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Electromagnetic cross sections for electron and nuclear research
N U C L E A R I N S T R U M E N T S A N D M E T H O D S 28 (I964) 1 9 9 - 2 o 4 ;
© NORTH-HOLLAND PUBLISHING
CO.
ELECTROMAGNETIC CROSS SECTIONS FOR ELECTRON AND NUCLEAR RESEARCH H. W. KOCH
National Bureau of Standards, Washington 25, D.C. Much of the extensive experimental data being generated by electron accelerators is difficult to interpret accurately because of the complexity of the electromagnetic cross section formalisms. The need for greater experimental and theoretical attention to
these characteristic complexities is illustrated with the cross section analyses employed in recent experiments on the total nuclear cross sections for 10 to 70 MeV X rays.
1. Introduction The National Bureau of Standards has contracted for a high intensity electron linear accelerator from the High Voltage Engineering Company. The accelerator installation is to begin in the spring of 1964 and hopefully will be completed sometime next summer. The accelerator will produce 100 MeV electrons with 40 kW in the beam and will be used for a variety of nuclear physics experiments, including elastic and inelastic electron scattering; photoproton and photoncutron studies; and mono-energetic photon studies made possible by annihilating magnetically analyzed positrons in flight. Research programs of these types have suffered in the past at our and other laboratories because of an inability to make, either, proper theoretical derivations, or, proper experimental applications of the electromagnetic cross sections that are basic to this research. This fundamental characteristic of electron research is the subject for my discussion at this conference. To be specific, much of the quantitative data that should be available from elastic and inelastic electron scattering has been unattainable because no one adequately understands how to correct for the radiation tail in the scattered electron spectrum. It is only because of the conscientious efforts of various experimenters such as Goldemberg, Isabelle, Bishop, Leiss, Hofstadter and Barber, and of such theoreticians as Walecka, Pratt, Drell, Schiff, Olsen and Maximon that substantial contributions are being made to this subject of real and virtual photon corrections to electron scattering. Similarly, the theories of bremsstrahlung production cannot predict an X-ray energy and angular distribution to better than a factor of two for incoming electron energies of the order of ½ MeV. 1) As a result, the spectra of X-ray machines operating in the very practical 50 kV to 5 MV range are quite unpredictable. Also, as a result, the thick target bremsstrahlung spectra used in the excellent experiments of Dr. Frank Firk, are not well known theoretically and are very difficult to
measure experimentally. Consequently, his photonuclear cross sections are difficult to evaluate accurately. Other numerous examples of the importance of understanding electromagnetic cross sections include the inaccuracies that have occurred in the measurement of high energy X-ray energies by the pair electron angles in emulsions due to the use of an incorrect theoretical expression / ) and the inaccuracies that have occurred in the predictions of X-ray attenuation coefficients at almost any X-ray energy. These examples serve to point up the paradox in the inaccuracies and uncertainties in that they do not result, surprisingly enough, from a lack of understanding of the electromagnetic interaction. Indeed, the electromagnetic interaction has been the best understood of the four basic kinds of force interactions: gravitation, electromagnetism, strong interaction, and weak interaction. That there should be this better understanding can be seen from the magnitude 3) of the electromagnetic coupling constant g = eZ/hc = 1/137. Thus, if the solution of a given electromagnetic cross section calculation can be expressed, as it has been, in terms of a matrix element which is expanded in powers of ea/hc, then it is obvious that a first order calculation obtained from the first term in the expansion will already be a 1% calculation. This accounts for the successes of the Bethe-Heitler calculation for light elements 4) which were performed in the Born approximation, a first order calculation. Additionally, the refined theories of quantum electrodynamics are able to predict accurately 5) the correct values for the Lamb shift, electron g values and the positronium fine structure which are 0.1% effects. For the purposes of this accelerator conference it was
199
1) E.g., H. W. Koch and J. Motz, Rev. Mod. Phys. 31 (1959) 920. 2) H. Olsen, private communication and Phys. Rev. 131 (1963) 406, 3) The relative magnitudes of the four coupling constants are strong ( ~ 1), electromagnetic (~ 1/137), weak ( ~ 10 14), and gravitational ( ~ 2 × 10-39). 4) W. Heitler, Quantum Theory of Radiation, 3rd ed. (Oxford Univ. Press, London, 1954) p. 257, form. 6. VI. E X P E R I M E N T A L TECHNIQUES
200
r~. w. KOCH
felt desirable to explore this seeming paradox and to provide examples of some recent experimental and theoretical conclusions on the nature o f specific electromagnetic cross sections. The specific cross sections and corrections that will be discussed are the pair production in the field o f electrons and the C o u l o m b corrections to pair production in the field of the nucleus. These cross sections have been basic to the evaluation of total photonuclear cross sections from the results of an X-ray attenuation experiment recently completed*). It will become evident from this discussion that the basic, completely differential cross sections can be written accurately for m a n y electromagnetic processes for all energies and angles. However, these are frequently in such a complicated form that it has been impossible to integrate over any of the variables in order to obtain a practical cross section prediction. F r o m an examination of a few examples it will be possible to demonstrate that detailed examination and understanding of the physics of the processes should make it possible for certain of the complications to be overcome. F r o m a more complete understanding of the practical electromagnetic cross sections is b o u n d to come a better realization of the potential of the so-called "wellunderstood" electromagnetic interactions with atoms in X-ray research or with nuclei in photonuclear research. 2. Pair production in the field of electrons The cleanest example of an electromagnetic process that should be calculable accurately is that of electron pair production in the field of an electron. In this process called triplet production (shown schematically in fig. 1) all of the elements" the incoming photon, the outgoing pair of electrons, and the recoiling electron,
.-7
'
///
/
¢/
(a)
Ib]
(c)
(d)
Fig. 1. Diagram of triplet production. interact by the electromagnetic process. Therefore, it ought to be possible in principle to predict the total integrated triplet cross section to high accuracies. Indeed, Votruba 7) has done a complete q u a n t u m electrodynamical calculation of triplet production in which he included many effects such as those due to the possible exchange of the recoil electron with the electron of the electron pair components, due to the retardation effects at the relativistic velocities of the recoiling
electrons, and due to the influence of each of the four contributing diagrams o f fig. 1. However, as Suh and Bethe 8 state, Votruba's completely differential cross section is so long and complicated that it is difficult to handle; consequently, in order to carry out a general analytic integration, approximations have had to be made which result in a cross section that does not include the contributions f r o m certain values o f the m o m e n t u m transfer, q, adequately. 100~
--'-T
I
"
=
~
T
~
"
r
T '
, ,
•
'
-- ~ r ~
....
I
l
TOTAL CROSS SECTION FOR PAIR PRODUCTION IN THE FIELD OF AN ELECTRON
,c
r~
BETHE - HEITLER BORSELLINO o., ,o
,o
,oo
,ooo
,oooo
PHOTON ENERGY, k ( M e V )
Fig. 2. Comparison of triplet theories. As a consequence, various predictions of the integrated triplet cross section have been uncertain by factors of two at an incoming p h o t o n energy of the order of 10 MeV. For example, fig. 2 compares the predictions of Votruba7), Borsellin09), and BetheHeitler ~) as taken from the review paper of Joseph and Rohrlich1°). Since the integrated triplet cross sections for elements such as carbon are about l/Z o f the integrated pair cross section, the pair production cross section for carbon including the triplet cross section has been uncertain by about 6 ~ depending on the choice o f the theoretical triplet cross section. The uncertainty for an element such as silicon is of the order of 4~o. This section will attempt to demonstrate by experimental and theoretical arguments what seems to be the best prediction of free-electron triplet cross sections. Next to Votruba7), Borsellino 9) has done the most 5) E.g., W. S. C. Williams, An introduction to elementary particles (Academic Press, New York, 1961) p. 227; B. T. Feld, Proc. Conf. on Photon Interactions in the GeV-Energy Range, MIT, 1963, pp. I.l and VIII.1. 6) j. M. Wyckoff, B. Ziegler, H. W. Koch and R. Uhlig, to be published. 7) V. Votruba, Phys. Rev. 73 (1948) 1468; Bul. intern, acad. tch6que sci., Prague 49 (1948) 19. 8) K. S. Suh and H. A. Bethe, Phys. Rev. 115 (1959) 674. 9) A. Borsellino, Helv. Phys. Acta 20 (1947) 136; Nuovo Cimento 4 (1947) 112. 10) j. Joseph and F. Rohrlich, Rev. Mod. Phys. 30 (1958) 354.
ELECTROMAGNETIC
CROSS SECTIONS
FOR ELECTRON
complete calculation by leaving arbitrary the mass, mN, in whose field the pair is produced and by including the retardation effects due to the recoiling nucleus which are important at relativistic energies, especially for recoiling electrons. However, he did not include all of the fig. 1 diagrams or the exchange diagrams that Votruba included. These effects, according to Suh and Bethe s) might not be large at high energies. Borsellino has also neglected screening and binding of the atomic electrons. Suh and Bethe s) have contributed considerably to the understanding of triplet production by analyzing the implications of Borsellino's results. They show by expanding Borsellino's formulas and by extending them that, in the limit of high photon energies and low q values, the magnitude of the recoiling mass, my, does not enter into the distribution. They also show that, to a good approximation, the integrated cross section will be the same for electron and proton recoiling masses, if (a) the photon energy, k ~> ½ MeV, and (b) screening can be neglected. This result should be accurate to order k -1 logk. For intermediate energies (approximately in the range of 5 MeV < k < 100 MeV) it is necessary to avoid the extreme relativistic approximation of a very large photon energy and the approximation of zero recoil energy. Since Borsellino did avoid these approximations, his momenta distributions and the approximate, but simpler, formulas of Suh and Bethe are useful. In addition, the integrated cross sections of Borsellino I 1) as shown on an earlier figure are the only accurate estimates available. However, at photon energies close to the triplet threshold of 2 MeV, the predictions of Votruba are probably more accurate. Therefore, the estimated dashed curve in fig. 2 is probably the best present guess of the integrated triplet cross section at low energies. Justification for the Borsellino integrated triplet cross section at high energies can be obtained from recent experiments t 2) employing hydr ogen-filled b ubble chambers and diffusion cloud chambers which are ideal for an accurate test of the triplet cross section predictions because the ratio of triplets to nuclear pairs in hydrogen is a maximum compared to other elements, because the atomic binding of the electron has little effect on the process and is negligible below a photon energy of 100 MeV, and because the effect of atomic screening has been accurately calculated by Wheeler and Lamb a3) using the exact hydrogen wavefunctions. Therefore, the experiments allow an accurate evaluation of the importance of exchange and radiative corrections which were omitted in Borsellino's calculations. However,
AND NUCLEAR
RESEARCH
201
they do not serve to test an additional source of uncertainty that becomes important at high photon energies: the incoherent scattering function required for the estimates of atomic excitation and ionization as calculated on the Thomas-Fermi atomic model for the low Z elements for which triplet production is important and for which the model is known to be inaccurate. More specific comment on the elements above hydrogen and the screening correction will be made later. The bubble and diffusion chamber measurements allow an examination of the differential cross sections for triplet production. In addition, the integrated cross sections for inelastic pair production (including triplet production) have been measured by the total X-ray attenuation technique in liquid hydrogen attenuators and other elemental attenuators12). The general conclusions from these various experiments, as discussed in detail in the review paper by Motz and Koch t4) are: a) The momenta distribution formulas of Borsellino as reinterpreted by Suh and Bethe for the large q and small q cases are reasonably accurate. There seems to be no good evidence for influence on these momenta distributions of the omission of exchange and the radiative corrections by Borsellino. b) The integrated cross section variation with energy in hydrogen is well accounted for by the Borsellino cross section values when corrected for atomic binding by the Wheeler-Lamb correction 15) which is discussed in the Motz-Koch paper14). 11) T h e specific numerical comparisor.s of the values for pair production in the field o f the n u c l e u s a n d in the field o f a n electron is given in the following table as t a k e n flora reference 9) : k(MeV) 2 2.2 2.6 3 5 10 25 50 t~g/Z2r2o/137 0.32 0.35 0.67 0.89 1.94 3.75 6.4 8.4 Cre/Z2r2o/137 0 0.0044 0.038 0.102 0.627 2.05 4.78 7.15 12) E. L. H a r t , G. Cocconi, V. T. Cocconi and J. M. Sellen, Phys. Rev. 115 (1959) 678; D. C. Gates, R. W. K e n n e y a n d W. P. Swanson, Phys. Rev. 125 (1962) 1310; J. D. A n d e r s o n , R. W. K e n n e y and C. A. M c D o n a l d , Jr., Phys. Rev. 102 (1956) 1626, 1632. 13) j. A. W h e e l e r a n d W. E. L a m b , Phys. Rev. 55 (1939) 858; a n d I01 (1956) 1836. 14) j. Motz a n d H. W. K o c h , to be published. 15) T h e influence of screening for heavier atomic nuclei t h a n h y d r o g e n c a n be included by a p r o c e d u r e suggested by Bethe a n d A s h k i n [E. Segt6, Experimental Nuclear Physics ( J o h n Wiley & Sons, Inc., N e w York, 1953) p. 260] w h o suggested taking the difference of the W h e e l e r - L a m b F o r m u l a with e = ~o a n d with e d e t e r m i n e d by the appropriate value of Z, k a n d E+. If screening is important, the screening correction so obtained c a n be subtracted f r o m the Borsellino free electron formula. T h u s O'triplet = o'Wheeler Lamb - - / O ' p a i r (Z = i) unscreened Bethe-Heitler -}- No" triplet (Z = i) Borsellino Vl. E X P E R I M E N T A L
TECHNIQUES
202
H. Wo K O C H
3. Pair production in the field of the bare nucleus
The simplifications in the calculations of triplet production due to the assumption of an infinite mass recoiling-field and due to the elimination of exchange, are considerable. These simplifications are permissible for the process of electron pair production in the field of a bare nucleus, as first calculated by Bethe and Heitler4). As suggested in the introduction, their theory performed in the Born approximation is very successful .......
2 c
I
........
I
sent the electrons. Their results (curve A) are excellent at very high energies. However, for photon energies below 20 MeV their results as taken from their paper diverge as shown in fig. 3. Based on the general conclusions from this figure various authors have been reluctant to place much reliability in the calculations of Davies et al. at energies below 50 MeV. For many practical applications at lower energies the results of these authors have been assumed to be non-usable, therefore. Davies, Bethe and Maximon have written their Coulomb - corrected, differential- in- positron - energy cross section
t--
I.E --
4 Z 2 r2 d E +
dae+ = 13ff o ~ j - {(E2+ + EZ-)[-4q~l(7) - ½1nZ - / ( Z ) ]
1.2--
- l l n Z - . f ( Z ) ] }.
+ 3E+E-[¼~2(7)
E
(1)
For small photon energies the screening corrections are small and eq. (1) becomes
J "60.8
--
b-
d~re+ -
0.4--
4 Z Z r 2 de+ I (EZ+ + e z2E+E_ ) 137 k3 [ +
( ln2E+~E~-- k ~~ o
1.0
[ IO
I00
f(Z)) }.
(2)
I I IIIII iooo
X-RAY ENERGY (MeV)
Fig. 3. Ratio of cross sections to tr Born for lead.
for those high energies and light nuclei for which the Born approximation is applicable. However, for photon energies less than 50 MeV and for high atomic number nuclei such as lead the Bornapproximation predictions are inaccurate by at least 10~. In order to illustrate this inaccuracy, fig. 3 shows a plot of the ratio of experimentally measured electron pair cross sections in lead to the Born-approximation predictions as a function of photon energyX6). This plot clearly shows that the Born-approximation predictions are about I 0 ~ too high down to energies of the order of 5 MeV. Also drawn on this plot are the Coulomb-corrected results of Davies e t a / . 17) who improved on the plane wave assumptions of Bethe and Heitler by using Sommerfeld-Maue relativistic wavefunctions to repre* q)1(7) and q~2(7) are screening functions with 7----IOOk/E+E_Zk. f ( Z ) = is the C o u l o m b correction function where = 1.2021 (Z/137)2 for low Z = 0.925 (Z/137) 2 for high Z = 0 for uncorrected differential cross section of BetheHeitler. All symbols and constants are based on energies in moc2 units.
If eq. (2) is integrated in the usual manner from k = 0 to E+, the result is O'N
=
Z ro
T37-
In 2k
-
-
.
(3)
At photon energies below 20 MeV, eq. (3) is plotted as the Davies, Bethe, Maximon result in curve A of fig. 3. As can be seen, it diverges logarithmically for small values of k. Unfortunately, the formula w i t h f ( Z ) = 0 also diverges and is also inaccurate as a representation of the Born-approximation values (curve B). Hough in 194818) showed that eq. (3) w i t h f ( Z ) = 0 was inaccurate even though the differential cross section eq. 2 was a reasonably accurate representation of the Born-approximation predictions. He pointed out that eq. (3) could be corrected by an empirical correction term as Z2r~ ( ~ 218 6.451 trN = ~ --lnZk- ~ff-+~k--I ' (4) which gives good estimates of the Born-approximate aN down to 3 MeV. 16) G. W. Grodstein, X-ray attenuation coefficients from 10 keV to 100 MeV, NBS Circular 583 (1957). 17) Davies, Bethe and Maximon, Phys. Rev. 93 (1954) 788 is) p. V. C. Hough, Phys. Rev. 73 (1948) 266.
ELECTROMAGNETIC
CROSS SECTIONS FOR ELECTRON
The basic importance of accurate triplet and nuclear pair cross section predictions for some experimental interpretations can be illustrated with fig. 4 that has plotted the total X-ray attenuation cross section for silicon. The results are taken from ref. 6). Since the total nuclear cross section is less than 5% of the total
218 In 2k - ~ - + 4 k
s (}k
-
2k
+ %
-
}
v)/(z)
203
RESEARCH
4. Total nuclear cross sections
Recently, Maximon ~9) has shown that the Hough correction is almost completely due to an improper integration between the limits of k = 0 to E0 instead of from k = ½ MeV to (E+ -- ½) MeV. I f one applies the Maximon argument to the eq. (1), the result is Z ro aN = T37-
AND NUCLEAR
co
12
[
~
SILICON
. n~ ~
I\
1,0{---~
,
_Z ~ \
After having made this basic improvement in the total cross section, one can then make the theoretical arguments that the remaining corrections have corrections of the order of
.J
~TOTAL ELECTRONICABSORPTION CROSSSECTION
TOTAL ABSORPTION CROSS SECTION
Z) ~
ALL PROCESSES
(~Ldt~O _O j 8p
--
/~./~_
~OMPTON CROSSSECTION~
co - X ~::O.E-- ~
/ ~-
-~'~TOTAL PAIR PRODUCTION CROSS SECTION
l n k (lnk) 2 (lnk) 3 l n k k' k ' k , k2'etc" nk-OA
g
In the evaluation of the results of the experiment of ref. 6), it has been assumed that the remaining correction terms are of a form
<1:02 _ J
,TRIPLET CROSSSECTION
©
AaN =
1-~
+ k (C1 + C2 In k + Ca(In k)3).
I0
The results of this fitting to the experimental data are: C, = - 9 6 6 . 3 barns MeV C2 = +804.3 barns MeV C3 = -125.1 barns MeV k = photon energy in MeV. This result is plotted as curve C in fig. 3 and is seen to be consistent with the earlier experimental data~6). The corrected formula also is a good representation of the iodine data given in ref. ~).
15
20
25
50
INCIDENT PHOTON ENERGY (MeV)
35
40
Fig. 4. C o m p o n e n t cross sections fol silicon,
cross section, it is important to obtain the total electronic cross section, including the triplet and nuclear pair cross section, to an accuracy of the order of ½ or better. By studying a wide range of elements in order to untangle the dependence of the electronic cross sections on atomic number and energy and by using the 19) L. M a x i m o n , private c o m m u n i c a t i o n .
6C
I
5C
,,
,
, I
4C Z
g d 20~
°
°~
°
°.
°
•
5 I
tr,p) ~0
Ill
J
t~,n) m INCIDENT
-~
20 PHOTON
ENERGY
z5 (MeVI
30
35
Fig. 5. Silicon total nuclear cross section. VI. E X P E R I M E N T A L
TECHNIQUES
204
H.W. KOCH
arguments presented in this paper, the total nuclear cross section for silicon has been obtained as given in fig. 5. These results are in excellent agreement in shape with the photoneutron cross sections obtained at Livermore2°). The results also show a large ratio of photoproton cross section to photoneutron cross section (44 millibarns versus 11 millibarns) when compared in magnitude to those of ref. 20). Other interesting data on differential and integrated total nuclear cross sections are given in ref. 6). These data were only made interpretable after some of the electronic cross section results were made consistent and understandable as described in this paper. 20) j. T. Caldwell, R. R. Harvey, R. L. Bramblett and S. C. Fultz, University of California UCRL-7424 (1963).
5. Conclusion The pair production process has been used to illustrate the frustrating and frequently trivial corrections that have been plagueing the practical applications of the results of quantum electrodynamical calculations. Other more subtle effects could also have been described and are under active examination at a number of laboratories. If one can master the complications of electron research analysis, and much of our research program at the National Bureau of Standards is attempting to do so, then the rewards of a well-understood interaction will be available. It will then be possible to fully apply the electromagnetic interaction as a lever in helping to understand other interactions and processes.
© NORTH-HOLLAND PUBLISHING
CO.
ELECTROMAGNETIC CROSS SECTIONS FOR ELECTRON AND NUCLEAR RESEARCH H. W. KOCH
National Bureau of Standards, Washington 25, D.C. Much of the extensive experimental data being generated by electron accelerators is difficult to interpret accurately because of the complexity of the electromagnetic cross section formalisms. The need for greater experimental and theoretical attention to
these characteristic complexities is illustrated with the cross section analyses employed in recent experiments on the total nuclear cross sections for 10 to 70 MeV X rays.
1. Introduction The National Bureau of Standards has contracted for a high intensity electron linear accelerator from the High Voltage Engineering Company. The accelerator installation is to begin in the spring of 1964 and hopefully will be completed sometime next summer. The accelerator will produce 100 MeV electrons with 40 kW in the beam and will be used for a variety of nuclear physics experiments, including elastic and inelastic electron scattering; photoproton and photoncutron studies; and mono-energetic photon studies made possible by annihilating magnetically analyzed positrons in flight. Research programs of these types have suffered in the past at our and other laboratories because of an inability to make, either, proper theoretical derivations, or, proper experimental applications of the electromagnetic cross sections that are basic to this research. This fundamental characteristic of electron research is the subject for my discussion at this conference. To be specific, much of the quantitative data that should be available from elastic and inelastic electron scattering has been unattainable because no one adequately understands how to correct for the radiation tail in the scattered electron spectrum. It is only because of the conscientious efforts of various experimenters such as Goldemberg, Isabelle, Bishop, Leiss, Hofstadter and Barber, and of such theoreticians as Walecka, Pratt, Drell, Schiff, Olsen and Maximon that substantial contributions are being made to this subject of real and virtual photon corrections to electron scattering. Similarly, the theories of bremsstrahlung production cannot predict an X-ray energy and angular distribution to better than a factor of two for incoming electron energies of the order of ½ MeV. 1) As a result, the spectra of X-ray machines operating in the very practical 50 kV to 5 MV range are quite unpredictable. Also, as a result, the thick target bremsstrahlung spectra used in the excellent experiments of Dr. Frank Firk, are not well known theoretically and are very difficult to
measure experimentally. Consequently, his photonuclear cross sections are difficult to evaluate accurately. Other numerous examples of the importance of understanding electromagnetic cross sections include the inaccuracies that have occurred in the measurement of high energy X-ray energies by the pair electron angles in emulsions due to the use of an incorrect theoretical expression / ) and the inaccuracies that have occurred in the predictions of X-ray attenuation coefficients at almost any X-ray energy. These examples serve to point up the paradox in the inaccuracies and uncertainties in that they do not result, surprisingly enough, from a lack of understanding of the electromagnetic interaction. Indeed, the electromagnetic interaction has been the best understood of the four basic kinds of force interactions: gravitation, electromagnetism, strong interaction, and weak interaction. That there should be this better understanding can be seen from the magnitude 3) of the electromagnetic coupling constant g = eZ/hc = 1/137. Thus, if the solution of a given electromagnetic cross section calculation can be expressed, as it has been, in terms of a matrix element which is expanded in powers of ea/hc, then it is obvious that a first order calculation obtained from the first term in the expansion will already be a 1% calculation. This accounts for the successes of the Bethe-Heitler calculation for light elements 4) which were performed in the Born approximation, a first order calculation. Additionally, the refined theories of quantum electrodynamics are able to predict accurately 5) the correct values for the Lamb shift, electron g values and the positronium fine structure which are 0.1% effects. For the purposes of this accelerator conference it was
199
1) E.g., H. W. Koch and J. Motz, Rev. Mod. Phys. 31 (1959) 920. 2) H. Olsen, private communication and Phys. Rev. 131 (1963) 406, 3) The relative magnitudes of the four coupling constants are strong ( ~ 1), electromagnetic (~ 1/137), weak ( ~ 10 14), and gravitational ( ~ 2 × 10-39). 4) W. Heitler, Quantum Theory of Radiation, 3rd ed. (Oxford Univ. Press, London, 1954) p. 257, form. 6. VI. E X P E R I M E N T A L TECHNIQUES
200
r~. w. KOCH
felt desirable to explore this seeming paradox and to provide examples of some recent experimental and theoretical conclusions on the nature o f specific electromagnetic cross sections. The specific cross sections and corrections that will be discussed are the pair production in the field o f electrons and the C o u l o m b corrections to pair production in the field of the nucleus. These cross sections have been basic to the evaluation of total photonuclear cross sections from the results of an X-ray attenuation experiment recently completed*). It will become evident from this discussion that the basic, completely differential cross sections can be written accurately for m a n y electromagnetic processes for all energies and angles. However, these are frequently in such a complicated form that it has been impossible to integrate over any of the variables in order to obtain a practical cross section prediction. F r o m an examination of a few examples it will be possible to demonstrate that detailed examination and understanding of the physics of the processes should make it possible for certain of the complications to be overcome. F r o m a more complete understanding of the practical electromagnetic cross sections is b o u n d to come a better realization of the potential of the so-called "wellunderstood" electromagnetic interactions with atoms in X-ray research or with nuclei in photonuclear research. 2. Pair production in the field of electrons The cleanest example of an electromagnetic process that should be calculable accurately is that of electron pair production in the field of an electron. In this process called triplet production (shown schematically in fig. 1) all of the elements" the incoming photon, the outgoing pair of electrons, and the recoiling electron,
.-7
'
///
/
¢/
(a)
Ib]
(c)
(d)
Fig. 1. Diagram of triplet production. interact by the electromagnetic process. Therefore, it ought to be possible in principle to predict the total integrated triplet cross section to high accuracies. Indeed, Votruba 7) has done a complete q u a n t u m electrodynamical calculation of triplet production in which he included many effects such as those due to the possible exchange of the recoil electron with the electron of the electron pair components, due to the retardation effects at the relativistic velocities of the recoiling
electrons, and due to the influence of each of the four contributing diagrams o f fig. 1. However, as Suh and Bethe 8 state, Votruba's completely differential cross section is so long and complicated that it is difficult to handle; consequently, in order to carry out a general analytic integration, approximations have had to be made which result in a cross section that does not include the contributions f r o m certain values o f the m o m e n t u m transfer, q, adequately. 100~
--'-T
I
"
=
~
T
~
"
r
T '
, ,
•
'
-- ~ r ~
....
I
l
TOTAL CROSS SECTION FOR PAIR PRODUCTION IN THE FIELD OF AN ELECTRON
,c
r~
BETHE - HEITLER BORSELLINO o., ,o
,o
,oo
,ooo
,oooo
PHOTON ENERGY, k ( M e V )
Fig. 2. Comparison of triplet theories. As a consequence, various predictions of the integrated triplet cross section have been uncertain by factors of two at an incoming p h o t o n energy of the order of 10 MeV. For example, fig. 2 compares the predictions of Votruba7), Borsellin09), and BetheHeitler ~) as taken from the review paper of Joseph and Rohrlich1°). Since the integrated triplet cross sections for elements such as carbon are about l/Z o f the integrated pair cross section, the pair production cross section for carbon including the triplet cross section has been uncertain by about 6 ~ depending on the choice o f the theoretical triplet cross section. The uncertainty for an element such as silicon is of the order of 4~o. This section will attempt to demonstrate by experimental and theoretical arguments what seems to be the best prediction of free-electron triplet cross sections. Next to Votruba7), Borsellino 9) has done the most 5) E.g., W. S. C. Williams, An introduction to elementary particles (Academic Press, New York, 1961) p. 227; B. T. Feld, Proc. Conf. on Photon Interactions in the GeV-Energy Range, MIT, 1963, pp. I.l and VIII.1. 6) j. M. Wyckoff, B. Ziegler, H. W. Koch and R. Uhlig, to be published. 7) V. Votruba, Phys. Rev. 73 (1948) 1468; Bul. intern, acad. tch6que sci., Prague 49 (1948) 19. 8) K. S. Suh and H. A. Bethe, Phys. Rev. 115 (1959) 674. 9) A. Borsellino, Helv. Phys. Acta 20 (1947) 136; Nuovo Cimento 4 (1947) 112. 10) j. Joseph and F. Rohrlich, Rev. Mod. Phys. 30 (1958) 354.
ELECTROMAGNETIC
CROSS SECTIONS
FOR ELECTRON
complete calculation by leaving arbitrary the mass, mN, in whose field the pair is produced and by including the retardation effects due to the recoiling nucleus which are important at relativistic energies, especially for recoiling electrons. However, he did not include all of the fig. 1 diagrams or the exchange diagrams that Votruba included. These effects, according to Suh and Bethe s) might not be large at high energies. Borsellino has also neglected screening and binding of the atomic electrons. Suh and Bethe s) have contributed considerably to the understanding of triplet production by analyzing the implications of Borsellino's results. They show by expanding Borsellino's formulas and by extending them that, in the limit of high photon energies and low q values, the magnitude of the recoiling mass, my, does not enter into the distribution. They also show that, to a good approximation, the integrated cross section will be the same for electron and proton recoiling masses, if (a) the photon energy, k ~> ½ MeV, and (b) screening can be neglected. This result should be accurate to order k -1 logk. For intermediate energies (approximately in the range of 5 MeV < k < 100 MeV) it is necessary to avoid the extreme relativistic approximation of a very large photon energy and the approximation of zero recoil energy. Since Borsellino did avoid these approximations, his momenta distributions and the approximate, but simpler, formulas of Suh and Bethe are useful. In addition, the integrated cross sections of Borsellino I 1) as shown on an earlier figure are the only accurate estimates available. However, at photon energies close to the triplet threshold of 2 MeV, the predictions of Votruba are probably more accurate. Therefore, the estimated dashed curve in fig. 2 is probably the best present guess of the integrated triplet cross section at low energies. Justification for the Borsellino integrated triplet cross section at high energies can be obtained from recent experiments t 2) employing hydr ogen-filled b ubble chambers and diffusion cloud chambers which are ideal for an accurate test of the triplet cross section predictions because the ratio of triplets to nuclear pairs in hydrogen is a maximum compared to other elements, because the atomic binding of the electron has little effect on the process and is negligible below a photon energy of 100 MeV, and because the effect of atomic screening has been accurately calculated by Wheeler and Lamb a3) using the exact hydrogen wavefunctions. Therefore, the experiments allow an accurate evaluation of the importance of exchange and radiative corrections which were omitted in Borsellino's calculations. However,
AND NUCLEAR
RESEARCH
201
they do not serve to test an additional source of uncertainty that becomes important at high photon energies: the incoherent scattering function required for the estimates of atomic excitation and ionization as calculated on the Thomas-Fermi atomic model for the low Z elements for which triplet production is important and for which the model is known to be inaccurate. More specific comment on the elements above hydrogen and the screening correction will be made later. The bubble and diffusion chamber measurements allow an examination of the differential cross sections for triplet production. In addition, the integrated cross sections for inelastic pair production (including triplet production) have been measured by the total X-ray attenuation technique in liquid hydrogen attenuators and other elemental attenuators12). The general conclusions from these various experiments, as discussed in detail in the review paper by Motz and Koch t4) are: a) The momenta distribution formulas of Borsellino as reinterpreted by Suh and Bethe for the large q and small q cases are reasonably accurate. There seems to be no good evidence for influence on these momenta distributions of the omission of exchange and the radiative corrections by Borsellino. b) The integrated cross section variation with energy in hydrogen is well accounted for by the Borsellino cross section values when corrected for atomic binding by the Wheeler-Lamb correction 15) which is discussed in the Motz-Koch paper14). 11) T h e specific numerical comparisor.s of the values for pair production in the field o f the n u c l e u s a n d in the field o f a n electron is given in the following table as t a k e n flora reference 9) : k(MeV) 2 2.2 2.6 3 5 10 25 50 t~g/Z2r2o/137 0.32 0.35 0.67 0.89 1.94 3.75 6.4 8.4 Cre/Z2r2o/137 0 0.0044 0.038 0.102 0.627 2.05 4.78 7.15 12) E. L. H a r t , G. Cocconi, V. T. Cocconi and J. M. Sellen, Phys. Rev. 115 (1959) 678; D. C. Gates, R. W. K e n n e y a n d W. P. Swanson, Phys. Rev. 125 (1962) 1310; J. D. A n d e r s o n , R. W. K e n n e y and C. A. M c D o n a l d , Jr., Phys. Rev. 102 (1956) 1626, 1632. 13) j. A. W h e e l e r a n d W. E. L a m b , Phys. Rev. 55 (1939) 858; a n d I01 (1956) 1836. 14) j. Motz a n d H. W. K o c h , to be published. 15) T h e influence of screening for heavier atomic nuclei t h a n h y d r o g e n c a n be included by a p r o c e d u r e suggested by Bethe a n d A s h k i n [E. Segt6, Experimental Nuclear Physics ( J o h n Wiley & Sons, Inc., N e w York, 1953) p. 260] w h o suggested taking the difference of the W h e e l e r - L a m b F o r m u l a with e = ~o a n d with e d e t e r m i n e d by the appropriate value of Z, k a n d E+. If screening is important, the screening correction so obtained c a n be subtracted f r o m the Borsellino free electron formula. T h u s O'triplet = o'Wheeler Lamb - - / O ' p a i r (Z = i) unscreened Bethe-Heitler -}- No" triplet (Z = i) Borsellino Vl. E X P E R I M E N T A L
TECHNIQUES
202
H. Wo K O C H
3. Pair production in the field of the bare nucleus
The simplifications in the calculations of triplet production due to the assumption of an infinite mass recoiling-field and due to the elimination of exchange, are considerable. These simplifications are permissible for the process of electron pair production in the field of a bare nucleus, as first calculated by Bethe and Heitler4). As suggested in the introduction, their theory performed in the Born approximation is very successful .......
2 c
I
........
I
sent the electrons. Their results (curve A) are excellent at very high energies. However, for photon energies below 20 MeV their results as taken from their paper diverge as shown in fig. 3. Based on the general conclusions from this figure various authors have been reluctant to place much reliability in the calculations of Davies et al. at energies below 50 MeV. For many practical applications at lower energies the results of these authors have been assumed to be non-usable, therefore. Davies, Bethe and Maximon have written their Coulomb - corrected, differential- in- positron - energy cross section
t--
I.E --
4 Z 2 r2 d E +
dae+ = 13ff o ~ j - {(E2+ + EZ-)[-4q~l(7) - ½1nZ - / ( Z ) ]
1.2--
- l l n Z - . f ( Z ) ] }.
+ 3E+E-[¼~2(7)
E
(1)
For small photon energies the screening corrections are small and eq. (1) becomes
J "60.8
--
b-
d~re+ -
0.4--
4 Z Z r 2 de+ I (EZ+ + e z2E+E_ ) 137 k3 [ +
( ln2E+~E~-- k ~~ o
1.0
[ IO
I00
f(Z)) }.
(2)
I I IIIII iooo
X-RAY ENERGY (MeV)
Fig. 3. Ratio of cross sections to tr Born for lead.
for those high energies and light nuclei for which the Born approximation is applicable. However, for photon energies less than 50 MeV and for high atomic number nuclei such as lead the Bornapproximation predictions are inaccurate by at least 10~. In order to illustrate this inaccuracy, fig. 3 shows a plot of the ratio of experimentally measured electron pair cross sections in lead to the Born-approximation predictions as a function of photon energyX6). This plot clearly shows that the Born-approximation predictions are about I 0 ~ too high down to energies of the order of 5 MeV. Also drawn on this plot are the Coulomb-corrected results of Davies e t a / . 17) who improved on the plane wave assumptions of Bethe and Heitler by using Sommerfeld-Maue relativistic wavefunctions to repre* q)1(7) and q~2(7) are screening functions with 7----IOOk/E+E_Zk. f ( Z ) = is the C o u l o m b correction function where = 1.2021 (Z/137)2 for low Z = 0.925 (Z/137) 2 for high Z = 0 for uncorrected differential cross section of BetheHeitler. All symbols and constants are based on energies in moc2 units.
If eq. (2) is integrated in the usual manner from k = 0 to E+, the result is O'N
=
Z ro
T37-
In 2k
-
-
.
(3)
At photon energies below 20 MeV, eq. (3) is plotted as the Davies, Bethe, Maximon result in curve A of fig. 3. As can be seen, it diverges logarithmically for small values of k. Unfortunately, the formula w i t h f ( Z ) = 0 also diverges and is also inaccurate as a representation of the Born-approximation values (curve B). Hough in 194818) showed that eq. (3) w i t h f ( Z ) = 0 was inaccurate even though the differential cross section eq. 2 was a reasonably accurate representation of the Born-approximation predictions. He pointed out that eq. (3) could be corrected by an empirical correction term as Z2r~ ( ~ 218 6.451 trN = ~ --lnZk- ~ff-+~k--I ' (4) which gives good estimates of the Born-approximate aN down to 3 MeV. 16) G. W. Grodstein, X-ray attenuation coefficients from 10 keV to 100 MeV, NBS Circular 583 (1957). 17) Davies, Bethe and Maximon, Phys. Rev. 93 (1954) 788 is) p. V. C. Hough, Phys. Rev. 73 (1948) 266.
ELECTROMAGNETIC
CROSS SECTIONS FOR ELECTRON
The basic importance of accurate triplet and nuclear pair cross section predictions for some experimental interpretations can be illustrated with fig. 4 that has plotted the total X-ray attenuation cross section for silicon. The results are taken from ref. 6). Since the total nuclear cross section is less than 5% of the total
218 In 2k - ~ - + 4 k
s (}k
-
2k
+ %
-
}
v)/(z)
203
RESEARCH
4. Total nuclear cross sections
Recently, Maximon ~9) has shown that the Hough correction is almost completely due to an improper integration between the limits of k = 0 to E0 instead of from k = ½ MeV to (E+ -- ½) MeV. I f one applies the Maximon argument to the eq. (1), the result is Z ro aN = T37-
AND NUCLEAR
co
12
[
~
SILICON
. n~ ~
I\
1,0{---~
,
_Z ~ \
After having made this basic improvement in the total cross section, one can then make the theoretical arguments that the remaining corrections have corrections of the order of
.J
~TOTAL ELECTRONICABSORPTION CROSSSECTION
TOTAL ABSORPTION CROSS SECTION
Z) ~
ALL PROCESSES
(~Ldt~O _O j 8p
--
/~./~_
~OMPTON CROSSSECTION~
co - X ~::O.E-- ~
/ ~-
-~'~TOTAL PAIR PRODUCTION CROSS SECTION
l n k (lnk) 2 (lnk) 3 l n k k' k ' k , k2'etc" nk-OA
g
In the evaluation of the results of the experiment of ref. 6), it has been assumed that the remaining correction terms are of a form
<1:02 _ J
,TRIPLET CROSSSECTION
©
AaN =
1-~
+ k (C1 + C2 In k + Ca(In k)3).
I0
The results of this fitting to the experimental data are: C, = - 9 6 6 . 3 barns MeV C2 = +804.3 barns MeV C3 = -125.1 barns MeV k = photon energy in MeV. This result is plotted as curve C in fig. 3 and is seen to be consistent with the earlier experimental data~6). The corrected formula also is a good representation of the iodine data given in ref. ~).
15
20
25
50
INCIDENT PHOTON ENERGY (MeV)
35
40
Fig. 4. C o m p o n e n t cross sections fol silicon,
cross section, it is important to obtain the total electronic cross section, including the triplet and nuclear pair cross section, to an accuracy of the order of ½ or better. By studying a wide range of elements in order to untangle the dependence of the electronic cross sections on atomic number and energy and by using the 19) L. M a x i m o n , private c o m m u n i c a t i o n .
6C
I
5C
,,
,
, I
4C Z
g d 20~
°
°~
°
°.
°
•
5 I
tr,p) ~0
Ill
J
t~,n) m INCIDENT
-~
20 PHOTON
ENERGY
z5 (MeVI
30
35
Fig. 5. Silicon total nuclear cross section. VI. E X P E R I M E N T A L
TECHNIQUES
204
H.W. KOCH
arguments presented in this paper, the total nuclear cross section for silicon has been obtained as given in fig. 5. These results are in excellent agreement in shape with the photoneutron cross sections obtained at Livermore2°). The results also show a large ratio of photoproton cross section to photoneutron cross section (44 millibarns versus 11 millibarns) when compared in magnitude to those of ref. 20). Other interesting data on differential and integrated total nuclear cross sections are given in ref. 6). These data were only made interpretable after some of the electronic cross section results were made consistent and understandable as described in this paper. 20) j. T. Caldwell, R. R. Harvey, R. L. Bramblett and S. C. Fultz, University of California UCRL-7424 (1963).
5. Conclusion The pair production process has been used to illustrate the frustrating and frequently trivial corrections that have been plagueing the practical applications of the results of quantum electrodynamical calculations. Other more subtle effects could also have been described and are under active examination at a number of laboratories. If one can master the complications of electron research analysis, and much of our research program at the National Bureau of Standards is attempting to do so, then the rewards of a well-understood interaction will be available. It will then be possible to fully apply the electromagnetic interaction as a lever in helping to understand other interactions and processes.