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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.

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 [

Shortly later, Debye [

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 [

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.

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

have been inferred by Einstein himself, whereas the appropriate statistical distribution law was introduced much later by Bose in 1920 [

The specific heat is expressed as a correction of the asymptotic classical quantity

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

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

In principle it is reasonable to guess that each

If

whence

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

the initial ratios

Also now the temperature still appears through the ratio

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

here

Also in this case

The Equations (3) and (10) differ for two reasons: because of their different

with notation emphasizing that the values are calculated at the inflexion points of the curves

Assessing comparatively the Equations (8) and (10) needs a general reasoning to guess the numerical values of the amounts

one half of

if so, then

In the following we take

so the Equation (8) results expressed as the sum of one zero point wave, function of

In principle, even without specifying the constant parameter

in practice, however, assessing the validity of the Equation (14) by direct comparison with the experimental data of various materials as a function of

A simple chance to assess the Equation (14) is to compare it with the Debye Equation (10): this is possible if

which yields

Of course this intentional choice of

The result reported in the

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

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

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 that

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

-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

-The

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

The notation emphasizes that the lattice volume defined in this way is just that including all atoms oscillating with wavelengths

then one obtains two equations

In the second equation,

In the Equation (17) the vibrational wavelengths determine the size of

being

Moreover, putting

which implies that

As expected, specific properties of the material appear in

being

The Equation (21) is crucial to explain the

The lattice energy was implemented by Debye via the number

Approaching quantitatively this kind of problem requires details about the thermodynamics of matter: this topic, inherent

To this aim put

being

The addends of the Equation (21) are now compared to understand in particular when

since both inequalities are in principle possible. The first inequality reads

The comparison is immediate considering preliminarily, for simplicity of notation only,

Put now purposely

i.e. it remains finite even at

Nevertheless the Debye approach, as it is, represents valuable enhancement of the early Einstein mode, particularly significant at low

As concerns the Equation (15), regard the second addend of the Equation (21) as

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

The left hand side of the first equation defines the element of solid angle

These positions are possible because all related quantities are arbitrary. Integrating both sides, one finds

indeed if

At this point, let us try to plug

in effect

whence

the value of

if

At this point integrating at constant volume it is possible to find the lattice internal energy

and entropy

from which one calculates the Helmholtz lattice free energy

As a closing remark, note that at very low

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

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

with the same physical meaning of degeneracy factor

Now the limit value of

As

owing to the coefficient

and then

The form of the function

1) the limit

2) the limit

3) the limit

Owing to the terms

in order that

The reason of having introduced in the Equation (32) the smallest one among the three

owing to the positions (32), this is a minimum zero point energy; hence

being

whereas

Despite the function

Owing to the zero order coefficient

In conclusion, replacing just the last form of

where the constant

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).

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

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

Tosto, S. (2016) Reappraising 1907 Einstein’s Model of Specific Heat. Open Journal of Physical Chemistry, 6, 109-128. http://dx.doi.org/10.4236/ojpc.2016.64011