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Specific Heat 8 Specific Heat more complicated, because not only standard repre- Electronic States of; Lattice Dynamics: Vibrational Modes; sentations but projective representations also occur. Periodicity and Lattices; Point Groups; Quantum Mechan- Finally, a basis for the full state space can be con- ics: Foundations; Quasicrystals; Scattering, Elastic structed as follows. The group of k is a subgroup of (General). the space group G. G can be decomposed according to PACS: 61.50.Ah; 02.30. À a G ¼ Gk þ g2Gk þ ? þ gsGk where the space group elements gi have homogeneous Further Reading parts Ri for which Rik ¼ ki. Then the basis is defined as C Cornwell JF (1997) Group Theory in Physics . San Diego: Aca- ij ¼ Tgi cj ½23 demic Press. Hahn Th. (ed.) (1992) Space-Group Symmetry. In: International The dimension of the representation is sd , where s is Tables for Crystallography. vol. A, Dordrecht: Kluwer. the number of points ki, and d the dimension of the Hahn T and Wondratschek H (1994) Symmetry of Crystals: In- point group representation D(Kk). The irreducible troduction to International Tables for Crystallography. vol. A . representation carried by the state space then is char- Sofia, Bulgaria: Heron Press. acterised by the so-called ‘‘star’’ of k (all vectors k ), Janssen T (1973) Crystallographic Groups. North-Holland: Am- i sterdam. and an irreducible representation of the point group Janssen T, Janner A, Looijenga-Vos A, and de Wolff PM (1999) Kk. This means that electronic states and phonons can Incommensurate and commensurate modulated structures. be characterized by k, v. Their transformation prop- In: Wilson AJC and Prince E, International Tables for erties under space group transformations follows from Crystallography, vol. C, Mathematical, physical and chemical this characterization. tables. ch. 9.8, Dordrecht: Kluwer. Aperiodic Crystals Nomenclature Apart from crystals with three-dimensional lattice a, b, c lattice basis vectors periodicity, there are materials with a diffraction aÃ, bÃ, cà reciprocal lattice basis vectors pattern with sharp Bragg peaks on positions A translation subgroup of a space group n D(R) matrix representation k aà exp( À ik Á r)U(r) Bloch form of a wave function ¼ hi i ðinteger hiÞ ½ 24 i¼1 E(3) Euclidean group X k 4 F( ) structure factor When n ¼ 3, the structure is periodic. If n 3, the G space group structure is aperiodic, but it is still considered as K point group crystal, because there is long-range order. Examples n lattice translation vector are modulated phases and quasicrystals. They may O(3) orthogonal group in three dimensions be described as intersections of physical space with a rj position of an atom in the unit cell higher-dimensional lattice periodic structure. The R orthogonal transformation symmetry of such structures is a space group in n {R|t} space group element dimensions, and in this case the theory of space Tg linear operator for group element g: # k r groups in arbitrary dimensions can be used. rð Þ Fourier component of r( ) r(r) density function See also: Crystal Structure; Electron–Phonon Interactions ci basis of a state space and the Response of Polarons; Group Theory; Insulators, c(r) wave function Specific Heat N E Phillips and R A Fisher , University of California at When heat is introduced under certain specified con- Berkeley, Berkeley, CA, USA ditions, it is a well-defined thermodynamic property & 2005, Elsevier Ltd. All Rights Reserved. that gives a measure of the increases in the entropy, the energy, and the enthalpy with increasing temper- ature. It is related to other thermodynamic proper- Introduction ties, for example, the thermal expansion, which is a The specific heat of a substance is the amount of heat measure of the pressure dependence of the entropy. required to raise the temperature by one degree. Specific-heat data make an important contribution to Specific Heat 9 the determination of the Gibbs and Helmholtz With other thermodynamic relations, based on energies, the thermodynamic properties that govern both the first and second laws, it can be shown that the direction of spontaneous change and the equilib- 2 rium conditions for chemical reactions and phase TV a CP À CV ¼ transitions. When interpreted in terms of microscop- k ic models and theories, they provide information on where a ¼ VÀ1ð@V=@ TÞ is the thermal expansion, the forces and interactions at the atomic and molec- P and k ¼ VÀ1ð@V=@ PÞ is the isothermal compress- ular level that determine the macroscopic properties. T ibility. Thermodynamic stability requires that both CV and k be positive. Although a can be either po- Thermodynamic Relations sitive or negative, it appears as the square, and CP À CV is always positive: when the heat is absorbed The specific heat of a substance is the amount of heat at constant volume, dw ¼ 0. In the constant-pressure required to increase the temperature by one degree: process there is, generally, a change in V, dwa0 and dq can be either positive or negative, but the increase in C dT U ensures that the heat absorbed is always greater than in the constant-volume process. Evaluation of a where dq, an inexact differential, is the quantity of and k for an ideal gas, for which PV ¼ RT , where R heat added and d T is the increase in temperature is the gas constant, gives CP À CV ¼ R. produced in the process. By the first law of thermo- dynamics, dq ¼ dU À dw, where U is the internal Microscopic Interpretation energy and dw is the work done in the process. Since U is a ‘‘thermodynamic property,’’ a quantity that Classical, High-Temperature Limit depends only on the thermodynamic state of the sys- In classical statistical mechanics, each term in the tem, dU is an exact differential and depends only on Hamiltonian for the total energy that is the square the initial and final states. However, dw, and there- of either a momentum or a coordinate, contributes fore both dq and C, depend on details of the way the (l/2)k T to the thermal energy, where k is the Boltz- process takes place as well as on the initial and final B B mann constant. For a particle moving freely in space, states. If the work is done by an external hydrostatic the translational kinetic energy includes three terms pressure, P, dw ¼ À P dV, where V is the volume. In in the square of a momentum, corresponding to the that case, three dimensions in space: U ¼ ð 3=2ÞkBT, and dU þ P dV dðU þ PV Þ À V dP CV ¼ ð 3=2ÞkB. For Avagadro’s number, NA, of such C ¼ ¼ dT dT particles, one mole of an ‘‘ideal gas,’’ CV ¼ NAð3=2ÞkB ¼ ð 3=2ÞR: For a particle bound by har- With the definition for the enthalpy, H U þ PV , monic forces to a lattice site in three dimensions, this leads to expressions for the constant volume and there are three terms in the square of a momentum in constant pressure specific heats, the kinetic energy and three in the square of a co- ordinate in the potential energy: U ¼ 3k T, and @U B C ¼ CV ¼ 3kB. At sufficiently high temperatures, quan- V @T V tum statistical mechanics gives the same results, the and classical or high-temperature limit. At temperatures @H for which kBT is of the order of, or smaller than, the C ¼ P @T spacing of the quantum mechanically allowed energy P levels, the higher energy levels are not fully accessi- thermodynamic properties, which can be expressed ble, and U and C do not reach the classical limit. in, for example, units of J K –1 mol–1 . If work is done The energies of the allowed translational states of by magnetic or electric forces, the relevant variables the particles of an ideal gas are proportional to are the magnetic induction, B, the magnetization, M, m–1/2 V2/3, where m is the mass of the particles and V the electric field, E, and the dielectric displacement, is the volume of the container. For gases of atoms and D B H . (In free space ¼ m0 , where m0 is the perme- molecules at ordinary densities, the energy levels are ability of free space and H is the magnetic field.) The so closely spaced that the classical limit applies at all expressions for the work are dw ¼ H dM and temperatures of interest, but for an electron gas at dw ¼ E dD, respectively. In other thermodynamic densities of the conduction electrons in a metal, that relations, H or E replaces the intensive variable P limit is reached only at temperatures B10 5 K. The and ÀM or ÀD replaces the extensive variable V. specific heat of a harmonic oscillator offers a typical 10 Specific Heat 1.0 Table 1 Characteristic temperatures associated with the vib- rational and rotational specific heats of some diatomic molecules and one linear, symmetric, triatomic molecule, CO 2 0.8 a Molecule yvðKÞ yvðKÞ 0.6 H2 6332 88 B D2 4487 43.8 /k ν HD 5500 66 C 0.4 HF 5955 30.2 HCl 4304 15.2 HBr 3812 12.2 0.2 HI 3321 9.4 N2 3393 2.88 O2 2274 2.08 0.0 F 1283 1.27 0.05 0.1 0.2 0.5 1 2 5 2 Cl 2 805 0.351 T/ ν Br 2 463 0.116 CO 3122 2.78 Figure 1 The contribution to the specific heat of a one-dimen- NO 2719 2.45 sional harmonic oscillator, Cv, displayed as Cv/k B, vs.
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