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Thermostructural and Elastic Properties of PbTe and Pb_{0.884}Cd_{0.116}Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects

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## Abstract

**:**

_{0.884}Cd

_{0.116}Te (i) at low temperatures (15 to 300 K) and (ii) at high pressures within the stability range of NaCl-type PbTe (up to 4.5 GPa). For crystal structure studies, powder and single crystal X-ray diffraction methods were used. Modeling of the data included the second-order Grüneisen approximation of the unit-cell-volume variation, V(T), the Debye expression describing the mean square atomic displacements (MSDs), <u

^{2}>(T), and Birch–Murnaghan equation of state (BMEOS). The fitting of the temperature-dependent diffraction data provided model variations of lattice parameter, the thermal expansion coefficient, and MSDs with temperature. A comparison of the MSD runs simulated for the PbTe and mixed (Pb,Cd)Te crystal leads to the confirmation of recent findings that the cation displacements are little affected by Cd substitution at the Pb site; whereas the Te displacements are markedly higher for the mixed crystal. Moreover, information about static disorder caused by Cd substitution is obtained. The calculations provided two independent ways to determine the values of the overall Debye temperature, θ

_{D}. The resulting values differ only marginally, by no more than 1 K for PbTe and 7 K for Pb

_{0.884}Cd

_{0.116}Te crystals. The θ

_{D}values for the cationic and anionic sublattices were determined. The Grüneisen parameter is found to be nearly independent of temperature. The variations of unit-cell size with rising pressure (the NaCl structure of Pb

_{0.884}Cd

_{0.116}Te sample was conserved), modeled with the BMEOS, provided the dependencies of the bulk modulus, K, on pressure for both crystals. The K

_{0}value is 45.6(2.5) GPa for PbTe, whereas that for Pb

_{0.884}Cd

_{0.116}Te is significantly reduced, 33.5(2.8) GPa, showing that the lattice with fractional Cd substitution is less stiff than that of pure PbTe. The obtained experimental values of θ

_{D}and K

_{0}for Pb

_{0.884}Cd

_{0.116}Te are in line with the trends described in recently reported theoretical study for (Pb,Cd)Te mixed crystals.

## 1. Introduction

#### 1.1. General Issues

_{1−x}Cd

_{x}Te, which is one of most studied thermoelectric materials.

#### 1.2. Cationic Substitutions

#### 1.3. PbX-CdX (X = Te, Se, S) Solid Solution

_{1−x}Cd

_{x}Te solid solution of the rocksalt type [12,13,14,15]. Therefore, this material attracts researchers’ attention, being suitable for design of thermoelectric devices. Metastable single-phase NaCl-type crystals have been reported with maximal Cd content depending on the preparation conditions, particularly on applied quenching conditions from high temperature or high pressure or on the annealing method.

_{1−x}Cd

_{x}Te decreases with the CdTe addition, as demonstrated experimentally [16,17,18,19] and theoretically [7]. The behavior is similar for PbSe and PbS matrices with the respective addition of CdSe and CdS. The linearity of lattice parameter variation in these three systems is illustrated in Figure 1, whereas the equations describing the lines are given in Table 1 (on the basis of the above references for PbTe, refs. [20,21,22,23] for PbSe and refs. [24,25,26,27] for PbS. (Some additional information is available in a study of the quaternary system [28].) The lines in Figure 1, representing the Vegard’s rule (linear behavior of lattice parameter) are based on cited studies. The highest reported content, x

_{max}, is shown for each system. Extrapolation to x = 1 gives the lattice parameters of hypothetical cadmium chalcogenides with a NaCl structure. The values of these parameters, of coefficients of Vegard’s equation (in one case, of the equivalent Zen’s equation), and the highest reported Cd content as well as the results of extrapolation are collected in Table 1, including the data from refs. [7,16,18,19,20,21,22,23,25,26,27].

#### 1.4. Knowledge on Thermostructural and Elastic Properties for PbTe and Pb_{1−x}Cd_{x}Te

_{1−x}Cd

_{x}Te, see refs. [66,68] (details are provided in Table 5).

#### 1.5. Pb-Te and PbTe-CdTe System

_{max}= 1197 K [70] (see refs. [71,72])

**.**The off-stoichiometry range for PbTe is extremely narrow [70,73,74,75,76].

_{1−x}Cd

_{x}Te solid solution of the rocksalt type can be prepared in a metastable form. A diffraction study on the structure of Pb

_{1−x}Cd

_{x}Te as a function of temperature has shown the decomposition process of metastable Pb

_{1−x}Cd

_{x}Te (x = 0.096) during heating [43]; these results led to evaluation of the maximum achievable Cd content in the metastable solid solution [13]. Some theoretical calculations based on first principles have been presented in ref. [68].

#### 1.6. Aim

_{0.884}Cd

_{0.116}Te. The literature data for PbTe are reviewed and taken into account in the comparative analysis of properties of these two crystals.

## 2. Materials and Methods

_{1−x}Cd

_{x}Te single crystals were obtained by the self-selecting vapor growth (SSVG) method described in refs. [88,89]. High purity polycrystalline PbTe and CdTe compounds were used as reaction components. The conditions were similar to those used in earlier work [19]. To produce the PbTe–CdTe solid solution, the synthesis was performed using a mixture of PbTe and CdTe enclosed in a sealed quartz ampoule located in a furnace with a gradient of about 1 deg/cm at a temperature of about 850℃. The process of growth of homogeneous (Pb,Cd)Te crystals lasted two weeks. Further details of the growth procedure can be found in refs. [19,90]. The Cd content, x = 0.116, was derived from the a(T) dependence reported in ref. [19].

_{1−x}Cd

_{x}Te. The in-situ low-temperature measurements were performed using synchrotron X-ray powder diffraction [92] at HASYLAB, Hamburg. The Debye–Scherrer geometry with monochromatic radiation (λ = 0.5385 Å) and an image plate detector [93] were applied. The incident beam size was 1 × 15 mm

^{2}. The measurements were performed in the 2θ range of 7–58° (corresponding d-spacing range is 4.410–0.555 Å), and for samples mounted in glass capillaries (Hilgenberg) of 0.3 mm diameter, the X-ray powder diffraction patterns were recorded with a 0.004° (2θ) step.

_{1−x}Cd

_{x}Te crystals and fine diamond powder (Sigma–Aldrich #48,359-1 synthetic powder), of ~1 μm monocrystalline grain size and purity of 99.9%). Addition of diamond powder served for both, (i) a diluent and (ii) an internal diffraction standard, avoiding the possible influence of wavelength instabilities (the use of such a standard has been proposed in ref. [94]). Low-temperature conditions (temperature range 15–300 K) were ensured by a closed-circuit He-cryostat. For the structural analysis, the Rietveld method [95,96] was applied using the refinement program, Fullprof.2k(v.7) [97]. In calculations, the pseudo-Voigt profile-shape function was assumed. The following parameters were refined: scale factor, lattice parameters, isotropic mean square displacement parameters, peak shape parameters, and systematic line-shift parameter. The background was set manually.

_{2}O (16:3:1) mixture as the pressure-transmitting medium. The pressure was calibrated with a Photon Control spectrometer by the ruby-fluorescence method [99], assuring a precision of 0.02 GPa. The experiments were conducted at a temperature of 296 K. High-pressure single-crystal X-ray diffraction data were collected at a four-circle KUMA X-ray diffractometer equipped with a graphite monochromator for the applied MoKα radiation. The gasket shadowing method was used for crystal centering and data collection [100]. The size of the diamond culets was 0.7 mm, the size of crystal for PbTe was 0.2 × 0.05 × 0.15 mm

^{3}, for Pb

_{0.884}Cd

_{0.116}Te was 0.23 × 0.05 × 0.17 mm

^{3}(only one crystal was loaded into DAC). UB-matrix determinations and data reductions were performed with the program CrysAlisPro [101]. The structures were solved by direct methods using the program ShelXS and refined by full-matrix least-squares on F

^{2}using the program ShelXT incorporated in Olex2 [102,103]. For high-pressure data analysis, the fitting procedures were conducted with the EoSFit7 program [104,105].

## 3. Results: Thermostructural and Elastic Properties of PbTe and Pb_{0.884}Cd_{0.116}Te

#### 3.1. Effect of Cd Substitution on Temperature Variation of Unit Cell Size, Thermal Expansion Coefficient and Cationic and Anionic Mean Square Displacements

#### 3.1.1. General Issues

_{1−x}Cd

_{x}Te (x = 0.116) in the 15–300 K temperature range and separately, in 0.1 MPa–4.5 GPa pressure range. Consequently, the properties measured at the same conditions for each of two crystals could be analyzed, leading to the understanding of the effect of Cd substitution on the crystal characteristics.

^{2}>(T) experimental variations, completed by the V(p) study. For the Pb

_{1−x}Cd

_{x}Te system, the detailed investigations at non-ambient temperature and pressure have been almost completely lacking.

_{0.884}Cd

_{0.116}Te by the Rietveld method yielded direct information on (i) the temperature dependencies of unit-cell size, a(T), and (ii) mean square displacements, <u

^{2}>(T), of both, cations and anions. Illustrative examples of structure refinement plots for PbTe and Pb

_{1−x}Cd

_{x}Te at temperatures 15 K and 300 K are given in Appendix A (Figure A1). Subsequent analysis of the a(T) data led to the derivation of the temperature variation of the thermal expansion coefficient, α(T). The modeling of the temperature variations of the studied quantities allowed for independent determination also of other properties, for both the cationic and anionic sublattices of studied crystals.

#### 3.1.2. Variation of Unit Cell Size of PbTe and Pb_{0.884}Cd_{0.116}Te with Temperature

_{B}= Boltzmann constant, θ

_{D}is the characteristic Debye temperature. The parameters Q, ${V}_{\left(T=0\right)}$ and b are obtained through fitting of experimental V(T) data modeled by Equation (1) (refined parameters are quoted in Appendix C, Table A5). For PbTe, the lattice parameter increases by 0.50% over the whole temperature range. The run of the a(T) (see inset in Figure 2b) is marginally different from the recent experimental data obtained in a wide temperature range (10–500 K) by neutron powder diffraction [29], and in the 105–300 K range by X-ray powder diffraction [31] (Figure 2a).

^{−4}Å). As for the value at 300 K, our result of data fitting is 6.46148(87) Å. It agrees perfectly with the average of the high quality records for PbTe stored in the ICSD database [26] (the quality is based on ICSD-staff evaluation). There are five such records; their a(293 K) values are 6.462(1) Å, 6.459(1) Å, 6.461(1) Å, 6.461(1) Å, and 6.460(1) Å; the average is 6.46060(15) Å. After temperature correction from 293 to 300 K the average increases by 0.00088 Å (based on present a(T) results) leading to the ICSD derived value at 300 K to be 6.46148(15) Å. This value is identical to the above-quoted present one. All these perfect agreements point out both, the high quality of the sample and precision of applied measurement approach, including the instrument calibration. This observation can justify recommendation of the present a(T) run as a reference for the PbTe lattice parameter as a function of temperature; particularly in the near-RT temperatures, through interpolation of the data of Table A2 (Appendix B). The recommended a(300 K) value at 300 K is 6.46148(87) Å (thsi result is quoted together with other ones in Table 7).

_{0.884}Cd

_{0.116}Te in the whole temperature range is reduced in respect to that of pure PbTe, and the reduction across the whole range is 0.53%, which is apparently larger than the value of 0.50% quoted at the beginning of this section for PbTe.

#### 3.1.3. Variation of Thermal Expansion Coefficient with Temperature

_{0.884}Cd

_{0.116}Te, the lattice parameter, according to the fitted model, increases from 6.37725(6) Å to 6.41133(116) Å. The difference in respect to PbTe in the slope of the cell-size dependence on temperature is visualized in Figure 2a, presenting the temperature variation of the cell volume for both studied crystals. The experimental dependence of the linear-thermal-expansion coefficient on temperature, α(T), was derived from the V(T) Grüneisen approximation (Equation (1)), using the equation:

_{V}(T)/3 = (dV/dT)/(3V(T))

^{−1}obtained in this work at 300 K matches very well the earlier reported experimental values of 19.80 MK

^{−1}[51] and 19.91 MK

^{−1}[29] (cf. Table 8) (this discrepancy is as low as 1.5%).

**.**The agreement is visualized through the difference curve, and it is worth noting that the little bump of 2% height, observed at this curve would be twice as small if the temperature axis of the theoretical curve was shifted by only −0.7 K. The consistency with other theoretical data is not as perfect, but the trends of these results are generally compatible with the experiments described herein and others. In particular, the present data marginally differ from theoretical ones ref. [46] up to 100 K, whereas the discrepancy markedly increases at higher temperatures.

_{0.884}Cd

_{0.116}Te, the increase of the thermal expansion coefficient, α(T), in the studied temperature range is more pronounced than that observed for PbTe (Figure 3b). At 300 K, the coefficient reaches the value of 20.7(8) MK

^{−1}, the increase in respect to PbTe being about 6.5% at this temperature (the rise is comparable at lower temperatures).

#### 3.1.4. Variation of Mean Square Displacements with Temperature

_{0.884}Cd

_{0.116}Te display an apparent monotonically increasing behavior with rising temperature (Figure 4).

^{2}>(T) = <u

^{2}>

_{dyn}(T) + <u

^{2}>

_{stat}

^{2}>

_{stat}in the same way as that used in ref. [110]. The first term at the right side, <u

^{2}>

_{dyn}(T), is the Debye function based on simplifying the assumption that takes into account the acoustic branches, whereas the optical branches are ignored:

_{D}for the Debye temperature, k

_{B}for the Boltzmann constant, and ħ for the reduced Planck constant. The second term in Equation (4), <u

^{2}>

_{stat}is an empirical term attributed to the temperature-independent static disorder that can be connected in unsubstituted crystals, e.g., with the presence of point defects [111] (the presence of such defects is known to influence the electrical and other properties of thermoelectric crystals [112]), and in substituted crystals-with the presence of foreign atoms at the cationic or anionic sites.

_{0.884}Cd

_{0.116}Te, the results of fitting of <u

^{2}>(T) defined by Equation (4) correctly describe the run of experimental points (the refined parameters of the model are provided in Table A5). The MSD values referring to temperatures near 0 K and near 300 K are compared with literature data in Table 9 (for values of fitted MSDs see Table A1, Appendix B).

^{2}>(T) curve shows (i) a characteristic nearly linear dependence at high temperatures, having a specific slope, and with (ii) a curvilinear behavior at low temperatures, characterized by a value of <u

^{2}>(T = 0). Each of these features has its own meaning. The given curve representing either the cationic or anionic site has its own characteristics determined by the fixed material parameter m, by the Debye temperature, θ

_{D}, and by the disorder term, <u

^{2}>

_{stat}. Basically, <u

^{2}>

_{stat}and θ

_{D}are fittable parameters, and m could also be fitted if the composition was not well specified.

- (1)
- The fitted <u
^{2}>_{stat}(T) curves for PbTe and Pb_{0.884}Cd_{0.116}Te behave differently. Namely:- (a)
- The MSDs at 0 K, <u
^{2}>(T = 0), increase significantly (by about 0.002–0.004 Å^{2}) with x rising from 0 to 0.116. We attribute this increase to the appearance of the static disorder expressed by the nonzero <u^{2}>_{stat}term resulting from fitting Equation (4) (the values of <u^{2}>_{stat}are quoted in Table 9). This effect is graphically presented in the insets of Figure 4a,b, where the variation of fitted <u^{2}>_{stat}values is displayed. Appearance of marginally small negative fitted value for anionic site in PbTe (instead of zero that represents the lack of disorder) is attributed to be the effect of imperfections of fitted <u^{2}>(T) data points. The quoted values (Table 9) show that the disorder in the anionic sublattice is considerably higher than that at the cationic site. Summarizing, an increase of the static disorder term, <u^{2}>_{stat}, in Equation (4), from approximately zero to a value of the order of 3 × 10^{−3}Å^{2}is observed for the mixed crystal in respect to PbTe crystal. Namely, the rise is from 0.38(4) × 10^{−3}Å^{2}to 2.03(6) × 10^{−3}Å^{2}for cations, and from −0.54(7) × 10^{−3}Å^{2}(a value marginally different from zero) to 3.4(1) × 10^{−3}Å^{2}for anions. - (b)
- At higher temperatures, the cationic MSDs are nearly equal for the two crystals, whereas the anionic ones differ markedly in the whole temperature range.
- (c)
- The slope of the cationic <u
^{2}>(T) curve decreases with rising x, whereas the anionic one apparently increases. The property of Equation (4) is that the slope of <u^{2}>(T) is governed at high temperatures by the Debye temperature (for high slope the Debye temperature is low and vice versa; the corresponding θ_{D}values are discussed in detail in Section 3.3 and Section 4).

- (2)
- The MSDs for the cationic and anionic sites behave differently for x = 0 than for x = 0.116.
- (3)
- Comparison of Figure 4a,b shows that the cationic and anionic MSDs of Pb
_{0.884}Cd_{0.116}Te are of comparable values in a broad temperature range. As this effect must depend on x, we expect that for x < 0.116, the <u^{2}> values of anions are lower than those of cations, whereas for x > 0.116 (if the structure is stabilized), the anionic ones are higher.

^{2}>(T) runs can be connected with differences in the defect structure of studied single crystals and polycrystals.

#### 3.2. Effect of Substitution of Cd in the PbTe Lattice on Variation of Unit-Cell Size and of Bulk Modulus with Pressure

_{0.884}Cd

_{0.116}Te single crystals at ambient conditions (T = 295 K and p = 0.1 MPa) was conserved at the applied high-pressure conditions. The structure refinement yielded the lattice parameter monotonically varying with increasing pressure (for values see Table A3 in Appendix B).

_{0}is the bulk modulus, and K′ is the pressure derivative of the bulk modulus, ${f}_{E}=[{({V}_{0}/V)}^{2/3}-1]/2$ is the Eulerian strain (V is the volume under pressure p, and V

_{0}is the reference volume). When K′ = 4, Equation (6) is reduced to a simpler, second order equation, applied in the present study (an equation of the second order has also been used in a recently reported experimental diffraction study of PbTe [83]).

_{0}is 45.6(2.5) GPa, which is consistent with previously reported values, in particular with those obtained from X-ray diffraction studies, 38.9 GPa [78,80] and 44(1) GPa [82] as well as with those from early ultrasonic wave velocity measurements of refs. [29,52,63,113], quoted in Table 10.

_{0.884}Cd

_{0.116}Te is found to be 33.5(2.8) GPa, providing the first experimental evidence that Cd substitution reduces the stiffness of the PbTe matrix. For both crystals, bulk modulus increases with pressure, in the range from 0.1 MPa to 4.5 GPa by about 50% (Figure 6, for numerical data see Table A4). For PbTe, the K(p) dependence is in line with the theoretical one reported in ref. [64].

#### 3.3. Effect of Cd Substitution on Values of Debye Temperature

^{2}>(T) and V(p), namely the V(T) variations using the second-order Grüneisen approximation (Equation (1)), the <u

^{2}>(T) variation involving the Debye expression (Equation (4)), and the V(p) variations using the BMEOS (Equation (6)) led to determination of the Debye temperature, θ

_{D}. In general, θ

_{D}is frequently considered as a quantity depending on temperature, but for PbTe, the reported θ

_{D}variations are weak and are observed mostly at cryogenic temperatures [53,61]. In most studies, including those based on diffraction, θ

_{D}is considered a temperature-independent quantity. For compounds of the NaCl structure, different θ

_{D}values are reported for the cation and anion sublattices. Such distinction is possible thanks to fitting of atomic displacements of the given (cationic or anionic) sublattice using Equation (4). Consequently, from the given experiment, we get a single overall θ

_{D}value from fitting V(T) and a pair of θ

_{D}’s from fitting of <u

_{C}

^{2}>(T) and <u

_{A}

^{2}>(T) (the corresponding symbols θ

_{DV}, θ

_{DUC}, and θ

_{DUA}are used here, respectively, to highlight the distinction between these three θ

_{D}definitions), whereas the overall θ

_{DU}denotes the average of θ

_{DUC}and θ

_{DUA}.

_{D}values for PbTe (θ

_{DV}and θ

_{DU}) are almost identical (135.2(3.8) K and 135.9(7) K; the average is ~135.5 K). The Debye temperature for cation and anion sublattices in PbTe is θ

_{DUC}= 102.8(3) K and θ

_{DUA}= 169(1) K. For PbTe, there are a number of articles reporting the Debye temperature values. Selected literature data are collected in Table 11 (experimental X-ray diffraction and neutron diffraction based data from refs. [29,31,41,55,56]). Non-diffraction-based experimental data quoted in Table 12 are taken from refs. [13,52,61,63,126,127,128,129,130,131,132]). For theoretical data, see Table 13 providing the values from refs. [9,29,47,57,133,134]).

_{DU}and θ

_{DV}values for the PbTe sublattices are in line with those determined in ref. [29] by both neutron powder and single crystal X-ray diffraction.

_{D}changes appearing with Cd substitution is indicated by θ

_{DV}reduction by 5.1 K. A small reduction of Debye temperature for Pb

_{0.884}Cd

_{0.116}Te (θ

_{DV}= 130.1(4.4) K) in comparison with that for PbTe, θ

_{DV}= 135.2(3.8) K, is observed.

_{D}values are also in line with the trends observed for those obtained by the non-diffraction methods in Table 12 (their average calculated for room-temperature data is 138.1 K, i.e., only 3 K larger than our value. The theoretical methods provided overall values with a higher average (Table 13) of 157.9 K, these data vary in an extended range.

_{DUC}= 99.6(2) K and θ

_{DUA}= 156.0(5) K [29] (the discrepancy is less than 8%). The results collected in Table 11 (the present one and those reported earlier) document that the cationic values determined in different laboratories are in very good agreement (between 95.5 and 102.8 K), whereas those for anions exhibit a larger scatter between 127 and 169 K. The θ

_{DUC}and θ

_{DUA}behave in an opposite way (the former rises, the latter decreases). Interestingly, the contribution of lighter Cd atoms at the Pb sites leads to a reduction of the difference between the cationic and anionic site from 66.2 K for the pure PbTe to 42.9 K in the mixed crystal.

_{0}(T)V

_{m}(T)/c

_{v}(T)

_{0}is the bulk modulus, and c

_{v}describes the isochoric heat capacity. In calculations, the α(T) and V

_{m}(T) based on experimental results obtained in this work were used. The K

_{0}(T) variation reported in ref. [29] for PbTe was rescaled to the present K

_{0}at room temperature equal 45.6(2.5) GPa for the PbTe sample and to 33.5(2.8) GPa for the Pb

_{0.884}Cd

_{0.116}Te sample (for Pb

_{0.884}Cd

_{0.116}Te we adopted the rescaled K

_{0}(T) dependence of PbTe of ref. [29]). For PbTe, the temperature variation of the molar isochoric capacity c

_{v}(T) was taken from ref. [29], whereas for the Cd substituted sample the theoretical c

_{v}(T) data of Pb

_{0.88}Cd

_{0.12}Te [68] were used. The dependencies obtained in this way are shown in Figure 7.

_{v}, therefore the reduction of γ below ~50 K displayed in Figure 7 may be questioned.

## 4. Discussion

_{0.884}Cd

_{0.116}Te solid solution, described in Section 3, are derived from X-ray diffraction data through fitting of Equations (1), (4) and (6). Temperature dependencies of the lattice parameter, a(T), the thermal expansion coefficient, α(T), and the mean square displacements, <u

^{2}>(T), are determined for both crystals from X-ray diffraction powder diffraction data. These results for PbTe are consistent with recent literature data, in particular with the most detailed ones [29,31]. Moreover, the diffraction study of the equation of state, V(p), provided the value of the PbTe bulk modulus dependence on pressure. The reliability of the present results is verified by the demonstrated close agreement of the a(T), α(T) and <u

^{2}>(T) dependencies, as well as of the Debye temperature and bulk modulus variation, for PbTe with earlier experimental and theoretical data. It is also worth noting that the fitted model curves for a(T), <u

^{2}>(T) and V(p) dependencies match well the experimental points, therefore we do not expect occurrence of significant systematic errors which could add to the statistical errors quoted in Table A3 and Table A4.

_{0.884}Cd

_{0.116}Te, the obtained results are novel, they describe the thermal characteristics of this crystal and indicate the direction and magnitude of variation of the considered temperature-dependent properties with rising content of Cd at the cationic site. In other words, the earlier unknown effect of sharing the cationic sites by Pb and Cd atoms on thermal properties is revealed.

_{0.884}Cd

_{0.116}Te sample shows a similar behavior with temperature. The a(T) run for Cd-substituted PbTe crystal depends on the amount of substituent (as can be deduced from a comparison with earlier results for lower Cd content [42,43,44]). A related influence of substituent on the a(T) runs is observed for Na and Eu substituted PbTe crystals [41]. In the above-cited results, which refer to temperatures exceeding the room temperature, the deviations from regular behavior indicate the decomposition of a metastable mixed crystal.

^{−1}for PbTe and 20.7(8) MK

^{−1}for Pb

_{0.884}Cd

_{0.116}Te: thus, the expansion rises by 6.2% at this temperature.

_{0.884}Cd

_{0.116}Te, the cationic MSDs are comparable to those of PbTe, except in the region of the lowest temperatures. The Cd substitution causes apparent increase of the anionic MSDs. This increase is expected to be proportional to the Cd content.

^{2}>

_{stat}term describing the static disorder was determined, for both, cationic and anionic sublattices, in the unsubstituted and substituted crystal. As expected, the fitting for PbTe gave <u

^{2}>

_{stat}a value close to zero, thus indicating that there is no significant disorder in this crystal (small values have also been reported in refs. [29,31]). We believe that the differences between the, values reported for pure PbTe by different groups can probably be attributed to differences in the defects’ kind and density.

^{2}>

_{stat}term after incorporating Cd into the PbTe lattice proves that alloying causes appearance of substitutional disorder in the mixed Pb

_{0.884}Cd

_{0.116}Te. crystal. We observe (see the insets in Figure 4) that the values of cationic and anionic <u

^{2}>

_{stat}terms describing the static disorder are markedly different in the Pb

_{0.884}Cd

_{0.116}Te. crystal. Namely, the anionic disorder is significantly larger in this crystal.

_{0.884}Cd

_{0.116}Te, at pressures ranging up to 4.5 GPa. The observed pressure variation is in line with a theoretical result reported in ref. [64]. Modeling of the BMEOS led to determination of the bulk modulus and its pressure variation. At 0.1 MPa, the bulk modulus value is 45.6(2.5) for PbTe, well coinciding within error bars with the value 44(1) GPa reported in the most recent diffraction study [83]. The bulk modulus value significantly decreases with rising Cd content; in other words, the Cd substitution leads to a crystal of lower stiffness.

_{0}value is larger than the calculated value of 46.61 GPa [9] for pure PbTe.

_{0}has been predicted to decrease from 46.61 GPa for PbTe to 46.42 GPa for Pb

_{0.969}Cd

_{0.031}Te. This leads to 45.90 GPa evaluated through extrapolation for Pb

_{0.884}Cd

_{0.116}Te. This evaluation differs from the experimental value obtained in the present study (33.5(2.8) GPa), but the direction of changes of K

_{0}with x is clear.

_{0}values for PbTe substituted with any cation are not available, except for the case of Ba substitution, where the mixing effect on K

_{0}consists of a 5% reduction of the PbTe value [6]. The assumption that K

_{0}(PbTe) equals 46.61 GPa leads to an evaluation of a (not explicitly reported) experimental value, 44.3 GPa for Pb

_{0.96}Ba

_{0.04}Te. Extrapolation of the theoretical value of 44.99 GPa for Pb

_{0.969}Ba

_{0.031}Te quoted in ref. [9] leads to K

_{0}= 44.5 GPa for Pb

_{0.96}Ba

_{0.04}Te. The excellent agreement between the values of calculated 44.3 GPa and experimental 44.5 GPa points out the reliability of both, the cited experiment and the calculation.

^{2}>(T) models led to the determination of values of Debye temperature, θ

_{D}, for both crystals. Together with the Cd substitution, a small reduction of the overall Debye temperature, θ

_{DU}, from 135.2(3.8) K to 130.1(4.4) K (i.e., a reduction of 5.1 K) is observed. Theoretical calculations predict reduction by 2.4 K for the composition of x = 0.031 [9]. Extrapolating this result to the composition of the mixed crystal studied in this work, (x = 0.116) gives a prediction of a 9 K (difference between θ

_{D}= 187.8 1 K and 178.8 K quoted in Table 13) reduction of the theoretical overall θ

_{D}. This theoretical result supports the observed trend of reduction of overall Debye temperature by increasing the cadmium content. Interestingly, the θ

_{D}values reported by different authors for the cationic sublattice are in perfect agreement, whereas those for the anionic one are scattered. The influence of Cd substitution on Debye temperatures of cationic and anionic sublattices, described in Section 3.3, is not uniform; these values differ markedly for PbTe, but the difference is reduced for the Cd substituted crystal.

_{0.03}Eu

_{0.03}Pb

_{0.94}Te [112]. The joint cationic/anionic substitution (Pb,Cu)(Se,Te) system has also been studied, providing another example of a dual system [135]. Mixed bi-cationic–bi-anionic systems such as Na

_{0.03}Eu

_{0.03}Pb

_{0.94}Te

_{0.9}Se

_{0.1}[112] are the subject of studies as well. It is noteworthy that upon replacing the Te anion by Se or S, the bond ionicity decreases (for the ionicity scale, see ref. [136]). Along the PbTe, PbSe, PbS series, some of the thermostructural/elastic properties (studied here for PbTe) vary monotonically; for example, the lattice parameter (see Figure 1 at x = 0), bulk modulus [46,85,114] and phase transition pressure [77,85].

## 5. Conclusions

_{0.884}Cd

_{0.116}Te, from X-ray diffraction data collected at varying temperature and pressure.

^{2}>(T), are determined for both crystals. For PbTe, these results and thermal expansion are fully consistent with results of earlier X-ray diffraction, neutron diffraction, dilatometric and other experimental studies, as well as with those of multiple theoretical investigations, and this agreement supports the reliability of the data collected.

_{0.884}Cd

_{0.116}Te crystal determined in the present study indicate the direction and magnitude of variation of the characteristics of Pb

_{1−x}Cd

_{x}Te system with rising x. The stiffness of the alloy is smaller than that of pure PbTe, the thermal expansion is larger throughout the whole temperature range, and the atomic mean-square displacements change with Cd substitution in a complex way, indicating (i) opposite variations of the Debye temperatures for both sublattices, as well as (ii) the appearance of substitutional disorder in the mixed crystal.

_{0.884}Cd

_{0.116}Te; such data are not yet available for alloys of the Pb

_{1−x}Cd

_{x}Te system. The obtained results show a consistent image of influence of the partial substitution of Pb ions by Cd ions, in the PbTe lattice, on the thermostructural properties. Namely, the obtained results show how the lattice parameter, the thermal expansion coefficient, the atomic mean-square displacements and other thermostructural properties (compressibility, Debye temperature, Grüneisen parameter and others) depend on the cadmium content. In particular it was found, that the Pb

_{0.884}Cd

_{0.116}Te lattice is less stiff than that of PbTe, whereas thermal expansion of the mixed crystal is discernibly larger. The described extension of the knowledge on the studied properties is expected to be profitable in a further work on the application of the fractionally substituted Cd lead telluride.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

APW | augmented plane-wave |

BMEOS | Birch-Murnaghan Equation of State |

CDM | crystal dynamics models |

CM | calorimetry |

DFPT | density functional perturbation theory |

DFT | density functional theory |

DM | dilatometry |

DPS | double parton scattering (nuclear inelastic scattering) |

EC | elastic constants |

FP | full potential |

FPBTF | first principles Boltzmann transport framework |

GGA | generalized gradient approximation |

GULP | computer program for the symmetry adapted simulation of solids, authored by Julian D. Gale |

HPM | heat-pulse method |

HS | hydrostatic conditions |

HSEsolSOC | revised Heyd-Scuseria-Ernzerhof functional + spin-orbit coupling |

LAPW | linearized augmented plane-wave |

LDA | local density approximation |

LDY | lattice dynamics calculations |

LEDPXRD | laboratory energy-dispersive X-ray diffraction |

LKF | Lin-Kleinman formalism |

LSCXRD | laboratory single-crystal X-ray diffraction |

MD | molecular dynamics |

MSD | mean square displacement |

n.a. | not available |

ND | neutron diffraction |

NNI | nearest-neighbor interaction model by Kagan and Maslow |

NPD | neutron powder diffraction |

NS | neutron scattering |

PAW | projector augmented wave method |

PBE | Perdew-Bucke-Ernzerhof exchange-correlation functional |

PBEsol | Perdew–Bucke–Ernzerhof exchange-correlation functional revised for solids |

PM | the paramagnetic resonance. θ_{D} estimated by the temperature-dependent hyperfine splitting constant A(T) |

PTW | plane temperature waves method |

QHA | quasiharmonic approximation |

QHS | quasi-hydrostatic conditions |

RT | room temperature |

SCXRD | single crystal X-ray diffraction |

SCXRDS | single crystal X-ray difraction at synchrotron |

SO | soft-constraint based online |

SME | slave mode expansion |

SP | spectroscopy |

SPXRD | synchrotron powder X-ray diffraction |

SV | sound velocity method |

TC | thermal conductivity |

TEC | thermal expansion coefficient |

THD | thermodynamic calculations |

UPE | ultrasonic pulse-echo method |

UIM | ultrasonic interferometry |

UWV | ultrasonic wave velocity |

UWVSC | ultrasonic wave velocity in single crystal |

XRD/ND/PDF | X-ray diffraction and neutron diffraction, analyzed with pair distribution function (PDF) method |

## Appendix A

**Figure A1.**Rietveld refinement results for PbTe (

**upper row**) and Pb

_{0.884}Cd

_{0.116}Te (

**bottom row**) at 15 K and 300 K. The experimental points are indicated by dots and the calculated patterns by the solid line. The positions of Bragg reflections of the sample are indicated as short vertical bars at the bottom (

**upper row**), and those of the diamond calibrant (

**lower row**). The difference patterns are displayed at the bottom of each subfigure.

## Appendix B

**Table A1.**Present a(T) and <u

^{2}>(T), obtained from Rietveld refinements for PbTe and Pb

_{0.884}Cd

_{0.116}Te.

T [K] | PbTe | Pb_{0.884}Cd_{0.116}Te | ||||
---|---|---|---|---|---|---|

a [Å] | <u_{C}^{2}> [Å^{2}] | <u_{A}^{2}> [Å^{2}] | a [Å] | <u_{C}^{2}> [Å^{2}] | <u_{A}^{2}> [Å^{2}] | |

15 | 6.4298(4) | 0.0020(1) | 0.0016(2) | 6.3775(5) | 0.0041(2) | 0.0057(3) |

20 | 6.4299(3) | 0.0027(1) | 0.0018(2) | 6.3773(4) | 0.0041(2) | 0.0055(3) |

25 | 6.4301(3) | 0.0024(1) | 0.0018(2) | 6.3776(5) | 0.0047(2) | 0.0055(3) |

30 | 6.4301(4) | 0.0028(1) | 0.0016(2) | 6.3777(5) | 0.0046(2) | 0.0052(3) |

35 | 6.4305(4) | 0.0030(1) | 0.0018(2) | 6.3783(5) | 0.0047(2) | 0.0055(3) |

40 | 6.4310(4) | 0.0037(1) | 0.0010(2) | 6.3787(7) | 0.0046(2) | 0.0059(3) |

45 | 6.4312(3) | 0.0036(1) | 0.0019(2) | 6.3792(6) | 0.0053(2) | 0.0058(3) |

50 | 6.4316(3) | 0.0038(1) | 0.0021(2) | 6.3793(6) | 0.0054(2) | 0.0060(3) |

55 | 6.4323(3) | 0.0044(1) | 0.0027(2) | 6.3800(7) | 0.0053(2) | 0.0066(3) |

60 | 6.4327(3) | 0.0047(1) | 0.0021(2) | 6.3805(5) | 0.0047(2) | 0.0067(3) |

65 | 6.4333(3) | 0.0052(1) | 0.0024(2) | 6.3810(7) | 0.0054(2) | 0.0064(3) |

70 | 6.4338(3) | 0.0059(1) | 0.0026(2) | 6.3817(6) | 0.0056(2) | 0.0071(3) |

75 | 6.4342(5) | 0.0057(2) | 0.0029(2) | 6.3823(5) | 0.0072(2) | 0.0072(3) |

80 | 6.4347(5) | 0.0058(2) | 0.0025(2) | 6.3829(6) | 0.0071(2) | 0.0074(3) |

85 | 6.4350(3) | 0.0062(1) | 0.0034(2) | 6.3833(5) | 0.0069(2) | 0.0084(3) |

90 | 6.4358(4) | 0.0064(2) | 0.0034(2) | 6.3840(4) | 0.0078(2) | 0.0077(3) |

95 | 6.4364(5) | 0.0072(2) | 0.0035(3) | 6.3848(4) | 0.0075(2) | 0.0087(3) |

100 | 6.4371(4) | 0.0074(2) | 0.0026(3) | 6.3855(4) | 0.0079(2) | 0.0089(3) |

110 | 6.4381(3) | 0.0083(2) | 0.0034(2) | 6.3863(5) | 0.0085(2) | 0.0079(4) |

120 | 6.4394(3) | 0.0092(2) | 0.0043(3) | 6.3875(4) | 0.0094(2) | 0.0092(4) |

130 | 6.4408(4) | 0.0095(2) | 0.0044(3) | 6.3891(4) | 0.0095(2) | 0.0094(3) |

140 | 6.4418(4) | 0.0100(2) | 0.0060(3) | 6.3903(4) | 0.0099(2) | 0.0097(4) |

150 | 6.4429(3) | 0.0111(2) | 0.0057(3) | 6.3915(4) | 0.0102(2) | 0.0104(4) |

160 | 6.4440(3) | 0.0117(2) | 0.0059(3) | 6.3931(5) | 0.0115(2) | 0.0115(4) |

170 | 6.4451(3) | 0.0120(2) | 0.0069(3) | 6.3941(5) | 0.0108(2) | 0.0129(4) |

180 | 6.4466(3) | 0.0118(2) | 0.0064(3) | 6.3953(5) | 0.0117(2) | 0.0103(4) |

190 | 6.4477(3) | 0.0133(2) | 0.0073(3) | 6.3969(5) | 0.0121(2) | 0.0129(4) |

200 | 6.4491(4) | 0.0144(2) | 0.0072(3) | 6.3981(5) | 0.0127(2) | 0.0132(4) |

210 | 6.4503(3) | 0.0147(2) | 0.0070(3) | 6.3997(4) | 0.0132(2) | 0.0131(4) |

220 | 6.4514(4) | 0.0142(2) | 0.0097(3) | 6.4007(5) | 0.0137(3) | 0.0138(4) |

230 | 6.4527(3) | 0.0167(2) | 0.0085(3) | 6.4020(4) | 0.0156(3) | 0.0137(4) |

240 | 6.4539(3) | 0.0170(3) | 0.0088(3) | 6.4035(3) | 0.0170(3) | 0.0135(4) |

250 | 6.4552(3) | 0.0166(2) | 0.0096(3) | 6.4045(3) | 0.0173(3) | 0.0153(4) |

260 | 6.4564(3) | 0.0171(2) | 0.0094(3) | 6.4062(3) | 0.0160(3) | 0.0148(4) |

270 | 6.4578(3) | 0.0180(3) | 0.0109(4) | 6.4071(3) | 0.0192(3) | 0.0160(5) |

280 | 6.4588(3) | 0.0192(3) | 0.0111(4) | 6.4086(4) | 0.0171(3) | 0.0165(5) |

290 | 6.4602(3) | 0.0178(3) | 0.0123(4) | 6.4100(4) | 0.0188(3) | 0.0176(5) |

300 | 6.4616(3) | 0.0195(3) | 0.0124(4) | 6.4116(4) | 0.0200(3) | 0.0166(5) |

**Table A2.**Present values of a(T), α(T) and <u

^{2}>(T), modeled by Equations (1) and (2) for PbTe and Pb

_{0.884}Cd

_{0.116}Te.

T [K] | PbTe | Pb_{0.884}Cd_{0.116}Te | ||||||
---|---|---|---|---|---|---|---|---|

a [Å] | α [MK^{−1}] | <u_{C}^{2}> [Å^{2}] | <u_{A}^{2}> [Å^{2}] | a [Å] | α [MK^{−1}] | <u_{C}^{2}> [Å^{2}] | <u_{A}^{2}> [Å^{2}] | |

0 | 6.42972(5) | 0 | 0.0021(1) | 0.0011(1) | 6.37725(6) | 0 | 0.0036(1) | 0.0052(1) |

10 | 6.42973(5) | 0.70(7) | 0.0022(1) | 0.0012(1) | 6.37726(6) | 0.9(1) | 0.0037(1) | 0.0053(1) |

20 | 6.42986(6) | 4.2(3) | 0.0025(1) | 0.0013(1) | 6.37742(7) | 5.0(4) | 0.0040(1) | 0.0054(1) |

30 | 6.43026(9) | 8.5(4) | 0.0030(1) | 0.0015(1) | 6.37788(11) | 9.8(6) | 0.0043(1) | 0.0056(1) |

40 | 6.43091(12) | 11.7(4) | 0.0035(1) | 0.0017(1) | 6.37861(15) | 13.2(6) | 0.0048(1) | 0.0059(1) |

50 | 6.43173(15) | 13.9(4) | 0.0041(1) | 0.0020(1) | 6.37952(18) | 15.4(6) | 0.0052(1) | 0.0062(1) |

60 | 6.43266(17) | 15.3(4) | 0.0047(1) | 0.0023(1) | 6.38054(22) | 16.9(5) | 0.0057(1) | 0.0066(1) |

70 | 6.43367(20) | 16.2(4) | 0.0053(1) | 0.0027(1) | 6.38165(25) | 17.8(5) | 0.0062(1) | 0.0070(2) |

80 | 6.43474(22) | 16.9(4) | 0.0059(1) | 0.0030(1) | 6.38281(28) | 18.5(5) | 0.0068(1) | 0.0074(2) |

90 | 6.43584(24) | 17.4(4) | 0.0066(1) | 0.0033(1) | 6.38400(31) | 19.0(5) | 0.0072(2) | 0.0079(2) |

100 | 6.43697(27) | 17.8(4) | 0.0072(1) | 0.0037(1) | 6.38523(35) | 19.3(5) | 0.0078(2) | 0.0083(2) |

110 | 6.43812(29) | 18.0(4) | 0.0079(1) | 0.0041(1) | 6.38647(38) | 19.6(5) | 0.0084(2) | 0.0087(2) |

120 | 6.43929(31) | 18.3(4) | 0.0085(1) | 0.0045(1) | 6.38773(41) | 19.8(5) | 0.0089(2) | 0.0091(2) |

130 | 6.44047(34) | 18.5(4) | 0.0092(1) | 0.0049(1) | 6.38900(44) | 20.0(5) | 0.0095(2) | 0.0096(2) |

140 | 6.44167(36) | 18.6(4) | 0.0098(1) | 0.0052(2) | 6.39028(47) | 20.1(5) | 0.0100(2) | 0.0100(2) |

150 | 6.44287(39) | 18.7(4) | 0.0105(1) | 0.0056(2) | 6.39157(51) | 20.2(5) | 0.0106(2) | 0.0105(2) |

160 | 6.44408(41) | 18.8(4) | 0.0111(1) | 0.0060(2) | 6.39287(54) | 20.3(6) | 0.0111(2) | 0.0109(2) |

170 | 6.44529(44) | 18.9(4) | 0.0118(1) | 0.0064(2) | 6.39417(58) | 20.4(6) | 0.0117(2) | 0.0113(2) |

180 | 6.44652(47) | 19.0(4) | 0.0125(1) | 0.0068(2) | 6.39547(62) | 20.4(6) | 0.0122(3) | 0.0118(2) |

190 | 6.44774(50) | 19.1(4) | 0.0131(1) | 0.0072(2) | 6.39678(66) | 20.5(6) | 0.0128(3) | 0.0122(2) |

200 | 6.44898(53) | 19.1(5) | 0.0138(1) | 0.0076(2) | 6.39809(70) | 20.5(6) | 0.0133(3) | 0.0127(3) |

210 | 6.45021(56) | 19.2(5) | 0.0144(1) | 0.0080(2) | 6.39941(74) | 20.6(6) | 0.0139(3) | 0.0131(3) |

220 | 6.45145(59) | 19.3(5) | 0.0151(1) | 0.0084(2) | 6.40073(78) | 20.6(7) | 0.0144(3) | 0.0136(3) |

230 | 6.45270(62) | 19.3(5) | 0.0158(1) | 0.0088(2) | 6.40205(82) | 20.6(7) | 0.0150(3) | 0.0140(3) |

240 | 6.45394(65) | 19.3(5) | 0.0164(1) | 0.0091(2) | 6.40337(87) | 20.7(7) | 0.0156(3) | 0.0145(3) |

250 | 6.45519(69) | 19.4(5) | 0.0171(1) | 0.0095(2) | 6.40469(91) | 20.7(7) | 0.0161(3) | 0.0149(3) |

260 | 6.45644(72) | 19.4(5) | 0.0177(1) | 0.0099(2) | 6.40602(96) | 20.7(7) | 0.0167(3) | 0.0154(3) |

270 | 6.45770(76) | 19.5(6) | 0.0184(1) | 0.0103(2) | 6.40734(100) | 20.7(8) | 0.0172(4) | 0.0158(3) |

280 | 6.45896(79) | 19.5(6) | 0.0191(1) | 0.0107(2) | 6.40867(105) | 20.7(8) | 0.0178(4) | 0.0163(3) |

290 | 6.46022(83) | 19.5(6) | 0.0197(2) | 0.0111(2) | 6.41000(110) | 20.7(8) | 0.0183(4) | 0.0167(3) |

300 | 6.46148(87) | 19.6(6) | 0.0204(2) | 0.0115(2) | 6.41133(116) | 20.7(8) | 0.0189(4) | 0.0172(3) |

**Table A3.**Present high-pressure V(p) data from single-crystal structure refinement for PbTe and Pb

_{0.884}Cd

_{0.116}Te.

PbTe | Pb_{0.884}Cd_{0.116}Te | ||
---|---|---|---|

p [GPa] | V [Å^{3}] | p [GPa] | V [Å^{3}] |

0.33(2) | 271.1(15) | 0.30(2) | 265.40(50) |

0.80(2) | 267.9(12) | 0.80(2) | 261.10(80) |

1.30(2) | 266.8(15) | 1.40(2) | 256.50(16) |

1.80(2) | 264.8(16) | 2.50(2) | 251.78(10) |

2.40(2) | 259.5(16) | 3.00(2) | 247.41(12) |

3.00(2) | 257.2(15) | 3.50(2) | 245.37(11) |

3.70(2) | 254.7(12) | 4.00(2) | 241.37(14) |

4.50(2) | 251.8(15) | 4.50(2) | 241.23(14) |

**Table A4.**Present unit cell volume and bulk modulus as a function of pressure, modeled using Equation (6) for PbTe and Pb

_{0.884}Cd

_{0.116}Te.

p [GPa] | PbTe | Pb_{0.884}Cd_{0.116}Te | ||
---|---|---|---|---|

V [Å^{3}] | K [GPa] | V [Å^{3}] | K [GPa] | |

0 | 273.25 | 45.6(2.5) | 273.25 | 33.5(2.8) |

0.5 | 270.33 | 47.6(2.5) | 270.33 | 35.5(2.8) |

1.0 | 267.57 | 49.6(2.5) | 267.57 | 37.5(2.8) |

1.5 | 264.93 | 51.5(2.5) | 264.93 | 39.4(2.8) |

2.0 | 262.42 | 53.5(2.5) | 262.42 | 41.3(2.8) |

2.5 | 260.02 | 55.4(2.5) | 260.02 | 43.2(2.8) |

3.0 | 257.72 | 57.3(2.5) | 257.72 | 45.1(2.8) |

3.5 | 255.52 | 59.1(2.5) | 255.52 | 47.0(2.8) |

4.0 | 253.41 | 61.1(2.5) | 253.41 | 48.8(2.8) |

4.5 | 251.37 | 62.9(2.5) | 251.37 | 50.7(2.8) |

5.0 | 249.41 | 64.8(2.5) | 249.41 | 52.5(2.8) |

## Appendix C

**Table A5.**Fitted values of the parameters of Equations (1), (4) and (6), obtained for PbTe and Pb

_{0.884}Cd

_{0.116}Te crystals.

Function | Fitted Equation | Parameters for PbTe | Parameters for Pb_{0.884}Cd_{0.116}Te |
---|---|---|---|

V(T) | Equation (1) | ${V}_{\left(T=0\right)}$ = 265.813(6) Å^{3},Q = 2.86(3) × 10 ^{−18},b = 1.3(6), θ _{D} = 135.2(3.9) K | ${V}_{\left(T=0\right)}$ = 259.358(7) Å^{3},Q = 2.63(4) × 10 ^{−18},b = 0.4(7), θ _{D} = 130.1(4.4) K |

<u^{2}>(T), for cationic site | Equation (4) | θ_{D} = 102.8(3) K,<u ^{2}>_{stat} = 0.00038(4) Å^{2} | θ_{D} = 114.5(5) K,<u ^{2}>_{stat} = 0.00203(6) Å^{2} |

<u^{2}>(T), for anionic site | Equation (4) | θ_{D} = 169.2(1.1) K,<u ^{2}>_{stat} = −0.00054(7) Å^{2} | θ_{D} = 158.1(1.3) K,<u ^{2}>_{stat} = 0.0034(1) Å^{2} |

V(p) | Equation (6) | V_{0} = 273.3(7) Å^{3}K _{0} = 45.6(2.5) GPaK′ = 4 (fixed) | V_{0} = 267.7(1.5) Å^{3}K _{0} = 33.5(2.8) GPaK′ = 4 (fixed) |

## Appendix D

**Table A6.**Reported theoretical bulk modulus and its derivative for PbTe and Pb

_{1−x}Cd

_{x}Te, x = 0.031, 0.116. In a number of papers (e.g., refs. [10,129]), multiple numerical approaches have been applied, so only selected representative values could be cited here. As a rule (with some exceptions), the experimental values refer to room temperature, whereas in the calculated ones (to 0 K), the temperature is marked if explicitly stated in the given reference.

Compound | K_{0} [GPa] | K′ | Method | Ref. | Year |
---|---|---|---|---|---|

PbTe | 45 | n.a. | LDA | ||

48 | n.a. | LDA | (a) | 1983 | |

51.7 | 4.52 | LAPW LDA | (b) | 1997 | |

51.44 (0 K) 40.30 (0 K) 49.82 (0 K) 39.5 (0 K) | 5.50 4.27 5.76 3.92 | FP-LAPW LDA FP-LAPW GGA FP-LAPW LDA+SO FP-LAPW GGA+SO | (c) | 2000 | |

41.4 51.4 | 3.352 4.080 | GGA LDA | (d) | 2002 | |

37.5 (0 K) | n.a. | FP-APW PBE | (e) | 2007 | |

40.4 (0 K) 50.3 (0 K) 30.7 | n.a. n.a. n.a. | LDA/GGA “ “ | (f) | 2009 | |

39.05 | 4.32 | FP-LAPW | (g) | 2011 | |

46.0 46.1 | 4.27 4.53 | LDA LDA+SO | (h) | 2012 | |

41.0 | n.a. | MD (GULP) | (i) | 2012 | |

39.1 | n.a. | PAW PBE | (j) | 2013 | |

47 (0 K) | n.a. | FP-LAPW | (k) | 2014 | |

38.54 (300 K) ~45.5 (0 K) | n.a. n.a. | PBEsol “ | (l) | 2014 | |

34.04 (0 K) | n.a. | LDA, GGA | (m) | 2014 | |

46.61 (0 K) | n.a. | PBEsol | (n) | 2015 | |

44.1(&) | n.a. | HSEsolSOC | (o) | 2016 | |

41.1 | n.a. | FP-LAPW | (p) | 2017 | |

36.19 (100 K) 37.52 (300 K) | n.a. n.a. | LDY “ | (q) | 2019 | |

48.242 (0 K) | 5.576 (0 K) | LDA | (r) | 2020 | |

43.6 | 4.6 | GGA-PBE | (s) | 2020 | |

Pb_{0.969}Cd_{0.031}Te | 46.42 | n.a. | GGA | (n) | 2015 |

Pb_{0.884}Cd_{0.116}Te | 45.90 | n.a. | GGA | ($) | 2021 |

_{0.969}Cd

_{0.031}Te. (&)—the authors presented results also for seven other approaches. Abbreviations are explained at the end of this study.

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**Figure 1.**Reported variations of the metastable-rocksalt-phase lattice parameter of ternary Pb

_{1−x}Cd

_{x}Te, Pb

_{1−x}Cd

_{x}Se and Pb

_{1−x}Cd

_{x}S. The plots present the linear equations reported in ref. [18] (1), ref. [19] (2), ref. [7] (theoretical data) (3), ref. [20] (4), and ref. [25] (5). The highest values of Cd content in quenched samples among the reported ones, x

_{max}(see Table 1), are marked with short vertical solid lines. The variations are extended above the achieved (in quenched crystals) Cd content towards x = 1, in order to indicate the extrapolated lattice parameters of rocksalt structures of the binary Cd chalcogenides. The open circle indicates the composition of the Pb

_{1−x}Cd

_{x}Te sample studied in this work.

**Figure 2.**(

**a**) Experimental variation of normalized unit-cell volume with temperature for PbTe (□) and Pb

_{0.884}Cd

_{0.116}Te (■) (the main figure). The corresponding dependencies of the absolute values of lattice parameters are shown in the inset. The uncertainties are smaller than the symbol size. The fits of Equation (1) (second-order Grüneisen approximation) are shown in the figure and in the inset using red (PbTe) and blue (Pb

_{0.884}Cd

_{0.116}Te) solid lines, respectively. (

**b**) Present temperature dependence of the PbTe lattice parameter, compared to selected literature data. Present experimental points (■) and the second-order Grüneisen approximation (thick solid line); experimental data of ref. [38] (△), ref. [31] ( ), ref. [29] (thin dashed line), ref. [39] (◇), ref. [34] ( ). The inset provides a comparison with theoretical data of ref. [45] (+ and dashed line), ref. [46] ( and dashed line), ref. [39] ( and dashed line), and ref. [50,108] (dot-dash line).

**Figure 3.**Variation of thermal expansion with temperature, α(T). (

**a**) The linear thermal expansion coefficient α(T) for PbTe and Pb

_{0.884}Cd

_{0.116}Te as a function of temperature, was obtained from the experimental-data fitting using a second order Grüneisen approximation (Equation (1)). The inset includes the magnified low-temperature part (0–50 K) of the α(T) run. (

**b**) Comparison of the present data for PbTe (black solid line) with previously reported experimental and theoretical data of PbTe. Selected data from literature (from ref. [51] (●), [29] (black dashed line) (experimental), (from ref. [45] (+); [46] ( ), from ref. [8] ( ) (theoretical)) are shown. (

**c**) Comparison of the present experimental data (solid line, also shown as (strongly overlapping) dotted line in (

**b**)), theory [60,108] (dashed line), difference curve (dash-dot line).

**Figure 4.**Temperature dependence of mean square displacements for cations (

**a**) and anions (

**b**), for PbTe ( ) and Pb

_{0.884}Cd

_{0.116}Te (■). The fitted Debye function is represented by red and blue solid lines, respectively. The insets show the static disorder term <u

^{2}>

_{stat}as a function of Cd content, x

_{Cd}(own results ( ) with uncertainties smaller than symbol size, and results of ref. [29] ( )).The horizontal dashed line visualizes the zero-disorder level.

**Figure 5.**Temperature dependence of mean square displacements for PbTe for cations (

**a**) and for anions (

**b**). Experimental data: this work (solid line, ■), ref. [55] (◇), ref. [37] ( ), ref. [38] (△), ref. [31] (▽), ref. [57] (□), ref. [34] (●), ref. [29] (dashed line), ref. [39] (+). Theoretical data: ref. [58] ( ), and ref. [39] (dotted line).

**Figure 6.**(

**a**) Relative unit-cell volume as a function of pressure, for PbTe (□), and Pb

_{0.884}Cd

_{0.116}Te (■). The solid lines correspond to the second-order BMEOS fit. Experimental literature data for PbTe are shown from ref. [83] (data ( ) and fit (dotted line), the theoretical ones from ref. [64] (△ and green dashed line). (

**b**) Bulk modulus, K, dependence on pressure for PbTe (■) and Pb

_{0.884}Cd

_{0.116}Te (□) crystals (present data). as dash-dot line shows the theoretical data of [64].

**Figure 7.**The temperature variation of the Grüneisen parameter for PbTe and Pb

_{0.884}Cd

_{0.116}Te in the ranges 10–300 K and 15–300 K, respectively. The thin and the thick lines refer to PbTe and Pb

_{0.884}Cd

_{0.116}Te, respectively.

**Table 1.**Linear equations for a(x) or V(x) (Vegard’s rule and Zen’s rule, respectively) for Pb

_{1−x}Cd

_{x}Te, Pb

_{1−x}Cd

_{x}Se and Pb

_{1−x}Cd

_{x}S solid solutions: equations, maximum reported Cd content, x

_{max}, and extrapolated lattice parameters, a

_{ex}, for rocksalt type CdTe, CdSe and CdS (at x = 1).

Compound | a(x) [Å] or V(x) [Å^{3}] | x_{max} | a_{ex} [Å] | Ref. | Year |
---|---|---|---|---|---|

Pb_{1−x}Cd_{x}Te | 6.459–0.30x | 0.20 (at 1139 K) | 6.159 | (a) | 1964 |

6.459–0.40x | 0.144 | 6.059 | (b) | 1980 | |

6.466–0.414x | 0.75 | 6.037 (*) | (c) | 1989 | |

6.462–0.433(5)x | 0.114 | 6.029 | (d) | 2009 | |

6.38–0.434x (&) | - | - | (e) | 2012 | |

Pb_{1−x}Cd_{x}Se | 6.127–0.42x | 0.26 (at 1213 K) | 5.707 | (f) | 1965 |

6.128–0.38x - | 0.03 (at 523 K) 0.18 (at 873 K) | - | (g) | 1968 | |

- | ~0.057 (at 673 K) | - | (h) | 1973 | |

6.1263–0.3025x | 0.04 | - | (i) | 2019 | |

Pb_{1−x}Cd_{x}S | 203.151–0.4389x ($) | 0.016 | - | (j) | 1971 |

5.9386–0.4302x | 0.40 | 5.5084 | (k) | 2014 | |

5.412 (&) | (l) | 2019 | |||

5.435, 5.45, 5.72 | (m) | 2021 |

_{1−x}Cd

_{x}Se

_{1−y}S

_{y}[28]. (*)–value for sample quenched from 2.5–3 GPa, 973–1473 K; this paper also gives 6.052

**Å**, derived from data for quenched Sn

_{1−x}Cd

_{x}Te. ($)–data refer to V(x) dependence. (&)–theoretical data.

**Table 2.**Temperature ranges for selected experimental and theoretical studies of the lattice parameter, a(T), of PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.013, 0.020, 0.056, 0.096, 0.116). In the case of experimental studies, the ranges refer to the lower and upper limit of the experiment.

Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|

Experiment | PbTe | 0–400 ($) | n.a. | (a) | 1971 |

PbTe | 120–298 | SCXRD | (b) | 1987 | |

PbTe | 15–500 | XRD/ND/PDF | (c) | 2010 | |

PbTe | 105–1000 | SPXRD | (d) | 2013 | |

PbTe | 105–600 | SPXRD | (e) | 2016 | |

PbTe | 10–500 | ND | (f) | 2016 | |

PbTe | 125–293 | SCXRDS | (g) | 2018 | |

PbTe | 50–600 | NPD | (h) | 2021 | |

PbTe | 20–622 | SCXRDS | (i) | 2021 | |

PbTe | 15–300 | SPXRD | this work | 2021 | |

Pb_{0.987}Cd_{0.013}Te, | 300–~600, ~900–1073, | SPXRD | (j) | 2009 | |

Pb_{0.944}Cd_{0.056}Te, | 300–~430, ~970–1073, | “ | “ | “ | |

Pb_{0.904}Cd_{0.096}Te | 300–~350 | “ | “ | “ | |

Pb_{0.98}Cd_{0.02}Te | 15–300 | SPXRD | (k) | 2011 | |

Pb_{0.884}Cd_{0.116}Te | 15–300 | SPXRD | this work | 2021 | |

Theory | PbTe | 0–300 | LDA, GGA | (l) | 2009 |

PbTe | 4–550 | PBEsol | (m) | 2014 | |

PbTe | 0–300 | LDA, GGA | (n) | 2014 | |

PbTe | 100–800 | QHA | (o) | 2018 | |

PbTe | 300–800 (*) | MD | (p) | 2018 | |

PbTe | 0–800 (&) | DFPT/LDA | (q) | 2019 |

**Table 3.**Temperature ranges for selected experimental and theoretical studies on the variation of the linear thermal expansion coefficient, α(T), of PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116). In the case of the experimental studies, the ranges refer to the lower and upper limit of the experiment.

Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|

Experiment | PbTe | 30-340 | DM | (a) | 1963 |

PbTe | 4–297 | CM | (b) | 1968 | |

PbTe | 10–500 | ND | (c) | 2016 | |

PbTe | 50–600 | NPD | (d) | 2021 | |

PbTe | 15–300 | SPXRD | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 15–300 | SPXRD | this work | 2021 | |

Theory | PbTe | 70–300 | CDM | (e) | 1966 |

PbTe | 0–300 | LDA, GGA | (f) | 2009 | |

PbTe | 0–300 | PBEsol | (g) | 2014 | |

PbTe | 0–300 | LDA, GGA | (h) | 2014 | |

PbTe | 0–350 | GGA | (i) | 2015 | |

PbTe | 0–300 | FPBTF | (j) | 2017 | |

PbTe | 0–800 | DFPT/LDA | (k) | 2019 |

**Table 4.**Temperature ranges for selected earlier studies of the experimental and theoretical mean square displacements <u

^{2}>(T) of PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116). In the case of experimental studies, the ranges refer to the lower and upper limit of the experiment.

Mode | Compound | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|

Experiment | PbTe | 78–400 | PXRD, SCXRD | (a) | 1973 |

PbTe | 100–300 | SCXRD | (b) | 1978 | |

PbTe | 120–298 | SCXRD | (c) | 1987 | |

PbTe | 15–500 | XRD/ND/PDF | (d) | 2010 | |

PbTe | 105–1000 | SPXRD | (e) | 2013 | |

PbTe | 8–500 | SPXRD | (f) | 2014 | |

PbTe | 105–600 | SPXRD | (g) | 2016 | |

PbTe | 10–500 | ND | (h) | 2016 | |

PbTe (*) | 100–450 | SCXRDS | (i) | 2018 | |

PbTe | 20–300 | SCXRDS | (j) | 2021 | |

PbTe | (40)–700 | NPD | (k) | 2021 | |

PbTe | 15–300 | SPXRD | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 15–300 | SPXRD | this work | 2021 | |

Theory | PbTe | 0–400 | LKF | (l) | 1968 |

PbTe | 0–700 | MD (SME) | (m) | 2014 | |

PbTe (*) | 100–450 | MD | (i) | 2018 | |

PbTe | 300–800 | MD | (n) | 2018 | |

PbTe | 0–800 | DFPT/LDA | (o) | 2019 |

**Table 5.**(Completing Table 2, Table 3 and Table 4). Pressure and/or temperature ranges for experimental and theoretical studies of structure-related variables for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116), reported as functions of pressure and/or temperature: volume on pressure dependence, V(p), bulk modulus, K(T), elastic constants, C(T), Debye temperature, θ

_{D}(p,T) or θ

_{D}(p), Grüneisen parameter, γ(T) and γ(p), thermal expansion, α(p,T), heat capacity, c

_{p}(T) or c

_{p}(T). For experimental studies, the ranges refer to the lower and upper limit of the experiment.

Mode | Compound | Variables | Pressure Range [GPa] | Temperature Range [K] | Method | Ref. | Year |
---|---|---|---|---|---|---|---|

Experiment | PbTe | c_{p}(T), c_{v}(T) | - | 20–260 | CM | (a) | 1954 |

PbTe | γ(T) | - | 30–340 | CM+XRD | (b) | 1963 | |

PbTe | K(T), C(T) | - | 4–297 | CM | (c) | 1968 | |

PbTe | c_{p}(T) | - | 300–700 | PTW | (d) | 1983 | |

PbTe | θ_{D}(p), γ(p) | amb.–15, amb.–10.5 | - | UIM | (e) | 2013 | |

PbTe | K(T), γ(T), c_{v}(T) | - | 10–300/300/260 | ND | (f) | 2016 | |

PbTe | V(p), K(p), γ(T) | amb.–4.5 | - | SCXRD | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | V(p), K(p), γ(T) | amb.–4.5 | - | SCXRD | this work | 2021 | |

Theory | PbTe | θ_{D}(T) | - | 0–200 | CDM | (g) | 1966 |

PbTe | K(T) | - | 0–300 | LDA, GGA | (h) | 2009 | |

PbTe | K(p), C(T) | 0–14 | - | LDA | (i) | 2012 | |

PbTe | a(p,T), α(p,T), K(p,T), θ_{D}(p,T), c_{v}(T) | 0–10 | 0–300 | LDA, GGA | (j) | 2014 | |

PbTe | K(T) | - | 0–600 | PBEsol | (k) | 2014 | |

PbTe | c_{v}(T) | - | 0–400 | GGA | (l) | 2015 | |

PbTe | K(T), C(T) | - | 100–500, 0–500 | LDY | (m) | 2019 | |

PbTe (*) | c_{p}(T), c_{v}(T) | - | ~20–1000 | THD | (n) | 2019 | |

PbTe | V(p) | - | - | LDA | (o) | 2020 | |

PbTe | c_{v}(p) | 0–6 | - | PBEsol | (p) | 2021 | |

Pb_{1−x}Cd_{x}Te (*) | c_{p}(T), c_{v}(T) | - | ~20–1000 | THD | (n) | 2019 |

**Table 6.**Present and recently reported values of experimental lattice parameter, a, near 0 K for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.013, 0.056, 0.116). For complete numerical data of a(T) of this work, see Table A1 and Table A2 (Appendix B).

Compound | T | a [Å] | Ref. | Year |
---|---|---|---|---|

PbTe | 1 | 6.42962 (*) | (a) | 2016 |

0 | 6.42972(5) (*) | this work | 2021 | |

15 | 6.42977(5) (*) | this work | 2021 | |

15 | 6.4298(4) | this work | 2021 | |

Pb_{0.98}Cd_{0.02}Te | 10 | 6.42114 (*) | (b) | 2011 |

Pb_{0.884}Cd_{0.116}Te | 0 | 6.37725(6) (*) | this work | 2021 |

Pb_{0.884}Cd_{0.116}Te | 15 | 6.37733(7) (*) | this work | 2021 |

Pb_{0.884}Cd_{0.116}Te | 15 | 6.3775(5) | this work | 2021 |

**Table 7.**Present and selected recently reported values of the experimental lattice parameter, a, at room temperature for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.013, 0.056, 0.116). For complete numerical data of a(T) of this work, see Table A1 and Table A2 (Appendix B).

Compound | T [K] | a [Å] | Ref. | Year |
---|---|---|---|---|

PbTe | 300 | 6.46179(3), 6.46201(4) | (a) | 2013 |

300 | 6.46255 (*) | (b) | 2016 | |

300 | 6.46040(4), 6.46054(4) | (c) | 2016 | |

293 | 6.4626(1) | (d) | 2018 | |

300 | 6.4651(*) | (e) | 2021 | |

300 | 6.459–6.462 ($), <a> = 6.46148(15) | (f) | 2021 | |

300 | 6.4616(3) | this work | 2021 | |

300 | 6.46148(87) (*) | this work | 2021 | |

Pb_{0.987}Cd_{0.013}Te | 300 | 6.457(2) | (g) | 2009 |

Pb_{0.944}Cd_{0.056}Te | 300 | 6.437(2) | (g) | 2009 |

Pb_{0.884}Cd_{0.116}Te | 300 | 6.41133(116) (*) | this work | 2021 |

Pb_{0.884}Cd_{0.116}Te | 300 | 6.4116(4) | this work | 2021 |

**Table 8.**Present and selected literature values of the experimental thermal expansion coefficient, α, at 300 K for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116). For full numerical data of α(T), see Table A2 (Appendix B).

Compound | T [K] | α [Å] | Ref. | Year |
---|---|---|---|---|

PbTe | 300 | 19.94 | (a) | 1964 |

300 | 19.91 | (b) | 2016 | |

300 | 19.36 (*) | (c) | 2019 | |

300 | 18.12 | (d) | 2021 | |

300 | 19.6(6) | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 300 | 20.7(8) | this work | 2021 |

**Table 9.**Present and selected reported experimental values of the mean square cationic and anionic displacements, <u

_{C}

^{2}>(T) and <u

_{A}

^{2}>(T), respectively, near 0 K and at room temperature for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116). The (independent on temperature) static components, <u

_{C}

^{2}>

_{stat}and <u

_{A}

^{2}>

_{stat}, are provided, where available. For detailed numerical data of the present study, see Table A1 and Table A2, Appendix B.

Temperature | Compound | T [K] | <u_{C}^{2}>(T) [Å^{2}] | <u_{A}^{2}>(T) [Å^{2}] | <u_{C}^{2}>_{stat}[Å ^{2}] | <u_{A}^{2}>_{stat}[Å ^{2}] | Ref. | Year |
---|---|---|---|---|---|---|---|---|

low | PbTe | 15 | 0.0037 | --- | (a) | 2010 | ||

temperature | PbTe | 0 | 0.0018 (*) | 0.0018 (*) | (b) (#) | 2014 | ||

PbTe | 1 | 0.00200 | 0.00315 | (c) | 2016 | |||

PbTe | 0 | 0.0021(1) (*) | 0.0011(1) (*) | 0.00038(4) | −0.00054(7) | this work (#) | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 0 | 0.0036(1) (*) | 0.0052(1) (*) | 0.00203(6) | 0.0034(1) | this work (#) | 2021 | |

room | PbTe | 300 | 0.0233(15) | 0.0209(14) | (d) | 1973 | ||

temperature | PbTe | 298 | 0.0204(3) | 0.0141(3) | (e) | 1987 | ||

PbTe | 300 | 0.0231 | --- | (a) | 2010 | |||

PbTe | 300 | 0.0098(2) | 0.01847(9) | small | small | (f) | 2013 | |

PbTe | 300 | 0.0238 | 0.0171 | (b) | 2014 | |||

PbTe | 300 | 0.0202 | 0.0136 | (g) | 2016 | |||

PbTe | 300 | 0.02155 | 0.01548 | 0.00031 | 0.00130 | (c) | 2016 | |

PbTe | 300 | 0.0260(2) | 0.0157(1) | (h) | 2018 | |||

PbTe | 300 | 0.0204(2) (*) | 0.0115(2) (*) | 0.00038(4) | −0.00054(7) | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 300 | 0.0189(4) (*) | 0.0172(3) (*) | 0.00203(6) | 0.0034(1) | this work | 2021 |

**Table 10.**Fitted parameters of experimental equation of state for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116) at various temperatures.

Type of Experiment | Compound | T [K] | BMEOS Parameters | Method and Remarks | Ref. | Year | ||
---|---|---|---|---|---|---|---|---|

V_{0} [Å^{3}] | K_{0} [GPa] | K’ | ||||||

X-ray diffraction | PbTe | RT | n.a. | 38.9(1) | 5.4 | LEDPXRD | (a) | 1984 |

PbTe | RT | 269.6(4) | 44(1) | 4 (fixed) | SPXRD (QHS) | (b) | 2013 | |

PbTe | 296 | 273.3(7) | 45.6(2.5) | 4 (fixed) | LSCXRD (t) (HS) | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | 296 | 267.7(1.5) | 33.5(2.8) | 4 (fixed) | LSCXRD (t) (HS) | this work | 2021 | |

other | PbTe | 0 | n.a. | 45.6(4) | - | UWVSC | (c) | 1968 |

PbTe | RT RT | n.a. n.a. | 39.76 38.39 | 5.171 4.891 | EC (s) UWV (t) | (d) | 1981 | |

PbTe | RT | n.a. | 40.5(7) | 3.8(2) | SV | (e) | 2013 | |

PbTe | 0 RT | n.a. n.a. | 44.89 41.26 | - - | UWVSC (*) | (f) | 2016 |

**Table 11.**Values of experimental Debye temperature θ

_{D}determined by XRD/ND for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.116) and earlier reported experimental values for PbTe: data θ

_{DUC}and θ

_{DUA}refer to values determined for cationic and anionic sublattices, respectively.

Compound | θ_{DUC} [K] | θ_{DUA} [K] | θ_{DU} [K] (*) | θ_{DV} [K] | Method | Ref. | Year |
---|---|---|---|---|---|---|---|

PbTe | 95.5(2.0) | 127(3) | 111.3(3.0) | - | PXRD/SCXRD | (a) | 1973 |

- | - | - | 107(2), 108(3) | LPXRD | (a) | 1973 | |

- | - | - | 111 | SCXRD | (b) | 1978 | |

- | - | 87(1) | - | XRD | (c) | 2013 | |

91(3) (&) | 175(5) (&) | - | 133(4) (&) | NPD+CM | (d) | 2016 | |

99.6(2) | 156.0(5) | 127.8(4) | - | NPD | (d) | 2016 | |

101.4 | 157.0 | 129.2 | - | SCXRD | (d) | 2016 | |

- | - | - | 129(2) | NPD | (e) | 2021 | |

102(1) | 163(2) | 132.5 | - | NPD | (e) | 2021 | |

102.0 | 161.4 | 131.7 | - | NPD/PDXRDS | ($) | 2021 | |

- | - | - | 135.2(3.8) | LPXRD | this work | 2021 | |

102.8(3) | 169(1) | 135.9(7) | - | LPXRD | this work | 2021 | |

Pb_{0.884}Cd_{0.116}Te | - | - | - | 130.1(4.4) | LPXRD | this work | 2021 |

Pb_{0.884}Cd_{0.116}Te | 115.1(5) | 158(1) | 136.6(8) | - | LPXRD | this work | 2021 |

_{DUC}and θ

_{DUA}; ($)—weighted average of literature values collected in ref. [41] (only those are taken into account for which both cationic and anionic MSDs have been reported). (&)—for details of the applied method see ref. [29]. Abbreviations are explained at the end of this study.

Compound | θ_{D} [K] | Method | Ref. | Year |
---|---|---|---|---|

PbTe | 127 (at 20 K), 125 at 200 K | CM | (a) | 1954 |

PbTe | 176.7(5) (at 0 K) (*) | UWV | (b) | 1968 |

PbTe | 110 | PM | (c) | 1975 |

PbTe | 168 | HPM | (d) | 1976 |

PbTe | 140 | SP | (e) | 1979 |

PbTe | 136 | n.a. | (f) | 1998 |

PbTe | 105 | TC | (g) | 2006 |

PbTe | 163 | UPE | (h) | 2011 |

PbTe | 136 | UWV | (i) | 2012 |

PbTe | 170(5) | DPS | (j) | 2013 |

PbTe | 143 | SV | (k) | 2013 |

PbTe | 95 | NS | (l) | 2014 |

PbTe | 128(1) | CM + NPD | (m) | 2016 |

**Table 13.**Reported theoretical values of Debye temperature, θ

_{D}, for PbTe and Pb

_{1−x}Cd

_{x}Te (x = 0.031, 0.116).

Compound | θ_{D} [K] | Method | Ref. | Year |
---|---|---|---|---|

PbTe | 167 at 0 K 131 at 300 K (&) | CDM | (a) | 1968 |

PbTe | 177(1) (&) | NNI | (b) | 1986 |

PbTe | 152 (&) | GGA | (c) | 2012 |

PbTe | 141.5 (&) ($) | LDA, GGA | (d) | 2014 |

PbTe | 187.8 (&) | GGA | (e) | 2015 |

Pb_{0.969}Cd_{0.031}Te | 185.4 | GGA | (e) | 2015 |

Pb_{0.884}Cd_{0.116}Te | 178.8 | GGA | (*) | 2021 |

_{D}with x. (&)—used for calculation of the average, 157.9 K. Abbreviations are explained at the end of this study.

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**MDPI and ACS Style**

Minikayev, R.; Safari, F.; Katrusiak, A.; Szuszkiewicz, W.; Szczerbakow, A.; Bell, A.; Dynowska, E.; Paszkowicz, W.
Thermostructural and Elastic Properties of PbTe and Pb_{0.884}Cd_{0.116}Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects. *Crystals* **2021**, *11*, 1063.
https://doi.org/10.3390/cryst11091063

**AMA Style**

Minikayev R, Safari F, Katrusiak A, Szuszkiewicz W, Szczerbakow A, Bell A, Dynowska E, Paszkowicz W.
Thermostructural and Elastic Properties of PbTe and Pb_{0.884}Cd_{0.116}Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects. *Crystals*. 2021; 11(9):1063.
https://doi.org/10.3390/cryst11091063

**Chicago/Turabian Style**

Minikayev, Roman, Fatemeh Safari, Andrzej Katrusiak, Wojciech Szuszkiewicz, Andrzej Szczerbakow, Anthony Bell, Elżbieta Dynowska, and Wojciech Paszkowicz.
2021. "Thermostructural and Elastic Properties of PbTe and Pb_{0.884}Cd_{0.116}Te: A Combined Low-Temperature and High-Pressure X-ray Diffraction Study of Cd-Substitution Effects" *Crystals* 11, no. 9: 1063.
https://doi.org/10.3390/cryst11091063