1. Introduction

The macroscopic and microscopic thermodynamic properties have the excellent statistical mechanical correspondence through the Helmholtz free energy. It may be further added that the temperature dependence of thermodynamical quantities, namely; Helmholtz free energy, lattice heat capacity at constant volume, vibrational entropy, etc., can be investigated with the help of phonon dynamics. The knowledge of the anharmonic contributions to the Helmholtz free energy of crystalline solids is necessary for understanding the physics of solids and the phenomenon of melting. Traditionally an ordering scheme was devised by Van Hove ^[1] in which an ordering parameter was introduced which in magnitude is equivalent to a typical atomic displacement divided by the nearest-neighbor distance. The lowest-order anharmonic contributions to the free energy have dependence. Later, several workers [2-7]^[2] have described the derivation of these contributions and their evaluation for a simple model. The free energy for a given crystal geometry has been taken up from the results of harmonic approximation ^{[3, 8, 9]} but the anharmonicity of the crystal potential causes the mode frequencies so obtained to be functions of the geometrical parameters, making the vibrational free energy a function of these parameters as well as that of temperature. Thus, the effects of anharmonicities cannot be ignored ^{[10, 11, 12]}.

The Helmholtz free energy can be resolved in to two components as follows:

(1)

where is the total electronic free energy evaluated at absolute zero of temperature (density functional theory may be used) while is the phonon contribution to free energy of the system considered as a function of volume and temperature and

(2)

being the phonon density of states and . In addition to anharmonicity, the presence of impurity dramatically modifies the phonon spectrum of the crystal and hence, substantial changes occur in the frequency dependent properties of the crystal ^{[13, 14, 15, 16, 17]}. Several workers, in the past worked out to evaluate the Cubic and quartic anharmonic contributions to the Helmholtz free energy of a crystal as functions of temperature for a general force constant model of the crystal. The resulting expressions are evaluated in the high and low temperature limits for the case of an over simplified model of monoatomic linear chain ^{[15, 16]}. However, it possesses all the characteristic difficulties of three-dimensional models, with main difference of occurrence of one-dimensional integrals rather than three-dimensional integrals. Further, the cubic and quartic anharmonic contributions to the free energy of a face centered cubic lattice with nearest neighbor central force interactions have been evaluated in the high and low temperature limits ^{[6, 7]}.

In the present work, we have adopted the quantum dynamical many body approach to investigate the lattice energy, partition function, free energy of an isotopically disordered anharmonic crystal. This method utilizes the evaluation of double time thermodynamic phonon Green’s functions with the help of a crystal Hamiltonian consisting the influence of the effects of mass and force constant change terms between the impurity and host lattice atoms (localized phonon fields) along with an anharmonic phonon fields up to quartic order of anharmonicity. This results in the simultaneously the involvement of the effects of (i) isotopic impurities (ii) cubic and quartic anharmonicities and (iii) electrons in a crystal to derive the expression for the Helmholtz free energy. The double time thermodynamic Green’s functions thus obtained enable us to develop the expressions for the energy shifts, widths, density of states, lattice energy, partition function, free energy and heat capacity.

2. Formulation of the Problem

Let us begin with the following well-known expression for Helmholtz free energy of a crystalline solid [2-7]^[2]

(3)

where is the crystal Hamiltonian and for a thermodynamic system the partition function is defined as with average lattice energy, which depends on the occupation number of the vibrating mode. We follow the quantum dynamical approach ^{[12, 17]} to express by its equivalent density of states expression

(4)

The density of states can be expressed in the Lehmann representation ^[19]

(5)

where is the phonon Green’s function. Obviously, the evaluation of becomes the most significant part of the problem which is defined as

(6)

3. The Hamiltonian and Green’s Functions

The method of quantum dynamics for a crystalline solid can be initiated via an almost complete Hamiltonian of the form

(7)

where subscripts ‘eo’, ‘po’, ‘ep’, ‘A’ and ‘D’, respectively stand for the unperturbed-electron, harmonic-phonon, electron-phonon, anharmonic and defect constituents of Hamiltonian and are given by ^{[11, 12, 19]}

(8a)

(8b)

(8c)

(8d)

(8e)

In the above equations are cubic- and quartic-anharmonic force constants, while and are the mass and force constant change parameters emerged because of the presence of impurities. and are phonon field and phonon momentum annihilation (creation) operators, respectively; while are electron annihilation (creation) operators. Also, and are the phonon and electron energies, respectively with as electron-phonon interaction parameter ^{[11, 12, 13, 17, 19, 20]}.

Now we adopt the many body quantum dynamical approach to write the equation of motion for the Green’s function via Hamiltonian Eq.(7) followed by the Dyson’s equation method and Fourier transformation to obtain

(9)

Here the energy of the renormalized mode is given by

(10a)

where

(10b)

The self-energy operator or response function appearing in equation (9) is given by

(11)

which can be evaluated via an equivalent zeroth-order renormalized Hamiltonian

(12)

One can express the response function in terms of its real and imaginary parts in the form

(13a)

Here, and are phonon energy line width at the half maximum (imaginary part of ) and the phonon energy shift (real part of ). The width can be obtained in the form

(13b)

(13c)

with

(14a)

(14b)

(14c)

(14d)

(14e)

(14f)

(14g)

Here, , , and can be obtained from , and after replacing by . The subscripts appearing throughout the paper stand for the contributions due to the interactions arising from cubic- and quartic anharmonicities and interaction of cubic and quartic anharmonicities with localized-phonon fields, respectively. In above equations, we have introduced the following symbols for convenience:

(15a)

(15b)

(15c)

4. The Lattice Energy

The anharmonicity introduces quantum corrections ^{[11, 21, 22, 23]} and it is observed that the energy as well as the density of states exhibit substantial changes due to the presence of lattice anharmonicity and defects. It is also useful to remark here that the sole effect of anharmonicity and defects is contained in the density of states which appears as a central problem to investigate the dynamical properties of the crystalline solids. Hence, the use of the traditional formula for the average lattice energy ^{[13, 18]} is quite reasonable ^{[24, 25, 26, 27, 28]}. A remarkable event occurs here is that, during the course of present derivations, the Planck’s occupancy automatically changes its frequency dependence, i.e., the renormalized mode frequencies enter in it. Hence, the average lattice energy can be obtained in the form ^[12]

(16)

where superscripts represent the diagonal and non-diagonal contributions, respectively. The non-diagonal term chiefly appear due to the presence of point impurities and vanishes in case of a pure crystal. Eq. (15) can be further simplified with the help of Breit-Wigner approximation which yields

(17a)

and

(17b)

The various contributions to the diagonal part can be expressed in the following forms:

(18a)

(18b)

(18c)

(18d)

(18e)

(18f)

where the various symbols appearing in above equations can be defined as

(19a)

(19b)

(19c)

(19d)

(19e)

Where and are the electron line width in the harmonic approximation and Fermi function, respectively. In the above expressions it is evident that the functions and are very important functions of renormalized phonon energies (frequencies) and and perturbed mode energy (frequency) and phonon occupancies . The dependence of and on renormalized energies (frequencies, ) (a general trend) is depicted in Figure 1 and Figure 2, which very clearly exhibits the very sensitive behavior of renormalized frequency and distribution functions. The drastic change like a step function in and (more pronounced in ) shows the crucial behavior of anharmonic terms. These figures certainly perpetuate the nature of the anharmonicities to the lattice energy.

Figure 1. Variation of with phonon energies and

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View next figure

Figure 2. Behavior of with phonon energies and

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

5. Partition Function

Substituting Eqs. (16) to (18) in Eq. (4) one can easily obtain the partition function in the form ^{[12, 27, 29]}

(20)

The various components of the partition function can be written in the form

(21a)

(21b)

(21c)

(21d)

(21e)

(21f)

These quantities have been utilized to evaluate the Helmholtz free energy in the following sections.

6. Free Energy

After obtaining the appropriate partition function for an anharmonic crystal containing substitutional impurities, we can calculate the Helmholtz free energy from partition functions described in Eqs. (20) and (21) as ^{[2, 10]}

(22)

where the free energy has been resolved into four types of contributions; namely, electron-phonon contribution , contribution due to the insertion of impurities , anharmonic contribution and impurity anharmonicity interference contribution which can be summarized as

(23a)

(23b)

(23c)

(23d)

(23e)

(23f)

Obviously, the contribution described in Eq. (23a) reveals a dependence on Fermi and Bose functions and , respectively, which simultaneously shows an electron and phonon energy dependence like while the defect contribution exhibits the phonon energy dependence like along with the mass and force constant change parameters. The terms related to the anharmonic and impurity-anharmonicity interactions chiefly depend on and terms and on respective anharmonic force constants: , and product of impurity change parameters and force constant change): (mass change), (force constant change) with and . Several attempts have been made by a number of groups to evaluate the anharmonic contributions to free energy [2-7,15,16,17,30,31,32] but simultaneous attempt to include the anharmonic and impurity contribution is almost missing.

7. Discussion and Conclusions

Let us carefully examine the Helmholtz anharmonic free energy for an impurity induced crystalline solid in various temperature ranges. In the low temperature regime , the various contributions to the Helmholtz anharmonic free energy can be obtained after some algebraic simplifications in the form

(24a)

(24b)

(24c)

(24d)

(24e)

(24f)

where

(25a)

(25b)

(25c)

(25d)

(26a)

(26b)

(26c)

(26d)

(26e)

(26f)

These results clearly establish that the free energy terms are very complicated terms constituting entire frequency spectrum of the crystal and groups of various phonon distribution functions etc., depending on energy scenario of the form , , , , etc., in the low temperature regime. The beauty of the present formulation is that the distribution functions automatically change their frequency dependence according to the demand of a particular interaction event. As a simple example, the graphics shown in Figure 3 clearly exhibits the complexity of the problem. In order to obtain the exact outcome of present theory one needs to adopt the detailed anharmonic lattice dynamics that involves a very heavy computation due to the involvement of a large number of frequencies and distribution functions.

Figure 3. Behavior of with phonon energies and

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

View next figure

However, the cubic and quadratic anharmonic contributions to the Helmholtz free energy of a crystal are evaluated in the high temperature limit for the particular case of a face centered cubic lattice with nearest neighbor central force interactions between atoms by Maradudin et al ^{[2, 15]}. Nevertheless, the present investigations on the anharmonic Helmholtz free energy for an impurity induced anharmonic crystal with various contributions in the classical hydrodynamic regime yield

(27a)

(27b)

(27c)

(27d)

(27e)

(27f)

Where

(28)

Evidently the results for Helmholtz free energy in the hydrodynamic regime also have very complex forms depending on and . The typical behavior of is depicted with the simultaneous changes in and shown in the contours of Figure 4.

Figure 4. Trend of with and

Download as

PowerPoint Slide

Larger image(png format)

Figures index

Veiw figure

View current figure in a new window

View previous figure

Similarly, the general trend of another physical quantities like , with varying of , and may also be investigated. In the present work, we have discussed the thermodynamic properties of an impure anharmonic crystal by considering the average lattice energy based on the density of states, which, in turn, depends on the imaginary part of the Green’s function that is quantum dynamically evaluated without using the quasiharmonic approximation. This gives substantial contribution through higher order anharmonic terms to the zero point energy to the crystal. The phonon–electron interaction has the fundamental longstanding role in quantum solid state physics, especially in the low temperature superconductivity (BCS theory) where the electron–phonon–electron interaction (Cooper pairing of electrons) seems to be the most beautiful theory ever in modern physics.

It emerges from the present study, that by considering anharmonicities in an isotopically disordered crystal, it is possible to investigate the advanced effects of crystal anharmonicity and impurities on the Helmholtz free energy. The separation of diagonal and non-diagonal lattice energy and the inclusion of simultaneous involvement of impurity and anharmonicity add a feather to the theory and may be more justifiable as compared to the traditional theories. A more detailed and careful analysis of the energy (frequency) spectrum through impurity induced anharmonic dynamics the free energy for a specific model crystal can enable one also to calculate density of states, local, gap, resonance and impurity-anharmonicity interference modes. This work may also be exploited to study the free energy of the low dimensional systems.

References

[1]	Van Hove, Quantum Theory of Many Particle Systems (Benjamin, New York, 1961).
	In article
[2]	A. A. Maradudin, P. A. Flinn, and R. A. Coldwell-Horsfall, Anharmonic Contributions to Vibrational Thermodynamic Properties of Solids Part II. The High Temperature Limit, Ann. Phys. (N.Y.) 15, 360-386, 1961.
	In article		CrossRef
[3]	R. C. Shukla and E. R.Cowley, Anharmonic Free Energy of an Anharmonic Crystal to O(λ4) Phys. Rev. B 3, 4055-4065. 1971.
	In article		CrossRef
[4]	R. C. Shukla, and E. R. Muller, On Renormalization of Einstein Phonons and Green’s Function, Ames. J. Phys. 39, 77-82. 1971.
	In article
[5]	R. C. Shukla, and E. R. Muller, On Renormalization of Einstein Phonons. II. The Diagrammatic Method and Helmholtz Free Energy, Ames. J. Phys. 40, 544-549, 1972.
	In article
[6]	R. C. Shukla, L. Wilk, Helmholtz Free Energy of an Anharmonic Crystal to O(λ4) II. Phy. Rev. B 10, 3660-3666. 1974.
	In article		CrossRef
[7]	R. C. Shukla, B. R.Taylor, Anharmonic Free Energy and Specific Heat at Constant volume of Sodium and Potassium, Phy. Rev. B 9, 4116-4120, 1974.
	In article		CrossRef
[8]	M. L. Klein and T. R. Koehler, Rare Gas Solids, Vol. 1 eds. M. L. Klein and J. A. Venables (Academic, New York, 1976).
	In article
[9]	M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford, 1954).
	In article
[10]	A. A. Maradudin, A. E. Fein, Scattering of Neutrons by an Anharmonic Crystal, Phys. Rev. 128, 2589-2608, 1963.
	In article		CrossRef
[11]	K. N. Pathak, Theory of Anharmonic Crystals, Phys. Rev. 139, A 1569-1580, 1965.
	In article
[12]	B. D. Indu, Theory of Lattice Specific Heat of an Isotopically Disordered anharmonic Crystal, Intern. J. Mod. Phys. B 4, 1379-1393. 1990.
	In article		CrossRef
[13]	B. D. Indu and R. P. Gairola, Five Phonon Interactions in Anharmonic Crystals, Ind. J. Theor. Phys. 33, 115-143, 1985.
	In article
[14]	A. A. Maradudin, Solid State Physics, Vols. 18 and 19, ed. F. Seitz and D. Turnbull, 273 & 1 (Academic Press, 1961).
	In article
[15]	A. A. Maradudin, P. A. Flinn and R. A. Coldwell-Horsfall, Anharmonic Contributions to Vibrational Thermodynamic Properties of Solids Part I. General Formulation and Application to Linear Chain, Ann. Phys. (N.Y.) 15, 337-359, 1961.
	In article		CrossRef
[16]	P. A. Flinn and A. A. Maradudin, Anharmonic Contributions to Vibrational Thermodynamic Properties of Solids Part III. The Low Temperature Limit, Ann. Phys. (N.Y.) 22, 223-238, 1963.
	In article		CrossRef
[17]	Ram Singh, B.D. Indu, R.P.Gairola and M.N.Sharma, Thermodynamics of Impure Anharmonic Crystals. Int.J.Theor. Phys. 32, 861-878, 1993.
	In article		CrossRef
[18]	J. D. Launay, Solids State Physics Vol. 2, eds. F.Seitz and D. Turnbull, 220 (Academic Press, 1956).
	In article
[19]	Vinod Ashokan and B.D. Indu, Thermal conductivity of Bi-Sr-Ca–Cu–O superconductors, Thin Solid Films 518, e 28-e 30, 2010.
	In article
[20]	Vinod Ashokan, Ashok K. Dimri and B.D. Indu, Signature of electron-phonon interaction in high temperature superconductors, AIP Advances 1, 032101-16, 2011.
	In article		CrossRef
[21]	R. Bahadur and P. K. Sharma, Specific Heat Due to Substitutional Defects in Crystals, Solid State Commun. 15, 621-624, 1974.
	In article		CrossRef
[22]	R. A. Cowley, Phonons in Perfect Lattice and in Lattices with Point Imperfection, ed. R.W. H. Stevenson 170 (Oliver and Boyd Ltd., Edinburgh, 1966).
	In article
[23]	R. E. Peierls, Quantum Theory of Solids, (Clarendon Press, 1955) Chap. II.
	In article
[24]	N. R. Werthamer. Self-Consistent Phonon Formulation of Anharmonic Lattice Dyanmics, Phys. Rev. B 1, 572-581, 1970.
	In article		CrossRef
[25]	J. C. K. Hui and P. B. Allen, Thermodynamics of Anharmonic Crystals with Application to Nb, J. Phys. C 8, 2923-2935, 1975.
	In article
[26]	A. A. Maradudin, P. A. Flinn, Anharmonic Contribution to the Debye-Waller Factor, Phys. Rev. 129, 2529-2547, 1963.
	In article		CrossRef
[27]	R. C. Shukla, Analysis of the Debye Waller Factor and Multi-Phonon Series contributions to the Phonon Limited Resistivity of Metals, Phys. Rev. B 22. 5810-5817. 1980.
	In article		CrossRef
[28]	N. H. March, W. M Young and S. Sampanthar, The Many-Body Problem in Quantum Mechanics 276 (Cambridge Univ. Press, 1967).
	In article
[29]	R. P. Gairola, Lattice Viscosity of Anharmonic Crystals with Impurities, Intern. J. Mod. Phys. B 5, 815-824, 1991.
	In article		CrossRef
[30]	C. Fan, D. D. Do, D. Nicholson and E. Ustinov, Chemical potential, Helmholtz free energy and entropy of argon with kinetic Monte Carlo simulation, Molecular Physics (2013).
	In article
[31]	Kuwabara, K. Matsunaga, and I. Tanaka, Lattice dynamics and thermodynamical properties of silicon nitride polymorphs, Phys. Rev. B 78, 064104-1-11, 2008.
	In article		CrossRef
[32]	S. L. Shang, Y. Wang, D. E. Kim, C. L. Zacherl, Y. Du, and Z. K. Liu, Structural, vibrational and thermodynamic properties of ordered and disordered Ni1-xPtx alloys from first-principles calculations, Phys. Rev. B 83, 144204-1-13, 2011.
	In article		CrossRef