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            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)

where  and   are the phase speeds of the transverse and longitudinal waves, respectively.

            Because the total number of vibrations (wave modes) of a crystalline lattice with N nodes is equal to 3N-6, 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 (Bose-Einstein) 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. 244-46 or V.I.Smirnov 'Course of higher mathematics' (in Romanian),vol.II, Ed.Tehnica,1954,p.440-41):

            , 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>>):

                                                            .                                                      (3.2)

            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<<):

                                                 .                                      (3.3)

            Because the contribution of the free electrons to the specific heat of a solid increases linearly with its temperature (see chapter 7), it results that the free electron contribution prevails at very low temperatures (see Figure 4).

                                                            Fig. 4


1.      Ch.Kittel “Introduction to Solid State Physics”, J.Wiley&Sons, New York, 1971.

2.      Gh.Zet, D.Ursu “Condensed Matter Physics. Applications in Engineering” (in Romanian), Technical Publishing House, Bucharest, 1989, chap.4.

3.      C.Motoc “Condensed Matter Physics”, Univ.”Politehnica” Publ. House, Bucharest, 1993, §2.2.

4.      Ion Munteanu “Condensed Matter Physics” (in Romanian), 1st part, Hyperion Publishing House, Bucharest, 1995, chapter 4.

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