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NATIONAL BUREAU OF STANDARDS - 1963
TWE"37.
1
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ORNE P-1600
Conf. 650929-2

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STUDIES OF COLLECTIVE ELECTRON OSCILLATIONS
OCT 6 1965
IN METALS BY OPTICAL METHODS*
MASTER
.
E. T. Arakawa, R. N. Hamm, W. F. Hanson, ** and T. M. Jelinek**
Health Physics Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee


** Th
F:-:":":
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7
TTT
IN HE
..
ACTS
ABSTRACT
.
.
..*
*
U
S
M
"Sir,'*..
Reflection of photons from opaque solids and photon emission
from electron-bombarded thin films have been investigated to determine
the plasma frequencies of free electrons in various metals, e.g., Al, cd,
Ti, and Zn. The energy loss functions -Ime and -Im(€ + 1)* for volume
and surface plasmons, respectively, were calcula ted from the opticai
constants n and k determined from reflectivity measurements and compared
with the results obtained by Robins and Powell from energy loss experiments.
Unambiguous identification of the plasma losses is made possible by a know-
ledge of the optical constants since these losses arise at frequencies where
peaks are found in the energy loss functions due to e, being small and
€, = 0 or -1, whereas other losses due to one-electron (or interband) transi-
tions are characterized by relative maxima in eg. Studies of photon emission
from electron-bombarded thin films give information not only on the volume
plasma frequency, but also on lifetimes of the plasma oscillations due to
radiation and to electronic damping.
*Research sponsored by the U.S. Atomic Energy Commission under
contract with the Union Carbide Corporation.
**AEC Health Physics Fellow, Vanderbilt University, Nashville,
Tennessee,
ORNI -AIC - OFFICIAL
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1. Introduction
toiminta
Since the first observation by Rudberg RUDBERG (1930) that
electron energy losses occur in discrete amounts when an electron penetrates
a solid, numerous investigations PINES (1963) have been made of the energy
Num
Tosses of electrons which have backscattered from thick targets or which
have penetrated thin solid films. The results of these investigations were
very often contradictory. This was due primarily to the lack of accuracy and
resolution in the measurements because of the necessity of using high energy
electrons to measure discrete losses in order of a few tens of volts.
The identification of these losses as due to surface plasmons,
volume plasmons, or interband transitions was also uncertain as it was made
simply upon the basis of whether the losses matched the free electron volume
plasma frequency
'.
... -,
.
..
or the surface plasma frequency
..
::
:
W3 =W,(2)*
An alternative method of measuring the characteristic energy losses was
A
.
suggested by Ferrell FERRELL (1958). He predicted the eraission of mono-
I
UKAT
.
i,
chromatic radiation at the plasma frequency due to the decay of plasma
oscillations. Since the detection of this radiation would be a direct measure
S.
of the energy loss, the optical method would provide a much more accurate
.
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1.
1
.
.,
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We
:
In addition to the above measurement, if one determines the
dielectric constants of the solid by optical methods, unambiguous identifi-
cation of the energy losses is made possible.
for several metals and the interpretation of the losses by the use of
optical data.
2. Experiment
The investigation of light emitted by electron-bombarded thin
foils has been described in detail elsewhere ARAKAWA et al. (1964). In
brief, thin, vacuum-evapora ted foils of the metal to be investigated are
bombarded by 25- to 100-keV electrons. The optical emission which results
from these electron interactions is analyzed with a vacuum uv spectrometer,
which can be rotated about the foil to study the angular and spectral distri-
butions of emitted photons.
The optical constants, n and k, of vacuum evaporated films of
cd, Tl, and Zn were determined from reflectance measurements of light
incident at 20°, 45°, and 70° from the foil normal HUEBNER et al. (1964).
The dielectric constants e, and e, were then calculated from the optical
constants, and the energy loss functions, -Imand -Im-t i for volume and
surface plasmons RITCHIE (1957), respectively, were determined. (The
€
+
1
essential features of the surface loss function are given by the -Im-
6 +
1
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part of the exact form im
+ 11 ) derived by Ritchie. )
.
he
1
L
.
3. Comparison of Results
Typical results obtained by the optical methods are shown in
figs. 1 and 2 for Ca and Zn, respectively. The energy loss functions for
surface and volume plasmons, -Im -Im respectively, calculated
from the experimentally determined dielectric constants and the intensity
of photon emission from electron-bombarded thin foils have been plotted
:
E +
1
vs. photon energy. The opticai emission spectra have been corrected for
the spectrometer response and an assumed uniform fluorescent quantum
efficiency of sodium salicylate WATANABE and INN (1953). The corresponding
energy loss spectra obtained from electron beam experiments by Powell
POWELL (1960) for Cd and Robins ROBINS (1961) for Zn are plotted on the
.
.
.
......
same energy scale. The breadth of the electron energy losses results from an
.
.
.
instrumental resolution of about 2. 0 eV, whereas a separation of 0.22 eV can
.
. .
.
be detected at 1500 Å by the optical method. Hence, the optical method not
.
. . .
only provides detail that cannot be obtained from electron energy loss spectra
..
.
.
but also makes possible a study of the inherent line widths.
...
it an inte noen
...
-
-
linin say that
The strong losses at 7.5 and 8. 6 eV in Cd and Zn, respectively
have been interpreted as surface plasma losses frorr. electron beam studies.
An inspection of figs. 1 and 2 shows that for both Cd and Zn, the peaks in
the surface and volume plasma loss spectra -Im-1 Im are close :
together. Thus, it appears that with the resolution available in the electron
were the most welchiorr
a
+
1
N2
energy loss experiments, it was not possible to separate these two losses and
ri
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that the peaks observed are the sums of both surface and volume losses. The
optical emission spectrum of cd confirms the results of the reflectivity data.
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Fig. 2. Comparison of surface and volume plasma loss functions for Zn with
... experimental electron energy loss and optical er.ission spectra. .
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loss function. No emission was seen corresponding to the peak in the surface
plasma loss function at 6.5 eV in agreement with theory. Ferrell FERRELL
(1958) has shown that the phase velocity of surface plasmons generated in a
planar foil does not become large enough to permit coupling with the trans- ..
verse electromagnetic field. Radiation from surface plasmons is possible
through the intermediary of electron-hole excitation in the metal. However,
since this is a secadorder process, its probability is very small compared
.
with the probability of radiation from volume plasmons. (Recent experimental
data indicate that radiation may occur from surface plasmons created in
non-planar foils bombarded by electrons at grazing incidence VON BLANCKEN-
HAGEN et al. (1964). The mechanism for this process is not clear at presented}
However, it appears that the emission peak observed at 8.6 eV corresponds to
the peak in the energy loss function at 9. 8 eV for volume plasmons.
The energy loss functions for surface and volume plasmons for ti
are shown in fig. 3 along with the electron energy loss spectrum obtained by
Robins ROBINS (1963). Again, it appears that the resolution of the electron
energy loss experiment is insufficient to separate the two losses. This loss,
ascribed to volume plasmons by Robins, appears to be the combination of both
surface and volume plasma losses. No optical emission data have yet been
obtained from Ti.
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Fig. 3. Comparison of surface and volume plasma loss functions of Tl with ... . i TI.
* ..... experimental electron energy loss spectrum of Robins. . .....
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4. Optical Emission
In the previous discussions, a tacit assumption has been mads
of a one to one correspondence between the energy loss functions and the
optical emission spectra. This is approximately correct because the electron
energy absorption is proportional to .cz/lel and the emission spectrum is
proportional to
for thin foils. Since prominent electron absorption
peaks occur at those energies for which both e, and e, are small, a peak in
the loss function will also give a peak in . However, the conditions for
which the approximate relationship above is valid require that the foil thickness
used in the optical emission experiment be very small. For the semi-infinite
foil, the generalized transition radiation theory shows a strong dependence
of the energy of peak emission upon angle of observation RITCHIE and ELDRIDGE
P
(1962). Photons originating in the foil interior will be absorbed for w<w,
and for w>w, photons striking the surface at angle e' with respect to the
foil normal will undergo refraction at the interface and will be bent toward the
normal. The exit angle e is related to e' by Snell's law e2 sin O' = sin e.
Setting o'= 1/2 (the maximum angle of photons in the interior) and
8 = 1-W5/w?, Snell's law reduces to w = w/cos & Thus, for a given
frequency w >wn, there is a value of e beyond which photons originating inside
p
the foil will not be observed. In the transition radiation theory, the maximum
in the photon intensity distribution should occur at w=w /cos e.
Typical recorder traces of the optical emission from an electron-
bombarded A1 foil 320 Å in thickness at angles of observation from 130° to 175°
NT 1
2
...
.
..-=
•
from the foil normal are shown in fig. 4. The shift in the wavelength at
which the peak in the optical emission occurs is shown in fig. 5. The peak
wavelength varies from 760 Å (16.3 eV) át 140° to 830 Å (14. 9 eV) at 170º.
As discussed above, the plasma wavelength 4, is determined from an
extrapolation of the curve of fig. 5 to 0 = 0°, and gives 1. = 832 Å, or 14.9 ev.
.
.
.
.
.
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The breadth of the emission peaks and the dependence of the intensity on angle
of observation shown in fig. 4 are in qualitative agreement with Ferrell's
descriptions. As @ is allowed to decrease toward 90°, the decay rate due to
radiation rises to excessive values, corresponding to a drastic broadening of
the plasmon energy. On the other hand, as e increases toward 180°, the
breadth of the emission peak decreases (fig. 6). Eventually, the radiative
decay rate decreases and becomes less than the damping rate due to interband
transitions, and the photon emission decreases. Ferrell suggested that it
should be possible to derive the intrinsic interband damping rate of the metal
from the characteristic angular dependence of both the photon intensity and
line breadth. The breadth of the emission lines is determined by the total
decay rate given by
where
is the decay rate due to radiation and Ta
is the damping rate due
to interband transitions. At 2 - 0 or 180°, 7" - 0, and therefore an extrapo-
lation of the half-width curve of fig. 6 to 8 = 0 or 180° will give the electronic
damping rate separately.' A value of 52 A is found for the half-width, which
corresponds to an electronic damping rate T, of . 71 x 10^5 sec. This value
agrees well with the results of Hunter HUNTER (1964); who found that the
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Fig. 4. Recorder traces of photon emission spectra from an electron-bombarded
:: Al foil 320 Å thick at various angles of observation. (Electron beam
12780 kev, 6. 2ua) .
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best fit by the Drude theory to his optical constant data for wavelengths
less than 800 Å was obtained when 1. was chosen to be 837 Å and Tg was:
0.6 x 10-45 to .7 x 10-15 sec. However, he found that this value did not
agree with the value of T, of 1. 1x 10 sec required to fit the data for
- 15
2000 Å to 6000 Å. The data in fig. 6 show that at 0 = 30° or 150°, M is
approximately 100 Å, i.e., that the radia tive lifetime T is equal to the decay
rate due to electronic damping at these angles.
The angular distribution of the photon intensity at the wavelength
of peak emission is shown in fig. 7. Also shown in the figure are curves
calculated from the transition radiation equation using the dielectric constants
of a free electron gas with 1 set at 835 Å and 1, varied from .44 x 10^" to
1.47 x 10° sec. The determination of the value of 1, from fig. 7 is not
quite as clear as the value derived above from the line width. The best fit
to the shape of the experimental angular distribution curve of photon intensity
is obtained for T, = .44 x 10 sec, and the best fit to the absolute intensity
is for T, = .88 x 10
sec. The average of these two values is .66 x 10"" sec,
again in good agreement with the values determined earlier. :)
5. Dielectric Constants
Both Cd and Zn are similar in electronic properties. Both have
two s electrons in the valence band with the closest bands being d bands at
approximately 10.5 eV. The next lowest bands are at 70-90 eV. Therefore,
in the region from 5 to 25 eV, there should only be the single interband
contribution at approximately 10.5 eV. If we write the dielectric constants
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;
of these metals in the form
... 2
,
www + igj + 8
w (w - ia)
w(W² + a²
+8
where a is the damping constant and 6 is the contribution of bound
electron, the real part of the dielectri: constant is
&j=1+81027 ?
In the region 5 to 25 eV, where a 2 is small compared to wa, we may plot
ur Vo
€,w vs. w' as shown in figs. 8 and 9. For both Cd and Zn, a straight line
is obtained in certain regions of the spectra whose slope is nearly one, i. e.,
(1 + 8,)=1 or the bound electron contribution is nearly zero. The bound
electron contribution
was calculated from the relation
€ = 1+ 0
where E
has the values 74 and 100 eV as determined from the y intercept
of curves (a) in figs. 8 and 9. The large peaks in 8, agree well with the soft
x-ray levels at 10.4 eV and 11. 0 eV labeled N, and N, in Cd and with the un- .
resolved Ma s level at 10.6 eV in Zn SANDSTRÖM (1957).
The dielectric constants of T1 are shown in fig. 10. The TI
data do not follow the same form as the Cd and Zn data, presumably due to.
the fact that there is a p electron in addition to the two s electrons in its

... Cd
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Fig.. 8. Graphical evaluation of the equat
... optical constants of Cd.
the equation o,=[1+%1015EPE for the
E (EV)?
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Fig. 9. Graphical evaluation of the equation ei
of the equation géº = (1 +31%)]E? - E for ti
optical constants of Zn. .,
for the
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Fig. 10. Dielectric constants of Ti vs. photon ene
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20
valence band which permit additional lower
y transitions. The curves
with E = 10, 11, and 12 eV are plotted in the figure and show that
En əll eV matches the low energy, free electron part of e, best. A bound
P
electron contribution appears at 12.2 eV but this value does not agree too
well with the value determined from soft x-ray spectroscopy for the ons
transition at 16 eV.
6. Conclusions
The dielectric constants of cd, Zn, and TI determined by optical
methods revealed that the volume plasma losses for charged particles should
occur at 8.5-, 9. 8-, and 10.2-eV and the surface plasma losses at 6.57, :7.1-,
and 8. 9-eV, respectively. Since the surface and volume plasmons are not very
widely separated in energy, it was not possible for these losses to be detected
separately with the resolution available in the conventional electron energy
loss experiments. The losses observed by Robins and Powell at 7.8-, 8. 6-,
and 9. 6-eV in Cd, Zn, and Ti, respectively, appear to have been due to the
unresolved surface and volume plasmons.
Interband transitions are also evident from the dielectric constant
at around 10.5 eV in Cd and Zn corresponding to the N,, N, and M,
levels
calculated by Sandström from soft x-ray data.
.
--
11.
:
The angular distribution of photon intensity and line breadth frrm
electron-bombarded Al foils yielded information on the lifetimes due to i
.
...
1
...
.......
21
radiation and to electronic damping in addition to the value of 14. 9 eV
for the volume plasma energy.
Finally, it should be pointed out that optical studies give
information only for processes with small momentum transfer. There-
foxe, dispersion studies still require the large momentum transfer of
electron beam experiments.
7. Acknowledgements
We wish to thank R. D. Birkhoff and R. H. Ritchie for
:
numerous discussions and suggestions in the preparation of this paper.
.
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References
.
Arakawa, E. T., N. O. Davis, L. C. Emerson, and R. D. Birkhoff, 1964,..
J. Phys. Radium 25, 129.
Ferrell, R. A., 1958, Phys. Rev. 111, 1214.
Huebner, R. H., E. T. Arakawa, R. A. MacRae, and R. N. Hamm, 1964,...
J. Opt. Soc. Am. 54, 1434.
Hunter, W. R., 1964, J. Phys. Radium 25, 154.
Pines, D., 1963, Elementary Excitations in Solids (W. A. Benjamin, Inc., . '
New York).
Powell, C. J., 1960, Proc. Phys. Soc. (London) A76, 593.
Ritchie, R. H., 1957, Phys. Rev. 106, 874.
Ritchie, R. H. and H. B. Eldridge, 1962, Phys. Rev. 126, 1935.
Robins, J. L., 1961, Proc. Phys. Soc. (London) A78, 1177.
Robins, J. L., 1963, Proc. Phys. Soc. (London) A79, 119.
Rudberg, E., 1930, Proc. Roy. Soc. (London) A127, 111.
Sandström, A. E., 1957, Handbuch der Physik (Springer-Verlag, Berlin) p. 226.
Von Blanckenhagen, P., H. Boersch, D: Fritsche, H. G. Seifert, and G. Sauerbrey,
1964, Phys. Rev. Letters 11, 296.
::
:
:::
Watanabe, H., and E. C. Y. Inn, 1953, J. Opt. Soc. Am. 43, 32.
...
:..
ORNI - AEC - OFFICIAL
.-
.
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.
23 :
Figure Captions
Fig. 1. Comparison of surface and volume plasma loss functions for ca
with experimental electron energy loss and optical emission spectra.
Fig. 2. Comparison of surface and volume plasma loss functions for Zn with
experimental electron energy loss and optical emission spectra.
Fig. 3. Comparison of surface and volume plasma loss furictions of T1 with
experimental electron energy loss spectrum of Robins.
. 4. Recorder traces of photon emission spectra from an electron- .
bombarded Al foil 320 Å thick at various angles of observations:
(Electron beam 80 keV, 6.2 MA)
Fig. 5. Wavelength of emission peak from Al vs. angle of observation. .
Fig. 6. Line width of Al emission vs. angle of observation.
Fig. 7. Photon intensity at wavelength of peak emission vs. angle of observation.
Fig. 8. Graphical evaluation of the equation 6, E’ = [1 + 8% (62)]E? - E, for
the optical constants of Cd.
Fig. 9. Graphical evaluation of the equation e, E? = [1 +63(87) DEC - E,- for
the optical constants of Zn.
Fig. 10. Dielectric constants of Ti vs. photon energy. :
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