R LOUDON. L. ALLEN. AND D. F. NELSON dinary momentum and pseudomomentum. The name"wave nomentum'' was introduced for this reason v×E While the inclusion of material deformation has played ar essential role in the clarification of what is momentum and what is pseudomomentum, it also acts as a barrier to simple V×B=j+ (24) physical understanding. The aim of the present paper is thus to find simplified versions of the conservation laws for mo- mentum and pseudomomentum, even after generalization of where the fields are functions of position and time previous work to include loss. This is achieved by the sim- E=E(r, t), and so on. The bound charge and current densi with cubic isotropy, essentially the single-resonance Lorentz time; they can be expressed in terms of the dielectric po i plification of the dielectric to a nonmagnetic diatomic crystal ties, p and j, respectively, are also functions of position model. The ions are assumed to be coupled to the electro- Ization P as magnetic field only by an electric-dipole interaction Before proceeding to the main calculations, we present a V. P in Sec. Il. An improved version of previous calculations of and the energy continuity equations and the velocity of energy propagation [5] leads to essentially the same results as be- (26) fore, but our method facilitates parallel discussions of the momentum propagation characteristics. It is found, however that the propagation of momentum involves both the center The electric displacement is defined in the usual way, of mass and relative coordinates of the diatomic dielectric whose proper treatment requires a Lagrangian formulation D=EoE+P current obey a conservation law when the center-of-mass and the magnetic field is given by momentum is included, but that the pseudomomentum, and hence the wave momentum, suffer dissipation on account of (28) he dielectric loss. The various electromagnetic densities de- Note that in the view implicit in these equations, E and b are rived in Secs. Il and Ill are evaluated for a steady-state the fundamental electromagnetic fields, while P describes the monochromatic wave in Sec. IV. where the velocities of propagation of energy and wave momentum are derived, and response of the matter, and Eqs. (2.7)and(2. 8)are constitu- for an optical pulse in Sec. V. The results are discussed in tive equations for D and H We consider a polar, diatomic, cubic, crystal lattice in which the relative spatial displacement field of the two ions in the unit cell is denoted s=s(r, t). The long-wavelength Il. SIMPLE THEORY OF ELECTROMAGNETIC ENERGY optic modes of vibration have a basic threefold degeneracy AND MOMENTUM PROPAGATION which is lifted by the long-range electrical forces to form a wofold-degenerate transverse mode and a nondegenerate The present section is devoted to a derivation of some longitudinal mode[10]. Then, if the frequency of the trans basic results for electromagnetic fields in a dielectric material verse mode is denoted or and its damping rate is denoted T, reated in the Lorentz model. We present a simple derivation the standard form of the lorentz equation for the ith Carte- of the equations that describe the propagation of energy, and sian component of the internal coordinate of the ionic motion show that the corresponding description of momentum IS propagation cannot be obtained by so simple a theory. The detailed derivations of the equations that describe momen- ms,+mrs+mo tum propagation and the identification of the different char- acters of the momentumlike contributions are given careful Here m is the reduced mass density of the two ions, of consideration in Sec. Ill masses Mi and M2, in the primitive unit cell of volume n2 A. Basic equations M1M2 1=(M1+M2) (2.10) The fundamental energy and momentum properties electromagnetic fields in matter are governed by maxwells nd the chars equations and by the equations of motion for the matter. we is given by ity s associated with the internal motion consider a nonmagnetic dielectric material that has no free charges or currents. The Maxwell-Lorentz forms of the equa s=e/① (2.11) tions in conventional notation and Systeme International (SI nits are then where e and -e are the charges on the two kinds of ion. The VE=p/Eo, (2. 1 polarization is expressed in terms of the internal coordinate V.B=0, (2.12)dinary momentum and pseudomomentum. The name ‘‘wave momentum’’ was introduced for this reason. While the inclusion of material deformation has played an essential role in the clarification of what is momentum and what is pseudomomentum, it also acts as a barrier to simple physical understanding. The aim of the present paper is thus to find simplified versions of the conservation laws for mo￾mentum and pseudomomentum, even after generalization of previous work to include loss. This is achieved by the sim￾plification of the dielectric to a nonmagnetic diatomic crystal with cubic isotropy, essentially the single-resonance Lorentz model. The ions are assumed to be coupled to the electro￾magnetic field only by an electric-dipole interaction. Before proceeding to the main calculations, we present a simplified discussion of energy and momentum propagation in Sec. II. An improved version of previous calculations of the energy continuity equations and the velocity of energy propagation @5# leads to essentially the same results as be￾fore, but our method facilitates parallel discussions of the momentum propagation characteristics. It is found, however, that the propagation of momentum involves both the center of mass and relative coordinates of the diatomic dielectric, whose proper treatment requires a Lagrangian formulation. Thus it is shown in Sec. III that the momentum density and current obey a conservation law when the center-of-mass momentum is included, but that the pseudomomentum, and hence the wave momentum, suffer dissipation on account of the dielectric loss. The various electromagnetic densities de￾rived in Secs. II and III are evaluated for a steady-state monochromatic wave in Sec. IV, where the velocities of propagation of energy and wave momentum are derived, and for an optical pulse in Sec. V. The results are discussed in Sec. VI. II. SIMPLE THEORY OF ELECTROMAGNETIC ENERGY AND MOMENTUM PROPAGATION The present section is devoted to a derivation of some basic results for electromagnetic fields in a dielectric material treated in the Lorentz model. We present a simple derivation of the equations that describe the propagation of energy, and show that the corresponding description of momentum propagation cannot be obtained by so simple a theory. The detailed derivations of the equations that describe momen￾tum propagation and the identification of the different char￾acters of the momentumlike contributions are given careful consideration in Sec. III. A. Basic equations The fundamental energy and momentum properties of electromagnetic fields in matter are governed by Maxwell’s equations and by the equations of motion for the matter. We consider a nonmagnetic dielectric material that has no free charges or currents. The Maxwell-Lorentz forms of the equa￾tions in conventional notation and Syste`me International ~SI! units are then “•E5r/«0 , ~2.1! “•B50, ~2.2! “3E52 ]B ]t , ~2.3! 1 m0 “3B5j1«0 ]E ]t , ~2.4! where the fields are functions of position and time, E[E~r,t!, and so on. The bound charge and current densi￾ties, r and j, respectively, are also functions of position and time; they can be expressed in terms of the dielectric polar￾ization P as r52“•P ~2.5! and j5 ]P ]t . ~2.6! The electric displacement is defined in the usual way, D5«0E1P, ~2.7! and the magnetic field is given by H5B/m0 . ~2.8! Note that in the view implicit in these equations, E and B are the fundamental electromagnetic fields, while P describes the response of the matter, and Eqs. ~2.7! and ~2.8! are constitu￾tive equations for D and H. We consider a polar, diatomic, cubic, crystal lattice in which the relative spatial displacement field of the two ions in the unit cell is denoted s[s~r,t!. The long-wavelength optic modes of vibration have a basic threefold degeneracy which is lifted by the long-range electrical forces to form a twofold-degenerate transverse mode and a nondegenerate longitudinal mode @10#. Then, if the frequency of the trans￾verse mode is denoted vT and its damping rate is denoted G, the standard form of the Lorentz equation for the ith Carte￾sian component of the internal coordinate of the ionic motion is ms¨i1mGs˙i1mvT 2 si5§Ei . ~2.9! Here m is the reduced mass density of the two ions, of masses M1 and M2 , in the primitive unit cell of volume V, m5 M1M2 V~M11M2! ~2.10! and the charge density § associated with the internal motion is given by §5e/V, ~2.11! where e and 2e are the charges on the two kinds of ion. The polarization is expressed in terms of the internal coordinate by P5§s. ~2.12! 1072 R. LOUDON, L. ALLEN, AND D. F. NELSON 55