Electron energy losses by excitation of transition radiation

Solid State Communications,Vol. 16, pp. 127—130, 1975.

Pergamon Press.

Printed in Great Britain

ELECTRON ENERGY LOSSES BY EXCITATION OF TRANSITION RADIATION J. Geiger and H. Katterwe Fachbereich Physik, Universität Trier Kaiserslautern, D-675 Kaiserslautern, Federal Republic of Germany (Received 4 July 1974 by M Cardona)

Inelastic scattering of keV electrons in thin films with energy losses corresponding to the infrared spectral region is discussed taking retardation into account, that is allowing for radiative modes. For sufficiently thin target films Cerenkov radiation does not play any role. Rather it turned out even for nonrelativistic electrons, that in the spectral region of negligible absorption, where for example excitation of phonons or surface phonons does not occur, the energy loss spectrum is exclusively due to the creation of transition radiation. By means of high resolution spectroscopy of electrons inelastically scattered from TiO2 films it is shown experimentally that this conclusion is indeed correct.

THE PROBABILITY of inelastic electron scattering in a solid slab taking into consideration retardation effects been~nd calculated for the casecase of aofvery thick slab by has Ritchie Eldridge,1 for the a thin film by Romanov2 using quantum field—theoretic methods and by Kroger3 on the basis of classical electrodynamics. Consideration of retardation is required if the coupling of the beam electrons with the electromagnetic radiation field is investigated. At first the interest was turned to energy losses caused by excitation of the Cerenkov radiation.4 These experiments are possible with medium energetic electron only if target materials are selected with a sufficiently high value of the dielectric constant. Moreover, excitation of Cerenkov radiation is a bulk effect, that means experiments have to be made with film thicknesses large compared with the wavelengths of t~ieradiation within the medium of the film, 5 have found the opVincent and Silcox tical Recently plasmon in the energy loss spectrum of oxidized aluminium foil. This mode, however, is excited rather weakly, and is difficult to separate because bulk and surface effects are lumped together.

following is concerned with energy losses in the infra. red region below 2 eV. The aim of the paper is to 6 on elucidate influence ofwhich the transition the energythe loss spectrum may playradiation a role even at electron velocities small compared with the velocity of light. The conditions, under which the coupling of the beam electron with the radiation field of the transition radiation can be made evident, will be explained by the following theoretical consideration. The original formula for the inelastic scattering of electrons in thin films is an expression which is lengthy and troublesome to handle. This expression can be considerably 51mph. fled, if we restrict ourself to electrons which are not too fast, so that the inequalities v2/c2 ~ I and e(v2/c2) ~ 1 are fulfilled, but stifi retardation is taken into account. e(w) = ej (c~,) ie 2(w) is the dielectric constant of the In target which is taken to be surrounded by vacuum. this film, approximation the scattering probability W can be written as —

W(c.., ~c)27rKd,cd(hw)=

The experiments mentioned above dealt with the energy loss range between 2 and about 20 eV. The

e2

2a

s~2h2v2

ek2 2

+~ 127

—

2,c2(e

—

ek4

sin2 (c..ia/v)

1)21

1x 0e + A tgh Xa

2~Kd~d(hw).

(1)

128

EXCITATION OF TRANSITION RADIATION

Here we used the abbreviations ________

(W

/

2

~

—;

2

2

=

~

K

a130

1)2

—

(4)

2 Consequently(W8)NR’~ the following conditions should be ob-

2

—

~2

v2’ k with K the component of the wave number transfer parallel to the surface of the foil. It follows from the theory and it has proved true by former measurements,7 =

(e

8)R

__________

~ c and the designations: 2a thickness of the film, hw energy loss, k wave number transfer during the collision. In particular we have = ~2

Vol. 16, No. 1

+

that in electron energy loss spectra with energy losses in the phonon region at small scattering angle and for film thicknesses below 1000 A the spectrum comes about mainly by surface modes. Hence it is only necessary to discuss the surface term which is contained within the squared brackets of equation (1). In order to make the influence of the transition radiation on the scattering probability recognizable, equation (1) will be developed for small ~ (which is justified under our experimental conditions) and the result without and with retardation will be compared.

served if energy losses due to excitation of transition radiation are to be studied: 1. The energy losses must be located left from the line A in the 0, for ~ diagam Fig. measurement; 1. R is the region which0 was used the present 2. The thickness of the foil should not be too small, but it has to be kept in mind that our approximations require aj3, aJ3 0 ~ 1; 3. The imaginary part of the dielectric constant 2 must be small, the real part e~should be as large as possible. That leads to the demand for light absorption being absent in the spectral range of interest. This is fulfilled for Ti02 in a sufficiently large range above 100 meV.

R~

L~1

L

3

~i

Neglecting retardation, 4c -+0, one has aX = aX0

R2

N

x~o

L2

=asc ~ 1 andwithel(w)~e2(~)onefindsforw~’wL >-

(WS)NR2KdKd(hw)

2 2 sin2 (wa/v) 2irK d,cd(hc~.,). 2e ~2v2h2k4 ae~

—P

z uJ

-

(2)

—;~—

——

c~ Ui U

a

the light cone bounded by A 0 = 0, compare Fig. 1), On other handiswithin the radiative (within and the if retardation included, with the region substitutions A0 =ij30

and

IONIC D

SLAB ioo - ioooA

X=i13

the following expression is obtained

0

2x

4x

Sx

SCATTERING ANGLE

(W8)R 27rIcdKd(hw) = 2K2~3o(e 1)2 2 sin2 (c~,,a/v)+ a2(J32 + e213~) e 2 v2h2k4 (~j3~)2 + a2(J32 + e2~3~)2 —

or

X 2lrKdKd(hw).

(3)

Here we have taken the dielectric constant as a real quantity. Using equations (2) and (3) the ratio between the retarded and unretarded scattering probability is roughly estimated to

i~iü~ ~—-----

FIG. 1. Dispersion diagram of the vibrational modes of a thin slab. The model used is Ti0 2,with restriction to the highest transversal phonon energy hWTXO = 0 is the light cone. Radiative modes are found in R. In L we have non-radiative modes. There are no modes in N. At the line F excitation of Cerenkov radiation could take place. The presentation is schematic, the distances between the parallel lines are strongly enlarged.

Vol. 16, No. 1

EXCITATION OF TRANSITION RADIATION

l0~

3 3,

~.

5x

Ox

9x10’

W Ui

0 5

1.0

1.5

~—

2.0

ENERGY LOSS [eV]

Using Ti02 as a target the ration (W8)R/(W3)NR canbeestimated.Withe1 =6,e2 =6X I0~,at3o= 0.098 (2a = 700 A, hw = 580 meV, 30 keV electrons, scattering angle 0 = 3 X 10-6) this ratio amounts to ~

SIR

—

1’.

S)NR

—

~ ~——— nonretarded _______________________________________________________

C)

~

0

FIG. 2. Retarded and non-retarded scattering probability (theory). Note that the curves start at 0.1 eV, which is above the discrete spectrum caused by pho. non creation,

~

calculated

I—

—•._...

t0~ 0.1

129

4000 ‘

which implies that in the region of small ~2 the entire energy, which was lost by the electron, is transferred to the radiative field of the transition radiation.

—

0:5

.0

1.5

2.0

ENERGY LOSS [eV~

FIG. 3. Measured and calculated integrated energy loss spectrum (angular range of integration is 0 ~ 0 ~ 1.1 X i0~rad). Primary electron energy is 30 keV, film thickness 700 A.

inside the light cone ho., = hi~c,where A0 becomes an imaginary quantity, a’step by several orders of magmtudes appears in the course of the differential scattering probability. The numerical result for the ratio of retarded to unretarded scattering probability is with the same data as above —

5093. So the estimation was not too bad. The very weak peak preceding the step in Fig. 2 is likely to be interpreted as the 0TH mode of Kliewer and Fuchs1°(see also Fig. 1). —

Equation (3) should be compared with the corresponding expression for ofthe transition radiation 8 emission Supposing sine in the applied to aofthin film. (3) is the dominant term, the numerator equation factors determining the spectral shape of the energy loss spectrum and of transition radiation are identically provided the radiation formula8 is rewritten in terms of~ and j3. Either formulae show a maximum at Ci = 0, the height of which depends periodically on the thickness of the film, leading to plasmon radiation in the case of a free electron gas or, in the case of lat. tice vibrations, to a “radiating longitudinal phonon”. For the 30 keV electrons used in the present cxperiment the more stringent expression3 for the differential scattering probability has been evaluated numerically rather than the approximation equation (1). This evaluation based on the dielectric constant of TiO 2 as9 quoted by Spitzer, and The result is shown Mifier, by Fig.Kleinman, 2. It corroborates Howarth. the features discussed already in connexion with the approximation equation (1): At an energy loss just

Experimentally the energy loss spectra for distinct scattering angles have not been measured but an energy loss spectrum integrated over an angular range up to 1.1 X l0~rad. Nevertheless as it is evident from Fig. 3 the integrated spectrum shows the crucial role of the transition radiation between 0.2 and 2 eV as mode of radiative excitation where the beam electrons deliver energy. Figure 3 presents the electron energy loss intensity with and without retardation. In the energy loss region, where the imaginary part of the dielectric constant nearly vanishes, the consideration of retardation increases the scattering probability by several order of The measured energy loss spectrum ofmagnitudes. TiO 2 has the expected behaviour. The measured relative intensity has been fitted to the theory at 95 meV in order to get absolute values. This

130

EXCITATION OF TRANSITION RADIATION

energy value for normalization is particularly favourable because at 95 meV the results of the retarded and unretarded theory are equal. On the other hand the wing of the primary line, which the measured spectrum must be corrected for, is sufficiently weak, and no additional error is introduced by this procedure. According

Vol. 16, No. 1

to Fig. 3 the measured spectrum and calculated spectrum with retardation included agree with each other concerning their spectral behaviour as well as the magnitude of the electron intensity. Acknowledgements A grant from Deutsche Forschungsgemeinschaft is gratefully acknowledged. —

REFERENCES 1.

RITCHIE R.H. and ELDRIDGE H.B.,Phys. Rev. 126, 1935 (1962).

2.

ROMANOVYu. A.,Radiophysika 7,828(1964); ROMANOVYu.A.,SovietPhys. JETF2O, 1424(1965).

3.

KROGER E., Z. Phys. 216, 115 (1968).

4.

VON FESTENBERG C., Z Phys. 227,453 (1969).

5.

VINCENT R. and SILCOXJ.,Phys. Rev. Lett. 31,1487(1973).

6. 7.

FRANK J.M., Soy. Ploys. Uspekhi 8, 729 (1966). GEIGER J., Elektronen und Festkörper, Kap. VIII, Vieweg, Braunschweig (1968).

8.

SILIN V.P. and FETISOV E.P., Phys. Rev. Lett. 7, 374 (1961).

9.

SPITZERW.G., MILLER R.C., KLEINMAN D.A. and HOWARTH L.E.,Phys. Rev. 126, 1710 (1962).

10.

KLIEWERK.L. and FUCHS R.,Phys. Rev. 150, 573 (1966).