Determining crystal force constants from measured X-ray thermal diffuse scattering: inversion limits, Raman-proper contributions, and practical obstacles (Born's 1942 program)

In 1942 Max Born derived an exact matrix-function relation between the thermal-displacement scattering tensor $S(\mathbf{q})$ and the dynamical matrix $D(\mathbf{q})$ of a crystal, proposed a three-step inversion of the diffuse X-ray intensity between Bragg spots (the 'background' scattering) into the harmonic interatomic force constants, and explicitly flagged what remained open: the mutual-deformation ('Raman-proper') coupling coefficients $\beta$ were unknown; the convergence of the formal inverse series was unassessed; and practical obstacles (noise, finite sampling, anharmonicity, defects, internal stresses, electronic contributions) stood between the in-principle scheme and real photographic data. Later literature settled the surrounding dispute in favor of the thermal theory, realized diffuse-scattering phonon spectroscopy (from Olmer's 1948 aluminum dispersion to modern momentum-resolved and benchtop measurements), and, in a formalism explicitly based on Born's S-matrix, proved that monatomic crystals are fully recoverable from one-phonon thermal diffuse scattering while polyatomic crystals irretrievably lose information in the kinematic single-scattering approximation; Bayesian, differentiable and global-optimization inversions now recover phonon dispersions and force constants from simulated and measured thermal diffuse scattering on benchmark crystals. What remains open is precise: quantitatively reliable, uncertainty-quantified inversion of measured thermal diffuse scattering for real polyatomic, anharmonic, imperfect crystals (including identifiability of long-range force constants under finite, noisy sampling), together with a systematic quantification of the Raman-proper contributions to X-ray diffuse scattering, which later literature has not pursued.

Background

When a monochromatic X-ray beam traverses a crystal, most of the diffracted intensity appears in sharp Bragg peaks, the constructive-interference spots predicted by the Laue conditions that make X-ray photographs of crystals usable for structure determination. A much weaker pattern of diffuse intensity spreads between and around those spots. In the early 1940s this background became the object of a public controversy: faint 'extra spots' on Laue photographs of diamond and other crystals were attributed by one school to quantum reflexions of a new type, and by the thermal-motion school (descending from Debye's and Waller's work) to scattering by thermally vibrating atoms. Max Born's three-part series 'The effect of thermal vibrations on the scattering of X-rays' (1941-1942; Part I co-authored with Kathleen Sarginson and Part II written by Sarginson alone) developed the thermal explanation quantitatively, and Part III (1942), the source of the present problem, asked what that explanation could deliver as a measurement tool.

In the harmonic approximation, where the crystal's potential energy is truncated at quadratic order in the displacements of atoms from equilibrium, the lattice supports phonons, the quantized normal modes with frequencies $\omega_j(\mathbf{q})$ at wave vector $\mathbf{q}$. All harmonic dynamics is encoded in the Born-von Karman interatomic force constants (IFCs), the second derivatives of the potential energy with respect to atomic displacements, indexed by atom pairs and cell separations; their mass-weighted Fourier transform over the lattice is the dynamical matrix $D(\mathbf{q})$, a $3s\times 3s$ Hermitian matrix for a unit cell of $s$ atoms whose eigenvalues are the squared phonon frequencies. Knowing the IFCs is knowing the harmonic lattice.

The thermally induced diffuse intensity between Bragg peaks, thermal diffuse scattering (TDS), is dominated at ordinary temperatures by one-phonon scattering, scattering events in which a single phonon is created or annihilated. Its distribution at each $\mathbf{q}$ is governed by a scattering tensor $S(\mathbf{q})$, a close relative of the displacement covariance matrix from which the Debye-Waller factors, the exponential attenuation of Bragg intensities by thermal motion, are built. Born's central result in Part III is the exact matrix-function relation

$$S(\mathbf{q})=\phi\big(D(\mathbf{q})\big),\qquad \phi(\omega^{2})=\frac{\hbar}{2}\,\omega^{-1}\coth\!\Big(\frac{\hbar\omega}{2kT}\Big),$$

where $T$ is the temperature and the scalar map $\phi$ is applied to the matrix by functional calculus (equivalently, as a sum over normal modes). Expanding $\phi$ in even powers of $\hbar\omega/kT$, a Bernoulli expansion that converges rapidly only when $\hbar\omega\ll kT$, gives the classical leading law $S\approx kT\,D^{-1}$ plus explicit quantum corrections, and inverting the series yields

$$D(\mathbf{q})=kT\,S^{-1}(\mathbf{q})\Big\{1+\frac{1}{12}\frac{\hbar^{2}}{kT}S^{-1}(\mathbf{q})+\frac{1}{180}\frac{\hbar^{4}}{k^{2}T^{2}}S^{-2}(\mathbf{q})+\cdots\Big\}.$$

Born then set out a three-step inversion program for experiment: (i) measure the normalized diffuse intensity at sets of symmetry-equivalent reciprocal-space points, points at which the periodic tensor components $S_{\alpha\beta}(\mathbf{q})$ take the same values while the known geometrical factors take distinct computable values, and solve the resulting quadratic equations for $S(\mathbf{q})$; (ii) apply the inverse series to obtain $D(\mathbf{q})$; (iii) Fourier-transform $D(\mathbf{q})$ over the reciprocal cell to recover the real-space force-constant tensors. He further treated the mechanism he called mutual deformation, the ordinary Raman effect, in which an atom's polarizability, and hence its X-ray scattering factor, depends on the instantaneous displacements of neighboring atoms: $f_k^l=f_k^0\exp[\sum\beta_{kk'}^{l-l'}\cdot u_{k'}^{l'}]$, with coupling vectors $\beta$ constrained by translational invariance and vanishing for identical-atom pairs, so that the effect drops out for monatomic lattices but can survive in polyatomic ones. Two scattering regimes matter throughout: the kinematic approximation, in which the X-ray is assumed to scatter at most once in the crystal (adequate for weak scattering), and the dynamical regime, in which multiple scattering must be treated.

The problem the paper left open is that each link of this chain was formal. The numerical values of the $\beta$ couplings were unknown for any material (obtainable, Born noted, only from optical Raman data or from microscopic calculation); the series inversions were manipulated without any convergence or truncation-error analysis; and the practical obstacles separating the scheme from real photographs, namely measurement noise, finite and discrete sampling, anharmonicity, defects and internal stresses, and contributions from electronic excitations, were named but not quantified. Two later facts frame the current status: the Raman-versus-thermal dispute over the extra spots was settled in favor of the thermal explanation (Nature letters, 1942-1948), and the theory of thermal X-ray scattering was systematized in monograph form (Krivoglaz, 1995 English translation).

Problem Statement

The problem, in the general form in which Born (1942) left it open, is to turn the inversion of measured X-ray diffuse scattering into a quantitatively reliable determination of a crystal's harmonic force constants. It has three linked components that jointly serve one common objective, recovered force constants with quantified uncertainties.

The objective has three linked components: (1) well-posedness of the inversion chain from $S(\mathbf{q})$ to $D(\mathbf{q})$ to the real-space force constants, covering the polyatomic case; (2) the mutual-deformation ($\beta$) contribution for a polyatomic crystal; and (3) a quantified error budget for the recovered force constants.

(1) Inversion well-posedness and limits. Establish when the inversion chain, from measured diffuse intensities at symmetry-equivalent reciprocal-space points (where $S_{\alpha\beta}(\mathbf{q})$ is fixed while the known geometrical factors vary) to the scattering tensor $S(\mathbf{q})$, then to the dynamical matrix $D(\mathbf{q})$ via the inverse function $D=\psi(S)$ (the series of Born's Eq. (3.8) or any equivalent exact inversion), and finally to real-space force constants by Fourier inversion (Eq. (3.9)), is well posed: the convergence or exact validity of the inversion, and the recoverability limits already established in part (full recovery for monatomic crystals; provable information loss for polyatomic crystals in the kinematic approximation, per Bosak & Chernyshov 2008), extended to conditions achievable in experiment.

(2) Raman-proper contributions. Determine, or rigorously bound, the mutual-deformation contribution to X-ray diffuse scattering, the terms controlled by the coupling vectors $\beta_{kk'}$ entering the displacement-dependent atomic scattering factor of Born's Eq. (4.2), for polyatomic crystals, so that these contributions are separated from the thermal background quantitatively rather than neglected by convention.

(3) Practical obstacles. Quantify and control the obstacles between the formal scheme and real data: measurement noise; finite, discrete and geometry-restricted reciprocal-space sampling; anharmonicity, which both shifts the modes and adds multi-phonon intensity, reaching a substantial fraction of the TDS signal at elevated temperature; defects, internal stresses and other departures from the ideal periodic harmonic lattice; and contributions from electronic excitations, so that force constants recovered from measured diffuse scattering carry a quantified error budget.

References

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