Symmetry Considerations and Origin of the Force Components

The distinct origins of the force components F|t and Fj^ can be clearly understood in terms of the symmetry

p r operties of the induced charge density. The general form of the 0 integrals in equations (5.48) may be written

277

I ( p ' / n ) de e x p [ikp'c o s (0- n )] f(cos0) (5.51)

W ith the variable change to 0-tt in the range (tt,2tt) followed by the change to tt — 0 in the range (*577,77), I(p',n) takes the form

I (p ' , n ) = 2[ i + ( P 1 , n ) + i _ (p ' , n) 1 (5.52)

w h ere

»*2 77

l + ( p ' , n ) d0 cos (Kp ' cos0cosn) cos (kp' sin0sin7i)

x [f(cos0) + f(-COS0)] (5.53)

and

r h v

I (p'fTi) = i I d0 sin(Kp 'cosecosn) cos (kp ' sin0sin7i)

J

x [f(cos0) - f(-COS0)] (5.54)

The following identities are easily verified:

I ± ( p \ n ) = I ± (p ' ,-n) , (5.55)

147

^2 TT

i + (o , n ) d0 [f (cos0) + f(-cos©) ] (5.57)

I (p ' , ^tt) I (0,n) 0 (5.58)

If f (c o s 0) is an even function of c o s 0 , then I_ vanishes and from equations (5.55) and (5.56) , 1 = 1 corresponds to a

symmetric density distribution with respect to inversion about both axes n = 0 and n = tt in any plane parallel to the

surface. In addition, on comparing equations (5.53) and (5.57), I has an absolute maximum at p' = 0, directly

beneath the moving charcje. When f(cos0) = - f (-cos0), then I = I is antisymmetric about, and vanishes on, the axis n = . The corresponding antisymmetric charge distribution will contribute no net charge.

X = k and <q respectively. When there are no singularities, f(cos0) is even and the symmetric density nj_ will give rise to Fj^. When singularities occur, only the imaginary parts

antisymmetric charge distribution n|( which in turn gives rise to F|( . Thus, for example,

In the expression for the surface and bulk charge

2 2 2 2 - 1

densities, f(cos0) is given by (w - k v c o s 0) where

. 2 2 2 2

i7Tsgn(cos0) (oo - k v c o s 0) contribute, leading to the A

p ^7T

X 0

d0 cos (kpic o s0costi) cos (kp'sin0sinn) , (5.59)

2 2 2 2

(a) - k y cos 0)

K

2 r00 00 Q / ck k y exp(y z-kz ) P I K r K O 2ne / ~ 2 2 2 x h , 2 2 2. % Jk (2oo + 3 k ) (k y - a) ) 2 s P K x sin(Kp1cosS^cosn) cos(xp1sinG^sinp) , (5.60)

where 0^ = cos 1 (oj^/Kr?) . Note also that the induced charge

~ens II (P ' ' 0 'z ) = en s\\ ^ P ' '77 ,z ^ t^ie same sign as Q in front of the moving charge. An equivalent analysis gives the bulk terms and with the corresponding limits and <7 (k) as in equations (5.32) and (5.37) for Fy and Fj_ .

n .. (p 1 ,r\fz) = - sll

*J.J CONCLUDING REMARKS

The approach of Takimoto {1966) and Muscat and Newns {1977) has been extended here to study the effects of spatial dispersion on the force on a charge moving with constant velocity parallel to a metal surface. The notable effects of the nonlocality of the dielectric response of the metal are the inclusion of the contributions from bulk plasmons and the introduction of the physically important

P

limits k s = o)g/[y(y-3)] 2 (compared to u> /y in the local approximation) on the surface plasmon wavenumbers and k = a) / {v2 - ß2 )"2 and q (k) = [k2 (i;2 - ß2 ) - w 2 ]^/ß for bulk plasmons, separating their contributions to the

longitudinal and transverse force components. These limits are connected with the results that F |( vanishes when y i ß,

with the induced density fluctuations in the metal then remaining symmetrically distributed beneath the moving

149

charge, and the shape of this symmetric distribution giving rise to the velocity dependence of . For v > ß the existence and origin of the force components Fj, and Fj_ can be understood in terms of the lack of symmetry of the induced density, rather than simply as resolved components of an 'image' force resulting from a charge distribution with centre of mass which lags behind the moving charge.

The results are easily extended to obtain the probability of plasmon excitation for very small angle grazing incidence as done by Barberän et al {1979) for the surface plasmon contributions. For the bulk contribution, equation (5.32a) must be used with the factor introduced into the q integral which then requires numerical

evaluation.

Although, as indicated above, the results of Barberan

et al {1979) only reproduce the surface contributions given here, and thus do not reduce to the same static limit (Newns 1 9 6 9 b ) r a clearer picture of the effects of single particle excitations on the results of this chapter is desired. In addition, the inclusion of retardation is expected to introduce some lag, and hence asymmetry, in the induced density so that there would no longer be a clear distinction for the contributions to F,| and Fj_ on the basis of plasmon phase velocities. However, the usefulness of the hydrodynamic model, in this case within the nonretarded

limit, is highlighted in this important problem in giving a clear physical basis on which these other effects can be evaluated.

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_,

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