1074 R LOUDON. L. ALLEN. AND D. F. NELSON and the relation between the frequency and wave vector has been taken in its free-space form, for the purpose of an order- of-magnitude estimate. Clearly it is important to include the where the charge and current densities are related to the di center-of-mass momentum of the crystal in any theory of electric polarization by Eqs. (2.5)and(2.6).However, when nomentum propagation through an absorbing dielectric the center-of-mass motion is included, the latter expression The transfer of momentum to the dielectric must be ac- should be augmented by inclusion of the Rontgen current ompanied by a growth in its kinetic energy density, whose [11], to give a total current density value for the momentum density given by Eq (2.23)is aP MR +V×(P×R (36) where r is again the continuum center-of-mass coordinate The rest-mass energy density Mc2 of the crystal is The interaction Lagrangian density(3. 5)can be converted ery much larger than the initial energy density Ni with the use of this expression to rystal kinetic energy is thus completely negligible cor to Nho. This justifies the neglect of center-of-mass motion C1=P(E+R×B) (3.7) in the theory of energy propagation given in Sec. II B, de spite its importance in the theory of momentum propagation. where some perfect space and time derivative terms, which make no contribution to the Lagrange equations of motion, IIL LAGRANGIAN THEORY OF ELECTROMAGNETIC have been discarded [10]. The material Lagrangian density MOMENTUM PROPAGATION for a rigid body is This section is devoted to a rigorous derivation of the CM=MR2+显ms2-显mos2 (38) various momentum densities associated with the propagation of electromagnetic waves through absorbing dielectrics. The where the dielectric parameters are as defined in Sec. IL. The basic dielectric model is the same as that used in Sec. II, but theory also needs to include a term that allows for damping it is necessary to generalize the model to include center-of- of the internal motion at a rate proportional to T. This mass motion in order to describe momentum propagation. It conveniently implemented by a Rayleigh dissipation func- is also necessary to distinguish the contributions of momen- tion of the form tum and pseudomomentum. The continuum mechanics back ground to the calculations is described in detail in Ref [10] R=m2, (3.9) It is assumed throughout that the dielectric material fills all of space the effects of crystal boundaries are excluded from which is incorporated into the Euler-Lagrange equations by the calculations an appropriate additional term [12] The equations of motion for the electromagnetic and ma- A. Lagrangian formulation terial field variables are obtained by the standard Lagrangian The system of dielectric material (M) and electron procedures. Thus the Maxwell-Lorentz equations(2. 1)and netic field (F) coupled by an electric-dipole interaction(n) (2. 4)are rederived straightforwardly, while Eqs.(2.2)and described by a Lagrangian density (2.3)are satisfied automatically from the definitions (3.3) and (3. 4)of the fields in terms of the potentials. It should L=LMt Lte (3. 1) however be noted that the Rontgen term in the current den- sity(3. 6)causes a generalization of relation(2.8)between where the Lagrangian itself is formed by integration over the magnetic field and magnetic induction to [see, for example, Lagrangian density in the usual way Eq(76. 11)of Ref [21 The Lagrangian density of the electromagnetic field is (3.10) The new term is a function of both the internal relative- where the electric and magnetic fields are determined by the displacement coordinate and the center-of-mass coordinate calar potential and the vector potential A in the usual of the dielectric material For the dielectric spatial displacement variables, the equa- tion of motion for the relative position of the two ions in the ⅴd-at (3.3) unit cell is obtained with the use of Eqs. (3. 7)-(3.9)as 5+mIs;+mo3s;=s(E1+(R×B)),(3,1) which is identical to Eq.(2.9)except for the addition of the B=V×A (3.4) term proportional to the center-of-mass velocity R. The equation of motion for the continuum center-of-mass coordi- The interaction Lagrangian density is is obtained similarlyand the relation between the frequency and wave vector has been taken in its free-space form, for the purpose of an order￾of-magnitude estimate. Clearly it is important to include the center-of-mass momentum of the crystal in any theory of momentum propagation through an absorbing dielectric. The transfer of momentum to the dielectric must be ac￾companied by a growth in its kinetic energy density, whose value for the momentum density given by Eq. ~2.23! is MR˙ 2 2 5N\v N\v 2M c2 . ~2.25! The rest-mass energy density M c2 of the crystal is always very much larger than the initial energy density N\v. The crystal kinetic energy is thus completely negligible compared to N\v. This justifies the neglect of center-of-mass motion in the theory of energy propagation given in Sec. II B, de￾spite its importance in the theory of momentum propagation. III. LAGRANGIAN THEORY OF ELECTROMAGNETIC MOMENTUM PROPAGATION This section is devoted to a rigorous derivation of the various momentum densities associated with the propagation of electromagnetic waves through absorbing dielectrics. The basic dielectric model is the same as that used in Sec. II, but it is necessary to generalize the model to include center-of￾mass motion in order to describe momentum propagation. It is also necessary to distinguish the contributions of momen￾tum and pseudomomentum. The continuum mechanics back￾ground to the calculations is described in detail in Ref. @10#. It is assumed throughout that the dielectric material fills all of space; the effects of crystal boundaries are excluded from the calculations. A. Lagrangian formulation The system of dielectric material (M) and electromag￾netic field (F) coupled by an electric-dipole interaction (I) is described by a Lagrangian density L5LM1LI1LF , ~3.1! where the Lagrangian itself is formed by integration over the Lagrangian density in the usual way. The Lagrangian density of the electromagnetic field is LF5«0 2 E22 1 2m0 B2, ~3.2! where the electric and magnetic fields are determined by the scalar potential f and the vector potential A in the usual way, E52“f2 ]A ]t ~3.3! and B5“3A. ~3.4! The interaction Lagrangian density is LI5j•A2rf, ~3.5! where the charge and current densities are related to the di￾electric polarization by Eqs. ~2.5! and ~2.6!. However, when the center-of-mass motion is included, the latter expression should be augmented by inclusion of the Ro¨ntgen current @11#, to give a total current density j5 ]P ]t 1“3~P3R˙ !, ~3.6! where R is again the continuum center-of-mass coordinate. The interaction Lagrangian density ~3.5! can be converted with the use of this expression to Ll5P•~E1R˙ 3B!, ~3.7! where some perfect space and time derivative terms, which make no contribution to the Lagrange equations of motion, have been discarded @10#. The material Lagrangian density for a rigid body is LM5 1 2 MR˙ 21 1 2 ms˙22 1 2 mvT 2 s 2, ~3.8! where the dielectric parameters are as defined in Sec. II. The theory also needs to include a term that allows for damping of the internal motion at a rate proportional to G. This is conveniently implemented by a Rayleigh dissipation func￾tion of the form R5 1 2 mGs˙2, ~3.9! which is incorporated into the Euler-Lagrange equations by an appropriate additional term @12#. The equations of motion for the electromagnetic and ma￾terial field variables are obtained by the standard Lagrangian procedures. Thus the Maxwell-Lorentz equations ~2.1! and ~2.4! are rederived straightforwardly, while Eqs. ~2.2! and ~2.3! are satisfied automatically from the definitions ~3.3! and ~3.4! of the fields in terms of the potentials. It should however be noted that the Ro¨ntgen term in the current den￾sity ~3.6! causes a generalization of relation ~2.8! between magnetic field and magnetic induction to @see, for example, Eq. ~76.11! of Ref. @2## H5 B m0 2P3R˙ . ~3.10! The new term is a function of both the internal relative￾displacement coordinate and the center-of-mass coordinate of the dielectric material. For the dielectric spatial displacement variables, the equa￾tion of motion for the relative position of the two ions in the unit cell is obtained with the use of Eqs. ~3.7!–~3.9! as ms¨i1mGs˙i1mvT 2 si5§„Ei1~R˙ 3B!i…, ~3.11! which is identical to Eq. ~2.9! except for the addition of the term proportional to the center-of-mass velocity R˙ . The equation of motion for the continuum center-of-mass coordi￾nate is obtained similarly as 1074 R. LOUDON, L. ALLEN, AND D. F. NELSON 55