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Instabilities and Self-Organization in MaterialsVolume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology$

Nasr Ghoniem and Daniel Walgraef

Print publication date: 2008

Print ISBN-13: 9780199298686

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199298686.001.0001

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(p.477) APPENDIX C DEBYE THEORY OF SOLIDS

(p.477) APPENDIX C DEBYE THEORY OF SOLIDS

Source:
Instabilities and Self-Organization in Materials
Publisher:
Oxford University Press

In his theory of solids, Debye [509] also assumed that a solid may be considered as an ordered array of atoms, each of them oscillating around their lattice site. However, contrary to Einstein's theory, these oscillators are not independent, and the classical Hamiltonian of this system may be written as:

(C.1)
H ( q N , p N ) = i = 1 3 N p i 2 2 m + i , j 3 N A i j q i q j
where m is the mass of the oscillators and the matrix A ij represents the interactions between oscillators. After diagonalization, individual oscillators are replaced by collective modes, that is, sound waves or phonons, which behave as independent oscillators, but with different frequencies [510]. The Hamiltonian then becomes
(C.2)
H ( Q N , P N ) = i = 1 3 N p i 2 2 m + i , j 3 N m ω i 2 2 Q i 2
where Q i and P i are the canonical coordinates of the collective modes, and the corresponding Hamiltonian operator is
(C.3)
H ^ = i = 1 3 N ω i ( N ^ i + 1 2 )
and the partition function becomes
(C.4)
Z N ( T , V ) = T r { exp ( β i = 1 3 N ω i ( N ^ + 1 2 ) ) }
and the Helmholtz free energy of a Debye solid is thus
(C.5)
A ( T , V ) = 2 i = 1 3 N ω i + k B T i = 1 3 N ln ( 1 e β ω i ) .
It is the fundamental equation of a harmonic solid. To evaluate this free energy, one has to know the frequency spectrum of the solid. The Debye theory neglects (p.478) the crystalline structure of the solid, and considers it as a continuous isotropic medium. In such a medium, the dispersion relation for sound waves is given by ω = kc, where c is the speed of sound and k = 2π/λ is the wavenumber of any possible vibration in the crystal. These vibrations are all standing waves, sustained by the crystal. These are the same as for a string pinned at both crystal ends. Hence, in one dimension, λ = 2L/n, with n = 1, 2,…, ∞, where L is the crystal length in this direction. In general, an arbitrary wave of this type will have the following dispersion relation
(C.6)
ω i 2 = ω x i 2 + ω y i 2 + ω z i 2 = c 2 { ( π n x i L x ) 2 + ( π n y i L y ) 2 + ( π n z i L z ) 2 }
where L x, L y and L z are the crystal lengths in the three space dimensions. Let us consider an isotropic crystal, and L x = L y = L z = L. Since n i are integers, the distance between adjacent points is, in the (ω x, ω y, ω z) space, πc/L in each space direction. The volume per point is thus (πc/L)3, and the number of points per unit volume is (L/πc)3. The total number n(ω) of allowed values of ω i < ω is then given by
(C.7)
n ( ω ) = 1 8 . 4 π ω 3 3 . L 3 π 3 c 3
and, going to a continuous description, which is justified by the huge number of available modes (3N ≫ 1), the number of points in the range (ω, ω + ), is given by
(C.8)
d n = L 3 2 π 2 c 3 ω 2 d ω .
Two remarks are now in order. First, there are usually two transverse sound modes and one longitudinal mode in a solid, and dn should in fact be written as
(C.9)
d n = L 3 2 π 2 ( 2 c t 3 + 1 c l 3 ) ω 2 d ω
where c t and c l are the transverse and longitudinal sound velocity, respectively. Second, since the number of modes in finite, though very large, there is a maximum available frequency. If we call this frequency ω D, it is given by the consistency relation
(C.10)
3 N = d n = 0 ω D L 3 2 π 2 ( 2 c t 3 + 1 c 3 ) ω 2 d ω = L 3 2 π 2 ( 2 c t 3 + 1 c 3 ) ω 3 3
or
(C.11)
ω D 3 = 18 N π 2 L 3 ( 2 c t 3 + 1 c l 3 ) 1
(p.479) where ω D is called the Debye frequency. We are now able to obtain the free energy, and, on replacing the summations by integrations on ω, one has
(C.12)
A ( T , V , N ) = 9 N 2 ω D 3 0 ω D ω 3 d ω + 9 N k B T ω D 3 0 ω D ln ( 1 e β ω ) ω 2 d ω .
The heat capacity becomes
(C.13)
C V = T ( S T V . N ) = 9 N k B T 3 θ D 3 0 θ D / T d x x 4 e x ( e x 1 ) 2
with x = βħω and where the Debye temperature is defined as θ D = ħω D/k B. The high- and low-temperature limit are easily obtained from equation (C.13). At high temperatures (T → ∞), one finds
(C.14)
C V 9 N k B T 3 θ D 3 0 θ D T x 2 d x = 3 N k B
which is the classical result. At low temperature (T → 0), one has
(C.15)
C V 9 N k B T 3 θ D 3 0 d x x 4 e x ( e x 1 ) 2 = 12 N k B π 4 5 T 3 θ D 3
which gives the experimentally observed T 3 decrease of the heat capacity at low temperature. The Debye results are in good agreement with experiment. However in real solids, the fitting between Debye results and experimental data shows a slight temperature dependence of the Debye temperature, which may be attributed to the remaining assumptions of the theory [511].