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K-shell ionization by extreme relativistic electrons
Volume 46A, number 1
PHYSICS LETI’ERS
19 November 1973
K-SHELL IONIZATION BY EXTREME RELATiVISTIC ELECTRONS G.R. DANGERFIELD School of Physics, University of Melbourne, Parkville, Victoria, Australia 3052 Received 13 September 1973 The results of a calculation of the electron K-ionization cross section which considers the influence of the atoms adjacent to the target atom are presented. The predicted energy dependence differs from that observed by Middleman, Ford and Hofstadter between 150 and 900 MeV.
Although the electron-induced K-shell ionization cross section has been measured extensively for electron energies below 1 MeV, there is only one published work describing measurements for energies above 10MeV, namely that of Middleman, Ford and Hofstadter [1] These latter results show a logarithmic increase in cross section with increasing electron energy, and were compared with a semiclassical virtual photon theory first given by Kolbenstvedt [2] and later revised for these higher energies [1]. This logarithmic increase is a feature of several theories, beginning with that of Bethe [3] but comparisons with them have been only relative and not absolute [4]. However, all these theories are calculated on the basis that the target atom is isolated and independent of its neighbours. In terms of the virtual photon theory, the range of the electron-electron interaction can extend to greater than several hundred times the atomic radius at high electron energies and therefore the influence of the adjacent atoms in condensed media should be considered [5]. Such effects are most readily incorporated with a virtual-photon calculation by describing the target medium, through which the virtual photons travel, by a general frequency-dependent complex dielectric constant e(w). This leads to a modified virtual photon spectrum, which, in the region of present interest, simplifies to .
,
~d W
x
— —
[ln(
L
2e2(w) i~ a dw r
(1)
‘~
\(l~78Wb 2(l_132Er(W))F 0)
~-i~2~ (w)1 r
.
This is comparable to eq. (1) of ref. [21 At frequen.
cies above the K-edge, the real part of the general Kallmann---Mark dielectric constant [6] reduces to the approximate form 2
e(w)~’l—-A/w where A is a constant, which depends only on the total electron density of the target and not on the detailed electronic structure. Substitution of this expression into eq. (1) shows that for “low” electron energies (dependent on A and w) the intensity spectrum increases with electron energy, as expected, but, for sufficiently high electron energies, the intensity spectrum approaches a constant value. Integration of this spectrum with the Born—Stobbe K-shell photoeffect cross section, following the normal virtual photon method as in ref. [2] results in a similar saturation in the calculated K-ionization cross section. The calculated cross section, with (BSa) and without (BSb) this density effect, is shown in fig. 1 for three elements, and is compared with interpolated results from ref. [1]. Cross sections calculated using the improved numerical photoeffect cross section of Schmickley and Pratt [7] in place of the Born Stobbe cross section are also shown in fig. 1, with (SPa) and without (SPb) the density effect included. Also shown in fig. I is the revised Kolbenstvedt result (dotted) [1] It follows most closely the result obtained using the Schmickley and Pratt photoelectric Thesection, energyand dependence of thethe cross section, incross not including density effect. cluding the density effect, differs markedly from ,
.
the simple logarithmic increase supported by the experimental results within ±2%,particularly for the lower atomic numbers. More results are needed in this energy range to determine more precisely 19
Volume 46A, number I
PHYSICS LETTERS
19 November 1973
160
Q
a
N~.
Ag
40-
600
.
20
SPb
BSb
too
5OO~~
400
~
-
80
a
BSb
40 Au
200
.
SPb
3f~ —----
I
tOO
200
I
300
400 500
700
900
EL EC/RON
100
200
EA/ERGY
I
300
I
4(0 500
700
900
(Me VI
Fig. 1. Total K-shell ionization cross sections (a) in barn as a function of electron energy. See text for description of the various curves.
both the absolute magnitude of the cross sections and the energy dependence.
References [1] L.M. Middleman, R.L. Ford and R. Hofstadter, Phys. Rev. A2 (1970) 1429.
20
121 H. Kolbenstvedt, J. App. Phys. 38 [31 H.A. Bethe, Ann. Physik 5 (1930)
[4] [5] [6] [7]
(1967) 4785. 325. C.A. Quarles, Phys. Lett. 39A (1972) 375. C.B.O. Mohr, Advances in atomic and molecular physics. Vol.4 (Academic Press, New York, 1968) p. 228. R.M. Sternheimer, Phys. Rev. 91(1953) 256. R.D. Schmickley and RI-I. Pratt, Phys. Rev. 164 (1967) 104.
PHYSICS LETI’ERS
19 November 1973
K-SHELL IONIZATION BY EXTREME RELATiVISTIC ELECTRONS G.R. DANGERFIELD School of Physics, University of Melbourne, Parkville, Victoria, Australia 3052 Received 13 September 1973 The results of a calculation of the electron K-ionization cross section which considers the influence of the atoms adjacent to the target atom are presented. The predicted energy dependence differs from that observed by Middleman, Ford and Hofstadter between 150 and 900 MeV.
Although the electron-induced K-shell ionization cross section has been measured extensively for electron energies below 1 MeV, there is only one published work describing measurements for energies above 10MeV, namely that of Middleman, Ford and Hofstadter [1] These latter results show a logarithmic increase in cross section with increasing electron energy, and were compared with a semiclassical virtual photon theory first given by Kolbenstvedt [2] and later revised for these higher energies [1]. This logarithmic increase is a feature of several theories, beginning with that of Bethe [3] but comparisons with them have been only relative and not absolute [4]. However, all these theories are calculated on the basis that the target atom is isolated and independent of its neighbours. In terms of the virtual photon theory, the range of the electron-electron interaction can extend to greater than several hundred times the atomic radius at high electron energies and therefore the influence of the adjacent atoms in condensed media should be considered [5]. Such effects are most readily incorporated with a virtual-photon calculation by describing the target medium, through which the virtual photons travel, by a general frequency-dependent complex dielectric constant e(w). This leads to a modified virtual photon spectrum, which, in the region of present interest, simplifies to .
,
~d W
x
— —
[ln(
L
2e2(w) i~ a dw r
(1)
‘~
\(l~78Wb 2(l_132Er(W))F 0)
~-i~2~ (w)1 r
.
This is comparable to eq. (1) of ref. [21 At frequen.
cies above the K-edge, the real part of the general Kallmann---Mark dielectric constant [6] reduces to the approximate form 2
e(w)~’l—-A/w where A is a constant, which depends only on the total electron density of the target and not on the detailed electronic structure. Substitution of this expression into eq. (1) shows that for “low” electron energies (dependent on A and w) the intensity spectrum increases with electron energy, as expected, but, for sufficiently high electron energies, the intensity spectrum approaches a constant value. Integration of this spectrum with the Born—Stobbe K-shell photoeffect cross section, following the normal virtual photon method as in ref. [2] results in a similar saturation in the calculated K-ionization cross section. The calculated cross section, with (BSa) and without (BSb) this density effect, is shown in fig. 1 for three elements, and is compared with interpolated results from ref. [1]. Cross sections calculated using the improved numerical photoeffect cross section of Schmickley and Pratt [7] in place of the Born Stobbe cross section are also shown in fig. 1, with (SPa) and without (SPb) the density effect included. Also shown in fig. I is the revised Kolbenstvedt result (dotted) [1] It follows most closely the result obtained using the Schmickley and Pratt photoelectric Thesection, energyand dependence of thethe cross section, incross not including density effect. cluding the density effect, differs markedly from ,
.
the simple logarithmic increase supported by the experimental results within ±2%,particularly for the lower atomic numbers. More results are needed in this energy range to determine more precisely 19
Volume 46A, number I
PHYSICS LETTERS
19 November 1973
160
Q
a
N~.
Ag
40-
600
.
20
SPb
BSb
too
5OO~~
400
~
-
80
a
BSb
40 Au
200
.
SPb
3f~ —----
I
tOO
200
I
300
400 500
700
900
EL EC/RON
100
200
EA/ERGY
I
300
I
4(0 500
700
900
(Me VI
Fig. 1. Total K-shell ionization cross sections (a) in barn as a function of electron energy. See text for description of the various curves.
both the absolute magnitude of the cross sections and the energy dependence.
References [1] L.M. Middleman, R.L. Ford and R. Hofstadter, Phys. Rev. A2 (1970) 1429.
20
121 H. Kolbenstvedt, J. App. Phys. 38 [31 H.A. Bethe, Ann. Physik 5 (1930)
[4] [5] [6] [7]
(1967) 4785. 325. C.A. Quarles, Phys. Lett. 39A (1972) 375. C.B.O. Mohr, Advances in atomic and molecular physics. Vol.4 (Academic Press, New York, 1968) p. 228. R.M. Sternheimer, Phys. Rev. 91(1953) 256. R.D. Schmickley and RI-I. Pratt, Phys. Rev. 164 (1967) 104.