The Field Theory of Specific Heat

TL;DR

This paper applies finite temperature quantum field theory and the heat kernel method to derive a new understanding of specific heat in crystal lattices, revealing a quartic temperature dependence at low temperatures and clarifying the role of geometry and lattice structure.

Contribution

It introduces a universal dimensionless constant for crystal lattices and derives a quartic low-temperature specific heat law, contrasting with the classical Debye cubic law, supported by experimental data.

Findings

Low temperature specific heat scales as T^4, not T^3.

The surface heat determines the true low temperature limit.

The Dulong-Petit law depends on lattice geometry and cut-off effects.

Abstract

Finite temperature quantum field theory in the heat kernel method is used to study the heat capacity of condensed matter. The lattice heat is treated a la P. Debye as energy of the elastic (sound) waves. The dimensionless functional of free energy is re-derived with a cut-off parameter and used to obtain the specific heat of crystal lattices. The new dimensionless thermodynamical variable is formed as the Planck's inverse temperature divided by the lattice constant. The dimensionless constant, universal for a class of crystal lattices, which determines the low temperature region of molar specific heat, is introduced and tested with the data for diamond lattice crystals. The low temperature asymptotics of specific heat is found to be the fourth power in temperature instead of the cubic power law of the Debye theory. Experimental data for the carbon group elements (silicon, germanium) and other materials decisively confirm the quartic law. The true low temperature regime of specific heat is defined by the surface heat, therefore, it depends on geometrical characteristics of a body, while the absolute zero temperature limit is geometrically forbidden. The limit on a growth of specific heat at temperatures close to critical points, known as the Dulong-Petit law, appears from the lattice constant cut-off. Its value depends on the lattice type and it is the same for materials with the same crystal lattice. The Dulong-Petit values of compounds are equal to those of elements with the same crystal lattice type, if one mole of solid state matter were taken as the Avogadro number of the lattice atoms. This means the Neumann-Kopp rule is valid only in some cases.

Tables

Table 1. Table 1: Properties of the carbon group elements and GaAs

material	a	$c_{11}$	$c_{12}$	$c_{44}$	$\rho$	$v_5$	$T_0$	$\theta$	$T_{\mathrm{cr}}$	$C_{\mathrm{DP}}$
diamond	3.567	1079	124.0	576.0	3.5156	11.659	173.3	1.44	3900	27.5
$\alpha$-Si	5.431	165.78	63.937	79.625	3.3291	4.6739	39.4	1.67	1685	29.16
$\alpha$-Ge	5.657	128.53	48.26	66.80	5.3256	2.7459	21.4	1.73	1210	28.76
$\alpha$-Sn	6.4892	66.7	36.5	30.2	5.7710	1.618∗	11.0∗	1.73∗	286	29.1∗
GaAs	5.6533	118.77	53.72	59.44	5.3175	2.4732	20.0	1.67	1513	29.08∗

Table 2. Table 2: Specific heat of germanium, Flubacher-PM1959

$T$	$C_M$	$T$	$C_M$	$T$	$C_M$	$T$	$C_M$	$T$	$C_M$
2.461	0.000537	13.008	0.1784	40.46	4.548	95.46	13.242	200.02	21.062
2.675	0.0007389	13.478	0.2069	42.43	4.8869	97.17	13.456	203.54	21.192
2.971	0.0009778	14.002	0.2439	44.29	5.1882	98.76	13.665	207.02	21.301
3.175	0.001259	14.536	0.2831	45.89	5.4894	100.27	13.845	210.49	21.418
3.481	0.001639	15.002	0.3200	47.87	5.8283	103.23	14.192	213.92	21.527
3.713	0.002040	15.192	0.3394	49.37	6.0793	106.87	14.652	217.34	21.644
3.961	0.002411	15.579	0.3693	51.90	6.5145	110.39	15.046	222.82	21.836
4.364	0.00332	15.945	0.4081	53.59	6.7864	113.82	15.406	226.53	21.891
4.471	0.003499	15.961	0.4112	55.63	7.1254	117.17	15.765	230.21	21.991
4.814	0.004506	16.578	0.4715	57.15	7.3848	120.44	16.104	233.87	22.117
4.963	0.004895	16.979	0.5151	59.02	7.6986	123.84	16.435	237.50	22.171
5.283	0.006050	17.579	0.5866	60.39	7.9287	127.36	16.761	241.11	22.280
5.503	0.006845	17.588	0.5853	62.35	8.2592	130.81	17.067	244.70	22.380
5.774	0.008012	18.051	0.6418	63.76	8.4893	134.18	17.364	246.4	22.380
6.024	0.009155	18.493	0.7008	65.81	8.8324	135.93	17.481	249.98	22.485
6.284	0.01051	18.628	0.7117	67.30	9.0751	139.22	17.761	253.51	22.552
6.506	0.01172	19.041	0.7640	69.22	9.3680	142.42	18.025	257.02	22.656
6.784	0.01359	19.632	0.8481	70.73	9.5939	145.75	18.255	262.81	22.748
7.131	0.01609	20.233	0.9439	72.54	9.8868	149.23	18.523	268.85	22.849
7.51	0.01953	21.449	1.1334	74.07	10.130	152.66	18.723	272.84	22.920
7.974	0.02409	23.186	1.4192	76.00	10.422	156.05	18.949	276.81	22.987
8.465	0.03021	24.548	1.6527	77.45	10.653	159.39	19.150	280.72	23.054
8.948	0.03745	25.86	1.8845	79.38	10.950	162.69	19.372	280.90	23.075
9.434	0.04619	27.22	2.1313	81.09	11.217	166.06	19.531	284.80	23.117
10.006	0.05820	28.46	2.356	82.75	11.477	169.50	19.715	286.55	23.167
10.442	0.07025	29.87	2.6221	84.36	11.719	172.90	19.887	288.66	23.205
11.015	0.08799	31.16	2.8614	85.97	11.962	176.26	20.067	292.38	23.251
11.458	0.1037	32.68	3.1418	87.50	12.180	178.49	20.184	292.48	23.263
12.011	0.1272	33.91	3.3677	89.06	12.439	183.82	20.418	296.23	23.351
12.366	0.1469	35.72	3.7020	90.70	12.615	189.31	20.631	296.24	23.313
12.471	0.1489	36.93	3.9229	92.22	12.791	192.91	20.782	300.01	23.468
12.902	0.1752	39.00	4.2928	93.98	13.054	196.48	20.916	300.05	23.430

Equations

$$\beta =\frac{\hslash v}{k_{B}T},$$

$$\square =g^{\mu \nu }{\nabla }_{\mu }{\nabla }_{\nu },$$

$$\Big{(}\frac{\partial}{\partial s}-\Box^{x}\Big{)}K(s|x,x^{\prime})=0,\ K(s|x,x^{\prime})|_{s\rightarrow 0}=\delta(x,x^{\prime}),$$

$$\mathrm{Tr}K(s)\equiv \int {\mathrm{d}}^{3}xK(s|x,x)=\frac{1}{(4\pi s{)}^{3/2}}\mathcal{V}+\frac{1}{(4\pi s)}\mathcal{S},$$

$$-F_{\tilde{a}}^{\beta }\equiv \tilde{A}{\int }_{{\tilde{a}}^2/4}^{\infty }\frac{\mathrm{d}s}{s}\text{Tr}{K}^{\beta }(s).$$

$$\text{Tr}{K}^{\beta }(s)=\frac{\beta }{(4\pi s{)}^{1/2}}\sum _{n=1}^{\infty }{\mathrm{e}}^{-\frac{{\beta }^2{n}^2}{4s}}\mathrm{Tr}K(s).$$

$$-F_{\tilde{a}}^{\beta }=\frac{\tilde{A}}{{\pi }^2}\frac{\mathcal{V}}{{\beta }^3}\sum _{n=1}^{\infty }{\int }_0^{{\alpha }^2}\mathrm{d}yy{\mathrm{e}}^{-y{n}^2}.$$

$$\alpha \equiv \frac{\beta }{\tilde{a}}=\frac{\hslash v}{\tilde{a}{k}_{B}T}.$$

$$-F^{\alpha}=\frac{{\tilde{A}}}{\pi^{2}}\,\frac{\mathcal{V}}{\tilde{a}^{3}}\,\sum_{n=1}^{\infty}\,\frac{1}{n^{4}\alpha^{3}}\Big{(}1-\exp(-\alpha^{2}n^{2})-n^{2}\alpha^{2}\exp(-\alpha^{2}n^{2})\Big{)}.$$

$$\frac{\partial F^{\alpha }}{\partial \alpha }=\tilde{A}\frac{3}{{\pi }^2}\frac{\mathcal{V}}{{\tilde{a}}^3}\Theta (\alpha ),$$

$$\Theta(\alpha)=\sum_{n=1}^{\infty}\,\frac{1}{n^{4}\alpha^{4}}\Big{\{}1-\exp(-\alpha^{2}n^{2})-n^{2}\alpha^{2}\,\exp(-\alpha^{2}n^{2})-\frac{2}{3}n^{4}\alpha^{4}\,\exp(-\alpha^{2}n^{2})\Big{\}}.$$

$${\Theta }_{\max }\approx 8.33\cdot 10^{-2}.$$

$$\frac{\partial F^{\alpha }}{\partial \alpha }=\tilde{A}\frac{{\pi }^2}{30}\frac{\mathcal{V}}{{\tilde{a}}^3}\frac{1}{{\alpha }^4},\alpha \to \infty .$$

$$\frac{\partial F^{\alpha }}{\partial \alpha }=\tilde{A}{\Theta }_{\max }\frac{3}{{\pi }^2}\frac{\mathcal{V}}{{\tilde{a}}^3},\alpha \to 0.$$

$$\frac{1}{v^2}\frac{{\partial }^2}{\partial t^2}u(x,t)=△u(x,t).$$

$$v_{1}=\Big{(}\frac{c_{11}}{\rho}\Big{)}^{1/2},\ v_{2}=\Big{(}\frac{c_{44}}{\rho}\Big{)}^{1/2},$$

$$v_{3}=\Big{(}\frac{c_{11}+c_{12}+2c_{44}}{2\rho}\Big{)}^{1/2},\ v_{4}=\Big{(}\frac{c_{44}}{\rho}\Big{)}^{1/2},\ v_{5}=\Big{(}\frac{c_{11}-c_{12}}{2\rho}\Big{)}^{1/2}.$$

$$\square =\frac{1}{v^2}\frac{{\partial }^2}{\partial t^2}+△.$$

$$F_{\mathrm{total}}=\sum _i{F}^{{\alpha }_i}[\mathcal{V}]+F^{{\alpha }_s}[\mathcal{S}].$$

$$C_V\equiv k_{\mathsf{B}}\frac{\partial F^{\alpha }}{\partial \alpha },$$

$$C_V=\tilde{A}{k}_{\mathsf{B}}\frac{3}{{\pi }^2}\frac{\mathcal{V}}{{\tilde{a}}^3}\Theta (\alpha ),$$

$$n=\frac{\mathcal{V}m}{a^3{N}_{\mathsf{A}}}.$$

$$C_M=C_V\frac{a^3{N}_A}{\mathcal{V}m}=\tilde{A}{k}_{\mathsf{B}}{N}_{\mathsf{A}}\frac{3}{{\pi }^2}\frac{a^3}{m{\tilde{a}}^3}\Theta (\alpha ).$$

$$\tilde{a}=Ba.$$

$$C_M=A{k}_{\mathsf{B}}{N}_{\mathsf{A}}\frac{3}{{\pi }^2}\frac{1}{m}\Theta (\alpha ),$$

$$\alpha =\frac{\hslash v}{B{k}_{\mathsf{B}}aT},$$

$$C_M=A{k}_{\mathsf{B}}{N}_{\mathsf{A}}\frac{{\pi }^2}{30}\frac{1}{m}\frac{1}{{\alpha }^4},\alpha \to \infty ,$$

$$C_M=A{k}_{\mathsf{B}}{N}_{\mathsf{A}}{\Theta }_{\max }\frac{3}{{\pi }^2}\frac{1}{m},\alpha \to 0.$$

$$C_M=\mathcal{R}{\Theta }_{\max },\alpha \to 0.$$

$${\alpha }_i\equiv T_i/T.$$

Full text

I Quantum field theory for condensed matter

When developing a physical theory of condensed matter, e.g., solid state matter, one usually starts with the atomistic view of matter, which consists of particles, e.g., atoms (ions) of the lattice and electrons Kittel-book1996 ; Landau-V5 . However, particles and quasiparticles in a physical theory are employed only at intermediate stages, while final expressions are some static functions, which describe integral properties of condensed matter systems, such as the heat capacity, electric conductivity, etc. One usually does not intend to and cannot describe the behaviour of individual particles. Furthermore, one normally starts with a system of non-interacting constituents, e.g., independent oscillators of A. Einstein Einstein-AdP1907 ; Kittel-book1996 or free electron gas of P. Drude Drude-AdP1900 ; Kittel-book1996 , and later builds in interactions between them to develop eventually physics of ’strongly interacting’ (correlated) systems. Building a theory this way follows the historical path of physics development and therefore relies on thermodynamics and statistical mechanics of gases. However, physical properties of solid and liquid matter are very different from those of gases. Indeed, a condensed matter system can be defined as a system of particles that form physical continuum. In fact, the view on condensed matter as continuous medium is well developed in such classic branches of physics as elasticity theory Landau-elastV7 ; Love-book1944 ; Royer-book2000 and hydrodynamics Landau-hydroV6 . These are phenomenological theories, which are not concerned with particle compositions of condensed matter systems, but they are capable of describing matter’s properties by summarizing them as physical laws. Elasticity theory describes mechanical phenomena of solid state matter, yet its mathematical apparatus can be used for the theory of thermal phenomena, as was first proposed by P. Debye Debye-AdP1912 .

Temperature is a key variable of any thermal theory. Statistical thermodynamics Landau-V5 was developed as physics of gas, which is viewed as a collection of massive particles moving in free space. Correspondingly, this physical theory used classical mechanics as its foundation. In statistical physics temperature is an emergent characteristic of a large number of particles. It appeared possible to use mechanics of particles (system’s constituents) to find a statistical distribution of their kinetic energies (velocities), which corresponds to an observed temperature. This was done first by J.C. Maxwell Maxwell-book2011 . Since the physical nature of temperature in a condensed matter system, which presents medium in a bounded space, is different, this procedure cannot work. In a quantum theory of matter temperature is introduced as an external (axiomatic) parameter. This is also the approach of the present work, but it is implemented in a geometrical language, since we believe that thermal and electronic properties of condensed matter can be better described by the methods of field theory and geometry.

In statistical mechanics, in order to derive macroscopic thermal quantities, the number of particles is taken to infinity, while the volume occupied by particles is taken to infinity as well Landau-V5 . Both limits are unphysical, but sought physical quantities are derived finite. In this limit, the influence of a system’s boundaries on its physical properties is infinitesimal and cannot be studied. However, finite size effects, e.g., in heat conductivity and heat capacity, are observed, usually at low absolute temperatures. For some condensed matter systems under certain conditions, effects of the body’s geometrical characteristics can become the leading. The Schwinger-DeWitt (geometric) formalism for quantum field theory (QFT) DeWitt-book2003 ; GAV-lectures2007 that we use naturally incorporates boundary effects due to its mathematical foundation, the heat kernel.

We will use the finite temperature QFT Gusev-FTQFT2015 to develop a theory of specific heats of crystal lattices that may also be applicable to other forms of condensed matter. Let us first recall its basic ideas. Since any condensed matter system exists within a certain three-dimensional domain, and its behaviour in time is not usually studied, the field theory can be defined in the Euclidean four-dimensional spacetime, i.e., the spacetime metric’s signature is not Minkowskian, but Euclidean (no time coordinate is singled out). The physical time is imaginary, and it is distinguished from other spacetime dimensions only by its closed topology, $\mathbb{S}^{1}$. The orbit’s length of the closed Euclidean time is expressed via the fundamental physical constants as the Planck’s inverse temperature,

[TABLE]

where $k_{B}$ is the Boltzmann’s constant and $\hbar$ is the reduced Planck’s constant. Note, that in contrast to the definition in Gusev-FTQFT2015 , there is no calibration coefficient in (1) because temperature $T$, or $\beta$ for that matter, is not an independent variable in the present setting. The characteristic velocity, $v$, enters the definition (1). For the electronic component of heat it is supposed to be the speed of light, while for the elastic (acoustic) component, contributed by the crystal lattice, it is the velocity of elastic (sound) waves.

In the Schwinger-DeWitt quantum field theory, the Laplace operator,

[TABLE]

defines a particular field model BarVilk-PRep1985 . It is constructed of the covariant derivatives, which may contain the metric (gravity) and gauge field (e.g., electromagnetic) connections. For the theory of lattice heat, gauge fields and gravity are not relevant, and the Laplacian (2) is trivially the second order partial derivatives. For the study of electronic properties, electromagnetic fields cannot be neglected, in particular, they should be important for the electronic specific heat.

The kernel of the heat equation DeWitt-book2003 ; GAV-lectures2007 ; BarVilk-PRep1985 ,

[TABLE]

is expressed via the proper time parameter, $s$. The trace of the kernel in a compact three-dimensional manifold with boundary is Gusev-FTQFT2015 ,

[TABLE]

where $\mathcal{V}$ is the volume and $\mathcal{S}$ is the boundary’s area of the bounded domain of manifold $\mathbb{R}^{3}$. This expression is valid at an arbitrary proper time. For the reason explained in Sect. II.4 we omit the boundary (’surface’) term and keep only the volume (’bulk’) term of this expression.

Let us compute the functional of free energy, defined in the usual way Gusev-FTQFT2015 , but with the positive lower limit of the the proper time integral. This is the standard QFT regularization used to remove the ultraviolet divergences BarVilk-PRep1985 . Our integral is finite, but the lower limit appears due to the wavelength cut-off caused by discreteness of the lattice. Because the length square, $\tilde{a}^{2}$, is proportional to the proper time, $s$, which parametrizes the geodesic DeWitt-book2003 ; BarVilk-PRep1985 , we introduce the integral’s lower limit as (the numerical coefficient is choosen to simplify expressions below),

[TABLE]

An observable quantity, such as specific heat, must certainly be independent of the regulator (cut-off parameter), and in Sect. II.2 we show that indeed it is. The overall numerical coefficient ${\tilde{A}}$ is to be calibrated by experiments. We substitute the three-dimensional heat kernel trace (4) into the thermal (3+1)-dimensional expression Gusev-FTQFT2015 ,

[TABLE]

After applying the change of variables, $y=\beta^{2}/(4s)$, free energy (5) looks like

[TABLE]

It is only different from the previously studied expression Gusev-FTQFT2015 by the integral’s upper limit, which is expressed as a new dimensionless variable,

[TABLE]

The computed expression (7) accepts the final form,

[TABLE]

This sum’s first term is the zeta function, $\zeta(4)={\pi^{4}}/{90}$. The two other terms cannot be analytically computed.

The new upper index of the functional $F^{\alpha}$ indicates a change in physical understanding of the free energy. We declare the dimensionless variable $\alpha$ be the thermal variable for condensed matter systems, i.e., field theory models with the short wavelength cut-off, that supersedes old variables, the absolute temperature, $T$ (Kelvin), and the Planck’s inverse temperature, $\beta$ (meter). Indeed, it is natural that the dimensionless functional depends on the dimensionless variable; the theory could be called conformal (or scale free) in the popular language.

Following the line of Gusev-FTQFT2015 , we take the derivative of (9) over the variable $\alpha$ to obtain,

[TABLE]

where the notation for the dimensionless thermal sum is introduced,

[TABLE]

We conjecture that $\Theta(\alpha)$ is a universal function of temperature scaling for condensed matter systems. The plot of $\Theta(\alpha)$, computed in Maple for the finite sum $n=400$, is shown in Fig. (1).

The maximum value of this quantity is found to be,

[TABLE]

It might be convenient to redefine (11) by $10^{2}$ especially because it usually has a rather large dimensionful factor, (29), see Sect. III.2. Three other terms of the sum have good behaviors in asymptotics, $\alpha\rightarrow 0$ (’high temperature’) and $\alpha\rightarrow\infty$ (’low temperature’). The limit $\alpha=0$ is topologically forbidden since it would be equivalent to the open manifold, $\mathbb{R}^{1}$, Gusev-FTQFT2015 . Even though a numerical estimate for the expression (10) becomes negative, when $\alpha$ goes to zero (note, the plot in Fig. 1 is from $\alpha=0.05$), its intermediate asymptotics is positive constant for any infinitesimal value of $\alpha$, when the sum’s upper limit tends to infinity. In fact, the leftmost bound of $\Theta(\alpha)$ is irrelevant, as in the regime, $\beta\ll a$ (the thermal wavelength of sound is much less than the lattice cut-off), the theory breaks down.

The expression (10) is valid at any values of its variables. We consider now expansions of this functional in its dimensionless thermal variable. The ’low temperature’ asympotics is easily found,

[TABLE]

The ’high temperature’ asymptotics can be explored numerically. The maximum value of $\Theta(\alpha)$ gives us the constant term,

[TABLE]

These simple derivations are quite general and can hopefully be used for different problems of thermal physics.

II The field theory of specific heat

II.1 From elasticity theory to the heat kernel

Let us apply this formalism to the physics of lattice heat of solid state matter. The theory we want to build can be viewed as a modified completion of the theory of specific heat of Peter Josef William Debye Debye-AdP1912 . One century after its creation, his theory is still widely used in practice. The key idea of P. Debye is that the lattice heat is energy of the standing sound waves in a solid body. The sound waves are elastic waves Love-book1944 , therefore, the velocity of sound in a solid body (crystal lattice), which is typically of the order of $10^{3}\ m/s$, enters the Planck’s inverse temperature (1) and correspondingly the $\alpha$-variable (8). This velocity’s magnitude lets us leave aside the electronic contribution, because it is separated from the lattice heat by the factor, $v/c\approx 10^{-5}$, taken to some power.

Free energy of finite temperature QFT (9) is applicable to this problem because the operator of the equations of elastic (sound) waves accepts the required form, Eq. (2). Indeed, let the three-dimensional vector of displacement, $u$, be a field. Two equations for the longitudinal and transverse elastic waves have the same form Landau-elastV7 ,

[TABLE]

The velocities of propagation of these waves are different and can be also expressed through the elasticity parameters Landau-elastV7 , which show that longitudinal velocities are always greater than transverse ones. The sound velocities can be also expressed through the elements of the elastic stiffness tensor (the elastic constants or moduli), $c_{ij}$, and the matter density, $\rho$, Royer-book2000 . There are, in general, three different velocities (one longitudinal and two transverse) for each principal crystallographic direction, for the cubic crystal lattices, which we consider as a simple example of this formalism. Due to the cubic lattice symmetries, there are only three independent elastic constants, $c_{11},c_{12},c_{44}$ Royer-book2000 . They determine velocities of the plane waves incident to the surface {100} (we use the notations of McSkimin-JAP1963 ),

[TABLE]

to the surface {110},

[TABLE]

For the transverse wave $v_{2}$ both polarizations have the same velocity. We accept the elasticity theory’s assumption that sound waves with different velocities are independent Kittel-book1996 and, therefore, they present independent contributions to the free energy. One can find sound velocities from the known elastic moduli $c_{ij}$, but in practice the velocities are experimentally determined using the ultrasound techniques, and the elastic moduli are derived. These high precision measurements were made by H.J. McSkimin and P. Andreatch Jr. for some relevant solids: diamond McSkimin-JAP1972 , silicon McSkimin-JAP1964 , germanium McSkimin-JAP1963 , gallium arsenide McSkimin-JAP1967 (where all the values are tabulated). The third-order elastic moduli were also measured McSkimin-JAP1967 , they introduce nonlinearity to the wave equations.

The solutions for standing sound waves are obtained via the wave equations Vladimirov-book1971 . This is done with help of the expansion over modes that are counted by the frequency, a variable reciprocal to time, but in the context of our problem the wavelength is an appropriate variable. However, in a physical theory of heat we do not really need actual solutions of differential equations. We want to obtain the total energy of sound waves as a function of temperature, i.e., to find a scalar function of one argument (with fixed geometrical charactristics of a body), then the time (or frequency) is a redundant variable. The heat kernel method allows us to obtain the trace of the heat kernel, which serves as a mathematical prototype for free energy, directly, by skipping explicit solutions of the field (wave) equations. Thus, usual derivations of the heat trace asymptotics from the Laplacian’s discrete spectrum can be avoided, if we seek solving physics problems in the field theory.

Since this computation is done in spacetime with the imaginary time, we first go from the wave equations (15) to the equivalent Laplace equations. The spacetime metric changes its signature from Lorentzian to Euclidean, which means that the l.h.s. of Eqs. (15) gains the minus sign. In QFT literature DeWitt-book2003 ; GAV-lectures2007 , this is commonly referred to as the Wick rotation. Thereby, we now have a Laplace equation with the four-dimensional operator,

[TABLE]

Then, instead of solving the Laplace equation we solve the corresponding heat equation (3). This way we reduce a mathematical problem of the Debye theory to the standard problem of the Schwinger-DeWitt QFT. Its operator (2) accepts the simplest form, where are no internal degrees of freedom (no matrix structure BarVilk-PRep1985 ), no potential term, and no connection in the covariant derivatives: these are just partial derivatives. The Laplace operator (18) allows the separation of variables and thus the computation of the heat kernel as the standard thermal sum (6).

Finite temperature QFT, modified for condensed matter physics in Sect. I, gives the volume contribution (7) for each independent velocity. The boundary contribution is defined by its own surface wave velocity, $v_{s}$. The total free energy is a sum of all these terms,

[TABLE]

For cubic lattices, the sum spans over nine $\alpha_{i}$, as the transverse wave contributions from the waves incident to the surfaces {100} and {111} double. The boundary contribution of the surface elastic waves, the Love or Rayleigh waves Love-book1944 , may be neglected so far because it is an order of magnitude smaller Gusev-FTQFT2015 in the experiments we consider below.

The significant restriction is that sound velocities should be independent of temperature. This condition was experimentally examined for the carbon group elements McSkimin-JAP1972 ; McSkimin-JAP1964 ; McSkimin-JAP1963 . For example, it was found that the sound velocities in diamond in the temperature range, $-195.8{}^{\circ}C$ to $+50{}^{\circ}C$, change relatively as little as $10^{-5}$. This is a good confirmation of the condition $v(T)=$const. Further experiments with germanium and silicon were done Hao-PRB2001 in the temperature range, $25K<T<70K$, and the relative changes in the longitudinal sound velocities were found to be about $10^{-4}$, with germanium showing the higher change. Experiments at temperatures beyond this range, which are important for the low temperature regime, were not done. Nevertheless, the obtained results gives us hope that the relative changes of velocities beyond the investigated range are also small.

The phase velocity of plane waves are used in the elasticity theory derivations Kittel-book1996 ; de-Launay-book1956 . In the heat kernel method, the measured velocity enters the Laplacian (2) and correspondingly the $\alpha$-variable, (8). Therefore, it is the velocity of energy transfer, i.e., the group velocity. This fact emphasizes the phenomenological nature of the proposed theory. There are many more issues with the use of elastic waves in thermal physics Kittel-book1996 that require further development of the basic ideas outlined here.

II.2 Molar specific heat

In Sect. I, we derived the dimensionless derivative (10), let us now propose the scaling hypothesis: scaling of a physical observable, e.g., heat capacity, with the change of a physical variable, e.g., thermodynamic temperature, is the same as scaling of the corresponding dimensionless functional, e.g., $\partial F^{\alpha}/\partial\alpha$, with the change of a dimensionless variable, when it is expressed with help of the fundamental physical constants. It is the postulate that will allow us to avoid solving the measurement problem.

Therefore, in order to obtain a physical observable, with the proper physical dimensionality, $JK^{-1}$, from the derived mathematical expression (10) we multiply it with the Boltzmann’s constant,

[TABLE]

where $C_{V}$ is the heat capacity at the constant volume. We get,

[TABLE]

where volume $\mathcal{V}$ is fixed by the used mathematics, Sect. I. However, in experiments the heat capacity should be measured at fixed pressure to avoid changing the elastic properties of the crystal due to an additional strain McSkimin-JAP1963 . Thus, we take the experimental values for $C_{P}$. It is the heat capacity per mole that is derived here any way. We do not treat the thermal expansion here, which could probably account for the higher temperature behaviour close to the critical points.

In order to translate the expression (20) to the normally used molar (atomic) specific heat, $C_{M}$, we should divide $C_{V}$ by the amount of substance, $n$ (mol), contained in the given volume. The molar quantity, $n$, can be found, with help of the Avogadro constant, $N_{A}$, if we know the number of atoms $N$, as $n=N/N_{A}$. The number $N$ can be determined from the system’s volume, if know the volume of a lattice unit cell, $V$, and the number of atoms per cell, $m$, as $N=m\mathcal{V}/V$. The lattice unit cell’s volume is defined by the lattice constants that correspond to a crystallographic lattice. For cubic lattices Kittel-book1996 ; Szwacki-book2010 ; Royer-book2000 , there is a single lattice constant, $a$, so, $V=a^{3}$. Finally, the amount of substance is,

[TABLE]

When Eq. (20) is divided by (22), the molar specific heat (or just the specific heat) becomes,

[TABLE]

This expression still depends on the unknown cut-off constant, $\tilde{a}$. However, $\tilde{a}$ is the length, which defines the limit of validity of the elastic model, so it can be declared proportional to the lattice constant,

[TABLE]

This gives us another calibration constant, $B$. As mentioned, we could introduce it into the Planck’s inverse temperature (1), but because temperature comes only as the combination (8), it does not matter if we assign the uncertainty to $\tilde{a}$. The overall combination of the calibration constants can be denoted, $A\equiv\tilde{A}/{B^{3}}$. Thus, as expected, we arrive at the regulator-free quantity,

[TABLE]

with

[TABLE]

which should be calibrated by experimental data in order to determine the parameters $A$ and $B$. These two calibration constants define scaling of the specific heat, $C_{M}$. The constant $A$ sets up a vertical scale, while the constant $B$ scales up $C_{M}$ horizontally.

The formula for the specific heat (25) is dimensionless, except for the factor of the molar gas constant, which is the only combination of fundamental physical constants, $R=k_{\mathsf{B}}N_{\mathsf{A}}$, that could make up the right dimensionality of specific heat. The expression for $C_{M}$ depends on the velocities of sound, $v$, and temperature, $T$, that enter (26). The sound velocities, in turn, are defined by the elastic constants and by the matter density. So, despite its simplicity, Eq. (25) embodies all elastic characteristics of crystalline bodies, yet, its functional behaviour is described by the universal thermal sum $\Theta(\alpha)$, Eq. (11) of the single variable, $\alpha$.

The ’low temperature’ asymptotics of the $\alpha$-derivative (13) generates the term, cf., Gusev-FTQFT2015 ,

[TABLE]

because it is a limit of the vanishing lattice constant, i.e., ideal (smooth) medium. The ’high temperature’ asymptotics (14) provides, via the $\Theta(\alpha)$ maximum value (12), the constant term independent of $\alpha$,

[TABLE]

The above expressions (25)-(28) are valid for cubic lattices, whose velocities are (16)-(17). For other types of lattices the final derivations should be redone starting from Eq. (22). This asymptotics can be also written in the form,

[TABLE]

In this form, it lets us obtain the solid state equivalent of the molar gas constant, $\mathcal{R}$, which is different for different lattices. This factor can be used in the main Eq. (25) instead of the complex combination.

Even with the known elastic moduli (or velocities) we still have to calibrate the theory. But after the main equations of the theory is derived, in principle, we can forget about elasticity theory. We could declare the variables $\alpha_{i}$ be the ratios of thermodynamic temperature and some characteristic temperatures, $T_{i}$,

[TABLE]

In such a form, the theory would be more similar to the Debye theory, Appendix A. However, this procedure would introduce many calibrating parameters in place of just one, $B$.

II.3 High temperature limit of specific heat, or On the Dulong-Petit law

In 1819 French natural philosophers Pierre Louis Dulong and Alexis Thérèse Petit published Dulong-ACP1819 the discovery that now bears their names. A.T. Petit and P.L. Dulong measured how fast solid bodies cooled down in low pressure air and found that the specific heat capacities of the bodies were inversely proportional to the atomic weights of the bodies’ chemical elements Kittel-book1996 ; Laing-JCE2006 . Therefore, the Dulong-Petit law states that the product of the molar mass, $M$, with the specific heat, $C_{V}$, of mono-atomic solid bodies, i.e., the molar specific heat, $C_{M}$, is approximately constant,

[TABLE]

The Dulong-Petit law was important not only in physics, but also in chemistry, where it helped Dmitry Mendeleev to discover the periodic table of chemical elements Laing-JCE2006 . Mendeleev used the Dulong-Petit law to correct wrong atomic weights of three elements (cesium, uranium and indium) by making new measurements of their specific heats.

Almost a century later, after the Avogadro constant, $N_{A}$, was introduced, Albert Einstein Einstein-AdP1907 used the Boltzmann’s equipartition theorem to propose that the molar specific heat is universally equal to,

[TABLE]

He attempted to explain this empirical rule (31) as the classical limit of a theory based on the new at that time Planck’s quantum hypothesis. The Dulong-Petit law in the form (32) was later re-derived in the Debye theory Debye-AdP1912 , and it eventually became a dogma of solid state physics Kittel-book1996 and thermochemistry Stoffel-ACIE2010 . However, this law does not hold strictly, and the Dulong-Petit (DP) values vary widely Kittel-book1996 . In many cases such variations arise from insufficient experimental data, as room temperature, assumed to be the high temperature regime for the Dulong-Petit law, is certainly not high enough for most substances. As is seen from the comprehensive analysis of numerous high temperature experiments specific heats, e.g., for iron and silicon Desai-FeSi1986 , do not becomes constant, but keep slowly increasing to the melting point. Thus, the melting (or, in general, critical) points should provide the DP values of specific heats.

The $\alpha\rightarrow 0$ asymptotics (28) derived in this field theory formalism corresponds to the Dulong-Petit limit. In finite temperature QFT for condensed matter, the Dulong-Petit limit emerges from the minimum length proportional to the lattice constants that cuts off the range of short wavelengths, when the field theory ceases to be valid. In other words, the approximation of condensed matter as continuous medium fails, when the elastic wavelengths become comparable to the average distance between the nodes of lattice.

The Dulong-Petit limit is a constant independent of temperature and elastic properties of the lattice, but not of its crystallographic type. This high temperature limit depends on the number of points in a unit cell, $m$, and the relative volume of a unit cell (one for cubic lattices). It should also depend on the total number of velocities of a lattice contributing to free energy (19). This means the Dulong-Petit limit should be the same for all substances that crystallize to the same lattice type.

The Avogadro constant and the Boltzmann’s constant enter the Dulong-Petit limit, because it is the only possible combination of fundamental physical constants with the right dimensionality for specific heat. Therefore, we argue that proportionality of the Dulong-Petit limit (31) to the molar gas constant (32) follows trivially from dimensional analysis, while its numerical factor close to 3 is merely a coincidence. The number three has never been well measured, and its appearance seems to be due to the magic of whole numbers that influenced theoretical developments. All the reasonings proposed as its proofs are based on a discrete model of independent oscillators Kittel-book1996 ; Einstein-AdP1907 ; Debye-AdP1912 , which apparently failed to correctly describe specific heat for other temperature ranges. However, we could expect that statistical physics, based on new mathematics, being developed by Victor Maslov Maslov-MN2013 should produce a correct version of the discrete theory for the lattice specific heat.

II.4 Finite size effects

Free energy in the finite temperature QFT based on the heat kernel (4) contains two terms defined by geometrical invariants of a compact domain, the volume and the area of boundary Gusev-FTQFT2015 . We focus in this work only on the volume (’bulk’) term as it gives the leading order contribution in experiments we want to consider. Let us sketch the reason for discarding the boundary term.

With a system’s effective size Gusev-FTQFT2015 , defined as the ratio of the body’s volume to its boundary’s area, $r={\mathcal{V}}/{\mathcal{S}}$, the acoustic kappa-factor is determined by the sound velocity, $v\propto 10^{3}\,m/s$,

[TABLE]

It shows that threshold for the appearance of the boundary (finite size) effects in the lattice specific heat is many orders of magnitude higher than in thermal radiation phenomena, where $\kappa\approx 3.3\cdot 10^{-4}K\,m$, Gusev-RJMP2014 . Assuming an experimental uncertainty is $1\%$, we expect that the finite size effects at low temperatures, e.g., $1<T<10\,K$, could appear only in systems of a nanometer size, i.e., with $r\propto 10^{-9}\,m$. In the specific heat experiments with ’macroscopic’ samples, Desnoyers-PhilMag1958 ; Flubacher-PM1959 ; Berg-PRSA1957 , the finite size effects can be seen only at sub-Kelvin temperatures. In general, the true low temperature asymptotics, $\beta\gg r$, depends on the shape and the surface curvatures of a condensed matter system, as well as on its material. Therefore, a universal low temperature asymptotics of the free energy (and of the specific heat) does not exist. Detailed analysis of finite size effects in the heat capacity certainly deserves a special study motivated by the recent experimental quest in nanotechnology.

II.5 Specific heat of compounds, or On the Neumann-Kopp law

The properties of specific heats of compounds (multi-atomic substances) have been studied in the 19th century by German physicist Franz Ernst Neumann and German chemist Hermann Franz Moritz Kopp. They formulated the rule (their independent works separated by a time interval) for specific heat of a chemical compound as equal to the sum of specific heats of its constituents. As every empirical law, the Neumann-Kopp rule does not hold strictly. When this rule was derived ambient conditions were considered to be sufficient to reach the Dolung-Petit limit, but they are not. The vast number of new compounds were discovered and studied. Extensive data of modern measurements show that many compounds do not obey the Neumann-Kopp rule, for example, the overview of solid binary antimonides Schlesinger-ChemRev2013 states that ”the Neumann-Kopp law is not a suitable substitute for experimentally obtained values of antimonide heat capacities”. Likely this is the case for other classes of compounds.

In general, the Neumann-Kopp rule would give n-times bigger a specific heat for an n-atomic compound, only if the Dulong-Petit law were exact and hold at ’ambient conditions’; this is not the case. Let us have a fresh look at this rule from the point of view of the geometrical formalism, where the heat capacity of a crystal is defined by the number of sound velocities and lattice characteristics. This means that specific heats of elements that make up a compound should be irrelevant.

The Neumann-Kopp rule for specific heats of crystal matter has been established, because it uses the definition of the amount of substance (mole) of solid state matter derived for gaseous matter. Thus, one mole of an n-atomic compound contains n-times more constituents (atoms as the lattice nodes) than all its composing elements in solid state contain. If the composing elements crystallize to the same type of lattice as the compound, then the Neumann-Kopp rule would be exact. This is, of course, a special case, but many common substances have similar specific heats under some commonly used conditions (close to the Dulong-Petit limit). It is clear that for many more substances, which were discovered and studied after the 19th century, under general conditions, the Neumann-Kopp rule is false.

Therefore, we suggest that one mole of crystal matter should not be equal to $N_{A}$ molecules, but rather to the Avogadro number of atoms (or ions) as true constituents of a crystal lattice. Indeed, separate molecules of a chemical compound do not exist in a crystal. Instead, the whole crystal lattice can be viewed as one large molecule of a compound, whose atoms take places at the lattice nodes. We propose to correct tabulated specific heats for n-atomic compounds by dividing their values by n. As an example, we will consider two-atomic compounds with the zinc-blende lattice, since it is a cubic lattice of the diamond type for two-atomic materials. We argue that the corrected Dulong-Petit limit of such substances is equal to the one for the carbon group elements with the diamond lattice.

III Experiments. Specific heat of the diamond lattice

III.1 The group IV elements

In order to examine the presented theoretical ideas, let us consider experimental determinations of specific heats of some solid substances. We will do the statistical analysis of available data for the elements from group IV (carbon group) of the periodic table of chemical elements that crystallize to the diamond type cubic lattice. They include diamond (carbon, C), silicon (Si), germanium (Ge), and gray tin ($\alpha$-Sn). These are $\alpha$-forms (diamond lattice) of Si and Ge, but their $\beta$-forms with tetragonal lattices exist under certain pressures Greenwood-book1998 . The properties of gray tin are little measured, so we try to make some predictions.

In 1950s, J.A. Morrison with collaborators performed at chemical laboratories of the National Research Council of Canada precise measurements of specific heats of many crystalline solids at a wide range of temperatures. These were traditional measurements done with the calorimeters. For some of these solids, the Morrison’s experiments remain a state of the art today, because no attempts were made to repeat them with higher precision. Morrison’s group measured specific heats of natural diamonds Desnoyers-PhilMag1958 and pure, commercially grown, single crystals of silicon and germanium Flubacher-PM1959 . They published full tables of the original experimental data together with the analysis. Among the three data sets available for the carbon group elements, the data for germanium and silicon Flubacher-PM1959 are the best. Likely this is because the diamonds used in the study were natural, thus, with uncontrolled defects and impurities. We use the germanium data as an instance of the diamond lattice although the silicon data are as good and produce the same physical results. All measurement data were converted to the SI units $J\,mol^{-1}\,K^{-1}$.

The number of atoms per unit cell, $m$, of the diamond lattice is eight Kittel-book1996 . The velocities of sound in crystals of the carbon group elements were measured in Refs. McSkimin-JAP1972 ; McSkimin-JAP1964 ; McSkimin-JAP1963 . The derived elastic constants are given according to these papers in the units of $GPa$, in Table 1, as a more concise and equivalent representation. The elastic constants of gray tin were implied from the neutron scattering experiments Price-PRB1971 , the standard technique to determine frequencies of crystals; these values are accepted in the later references like Adachi-book2005 . The lattice constants, $a$, in Ånsgtrom, $10^{-10}\,m$, are from the reference book Adachi-book2005 . The densities, $\rho$, also from Adachi-book2005 , are expressed in the commonly used units of $g/cm^{3}$. The lowest velocities of sound, $v_{5}$, in units of $10^{3}\,m/s$, are also given (it is computed for $\alpha$-tin). Critical temperatures, $T_{\mathrm{cr}}$, are the sublimation temperature for diamond Pierson-book1993 , the melting points of Si and Ge Adachi-book2005 , and the $\alpha\rightarrow\beta$ lattice transition (its reverse is the ’tin pest’ effect Greenwood-book1998 ) for $\alpha$-tin. The low-$T$ characteristic temperature, $T_{0}$, is explained below. The values not directly measured yet (predicted) are denoted with the asterisk sign. The elastic moduli for GaAs are from Ref. McSkimin-JAP1967 . Its highest specific heat, given in Ref. Glazov-IM2000 , is corrected by the factor 1/2. Some of the basic properties of these and other semiconductors are also available online at Ioffe Institute IoffeInst-2001 .

III.2 The Dulong-Petit values

We need to know the high temperature limit (the Dulong-Petit value) of specific heat of the carbon group elements with the diamond lattice (or any other elements with this lattice) for calibrating the model. At the same time we check the hypothesis that the Dulong-Petit values of all such elements are the same as suggested by Eq. (28).

The Dulong-Petit value for diamond is the most difficult to obtain, because its $\Theta(\alpha)$ spreads wide, thus, its low temperature regime is high, $T<170K$. The only reliable number is $C_{\mathrm{DP}}=22.10\,J/(mol\,K)$ at 1100K in Ref. Victor-JCP1962 , but its temperature is too low for the anomalous thermal behaviour of diamond. Two more numbers, 24.7 and 26.3 $J/(mol\,K)$, at correspondingly 1800 and 3000K, are given in the reference book on carbon Pierson-book1993 , but their original source is not clear. Since the monotonic growth of specific heats is almost linear, we make an extrapolation to the critical (sublimation) temperature 3900K to get $C_{\mathrm{DP}}\approx 27.5,\ J/(mol\,K)$. The reliably measured DP limit of diamond is apparently not known.

The DP value for silicon, 29.199 J/(mol  K) at its melting point is taken from the reference book Barin-book1995 . Ref. Glazov-HT2001 cites the book Gurvich-book1990 for the recommended polynomial fit, which gives 29.114 J/(mol  K). We take their average as $C_{\mathrm{DP}}$.

The specific heat of germanium near the melting point is given in the reference book Barin-book1995 : $C_{\mathrm{DP}}=28.756J/(mol\,K)$. It lets us fix the overall constant of the model, (29), for the diamond lattices:

[TABLE]

The Dulong-Petit value of gray tin could be measured at rather low temperatures. Its critical temperature is marked by transition to $\beta$-Sn. We found the $\alpha$-Sn specific heat $0.278\,J/(g\,K)$ at 100K in the reference book Adachi-book2005 , which translates to $33.00\,J/(mol\,K)$. However, it is not measured, but implied from the elastic constants. It is clearly higher then other $C_{\mathrm{DP}}$ values in the table. According to the proposed model, we conjecture, that the true value of the Dulong-Petit limit of specific heat of gray tin is the same as the one for silicon and germanium, namely, approximately $29\,J/(mol\,K)$.

Summarizing, it is likely, when experimental inconsistencies are cleared up, the lattice specific heats of the group IV elements with the diamond lattice would be the same. This value is close to $29J\,mol{}^{-1}\,K{}^{-1}$ and is certainly away from the presumed exact value (32). We will further check if this statement is true for any substance with the diamond lattice.

III.3 Low temperature asymptotics

Let us explore the low temperature asymptotics of specific heats. It is common in the literature Kittel-book1996 to plot the data not for specific heat, but for the implied (effective) Debye temperature (43). It is experimentally known that the effective Debye temperature is not constant, as it should be if the Debye theory were correct, instead it exhibits some complex behaviour, especially at low temperature Kittel-book1996 ; Gopal-book1966 ; Flubacher-PM1959 ; Berg-PRSA1957 . We use a similar combination, $T(R/C_{M})^{1/3}$, because it clearly distinguishes different temperature regimes at low temperatures. The gas constant merely allows us to have the units of Kelvin on both axes, while numerical values on the vertical axis are irrelevant. The resulting graph for germanium in Fig. 2 resembles typical curves for the Debye temperatures Flubacher-PM1959 ; Berg-PRSA1957 .

Because we propose that the low temperature behaviour of specific heat is $T^{4}$, we interpret the descending branch of this graph as the power law $T^{-1/3}$. Temperature at the minimum value of this graph, denoted as $T_{0}$ , indicates the limit of the low temperature regime. It is the characteristic temperature for any given material. For germanium, the characteristic temperature is $T_{0}=21.4\,K$. Characteristic temperatures for diamond and silicon are given in Table 1. The region of low temperatures for specific heat is limited from below as well, because the relative contribution of the surface heat grows in addition to the experimental uncertainty, which shows up in these data.

Let us show that the characteristic temperature of a condensed matter system that specifies the condition $\alpha\gg 1$, also gives a universal heat capacity constant. Skipping the unessential calibration parameter, the explicit form of the low-$T$ asymptotics is,

[TABLE]

We take the lowest velocity of sound, $v_{5}$, since it gives the leading contribution at low temperatures. The threshold for this condition to hold is given by the characteristic temperature, $T_{0}$. We conjecture that the l.h.s. of (35), taken at this temperature, is a dimensionless constant, which is the same for all materials with the same type of crystal lattice,

[TABLE]

The values of $\theta$ are given Table 1, which shows that they are indeed very close, with the diamond’s value somewhat apart. Let us take now the computed sound velocity, $v_{5}$, for $\alpha$-tin to derive $T_{0}$ for gray tin from this constant (36), taken to be 1.73:

[TABLE]

This is a testable prediction for future determinations of the specific heat of single crystals of gray tin.

Let us now test the $T^{4}$-hypothesis (27) vs. the Debye cubic law (42) statistically and graphically, without calibrating the full theory, as we need this for the second calibration parameter. We take the obtained low temperature range, $T=4.364\ldots 20.233\,K$, and fit its specific heat data with two testing functions, $aT^{4}$ and $bT^{3}$. The resulting least-square fits obtained with Maple are $a=6.297\cdot 10^{-6}\ J/(molK^{5})$ and $b=9.085\cdot 10^{-5}\ J/(molK^{4})$. The chi-square statistics for both hypotheses do not reject either, i.e., both laws are statistically allowed. However, the $\chi^{2}$-statistic for the cubic law is 0.162 vs. 0.002 for the quartic function, with both p-values equal to one; that shows the $T^{4}$-function is favored.

The more apparent evidence can be seen, when two obtained fits are plotted together with the measured data, in a popular graph, $C_{M}/T$ vs. $T^{2}$, Fig. 3, often used to demonstrate validity of the Debye cubic law Kittel-book1996 ; Flubacher-PM1959 ; Berg-PRSA1957 ,

It is obvious that in the selected temperature range the specific heat of germanium does not obey the cubic law.

Another popular graph in the solid state physics literature is $C_{M}/T^{3}$ vs. $T$, Fig. 4.

This graph looks even more compelling, as the observed data are nowhere close to a constant (horizontal line) expected from the Debye $T^{3}$-law. Instead they can be approximated by a linear function (obtained above by fitting the original data, not this scaled data), which signifies a $T^{4}$-law. Note, these fits assumption, $C_{M}=0$ at $T=0$, which is excluded in the geometrical formalism, is only a reasonable approximation here, as $C_{M}$ is infinitesimal (but never zero due to the surface heat) at $T<1\,K$.

The results of the power law fits and the $\chi^{2}$-statistics as well as the graphs made for the silicon data Flubacher-PM1959 are qualitatively the same, they confirm the quartic power law.

The diamond data Desnoyers-PhilMag1958 gave inconclusive results, both the power laws are equal at approximating the low-T specific heat. The reasons could be not only the anomalously big region of low temperatures, but also the presence of impurities and defects in natural diamonds, or catalytic atoms in synthetic diamonds (also see Discussion).

In general, it is difficult to make a selection of the two tested power laws using original (not scaled) data. However, this is not in the case for germanium as seen from Fig. 5.

However, in the solid state physics literature, such fits are done not to the cubic law, but to the extended polynomial, $C_{M}=a\,T^{3}+b\,T^{5}+c\,T^{7}$. Various arguments were given for its introduction, but its two higher power coefficients usually have low or no statistical significance. Probably because of the use of such extended polynomials and because of the firm belief in the Debye cubic law, its validity has not been questioned, even though experimental data contradicting to it are abundant.

III.4 Specific heat of gallium arsenide

The ideas about the Kopp-Neumann rule for multi-atomic substances suppose that compounds have the same properties as elements if their crystal lattice types are the same. In the diamond lattice of a two-atomic compound (zinc-blende lattice), half of its atoms belong to each composing element Kittel-book1996 ; Szwacki-book2010 . Many semiconductors crystallize to the zinc-blende lattices IoffeInst-2001 , thus, their properties are well measured Adachi-book1992 .

Let us take as gallium arsenide, GaAs, an example. According to the conjecture of Sect. II.5, the Dulong-Petit limit of GaAs should be the same as the DP value of any of the carbon group elements with the diamond lattice, if we would take one mole of its atoms, not molecules. Indeed, this is true within experimental uncertainty: the GaAs specific heat given at $1500\,K$ in Glazov-IM2000 is $C_{\mathrm{DP}}=58.17/2=29.08\ J/(molK)$.

We can also look at the low temperature asymptotics of the GaAs specific heat. The study Cetas-PR1968 provides the sufficient number of data points for specific heat of gallium arsenide. Using it we made a plot of the type of Fig. 2 that determined the low temperature region by $T_{0}=20.0\,K$, which produced the $\theta$-constant as 1.67, confirming its universal nature. Further statistical and graphical analysis performed according to the described above procedures gave the same conclusions as for the group IV elements: the low temperature asymptotics of the GaAs specific heat is a $T^{4}$-function.

Therefore, we can conclude that the conjecture about similarity of thermal characteristics of compounds and elements with the same lattice type holds. In fact, their elastic properties are also similar as they should be. In particular, gallium arsenide resembles germanium, as is seen from Table 1. Such a similarity was noted first in McSkimin-JAP1967 , and it is likely the fundamental reason for a near coincidence of characteristics of the considered specific heats.

IV Summary

Chief applications of finite temperature quantum field theory lie in condensed matter physics. As a first element of this program we have implemented the idea of P. Debye about lattice heat as energy of sound waves in elastic bodies. The proposed thermodynamics of solid state matter presents the theory of specific heat formulated in the field theory language, without reference to the Planck’s distribution and quantum oscillators. The summary is here.

• •

Thermodynamics of condensed matter is a theory different from classical thermodynamics of gases.

• •

The dimensionless combination of thermodynamic temperature and the lattice constant is a proper thermal variable in condensed matter physics.

• •

The dimensionless functional of free energy is defined by the sound velocities, lattice constants, crystallographic type and geometrical characteristics of a condensed matter system.

• •

The measurement postulate is expressed as the scaling hypothesis.

• •

Specific heat of any condensed matter system is determined by the universal thermal sum of the new thermodynamic variable.

• •

The specific heat near a critical point (the Dulong-Petit vale) is the same for all substances with the same type of crystal lattice. The Dulong-Petit limit appears due to the lattice cell size cut-off.

• •

One mole of an n-atomic compound, in a crystal form, should be equal to the Avogadro number of its atoms. The Neumann-Kopp law holds only in special cases due to the old definition of the solid matter mole.

• •

The lattice specific heat at low temperatures is the $T^{4}$ law, not the Debye $T^{3}$ law. Experimental data of J.A. Morrison group and some other data decisively confirm the quartic law. Other power laws may appear due to the electronic, surface and polycrystalline properties of solid matter.

• •

Universal thermal asymptotics of a solid body towards the absolute zero temperature do not exist, because they depend on the body’s shape and material.

V Discussion

The central object of spectral geometry Gilkey-book2004 is the trace of the heat kernel. Its leading (under the ’high-temperature’ conditions Gusev-FTQFT2015 ) terms are expressed by two geometrical invariants of a system’s domain, volume and the boundary’s area, (4). This theorem of mathematical physics has been re-derived by various methods starting from the pioneering works of P. Debye Debye-AdP1912 and H. Weyl Weyl-RCMP1915 . The heat kernel removes the need to study actual spectra of condensed matter systems. The spectrum of lattice frequencies, as studied in the neutron scattering experiments and in the phonon theory, may be complex, nevertheless, its heat kernel trace is always proportional to the system’s volume, (4).

We need to know the function’s values at two temperature points in order to finally fix it for a specific class of materials, e.g., diamond lattice crystals. Thus, the model should be calibrated anew for every type of crystal lattices, which means there is its own thermodynamics for every crystallographic type. This is not surprising at all. Classic thermodynamics was developed as the physics of rarefied gases, and the universal feature of gas is its homogeneity. In contrast, solid state matter is inhomogeneous and anisotropic, i.e., it possesses internal scales along different directions. These geometrical properties determine the thermodynamics of condensed matter systems.

We have not completed the full calibration of this model. In the low temperature regime, the model has a simple power behaviour, which is defined by the lowest velocity of sound. However, as seen from the $\Theta(\alpha)$ behaviour, Fig. 1, the one-velocity model can only be correct in the low temperature limit. According to the assumption about the total free energy (19), the sum over all allowed velocities should be performed. However, the naive sum without independent weights, i.e., with equal weights, $\mathcal{R}/9$, does not match the full data set. Obviously, by varying weights in the sum we could fit the data well, but this is an artificial procedure that would multiply the number of calibration parameters. This means we have to acquire more information about physics of elastic waves to finish this work.

Experimental testing of this model has only begun. The data for natural Desnoyers-PhilMag1958 and synthetic Atake-RRLEM1991 diamonds do not confirm the quartic law. Instead both power laws are statistically allowed, but they are equally bad at describing the natural diamonds data, while the Debye cubic law is selected for the synthetic diamonds. However, diamond is the single thermophysical anomaly among all substances in the Nature; more precise measurements are needed. Furthermore, available data sets for metals, copper, Cu Cetas-PR1968 and aluminum, Al Berg-PhysRev1968 , show the $T^{3}$ behaviors at low temperatures when treated with the algorithms above. Whether this is a property of metals with the electronic heat contribution or the characteristics of polycrystalline solids, which apparently formed the used ingots, should be examined. At the same time we can observe the $T^{4}$-power law in the specific heat data of some two-atomic compounds: gallium antimonide, GaSb Cetas-PR1968 , silver chloride, AgCl Berg-PRB1976 . The critical experimental analysis and the theory development should continue.

Physical properties of polycrystalline bodies are different from those of single crystals. A polycrystalline body can be considered as a composite of grains (domains) of single ideal crystals joined by their common faces (boundaries). Therefore, it is reasonable to assume that each grain has its own volume free energy as well as its own surface free energy, while the contributions from the edges and vertices of the grains could be significant. We know that the contribution of a boundary may exceed the volume one in the ’low temperature’ regime Gusev-FTQFT2015 . Thus, the total heat capacity of polycrystalline bodies may be complex.

In his work Einstein-AdP1907 A. Einstein introduced the quantum hypothesis of M. Planck to condensed matter theory. It has been admitted that this innovative approach was not correct in the method and the result. The Einstein theory seemingly produced the Dulong-Petit law, but not the low temperature asymptotics. We have argued that the DP limit could also not be derived this way. The non-success was caused by the use of mechanistic methods, inherited through statistical mechanics (oscillators and the equipartition theorem), in an entirely new realm of physics. In this early work Einstein did not get rid of classical mechanics altogether, but later he criticized a mechanical view of physics and advocated dispensing it later Einstein-book1967 . P. Debye chose the right approach, but the Planck’s empirical formula Planck-book1914 remained the only way to connect elasticity theory and thermal physics. More critique of his theory is in Appendix A.

The dynamical theory of crystal lattices developed for specific heat by Max Born and Theodore von Karman Kittel-book1996 ; Born-book1962 was originally close to the Debye theory Born-PZ1913 in its use of elasticity theory. In modern textbooks the Born-von Karman theory is considered the first principles theory based on quantum mechanics. The obvious and often criticized fault of this theory is its periodic (cyclic) boundary conditions (the Born-von Karman boundary conditions), which define a condensed matter system’s space as compact without a boundary. Thus, this theory cannot deal with boundary effects at all. Furthermore, it is important to realize that topology of the Born-von Karman theory’s space is non-trivial. Its boundary conditions effectively substitute topology of the open space, $\mathbb{R}^{3}$, with topology of the three-dimensional torus, $\mathbb{S}^{1}\times\mathbb{S}^{1}\times\mathbb{S}^{1}$. This makes it a different mathematical problem, whose solutions are also different from the original problem’s ones. The comparisons of the Born theory’s predictions with experimental data were done and they demonstrated de-Launay-book1956 that the Born-von Karman theory had failed to agree with the available at that time experimental data. Later experimental studies Victor-JCP1962 also cited significant disagreements of measured specific heats with theoretical results of the this theory. Therefore, we are forced to conclude that the Born-von Karman theory is false.

Another theory of the lattice specific heat is a phenomenological theory of Chandrasekara Venkata Raman Raman-PIASA1956 . His theory of the crystal lattice dynamics was based on the frequencies of a basic block of atoms, e.g., in the diamond lattice Raman-PIASA1956-2 , and it is apparently good at describing some properties of diamond’s specific heat Victor-JCP1962 . However, Raman’s theory is not working at low temperatures Victor-JCP1962 . The likely reason is that in this regime the theory should take into account the system’s spatial domain as a whole, i.e., the full lattice with its boundary, and the discreteness of its frequency spectrum.

Dissatisfaction with the Debye theory brought up attempts to improve its agreement with experiment by introducing more parameters to fit the empirical equations to the observation data. Sometimes large discrepancies between the Debye theory and experiments are assigned to peculiar properties of a specific state of matter, such as glass Matsuda-JNCS2011 . However, the shape of the low-temperature graphs in Ref. Matsuda-JNCS2011 looks similar to Fig. 4. Thus, it could likely be a consequence of the universal power law (27), which makes its appearance in many instances of condensed matter.

Diamond is one of the most experimentally studied crystals. It has fascinated people and researchers for a long time, Einstein-AdP1907 ; Raman-PIASA1956 ; Desnoyers-PhilMag1958 ; Atake-RRLEM1991 . Diamond is indeed a thermal anomaly among elements, and it proved its character again by refusing to reveal its low temperature power law. The belief in the anomalously low Dulong-Petit limit of diamond Kittel-book1996 was false as its specific heat maximum is simply reached at very high temperatures. Diamond’s remarkable thermal properties are explained by its unusual elastic characteristics. Namely, the high velocities of sound and the dense lattice provide the anomalously large combination $v/a$ in (26), which spreads the functions $\Theta(\alpha)$ along the temperature axis. Thus, the diamond gets the longest range of low temperatures. We believe the parameters in Table 1 found in experiments should be different if ideal diamond crystals were available for measurement. Neither natural, nor currently produced synthetic (due to the used technology) diamonds can be satisfactory. In particular, the DP value should be the same as for other elements, about 29.1 J/(mol K), and the $\theta$-parameter be equal to 1.7. The importance of carbon for the technology of high temperature resistant materials gives us a hope that these numbers will be verified soon.

It is derived in statistical mechanics that the specific heat and the heat conductivity are intrinsically related. Therefore, often instead of direct determinations, the heat conductivity is measured while the specific heat is implied from the Fourier’s thermal conductivity equation Kittel-book1996 . Since we have not derived yet thermal conductivity in the present formalism, we have focused only on the calorimeter measurements. For this reason, we place the discussion of specific heat of solid argon Finegold-PR1969 ; Flubacher-PPS1961 , the textbook’s topic, Chap. 5 of Kittel-book1996 , to Appendix B.

To a great surprise, basic thermal and elastic properties of $\alpha$-tin, in single crystals, were not studied yet (at least we failed to find such studies and data). Nor specific heat, neither sound velocities were directly measured in gray tin. The technique to produce single crystals of $\alpha$-tin is described in Ref. Styrkas-IM2005 . Now these crystals await an experimentalist to have their properties measured.

A special statement about experimental data is in order. It is regretful that starting from around 1980s, many reports on physics experiments stopped including essential measurement data in their publications. Instead, data are analyzed by experimental teams, sometimes in collaboration with theoreticians, and a result of such an analysis is presented. If data are presented, it is done in plots, which makes them useless for other researchers. This situation creates a serious danger to physics as an experimental science by removing access to valuable information that should belong to the whole science community. Appendix C of this paper reproduces the specific heat data for germanium, Ref. Flubacher-PM1959 , that were obtained at the publicly funded laboratories of National Research Council of Canada. Let us mention that the results of another great work of J.A. Morrison’s group Berg-PRSA1957 were essentially lost, as we failed to recover its full data, deposited at the Royal Society (U.K.) archives, despite the efforts of both countries’ agencies.

When the present work was done and the paper was written, we discovered the comprehensive series of works of Roland Pässler, e.g., Passler-PSSB2008 ; Passler-PSSB2007 . In these works a new model for the description of specific heats over an entire range of measured temperatures is developed. The model contains several characteristic temperatures and fits the data with the sums of exponential functions, whose arguments are quadratic in the temperature. The fact that data of Ref. Flubacher-PM1959 are used to calibrate the model further emphasizes the crucial role of experimental data for theory development. A careful comparison is required, but it is already clear that the empirical representation derived by Pässler is similar in some aspects to the axiomatic proposal above.

Thermodynamics was created as physics of gaseous state of matter, and its main practical application was the development of heat engines. Therefore, in its original form classical thermodynamics is limited in the range of physical phenomena it can describe, and its extension to continuous medium requires further development. Thermodynamics is incomplete until a body’s boundaries included into a physical theory. In classical thermodynamics, boundaries are present, but only implicitly as a means to keep volume finite and to transfer heat energy from gas to external media and reverse. Thermodynamics with boundaries included explicitly arises from the finite temperature quantum field theory.

The opening paragraphs of the paper by James Clerk Maxwell ’On the dynamical theory of gases’ Maxwell-book2011 read: ”Theories of the constitution of bodies suppose them either to be continuous and homogeneous, or to be composed of a finite number of distinct particles or molecules.

In certain applications of mathematics to physical questions, it is convenient to suppose bodies homogeneous in order to make the quantity of matter in each differential element a function of the co-ordinates, but I am not aware that any theory of this kind has been proposed to account for the different properties of bodies. Indeed the properties of a body supposed to be a uniform plenum may be affirmed dogmatically, but cannot be explained mathematically”.

The present work is only a first step towards the mathematical theory of ’homogeneous bodies’, i.e., the field theory of condensed matter. Let us hope it will motivate some critical development in condensed matter physics.

Acknowledgment

Let us acknowledge Grigori Vilkovisky’s insight into the the value of the heat kernel for theoretical physics. We are grateful to Hermann Nicolai for encouraging new research directions at AEI. This work was partially done during the 2014 visit to Institut des Hautes Études Scientifiques. We thank Veselin Jungic at SFU’s IRMACS Centre for the electronic information support. We acknowledge emergency help of Wikipedia as a guide in the information Universe.

Appendix A: Review of the Debye theory

The generating functional of the Debye theory is not dimensionless, but it is energy (J) Debye-AdP1912 . The Debye theory contains one calibration parameter, the Debye temperature, $T_{\mathrm{Debye}}$, which gives the dimensionless variable of the theory,

[TABLE]

The scaling function of the Debye theory has the form,

[TABLE]

which gives the specific heat as,

[TABLE]

The overall factor of the Debye specific heat is fixed by the belief in the exact Dulong-Petit law,

[TABLE]

The low temperature asymptotics

[TABLE]

and $C_{\mathrm{Debye}}$ is zero at $T=0$.

The Debye temperature is an experimentally determined constant for every material. It is found from the low-T limit of the specific heat (42) as,

[TABLE]

The Debye temperature is related to the elasticity constants through the sound velocities Debye-AdP1912 ; Gopal-book1966 ; Kittel-book1996 ,

[TABLE]

The general deficiency of the Debye theory is its single characteristic temperature that could not be physically realistic for describing specific heats, as is recognized in the literature. With one Debye temperature, there can be only one velocity. It is the average velocity $v_{m}$ of the longitudinal and transverse velocities,

[TABLE]

In fact, at low temperatures, this expression is equivalent to using the transverse velocity: $v_{t}$ is typically 1.5-2.0 times smaller than $v_{l}$, so, $v_{m}\propto v_{t}$. This fact may explain why the values of the implied Debye temperature (43) and the elastic Debye temperature (44) are so close.

It is described in the literature Gopal-book1966 ; Kittel-book1996 , that the Debye theory of specific heat works relatively well at temperatures below $T<T_{\mathrm{Debye}}/50$ and above $T>T_{\mathrm{Debye}}/2$, with different values for the Debye temperature in each range. Thus, it does not work for the large and important range of temperatures. Above we have argued that the Debye theory also fails to work at the high and low temperature asymptotics.

The reason for poor performance of the Debye theory is its two intrinsic problems. First, in order to deal with the divergence of total energy in the high frequency limit, the continuous medium model based on elasticity theory was replaced by the discrete model of quantum oscillators, which were identified with the lattice nodes Kittel-book1996 ; Debye-AdP1912 . Here Debye used Einstein’s idea Einstein-AdP1907 to explain the Dulong-Petit law by postulating that the number of quantum oscillators is equal to the number of lattice nodes of a crystal lattice, $N$. Then he equated the total energy of these oscillators, found according to the equipartition theorem, to the total energy of elastic standing waves, in the volume ${\mathcal{V}}$, up to certain maximum frequency, $\nu_{\mathrm{max}}$. He derived from the elasticity theory that the total energy of elastic waves is proportional to the third power of this maximum frequency Debye-AdP1912 ,

[TABLE]

where ${\mathsf{F}}$ is given by the elasticity parameters. This equation enforces the fixed upper limit of a body’s elastic spectrum. The maximum value $\nu_{\mathrm{max}}$ obtained from (46) serves as the upper cut-off of the frequency integral.

However, this explanation could not have been acceptable, because the Debye theory is a field model, while the discrete model of a crystal lattice presents an alternative description of the same physical phenomena. This creates a mix-up of two entirely different formalisms, and the resulting hybrid theory is not self-consistent. Second, in his derivation of specific heat P. Debye computed an integral of the density of eigen-states over frequency. Thereby, he replaced the discrete modes by a continuous variable. However, in the low temperature regime (42), the deviation from the true discrete distribution inside a compact domain can be significant. In fact, this substitution procedure in general is ambiguous and serves as a cause of confusion.

Appendix B. Specific heat of solid argon

The popular solid state physics textbook of Charles Kittel Kittel-book1996 gives a graphic proof of the $T^{3}$ law by plotting the specific heat of solid argon vs. $T^{3}$, Fig. 9, p. 125. The figure is given with no reference, but the names of experimental data’s authors let us trace its published source, which is the work of Leonard Finegold and Norman Phillips Finegold-PR1969 . These experiments were done, at temperatures between 0.4 and 12K, with the heat pulse technique, i.e., by determining the speed of heat propagation in matter, as different from the calorimeter measurements. Beside the crystal lattice of solid argon is face-centered cubic, not the diamond one as in the examples above. However, due to significance given to this plot in the literature we still analyzed these references and the specific heat data for solid argon.

Using the data of Ref. Finegold-PR1969 , the graph of $C_{M}$ vs. $T^{3}$, similar to Fig. 9 of Chap. 5 in the textbook Kittel-book1996 , can be plotted, and it does look like, at the first glance, as a straight line. However, this figure is really deceiving, because the use of such a scale on the horizontal axis does not reveal the data’s power law. This apparently straight line is determined by a few points at higher temperatures $7\,K<T<11\,K$, while nothing quantitative can be concluded about the majority of data points in the few Kelvin region of interest.

We performed on these data the same analysis as done for the carbon group elements, and the conclusions are the same. If the statistical analysis is done with the raw data, $C_{M}(T)$, it confirms the quartic law at lower temperatures, but the cubic law is not statistically excluded. Using the plot, $T(R/C_{M})^{1/3}$ vs. $T$, Fig. 6, the characteristic temperature of solid argon is found, $T_{0}=8.0K$.

If we plot $C_{M}/T^{3}$ vs. $T$, Fig. 7, as done by L. Finegold and N. Phillips Finegold-PR1969 , we do not get a horizontal line predicted by the $T^{3}$-law, neither Finegold and Phillips did.

This graph, Fig. 7 (Fig. 2 in Ref. Finegold-PR1969 ), varies significantly, likewise the krypton data’s plot Finegold-PR1969 . The data points below 1.5 K should be excluded due to apparent experimental uncertainties. In the remaining range of temperatures, from 1.5 to 8 K, this graph Fig. 7 is almost a linear function, $C_{M}/T^{3}=a\,T$, which implies the $T^{4}$ power law in the specific heat data of solid argon (as well of solid krypton).

These experimental data are similar to and improve on the results of J.A. Morrison’s group Flubacher-PPS1961 , as compared in Finegold-PR1969 . The experimentalists, being unable to explain these discrepancies with the theory, concluded their 1969 paper by calling for better theories Finegold-PR1969 : ”whereas some years ago the theories were more advanced than the experimental data available, now the need is for a better understanding of the interatomic forces as well as for improved anharmonic theories”. The old experimental data are still quite sufficient for modern use, but instead of building a tower of improvements upon the old theory we suggest to explore different theoretical ideas.

Appendix C. Specific heat of germanium, Flubacher-PM1959

The temperature is given in the units of Kelvin, and the specific heat of germanium is converted from calories of the original table to Joules of the SI system, $J\,mol^{-1}\,K^{-1}$.