Technical Physics, Vol. 48, No. 7, 2003, pp. 931–933. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 73, No. 7, 2003, pp. 136–138. Original Russian Text Copyright © 2003 by Bodryakov, Povzner.
BRIEF COMMUNICATIONS
Self-Consistent Thermodynamic Approach to Calculating the Grüneisen Parameters of the Crystal Lattice in Solids V. Yu. Bodryakov and A. A. Povzner Ural State Technical University, ul. Mira 19, Yekaterinburg, 620002 Russia e-mail:
[email protected] Received December 17, 2002
Abstract—For a nonmetallic isotropic paramagnetic solid, a set of generalized Grüneisen parameters γi that are suitable in applied computations is introduced. Exact thermodynamic expressions for the temperature dependences of the Grüneisen parameters γθ and γ *θ are derived, and basic factors responsible for these dependences are found. Thermodynamic conditions under which the parameter γθ changes sign and the Invar effect shows up in the material are determined. © 2003 MAIK “Nauka/Interperiodica”.
(1) The dimensionless Grüneisen parameter Γ, together with the characteristic Debye temperature θ, is a basic parameter in the solid-state theory. It is defined as the isothermal logarithmic derivative of the Debye temperature with respect to volume V [1, 2]: ∂ ln θ Γ = – ------------ . ∂ ln V T
(1)
Generally speaking, the parameter Γ characterizes the average “distortion” of the phonon spectrum of acoustic oscillations in the lattice by an applied voltage, which changes the solid volume. For most solids, the Grüneisen parameter is positive and close to unity. In some cases, however, Γ may change sign and become negative. It is believed (see, e.g., [2]) that such a reversal may be responsible for the unusual behavior of the thermal expansion coefficient in a number of materials, such as Si and Ge. In these materials, the thermal expansion coefficient changes sign, becoming negative in a rather wide range of low temperatures (the socalled Invar anomaly). This fact has been known for a long time and is explained [2] by the features of the phonon spectrum. However, an adequate thermodynamic interpretation of this phenomenon is still lacking. Even thermodynamic conditions under which the Grüneisen parameter changes sign remain unclear. Thermodynamic properties whose anomalous behavior is responsible for the Invarness of the materials mentioned above are also unknown. The aim of this work is to calculate the Grüneisen parameters of the first and second order for a nonmetallic isotropic paramagnet by using a self-consistent thermodynamic approach. This study elaborates upon the authors’ ideas put forward in previous publications [3−6].
Let us introduce for convenience generalized Grüneisen parameters γi of the first (γi) and second ( γ i* ) order, which are defined as V ∂i γ i = --- ------- , i ∂V T 2 2 V ∂ i * -------- , ----= γ i i ∂V 2 T
(2)
where i = θ, θl, θt, K, K0 , Kph, σ, Ξ, Ξl, and Ξt (see the text). It is easy to check that the Grüneisen parameter γθ from (2) coincides with conventional definition (1) up to the sign. As was already mentioned, we are interested in the temperature dependences of the parameters γθ and γ *θ . (2) The parameter specifying the value of the Grüneisen parameter γθ is the Debye temperature, which is conveniently represented as the average over one longitudinal and two transverse branches of the phonon spectrum of acoustic oscillations in the crystal lattice [1]: 1/3 3 -3 . θ = -------------------------3 1/θ + 2/θ l t
(3)
The partial Debye temperatures θl and θt that correspond to longitudinal and transverse acoustic oscillations, respectively, are given by 4 1/3 K + -2 -G ប ( 6π N A ) 3 1/6 θ l = ------------------------------ ------------------ V ; kB µ
1063-7842/03/4807-0931$24.00 © 2003 MAIK “Nauka/Interperiodica”
(4)
BODRYAKOV, POVZNER
932 1/3
ប ( 6π N A ) G 1/6 - ---- V . θ t = ----------------------------kB µ 2
K = K 0 + K ph (5)
In (4) and (5), ប, NA, and kB are the Planck constant, the Avogadro number, and the Boltzmann constant, respectively; µ and V are the molar weight and volume, respectively; and K and G are the bulk and shear moduli, respectively. The well-known relationships of the elasticity theory [7], 4 1–σ K + --- G = 3 ------------K, 3 1+σ 1 – 2σ G = 3 ---------------K, 1+σ
3R 3 2 = K 0 + ------- --- γ *θ θ – T [ γ θ C VR ( z ) – γ *θ D ( z ) ] . V 8
(9)
In (8) and (9), V0 and K0 are the initial molar volume and bulk modulus extrapolated to T = 0; Vph and Kph are the phonon (lattice) contributions to related quantities; z = θ/T; and D(z) and CVR(z) are the tabulated Debye functions and the Debye heat capacity at constant volume normalized to unity. The baric isothermal derivative of the Debye temperature in (9) can be expressed through γθ by means of the well-known thermodynamic relationships [1]: ∂θ θ -----= – ---- γ θ . ∂P T K
(10)
allow us to express the so-called longitudinal elastic modulus K and the transverse elastic (shear) modulus G through the Poisson’s ratio σ and the bulk modulus. Since the Poisson’s ratio lacks the thermodynamic definition and there is no reliable data for its temperature dependence, we will assume it to be a temperature independent material constant. Then, the temperature dependences of the partial Debye temperatures given by (4) and (5) are related largely to the temperature dependences of the bulk modulus and molar volume.
Eventually, we will arrive at a consistent set of expressions that relate the thermodynamic parameters V, K, σ, θ, γθ, and γ *θ to each other. Next, one may devise an iterative scheme to find consistent values of these parameters in a sufficiently wide temperature range by the method of successive approximations and determine the temperature dependences of γθ and γ *θ .
In view of the aforesaid, the partial Debye temperatures can be expressed in the mathematically convenient form
(3) Omitting simple mathematical manipulations based on (2) and (3), we write averaging expressions for γθ and γ *θ :
θ i = constΞ i K V , 1/2
1/2
1/6
(6)
where the subscript i = l, t for the longitudinal and transverse phonon oscillation modes, respectively. The constant in (6), which is of no significance for the subsequent calculations, is given by const =
1/3 3 2 --- ប ( 6π N A ) /k B , µ
(7)
Thus, expression (6) contains in explicit form the thermodynamic parameters that are functions of volume (Ξ, K, and V) and those depending on temperature (K, V). For the molar volume and bulk modulus, we will employ the thermodynamically exact expressions derived previously: V = V 0 + V ph
3 D ( z ) ∂θ = V 0 + 3R --- + ------------ ------ , 8 z ∂P T
3
3
(11)
With (6)–(10), it is easy to obtain a set of four * , and γ θt *: expressions for γθl, γθt, γ θl
and 1–σ ------------, i = l 1 + σ Ξi = – 2σ 1---------------, i = t. 1+σ
γ θl /θ l + 2γ θt /θ t γ θ = -------------------------------------, 3 3 1/θ l + 2/θ t 3 3 γ *θl /θ l + 2γ *θt /θ t * -------------------------------------. γ = θ 3 3 1/θ l + 2/θ t
(8)
1 1 1 γ θi = --2- γ Ξi + --2- γ K + --6- , 1 1 2 γ *θi = – --4- ( γ Ξi – γ K ) + --6- ( γ Ξi + γ K ) 1 5 + --- ( γ *Ξi + γ *K ) – ------ ; i = l, t. 36 2
(12)
Omitting mathematical manipulations again, we write the basic relationships for γΞl and γΞt: 2 γ Ξl = – ----------------------------------γ , (1 + σ)(1 – σ) σ 3 γ = – --------------------------------------γ , Ξt ( 1 + σ ) ( 1 – 2σ ) σ TECHNICAL PHYSICS
Vol. 48
(13)
No. 7
2003
SELF-CONSISTENT THERMODYNAMIC APPROACH
parameter γθ changes sign subsequent to γθl and γθt; namely,
for γ *Ξl and γ *Ξt : 2σ 2 - [ 2σγ σ – ( 1 + σ )γ *σ ], γ *Ξl = – -----------------------------------2 (1 + σ) (1 – σ) (14) 3σ 2 γ * = – ---------------------------------------[ 2σγ σ – ( 1 + σ )γ σ* ], 2 Ξt ( 1 – 2σ ) ( 1 + σ ) and for γK: γK
933
γ K 0 K 0 + γ K ph K ph = ---------------------------------------. K 0 + K ph
(15)
Finally, the parameter γKph is given by the awkward expression 3PT 3 2 ' ( z ) zγ θ γ K ph + 1 = ----------- --- + D' ( z ) γ *θ – γ θ C VR V 8 3 D(z) + --- + ------------ ( 2γ *θ – γ θ γ *θ + γ ** θ ) 8 z
(16)
2 - – 2γ θ C VR ( z ) ( γ θ + γ θ* – γ θ ) .
Here, the prime means differentiation in respect to the argument 3
Thus, expressions (12)–(16), together with (3)–(5), (8), and (9), completely determine the temperature dependence of the Grüneisen parameters γθ and γ *θ . In the iterative process, the parameters σ, γσ, γ *σ , γK 0, and γ θ** may be used as variable material constants if experimental data for them are absent. Such data may be found, e.g., for σ or γK 0. The latter parameter depends on the baric derivative of the bulk modulus and in principle can be measured. It is seen from the above relationships that the temperature dependence of the Grüneisen parameters γθ is specified largely by the temperature dependences of the thermodynamic quantities Kph and γKph. Expression (12) clarifies the thermodynamic conditions under which the usually negative Grüneisen
TECHNICAL PHYSICS
Vol. 48
No. 7
and/or 1 γ Ξt + γ K = – --- . 3 More detailed calculations (for example, the calculation of γ ** θ ) can be made following the same scheme; however, they seem to be unjustified in the light of the experimental accuracy. The results of this work can be summarized as follows. A set of generalized Grüneisen parameters γi that are convenient in applied computations is introduced. As far as we know, the thermodynamically exact temperature dependences of the Grüneisen parameters γθ and γ *θ for a nonmetallic isotropic paramagnetic solid are derived for the first time. The main reasons for the temperature dependences of γθ and γ *θ are established correctly in terms of thermodynamics. Thermodynamic conditions under which the parameter γθ changes sign and, accordingly, a nonmetallic isotropic paramagnet acquires the Invar properties are established. REFERENCES
V ∂ θ = ------ ---------3 . γ ** θ θ ∂V T 3
1 γ Ξl + γ K = – --3
2003
1. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 5: Statistical Physics (Nauka, Moscow, 1976; Pergamon, Oxford, 1980), Part 1. 2. S. I. Novikova, Thermal Expansion of Solids (Nauka, Moscow, 1974). 3. V. Yu. Bodryakov, A. A. Povzner, and O. G. Zelyukova, Fiz. Tverd. Tela (St. Petersburg) 40, 1581 (1998) [Phys. Solid State 40, 1433 (1998)]. 4. V. Yu. Bodryakov, A. A. Povzner, and O. G. Zelyukova, Metally, No. 2, 79 (2000). 5. V. Yu. Bodryakov, V. V. Petrushkin, and A. A. Povzner, Fiz. Met. Metalloved. 89 (4), 5 (2000). 6. V. Yu. Bodryakov and A. A. Povzner, Fiz. Met. Metalloved. 89 (6), 21 (2000). 7. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 7: Theory of Elasticity (Nauka, Moscow, 1987; Pergamon, New York, 1986).
Translated by V. Isaakyan
