Fundamentals of the Physics of Solids / 11-Dynamics of Crystal Lattices
.pdf11.5 Localized Lattice Vibrations |
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they are shifted significantly, almost to the frequency just below. Therefore somewhere around the middle of the spectrum frequencies are denser than elsewhere. Every atom of the crystal participates in these vibrations, however the displacement amplitude is much larger for the impurity than for distant atoms. It looks as if the impurity were resonating, therefore such states are called resonances.
11.5.2 Impurities in a Three-Dimensional Lattice
The previous method can now be generalized to three-dimensional lattices in which one atom is substituted by another atom of di erent mass, and thus force constants around the impurity are di erent from those in a regular crystal. The classical equation of motion (11.1.28) is replaced by a system of equations in which the mass Mm,μ of the μth atom in the mth primitive cell depends on the cell index m as well, and owing to the breakdown of translational symmetry force constants do not depend only on the separation of lattice points any more:
Mm,μu¨α(m, μ, t) = − |
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Φαβμν (m, n)uβ (n, ν, t) . |
(11.5.15) |
n,ν,β
Fourier transformation with respect to time gives
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ω2Mm,μuα(m, μ, ω) = |
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(m, n)uβ (n, ν, ω) . |
(11.5.16) |
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n,ν,β
Rewriting this in the form
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n,ν,β ω2Mm,μδmnδμν δαβ − Φαβμν (m, n) uβ (n, ν, ω) = 0 , |
(11.5.17) |
the equation has nontrivial solutions if the matrix |
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Lm,μ,α;n,ν,β (ω2) = ω2Mm,μδmnδμν δαβ − Φαβμν (m, n) |
(11.5.18) |
made up of the coe cients of uβ (n, ν, ω) has a vanishing determinant. In lattices without impurities, where translational invariance may be exploited, after a Fourier transformation with respect to the lattice points only a 3p ×3p matrix needs to be diagonalized, since the equations for the Fourier coe cients of di erent qs are not coupled. In contrast, when the crystal has an impurity, one has to deal with a 3pN ×3pN matrix. Nevertheless similar statements can be made about the main features of the excitations as in the one-dimensional case.
For notational simplicity, we shall assume that the basis of the crystal is monatomic and that the impurity atom of mass M0 sits at the lattice point Rm = 0; moreover, we shall again neglect any modifications in the force constants. The equation of motion for the impurity atom is then
382 11 Dynamics of Crystal Lattices
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M0u¨α(0) = − Φαβ (n)uβ (n) . |
(11.5.19) |
n,β
Using the notation M = M0 − M , the equation of motion can be rewritten as
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M u¨α(0) + Φαβ (n)uβ (n) + M u¨α(0) = 0 . |
(11.5.20) |
n,β
Together with the equations of motion for other atoms, this can be written in the common form
M u¨α(m) + |
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Φαβ (m − n)uβ (n) + δm0 M u¨α(0) = 0 |
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(11.5.21) |
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or, using the Fourier transforms with respect to time, |
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ω2M δmnδαβ − Φαβ (m − n) uβ (n) + δm0ω2 |
M uα(0) |
= 0 . |
(11.5.22) |
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n,β
The expression for matrix L is then
Lm,α;n,β (ω2) = ω2M δmnδαβ − Φαβ (m − n) + δαβ δm0ω2 M . (11.5.23)
The first two terms on the right-hand side are recognized as the matrix L0 that governs the vibrations of pure crystals. Separating this leads to
Lm,α;n,β(ω2) = Lm,α0 ;n,β(ω2) + δLm,α;n,β(ω2) , |
(11.5.24) |
where |
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δLm,α;n,β (ω2) = δm0δαβ ω2 M . |
(11.5.25) |
Next the inverse of L0 is introduced through the definition |
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Rm,α;m ,α (ω2)Lm0 ,α ;n,β (ω2) = δmnδαβ . |
(11.5.26) |
m ,α
Since the eigenvectors of matrix L0 are the polarization vectors and its eigenfrequencies are the vibrational frequencies of the pure crystal, it is easily seen that
Rm,α;n,β (ω2) = 1
M N
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eiq·(Rm −Rn ) . |
(11.5.27) |
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The equation of motion (11.5.22) may be written in the concise form (L0 + δL)u = 0. Multiplying it from the left by R,
(1 + R · δL)u = 0 |
(11.5.28) |
is obtained. Using the above forms of R and δL, the equation governing the displacement of the impurity atom is
11.6 The Specific Heat of Classical Lattices |
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1 + M N |
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uα(0) = 0 . |
(11.5.29) |
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The condition of self-consistency is now
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= 1 . |
(11.5.30) |
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Its structure is very similar to that of the condition obtained for a onedimensional chain with an impurity. Using similar graphical methods for determining the solutions, vibrational frequencies are usually found to lie inside the original quasicontinuum, slightly shifted with respect to the unperturbed values. For heavy impurities the deformation of the spectrum is such that a resonance may show up inside it, while for light impurities a frequency that corresponds to a localized vibration may appear above the continuum. If the Debye spectrum (to be discussed in the next chapter) is used for characterizing the unperturbed lattice, such a localized vibration is observed to appear for any M < 0. Numerical calculations show that when a more realistic spectrum is chosen, localized vibrations show up only above a certain mass di erence.
11.6 The Specific Heat of Classical Lattices
Having established that the thermal motion of ions about their equilibrium positions may be described in terms of normal modes, it is a relatively simple matter to determine the thermal properties of a classical crystal. According to statistical mechanics, the thermal energy of the crystal may be derived from the formula
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dΓ He−βH , |
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> dΓ e−βH |
(11.6.1) |
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where integration is over the |
phase space spanned by the variables u(m, μ) |
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and P (m, μ), that is |
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dΓ = |
du(m, μ) dP (m, μ) . |
(11.6.2) |
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m,μ
Calculations are performed more easily when displacements and momenta are expressed in terms of normal coordinates. Using the form (11.3.30) for the Hamiltonian, the volume element in the phase space (Q, P ) is
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dΓ = d|Qλ(q)| d|Pλ(q)| . |
(11.6.3) |
q,λ
The total energy is the sum over all independent modes,
384 11 Dynamics of Crystal Lattices |
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E = |
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dΓλ(q) Hλ(q)e−βHλ (q) , |
(11.6.4) |
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dΓλ(q) e−βHλ (q) |
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where, in accordance with (11.3.30), |
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Hλ(q) = 21 |
|Pλ(q)|2 + ωλ2 (q)|Qλ(q)|2! . |
(11.6.5) |
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Owing to their quadratic form, the contributions of the kinetic and potential energies of a vibrational mode are both kBT /2, as established by the equipartition theorem in statistical physics. Since each of the 3N p possible modes contribute kBT ,
E = 3N p kBT . |
(11.6.6) |
The contribution of ionic vibrations to the specific heat is then
CV = |
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= 3N p kB . |
(11.6.7) |
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For the molar heat (specific heat per mole) of monatomic solids this gives the classical result expressed by the Dulong–Petit law :9
CV = 3pR = 24.943 J mol−1 K−1 = 5.958 cal mol−1 K−1. |
(11.6.8) |
Around and above room temperature, the measured value of the specific heat is close to the Dulong–Petit value in most cases. By the end of the 19th century it had become clear that at lower temperatures the specific heat drops. Today it is also known that it vanishes at the absolute zero of temperature, as shown in Fig. 11.15.
This departure from the theoretical value casts doubts on the applicability of the used approximations. Two important approximations were made: the
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Sn
3NkB
Cu
100 200 300 T (K)
Fig. 11.15. Temperature dependence of the specific heat for (a) noble-gas crystals; (b) some metals
9 P. L. Dulong and A. T. Petit, 1819.
11.6 The Specific Heat of Classical Lattices |
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harmonic expansion of the potential and the classical treatment of the vibrations. In the next chapter we shall first examine the role of quantum e ects, and then analyze the role of the terms beyond the harmonic approximation.
Further Reading
1.M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford Classic Texts in the Physical Sciences, Oxford University Press, Oxford (1998).
2.P. Brüesch, Phonons: Theory and Experiments I, Lattice Dynamics and Models of Interatomic Forces, Springer-Verlag, Berlin (1982).
3.A. A. Maradudin, E. W. Montroll, G. H. Weiss, I. P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Second Edition, in: Solid State Physics, Supplement 3, Academic Press, New York (1971).
4.J. A. Reissland, The Physics of Phonons, John Wiley & Sons Ltd., London (1973).