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Dynamical study for the Gruneisen parameters in FCC metals
Abstract
By using a non-central force model proposed by the authors, the microscopic Gruneisen parameters gamma q,j and the effective value gamma G at different temperatures are calculated for silver, gold and aluminium in terms of elastic constants and their pressure derivatives. The microscopic Gruneisen parameters gamma q,j are calculated as a function of the wavevector q along the principal symmetry directions (100), (110), (111), (210), (211) and (221) and the values are averaged over a wavevector space in the first Brillouin zone by using the modified Houston's method for evaluation of integrals. The results obtained for gamma G show a fairly good agreement with the experimental values.
... The Grüneisen parameters play a significant role in the study of thermoelastic properties. It has its fundamental importance to the equation of state of a system and related thermodynamic properties of the solids [2]. The calculation of anharmonic effects in solids such as thermal expansion or the interaction of acoustic and thermal phonons involves Grüneisen parameters, which describe the volume and strain dependence of the lattice vibrational frequencies. ...
In this paper, we have investigated the temperature dependence of the ultrasonic parameters like ultrasonic velocities and Grüneisen parameters in californium monopnictides CfY (Y: N, As and Sb) for longitudinal and shear waves along <100>, <110> and <111> crystallographic directions in the temperature range 100–500K. For the same evaluation the second- and third- order elastic constants have also been computed for these monopnictides using Coulomb and Born-Mayer potential upto second nearest neighborhood. The mechanical properties and stability of CfN is best, because of its high valued elastic constants. Ultrasonic velocity is found to be highest for CfAs along all chosen directions, so CfAs will be most suitable compound for wave propagation. The obtained results of present investigation are discussed in along with identified thermophysical properties.
... Using the bulk modulus value of silver B = 100.7 GPa [22] and a mean diameter D = 4 nm one deduces a compressive strain a/a ≈ −0.43%. Since the Grüneisen parameters in Ag have been estimated as γ ≈ 2.5 [63], the expected relative frequency shift is ν/ν = −3γ a/a ≈ 3.2%. A homogeneous strain in each NC, as shown here by the GPA technique, with an absolute value a/a ≈ −0.43% would induce a shift around 0.16 THz for its higher frequency part of the VDOS in good agreement with the blueshift observed experimentally for the LA band in Fig. 8(b). ...
An investigation of the vibrational density of states (VDOS) in silver nanocrystals is performed using Raman
scattering. A specific sample architecture, setup configuration, and original elaboration process are used in
order to take simultaneously advantage of spectrally and spatially localized surface plasmon resonance, optical amplification, and dark-field spectroscopy. Disentangling the contributions of atom vibrations and electron-hole excitations (i.e., the so-called “background” in surface-enhanced Raman scattering) is performed. The extracted VDOS is successfully compared with theoretical ones obtained by atomic scale simulations. The effects of size, strain, and disorder on the VDOS are analyzed; in particular, the strain effect is investigated experimentally using the geometrical phase analysis coupled with high-resolution transmission electron microscopy. This work offers an opportunity to examine thermodynamic properties, like specific heat, at the nanoscale.
The temperature dependence of the ultrasonic parameters like ultrasonic velocities and Grüneisen parameters in americium monopnictides AmY (Y: N, P, As, Sb and Bi) have been studied for longitudinal and shear waves along <100>, <110> and <111> crystallographic directions in the temperature range 100 K -500 K. The second-and third-order elastic constants have also been evaluated for these monopnictides using Cou-lomb and Born-Mayer potential. The values of elastic constants are the highest for AmN. Hence the me-chanical properties of AmN are better than other monopnictides AmP, AmAs, AmSb and AmBi. Ultrasonic velocity is found large for AmP. So the ultrasonic wave propagation will be much better than others in AmP. Obtained results are compared with available results of same type of materials.
The temperature dependence of the ultrasonic parameters like ultrasonic velocities and Grüneisen parameters in americium monopnictides AmY (Y: N, P, As, Sb and Bi) have been studied for longitudinal and shear waves along <100>, <110> and <111> crystallographic directions in the temperature range 100 K-500 K. The second-and third-order elastic constants have also been evaluated for these monopnictides using Cou-lomb and Born-Mayer potential. The values of elastic constants are the highest for AmN. Hence the mechanical properties of AmN are better than other monopnictides AmP, AmAs, AmSb and AmBi. Ultrasonic velocity is found large for AmP. So the ultrasonic wave propagation will be much better than others in AmP. Obtained results are compared with available results of same type of materials.
Observations of the linear thermal expansion of a potassium chloride crystal have been made between 2° and 30°k, 55° and 90°k, and 273° and 293°k. At the lowest temperatures (3° < T < 9°k), the linear coefficient of expansion α≃5·4±0·3 × 10−11 T3 per °c, which corresponds to a Grüneisen γ value of 0·39 ± 0·2. γ increases with temperature, particularly in the range 15° to 70°k (cf. θD≃235°k), being about 0°9 at 30°k, 1·25 (60°k) and 1·5 above 200°k. This increase is of the form suggested theoretically by Barron and by Blackman for ionic cubic solids but is greater than they expected. It is not very different from that predicted by Sheard from the pressure variation of the elastic constants.
Following recent experiments in which Grüneisen's γ was observed to decrease at temperatures below 0·3 Θ, an investigation is made of the variation of γ with temperature according to the lattice dynamics of Born and von Kármán. γ is defined as a function of T, and a general analysis applicable to all lattices is used to derive the low and high temperature limits γ 0 and γ ∞. By expressing γ as a quotient of power series in T-1 it is shown that the main variation between these limits occurs at temperatures of the order of 0·2 Θ. Calculations are carried out for a cubic close packed lattice with central forces between nearest neighbours; it is found that γ ∞-γ 0=0·3, and that the behaviour of γ is qualitatively similar to that found experimentally in monatomic metals. The amount of variation in γ is decreased when interaction between more distant neighbours is taken into account, and a model representing the heavier inert gas solids gives γ ≈ 3 and γ ∞-γ 0 ≈ 0·15.
The thermal expansion of copper has been determined at temperatures down to 2 °K by means of an electrical capacitance technique. By measuring three-terminal capacitances in a ratio-transformer bridge length changes smaller than 10 ⁻⁹ cm can be detected in the capacitor-expansion cell. Below 10 °K, the linear expansion coefficient, α , may be represented by α = (1·4 5 ± 0·1 5 ) x 10 ⁻¹⁰ T + (2·8 ± 0·1) x 10 ⁻¹¹ T ³ per degK. These terms, identified respectively as electronic ( e ) and lattice ( l ) in origin, lead to the following values of the Grüneisen parameter: γ e = 0·63 ± 0·06, γ l = 1·72 ± 0·03 ( = γ o ). γ e is compared with a theoretical free-electron value of 2/3, and the low temperature value of γ l is compared with values of 1·75 to 1·77 calculated from elastic constants. At room temperature γ (≈ γ ∞ ) is 2·00, so that γ ∞ – γ 0 ≈ 0·28, which agrees well with the theoretical estimates of Barron for a close-packed cubic lattice.
A model for the lattice dynamics of metals propounded by us earlier is applied to copper, a representative of face‐centered cubic metals. The force constants appearing in the secular equation for the lattice frequencies are estimated from the knowledge of the elastic constants. The phonon dispersion curves for the three symmetry directions [100], [110], and [111] are presented and are found to be in fair agreement with the neutron spectroscopic measurements. The frequency distribution function has been calculated. The specific heat computed from the frequency distribution function is found to be in good agreement with the experimental measurements.
A new model is proposed for the calculation of dispersion curves in metals. The conduction electrons are taken into account with the Thomas-Fermi interaction. In this model, the requirements of symmetry are satisfied and no external forces are needed in order to maintain static equilibrium.
The dispersion relations for copper in the three symmetry directions have been calculated using a modified angular force model which takes into account the effect of electron-ion interaction. There is found to be good agreement with neutron scattering results.
The pressure derivatives of the elastic stiffness constants of aluminum and magnesium have been measured over a pressure range from 0 to 6500 bars, by a modified ultrasonic pulse-echo method. The values found are: dC/dP dC′/dP dBs/dP Al 2·31 1·62 5·19 dC11/dPdC33/dPdC'/dPdC'/dPdCH/dP Mg 6·11 7·22 1·58 1·36 13·7 The notation C = C44, C′ = (C11−C12)/2, Bs = (C11 + 2C12)/3, and CH = C11+C12+2C33− 4C13 has been used. The pressure data for aluminum have been interpreted in terms of Leigh's theory for the elastic stiffnesses themselves, and the magnesium data in terms of the similar theory of Reitz and Smith. Variation of the effective charge density at the boundary of the atomic polyhedron under volume strain causes the electrostatic contribution to the shear stiffnesses to vary approximately as r−8 instead of r−4 for both metals. Electron transfer from states overlapping one type of Brillouinzone face to states overlapping another type of face under volume strain contributes appreciably to the variation of the Fermi shear-stiffness contribution with pressure.
An apparatus for measuring the thermal expansion of solids over the temperature range 15-300°K. by a differential method making use of a Fizeau interferometer is described. Experimental data for copper are presented and are applied to the Grüneisen relations.
A Born-von Karman model has been used to determine the frequency spectrum for a body-centered cubic lattice. The model used contains noncentral angular stiffness forces as well as central forces between nearest and next-nearest neighbors. Frequency distributions have been calculated for vanadium and compared with experimental frequency distributions available from slow neutron experiments. Qualitative agreement was found between the shapes of the experimental and theoretical distributions. The calculated maximum frequency for vanadium was within 2% of that derived from the experimental frequency distribution. Dispersion curves for iron were also calculated, and were found to be in good agreement with experiment.
A four parameter model for the dynamical study of fcc metals is proposed by combining the ion-ion angular interaction with the Chéveau model. The computed phonon frequencies for platinum compare well with the experimental results.
The three elastic constants C44, 12(C11-C12), and 12(C11+C12+2C44) have been directly measured for silver and gold in the temperature range between 4.2 and 300°K. The various contributions to the values of the 0°K constants are analyzed in terms of a simple model which quite successfully describes copper. It is concluded that such a model is unsatisfactory when applied to the heavier noble metals because these appear to have large noncentral forces contributing to their constants. Combined with pressure data the present results show the elastic constants to be explicit functions of temperature. The Debye characteristic temperatures calculated from the 0°K elastic constants are shown to be in substantial agreement with the results from calorimetry.
The third-order elastic constants of copper, silver, and gold have been measured at room temperature using high-purity single crystals. A sensitive ultrasonic interference method was employed to measure small changes of wave velocities in crystals deformed both uniaxially and hydrostatically at low stress levels. The results of the measurements were analyzed using finite elasticity theory to obtain the complete sets of six third-order constants. The values of C111 are largest (of the order of 1013 dyn/cm2), C112 and C166 are about half of C111 in magnitude, and all are negative in sign. Almost all of the values of C123, C144, and C456 are negative and small compared with the other constants. It is shown that the closed-shell repulsive interaction between nearest-neighbor atoms makes the dominant contribution to the higher order elastic constants in noble metals. In this sense, the anharmonic properties are simpler than the harmonic properties of these materials. The fourth-order elastic constants are also estimated on this basis.
The experimental phonon-dispersion curves of aluminum at 80°K and at 300°K have been analyzed in terms of axially symmetric Born-von Kármán force-constant models, including 8 nearest neighbors. The resulting models have been used to compute a frequency distribution function g(omega) at each temperature from which various thermodynamic properties have been derived. The specific-heat curve predicted by the g(omega) appropriate to 80°K fits excellently the experimental results in the temperature range 20 to 80°K. At higher temperatures the experimental results deviate from this calculated curve and approach the curve appropriate to g(omega) at 300°K. Similar behavior is found for the experimental Debye-Waller coefficient in the range above 100°K. It is concluded that inelastic-neutron-scattering data and thermodynamic data are compatible in the range of sufficiently low temperatures where deviations from the quasiharmonic approximation are small, provided a good force-constant model as well as a statistically adequate g(omega) are available. There is evidence that the quasiharmonic approximation in aluminum is invalid at room temperature at least for the extreme low-frequency part of g(omega).
The dispersion relations for aluminum have been determined at 80 and 300°K by neutron spectrometry, using a three-axis crystal spectrometer. Particular attention was paid to precision, in order to investigate small effects-e.g., Kohn anomalies, phonon frequency widths, frequency shifts with temperature-and to establish an accurate experimental routine. A focusing method used throughout to optimize resolution is described, as well as a method for calculating energy resolution and extracting phonon widths from observed one-phonon resonances. Results are presented as dispersion curves for phonons in the three principal directions, accompanied by phonon widths, and as contour maps of phonon frequency on the surface of an elemental tetrahedron in q space. Measurements at points off the principal directions are utilized in the latter maps. Kohn anomalies have been observed, but are reported elsewhere. Phonon widths at 80°K are interpreted semiquantitatively in terms of the interaction between phonons and conduction electrons. Exceptionally large phonon widths in two regions may be due to singularities in the phonon-phonon interaction: the conservation rules for decay of a phonon into two phonons suggest a source for such singularities, but the suggestion has not been confirmed by computation.
Houston's formula first derived for an approximate determination of the frequency spectrum of cubic crystals, essentially approximates the integral over the surface of a unit sphere of any function I(theta, varphi) which is invariant under the operations of the cubic symmetry group in terms of the values of I(theta, varphi) along the three directions (100), (110), and (111). In this paper approximate formulas are given for the integral in terms of the values of I(theta, varphi) along the above three and any or all of the (210), (211), and (221) directions. As an application, Debye temperatures Theta are calculated for nine cubic crystals. It appears that the Theta values calculated from the formula containing the values of I along all the above six directions may be expected to be correct to about 1% for crystals for which 0.25
The heat capacity, at constant (0 degrees K) volume, of face-centred cubic crystals is evaluated by a modified Houston (1948) approximation. There are four principal changes. (a) Instead of three directions in the reciprocal lattice, we use five, namely, the (100), (110), (111), (210) and (211) directions. (b) We replace the first Brillouin zone by a sphere of equal volume. In this way we avoid the difficulty, at low and medium temperatures, of the appearance of the characteristic normalization factor in Houston's method. At the same time we overcome the difficulty that the latter is only applicable to the integration of functions defined over a sphere. (c) We do not concern ourselves with the frequency spectrum. This is due to a change from integration over frequency to integration over wave-number in the expression for Cv. The reason for the change is that in this way we obtain integrands, in the five directions, that contain no singularities, vary smoothly, and are convenient for numerical evaluation on a digital computer. (d) At higher temperatures we give an expression for Cv in terms of the first three even moments that we prove is in error by less than 0\cdot 1%. We are able to ensure this accuracy by fitting an even polynomial to the Einstein function over the range 0
De Launay type angular force model has been applied to sodium to study the dispersion relations in the three symmetry directions. Electron-ion interaction has also been taken into account. The theoretical results are compared with the experimental ones obtained from inelastic neutron scattering and an excellent agreement has been obtained. The Cauchy discrepancy is attributed to the presence of both types of non-central interactions.
The thermal expansion of a solid is due to anharmonicity of the interatomic forces. It may therefore be calculated, on the basis of a continuum model, from the observed elastic anharmonicity, i.e. the third-order elastic constants. Using Lazarus' (1949) measurements on the pressure variation of the elastic moduli of several cubic crystals, the Grüneisen γ is calculated for the limiting cases of low and high temperatures. For KCl, NaCl the high temperature values agree with those obtained from thermal expansivities, and a decrease in γ at low temperatures is predicted. However, Lazarus' data on metals cannot be reconciled with the thermal expansions, which suggests that the continuum model is inapplicable to these cases.
A simple lattice dynamical model for metals is presented. A body‐centered cubic lattice of ion cores interacting with nearest and next‐nearest neighbors imbedded in a uniform background of electron gas is considered. A secular equation determining the frequencies of normal modes of vibration is derived. This equation contains three force constants which can be written in terms of elastic constants. The dispersion relations are presented for propagation vectors in the [100], [110], and [111] directions. Comparison is made in the case of sodium with the experimental curves obtained from neutron spectrometry and also with theoretical calculations of Toya. Our model is found to give a satisfactory description of the lattice dynamics of sodium.
The pressure derivatives of the elastic constants of the homologous series of metals, copper, silver, and gold have been measured over the pressure range from 0 to 10 000 bars, using a modified ultrasonic pulse-echo method. Means have been devised to measure the change of elastic constant with pressure as directly as possible. The values found for the pressure derivatives of the elastic constants are as follows: Cu Ag Au dBsdP 5.59 6.18 6.43 dCdP 2.35 2.31 1.79 dC'dP 0.580 0.639 0.438 The notation C=C44, C'=(C11-C12)2, and Bs=(C11+2C12)s3 has been used. The data for each metal, of the three elastic constants and their pressure derivatives, have been interpreted in terms of conventional theory. The theoretical contributions of long-range interactions have been subtracted off and the remainder attributed to short-range nearest-neighbor interaction. The analysis indicates that these must be non-central, many-body interactions in order to account for the shear constants and especially their pressure derivatives. The many-body character of the interactions is of rapidly increasing importance in the sequence copper, silver, and gold.
Extremely accurate determinations of the linear thermal expansions have been made interferometrically from room temperature to the temperature of liquid nitrogen for Mo, Pd, Ag, Ta, W, Pt, and Pb. Comparisons are made with the Grueneisen theory.
Extremely accurate determinations of the linear thermal expansions have been made interferometrically from - 196°C to temperatures about +400°C for Al and +700°C for Fe, Ni, Cu and Au. The relationship between true thermal coefficient of expansion and temperature conforms very well to the Grueneisen-Debye theory when values are chosen for the Debye characteristic temperatures which turn out to agree well with those chosen to achieve agreement with the Debye theory of specific heats. Our values for these characteristic temperatures are: 410°K for Ni, 420°K for Fe, 400°K for Al, 325°K for Cu, and 190°K for Au. The magnetic Curie temperature for Ni is found to be 352°C. In plotting true coefficient of thermal expansion versus temperature Simon and Bergmann found a horizontal plateau at about 175° to 235°K for Ni and Fe; but we do not confirm this.
From measurements of the inertial thermoelastic stress produced by pulse heating a portion of a sample, the Grüneisen parameters of silicon and aluminum as a function of temperature were obtained directly between 5 and 290 °K. Pulses of 1.5-MeV average-energy electrons, lasting approximately 40 nsec, were used as the heating agent, and the stresses were recorded with a quartz gauge bonded to the back face of the sample. For silicon, the low-temperature stress measurements indicated a limit of the Grüneisen parameter of about 0.2, which is lower than the value calculated from thermal-expansion data but in agreement with that calculated from pressure derivatives of elastic constants. For aluminum (6061 alloy-97% Al), the low-temperature limit of the lattice Grüneisen parameter was found to be 1.7 and the electronic contribution was estimated to be 1.7 as well, agreeing within experimental uncertainty (approximately 25%) with published values based on thermal-expansion measurements on pure aluminum.
The temperature variation of the Grüneisen parameter γ is calculated for copper and germanium on the basis of the anisotropic continuum dispersive model, from the observed elastic anharmonicity. The detailed nature of the frequency spectrum does not seem to have any pronounced effect on the thermal expansion of a solid.
The temperature variation of the Grüneisen γ for Cu, Ag, Au, Al, Na, KCl, NaCl, Si, Ge is calculated from experimental values of the pressure dependence of the elastic constants, using the ‘anisotropic continuum model' of Sheard (1958). Except for KCl, a simple summation over six symmetry directions of the crystal is sufficiently accurate to replace numerical integration. The theoretical results agree tolerably with experiment, except for Si and Ge.
It is shown that the Grüneisen constant derived from the elastic properties of a solid can be negative in certain cases. The lattice models examined are (a) ionic lattices with the zinc blende structure, (b) ionic lattices with the rocksalt structure. In the first case it is found that a negative value is obtained over the whole range of parameters examined. In the second, negative values are obtained when the shear modulus of the lattice is sufficiently small. In all these cases the volume expansion coefficient of the cubic lattice would be negative at sufficiently low temperatures.
Interferometric determinations of the expansion coefficient of synthetic rock salt were made from 20 to 300°K. Similar measurements for natural rock salt were made from 90 to 300°K. Derivatives for the expansion coefficient were calculated at a variety of temperatures. Grüneisen coefficients were computed as a function of temperatures using measured compressibilities and both measured heat capacities and those computed from Kellerman's frequency spectrum. A correlation of these Grüneisen coefficients with those expected from a nearest neighbor cubic closest packed structure and for a sodium chloride type lattice has been made.
The frequency against wave vector dispersion relations are studied for waves travelling along three symmetry directions (100), (110) and (111) for copper, silver and gold using a modified non-central force model, which takes into account the effect of electron-ion interaction also. The theoretical results are compared with the inelastic neutron scattering results for copper and silver, and are found to be in good agreement. A good agreement for gold is also anticipated. The contribution of angular interactions to the Cauchy discrepancy has been emphasized.
The temperature variation of the Grüneisen parameter γ for copper is calculated from the observed pressure derivatives of the elastic constants using the electron gas model proposed by one of the present authors. Microscopic Grüneisen parameters, γq,i, are calculated as a function of the wave number q along the six directions [100], [110], [111], [210], [211], and [221] in reciprocal space and are averaged by a modification of Houston's spherical six-term integration method. The calculated value of γ is found to be in good agreement with the experimental values of Rubin et al.
Die Änderung des Grüneisenparameters γ mit der Temperatur für Kupfer wird aus den gemessenen Druckkoeffizienten der elastischen Konstanten berechnet, wobei das Elektronengasmodell benutzt wird, das von einem der Autoren vorgeschlagen wurde. Die mikroskopischen Grüneisenparameter γq, i werden als Funktionen der Wellenzahl q längs den sechs Richtungen [100], [110], [111], [210], [211] und [221] im reziproken Gitter berechnet und durch eine modifizierte, sphärische Sechs-Term-Integrationsmethode nach Houston gemittelt. Das berechnete γ ist in guter Übereinstimmung mit den experimentell bestimmten Werten von Rubin et al.
Der Henningsche Apparat wird fr Ausdehnungsmessungen bis zu der Temperatur des flssigen Wasserstoffs hergerichtet. Der Ausdehnungskoeffizient bis zu dieser Temperatur ist an verschiedenen Metallen, Glsern und Porzellanen, sowie Glimmer bestimmt. Die bei ersteren gefundenen Ergebnisse sind mit einer ans der Theorie der festen einatomigen Krper gefolgerten Gleichung verglichen.
The adiabatic elastic moduli of single crystals of aluminum have been measured from 4.2° to 300°K by using the ultrasonic pulse‐echo technique. The values obtained by extrapolation to 0°K are C 11 =11.430, C 12 =6.192, and C 44 =3.162 in units of 1011 dyn/cm2. The Debye temperature obtained from the 0°K modulus values is 430.3°K, in excellent agreement with the value from heat‐capacity measurements.
The value of Grüneisen's parameter γ and its variation with temperature are calculated for some simple crystalline models; for comparison the value of γS given by Slater's formula is also calculated (, where χ is the compressibility). There is no unique measure of the anharmonicity of a crystal, and in particular there appears to be no interatomic potential that will give a crystal potential energy that is ideally harmonic. For example, a close-packed cubic lattice in which each pair of nearest neighbors interacts with a harmonic potential has a negative thermal expansion, the values of γ at high and low temperatures being and ; on the other hand for this model.A model representing an ionic crystal of sodium chloride structure with no polarization of the ions gives and ; the values of γ∞ and γ0 are lower than much earlier estimates by Born. In the calculation of γ0, the γj corresponding to individual long acoustic waves in different crystal directions are derived, and are found to cover a wide range of values, some of them negative.In both these theoretical models, and also in close packed crystals with more general central forces, γS is appreciably larger than γ∞. Most of the experimental data is uncertain, but it would indicate that, in agreement with theory, γS > γ∞ for most alkali halides. For metals, on the other hand, a revised interpretation of experimental data given recently by Gilvarry shows that usually γS < γ∞. An artificial model is put forward as an example of how this might happen.In conclusion a summary is given of the main results of this paper and its predecessors.