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THERMAL PROPERTIES OF SOLIDS
The most important thermal parameters of solids are: the specific heat, the thermal conductivity, the thermal expansion (dilatation) coefficient a.o. We will study here the quantum theory of the specific heat. As we will find here, the main contributions to the internal energy (and to the specific heat, implicitly) of solids are due to the crystalline lattice vibrations and to the free electrons, respectively.
Owing to the very different acoustic impedances of the studied solid sample (specimen) and of its surrounding medium, the vibrations of the crystalline lattice lead to standing waves, the opposite faces of the parallelepipedic sample having the same (node or of maximum amplitude) character.
§1.The spectral density of the number of independent standing waves
In order to simplify the next calculations, we will consider a cubic solid specimen, of side a (see Fig.1).
If the cube faces correspond to maxima of the standing waves, the time t and the x,y,z coordinate dependencies of the displacements u can be written as:
_{} , (1.1)
where _{} are the components of the wavevector _{}. Due to the boundary conditions: _{} , (1.2) Fig. 1
one obtains: _{} and: _{} , (1.3)
where _{} are arbitrary (negative or positive) integers.
It results the quantization of the standing waves:
_{} , (1.4)
where: _{} . (1.5)
Because the negative values (_{}) of the quantum numbers _{} correspond to phase jumps, but not to different standing waves (1.1), it results that the number _{} of the different standing waves of frequencies between _{} and _{}, corresponds to the number of points with positive integer coordinates in the space of real numbers _{} (see Fig.2).
Taking into account that one can build a cube of side 1 around each point of integer coordinates and these cubes (of volume 1) fill without holes or crossings the crown between the spheres of radii n and n+_{}n (where, according to relation (1.4): _{}), one finds that:
_{}_{}. (1.6)
Finally one finds that the volume spectral density of the number of independent wave states is: _{} , (1.7)
where V is the phase speed of the considered waves.
§2. The quantum (Debye's) theory of the contribution of crystalline lattice vibrations to the internal energy of solids
Taking into account that to each wave propagation direction (Oz) there correspond 2 independent (along the Ox and Oy axes) transverse waves and a (third) longitudinal one (see Figure 3), it results that the total volume spectral density of the number of independent wave states is: _{} , (2.1)
Because the total number of vibrations (wave modes) of a crystalline lattice with N nodes is equal to 3N6, one can deduce the maximum frequency _{}of the standing waves of this solid (of volume _{}) starting from the expression:
_{} .
For usual solid sample: N>>1, therefore one obtains:
_{} . (2.2) Taking into account that the sound waves are quantized similarly to the electromagnetic waves, and their quanta  the phonons satisfy the same quantum (BoseEinstein) distribution as the photons, it results that the contribution of the vibration modes (of frequencies between _{} and _{}) of the crystalline lattice nodes to the internal energy of the considered solid sample can be expressed by means of the average occupation number _{} of the energy states h_{} by bosons as: _{} . (2.3)
Using the symbols: _{} and: _{} (the Debye's temperature), the total contribution of the vibration modes of the crystalline lattice nodes to the internal energy of the considered solid sample can be written as:
_{} . (2.4)
Defining the Debye's function as: _{} , (2.5)
one can write the contribution of the crystalline lattice vibrations to the internal energy of a solid sample including _{} moles as:
_{} . (2.6)
For relatively high work temperatures (T>>_{}, therefore _{} and: _{}), the asymptotic expression of the Debye's function is:
_{} , (2.7)
therefore the corresponding asymptotic expression of the contribution of the crystalline lattice vibrations to the internal energy of the considered solid sample is:
_{} , (2.8)
which coincides with the experimental Dulong and Petit law (that agrees also with the theoretical results of the classical Statistical Physics).
Conversely, for relatively low work temperatures (T<<_{}), the Debye's function tends to the limit: _{} . (2.9)
Problem 1: Deduce the values of the sums _{} .
Solution: Starting from the series expansion of the complex variable function z.cotan z (see e.g. Th.Angheluta 'Course of the theory of complex variable functions' (in Romanian), Ed.Tehnica, 1957, p. 24446 or V.I.Smirnov 'Course of higher mathematics' (in Romanian),vol.II, Ed.Tehnica,1954,p.44041):
_{}, where the Bernoulli's numbers _{} are defined by the series expansion:
_{} ,
which implyes the recurrence relation with the symbolic expression: _{} . (2.10)
From the recurrence relations for n =2, 3, 4 and 5:
_{} _{} and: _{}, one obtains successively the values: _{} and: _{} .
Comparing the expressions of the general terms from the 2 series expansions of the function z.cotanz, one obtains: _{} , (2.11)
therefore for n =2: _{} . (2.12)
The recurrence relation (2.10) allows the successive determination of Bernoulli's numbers _{} and _{} for n =3, n =4 a.s.o., while the relation (2.11) will lead to the corresponding expressions of the sums _{}.
From relations (2.6),(2.9) and (2.12), it results the asymptotic expression  for relatively low work temperatures (T<<q )  of the contribution of the crystalline lattice vibrations to the internal energy of the considered solid sample:
_{} . (2.13)
Problem 2: Starting from the values of the longitudinal (Young) and shear elasticity moduli, respectively: _{} and _{} for copper (A=63.54, _{}), determine: a) the Poisson's coefficient _{}, as well as the propagation velocities of longitudinal and shear sound waves, respectively, in copper; b) the maximum frequency _{} of the crystalline lattice vibrations (of phonons) and the Debye's temperature of the copper.
Solution: a) From the relations: _{} and: _{} ,
one obtains: _{} .
b) The number of copper atoms in the volume unit of the considered rod is:
_{} .
Finally, one finds:
_{}_{} and: _{} .
§3.The quantum (Debye) theory of the contribution of the crystalline lattice vibrations to the specific heat of solids
Starting from the definition of the molar specific heat and neglecting the dilatation of solids, one finds: _{} . (3.1)
Using the relations (2.7) and (2.13), one obtains the asymptotic quantum (Debye) expressions of the crystalline lattice contributions to the specific heat of solids.
a) The case of relatively high work temperatures (T>>_{}):
_{} .
One finds that this result coincides with the classical expression (the Dulong and Petit's law) of the molar specific heat. It results that the contribution of the crystalline lattice vibrations to the specific heat prevails at high work temperatures (T>>_{}).
b) The case of relatively low work temperatures (T<<_{}):
_{} .
Fig. 4
REFERENCES
Ch.Kittel
“Introduction to
Gh.Zet,
D.Ursu “Condensed Matter Physics. Applications in Engineering” (in Romanian),
Technical Publishing House,
C.Motoc “Condensed Matter Physics”,
Univ.”Politehnica” Publ. House,
Ion
Munteanu “Condensed Matter Physics” (in Romanian), 1^{st} part,
Hyperion Publishing House,
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