Reappraising 1907 Einstein ’ s Model of Specific Heat

This article emphasizes that the Einstein and Debye models of specific heats of solids are correlated more tightly than currently acknowledged. This correlation is evidenced without need of additional hypotheses on the early Einstein model. The results are also extensible to the case of a system of fermions; as an example, the specific heat of the electron sea in metals is inferred in the frame of the proposed approach only.


Introduction
Einstein's aims are summarized by one of his most celebrated sentences: "I want to know all God's thoughts; all the rest are just details".With this intent, he set about elaborating a model of specific heat of solids to test the new Planck idea of energy quantization.For this reason Einstein implemented the quantization hypothesis of independent harmonic oscillators vibrating in a crystal lattice with a unique frequency.Of course he knew that this was an oversimplification of the problem; yet his primary attention was focused on the new born energy quantization, rather than on the actual vibrational spectrum of coupled oscillators.The Einstein naive model [1] is so well known that any further remark is superfluous: it is only worth quoting that the result was a brilliant validation of the energy quantization, able to predict the vanishing of specific heat at low temperatures and the empirical Dulong-Petit law of classical mechanics at high temperatures.
Shortly later, Debye [2] added the necessary "details" to the elementary Einstein approach: he refined the model introducing the statistical distribution of allowed vibra-tional frequencies, reasonably expected on the basis of the Born-Von Karman ideas [3]: their model of proper oscillations of linear chains of atoms with coupled motion implied the existence of lattice waves with periodic boundary condition.The group velocity of these waves introduced next the concept of phonon and the band structure of solids, which are in fact the most interesting consequences of these early studies; the characteristic temperature Θ is the key concept to correlate the elastic oscillations with the thermal, optical and electric properties of solids.
Next, the Fermi statistics extended these achievements to the electrons of the lattice.
A comprehensive exposition of these seminal papers and their subsequent evolution are found in several textbooks, for example [4].
The present article concerns in particular the first step of the path shortly outlined, i.e. that from Einstein to Debye.Usually the former model is acknowledged as a crucial contribution to the birth of the quantum physics; the latter model is a significant step forward not only for the accuracy with the specific heat which is calculated at low temperatures but also mostly for emphasizing the correlation between oscillation frequencies and elasticity constants of solids.Yet, simple considerations show that actually these models are more interconnected than their standard assessment taken for granted.The importance of elucidating this correlation is clear: Einstein's reasoning has essentially quantum basis, as it is also emphasized below in this paper that, Debye's reasoning regards a continuum body of solid matter described according to the classical elasticity theory.If these models could be someway linked, then even the oscillator frequency spectrum would automatically result entirely as a consequence of quantum principles.Just these considerations highlight the motivations of the present paper: -to infer the Debye specific heat directly from that of the Einstein model without need of additional "ad hoc" hypotheses; -to show that the present approach can be also extended to a system of fermions.
For sake of simplicity, the present paper assumes a monoatomic lattice of any symmetry.

Einstein's Theoretical Model and Its Extension
In the Einstein model of monoatomic perfect lattice, the energy of an oscillator is in fact nothing else but the mere BE energy statistical distribution ( ) where E β is the degeneracy factor of the distribution function.As the frequency is unique by assumption, the degeneracy is in fact given by the number of oscillators in the lattice; at the thermal equilibrium, all of them have the same energy.So, as the number of freedom degrees yields the number of possible oscillations, it follows 3 6 3 − ≈ , being N the number of lattice atoms.Here the Equation ( 1) is reasonably regarded as a starting point because of its general validity, direct manifestation of the quantum statistics.It is worth emphasizing that actually the lattice energy and specific heat at constant volume have been inferred by Einstein himself, whereas the appropriate statistical distribution law was introduced much later by Bose in 1920 [5].The Equation ( 2) are conveniently rewritten as follows for Avogadro's number of oscillators The specific heat is expressed as a correction of the asymptotic classical quantity 3R via a function of the parameter ζ only; also, E Θ is uniquely defined by the given ν .
Actually, however, the real lattice consists of coupled oscillators.To introduce the coupling mechanism, consider the vibration of one atom that propagates to the first neighbors by direct interaction, and then from these latter to the next neighbors and so on.In general one atom triggers a cooperative vibrational process that involves progressively an increasing number neighbor atoms; so the progressive coupling of oscillators is described by the number of neighbors involved and by the time necessary to spread the initial perturbation, which define the wavelength of the resulting collective wave and its propagation rate throughout the lattice.Indeed the frequency v ν λ = is related to the modulus of velocity v defining the wavelength λ ; it suggests that just this v describes the propagation velocity with which the vibrational interaction spreads an oscillation wave throughout the whole crystal lattice.The fact that the velocity is a vector explains intuitively the progressing of the initial vibrational perturbation all around the trigger atom without additional hypotheses.All this is compatible with the unique frequency ν simply rewriting that in turn splits into the three components of the respective vectors ( ) The components of the former Equation ( 4) define three orthogonal waves having different wavelengths i λ and propagating through the lattice along orthogonal direc- tions with related rates i v and frequencies i ν .In fact this is nothing else but the actu- alization of the previous reasoning: the vibration of the reference atom perturbs next neighbors regularly aligned along three space regular sequences of the lattice, as realistically expected in a 3D model.So, no further hypothesis has been actually introduced with respect to the original Einstein approach: in the unique frequency ν are actually hidden three frequencies related to the components inherent its propagation rate vector throughout the lattice.
In principle it is reasonable to guess that each i λ is related to the cell parameters of the crystal lattice along the respective direction; the extent of i λ corresponds thus to the number of elementary cells whose atoms concur to propagate the vibrational mo-tion.On the one hand i λ are expectedly different, being related to the different spac- ing of crystal planes along the propagation directions of the respective vibrational waves; on the other hand, even the respective propagation rates i v are in general dif- ferent depending on the symmetry properties of the crystal lattice.Consequently, owing to the different energies inherent the respective i ν of the three lattice waves, it is rea- sonable to rewrite the Equation (3) as the sum of three energy equations corresponding to the components of v ; hence ( ) .Proceeding exactly as before, the first Equation ( 2) turns now into ( ) ) in this way the early Einstein equation turns into a linear combination of three functions having the same form and weighed by the arbitrary coefficients i β coming from the unique E β , whereas the Equation (3) are a particular case of these equations for i Θ and i β all equal.To give this result a reasonable physical meaning, consider that if the crystal lattice is homogeneous and isotropic there is no reason to expect that the three orthogonal waves should appear with different coefficients in the linear combination; this would mean assigning "a priori" a preferential statistical weight to one of them, i.e. to one propagation direction of the initial vibrational perturbation along one specific sequence of lattice atoms.This however seems unjustifiable.Rather, considering that the lattice directions involved by the vibrational perturbation are physically equiprobable as concerns the total energy, the hypothesis that 1 is in principle rigorous for an infinite single crystal with perfect lattice; however it is reasonably assumed true, at least statistically, even for real polycrystalline materials with point and line lattice defects and grain boundaries.Moreover, defining the initial ratios i T Θ turn into an average quantity T Θ times the direction de- pendent quantities i ξ , which result proportional to the ratios i ν ν ; also, the Equa- tion (1) splits into a new equation where the three frequencies that define ν of E ε appear explicitly.The explicit expression of the specific heat results thus to be ( ) ( ) ( ) ( ) Also now the temperature still appears through the ratio T Θ times the quantities i ξ expectedly different for the three waves.In practice this expression is defined as the sum of functions of i ξ ζ , with ζ regarded as arbitrary parameter; yet it is explicitly calculable as a function of T , and thus comparable with the experimental data, once knowing the three values of i ξ .

Comparison with the Debye Model
A possible way to assess these results, is to compare the Equation (8) with the specific heat of the Debye model.After the early approach of Einstein, who did not introduce the frequency spectrum actually allowed in the lattice, this is the most famous and simplest model to calculate the specific heat of solids.As it is known, in this model the unique Einstein lattice frequency ν determining ε is replaced by a frequency distri- bution according to the 2 ν law; also, the interval of frequencies allowed to ν is up- per bounded by a postulated maximum frequency m ν , i.e. 0 m ν ν < ≤ .This model, in agreement with that based on the Born-Von Karman theoretical background almost simultaneously developed, is so known that the basic results only are reported here without further comments.Owing to the actual existence of several frequencies in principle admissible, the lattice vibrational energy is obtained integrating E ε over the giv- en frequency spectrum ( ) ) here D β is still the degeneracy factor of the BE distribution.Trivial manipulations of these formulas yield thus the well known result ( ) ( )  ( ) The Equations ( 3) and ( 10 On the other hand, however, the question arises: does the modified Equation (8) overcome both discrepancies without need of any "ad hoc" hypothesis, e.g.taking advantage of the fact that expectedly Assessing comparatively the Equations ( 8) and ( 10  one half of In effect this conclusion is expectable at 0 T > , because the lattice can be in the plain zero point energy state at the absolute zero only.So the actual vibrational level of the lattice is still one only, 1 ν , exactly as in the early Einstein mod- el; yet it appears here with the zero point energy too, as it is reasonably understandable.Since a free oscillator is characterized by the frequencies 2 n ν ν + , assume that the frequencies i ν fit the condition (7) putting if so, then In the following we take 1 n = to implement in the next calculations the condition of minimum vibrational energy.The third Equation (7) reads thus so the Equation ( 8) results expressed as the sum of one zero point wave, function of ξ , and two identical vibrational waves, both at the ground energy level, functions of In principle, even without specifying the constant parameter 1 ξ Θ , a universal curve of specific heat is still obtained as a function of the ratio ) in practice, however, assessing the validity of the Equation ( 14) by direct comparison with the experimental data of various materials as a function of T requires knowing 1 ξ Θ .In this respect, note that for the parameter 1 ξ ζ holds the boundary condi- A simple chance to assess the Equation ( 14) is to compare it with the Debye Equation (10): this is possible if 1 ξ is known, so that both equations are calculable as a function of ζ only.To this aim regard preliminarily 1 ξ as best fit parameter, whose numeri- cal value will be justified in the next section together with the physical meaning of 0 ν and ν to which is related Θ .Put therefore  temperature range where the reliability of the Debye model is well acknowledged.
Note that this agreement does not represent a mere numerical result of best fit between the linear combination of three Einstein functions (6) and the Debye function, for at least four reasons: 1) since the Equation ( 15) is justifiable, see the next Equation ( 26), the Equation (13) express a specific physical idea, rather than fulfilling a mere numerical purpose; 2) the coefficients i β have not been calculated according to standard best fit algo- rithms; 3) no "ad hoc" physical hypothesis has been purposely introduced to force this result, which has full theoretical character; 4) since the unique Einstein frequency waives the vibrational spectrum of the Debye model, the mere elaboration of the Equation (1) that yields (14) has in fact nothing to do with the elasticity theory.
Hence the conversion from the Equation (3) to the Debye-like Equation (8) cannot have numerical worth only, as it will be stressed in the next section.With these clarifications, the Equation ( 14) has its own self-contained physical meaning.The comparison with the Equation (10) has validation purpose only.The next section clarifies the reasons of this general agreement, while explaining also the small deviation between the curves observable in the Figure 2

Discussion
The Equation ( 15) is definable in the frame of the present quantum model only, and not outside it e.g.via ancillary considerations involving classical hints.According to the Equation (13), one oscillator is actually an atom randomly delocalized in a crystal plane and vibrating normally to this plane: so, as expected, its zero point energy is direct consequence of its position uncertainty on one plane of the elementary cell.Consider , being from a mere dimensional standpoint 3 V λ = .Strictly speaking, however, is more appropriate to regard 2 λ for example as x y λ λ , in which case the atom vibrates along the z direction.Actually the crystal plane defining the zero point energy is not rigidly fixed or uniquely definable; rather, the cyclic permutation of the indexes , , x y z implies three delocalization chances on the planes xy , xz and yz and corresponding orthogonal vibration di- rections, in agreement with the fact that the number of oscillators is three times that of the atoms.Moreover, as the three planes describe in fact the lattice volume of the elementary cell, the zero point energy fulfills both properties inherent its quantum nature: on the one hand it is naturally associated to the vibrational energy of the propagating wave, in agreement with the standard description of the quantum oscillators; on the other hand it also appears as expected consequence of the confinement of any particle in a volume of space, whose size is defined by the distance between neighbor atoms consistently with the lattice parameters of the elementary cell.
Four remarks summarize the present outcomes.
-The Equation (14) has general validity: no specific hypothesis about the kind of material has been introduced; the features of the lattice oscillator are defined by three i ξ only, in order to account for its zero point energy and ground vibrational level regardless of the specific kind of atom.
-The conversion of the plain Equation (3) into the form (8) does not require assuming a continuous body of matter and does not involve thermodynamic quantities, like for example the compressibility, which are unnecessary and bypassed.
-The Debye-like formula (14) has full quantum meaning, without reference to classical concepts; rather, reverting the conceptual path of Debye, it is possible to infer as a corollary of this equation his background considerations about elastic constants of the material and vibrational spectrum.In effect the position (15) yields D Θ ≡ Θ , as it appears comparing the Equations ( 15) and ( 16).
-The Figure 2 does not require the detailed analysis about the longitudinal or transversal character of the lattice waves nor about the microscopic interaction mechanism between atoms introduced by the Equation (4); it is enough to admit that the coupling of lattice oscillators is induced by propagating the perturbation of one trigger atom, in turn due to its position uncertainty in the lattice site.
The next considerations of this section highlight further these positions.The Debye approach refined the early Einstein model of specific heat at the conceptual cost of several approximations, first of all postulating an upper cutoff frequency m ν to bypass the consequence of an infinite number of proper oscillations in principle admissible in a continuous body of matter.Clearly the key assumption of "continuum" is senseless for high frequencies, whose vibrational wavelengths are so short to be smaller than or comparable with the crystal spacing of atom lattice; also, the 2 ν spec- trum is sensible for low vibrational frequencies only, i.e. whose wavelengths are so extended to involve several atoms.Eventually, the lower integration limit = 0 ν is of course a numerical extrapolation rather than a real physical value.Yet the successful intuition of Debye was that just these low energy waves that overcome the interatomic distances in the lattice are related to the elastic properties of solids: at low T the den- sity of spectral lines described by the frequency distribution function is a satisfactory approximation.This point, well discussed in the Debye original paper, is shortly reappraised here considering the amount m of matter contained in a volume of lattice given by 1 .
The notation emphasizes that the lattice volume defined in this way is just that including all atoms oscillating with wavelengths j λ , i.e. it is the volume including coupled oscillators.As previously highlighted, the wavelengths are reasonably related to the lattice parameters of the elementary cell via the respective average velocities j v describing the displacement rate of atoms around the equilibrium lattice sites; κ is an appropriate proportionality factor added for sake of generality.If, for example, κ is defined by the product of three integers 1 , then one obtains two equations 3 , , .
In the second equation, j a are the shortest lengths that repeated j n times repro- duce the j λ periodicity; these positions account for the actual extent of V λ via the arbitrary j n , as in ( )( )( ) In the Equation (17) the vibrational wavelengths determine the size of V λ , which therefore defines the local density ρ and energy density η due to all lattice oscillators involved by the extent of j λ .So V λ represents the size of the coupling volume.
Let be then being m the total mass in V λ and , the lattice energy is ( ) As expected, specific properties of the material appear in ε ; thus, whatever 0 v might be, δ ν δ ν ξ = according to the Equations (13).Clearly δ ν is the frequency range around the average value ν to which is related, at the first order, the given interval δ  of local lattice energy.
The Equation ( 21) is crucial to explain the Figure 2: assuming that the former addend is negligible with respect to the second one, i.e. if y is approximately independent of ν or even constant, then δ  reduces to the form of the integrand in the Equation (9).In effect the frequency spectrum 2 ν δν , here regarded as 2 ν δ ν , agrees with that of the Debye model, but implies that the dynamics of lattice atoms is approximately independent of the average wave frequency.This is intuitively reasonable only for vibrational wavelengths comparable or small enough with respect to the lattice parameters, when in effect the interaction between neighbor atoms in contiguous lattice sites becomes negligible; the fact that in this case V λ concerns independent Einstein oscillators explains why at high T the curves VE c and VD c tend to merge into the unique classical limit.In effect the most significant deviation of the Einstein curve with respect to the Debye curve is at low T , when the low vibrational energies become significant: as previously emphasized, long range wavelengths necessarily imply by definition coupled oscillators.This appears in the Equation ( 17): on the one hand V λ decreases at increasing ν , i.e.T , until its size is of the order of the volume of the elementary cell, on the other hand m ν corresponds just for this reason to this minimum value of V λ .
The lattice energy was implemented by Debye via the number 3 z FVν = of oscillators with a given frequency inferred from the elasticity theory, as previously found in the Equation (18); normalizing then ν ν law via the elasticity theory, while acknowledging however its inherent approx- imate character.All this implies that the 3 T dependence of the specific heat is to be expected only in the range of temperatures where holds the Debye spectrum; in the range of temperatures where prevails the ( ) frequency spectrum, however, the 3 T law does not hold.The Figure 2 agrees with the idea that this law is actually not extensible down to the absolute zero.
Nevertheless the Debye approach, as it is, represents valuable enhancement of the early Einstein mode, particularly significant at low T ; its acknowledged accuracy represents therefore reliable reference to assess the physical significance of the steps from the Equations ( 3) to (8).
As concerns the Equation ( 15), regard the second addend of the Equation (21) as 2 const ν δ ν only; introducing it in the Equation ( 1), including const into E β and then integrating as done in the Equation ( 9) means just replicating the Debye approach, for which holds therefore the second Equation ( 11).This is indeed an obvious condition to overlap successfully the Equations ( 9) and ( 14).Next, solving Despite these steps from (17) to (21) do not involve classical hints, this way to infer the Equation ( 15) is however indirect: it requires implementing the link just exposed of the Equation ( 14) with the Debye Equation (10), and is thus unsatisfactory.Below, a more fundamental way is proposed to show that the physical background of the second Equation (11) has full quantum base directly related to the Equation (13) regardless of the frequency distribution spectrum: in this way the Equation (15) shows its inherent physical meaning, rather than being mere numerical result of calculations.
To describe how the lattice atom interacts with the neighbors, let us introduce its momentum transferred towards an arbitrary surface surrounding the equilibrium lattice site: the momentum exchanged with neighbor atoms accounts for its coupling and shows that the ratio 0 ν ν of the Equations ( 12) and ( 13) is related just to the con- cept of coupling process.According to the Equations ( 17) and (18), let v be the aver- age displacement velocity of an atom oscillating around its equilibrium position and The left hand side of the first equation defines the element of solid angle the fact that it is arbitrary, and thus purposely definable, avoids the difficulty of regarding it as new frequency hardly explainable in the present frame.Accordingly, link this result with 1 ξ ζ of the Equation ( 14), of interest for the present purposes and expected recalling the form of the Equation (13).Let be at the inflexion point of the curve in effect 1 2π is just the value of the second Equation (11), whereas the second posi- tion follows thinking that in general the zero point energy 0 2 hν is merely due to the random delocalization of any particle confined in a region of space.As , then pv pv = would yield 1 2 ξ = ; the fact that actually 1 0.905 ξ = shows that the momentum p is not uniformly transferred all around the oscillating atom.This is reasonable: the interaction driven coupling preferentially points towards specific lattice directions where other atoms are found to which the vibrational momentum is effectively transferred.So the integration of momentum transfer across the surface element defined by d ′ Ω is smaller than that expected for a continuous solid or amorphous microstructure.
At this point integrating at constant volume it is possible to find the lattice internal energy from which one calculates the Helmholtz lattice free energy U TS U S As a closing remark, note that at very low T the Equation ( 14) can be also rewritten thinking one vibration wave only and two zero point terms; in other words, it is also physically admissible the following formula of specific heat, obtained simply moving the coefficient 2 from the second to the first addend, ( to which are related the energy ( ) and entropy ( ) The notation reflects preliminary indications, according which these quantities could be related to the superfluid state, of course with different 1 ξ and ν and Θ .
Further investigation is in progress on this possible implication of the present model.

Further Implications of the Present Model
The main purpose of the present paper was to highlight that the Einstein and Debye approaches are directly correlated when accounting appropriately for the zero point energy of the crystal lattice: the Equation ( 5) describes the thermal oscillators of the lattice introducing the Einstein initial Equation (1) as a straightforward consequence of the Bose statistics and allows to infer the Debye-like Equation ( 14) calculable as a function of T .Yet a similar kind of approach, owing to its generality, should be in prin- ciple adequate to describe even a system of fermions, for example the free electron gas in the lattice.Indeed this section shows that to this purpose it is enough to start from ( ) with the same physical meaning of degeneracy factor F β and still with the unique frequency ν .Besides the intrinsic importance of this topic, the following considera- tions are significant to confirm further the validity of the steps leading from the Equation (1) to the Equation (5).In fact the extension proposed here of the present approach implies merely finding how the physical differences between either statistical distribution compel reformulating the Equations ( 6) to (8) once having replaced the Equation ( 1) with the Equation ( 27).Let us rewrite first the Equation ( 5) with the same notations as ( ) RT β ∑ ; so, without chance of obtaining this limit case, is also missing the necessity of and thus 1 i β = .Furthermore fails also the idea of replacing i Θ with a unique Θ , as the three waves must be in different quantum states: this of course implies different i Θ .
Instead still holds the idea of equivalence of the three space directions along which is transmitted the interaction between particles; it is irrelevant in this respect that, actually, in this case the Coulomb electron interaction replaces vibrational interaction of lattice oscillators.
and then ( ) The form of the function el β must fulfill three boundary conditions: 1) the limit ( ) ( ) in order that ( ) ( ) ( ) The reason of having introduced in the Equation (32) the smallest one among the three i Θ is coherent with the chance of defining uniquely el β independently of the in- dex i ; so, for 0 T → in one term only of the sum the ratio is different from zero, whereas the remaining two vanish according to ( ) ( ) . Also now one wave accounts for the zero point energy.Hence ( ) ( ) ( ) owing to the positions (32), this is a minimum zero point energy; hence el β ′ does not vanish at 0 T = .It is formally convenient now to express q and 3 q multiplicative proportionality constants to express 2 Θ and 3 Θ ; both constants are by definition > 1 according to the Equation (32).So the first Equa- tion (34) reads explicitly whereas VF c follows by consequence.Despite the function el β ′ is still unknown, let us express it through its series expan- T is in principle admissible, we have three chances to check.Trivial calculations yield: Owing to the zero order coefficient a , the zero point energy is correct in all cases.The first case yields sensible ( ) VF c ∞ but wrong ( ) 0 VF c .In the third case holds exactly the contrary.In the second case both limits are wrong.As the term 2 T only accounts well for VF c at low temperatures, whereas T only accounts well for its high temperature limit, these results suggest that the correct form of el β ′ is the one that interpolates ap- propriately both chances: where the constant c plays the role of reduced temperature.In conclusion, the Fermi statistics compels at the first order the linear T dependence of the specific heat, as it is known.Of course several considerations are possible about how the constants appearing in this expression are related to the physical properties of metals.However these considerations, going back to Fermi's time, are omitted here for brevity: the purpose of this extension is simply to demonstrate that the Equation (27) only is inherently enough to obtain the well known Equation (36), likewise as the Equation (1) only is inherently enough to obtain the well known Debye-like Equation (14).

Conclusions
Unfortunately neither Einstein nor Debye realized that actually their theoretical models coincide merely handling appropriately a unique lattice frequency, even without need of implementing the classical theory of elasticity.
In this respect, the key point to improve the early Einstein result is not the frequency distribution spectrum 2 ν but the quantum zero point energy, being actually the for- mer, a consequence of the latter.
The conceptual basis of all Debye considerations and its implications on the link between thermodynamic and elastic properties of solids have actually full quantum origin, once regarding the latter as a mere corollary of the Equation ( 14).
The Equations ( 5) and ( 6) are easily generalizable to the case of a system of fermions.
Eventually it is noted that the model is easily generalizable to describe phenomena like the superfluidity as well, simply admitting that in the case of a fluid 1 ξ of the Eq- uation (15) is not necessarily a constant; more specifically, the Equation (14) shows that if 1 ξ decreases then V c′ increases.If a good physical reason exists to demonstrate that the right hand side of the first equation tends to i i kT β ∑ ; so the second equation ensures the consistency with the asymptotic behavior of the Equation (1) for T h k ν 

1 DT
) differ for two reasons: because of their different ζ pro- files, especially at Θ  , and because of their different inflexion temperatures with notation emphasizing that the values are calculated at the inflexion points of the curves VE c and VD c vs ζ .In general the inflection point of V c vs ζ has significant physical meaning, as it marks the transition between quantum and classical behavior of the oscillators: correspondingly, the rising rate of specific heat V it decreases and tends to vanish asymptotically when V c reaches the classical Dulong-Petit limit for ζ → ∞ .The Figure 1 emphasizes that the respective curves not only have different ζ profiles but are also shifted along the ζ axis, although tending both to the same asymptotic limit for 1 ζ  .On the one hand, the physical reasons of the intrinsic disagreement of VE c and VD c especially at low T are well known, in particular as concerns D E Θ ≠ Θ .
) needs a general reasoning to guess the numerical values of the amounts i ξ and to calculate both equations for arbi- trary values of the free parameter ζ only; this latter becomes thus the unique variable as a function of which are compared VE c′ and VD c .Assuming that the energies of the oscillators are quantized, it is reasonable to expect that one of them, say 3 ν , accounts for the zero point energy in the lattice, whereas the remaining two, 1 ν and 2 ν , ac- count for the ground vibrational energy level of the lattice; i.e.

3 hν should beFigure 1 .
Figure 1.Specific heats V c of the Debye and Einstein models vs T ζ = Θ .
course this intentional choice of 1 ξ makes Θ consistent with the Debye temperature D Θ in the Equation (11), whence the importance of justifying below the Equation (15) from a theoretical point of view.The Equations (8) and (14) calculated with the given value of 1 ξ have been plotted together in the range 0.01 1 ζ ≤ ≤ , which seems appropriate to compare lattice specific heats and experimental data.As D Θ ranges typically between about 100 K (alkali metals) and 2000 K (diamond), 0.01 ζ ≈ corresponds to T not greater than a few tens K degrees only; thus below 0.01 ζ ≈ the electron specific heat becomes comparable or predominant with respect to the mere lattice contribution.The result reported in the Figure 2 shows that V c′ overlaps reasonably well the whole Debye curve, at least in the range of temperatures where the lattice specific heat alone realistically represents the experimental data.It appears in particular that even the 3 T dependence of lattice specific heat is correctly reproduced by the Equation (14) in the range inf 0.1 ζ ζ < <   , directly comparable with the experimental data: 0.1 ζ ≈ corresponds indeed to T of the order of about ten to hundred K degrees, where the lattice specific heat overcomes in general the electron contribution.The Figure 2 makes superfluous the direct comparison of the Equation (14) with the experimental data at
appears in a natural way.In the Debye model 3 V F ν (with Author's notation) defines the number of proper frequencies below the upper threshold m ν via the func- tion F of elasticity constants and density ρ : in effect, putting F has physical dimensions 3 velocity − and normalizing x y z n n n to 3N for m supports the validity of the Equation (17).

2 vwhereas 2 v
the average square velocity of vibrational displacement of lattice atoms in the coupling volume; ε takes into account the virial theorem, links these considerations to the lattice T .Note now that 2 x y z v v v v has physical dimensions of reciprocal velocity, so that the energy density can be rewritten as .Eventually, since the energy density per unit volume of solid is by definition proportional to the number of oscillators contained in V , which is in turn representative of the amount of oscillating mass and thus related to 2 v ρ


essarily constant stimulates comparing the approximations of dz either with 2 this last term can be calculated knowing the function y .Accordingly, once normalizing again alternative to that of Debye.In principle, when significant deviations from the 2 ν law are expected, the first addend of the Equation (21) should provide the appropriate correction to the approximate Debye spectrum.As actually is realistically expectable in general a concurrent contribution of both terms, these considerations explain the small discrepancy between VD c and V c′ visible in the Figure2at very low values of ζ , where however in most cases the lattice specific heat alone does not represent the experimental specific heat, e.g. in metals.Approaching quantitatively this kind of problem requires details about the thermodynamics of matter: this topic, inherent y , leads to the domain remarkably explored is still to define o ν as that where the right hand side vanishes, but not the left hand side.Hence around the boundaries of the range a small range of frequencies ν around D ν that defines D y .Un- fortunately Debye did not find both alternatives because of his classical way to infer the 2 d

1 ξ, one finds that 1 ξ
, which is now the only unknown once having calculated inf 0.164 ζ = is of course just that of the Equation (15): indeed just this latter yields the Equation (16).
momentum; i.e. the components of v are that previously introduced to define 2 v of the Equation (19).Also, consider the surface element ds δ = s n around the equilibrium site; the local unit vector n is oriented outwards normally to the local surface.As d d dividing both sides by λ one finds that ( ) ( ) function of m and * ν a frequency that by de- finition does not depend upon dΩ , as instead ν does by reasons that will appear just below.Therefore the first Equation (23) yields These positions are possible because all related quantities are arbitrary.Integrating both sides, one finds this point, let us try to plug * ν in the frame of the model hitherto formulated;

1 ξ
the propagation directions of the respective waves, according to the Equation (12) one concludes that 3 results to be just that previously introduced in the Equation (15).This shows that in effect both positions (25) fit 1ξ and * ν to the three vibrational and ze- ro point i ξ (13) without introducing further frequencies.Moreover note that in gen- eral h hv pv ν λ = = , with p and v defined in the Equation (23); hence the Equa- tion (26) reads

1
29)owing to the coefficient el β , appear in the Equation (28) both i ν ′ and i ν .It is inter- of oscillators is changed with respect to the initial N .Nevertheless the primed and unprimed quantities are correlated, the correlation function being just el β .Let be therefore( ) , via appropriate coefficients , , a b c of the series.Now let us account for the three boundary conditions.It is clear that the highest power cannot be 2 > , otherwise 2 el i β ζ in the second addend of the Equation (31) would diverge with T .Since the power up to 2

0 T
In conclusion, replacing just the last form of el β ′ in the Equation (35) and deriving with respect to T , one should find the correct form of F ε and thus of VF c .This formula is not quoted explicitly for brevity.To check this conclusion in a crucial case, i.e. at low T , it is enough to expand in series this VF c calculated from the Equation (35), taking advantage of the fact that the limits for 0 = are in fact finite in the present model: i.e.

1 ξ
is allowed to decrease at an appropriate value of * * T ζ = Θ , then simple calcula- tions show that at this particular *T the Equation (14) is in fact compatible with the analytical form of the so called λ -point.In principle this conclusion is physically expectable in the present model without additional hypotheses; it simply follows from the chance of defining the transition between a thermodynamic state with energy and entropy U and S to another state characterized by ( ) zero point energy and vibrational wave energy consistent with either chance of regarding the components of the position (4) could be the key concept to describe the low T behavior of3 He and4 He , of course in agreement with the va- nishing of both V c and ( ) sf V c at the absolute zero.Activity is in advanced progress on this topic.
As el β is no longer a constant, it can be nothing else but a function of the only quantity, kT , that does not depend on the index i .So el β ≠ ; should hold by consequence the position ( ).
be zero, because of the zero point energy; 2) the limit( )