The generalized temperature factor

Maslen, E. N.; Fox, A. G.; O'Keefe, M. A.

doi:10.1107/97809553602060000600

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 585-590

Section 6.1.1.6. The generalized temperature factor

E. N. Maslen,^e ^‡ A. G. Fox^b and M. A. O'Keefe^c

6.1.1.6. The generalized temperature factor

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The Gaussian model of the probability density function (p.d.f.) $[\rho_o({\bf u})]$ for atomic thermal motion defined in (6.1.1.30) is adequate in many cases. Where anharmonicity or curvilinear motion is important, however, more elaborate models are needed.

In the classical (high-temperature) regime, the generalized temperature factor is given by the Fourier transform of the one-particle p.d.f: $[\rho({\bf u})=N^{-1}\exp[-V({\bf u})/kT],\eqno (6.1.1.36)]$ where $[N=\textstyle\int\exp[-V({\bf u})/kT]\,{\rm d}{\bf u} \eqno (6.1.1.37)]$

In the cases where the potential function V(u) is a close approximation to the Gaussian (harmonic) potential, series expansions based on a perturbation treatment of the anharmonic terms provide a satisfactory representation of the temperature factors. That is, if the deviations from the Gaussian shape are small, approximations obtained by adding higher-order corrections to the Gaussian model are satisfactory.

In an arbitrary coordinate system, the number of significant high-order tensor coefficients for the correction is large. It may be helpful to choose coordinates parallel to the principal axes for the harmonic approximation so that $[V({\bf u})/kT=1/2\textstyle\sum\limits^3_{i=1}(B_iu_i){}^2,\eqno (6.1.1.38)]$ in which case (6.1.1.36) may be written as $[\rho_o({\bf u})={1\over N_0}\exp\bigg[-1/2\sum_i(B_iu_i){}^2\bigg],\eqno (6.1.1.39)]$ where $[N_0={B_1B_2B_3\over8\pi^3}.\eqno (6.1.1.40)]$ The harmonic temperature factor is $[T_o({\bf S})=\exp\bigg[-1/2\textstyle\sum\limits_i(b_iS_i){}^2\bigg],\eqno (6.1.1.41)]$ where [b_i] and [B_i] are related by the reciprocity condition $[b_iB_i=1. \eqno (6.1.1.42)]$

6.1.1.6.1. Gram–Charlier series

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In the Gram–Charlier series expansion (Kuznetsov, Stratonovich & Tikhonov, 1960), the general p.d.f. $[\rho({\bf u})]$ is approximated by $[\bigg[1-c\,^jD_j+{c\, ^{jk}\over2!}D_jD_k-\ldots+(-)^p{c\,^{jk\ldots\zeta}\over p!}D_\alpha D_\beta\ldots D_\zeta\bigg]\rho_o({\bf u}).\eqno (6.1.1.43)]$ The operator $[D_\alpha D_\beta\ldots D_\zeta]$ is the pth partial (covariant) derivative $[\partial\,^p/\partial u_\alpha\partial u_\beta\ldots\partial u_\zeta]$ , and $[c\,^{jk\ldots\zeta}]$ is a contravariant component of the coefficient tensor. The quasi-moment coefficient tensors are symmetric for all permutations of indices. The first four have three, six, ten, and fifteen unique components for site symmetry 1. The third- and fourth-order terms describe the skewness and the kurtosis of the p.d.f., respectively.

The Gram–Charlier series may be rewritten using general multidimensional Hermite polynomial tensors, defined by $[\eqalignno{H_{\alpha\beta\ldots\zeta}({\bf u})&=(-){}^p\exp(\,\textstyle{1\over2}\sigma_{jk}^{-1}u\,^ju^k)\cr &\quad\times D_\alpha D_\beta\ldots D_\zeta\exp(-\textstyle{1\over2}\sigma_{jk}^{-1}u\,^ju^k).& (6.1.1.44)}]$ For $[w_j=\sigma_{jk}^{-1}u^k]$ , and with $[\sigma_{jk}^{-1}=\sigma_{kj}^{-1}]$ and [w_jw_k=w_kw_j] , the first few general Hermite polynomials may be expressed as $[\eqalign{^0H({\bf u})&=1\cr ^1H_j({\bf u})&=w_j\cr ^2H_{jk}({\bf u})&=w_jw_k-\sigma_{jk}^{-1}\cr ^3H_{jk}({\bf u})&=w_jw_kw_l-w_j\sigma_{kl}^{-1}-w_k\sigma_{l\,j}^{-1}-w_l\sigma_{jk}^{-1}\cr &=w_jw_kw_l-3w_{(\,j}\sigma^{-1}_{kl)}\cr ^4H_{jklm}({\bf u})&=w_jw_kw_lw_m-6w_{(\,j}w_k\sigma^{-1}_{lm)}+3\sigma^{-1}_{j(k}\sigma^{-1}_{lm)}.} (6.1.1.45)]$ Indices in parentheses indicate terms to be averaged over all unique permutations of those indices.

The Gram–Charlier series is then $[\rho_o({\bf u})\bigg[1+{1\over3!}c\,^{jkl}H_{jkl}({\bf u})+{1\over4!}c\,^{jklm}H_{jklm}({\bf u})+\ldots\bigg],\eqno (6.1.1.46)]$ in which the mean and the dispersion of $[\rho_o({\bf u})]$ have been chosen to make $[c\,^j]$ and $[c\,^{jk}]$ vanish.

The Fourier transform, after truncating at the quartic term, gives an approximation to the generalized temperature factor: $[T({\bf S})=T_o({\bf S})\bigg[1+{i^3\over3!}c\,^{jkl}S_jS_kS_l+{i^4\over4!}c\,^{jklm}S_jS_kS_lS_m\bigg],\eqno (6.1.1.47)]$ i.e. the Fourier transform of the Hermite polynomial expansion about the Gaussian p.d.f. is a power-series expansion about the Gaussian temperature factor with even-order terms real and odd-order terms imaginary.

Because of the symmetry of the relationship between the Fourier transform of a real function and its inverse, the functional form of the p.d.f. and that of the temperature factor can be interchanged. Exchanging the role of the Hermite polynomials and the power series from the Gram–Charlier expansion has been studied by Scheringer (1985), with the objective of obtaining the one-particle potentials more directly.

6.1.1.6.2. Fourier-invariant expansions

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When truncated, an expression for a multipole expansion, p.d.f. or temperature factor must retain those terms essential to the accuracy required of the expansion. Some authors (e.g. Stewart, 1980b) strongly favour classes of truncated expansion that retain symmetry properties appropriate to particular classes of transformation, such as rotation or Fourier inversion. Others, emphasizing simplicity, retain the minimum set of terms required to preserve the accuracy needed in the expansion. In either case, it is desirable for the expansion to converge rapidly, and to have a form related to physical theory.

In principle, the one-particle potential may be expanded in any complete set of functions. Harmonic oscillator functions simplify simultaneous interpretation of the probability distribution in real and reciprocal space because their form does not change under Fourier inversion (Kurki-Suonio, Merisalo & Peltonen, 1979).

If both anharmonicity and anisotropy are small, the p.d.f. may be expressed as a rapidly converging expansion in spherical polar coordinates $[u,\theta,\varphi]$ : $[\rho({\bf u})=\rho_o({\bf u}){N_0\over N}\bigg[1-\sum_{n,l,m,p}a_{nlmp}R_{nl}(Bu)Y_{lmp}(\theta,\varphi)\bigg]\eqno (6.1.1.48)]$ for non-cubic and $[\rho({\bf u})=\rho_o({\bf u}){N_0\over N}\bigg[1-\sum_{n,l,\,j}a_{nlj}R_{nl}(Bu)K_{l\,j}(\theta,\varphi)\bigg]\eqno (6.1.1.49)]$ for cubic site symmetry. The radial term may be written as $[R_{nl}(x)=x^lL^{l+1/2}_{(n-l)/2}(x^2),\eqno (6.1.1.50)]$ where the associated Laguerre polynomial is $[L_k^\alpha(t)=\sum^k_{\nu=0}\bigg(\matrix{k+\alpha\cr k-\alpha\cr}\bigg)\displaystyle{(-t){}^\nu\over\nu!}\eqno (6.1.1.51)]$ with $[\bigg(\matrix{p\cr q\cr}\bigg)={{\Gamma}(p+1)\over[{\Gamma}(q+1){\Gamma}(p-q+1)]}\eqno (6.1.1.52)]$ and the normalizing factor $[N={8\pi^3\over B^3}\bigg[1-\sum_\nu(-)^\nu\displaystyle{(2\nu+1)!\over2^{2\nu}(\nu!)^2}a_{2\nu00+}\bigg].\eqno (6.1.1.53)]$

The real spherical harmonics $[Y_{lmp}(\theta,\varphi)]$ and the cubic harmonics $[K_{lj}(\theta,\varphi)]$ are as defined in Subsection 6.1.1.4. As in the case of multipole expansions, the non-zero coefficients in these expressions are limited by the site symmetry. The restrictions on the temperature factor are identical to those for the generalized scattering factor listed in Tables 6.1.1.7 and 6.1.1.8.

From the Fourier invariance of harmonic oscillator functions, $[\eqalignno{T({\bf S})&={N_0\over N}\exp(-b^2S^2/2)\cr &\quad\times\bigg[1-\sum_{n,l,m,p}a_{nlmp}i^nR_{nl}(bS)Y_{lmp}(\theta_S,\varphi_S)\bigg]& (6.1.1.54)}]$ and $[\eqalignno{T({\bf S})&={N_0\over N}\exp(-b^2S^2/2)\cr &\quad\times\bigg[1-\sum_{n,l,\,j}a_{nlj}i^nR_{nl}(bS)K_{l\,j}(\theta_S,\varphi_S)\bigg]& (6.1.1.55)}]$ for non-cubic and cubic site symmetries, respectively. $[\theta_S]$ and $[\varphi_S]$ are polar coordinates in reciprocal space.

With an appropriate choice of origin, the first-order (110+) and (111 $[\pm]$ ) terms vanish. The isotropic harmonic (200+) and constant (000+) terms have been removed from the summation. If coordinate axes are chosen coincident with the principal axes for the harmonic approximation, (221 $[\pm]$ ) and (222−) vanish. (220+) indicates the prolateness and (222+) the non-axiality in the harmonic approximation (Kurki-Suonio, 1977). Terms with $[n\ge2]$ describe the anharmonicity.

The approximations in (6.1.1.48) to (6.1.1.55) fail if the anisotropy, indicated by the size of the (220+) and (222+) terms, or the anharmonicity is large. If the anharmonicity and non-axiality are small, one can invoke Fourier-invariant expansions in cylindrical polar coordinates $[u_r,u_z,\varphi]$ : $[\specialfonts\eqalignno{\rho({\bf u})&=\rho_o({\bf u}){N_0\over N}\cr &\quad\times\bigg[1-\sum_{n_z,n,m,p}b_{n_znmp}H_{n_z}(B_zu_z){\bsf P}_{nm}(B_ru_r)\Phi_{mp}(\varphi)\bigg]\cr&& (6.1.1.56)}]$ and $[\specialfonts\eqalignno{T(S)&={N_0\over N}\exp[-\textstyle{1\over2}(b^2_rS^2_r+b^2_zS^2_z)]\cr &\quad\times\bigg[1-\sum_{n_z,n,m,p}b_{n_znmp}H_{n_z}(b_zS_z){\bsf P}_{nm}(b_rS_r)\Phi_{mp}(\varphi_S)\bigg],\cr&&(6.1.1.57)}]$ where $[S_r,S_z,\varphi_S]$ are cylindrical coordinates for S. $[\specialfonts{\bsf P}_{nm}(x)= x^mL^m_{(n-m)/2}(x^2),\quad\Phi_{m\pm}(\varphi)=\matrix{\cos m\varphi\cr \sin m\varphi\cr} \eqno (6.1.1.58)]$ and $[N={8\pi^3\over B^2_rB_z}\bigg[1-\sum_{\mu\nu}(-)^\nu\displaystyle{(2\mu)!\over\mu!}b_{2\mu2\nu0+}\bigg].\eqno (6.1.1.59)]$ The indices allowed for the site symmetrical basis are as indicated in Table 6.1.1.10.

Table 6.1.1.10| top | pdf |
Indices nmp allowed by the site symmetry for the functions $[H_n(z)\Phi_{mp}(\varphi)]$ ; μ, ν and j are integers such that m, n ≥ 0; (−)ⁿ implies p = − for n odd and p = + for n even

Site symmetry	Coordinate axes	Indices
1	Any	$[{\rm All}\,\,(n,m,p)]$
$[\bar1]$	Any
2	$[2\parallel x]$	$[(n,m,(-)^{n})]$
	$[2\parallel y]$	$[(n,m,(-)^{n-m})]$
	$[2\parallel z]$	$[(n,2\nu,p)]$
m	$[m\,\bot\, x]$	$[(n,m,(-)^{m})]$
	$[m\,\bot\, y]$
	$[m\,\bot\, z]$	$[(2\mu,m,p)]$
	$[2\parallel x,m\,\bot\,x]$	$[(m+2j ,m,(-)^{m})]$
	$[2\parallel y,m\,\bot\,y]$
	$[2\parallel z,m\,\bot\,z]$	$[(2\mu,2\nu,p)]$
222	$[2\parallel z,2\parallel y]$	$[(n,2\nu,(-)^n)]$
	$[2\parallel x,m\,\bot\,z]$	$[(2\mu,m+)]$
	$[2\parallel y,m\,\bot\,z]$	$[(2\mu,m,(-)^m)]$
	$[2\parallel z,m\,\bot\,y]$	$[(n,2\nu,+)]$
	$[m\,\bot\,z,m\,\bot\,y,m\,\bot\,z]$	$[(2\mu,2\nu,+)]$
4	$[4\parallel z]$	$[(n,4\nu,p)]$
$[\bar4]$	$[\bar4\parallel z]$
	$[4\parallel z,m\,\bot\,z]$	$[(2\mu,4\nu,p)]$
422	$[4\parallel z,2\parallel y]$	$[(n,4\nu,(-)^n)]$
	$[4\parallel z,m\,\bot\,y]$	$[(n,4\nu,+)]$
$[\bar42m]$	$[\bar4\parallel z,2\parallel x]$
	$[\bar4\parallel z,m\,\bot\,y]$
	$[4\parallel z,m\,\bot\,z,m\,\bot\,x]$	$[(2\mu,4\nu,+)]$
3	$[3\parallel z]$	$[(n,3\nu,p)]$
$[\bar3]$	$[\bar3\parallel z]$	$[(m+2j,3\nu,p)]$
32	$[3\parallel z,2\parallel y]$	$[(n,3\nu,(-)^{n-m})]$
	$[3\parallel z,2\parallel x]$	$[(n,3\nu,(-)^{n})]$
	$[3\parallel z,m\,\bot\,y]$	$[(n,3\nu,+)]$
	$[3\parallel z,m\,\bot\,x]$	$[(n,3\nu,(-)^m)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,y]$	$[(m+2j,3\nu,+)]$
$[\bar3m]$	$[\bar3\parallel z,m\,\bot\,x]$	$[(m+2j,3\nu,(-)^m)]$
6	$[6\parallel z]$	$[(n,6\nu,p)]$
$[\bar6]$	$[\bar6\parallel z]$	$[(2\mu,3\nu,p)]$
	$[6\parallel z,m\,\bot\,z]$	$[(2\mu,6\nu,p)]$
622	$[6\parallel z,2\parallel y]$	$[(n,6\nu,(-)^n)]$
	$[6\parallel z,m\,\bot\,y]$	$[(n,6\nu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,y]$	$[(2\mu,3\nu,+)]$
$[\bar6m2]$	$[\bar6\parallel z,m\,\bot\,x]$	$[(2\mu,3\nu,(-)^m)]$
	$[6\parallel z,m\,\bot\,z,m\,\bot\,y]$	$[(2\mu,6\nu,+)]$

Again, the first-order (100+) and (011 $[\pm]$ ) terms vanish with the appropriate choice of origin. For coordinate axes coinciding with the principal axes of the harmonic approximation, (111 $[\pm]$ ) and (022−) vanish. (020+), (200+), and (000+) have been removed from the summation.

Equations (6.1.1.56) and (6.1.1.57) apply accurately to non-cubic symmetries with rotation axes higher than twofold where non-axiality vanishes. Where non-axiality is large, it is preferable to use the Cartesian Fourier invariant expansion $[\eqalignno{\rho({\bf u})&={N_0\over N}\exp\bigg[-1/2\sum_iB^2_iu^2_1\bigg]\cr &\quad\times\bigg[1-\sum_{n_x,n_y,n_z}c_{n_xn_y,n_z}H_{n_x}(B_xu_x)H_{n_y}(B_yu_y)H_{n_z}(B_zu_z)\bigg]\cr && (6.1.1.60)}]$ and $[\eqalignno{T({\bf S})&={N_0\over N}\exp\bigg[-1/2\sum_ib^2_iu^2_1\bigg]\cr &\quad\times\bigg[1-\sum_{n_x,n_y,n_z}c_{n_xn_y,n_z}H_{n_x}(b_xu_x)H_{n_y}(b_yu_y)H_{n_z}(b_zu_z)\bigg], \cr&&(6.1.1.61)}]$ where $[N={8\pi^3\over B_xB_yB_z}\bigg[1-\sum_{\lambda\mu\nu}\displaystyle{(2\lambda)!(2\mu)!(2\nu)!\over\lambda!\mu!\nu!}c_{2\lambda2\mu2\nu}\bigg].\eqno (6.1.1.62)]$ The indices allowed under the site symmetry are listed in Table 6.1.1.11.

Table 6.1.1.11| top | pdf |
Indices n_x, n_y, n_z allowed for the basis functions H_{n_x}(Ax)H_{n_y}(By)H_{n_z}(Cz); λ, μ and ν are non-negative; conditions for other choices of axes are derived by cyclic permutation

Symmetry	Coordinate axes	Allowed indices
1	Any	$[{\rm All}\,\,(n_x,n_y,n_z)]$
$[\bar1]$	Any	$[n_x+n_y+n_z=2\lambda]$
2	$[2\parallel z]$	$[n_x+n_y=2\lambda]$
m	$[m\,\bot\, z]$	$[n_z=2\nu]$
	$[2\parallel z,m\,\bot\,z]$	$[n_x+n_y=2\lambda,n_z=2\nu]$
222	$[2\parallel z,2\parallel y]$	all even or all odd
mm2	$[2\parallel z,m\,\bot\,y]$	$[n_x=2\lambda,n_y=2\mu]$
mmm	$[m\,\bot\,z,m\,\bot\,y,m\,\bot\,z]$	$[n_x=2\lambda,n_y=2\mu,n_z=2\nu]$

The first-order terms vanish with suitable choice of origin. (110), (101), and (011) vanish if the coordinates coincide with the principal axes for the harmonic approximation, and (200), (020), (002), and (000) are removed from the summation. Only anharmonic terms remain.

6.1.1.6.3. Cumulant expansion

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In a cumulant expansion (Johnson & Levy, 1974), the entire series is expressed in exponential form. The cumulant expansion about S = 0 for the generalized temperature factor is $[\eqalignno{T({\bf S})&=\exp\bigg[1+i\kappa^jS_j+{i^2\over2!}\kappa^{jk}S_jS_k+{i^3\over3!}\kappa^{jkl}S_jS_kS_l\cr &\quad+{i^4\over4!}\kappa^{jklm}S_jS_kS_lS_m+\ldots\bigg],& (6.1.1.63)}]$ where the coefficient tensor $[\kappa^{\alpha\beta\ldots\zeta}]$ , a symmetric tensor of order p, is the pth-order cumulant. The inverse Fourier transform is the Edgeworth expansion around the Gaussian p.d.f. Cumulants can be expressed in terms of moments and vice versa. The pth moment $[\mu^{\alpha\beta\ldots\zeta}]$ (if it exists) of a general p.d.f., ρ(x), is a symmetric tensor defined as $[\mu^{\alpha\beta\ldots\zeta}({\bf x})=\textstyle\int\limits^\infty_{-\infty}x^\alpha x^\beta\ldots x^\zeta\rho({\bf x})\,{\rm d}{\bf x}.\eqno (6.1.1.64)]$ The relations between the lower-order moments and cumulants are $[\eqalign{\mu^j&=\kappa^j\cr \mu^{jk}&=\kappa^{jk}+\kappa^j\kappa^k\cr \mu^{jkl}&=\kappa^{jkl}+\kappa^j\kappa^{kl}+\kappa^k\kappa^{lj}+\kappa^l\kappa^{jk}+\kappa^j\kappa^k\kappa^l\cr &=\kappa^{jkl}+3\kappa^{(j}\kappa^{kl)}+\kappa^j\kappa^k\kappa^l\cr \mu^{jklm}&=\kappa^{jklm}+3\kappa^{j(k}\kappa^{lm)}+4\kappa^{(j}\kappa^{klm)}\cr &\quad+6\kappa^{(j}\kappa^k\kappa^{lm)}+\kappa^j\kappa^k\kappa^l\kappa^m} \eqno(6.1.1.65)]$ and, conversely, $[\eqalign{\kappa^j&=\mu^j\cr \kappa^{jk}&=\mu^{jk}-\mu^j\mu^k\cr \kappa^{jkl}&=\mu^{jkl}-3\mu^{(j}\mu^{kl)}+2\mu^j\mu^k\mu^l\cr \kappa^{jklm}&=\mu^{jklm}-3\mu^{j(k}\mu^{lm)}-4\mu^{(j}\mu^{klm)}\cr &\quad+12\mu^{(j}\mu^k\mu ^{lm)}-6\mu^j\mu^k\mu^l\mu^m.} \eqno(6.1.1.66)]$ In the Gram–Charlier and Fourier-invariant expansions, the Fourier-transform relationship between the p.d.f. and the temperature factor to given order can be made exact. Each cumulant $[\mu^{jkl}]$ contributes to all higher-order quasi-moment terms and vice versa. Hence, a given cumulant expansion is to an extent arbitrarily truncated (Kuhs, 1983). Care is required when interpreting the coefficients (Zucker & Schulz, 1982).

On the other hand, the cumulant expansion has the advantage of yielding tractable expressions for the one-particle potential in the quantum regime (Mair, 1980a). In that regime, equation (6.1.1.36) for the one-particle potential is invalid, and the expressions relating V(u) to ρ(u) in the Gram–Charlier and Fourier-invariant expansions are cumbersome (Mair & Wilkins, 1976).

Coefficients obtained by applying least-squares methods to structure-factor equations related to the truncated cumulant expansions do not necessarily yield non-negative p.d.f.'s nor are the linear-term coefficients necessarily faithful representations of the mean. Caution must be exercised in interpreting the results.

All the methods are satisfactory in the case of rapidly converging potential series. The methods are equivalent up to λ² in the van Hove order parameter (Mair, 1980b). Difficulties are encountered with convergence of the series in the case of strong anharmonicity, in which case numerical or alternative analytical models may be necessary. If the anharmonicity is such that the difference between the expansions is significant, it may be preferable to evaluate the Fourier transforms directly, as recommended by Mackenzie & Mair (1985).

6.1.1.6.4. Curvilinear density functions

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For groups of atoms moving on the surface of a circle or sphere, perturbation expansions in Cartesian coordinates may converge slowly. Methods of representing curvilinear density functions that are multimodal or have large amplitude are described by Press & Hüller (1973).

For atoms constrained to rotate about a single axis, $[a({\bf u})={1\over2\pi\tau}\delta(r-\tau)\delta(z) f(\varphi),\eqno (6.1.1.67)]$ where $[r,z,\varphi]$ are cylindrical coordinates for the displacement u. Setting $[f(\varphi)=\textstyle\sum\limits_{m=0}c_m\exp(im\varphi)+c_m^*(-im\varphi)\eqno (6.1.1.68)]$ and $[\exp(i{\bf S}\cdot{\bf r})=\exp(iS_zz)\exp[iS_rr\cos(\varphi_S-\varphi)]\eqno (6.1.1.69)]$ and using $[\exp[iS_rr\cos(\varphi_S-\varphi)]=\textstyle\sum\limits_{l=0}\,(2-\delta_{l0})i^lJ_l(S_rr)\cos[l(\varphi_S-\varphi)]\eqno (6.1.1.70)]$ yields $[T({\bf S})=\textstyle\sum\limits_{l=0}i^lJ_l(S_r\tau)[c_l\exp(il\varphi_S)+c_l^*\exp(-il\varphi_S)]. \eqno (6.1.1.71)]$

For atoms moving on the surface and a sphere, the density function may be written $[\rho({\bf u})=\textstyle\sum\limits^\infty_{l=0}\sum\limits^{2l+1}_{j=1}a_{l\,j}(u)K_{l j}(\theta,\varphi),\eqno (6.1.1.72)]$ where $[u,\theta,\varphi]$ are spherical polar displacement coordinates and the $[K_{l j}]$ are cubic harmonics. Thus, for a rigid molecule, the density function for nuclei confined to move on a spherical shell of radius τ is $[a_{l j}({\bf u})=c_{l j}\delta(u-\tau)/u^2.\eqno (6.1.1.73)]$ Expansion of $[\exp(i{\bf S}\cdot{\bf r})]$ in cubic harmonics $[\exp(i{\bf S}\cdot{\bf r})=4\pi\textstyle\sum\limits_{l,j}i^lj_l(Sr)K_{l j}(\theta_S,\varphi_S)K_{l j}(\theta,\varphi)\eqno (6.1.1.74)]$ leads to $[T({\bf S})=4\pi\textstyle\sum\limits_{l,j}i^lc_{l j} j_l(S\tau)K_{l j}(\theta_S,\varphi_S).\eqno (6.1.1.75)]$

Equations (6.1.1.71) and (6.1.1.75) are useful when the p.d.f.'s (6.1.1.67) and (6.1.1.72) can be approximated by a limited number of significant terms. They are readily adapted to the case of oscillations about axes of symmetry (Press & Hüller, 1973).

6.1.1.6.5. Model-based curvilinear density functions

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For rotational oscillations, which are the curvilinear coordinate analogues of the p.d.f.'s approximating harmonic rectilinear motion, techniques for evaluating the temperature factor are described by Johnson & Levy (1974).

The p.d.f. for an atom in a group of atoms undergoing large-amplitude rotational oscillation (libration) can sometimes be approximated satisfactory by a standard p.d.f. on the circle or on the sphere. The closest analogues of the rectilinear Gaussian p.d.f. are the Brownian-diffusion p.d.f.'s defined on the closed spaces of the circle and the sphere. For statistical analysis, two other p.d.f.'s, the von Mises `circular normal' and the Fisher `spherical normal', are often substituted for the Brownian-diffusion density functions because of their simpler forms.

The p.d.f. for Brownian diffusion on a circle, also called the `wrapped normal' p.d.f. (Feller, 1966; Lévy, 1938), is given by $[\rho(\theta)={1\over(2\pi){}^{1/2}\sigma}\sum^\infty_{n=-\infty}\exp[-(\theta-2n\pi)^2/2\sigma^2],\eqno (6.1.1.76)]$ which may be transformed (Bellman, 1961) into $[\rho(\theta)={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\exp(-m^2\sigma^2/2)\cos(m\theta).\eqno (6.1.1.77)]$ The von Mises p.d.f. (Gumbel, Greenwood & Durand, 1953; Mardin, 1972; von Mises, 1918) is $[\rho(\theta)={\exp(k_c\cos\theta)\over2\pi I_o(k_c)}={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\displaystyle{I_m(k_c)\over I_0(k_c)}\cos(m\theta).\eqno (6.1.1.78)]$ [I_m(x)] is the mth-order Bessel function of the first kind with imaginary argument. The parameter $[\sigma^2]$ is the variance; [k_c] is a measure of concentration such that when [k_c] is zero the probability density is uniformly distributed over the circle, and when [k_c] is large the density is concentrated around the modal vector at θ = 0. An approximate relation between $[\sigma^2]$ and [k_c] can be obtained by equating expressions for the centres of mass of the circular Brownian diffusion and von Mises p.d.f.'s (Stephens, 1963), $[\exp(-\sigma^2/2)={I_1(k_c)\over I_0(k_c)}.\eqno (6.1.1.79)]$ For small $[\sigma^2]$ (large [k_c] ), we find that $[\sigma^2\simeq1/k_c.\eqno (6.1.1.80)]$

Equations (6.1.1.76) to (6.1.1.78) can be generalized to describe multimodal density functions with modes (maxima) arranged symmetrically about the circle. The p.d.f. for the s-modal Brownian-diffusion p.d.f. with one of the s modes at θ = θ₀ is $[\eqalignno{\rho(\theta)&={1\over\sqrt{2\pi} s\sigma}\sum^\infty_{m=-\infty}\exp[-(\theta-\theta_0-2\pi m/s)^2/2\sigma^2]\cr &={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\exp[-(ms\sigma)^2/2]\cos[ms(\theta-\theta_0)]. \cr&&(6.1.1.81)}]$ The two-dimensional Fourier transform (Chidambaram & Brown, 1973) of the last equation in terms of the polar coordinates $[(S,\theta)]$ of the reciprocal-space vector S relative to an origin at the centre of the circle is $[T({\bf S})=\textstyle\sum\limits^\infty_{j=0}(2-\delta_{j 0})i^{js}J_{js}(Sr)\exp[-(js\sigma)^2/2]\cos js\theta_0,\eqno (6.1.1.82)]$ where [J_n(x)] is the Bessel function of the first kind of order n with real argument. Corresponding equations for the von Mises s-modal density function (Atoji, Watanabe & Lipscomb, 1953; King & Lipscomb, 1950; Mardin, 1972) are $[\eqalignno{\rho(\theta)&={1\over2\pi I_o(K_c)}\exp[K_c\cos s(\theta-\theta_0)]\cr &={1\over2\pi}\sum^\infty_{m=0}(2-\delta_{m0})\displaystyle{I_m(K_c)\over I_o(K_c)}\cos ms(\theta-\theta_0)\cr&&(6.1.1.83)}]$ and $[T({\bf S})=\sum^\infty_{j=0}(2-\delta_{j 0})i ^{js}J_{js}(Sr)\displaystyle{I_j(K_c)\over I_0(K_c)}\cos js\theta_0,\eqno (6.1.1.84)]$ where [K_c] , a measure of concentration over 1/sth of the circle about $[\theta_0]$ , is substituted for the [k_c] parameter of the unimodal von Mises density function and [K_c] is related to [k_c] approximately by $[I_1(k_c)/I_0(k_c)=I_s(K_c)/I_0(K_c).\eqno (6.1.1.85)]$

For symmetrical Brownian diffusion on a sphere (Furry, 1957; Lévy, 1938; Mardin, 1972; Perrin, 1928), the p.d.f. in terms of the angular displacement θ from the pole is $[\rho(\theta)=\sum^\infty_{n=0}\displaystyle{2n+1\over4\pi}\exp[-n(n+1)V]P_n(\cos\theta)\sin\theta,\eqno (6.1.1.86)]$ where [P_n(x)] is the nth-order Legendre polynomial. The Fisher (1953) `spherical normal' p.d.f. (Mardin, 1972) is a similar density function given by $[\eqalignno{\rho(\theta)&={k_s\over4\pi\sinh k_s}\exp(k_s\cos\theta)\sin\theta\cr &=\sum^\infty_{n=0}\displaystyle{(2n+1)\over 4\pi}{I_{n+1/2}(k_s)\over I_{1/2}(k_s)}P_n(\cos\theta)\sin\theta.& (6.1.1.87)}]$ The parameters V (variance) and [k_s] are measures of concentration analogous to those for the circle and may be related (Roberts & Ursell, 1960) by an equation analogous to (6.1.1.79), $[\exp(-V/2)=\coth k_s-{1\over k_s}={I_{3/2}(k_s)\over I_{1/2}(k_s)},\eqno (6.1.1.88)]$ the small V approximation being $[V\simeq2/k_s.\eqno (6.1.1.89)]$ Equations (6.1.1.86) and (6.1.1.87) are generalized to place the mode of the density at $[(r,\theta',\varphi')]$ by replacing $[\cos\theta]$ by $[\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos(\varphi-\varphi')]$ and by replacing $[P_n(\cos\theta)]$ by $[\eqalign{&P(\cos\theta)P_n(\cos\theta')+2\sum^n_{m=1}\displaystyle{(n-m)!\over(n-m)!}\cr &\quad\times P^m_n(\cos\theta)P^m_n(\cos\theta')\cos m(\varphi-\varphi').}]$

The three-dimensional Fourier transform of the generalized form of (6.1.1.86) in terms of S in spherical coordinates $[(S,\theta_S,\varphi_S)]$ is $[\eqalignno{T({\bf S})&=\sum^\infty_{q=0}i^q {(2q+1)\over r^2}\exp[-q(q+1)V]\cr &\quad\times\sum^\infty_{s=0}{4\over2p+1}Y_{qs+}(\theta',\varphi')Y_{qs+}(\theta_S,\varphi_S)j_q(Sr),& (6.1.1.90)}]$ where r is the radius of the sphere, and [j_n] is the nth-order spherical Bessel function of the first kind. The real spherical harmonics $[Y_{lmp}]$ are normalized as in (6.1.1.22).

The Fourier transform of the generalized form of (6.1.1.87) is identical to (6.1.1.90) except that the term $[\exp[-q(q+1)V]]$ in (6.1.1.90) is replaced by $[I_{q+1/2}(k_s)/I_{1/2}(k_s).]$

The foregoing equations describe isotropic distributions on a sphere. The p.d.f. for general anisotropic Brownian diffusion (or rotation) on a sphere is not available in a convenient form. However, some of the results of Perrin (1934) and Favro (1960) on rotational Brownian motion are applicable to thermal motion. For example, the centre of mass of a p.d.f. resulting from anisotropic diffusion on a sphere is given by equation (6.8) of Favro (1960). The following equation valid in Cartesian coordinates is obtained if the diffusion tensor D of Favro's equation is replaced by the substitution L = 2D $[\eqalignno{\langle{\bf x}\rangle&=\exp[-\textstyle{1\over2}({\rm tr}({\bf L}){\bf I}-{\bf L})]{\bf r}\cr &={\bf r}-\textstyle{1\over2}[{\rm tr}({\bf L}){\bf I}-{\bf L}]{\bf r}+{1\over8}[{\rm tr}({\bf L}){\bf I}-{\bf L}]^2{\bf r}-\ldots, & (6.1.1.91)}]$ where r is the vector from the centre of the sphere to the mode of the p.d.f. on the sphere and $[\langle{\bf x}\rangle]$ is the vector to the centre of mass. This equation, which is valid for all amplitudes of libration L, can be used to describe the apparent shrinkage effect in molecules undergoing librational motion.

6.1.1.6.6. The quasi-Gaussian approximation for curvilinear motion

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The p.d.f.'s defined by (6.1.1.77), (6.1.1.78), (6.1.1.86) and (6.1.1.87), and their Fourier transforms given in §6.1.1.6.5 may be considered `inverted series' since zero-order terms describe uniform distributions. The inverted series converge slowly if the density is concentrated near the mode. If $[\sigma^2]$ in (6.1.1.76) is sufficiently small, the cyclic overlap on the circle becomes unimportant and the summation for $[n\ne0]$ can be neglected. In this limiting case, the p.d.f. assumes the same form as a one-dimensional rectilinear Gaussian density function except that the variable is the angle $[\varphi]$ . A similar relation must exist between the p.d.f. on the sphere and the two-dimensional Gaussian function. This `quasi-Gaussian' approximation is the basis for a number of structure-factor equations for atoms with relatively small amplitude components of curvilinear motion (Dawson, 1970; Kay & Behrendt, 1963; Kendall & Stuart, 1963; Maslen, 1968; Pawley & Willis, 1970).

References

Atoji, M., Watanabe, T. & Lipscomb, W. N. (1953). The X-ray scattering from a hindered rotator. Acta Cryst. 6, 62–66.Google Scholar

Bellman, R. (1961). A brief introduction to theta functions. New York: Holt, Reinhart and Winston.Google Scholar

Chidambaram, R. & Brown, G. M. (1973). A model for a torsional oscillator in crystallographic least-squares refinement. Acta Cryst. B29, 2388–2392.Google Scholar

Dawson, B. (1970). Neutron studies of nuclear charge distributions in barium fluoride and hexamethylenetetramine. Thermal neutron diffraction, edited by B. T. M. Willis, pp. 101–123. Oxford University Press.Google Scholar

Favro, L. D. (1960). Theory of the rotational motion of a free rigid body. Phys. Rev. 119, 53–62.Google Scholar

Feller, W. (1966). An introduction to probability theory and its applications, Vol. II, Chap. 19. New York: John Wiley.Google Scholar

Fisher, R. (1953). Dispersion on a sphere. Proc. R. Soc. London Ser. A, 217, 295–305.Google Scholar

Furry, W. H. (1957). Isotropic rotational Brownian motion. Phys. Rev. 107, 7–13.Google Scholar

Gumbel, E. J., Greenwood, J. A. & Durand, D. (1953). The circular normal distribution: theory and tables. J. Am. Stat. Assoc. 48, 131–152.Google Scholar

Johnson, C. K. & Levy, H. A. (1974). Thermal motion of independent atoms. International tables for X-ray crystallography. Vol. IV, pp. 317–319. Birmingham: Kynoch Press.Google Scholar

Kay, M. I. & Behrendt, D. R. (1963). The structure factor for a harmonic quasi-torsional oscillator. Acta Cryst. 16, 157–162.Google Scholar

Kendall, M. G. & Stuart, A. (1963). The advanced theory of statistics, Vol. 1, Chaps 2, 3 and 6. London: Griffin.Google Scholar

King, M. V. & Lipscomb, W. N. (1950). The X-ray scattering from a hindered rotator. Acta Cryst. 3, 155–158.Google Scholar

Kuhs, W. F. (1983). Statistical description of multimodal atomic probability densities. Acta Cryst. A39, 149–158.Google Scholar

Kurki-Suonio, K. (1977). Electron density mapping in molecules and crystals. IV. Symmetry and its implications. Isr. J. Chem. 16, 115–123.Google Scholar

Kurki-Suonio, K., Merisalo, M. & Peltonen, H. (1979). Site symmetrized Fourier invariant treatment of anharmonic temperature factors. Phys. Scr. 19, 57–63.Google Scholar

Kuznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Quasi-moment functions in the theory of random processes. Theory Probab. Appl. (USSR), 5, 80–97.Google Scholar

Lévy, P. (1938). C. R. Soc. Math. Fr. p. 32. Also Processus stochastiques et mouvement Brownian, p. 182. Paris: Gauthier-Villars.Google Scholar

Mackenzie, J. K. & Mair, S. L. (1985). Anharmonic temperature factors: the limitations of perturbation-theory expressions. Acta Cryst. A41, 81–85.Google Scholar

Mair, S. L. (1980a). Temperature dependence of the anharmonic Debye–Waller factor. J. Phys. C, 13, 2857–2868.Google Scholar

Mair, S. L. (1980b). The anharmonic Debye–Waller factor in the classical limit. J. Phys. C, 13, 1419–1425.Google Scholar

Mair, S. L. & Wilkins, S. W. (1976). Anharmonic Debye–Waller factor using quantum statistics. J. Phys. C, 9, 1145–1158.Google Scholar

Mardin, K. V. (1972). Statistics of directional data. New York: Academic Press.Google Scholar

Maslen, E. N. (1968). An expression for the temperature factor of a librating atom. Acta Cryst. A24, 434–437.Google Scholar

Mises, R. von (1918). Über die `Ganzahligheit' der Atomgewichte und verwandte Fragen. Phys. Z. 19, 490–500.Google Scholar

Pawley, G. S. & Willis, B. T. M. (1970). Neutron diffraction study of the atomic and molecular motion in hexamethylenetetramine. Acta Cryst. A26, 263–271.Google Scholar

Perrin, F. (1928). Etude mathématique du mouvement Brownien de rotation. Ann. Ecole. Norm. Suppl. 45, pp. 1–23.Google Scholar

Perrin, F. (1934). Mouvement Brownien d'un ellipsoïde (I). Dispersion diélectrique pour des molécules ellipsoïdales. J. Phys. Radium, 5, 497.Google Scholar

Press, W. & Hüller, A. (1973). Analysis of orientationally disordered structures. I. Method. Acta Cryst. A29, 252–256.Google Scholar

Roberts, P.-H. & Ursell, H. D. (1960). Random walk on a sphere. Philos. Trans. R. Soc. London Ser. A, 252, 317–356.Google Scholar

Scheringer, C. (1985). A general expression for the anharmonic temperature factor in the isolated-atom-potential approach. Acta Cryst. A41, 73–79.Google Scholar

Stephens, M. A. (1963). Random walk on a circle. Biometrika, 50, 385–390.Google Scholar

Stewart, R. F. (1980b). Multipolar expansions of one-electron densities. Electron and magnetisation densities in molecules and crystals, edited by P. Becker, pp. 405–425. New York: Plenum.Google Scholar

Zucker, U. H. & Schulz, H. (1982). Statistical approaches for the treatment of anharmonic motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563–568.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 6.1, pp. 585-590