Double plasmon oscillations during X-ray emission

Physica 145B (1987) 45-49 North-Holland, Amsterdam

DOUBLE PLASMON OSCILLATIONS DURING X-RAY EMISSION

O.K. HARSH

Department of Post-Graduate Studies and Research in Physics, Feroze Gandhi College, Rae Bareli (U.P.), India B.K. AGARWAL

X-Ray Laboratory, Physics Department, University of Allahabad, Allahabad (U. P.), India Received 11 November 1986

The role of double plasmon oscillations in X-ray emission can be theoretically explained by obtaining expressions for the dispersion relation and relative intensity. The calculated values of relative intensities from our expression are fairly close to the observed relative intensities for Na and Cu. The present expression for the relative intensity has been compared with the Langreth and Pardee formulae. It is shown that for high electron density our expression turns into the Pardee et al. expression. It is also shown that the present expression works best for intrinsic double plasmon satellites.

I. Introduction

Ashley and Ritchie [1] were first with calculating the probability of the second-order process of the double plasmon single scattering event for AI metal. The Spence and Spargo experiment [2] on the characteristic loss spectrum of A1 also supports the idea of a double plasmon oscillation process. Multiple plasmon peaks have been observed by various workers [3-5]. During the last few years, several workers [6-11, 22] have observed and explained X-ray satellites at an energy distance of 2hWp (hWp = plasmon energy) on the high or low energy side from the main line. These satellites are known as double plasmon high or low energy satellites. To understand the high energy satellites, which have been observed at twice the energy of the characteristic electron energy loss values due to double plasmon oscillations, we derive in this paper expressions for the dispersion relation and the relative intensity for the double plasmon satellites. The calculated values of the relative intensities of double plasmon satellites for Na and Cu will be compared with the experimental values of Arakawa and Jenkins [6, 7]. The dispresion relation derived here holds good in the case of X-ray emission spectra as well.

2. Dispersion relations

Following Bohm and Pines [12], the general Hamiltonian for a system of coupled particles, fields and their interactions is given by 2

~i Pi

H = " ~-~m +

k~
+

+

; Q~ Qk

+Wp

2

27rNe2~ -£7 ]+Hsr+Hint + U ,

(1)

c

where It~, = E

k>kc

[2ere 2 ~, + ~ T - - J t P ~ Pk - n ) ,

0378-4363/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

(2)

46

o.K. Harsh and B.K. Agarwal / Double plasmon oscillations during X-ray emission

Hint = i i,k
i

~m MkQk e-ik.r, '

U - 2~'e2 ~ Q+kOk ~k 7" -k t m i,k~k'

e -i(k+k')'ri

(3) (4)

•

The first term in eq. (1) represents the kinetic energy of the particles (electrons), the second term in brackets shows the kinetic energy of the field oscillators and field particle interactions, Hsr is the short range interaction between the electrons, Hi. t is the linear interaction between electrons and plasmons, and U is the bilinear interaction between electrons and plasmons. Following Brouers [13] Hamiltoman (1) can be written in the second quantized form as 1 ~,

1

2+

H= m

Z

2 + Mk(Cp+kCp, k C p,Cp

N)

p,p'

(5)

- E M2Cp+kCpb+(R)b(R) e-~k'" + E V ( I ) C , b ( R ) L +. k,p

l

The last term in eq. (5) is the X-ray emission term, C¢ is the electron annihilation operator, b(R) is the hole annihilation operator and L + is the photon creation operator. We can separate the collective behaviour from the individual particle behaviour and some other field-particle interactions, and particle-particle interactions. For this we perform on Hamiltonian (5) three successive canonical transformations [13] generated by the operators, S, = - i

+

~

(6)

MkQkCp+kC p,

p,k
$2=_

~

(7)

MkQkb+b,

k
and

( - - - - k - . p L - hk2----] e [ W - m +2mm]

-

k'pg [W---~+

hk:] 2md

(8)

On account of the bilinear term U, eq. (4), the transformed Hamiltonian contains an additional term compared to the Hamiltonian of Brouers [13] and Bohm-Pines [14] which is given as 1 ~ V ( p + k + k')MkMk,QkQg,CpbL + . Hadd - 2h2 k,k,
(9)

Eq. (9) can also be expressed in terms of plasmon creation (A~) and annihilation (Ak) operators as Hadd

1

-

-

(4hWp) k,*'
(lo)

where A~ and A k are defined as ~.k = ( ~ / 2 w ) l / 2 ( d k Pk = i(hW/2)'/2(

A+k

A+-k) , + A-k)"

(11)

O.K. Harsh and B.K. Agarwal / Double plasmon oscillations during X-ray emission

47

The additional term (eq. (9) or eq. (10)) is due to the effect of canonical transformations on the extended Bohm-Pines second order term. This term involves the double plasmons or two plasmons with the X-ray emission. The bilinear term U (eq. (4)) is responsible for the double plasmon oscillations, as shown by Pines [12]. This term involves the emission or absorption of two plasmons. To eliminate this bilinear coupling term, we make a fourth canonical transformation [15] (which separates the double plasmon oscillation term during X-ray emission from the X-ray emission term) generated by the function,

$4

_

e 1 ~ m

i

rrh k--W (Ek'Ek)

k'
--e

-i(k+k').X

[2 W

k'*k

A k A k, ei(k+k,).xi (k+k')'p, h(k+k') 2 --

f + -e-2., Z -77"h ~ (Ek'Ek) m i,k
AkA k, k).p, -

~,~

e -i(k'

k)'Xi

A+

_

m

h(k'

-

-

k) 2

~m

kAk'

k) . pi

ei(k,_k).xi

h(k'

m

(k' -

2m

AkAk ] (k + k')'Pi + h(k + k')2 m 2m

i

2W

--

-~-

m

- k) 2

2m

1

(12)

"

As a result of transformation (12), eq. (9), which represents the double plasmon oscillation, can be written as

V(p + k + k ' ) M k M k

1

• (2hWp)2 [2Wp Hadd = HehWp= 21 P'k"k
(k+k').pi ~(k+k')2] m

(13)

2m

To a first appproximation, we take k = k', that is, the wave vectors of the two plasmons are identical. Then the dispersion relation for the double plasmon can be written as 2

l = ( 1 6 r rme 2 ) ~/ [ (

h2k4

kruPP)~m2J

] - 1

"

(14)

Eq. (14), when expanded in powers of k, can also be written as

h2k4

W~ = 4Wep +

2 - . k 2V F + -8m 2

(15)

Eq. (15) is the required dispersion relation for the double plasmon oscillations. The second and third terms on the right hand side of (15) are very small and may be neglected to a first approximation. Thus (15) becomes W~=4W2p

which gives

Wk = 2 W p .

48

O.K. Harsh and B.K. Agarwal / Double plasmon oscillations during X-ray emission

The second and third terms on the right hand side of (15) are due to electron-plasmon coupling and electron-electron coupling.

3. Relative intensities

The transition probability per unit time [16] for the interaction hamiltonian H2hwp, eq. (13), is given by

o-= ( ~ ) (mlH2~wpll>2p(E) .

(16)

If one assumes that the matrix element varies slowly with p, then the relative transition probability or relative intensity, can be written as Kc +1 2

I2t~Wp(Sat)(Ep) Io(Ep)

1 1 x~f (4hWp) 2 (2¢r) 3 l0

1

2

MeMo,

f 2K2dkd(cosO) 2Wp

(k+k')'Pi+ h(k -~ kt)2 ] m

(17)

2m

Simplification of eq. (17) gives the expression for the relative intensity I / I o of the high energy double plasmon satellites as I

O~

i2 - I o

2

~ tan(h

1V~),

(18)

where (see refs. 12, 17, 18, 23) a =0.12r s ,

z xl/2 Kc/K F = /3 =0.471.rs)

and

r S = (47.11/hWp) 2/3 .

(19)

Using (18) and (19) we have calculated the relative intensity of high energy double plasmon satellites of Na and Cu. A fair agreement has been obtained between the calculated and observed values (see table I).

4. Discussion

Dispersion relation (15) can be compared with the Bohm-Pines dispersion relation [12] for the single or bulk plasmon satellites, which may be expressed as (4rre 2 ]

2

Table I Element hWp (eV) Na 5.7 Cu 10.5

h2k4] 1

rs [12] 4.087 2.720

a [201 0.4905 0.3264

18 [12] 0.95 0.77

i2(cal.) i2(obs.) (eq. (18)) (ref.) 0.023 0.02 [61 0.011 0.016 [71

O.K. Harsh and B.K. Agarwal / Double plasmon oscillations during X-ray emission

49

When expanded in powers of k, it gives

W~

23 =Wp

2 2 h2k4 ~ k V v+-4m 2 .

(21)

To a first approximation (21) may be written as Wk---~Wp, that is, there is a free longitudinal oscillation of the electron gas at the plasmon frequency. While in case of double plasmon oscillations, two plasmons are involved. Langreth [19] and Pardee et al. [20] have obtained expressions for the relative intensity of double plasmon satellites as (O.06a 2) 2e_ ~ ,

i2 -

O~

(22)

2

i 2 = -~--,

(23)

where 13 is the coupling constant defined above and a = 0.12r S (r s is the dimensionless parameter). It was confirmed by Langreth [19] that his expression involves both intrinsic and extrinsic effects. The expression derived by us contains only intrinsic effects. For the high electron density our expression for the relative intensity of double plasmon satellites turns into the Pardee et al. [20] expression, that is, ½a 2. The Pardee et al. [20] formula is more valid in the case of photoemission, since in the case of photoemission the coupling between the valence electron and the hole (created) is absent [21].

References [1] J.C. Ashley and R.H. Ritchie, Phys. Stat. Sol. 38 (1970) 425. [2] J.C.H. Spence and A.E.C. Spargo, Phys. Rev. Lett. 26 (1971) 895. [3] V.E. Henrich, Phys. Rev. B 7 (1973) 3512. [4] C.V. Von Koch, Phys. Rev. Lett. 25 (1970) 792. [5] L.H. Jenkins, D.M. Zehner and M.F. Chung, Surf. Sci. 38 (1973) 327. [6] E.T. Arakawa, private communication. [7] L.H. Jenkins and M.F. Chung, Surf. Sci. 26 (1971) 151. [8] M.O. Krause and J.G. Ferreira, J. Phys. B 89 (1975) 2007. [9] L.H. Jenkins and M.F. Chung, Surf. Sci. 24 (1971) 649. [10] M.F. Chung and L.H. Jenkins, Surf. Sci. 33 (1972) 159. [11] M.F. Chung and L.H. Jenkins, Surf. Sci. 26 (1971) 649. [12] D. Pines, Elementary Excitations in Solids (Benjamin, New York, 1964), p. 112. [13] F. Brouers, Phys. Stat. Sol. 22 (1967) 313. [14] D. Bohm and D. Pines, Phys. Rev. 92 (1953) 609. [15] D.N. Tripathi and S.S. Mandal, Phys. Rev. B 12 (1975) 3652. [16] J.J. Sakurai, Advanced Quantum Mechanics (Addison Wesley, London, 1967), p. 41. [17] K.S. Srivastava, R.L. Shrivastava, O.K. Harsh and V. Kumar, Phys. Rev. B 19 (1979) 4336. [18] O.K. Harsh, J.P. Goel and M. Hussain, to be published in J. Sci. Res. [19] D.C. Langreth, Initial collective properties of physical systems, Nobel Symp. 24 (1973) 210. [20] W.J. Pardee, G.D. Mahan, D.E. Eastman, B.A. Pollak, L. Lay, F.R. McFeely, S.P. Kowalcyzk and D.A. Shirley, Phys. Rev. B 11 (1975) 3614. [21] J.P. Goel, O.K. Harsh and K.S. Srivastava, Physica 144 B (1987) 190. [22] K.S. Srivastava, R.L. Shrivastava and O.K. Harsh, Curr. Sci. 18 (1981) 795. [23] O.K. Harsh, J.P. Misra, J.P. Goel and M. Hussain, J. Sci. Research 7 (1985) 35.