Full elasticity tensor from thermal diffuse scattering
Björn Wehinger, Alessandro Mirone, Alexeï Bosak, Michael Krisch
FFull elasticity tensor from thermal diffuse scattering
Bj¨orn Wehinger,
1, 2, ∗ Alessandro Mirone,
3, †
Michael Krisch, and Alexe¨ı Bosak Department of Quantum Matter Physics, University of Geneva,24, Quai Ernest Ansermet, CH-1211 Gen`eve, Switzerland Laboratory for Neutron Scattering and Imaging,Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland European Synchrotron Radiation Facility, 71,Avenue des Martyrs, F-38000 Grenoble, France (Dated: November 5, 2018)
Abstract
We present a method for the precise determination of the full elasticity tensor from a single crystal diffrac-tion experiment using monochromatic X-rays. For the two benchmark systems calcite and magnesium oxidewe show that the measurement of thermal diffuse scattering in the proximity of Bragg reflections providesaccurate values of the complete set of elastic constants. This approach allows for a reliable and model freedetermination of the elastic properties and can be performed together with crystal structure investigation inthe same experiment. a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug . INTRODUCTION The elasticity tensor is the fundamental quantity for describing elastic waves and determineseventual anisotropic compression and sound velocities in crystalline materials . The elastic prop-erties define mechanical properties of materials and influence phase stability . Accurate mea-surements of the elasticity are of principal interest for the description of seismological wavesand their discontinuities, which allow for decisive conclusions on composition, temperature andpressure of Earth’s interior . In condensed matter physics elastic constants are important inthe study of quantum phase transitions in systems with pronounced interaction of phonons withother quasi-particles. Such interaction may include electron-phonon coupling with applicability tosuperconductors as well as spin-phonon coupling with interesting anomalies in low dimensionalspin systems . The most commonly used experimental techniques to determine the coefficientsof the elasticity tensor – the so-called elastic constants – are ultrasound measurements or Brillouinscattering. Ultrasound measurements are limited to relatively large crystals with well-definedfaces, measurements at extreme conditions such as high pressures or high magnetic fields are verychallenging. Brillouin scattering can be performed at high pressures but is difficult for opaquematerials. Alternatively, the elasticity tensor can be extracted from inelastic x-ray or neutron scat-tering, which becomes efficient for crystals of low symmetry if combined with calculations fromfirst principles . Inelastic x-ray scattering can be performed at extreme pressures using diamondanvil cells and can be combined with low or high temperatures .A complete formalism to derive ratios of elastic constants from thermal diffuse scattering (TDS)measured with energy integrating detectors has been derived in the 1960ies . Historically, thethermal nature of diffuse scattering was noticed already in the 1920ies and the first phonondispersion relation were in fact determined from TDS of X-rays . The currently available highflux and brilliant x-ray beams from synchrotrons in combination with bi-dimensional single photoncounting x-ray detectors with good quantum efficiency and no readout noise attracted new interestin TDS studies . By the use of force constant models it is possible to determine the phonondispersion relations, while a model-free reconstruction of the lattice dynamics can be realized onlyfor mono-atomic crystals .In this Letter, we present a method for the precise determination of the full elasticity tensorfrom a single crystal diffraction experiment using a model free data analysis for arbitrary crys-tal symmetries. For the two benchmark systems calcite and magnesium oxide we show that the2easurement of thermal diffuse scattering at two close temperatures is sufficient to obtain the fullelasticity tensor in absolute units within remarkable accuracy. We discuss the influence of addi-tional contributions to the diffuse scattering and evaluate multiple phonon scattering with help offirst principle calculations using density functional perturbation theory. II. EXPERIMENTAL DETAILS
Diffuse x-ray scattering intensities were collected in transmission geometry using a single pho-ton counting pixel detector with no readout noise and large dynamical range. High quality singlecrystals were prepared by mechanical cutting and polishing; surface defects were removed by gen-tle etching. Temperature was controlled by a nitrogen cryostream. The experiment on calcite wasconducted on ID29 at the European Synchrotron Radiation Facility (ESRF) . The sample was arectangular bar with dimensions of approximately 50 µ m × µ m × λ = . µ m × µ m were used. The sample was rotated from 0 to 100 ◦ orthogonal to the beam withangular steps of 0.1 ◦ . Scattering intensities were collected with a PILATUS 6M detector (Dectris,Baden, Switzerland), equipped with 300 µ m thick Si pixels of size 172 µ m , at a sample-detectordistance of 300 mm. Measurements on magnesium oxide were performed at the Swiss NorwegianBeamline BM01A at the ESRF where we used a cubic single crystal with edge length of 2 mm.The sample was measured at λ = . ◦ with the same angular step as for calcite. Scattering intensities were recorded witha PILATUS 2M detector at a distance of 244 mm from the sample. III. FORMALISM
Assuming the validity of both, adiabatic and harmonic approximations, the intensity of x-rayscattering from phonons for single and two phonon processes is given by I ( Q ) = (cid:126) NI inc ∑ ν Ω q , ν (cid:12)(cid:12) ∑ a f a ( Q ) √ m a e − W a , Q ( Qe Q , ν , a ) e − i Qτ a (cid:12)(cid:12) (1)and 3 ( Q ) = (cid:126) NV I inc (cid:90) d q ( π ) ∑ ν , ν (cid:48) Ω q , ν Ω Q − q , ν (cid:48) × (cid:12)(cid:12) ∑ a f a ( Q ) m a e − W a , Q e − i Qτ a ( Qe q , ν , a )( Qe Q − q , ν , a ) (cid:12)(cid:12) (2)with Ω q , ν = ω q , ν coth (cid:16) (cid:126) ω q , ν k B T (cid:17) , (3)respectively; see Refs. for a description in modern notation. Here, N is the number of unitcells, I inc the incident beam intensity, f the atomic scattering factor of atom a with mass m and(anisotropic) Debye Waller factor W . ω denotes the eigenfrequency and e the eigenvector of thephonon with wavevector q (equal to the reduced momentum transfer) and branch ν . Q is the totalscattering vector, τ the atomic basis vector within the unit cell and V the unit cell volume, T thetemperature and k B the Boltzmann constant. For small q the scattering intensities I ( Q ) ∼ / ω q , ν and single phonon scattering is thus dominated by the contribution of the acoustic phonons.Within the theory of elastic waves in crystals the equation of motion is given by ρω u i = c i jlm k j k l u m , (4)where c i jlm is the tensor of elastic constants, ρ the mass density and k = k n and ω are the wavevector and the frequency of the elastic waves, respectively .For fitting the elastic constants to the experimental intensities in the vicinity of Bragg reflec-tions we calculate the scattering intensities as the sum of the contributions from the three acousticbranches. We therefore solve the equation of motion (Eq. 4) for a given crystal symmetry and cal-culate the scattering intensities by summing over the three acoustic phonon branches in Eqs. 1 and2. The thus obtained intensities are renormalized by a vector g ( Q ) that accounts for absorption,polarisation and geometrical corrections. A vector b ( Q ) is added for the background.Given a set of experimental intensities I exp Q , T measured at T over a set of reciprocal space points Q ∈ (cid:8) Q exp (cid:9) in the proximity to Bragg reflections, we find the elasticity tensor c by solving the4ptimization problem c , b , g = argmin c (cid:48) , b (cid:48) , g (cid:48) (cid:32) ∑ Q (cid:16) I calc Q , T ( c (cid:48) , b (cid:48) , g (cid:48) ) − I exp Q , T (cid:17) (cid:33) , (5)where I calc Q , T is the calculated intensity containing contributions from one-phonon and eventually ( n > ) -phonon processes. c , b and g are the fit parameters. In order to reduce the free parameters c is constrained to the crystal symmetry. b and g are kept constant in the vicinity of individualBragg reflections. Such approximation for b is justified by the fact that diffuse scattering dueto additional contributions varies much less across reciprocal space than TDS for small q . Thevariation of corrections for absorption, polarisation and planar projection is small for the employedscattering geometry at small q and thus justifies the approximation for g .Solving the minimization problem Eq. 5 requires that the diffuse scattering is due to phononscattering only. It can provide absolute values of the elastic constants if absolute intensities areknown. If not, the elastic tensor is determined upon a single scaling factor and the absolute valuesof the elastic constants can be obtained if constraint to the adiabatic bulk modulus via the Reuss-Voigt-Hill relation.Another option consists of measuring scattering intensities at slightly different temperatures. Infact, diffuse scattering from static disorder, air scattering, and fluorescence displays a much smallertemperature dependence than TDS, and can therefore be isolated in a difference measurement.For experimental data at two slightly different temperatures T and T measured in the samegeometry we can solve the minimization problem c , b , b , g = argmin c (cid:48) , b (cid:48) , b (cid:48) , g (cid:48) (cid:32) ∑ Q (cid:32)(cid:16) I calc Q , T ( c (cid:48) , b (cid:48) , g (cid:48) ) − I calc Q , T ( c (cid:48) , b (cid:48) , g (cid:48) ) (cid:17) − (cid:16) I exp Q , T − I exp Q , T (cid:17)(cid:33) , (6)which we call multi-temperature method. It neglects the variation of the elastic constants overthe temperature interval T , T . The intensity difference must be compared to the variations of theBose factor coth (cid:16) (cid:126) ω q , ν kT (cid:17) , which is approximately proportional to the temperature at high temper-atures ( (cid:126) ω > kT ). This implies that, at constant c , the intensities measured at T and T are almostproportional to each other if one-phonon processes are predominant. Therefore, at such high tem-peratures, only ratios of elastic constants can be obtained if the scattering intensities are unknownon an absolute scale. At lower temperatures the acoustic branches span two regions, a low fre-5uency one with (cid:126) ω (cid:28) kT and one with higher frequencies (cid:126) ω > kT . In such case the intensitiesof the two measurements become linearly independent. The knowledge of T determines the ab-solute scale to which ω compares, and, as a consequence, absolute values of the elastic constantscan be obtained even for unknown absolute intensities. The temperature must be low enough sothat the condition (cid:126) ω > kT is realized in regions of q where the elastic approximation is fulfilled. IV. LATTICE DYNAMICS CALCULATIONS
Lattice dynamics calculations were carried out from first principles employing density func-tional perturbation theory as implemented in the CASTEP code . For both, calcite and mag-nesium oxide we used the local density approximation within a plane wave basis set and pseudo-potentials of the optimized form . The sampling of the electronic structure and the plane wavecut-off energy were chosen to ensure the convergence of internal forces to < − eV/ ˚A. Theacoustic sum rule was enforced to ensure translational symmetry. Eigenfrequencies and eigenvec-tors of the acoustic branches were replaced in order to reproduce the elastic approximation closeto Γ . Fourier interpolation was employed for the computation of dynamical matrices at arbitrary q .The calculated dynamical matrices were used to compute the Debye Waller factors and scatteringintensities were calculated via Eqs. 1 and 2. The calculations are used for the correct selectionof q -values to be fitted and for the evaluation of multiple phonon scattering and contribution ofoptical phonons. V. RESULTSA. Calcite at 170 Kelvin
The determination of the elastic constants from TDS measured at a single temperature isdemonstrated for a small calcite single crystal. For the fit we consider the diffuse scattering inten-sity in the proximity of the 48 most intense Bragg reflections. The regions of interest (ROI) wereparametrized by the absolute value of the reduced wave vector q in Cartesian coordinates with q i = | ∆ Q i | / | a ∗ | and the data points selected according to the criteria q ∈ [ q min , q max ] . The regionsbelow q min are excluded to minimize the contribution from elastic scattering while the regionsabove q max are excluded in order to ensure the validity of the elastic approximation.6 alcite, T = T = I ( ) − I ( ) FIG. 1. Graphical rendering of measured diffuse scattering and calculated TDS. Considered cases are:Calcite at T = 170 K for a ROI of q ∈ [ . , . ] (top panels), MgO with q ∈ [ . , . ] at T = 90 K(middle panels) and with the multi-temperature method at T = 90 K, 120 K (bottom panels). Each imageshows a cross section of the reciprocal space, in the neighborhood of the select Bragg peak, and for a given q z = | ∆ Q z | / | a ∗ | . The data are grouped by pairs, with the experimental and calculated intensity distributionon the left and right side, respectively. The scattering intensity is shown on a linear color scale from black(zero) to white (given maximal intensity I max ). The reverse interpolation, which is only used for graphicalrendering, results in some artifacts, for example some intensity remains outside the ROI. The stripes visibleat hkl = ¯2 , ¯2 , , ¯2 , ABLE I. Elastic constants of calcite at 170 K. The values derived from fitting TDS are obtained for q ∈ [ . , . ] and normalized by the adiabatic bulk modulus. Experimental reference values from ultrasoundmeasurements (Ref. ) and the relative difference between the values derives from TDS and ultrasound arereported for comparison. Theoretical values were derived from fitting calculated TDS intensities within thesame ROI ( q ∈ [ . , . ] ) to the contribution of (i) acoustic branches only, (ii) all phonon branches and(iii) acoustic branches including two-phonon scattering.TDS Ref. rel. diff. Calc. (i) Calc. (ii) Calc. (iii) c
156 155.1 0.6% 154.3 153.5 154.2 c c c c c Experimental scattering intensities and calculated TDS from the fitted elastic constants in theelastic approximation for first order scattering are shown for selected regions of reciprocal space inFig. 1. The plotted values are obtained from the irregular experimental data grid by inverting theinterpolation equation. We do this using a few iterative steps and a Thikonov regularisation term.Diffraction patterns that contain Bragg peaks were removed from the data treatment, because suchimages are affected by secondary scattering effects .The elastic constants as obtained from fitting intensities of an ensemble of individual pixelsare reported in Table I. Eq. 5 was used to determine c upon a single scaling factor and the abso-lute values were obtained by normalization to the known adiabatic bulk modulus K . Remarkableagreement with literature values determined from ultrasound measurements are obtained for aROI q ∈ [ . , . ] , which corresponds to approximately 2 . × intensity points. The differ-ence is in the order of 1 % for all elastic constants with exception of c . The higher limit q max of the ROI is verified by ab initio calculations. We therefore compute the scattering intensitiesfrom the calculated dynamical matrices and fit c to it for different choices of ROI and compare theresult to the expected value for lim q →
0. The lower limit q min of the ROI must be chosen care-fully, because very small momentum transfers might be affected by elastic scattering, as discussedfurther below. The contribution of optical phonons and multiple phonon scattering is evaluated8 ABLE II. Elastic constants of MgO measured at T = 90 K and T = 120 K. The elastic constants were fittedfor q ∈ [ . , . ] applying the multi-temperature method for the two temperatures ( I ( T − T ) ) and for T only (Fit I ( T ) , values re-scaled to the reference value of c ). Ultrasound data for the two temperaturesand the relative difference between the average reference values and the results of the multi-temperaturemethod are listed for comparison.Fit I ( T ) − I ( T ) Fit I ( T ) Ref. Ref. c rescaled T = 90 K T = 120 K rel. diff. c
300 306 306.1 305.4 2.0% c
151 194 157.2 156.9 3.7% c
89 39 94.07 94.26 5.7% by computing I ( Q ) and I ( Q ) including all phonon branches. A fit of elastic constants to thesecomputed scattering intensities for the same ROI results in a maximal relative deviation of 1.2 % ifoptical phonons are considered and only 0.8 % maximal relative deviation if second order phononscattering is included (see Table I). B. Magnesium oxide measured at two temperatures
The diffuse scattering in MgO at T = 90 K is much less structured than the one of calcite dueto the higher cubic crystal symmetry, see Fig. 1. For fitting the elastic constants both methods, Eq.5 and 6, were employed considering diffuse scattering in the proximity of 78 of the most intenseBragg reflections. The results are reported in Table II. Fitting a single temperature with Eq. 5 isperformed to obtain c upon a uniform scaling factor. Due to the large discrepancy in c the valuesare scaled to the experimental value of c from ultrasound measurements instead of applying ascaling to the bulk modulus. The results are rather unsatisfactory compared to literature data .This demonstrates the influence of additional diffuse scattering due to elastic scattering. We thusemploy the multi-temperature method (Eq. 6) and fit the intensity differences of diffuse scatteringmeasured at two close temperatures, 90 and 120 K, see Fig. 1. The data seems noisy, but fittingthe ensemble of approximately 1 . × pixels is sufficient for a well converged result. Using thisstrategy we obtain accurate values of the full elasticity tensor in absolute units, presented in TableII. 9 I. DISCUSSION
The results on calcite and magnesium oxide show that the full elasticity tensor can be measuredwith high accuracy by a rigorous data treatment of diffuse scattering in the proximity of Bragg re-flections. The availability of single photon counting detectors with no readout noise together witha large number of independent intensity points are crucial for successful experiments. For sin-gle crystals of extreme high quality such as our investigated calcite crystal the measurement ofTDS at a single temperature can be sufficient to obtain the correct ratio of all elastic constants.The absolute values may then be obtained by a normalization to the adiabatic bulk modulus. Theoptimized choice of the ROI to be fitted is very important to ensure the validity of the elastic ap-proximation and allows minimizing the contribution of elastic scattering. The best ROI can befound by a successive adjustments of q min and q max and is verified here with help of lattice dynam-ics calculations. Scattering contributions that vary slowly in reciprocal space can be taken intoaccount in good approximation by a constant background to the diffuse scattering in the vicinityof individual Bragg reflections. Such contributions may include air scattering, fluorescence andCompton scattering. This includes as well the contribution of optical phonons and higher orderphonon scattering as shown by evaluating calculated dynamical matrices. A de-convolution pro-cedure for air scattering and beam shape may be envisaged in order to extend the ROI to smaller q values, but this generally increases the noise level of the data. Secondary scattering effects likeBragg-diffuse scattering, where the Bragg reflected beam acts as a source of secondary diffusescattering, are more difficult to treat and its contribution and scattering conditions are discussedelsewhere . Here, we exclude all diffraction patterns that may be affected by such effects. If thediffuse scattering close to Bragg reflections is affected by elastic scattering the multi-temperaturemethod might be a good choice. This method is based on the fact that TDS has a much strongertemperature dependence than other sources of diffuse scattering. Elastic and quasi-elastic contri-butions can thus be subtracted to a good approximation. Absolute values of the elasticity tensorcan finally be obtained if the temperature interval is chosen such that the intensities become linearindependent. This is shown for MgO. In addition to TDS we observe elastic diffuse scatteringwhich we attribute to crystal defects. A careful measurement of absolute intensities may be apromising alternative to extract absolute values of the elastic constants but is likely less practical.In this study we compute the Debye Waller factors from first principles. However, they can alsobe obtained from experiment using x-ray diffraction employing the same scattering geometry.10 II. CONCLUSION AND OUTLOOK
In summary, we have shown that accurate values of the full elasticity tensor can be obtainedfrom a simple diffraction experiment on single crystals. Our method opens the perspective to de-termine elastic properties together with crystal structure investigations and thus under the sameexperimental conditions. This implies a broad applicability in material science, geophysics and inthe investigation of sound wave anomalies due to fundamental interactions in condensed matterphysics. The achieved accuracy can compare with the standard methods such as ultrasound mea-surements and Brillouin scattering with the advantage of applicability to very small and opaquecrystals of arbitrary shape. The proposed methodology can be extended to measurements at ex-treme conditions such as high pressures, high or low temperatures or high magnetic fields. Thecontribution from the sample environment might be treated with the multi-temperature methodtogether with a deconvolution procedure for the treatment of temperature independent contribu-tions. TDS from diamond in high pressure cells, for example, might be modeled or measured andthen separated from the data by deconvolution. The application to high pressures is particularlyinteresting for the establishment of absolute pressure scales in a single experiment. At very lowtemperatures the proposed strategy is expected to work well and potentially very useful in thestudy of spin-lattice coupling. At temperatures relevant for geophysical processes the scatteringintensities will not be linear independent but absolute values can be obtained if the adiabatic bulkmodulus is known.
VIII. ACKNOWLEDGMENT
We thank Dmitry Chernyshov, Daniele de Sanctis and Harald Reichert for providing beamtime and fruitful discussions on measurement and data treatment of diffuse scattering. Gael Goretis acknowledged for help in programming the first versions of the reciprocal space reconstructionroutines. This work was supported by the European Community’s Seventh Framework Programme(FP7/2007-2013) under grant agreement no. 290605 (PSI-FELLOW/COFUND) and resources ofthe European Synchrotron Radiation Facility. ∗ [email protected] [email protected] F. I. Fedorov,
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