## research papers

## On the Einstein model for

parallel and perpendicular mean-square relative displacements^{a}Università degli Studi di Verona, Dipartimento di Informatica, Strada le Grazie 15, I-37134 Verona, Italy^{*}Correspondence e-mail: sanson@sci.univr.it

The correlated Einstein model for

parallel and perpendicular mean-square relative displacements (MSRDs) is discussed. By means of dynamical simulations on different crystalline structures, the error owing to the Einstein-fit model on the MSRDs is estimated as a function of the standard deviation of the density of vibrational states. This error should be taken into account to improve the accuracy of the MSRDs.Keywords: EXAFS; mean-square relative displacement; Einstein model.

### 1. Introduction

Extended X-ray absorption fine structure (EXAFS) is known as a powerful probe for investigating disordered systems and, thanks to accurate temperature-dependent measurements, for obtaining original information on the local dynamics of crystals, such as the parallel and perpendicular mean-square relative displacements (Brewe *et al.*, 1997; Fornasini *et al.*, 2004). Indicating by and the thermal displacement of the absorber and backscatterer atoms of respectively, the parallel mean-square relative displacement (MSRD_{∥}) is defined as

and the perpendicular mean-square relative displacement (MSRD_{⊥}) as

where is the unit vector of the _{∥} (Beni & Platzman, 1976). The difference between the interatomic distance 〈*r*〉 = 〈 − 〉 probed by and the crystallographic distance *R* = − between average positions allows measurement of the temperature dependence of MSRD_{⊥} (Dalba *et al.*, 1999).

To find accurate values of MSRD_{∥} and MSRD_{⊥} by an accurate knowledge of the scattering amplitudes, phase shifts and inelastic terms is required (Teo, 1986). These parameters can be determined by theoretical calculations, for example using *FEFF8* code (Ankudinov *et al.*, 1998). However, the uncertainties on these calculations affect the accuracy on the absolute value of MSRD_{∥} and MSRD_{⊥}. An alternative procedure consists of a separate analysis of phase and amplitude of the filtered signal *via* the ratio method (Bunker, 1983; Tranquada & Ingalls, 1983) taking the lowest-temperature spectra as reference for backscattering amplitudes, phase shifts and inelastic terms. However, only the relative differences δMSRD_{∥}(*T*) = MSRD_{∥}(*T*) − MSRD_{∥}(*T*_{0}) and δMSRD_{⊥}(*T*) = MSRD_{⊥}(*T*) − MSRD_{⊥}(*T*_{0}) with respect to the reference temperature *T*_{0} can be obtained from this second procedure. In both procedures it is customary to fit the temperature dependence of δMSRD_{∥}(*T*) and δMSRD_{⊥}(*T*) to a correlated Einstein model in order to obtain with reasonable accuracy the absolute value of MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) (Dalba & Fornasini, 1997).

In the correlated Einstein model, which consists of substituting the phonon ρ(ν) with a delta function centred at ν_{E}, one obtains the following expressions for MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) (Sevillano *et al.*, 1979; Vaccari & Fornasini, 2006),

To obtain the absolute values of MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) from the experimental temperature-dependence of δMSRD_{∥}(*T*) and δMSRD_{⊥}(*T*), one uses the following fit expressions,

where *a*_{∥}, *a*_{⊥} and the Einstein frequencies ν_{∥}, ν_{⊥} are the only fitting parameters. The absolute values of MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) are obtained, respectively, by shift

Although the Einstein model is based on a strong assumption (phonon spectrum approximated to a single frequency), it is a matter of experience that the temperature dependence of the experimental MSRDs can be reasonably fitted by this model. However, the error deriving from this procedure has never been estimated. This is the aim of this work: to estimate, for the first time, the error on the absolute value of the

MSRDs owing to the use of the Einstein-fit model.The paper is organized as follows: in §2 the procedure of estimating the MSRDs error is briefly described; in §3 the results are reported and discussed; a remark on the connection between Einstein-model and crystal dynamics is made in §4; §5 is dedicated to conclusions.

### 2. Procedure

As a first step, the temperature dependences of MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) have been calculated (between 0 and 1000 K) in different crystals by means of dynamical simulations. The dynamical calculations have been performed within the Born–Von Karman approximation (Maradudin *et al.*, 1971) on the following crystalline structures: face-centered cubic, diamond structure, wurtzite and cuprite (one, two, four and six atoms in the elementary cell, respectively) with arbitrary and atomic masses, in all 27 crystals. The parallel and the perpendicular MSRD(*T*) are calculated in terms of eigenfrequencies ν(λ,) and eigenvectors of the dynamical matrix (for more details see Maradudin *et al.*, 1971; Vaccari & Fornasini, 2006) as

and

As a second step, the relative differences with respect to 0 K, *i.e.* δMSRD_{∥}(*T*) = MSRD_{∥}(*T*) − MSRD_{∥}(0) and δMSRD_{⊥}(*T*) = MSRD_{⊥}(*T*) − MSRD_{⊥}(0), have been fitted with the correlated Einstein model in agreement with equations (5) and (6), respectively. As a result, the fitting parameters *a*_{∥} and *a*_{⊥} (as well as ν_{∥} and ν_{⊥}) have been obtained for each crystal.

For the third step, if the Einstein-fit error is zero, then *a*_{∥} = MSRD_{∥}(0) and *a*_{⊥} = MSRD_{⊥}(0), and so δMSRD_{∥}(*T*) + *a*_{∥} and δMSRD_{⊥}(*T*) + *a*_{⊥} must give MSRD_{∥}(*T*) and MSRD_{⊥}(*T*), respectively. As a consequence, the percentage errors on MSRD_{∥} and MSRD_{⊥} at temperature *T* have been estimated, respectively, as

Fig. 1 shows an example of calculated MSRD_{∥}(*T*), the relative difference δMSRD_{∥}(*T*) with respect to 0 K, and the corresponding Einstein fit by equation (5). The resulting shift between MSRD_{∥}(*T*) and δMSRD_{∥}(*T*) + *a*_{∥} (open and full circles, respectively) is the error on the absolute MSRD_{∥}(*T*) owing to the Einstein fit.

It can be expected that the error owing to the Einstein fit increases with an increase in the standard deviation of the density of vibrational states (DOS) of the crystal. In a `perfect Einstein crystal', where DOS is a δ function and the DOS standard deviation is zero, the Einstein-fit error is zero. As a consequence, in §3, for each crystal the Einstein-fit error has been plotted against the DOS standard deviation. Fig. 2 shows three examples of the calculated DOS with different standard deviations.

### 3. Results and discussion

According to equations (5)–(6) and (9)–(12), the percentage errors on MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) at 0, 300 and 600 K have been calculated and plotted against the DOS standard deviation of the crystal (Figs. 3 and 4). The Einstein fit has been performed over a typical experimental range, between 0 and 500 K. To first approximation, the correlation between the Einstein error and the DOS standard deviation is independent of the crystal structure.

From Figs. 3 and 4 it can be observed that (i) the Einstein fit gives an overestimation of both MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) (%Err_{∥} and %Err_{⊥} are always greater than zero); (ii) the error on MSRD_{⊥}(*T*) is about two times the error on MSRD_{∥}(*T*) (%Err_{⊥} ≃ 2%Err_{∥}); (iii) on average, the error is within 5% on MSRD_{∥}(*T*) and within 10% on MSRD_{⊥}(*T*), at least if 0–500 K is considered as the fitting interval.

It is obvious that the error owing to the Einstein fit depends on the fitting interval. Since at high temperatures equations (5)–(6) are approximated by a classical linear behaviour, problems are mainly expected at low temperatures. Fig. 5 reports the percentage error in the diamond structure obtained with four different fitting intervals: 0–300 K, 0–500 K, 0–1000 K and 200–1000 K. The error is reduced when the high-temperature range prevails against the low-temperature range in the fitting procedure. Analogous results have been obtained for the other three crystalline structures.

From this result one finds that it is more convenient to fit the experimental δMSRD_{∥}(*T*) and δMSRD_{⊥}(*T*) neglecting the low-temperature intervals. However, this is true only in ideal situations. In real situations, where δMSRD_{∥}(*T*) and δMSRD_{⊥}(*T*) are affected by experimental uncertainties, it is probably more convenient to fit the whole available temperature range in order to maximize the experimental information. In any case, the high-temperature ranges reduce the Einstein error on the absolute MSRDs.

Only recently expression (4) for perpendicular MSRD has been derived by Vaccari & Fornasini (2006). In previous works (for example, Dalba *et al.*, 1999; Fornasini *et al.*, 2004) the absolute MSRD_{⊥}(*T*) was estimated from the correlated Einstein model for parallel MSRD, *i.e.* using equation (5) instead of equation (6). The use of equation (5), which differs by a factor of two from equation (6), is equivalent to inserting an incorrect value of the μ (twice the real value) in the fitting procedure of MSRD_{⊥}(*T*). I have estimated the error owing to the utilization of equation (5) instead of equation (6): as shown in Fig. 6, equation (5) gives an underestimation (%Err_{⊥} < 0) on the absolute value of MSRD_{⊥}(*T*), and the error is about twice that resulting from equation (6). As a consequence, equation (6) is fundamental in the fitting of perpendicular MSRD. For the first time it has been used by Vaccari *et al.* (2007) on cuprous chloride.

In the practical ). From Figs. 3 and 4, the Einstein fit between 0 and 500 K overestimates MSRD_{∥} by about 2% at 300 K, and MSRD_{⊥} by about 3%. In the case of the Einstein fit performed between 0 and 300 K, the error increases by about 1.5 times (as evident by Fig. 5), and the overestimation on MSRD_{∥} and MSRD_{⊥} increases by about 3% and 5%, respectively. Accordingly, the resulting MSRDs should be properly corrected to improve their accuracy.

### 4. Einstein model and crystal dynamics

Before the conclusion, let us make a final consideration on the Einstein model and on the corresponding crystal dynamics.

Let us indicate by γ(*T*) the ratio

which measures the anisotropy of relative thermal vibrations. If MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) can be described by a correlated Einstein model, then by inserting equations (3) and (4) into equation (13) one obtains

and

and, by comparing the last two equations,

As a result, in the correlated Einstein model the temperature dependence of γ(*T*) follows the condition expressed by equation (16). This is the case, for example, of perfect isotropy, where γ(*T*) = 2 at all temperatures.

Fig. 7 shows the ratio [2γ(*T* → +∞)]^{1/2}/γ(*T* → 0) calculated in the 27 crystals of this work [with γ(*T* → +∞) approximated to γ at 1000 K]: in most cases this ratio is closer to unity. However, this does not mean that the Einstein model works well: equation (16) is a necessary but not sufficient condition of the crystal dynamics to describe MSRD_{∥}(*T*) and MSRD_{⊥}(*T*) by the correlated Einstein model.

### 5. Conclusions

In this work the error owing to the Einstein fit on the absolute value of has been demonstrated.

MSRDs has been estimated by means of dynamical simulations. On average the error is less than ∼5% for the parallel MSRD, less than 10% for the perpendicular MSRD, but it should be taken into account to improve the MSRDs accuracy. The fitting error depends on the fitting range and, in principle, decreases neglecting the low temperatures. Finally, in the fitting of perpendicular MSRD, the importance of equation (6)### Acknowledgements

I would like to thank P. Fornasini and M. Vaccari for helpful discussions.

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