**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 [i*kp

**'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**

**J**

**0**

**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 *n*j_ 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** * r*00
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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