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On the possibility of observing characteristic energy gain by fast electrons in solids
Volume24.A, number 7
THE
ON
PHYSICS LETTERS
27 March 1967
POSSIBILITY OF OBSERVING CHARACTERISTIC BY FAST ELECTRONS IN SOLIDS *
ENERGY
GAIN
R. H. RITCHIE Health Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
Received 13 February 1967
It is shown that photon absorption by a fast electron may occur with appreciable probability through the intermediary of a virtual plasmon. An estimate shows that it may be feasible to observe this process experimentally. F e r r e l [1] has p r e d i c t e d that Coulomb s t i m u l a t e d p l a s m o n s in thin f i l l s m a y undergo r a d i a tive decay. That t h i s phenomenon can be viewed a l t e r n a t i v e l y a s a s p e c i a l c a s e of t r a n s i t i o n r a d i ation [2] was pointed out by o t h e r s [3]. C o n s i d e r a b l e e x p e r i m e n t a l work on this phenomenon has been c a r r i e d out s u b s e q u e n t l y [4, 5]. T h i s work, t o g e t h e r with m e a s u r e m e n t s on c h a r a c t e r i s t i c e n e r g y l o s s e s by f a s t e l e c t r o n s [6], has been u s e ful in i n v e s t i g a t i n g the v a l e n c e e l e c t r o n d y n a m i c s of many different s o l i d s . The p u r p o s e of t h i s l e t t e r is to point out that it m a y be p o s s i b l e to o b s e r v e the i n v e r s e of the p l a s m o n decay p r o c e s s in a c h a r a c t e r i s t i c e n e r g y l o s s e x p e r i m e n t . This phenomenon involves the p h o t o - e x c i t a t i o n of a p l a s m o n , which d e - e x c i t e s by d e l i v e r i n g i t s e n e r g y to a f a s t e l e c t r o n . Thus, e n e r g e t i c e l e c t r o n s p a s s i n g through a thin foil which is i r r a d i a t e d s i m u l t a n e o u s l y by an i n t e n s e light b e a m of the p r o p e r f r e q u e n c y should exhibit e n e r g y gain in quanta of the p l a s m o n field. An e s t i m a t e of the p r o b a b i l i t y of this p r o c e s s i s given below. It a p p e a r s f e a s i b l e to a t t e m p t obs e r v a t i o n of such " c h a r a c t e r i s t i c e n e r g y g a i n s " using a p u l s e d l a s e r s o u r c e and s u i t a b l e f r e q u e n c y d i s c r i m i n a t i o n techniques. The p r o b a b i l i t y of this p r o c e s s is e s t i m a t e d using an a p p r o a c h equivalent to that of F e r r e l l [1]. One q u a n t i z e s the e l e c t r o m a g n e t i c f i e l d and the r a d i a t i v e p l a s m o n f i e l d using conventional t e c h niques. The i n t e r a c t i o n Hamiltonian for t h e s e two f i e l d s m a y be w r i t t e n
if the foil t h i c k n e s s a << the photon wavelength. The foil is taken p a r a l l e l with the x - y plane and with a r e a L 2, and ~^ is a unit v e c t o r n o r m a l to the foil. As usual, e / ~ is the unit p o l a r i z a t i o n v e c t o r of a photon with f r e q u e n c y ¢ok, wave v e c t o r k, and p o l a r i z a t i o n index ~, while k± is the component of k p e r p e n d i c u l a r to ~. The (creation, annihilation) o p e r a t o r s (aj~k, akA ) of the r a d i a t i o n ÷ f i e l d and (bE, bK) of the p l a s m o n field s a t i s f y the u s u a l Boson c o m m u t a t i o n r e l a t i o n s . The wave v e c t o r K of the p l a s m o n field i s p a r a l l e l with the x - y plane. The Hamiltonian d e s c r i b i n g the i n t e r a c t i o n b e tween the p l a s m o n f i e l d and the e l e c t r o n field m a y be w r i t t e n 1
H p l - e = -e ( 8 ~ i w p / a L 4 ) ~ × × ~
sin(½aa-~.)
2
(2)
+
+
{ha±÷ b-o±ICg÷ock'
w h e r e it is a s s u m e d that the e l e c t r o n wave field m a y be expanded in m o m e n t u m eigenfunctions n o r m a l i z e d in a volume L 3. The (c~, ck) a r e ( c r e a t i o n , annihilation) o p e r a t o r s c o r r e s p o n d i n g to an e l e c t r o n s t a t e of wave v e c t o r k. Eq. (2) is d e r i v e d under the a s s u m p t i o n that Ka << 1. By s t a n d a r d m e t h o d s one c o m p u t e s P(COk), the p r o b a b i l i t y p e r incident photon that a photon s h a l l e x c i t e a p l a s m o n which then d e - e x c i t e s by i m p a r t ing e n e r g y to a f a s t incident e l e c t r o n . In the neighborhood of the r e s o n a n c e , one finds
1
Hph-pl
=
×
_
348
(1)
{akX b k± + akk bk±- akk b ~ - akA b k~ }
* Research sponsored by the U.S.Atomic Energy Commission under contract with the Union Carbide Corporation.
Volume24A, number 7
P(¢ok)
~aneV3
PHYSICS LETTERS
w~ sin 2 0 sin2 ¢opa wk) 2 + ¼~2 cos 0 2v
W3p (Wp-
(3) w h e r e n e is the d en s it y of f a s t e l e c t r o n s with v e l o c i t y . v , a = e2/~ic, ~9= a r c c o s (k.7./k), and ~, is the damping r a t e of the i n t e r m e d i a t e state. The photon is a s s u m e d to be p o l a r i z e d in the plane o f incidence. One notes the fluctuating dependence of eq. (3) upon foil t h i c k n e s s ; analogous fluctuations in the y i e l d of p l a s m o n decay photons [1] have been c o n f i r m e d e x p e r i m e n t a l l y by A r a kawa et al. [5]. An e x p e r i m e n t a l i n v e s t i g a t i o n of this effect mi ght u s e a b e a m of e l e c t r o n s n o r m a l l y incident on a thin Ag foil. A s s u m e that the b e a m a r e a is 10 -2 cm 2 and that the s a m e a r e a is i r r a d i a t e d by light f r o m a l a s e r at an angle of 45 ° with r e s p e c t to the foil n o r m a l . T h e l a s e r is a s s u m e d to e m i t 3300 .~ photons, c o r r e s p o n d i n g to the p l a s m a f r e quency in Ag m e t a l . One w i s h e s to m a x i m i z e the joint e l e c t r o n - p h o t o n density in the foil without m e l t i n g it. An Ag foil 600 A thick m a y be b o m b a r d e d by a 10 ~zA b e a m of 100 keV e l e c t r o n s without a p p r e c i a b l e d e t e r i o r a t i o n ; this c o r r e sponds to an e n e r g y d i s s i p a t i o n r a t e of ~ 1.5 mW o v e r the b e a m a r e a [7]. A s s u m e that s i m u l t a n e o u s i r r a d i a t i o n by light of i n t e n s it y c o r r e s p o n d i n g to e n e r g y d e p o s i t i o n at the s a m e r a t e m a y be t o l e r ated without d am ag e . If one a s s u m e s that the photon a b s o r p t i o n p r o b a b i l i t y in the foil is ~ 2, a m a x i m u m b e a m i n t e n s i t y of 6 mW is obtained, c o r r e s p o n d i n g to Np ~ 1016 p h o t o n s / s e c c r o s s i n g the foil. Ne, the n u m b e r of e l e c t r o n s p e r s e c o n d which a c q u i r e the v o l u m e p l a s m o n e n e r g y ~Wp, is then gi ve n by
o)pa
N e = NpP(wp) = 7raneN p 4¢o~ 3 sin 2 0 sin2 ~2 w 3 c o s 0 2v " P
(4)
T he f a c t o r sin2(~0pa/2v) ~ 1 in the p r e s e n t c a s e . One u s e s ~, = ~'d + ~'r, w h e r e ~'d is the d e c a y r a t e of a r a d i a t i v e s u r f a c e p l a s m o n due to e l e c t r o n i c
27 March 1967
damping in the m e t a l , and Tr = ¢°2a sin2 0/2c cos 0 is the p l a s m o n r a d i a t i v e damping r a t e [1]. Taking Td/COp ~ E2(Wp) , w h e r e e 2 is the i m a g i n a r y p a r t of the d i e l e c t r i c constant of Ag, f r o m the work of Huebner et al. [8] to be 0.4 and e v a l u a t i n g eq. (4) using the v a l u e s a s s u m e d above, one finds that of the o r d e r of 104 e l e c t r o n s / s e c w i l l u n d er g o c h a r a c t e r i s t i c e n e r g y gain f r o m photons v i a the i n t e r m e d i a t e p l a s m o n state. This c a l c u l a t i o n has been m a d e a s s u m i n g steady s t a t e l a s e r operation; o p e r a t i o n in a pulsed m o d e at the s a m e a v e r a g e power l e v e l should allow one to u se f r e q u e n c y d i s c r i m i n a t i o n t ech niques so that u n d e s i r e d background f r o m the nol o s s component of the e l e c t r o n b e a m m ay be m i n imized. The author would like to e x p r e s s his thanks to D r s . E. T. A r a k a w a and R. D. Birkhoff f o r s e v e r a l helpful c o n v e r s a t i o n s in this connection.
Refe~'enc~s 1. R . A . F e r r e l l , Phys. Rev. 111 (19585 1214. 2. V. L.Ginzburg and I.M.Frank, J. Phys. USSR 9 (1945) 353. 3. R.H. Ritchie and H. B. Eldridge, Bull. Am. Phys. Soc. 4 (1959} 384; Phys. lqev. 126 (1962} 1935; E.A.Stern, Phys.lqev. Letters 8 (19625 7. 4. W.Steinman, Phys. Rev. Letters 5 (19605 470; R. W. Brown, P. Wessel and E. P. Trounson, Phys. Rev. Letters 5 (1960) 472; E. T. Arakawa, R.J. Herickhoff and R.D. Birkhoff, Phys. Rev. Letters 12 (19645 319; R. J. Herickhoff, W.F. Hanson, E.T. Arakawa and R. D. Birkhoff, Phys. Rev. 139 (1965) A1455. 5. E.T. Arakawa, N.O. Davis and R. D. Birkhoff, Phys. lqev. 135 (19645 224. 6. Reviews of the extensive work in this field have been made by R.D.Birkhoff, in Physical processes in radiation biology (Academic Press, New York, 1964} ; O. Klemperer and J. P. G. Shepherd, Advances in Phys. 12 (1963) 355; H. Raether, in Springer tracts in modern physics (Springer-Verlag, Berlin, 1965) Vol.35. 7. E.T.Arakawa, private communication. 8. R.H. Huebner, E . T . Arakawa, R.A. Maclqae and R.N.Hamm, J.Opt. Soc. Am.54 (1964) 1434.
349
THE
ON
PHYSICS LETTERS
27 March 1967
POSSIBILITY OF OBSERVING CHARACTERISTIC BY FAST ELECTRONS IN SOLIDS *
ENERGY
GAIN
R. H. RITCHIE Health Physics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee
Received 13 February 1967
It is shown that photon absorption by a fast electron may occur with appreciable probability through the intermediary of a virtual plasmon. An estimate shows that it may be feasible to observe this process experimentally. F e r r e l [1] has p r e d i c t e d that Coulomb s t i m u l a t e d p l a s m o n s in thin f i l l s m a y undergo r a d i a tive decay. That t h i s phenomenon can be viewed a l t e r n a t i v e l y a s a s p e c i a l c a s e of t r a n s i t i o n r a d i ation [2] was pointed out by o t h e r s [3]. C o n s i d e r a b l e e x p e r i m e n t a l work on this phenomenon has been c a r r i e d out s u b s e q u e n t l y [4, 5]. T h i s work, t o g e t h e r with m e a s u r e m e n t s on c h a r a c t e r i s t i c e n e r g y l o s s e s by f a s t e l e c t r o n s [6], has been u s e ful in i n v e s t i g a t i n g the v a l e n c e e l e c t r o n d y n a m i c s of many different s o l i d s . The p u r p o s e of t h i s l e t t e r is to point out that it m a y be p o s s i b l e to o b s e r v e the i n v e r s e of the p l a s m o n decay p r o c e s s in a c h a r a c t e r i s t i c e n e r g y l o s s e x p e r i m e n t . This phenomenon involves the p h o t o - e x c i t a t i o n of a p l a s m o n , which d e - e x c i t e s by d e l i v e r i n g i t s e n e r g y to a f a s t e l e c t r o n . Thus, e n e r g e t i c e l e c t r o n s p a s s i n g through a thin foil which is i r r a d i a t e d s i m u l t a n e o u s l y by an i n t e n s e light b e a m of the p r o p e r f r e q u e n c y should exhibit e n e r g y gain in quanta of the p l a s m o n field. An e s t i m a t e of the p r o b a b i l i t y of this p r o c e s s i s given below. It a p p e a r s f e a s i b l e to a t t e m p t obs e r v a t i o n of such " c h a r a c t e r i s t i c e n e r g y g a i n s " using a p u l s e d l a s e r s o u r c e and s u i t a b l e f r e q u e n c y d i s c r i m i n a t i o n techniques. The p r o b a b i l i t y of this p r o c e s s is e s t i m a t e d using an a p p r o a c h equivalent to that of F e r r e l l [1]. One q u a n t i z e s the e l e c t r o m a g n e t i c f i e l d and the r a d i a t i v e p l a s m o n f i e l d using conventional t e c h niques. The i n t e r a c t i o n Hamiltonian for t h e s e two f i e l d s m a y be w r i t t e n
if the foil t h i c k n e s s a << the photon wavelength. The foil is taken p a r a l l e l with the x - y plane and with a r e a L 2, and ~^ is a unit v e c t o r n o r m a l to the foil. As usual, e / ~ is the unit p o l a r i z a t i o n v e c t o r of a photon with f r e q u e n c y ¢ok, wave v e c t o r k, and p o l a r i z a t i o n index ~, while k± is the component of k p e r p e n d i c u l a r to ~. The (creation, annihilation) o p e r a t o r s (aj~k, akA ) of the r a d i a t i o n ÷ f i e l d and (bE, bK) of the p l a s m o n field s a t i s f y the u s u a l Boson c o m m u t a t i o n r e l a t i o n s . The wave v e c t o r K of the p l a s m o n field i s p a r a l l e l with the x - y plane. The Hamiltonian d e s c r i b i n g the i n t e r a c t i o n b e tween the p l a s m o n f i e l d and the e l e c t r o n field m a y be w r i t t e n 1
H p l - e = -e ( 8 ~ i w p / a L 4 ) ~ × × ~
sin(½aa-~.)
2
(2)
+
+
{ha±÷ b-o±ICg÷ock'
w h e r e it is a s s u m e d that the e l e c t r o n wave field m a y be expanded in m o m e n t u m eigenfunctions n o r m a l i z e d in a volume L 3. The (c~, ck) a r e ( c r e a t i o n , annihilation) o p e r a t o r s c o r r e s p o n d i n g to an e l e c t r o n s t a t e of wave v e c t o r k. Eq. (2) is d e r i v e d under the a s s u m p t i o n that Ka << 1. By s t a n d a r d m e t h o d s one c o m p u t e s P(COk), the p r o b a b i l i t y p e r incident photon that a photon s h a l l e x c i t e a p l a s m o n which then d e - e x c i t e s by i m p a r t ing e n e r g y to a f a s t incident e l e c t r o n . In the neighborhood of the r e s o n a n c e , one finds
1
Hph-pl
=
×
_
348
(1)
{akX b k± + akk bk±- akk b ~ - akA b k~ }
* Research sponsored by the U.S.Atomic Energy Commission under contract with the Union Carbide Corporation.
Volume24A, number 7
P(¢ok)
~aneV3
PHYSICS LETTERS
w~ sin 2 0 sin2 ¢opa wk) 2 + ¼~2 cos 0 2v
W3p (Wp-
(3) w h e r e n e is the d en s it y of f a s t e l e c t r o n s with v e l o c i t y . v , a = e2/~ic, ~9= a r c c o s (k.7./k), and ~, is the damping r a t e of the i n t e r m e d i a t e state. The photon is a s s u m e d to be p o l a r i z e d in the plane o f incidence. One notes the fluctuating dependence of eq. (3) upon foil t h i c k n e s s ; analogous fluctuations in the y i e l d of p l a s m o n decay photons [1] have been c o n f i r m e d e x p e r i m e n t a l l y by A r a kawa et al. [5]. An e x p e r i m e n t a l i n v e s t i g a t i o n of this effect mi ght u s e a b e a m of e l e c t r o n s n o r m a l l y incident on a thin Ag foil. A s s u m e that the b e a m a r e a is 10 -2 cm 2 and that the s a m e a r e a is i r r a d i a t e d by light f r o m a l a s e r at an angle of 45 ° with r e s p e c t to the foil n o r m a l . T h e l a s e r is a s s u m e d to e m i t 3300 .~ photons, c o r r e s p o n d i n g to the p l a s m a f r e quency in Ag m e t a l . One w i s h e s to m a x i m i z e the joint e l e c t r o n - p h o t o n density in the foil without m e l t i n g it. An Ag foil 600 A thick m a y be b o m b a r d e d by a 10 ~zA b e a m of 100 keV e l e c t r o n s without a p p r e c i a b l e d e t e r i o r a t i o n ; this c o r r e sponds to an e n e r g y d i s s i p a t i o n r a t e of ~ 1.5 mW o v e r the b e a m a r e a [7]. A s s u m e that s i m u l t a n e o u s i r r a d i a t i o n by light of i n t e n s it y c o r r e s p o n d i n g to e n e r g y d e p o s i t i o n at the s a m e r a t e m a y be t o l e r ated without d am ag e . If one a s s u m e s that the photon a b s o r p t i o n p r o b a b i l i t y in the foil is ~ 2, a m a x i m u m b e a m i n t e n s i t y of 6 mW is obtained, c o r r e s p o n d i n g to Np ~ 1016 p h o t o n s / s e c c r o s s i n g the foil. Ne, the n u m b e r of e l e c t r o n s p e r s e c o n d which a c q u i r e the v o l u m e p l a s m o n e n e r g y ~Wp, is then gi ve n by
o)pa
N e = NpP(wp) = 7raneN p 4¢o~ 3 sin 2 0 sin2 ~2 w 3 c o s 0 2v " P
(4)
T he f a c t o r sin2(~0pa/2v) ~ 1 in the p r e s e n t c a s e . One u s e s ~, = ~'d + ~'r, w h e r e ~'d is the d e c a y r a t e of a r a d i a t i v e s u r f a c e p l a s m o n due to e l e c t r o n i c
27 March 1967
damping in the m e t a l , and Tr = ¢°2a sin2 0/2c cos 0 is the p l a s m o n r a d i a t i v e damping r a t e [1]. Taking Td/COp ~ E2(Wp) , w h e r e e 2 is the i m a g i n a r y p a r t of the d i e l e c t r i c constant of Ag, f r o m the work of Huebner et al. [8] to be 0.4 and e v a l u a t i n g eq. (4) using the v a l u e s a s s u m e d above, one finds that of the o r d e r of 104 e l e c t r o n s / s e c w i l l u n d er g o c h a r a c t e r i s t i c e n e r g y gain f r o m photons v i a the i n t e r m e d i a t e p l a s m o n state. This c a l c u l a t i o n has been m a d e a s s u m i n g steady s t a t e l a s e r operation; o p e r a t i o n in a pulsed m o d e at the s a m e a v e r a g e power l e v e l should allow one to u se f r e q u e n c y d i s c r i m i n a t i o n t ech niques so that u n d e s i r e d background f r o m the nol o s s component of the e l e c t r o n b e a m m ay be m i n imized. The author would like to e x p r e s s his thanks to D r s . E. T. A r a k a w a and R. D. Birkhoff f o r s e v e r a l helpful c o n v e r s a t i o n s in this connection.
Refe~'enc~s 1. R . A . F e r r e l l , Phys. Rev. 111 (19585 1214. 2. V. L.Ginzburg and I.M.Frank, J. Phys. USSR 9 (1945) 353. 3. R.H. Ritchie and H. B. Eldridge, Bull. Am. Phys. Soc. 4 (1959} 384; Phys. lqev. 126 (1962} 1935; E.A.Stern, Phys.lqev. Letters 8 (19625 7. 4. W.Steinman, Phys. Rev. Letters 5 (19605 470; R. W. Brown, P. Wessel and E. P. Trounson, Phys. Rev. Letters 5 (1960) 472; E. T. Arakawa, R.J. Herickhoff and R.D. Birkhoff, Phys. Rev. Letters 12 (19645 319; R. J. Herickhoff, W.F. Hanson, E.T. Arakawa and R. D. Birkhoff, Phys. Rev. 139 (1965) A1455. 5. E.T. Arakawa, N.O. Davis and R. D. Birkhoff, Phys. lqev. 135 (19645 224. 6. Reviews of the extensive work in this field have been made by R.D.Birkhoff, in Physical processes in radiation biology (Academic Press, New York, 1964} ; O. Klemperer and J. P. G. Shepherd, Advances in Phys. 12 (1963) 355; H. Raether, in Springer tracts in modern physics (Springer-Verlag, Berlin, 1965) Vol.35. 7. E.T.Arakawa, private communication. 8. R.H. Huebner, E . T . Arakawa, R.A. Maclqae and R.N.Hamm, J.Opt. Soc. Am.54 (1964) 1434.
349