Specific Heat of Solids

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Specific heat of solids

Contributed by: Robert O. Pohl Publication year: 2014

The specific heat (short for specific heat capacity) of a solid is the amount of heat required to increase the temperature of a unit mass of the solid by a unit amount. It is determined by the vibrations of its atoms or excitations of its electrons, and also by a variety of phase transitions. See also: SPECIFIC HEAT.

Contribution of atomic vibrations

If a solid object, say a piece of rock salt, is said to have a certain temperature, it means that its constituent atoms perform motions around their equilibrium positions. The larger the amplitude of these motions, the larger the temperature.

Classical case. In the classical picture of this thermal motion, each atom is considered to be a harmonic, three-dimensional oscillator with six degrees of freedom, three of kinetic and three of potential energy. Each degree of freedom contains on average the energy (1∕2)k,B T,whereT is the absolute temperature [measured in ,−23 ⋅ ,−1 kelvins (K)] and k,B is Boltzmann’s constant (k,B = 1.38 × 10 J K ). Thus, an object containing 1 mole or N,A = 6.02 × 10,23 atoms, contains the thermal energy Q in the form of atomic vibrations given by Eq. (1),

Image of Equation 1 (1)

and has the molar heat capacity c,v given by Eq. (2),

Image of Equation 2 (2)

⋅ ,−1 ⋅ ,−1 where R is the molar gas constant, and R = N,A k,B = 8.31 J mol K . The molar heat capacity can be converted to specific heat capacity by dividing the molar heat capacity by the mass of 1 mole of the substance. Equation (2) is called the rule of Dulong and Petit, proposed in 1819 on the basis of measurements near room temperature. See also: DEGREE OF FREEDOM (MECHANICS); KINETIC THEORY OF MATTER; TEMPERATURE.

Quantized atomic vibrations. When H. F. Weber (1875) extended the measurements over a wide temperature range (220 to 1300 K or −84 to 1880◦F), he found that Eq. (2) holds only in the limit of high temperatures. Figure 1 shows measurements on copper, extended to even lower temperatures: at ∼20 K (−424◦F) the specific heat AccessScience from McGraw-Hill Education Page 2 of 7 www.accessscience.com

WIDTH:BFig. 1 Specific heat capacity for copper (data points), and predictions according to theories of Einstein and Debye (curves), on logarithmic scales. Above ∼300 K (80◦F), the specific heat capacity approaches the value based on the rule of Dulong-Petit, which for copper is J⋅g,−1⋅ K,−1. The molar heat capacity of 3R for copper (molecular weight 63.546) is indicated.

capacity has dropped to ∼3% of its room-temperature value which agrees with the value based on the rule of Dulong-Petit; the arrow on the right ordinate of Fig. 1 indicates the molar heat capacity of 3R. The explanation for the decrease at low temperatures was offered by Albert Einstein (1907). He suggested that the atomic vibrations are quantized; the atoms cannot vibrate with continuously variable amplitude (or energy). Rather, the atoms can only have energies varying in integer steps of hv,whereh is Planck’s constant, h = 6.63 × 10,−34 J ⋅ s, and v is the classical frequency of vibration. As the temperature decreases, an increasing fraction of the atoms will not be excited at all; they will be frozen out. This leads to an exponential decrease of the specific heat capacity. Figure 1 ,12 ,−1 shows Einstein’s fit to the data, which he obtained by using the frequency v,E = 4.6 × 10 s (Einstein frequency for copper) as an adjustable parameter. Through this work, the validity of the quantum concept was demonstrated for the motion of material particles. See also: PLANCK’S CONSTANT; QUANTUM MECHANICS.

Quantized elastic waves. In 1911, Einstein realized a shortcoming in his theory: not only did it lead to a drop-off of the specific heat capacity at low temperatures faster than observed experimentally (Fig. 1), but it also led to a thermal conductivity of the wrong magnitude and temperature dependence. He also realized the cause for this problem: he had assumed that each atom vibrates independently of its neighbors; however, every atom, as it vibrates, will push and pull on its neighbors, and this will lead to a coupling between neighboring atoms. M. Born and T. von K´arm´am (1912) showed that this coupling will lead to elastic waves, which will propagate with the velocity of sound through the solid. Using the analogy with electromagnetic waves, P. Debye (1912) showed that these elastic waves lead to a slower temperature dependence of the specific heat capacity at low temperatures. AccessScience from McGraw-Hill Education Page 3 of 7 www.accessscience.com

< According to Debye, the molar heat capacity is given at low temperatures (T 0.1Θ,D) by Eq. (3),

(3) Image of Equation 3

where Θ,D is called the Debye characteristic temperature and is determined by the elastic constants of the solid, which are measured.

Figure 1 shows the specific heat capacity of copper predicted by Debye’s theory. Another comparison is shown in Fig. 2, for crystalline quartz. The agreement below 10 K (−442◦F) for quartz is excellent and demonstrates the validity of the theoretical model. The discrepancy between theory and experiment above this temperature is also well understood: Not all the elastic waves in quartz have the same speeds of sound; the high-frequency ones, which will be excited predominantly at high temperatures, are particularly variable. This variation of the speeds of sound leads to variations in the number density of the elastic waves that can be thermally excited, and hence will lead to variations in the specific heat. See also: LATTICE VIBRATIONS; THERMAL CONDUCTION IN SOLIDS.

Difference of specific heats. While in gases the difference between the specific heat capacity at constant pressure and that at constant volume is significant, in solids the difference is relatively small. Therefore, the specific heat is usually measured under constant pressure p (usually p = 0, vacuum). In the example shown in Fig. 3,themolar heat capacity under constant pressure (c,p) of aluminum above room temperature does not saturate at the value given by the Dulong-Petit rule, which is strictly valid only for c,v; the gradual increase of c,p above room temperature is the result of the thermal expansion of the solid. Above 700 K (800◦F), there is also a sudden ◦ increase of the specific heat as the melting point T,m (933.2 K or 1220.1 F) is approached. See also: THERMAL EXPANSION.

Electronic specific heat

A typical metal like copper or aluminum contains roughly as many free electrons as atoms. According to classical 1 theory, each electron should contain a thermal energy of ( ∕2)k,B T for each degree of freedom, three in the case of translational motion. It is, therefore, remarkable that the Debye theory yields such excellent agreement with the experiment for copper, a metal (Fig. 1), considering that this theory completely ignores an electronic contribution. In fact, free electrons are noticed only at lower temperatures. In aluminum, the lattice specific heat according to Debye’s theory, shown in Fig. 3, agrees well with the measurements above 20 K (−424◦F). Below 1 K(−458◦F), the experimental molar heat capacity approaches a linear temperature dependence, also shown in AccessScience from McGraw-Hill Education Page 4 of 7 www.accessscience.com

WIDTH:BFig. 2 Specific heat capacity of silicon dioxide (SiO,2) in its crystalline and amorphous (glass) phases. The Debye characteristic temperatures Θ,D, determined from elastic constants, are so similar that the Debye specific heat prediction is shown as a single curve for both solids.

Fig. 3, that is given by Eq. (4).

(4) Image of Equation 4

At 1.163 K, aluminum becomes superconducting: its electrical resistivity suddenly drops to zero. By applying a magnetic field (∼100 gauss or 0.01 tesla), this transition can be suppressed, and the linear temperature dependence of the electronic specific heat can be observed to the lowest temperatures. The electronic specific heat in the normal (that is, nonsuperconducting) state can be understood on the basis of the Pauli exclusion AccessScience from McGraw-Hill Education Page 5 of 7 www.accessscience.com

WIDTH:BFig. 3 Molar heat capacity at constant pressure (c,p) of aluminum. The specific heat capacity is obtained by dividing the molar heat capacity by the molecular weight of aluminum (26.982).

principle: each quantum state can be occupied by at most one electron, and thus only electrons occupying the uppermost energy states have unoccupied energy states nearby, to which they can move by picking up thermal energy. According to a theory due to A. Sommerfeld, this leads to a molar specific heat given by Eq. (5),

(5) Image of Equation 5

,5 where T,F is a characteristic temperature called the Fermi temperature, which is of the order of 10 K. In deriving this theory, it has been assumed that each atom in the metal supplies one conduction electron. See also: EXCLUSION PRINCIPLE; FERMI-DIRAC STATISTICS; FREE-ELECTRON THEORY OF METALS; SUPERCONDUCTIVITY.

When the metal sample is cooled in the absence of a magnetic field below its superconducting transition temperature T,c (1.163 K in aluminum), the specific heat rises abruptly as the electronic energy states are rearranged. Upon further cooling, the specific heat decreases exponentially. However, even at 0.2 K the specific AccessScience from McGraw-Hill Education Page 6 of 7 www.accessscience.com

heat of the electrons exceeds that of the atomic vibrations in aluminum (according to the Debye theory) by at least tenfold.

Specific heat of phase transitions

The transition from the normal to the superconducting state shown in Fig. 3 is an example of a phase transition occurring among the metal electrons. Other phase transitions in the solid state are, for example, those from magnetic disorder (paramagnetic) to magnetic order (ferro-, ferri-, or antiferromagnetic), or from electric dipolar disorder (paraelectric) to an ordered phase (for example, ferroelectric). The transition from an ordered to a disordered phase ordinarily occurs at a critical temperature T,c, near which fluctuations between regions of order and disorder cause an increase of the specific heat capacity. Thus a graph of specific heat capacity versus temperature T has a sharp peak at T = T,c. See also: CRITICAL PHENOMENA; PHASE TRANSITIONS.

While much progress has been achieved in the understanding of the approach to phase transitions in the solid state, including the specific heat near the critical temperature T,c, the solid-liquid phase transition (for example, the rapid rise of the specific heat of aluminum near the melting point, shown in Fig. 3) is still only poorly understood.

Amorphous solids

In contrast to crystals, these solids lack long-range order. Their structure resembles that of frozen-in liquids. The specific heat of these solids exceeds that of the Debye prediction, even if they are electrical insulators, notably at the lowest temperatures; below 1 K, it varies approximately linearly with the temperature and has almost the same magnitude (to within a factor of 10) in all amorphous solids. (In amorphous metals, the low-temperature specific heat will be increased further by the contribution by the electrons.) Figure 2 shows a comparison of the specific heat capacity of the electric insulator silicon dioxide (SiO,2) in the amorphous (silica) and the crystalline (α-quartz) phases. The Debye theory predicts nearly the same heat capacity for both phases, which is therefore shown as a single curve. Obviously, the Debye model is only a poor approximation in the amorphous phase. The physical nature of the excitations giving rise to the linear specific heat anomaly is not yet understood. A particular challenge is the universality of these excitations. See also: AMORPHOUS SOLID. Robert O. Pohl

Bibliography

N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1976

A. Cezairliyan (ed.), Specific Heat of Solids, 1988

E. S. R. Gopal, Specific Heats at Low Temperatures, 1966 AccessScience from McGraw-Hill Education Page 7 of 7 www.accessscience.com

C. Kittel, Introduction to Solid State Physics, 8th ed., 2005

Y. S. Touloukian et al. (eds.), Thermophysical Properties of Matter, vols. 4–6, 1970

G. K. White and P. J. Meeson, Experimental Techniques in Low-Temperature Physics, 4th ed., 2002

Additional Readings

J. Gallant, Doing Physics with Scientific Notebook: A Problem Solving Approach, 1st ed., John Wiley & Sons, Chichester, West Sussex, UK, 2012

D. M. Kim, Introductory Quantum Mechanics for Semiconductor Nanotechnology, Wiley-VCH Verlag, Weinheim, Germany, 2010

G. Ventura and L. Risegari, The Art of Cryogenics: Low-Temperature Experimental Techniques, 1st ed., Elsevier, Oxford, UK, 2008

M. Wyart, Scaling of phononic transport with connectivity in amorphous solids, Europhys. Lett., 89(6):64001, 2010 DOI: http://doi.org/10.1209/0295-5075/89/64001