Study of anharmonic elastic properties of mixed diatomic crystals AgCl−AgBr and NaCl−NaBr

Summary An interionic-force model has been employed to analyse the anharmonic elastic properties of mixed-diatomic-crystal systems with varying compositions. The interaction system consists of long-range Coulomb and three-body interactions within the framework of Hafemeister-Zarht (HZ) short-range repulsion, effective up to second-neighbour ions. This interionic potential has been used for predicting the third-order elastic constants, the first pressure derivative of second-order elastic constants and the Anderson-Grüneisen (AG) parameter in NaCl−NaBr and AgCl−AgBr mixed crystals. The values of the hardness parameter between nearest neighbours as well as between second nearest neighbours have been evaluated by using the overlap integral method. The results obtained in the present study are generally in good agreement with the available experimental data as compared to those obtained by earlier investigations. The experimental values of the Anderson-Grüneisen parameter have been calculated by using Chang’s relation.

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Riassunto Un modello di forza interionica è stato usato per analizzare le proprietà anarmoniche elastiche di sistemi di cristalli diatomici misti con composizioni variabili. Il sistema d’interazione consiste in interazioni di Coulomb a largo raggio e a tre corpi nell’àmbito della repulsione a corto raggio di Hagemeister-Zorht (HZ), efficaci fino a ioni secondi vicini. Questo potenziale interionico è stato usato per prevedere costanti elastiche di terz’ordine, la prima derivata di pressione delle costanti elastiche di second’ordine e il parametro di Anderson-Grüneisen (AG) in cristalli misti NaCl−NaBr e AgCl−AgBr. I valori del parametro di durezza tra vicini prossimi e tra secondi vicini prossimi sono stati calcolati usando il metodo dell’integrale di sovrapprosizione. I risultati ottenuti in questo studio sono generalmente in buon accordo con i dati sperimentali disponibili se confrontati con quelli ottenuti da precedenti ricerche. I valori sperimentali del parametro di Anderson-Grüneisen sono stati calcolati usando la relazione di Chang.

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Резюме Используется модель междуионной силы для анализа ангармонических упругих свойств смешанных двухатомных кристаллических систем с изменяющимся составом. Взаимодействие системы состоит из длиннодействующего кулоновского взаимодействия и взаимодействий трех тел в рамках короткодействующего отталкивания Хафемайстера-Зарта, с учетом взаимодействий с ионами, следующими за соседями. Предложенный междуионный потенциал используется для предсказания упругих постоянных третьего порядка, первой производной упругих постоянных второго порядка по давлению и параметра Андерсона-Грюнайзена в смешанных кристаллах NaCl−NaBr и AgCl−AgBr. Оцениваются параметры жесткости между соседними ионами, а также между ионами, следующими за соседними. Полученные результаты хорошо согласуются с имеущимися экспериментальными данными. Используя соотношение Чанга, вычисляются экспериментальные значения параметра Андерсона-Грюнайзена.

     

Cohesive and anharmonic elastic properties of mixed fluorite crystals

Summary The cohesive and anharmonic elastic properties of four mixed fluorite crystals (Ca _x Sr_1−x F₂, Sr _x Ba_1−x F₂ Ba _x Ca_1−x F₂ and Cd _x Pb_1−x F₂) have been investigated by means of a three-body potential (TBP) model which consists of the long-range Coulomb and three-body interactions and the short-range van der Waals attraction and overlap repulsion effective up to the second-neighbour ions. Due to the lack of measured data on cohesive energy, third-order elastic constants and pressure derivatives of the secondorder elastic constants of mixed fluorites, the accuracy of the present results has been tested by comparing them with the so-called experimental results generated by the application of Vegard’s law to their corresponding experimental values for the host fluorites.

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Riassunto Sono state ricercate le proprietà elastiche coesive e anarmoniche di quattro cristalli misti di fluorite (Ca _x Sr_1∮x F₂, Sr _x Ba_1−x F₂, Ba _x Ca_1−x F₂ e Cd _x Pb_1−x F₂) per mezzo di un modello del potenziale a tre corpi (TBP) che consiste di interazioni di Coulomb a lungo raggio e a tre corpi, di attrazione di van der Waals a corto raggio e di repulsione di sovrapposizione efficace fino agli ioni del secondo vicino. A causa della mancanza di dati misurati sull’energia coesiva, di costanti elastiche di terz’ordine e di derivate di pressione delle costanti elastiche di secondo ordine, l’accuratezza dei risultati attuali è stata provocata confrontandoli con i cosidetti risultati sperimentali generati dalla applicazione della legge di Vegard ai loro valori sperimentali corrispondenti per i fluoriti ospiti.

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Резюме С помощью трех-частичной потенциальной модели исследуются когезионные и ангармоничные упрыгие свойства четырех смешанных кристаллов флюорита (Ca _x Sr_1−x F₂, Sr _x Ba_1−x F₂, Ba _x Ca_1−x F₂ и Cd _x Pb_1−x F₂). Используемая модель состои т из длиннодейтсвующего кулоновского и трех-частичного веаимодействий и короткодействующего притяжения Ван дер Ваальса и перекрывающегося отталкивания, эффективного вплоть до вторых соседних ионов. Из-за недостатка зкспериментальных данных для когезионной энергии упругих постоянных третьего порядка и производных по давлехию упрутих постоянных второго поядка для смешанных флюоритов, проверяется точность полученных результатов посредством сравнения их с так называемыми экспериментальными результатами, образованными с помощью применения закона Вегарда к соответствующим экспериментальным величинам для первичных кристаллов флюоритов.

     

The elastic constants and their temperature and pressure derivatives of AgBr-AgCl mixed crystals

The three independent second-order elastic constants and their temperature and pressure derivatives have been measured for four AgBr-AgCl mixed crystals, with 19.5, 39.1, 56.6 and 78.7 mole % AgCl, using the ultrasonic pulse-echo technique at room temperature. The explicit temperature dependence of the elastic constants is calculated and is found to be much larger than that of other NaCl structure crystals. The violation of the Cauchy relation C₁₂ = C₄₄ is found to be significant and increases between AgBr and AgCl. The high temperature limit of the Gruneisen parameter is calculated from the elastic data. A comparison is made between the elastic properties of the silver halides and the alkali halides.     

Dielectric properties of RbCl-RbBr mixed crystals

A systematic measurement of dielectric constant and loss on RbCl-RbBr mixed crystals in various compositions has been carried out in the frequency range 100 Hz to 100 kHz and in the temperature range from room temperature to 320°C. From these measurements the static dielectric constant, the Szigeti charge, the conductivity and the activation energy for conduction are evaluated. All these properties show a nonlinear composition dependence. Semiempirical equations proposed earlier are employed to evaluate the dielectric constant as a function of composition. The validity of these relations is also discussed.     

Cohesive and elastic properties of alkaline earth chalcogenide crystals

In the present study an analysis of the cohesive and elastic properties has been performed in fourteen alkali chalcogenide crystals using the composite potential forms based on the Drude oscillator model. The short-range overlap repulsion has been considered up to the second nearest neighbour interactions. The van der Waals coefficients have been estimated from the Kirkwood-Muller formulae. Cohesive energies, bulk-modulus and its pressure derivatives calculated for the entire family of crystals under study are found to present a good agreement with available experimental data, better than those obtained by previous workers using the traditional Born potential model.On leave from theDepartment of Physics, G. B. Pant University, Pantnagar-263 145, India.Authors are thankful to Mr. Saleh O. Harraga for the computational help.     

Theories of elastic properties and elastic waves of equal phase velocities in crystals

A survey is made of pairs of heteronormal planar elastic waves, which can propagate in an unlimited crystal medium and have equal theoretical values of the phase velocity, using Voigt elastic moduli. Such pairs are chosen for which it is possible that the theoretical equality of velocities will be disturbed when using another theory of the elastic properties of crystals with generally different tensors of material constants (for example Kuvshinskij J. V., Aero E. L.: Fiz. tv. tela5 (1963), 2591). In such a case, a comparison of the measured velocities of a suitable pair of waves can help to decide between the two theories. Pairs of waves are found for 12 non-piezoelectric crystal classes. Among such pairs it is easy to find pairs of waves of the same properties for piezoelectric crystal classes.     

Far infrared properties of SbSI-SbSeI mixed crystals

Far infrared reflectivity of eight SbS_1–x Se _x I mixed single crystals was measured in 50–400 cm^–1 region in polarized light at 25 and –170 C. While the low-frequency ferroelectric mode shows clear one-mode behaviour the second B_1u mode in 126–180 cm^–1 region shows two-mode behaviour. The existence of local and gap modes in this case is discussed.The author thanks ing. K.Nejezchleb from College of Chemical Technology in Pardubice for the crystals, L.Novák for helpful discussions, Z.Híová for computations and N.Dvoáková for help and figure drawing.     

Study of static and thermophysical properties of alkaline earth fluoride crystals

An analysis for cohesive energy, bulk modulus and its pressure derivatives in BaF₂, SrF₂ and CaF₂ crystals has been carried out by means of interionic potential model. Interionic potential consist of long range coulomb and three-body interactions along with short-range attractive van der Waals and repulsive Hafemeister-Zarht type interactions. The three body force parameter $\text{?(r)}$ and vdWI coefficients C and D are determined using the Cochran's formula and Slater-Kirkwood variational (SKV) integrals. The calculated values of cohesive energy, bulk modulus and its pressure derivatives for solids under study are found generally in good agreement with available experimental data. The results achieved by other investigations [13, 14] have also been compared with results calculated in the present study.     

The band structure and optical properties of the CdxHg1-xTe mixed crystals

The band structure of the Cd_xHg_1?xTe mixed crystals have been computed on the basis of tight-binding interpolation scheme and virtual crystal approximation. The bowing parameters of main energy gaps have been calculated as well as composition dependence of position of E₁ and E₁ + Δ₁ maxima in reflectivity spectra. Good agreement with previously reported experimental data has been found.     

The influence of local distortions on the static and dynamic properties of copper and silver eightfold-coordinated Jahn-Teller complexes in mixed crystals with a fluorite-type structure

The influence of local distortions on the structure and properties of copper and silver impurity Jahn-Teller complexes in mixed crystals, namely, Ca_xSr_1?x F₂: Me ²⁺ and Sr_1?x Ba_xF₂: Me ²⁺ (0≤x≤1, Me ²⁺=Cu²⁺ or Ag²⁺), is investigated using electron paramagnetic resonance (EPR) spectroscopy at frequencies of 9.3 and 37 GHz in the temperature range 4.2–250 K. Local distortions of the tensile and compressive types are induced by Ca²⁺, Sr²⁺, and Ba²⁺ impurity ions incorporated into the first or second coordination sphere of the cationic environment of the Me ²⁺ impurity ion during crystal growth.     

Refractive indices and related properties of some potential mixed crystals

The high frequency refractive indices of some binary, ternary and quaternary mixed crystals have been evaluated from the knowledge of plasmon energy and the lowest gap energy of the crystals for their applications in heterojunctionled and solar cells. The Fermi energy screening factor correction has been applied to effect accuracy in prediction. The model has been used to study the temperature and pressure dependence of refractive index. The calculated value agrees with experiment (within a few percent) justifying the validity of the model.     

The influence of excess oxygen on the elastic properties of non-stoichiometric SrO crystals

Measurements have been carried out of the elastic constants of SrO in the virgin undoped state and of the changes produced in them by equilibrium doping with oxygen at ? 1200°C and oxygen partial pressure of 0.95 atm. The method used was Papadakis' pulse-echo overlap technique in conjunction with thermogravimetric analysis (T.G.A.) to determine mass and density changes due to oxygen doping.The values obtained for C₁₁, C₁₂ and C₄₄ of the virgin crystal at 23°C are

$\text{C}{}_{11}\text{= 17.60 ± 0.03 × 10}{}^{11}\text{dynes/cm}{}^2$

;

$\text{C}{}_{12}\text{= 4.808 ± 0.007 × 10}{}^{11}\text{dynes/cm}{}^2$

;

$\text{C}{}_{44}\text{= 5.577 ± 0.008 × 10}{}^{11}\text{dynes/cm}{}^2$

.(These values are in very good agreement with those of Son and Bartels [2].)Values for $\text{δC}{}_{11}\text{C}{}_{11}$ and $\text{δC}{}_{44}\text{C}{}_{44}$ were found to be ?1.74% and ?0.86% respectively. Accurate valu $\text{δC}{}_{12}\text{C}{}_{12}$ could not be obtained because of sample size limitations after quenching. However, C₁₂ was shown to definitely increase due to doping.Analysis of the results indicate that the elastic modulus changes can only be attributed to the formation of cation vacancies during doping. Analysis of the T.G.A. behavior indicates that this cation vacancy formation is probably associated with the presence of various tripositive cation and uninegative anion species depending upon the impurity concentrations of the sample. This implied impurity-controlled cation vacancy concentration is consistent with the earlier observed extrinsic nature of cation diffusion in SrO at 1200°C.     

Thermal expansion and elastic properties of the crystals with modulated structure

An experimental study of structure characteristics and elastic properties of the incommensurate crystals is reported.     

Bulk and surface phonons in superlattices of diatomic crystals

The vibrations of a superlattice are studied within a model of diatomic linear chain crystals. For such an infinite or a semi-infinite superlattice the Green's function is calculated in explicit form. The dispersion of the bulk phonons are then obtained analytically, as well as the complex wavevector inside the gaps separating the bulk bands. Localized surface modes, decaying inside the superlattice, may exist in these gaps for a semi-infinite system. The model is applied to bulk and surface longitudinal phonons in GaAsAlAs superlattices.     

The elastic behaviour of InBi single crystals

Ultrasonic wave velocities have been measured by the pulse echo technique at 15MHz in single crystals of InBi grown by zone-melting. The elastic stiffness constant set has been computed from the velocity data by a least-mean-squares procedure. The elastic behaviour of this metallic compound is quite different in kind from that of the semiconducting III-V crystals; the volume compressibility does not follow Keyes' generalisation for the other III-V compounds. The elastic properties of InBi have been found to show the characteristics of a layer-like crystal with weak interlayer binding; this finding is illustrated by the linear compressibilities and by a compilation of cross-sections of the wave velocity and Young's modulus surfaces. The Debye temperature is 115°K.     

Propagation properties of elastic waves in semi-infinite phononic crystals and related waveguides

The transmission and reflection coefficients of two-dimensional semi-infinite solid-solid phononic crystal systems and fluid-fluid phononic waveguide structures have been investigated. The numerical results show that the transmission spectra for longitudinally and transversally polarized incident waves are different, and the spectra of the transmission and reflection coefficients of the semi-infinite system agree well with the band structure. The numerical results show that when a guided wave incident, localized modes are excited, and different polarities have different coupling efficiencies with the incident guided wave. At the same time, far from the cutoff frequency, the guided wave couples out of semi-infinite waveguide highly efficiently.