto, and K. Oikawa, Mater. Sci.
Forum, in press.
[11] G. Caglioti, A. Paoletti, and F. P. Ricci, Nucl. Instrum. Methods, 35, 223 (1958).
[12] G. S. Pawley, J. Appl. Crystallogr., 14, 357 (1981).
[13] A. Le Bail, H. Duroy, and J. L. Fourquet, Mater. Res. Bull., 23, 447 (1988).
[14] H. Toraya, J. Appl. Crystallogr., 23, 485 (1990).
[15] W. Parrish, “International Tables for Crystallography,” Vol. C, ed by A. J. C. Wilson,
Kluwer, Dordrecht (1992), pp. 42–79.
[16] T. Ikeda, K. Miyazawa, F. Izumi, Q. Huang, and A. Santoro, J. Phys. Chem. Solids, 60,
1531 (1999).
[17] T. Ikeda, F. Izumi, T. Kodaira, and T. Kamiyama, Chem. Mater., 10, 3996 (1998).
[18] P. Thompson, D. E. Cox, and J. B. Hastings, J. Appl. Crystallogr., 20, 79 (1987).
[19] L. W. Finger, D. E. Cox, and A. P. Jephcoat, J. Appl. Crystallogr., 27, 892 (1994).
[20] M. Sakata, R. Mori, S. Kumazawa, M. Takata, and H. Toraya, J. Appl. Crystallogr., 23,526 (1990).
[21] H. Toraya, “The Rietveld Method,” ed by R. A. Young, Oxford University Press, Oxford(1995), Chap. 14.
[22] M. Takata, B. Umeda, E. Nishibori, M. Sakata, Y. Saito, M. Ohno, and H. Shinohara,

14

Nature (London), 377, 46 (1995).
[23] S. Kumazawa, Y. Kubota, M. Takata, M. Sakata, and Y. Ishibashi, J. Appl. Crystallogr.,
26, 453 (1993).
[24] S. Kumazawa, S. Yamamura, E. Nishibori, M. Takata, M. Sakata, F. Izumi, and Y. Ishii,
J. Phys. Chem. Solids, 60, 1407 (1999).
[25] S. Kumazawa, M. Takata, and M. Sakata, Acta Crystallogr., Sect. A, 51, 651 (1995).
[26] T. Sasaki, F. Kooli, M. Iida, Y. Michiue, S. Takenouchi, Y. Yajima, F. Izumi, B. C.
Chakoumakos, and M. Watanabe, Chem. Mater., 10, 4123 (1998).
[27] F. Izumi, T. Ikeda, T. Sasaki, and S. Kumazawa, Mol. Cryst. Liq. Cryst., in press.
[28] F. Izumi, A. Yamamoto, N. R. Khasanova, S. Kumazawa, W.-Z. Hu, and T. Kamiyama,
Physica C (Amsterdam), in press.

15

Figure captions
Fig. 1. Observed (plus marks), calculated (solid line), and difference (bottom) patterns of
Na-LTA in a low-2θ region. X-Ray diffraction data were measured with CuKα and a pair ofSoller slits having angular apertures of (a) 5° and (b) 1°.
Fig. 2. Rietveld-refinement patterns for hydrated Na-X. The X-ray diffraction pattern
between 9° and 35° is magnified in the inset.
Fig. 3. Observed, calculated, and difference patterns for AlPO4 -5 incorporating triethylamine(λ = 2.0785 Å). The neutron diffraction pattern between 33° and 90° is magnified in the inset.
Fig. 4. Rwp ’s (%) resulting from conventional and profile-relaxed Rietveld refinements offive inorganic compounds. KTLO: KxTi2–x/3 Lix/3 O4 (x = 0.8), Apatite: Ca5 (PO4 )3 F. Synchrotron
X-ray diffraction data were measured with a wavelength of 0.9997 Å for Nd2 CuO4 while
CuKα radiation was used for the other compounds.
Fig. 5. Flow chart of structure refinement by Rietveld analysis followed by iterative MEMbased whole-pattern fitting.
Fig. 6. Programs and files used in Rietveld analysis, MEM-based whole-pattern fitting, andFourier synthesis.
Fig. 7. Electron-density distribution on the (110) section in dehydrated Na-LTA. (a) Fourier
synthesis, range: –1 to 5 e/Å3 , step: 0.1 e/Å3 . (b) MEM analysis, range: 0.1 to 5 e/Å3 , step: 0.12
e/Å3 .
Fig. 8. Nuclear-density distribution projected on the (001) plane in the paraelectric phaseof KH2 PO4 at room temperature. Contours were plotted for 2.5×1.5n fm/Å3 (n = 0–20).
Fig. 9. Electron-density distribution on the (100) section of KxTi2–x/3 Lix/3 O4 (x = 0.8). Range:
0.2–8 e/Å3 .
Fig. 10. (a) Nuclear- and (b) electron-density maps of the (100) plane in HgBa2 CuO4+δ.
Arrows denote directions along which thermal motion is more significant.

16

(a)
220

622

200

222

20000

10000

420

30000

0

5

10

15

20

25

622

20000

420

222

30000

(b)
220

40000

200

50000

10000

0

5

10

15

2

20

25

q / degree

Fig. 1Intensity

0

20000

40000

60000

80000

10

111

20

0

5000

10000

30

10

40

15

2

q / degree

50

20

60

70

25

80

30

100

Fig. 29035

Intensity

0

5000

10000

15000

100

20

40

1500

2000

2500

3000

60

40

2

q / degree

80

50

60

100

70

120

80

Fig. 3

140

90

15

10

5

0

Nd2CuO4

KTLO

ApatiteNa-LTA

ZSM-5

Conventional Rietveld refinement

Profile-relaxed Rietveld refinement

Fig. 4

Powder diffraction data

Initial structural model

Rietveld analysis

RevisedRIETAN

structural model

Visualization of

Fo (Rietveld)

density distribution

Mevius

MEM analysis

Electron/nuclear

MEED

densities

Fc (MEM)
Whole-pattern fitting

RIETANF o(w.p.f.)

REMEDYcyclesFig. 5

Fousyn

Fo (Rietveld)

2

Rietveld analysis 1

Revised structural model

Visualization 4

MeviusRIETAN

MEEDFc (MEM)

Fig. 6

MEM analysis 3

Fo (w.p.f.)2

Whole-pattern fitting 6

Powder diffraction data

Electron/nuclear density

Fourier synthesis

5

File

Program(a)(b)

Fig. 7

010

P
O

H

K

O

1 

Fig. 8

Ti/LiK

O1

O2

Fig. 9

b

c

(a)

(b)
Hg

O2

O1

Fig. 10Cu