Table Of ContentCOPYRIGHTED
BY
HENRY JAMES BOWLDEN
195?.
THE HIGH FREQUENCY END OF THE
BREMSSTRAHLUNG SPECTRUM
BY
HENRY JAMES BOWLDEN
B.A., McMaster University, 1946
A.M., University of Illinois, 1947
THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS. 10IU
URBANA. ILLINOIS
UNIVERSITY OF ILLINOIS
THE GRADUATE SCHOOL
September.15, 1951
I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY
SUPERVISION BY HENRY JAMES BOWLDEN
ENTITLED THE HTGF FREQUENCY END OF THE BREW.SSTRAHLHML
SPECTRUM
BE ACCEPTED* AS FULFILLING THIS PART OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY TN PHYSTOR
CU^rU ^VJLIUJL'JU
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HeadXof Department
Recommendation concurred in|
Committee
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Final Examinationf
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•Subject to successful final examination in the case of the doctorate.
fRequired for doctor's degree but not for master's.
50 0—1-44—27421
TABLE OF CONTENTS
I Introduction
1.1 History and Discussion of the Problem
1.2 The Method of Attack
II Formulation of the Problem
2.1 The Incident Beam
2.2 The Outgoing Electron
2.3 The Matrix Element
III Mathematical Manipulations
3.1 The Space Integration
3.2 Contributions to the Cross-Section
3.3 Evaluation of I
3.4 Special Cases. The "Tail" and the case p
IV Results and Conclusions
4.1 The Formula for the Cross-section
4.2 Results
Appendix A Angular Dependence,
Appendix B The Limit p£ 0
Appendix C Tables of Hypergeometric Functions
iv
PREFACE
This paper contains a calculation of the extreme high-
frequency tip of the bremsstrahlung spectrum that is obtained
by bombarding heavy elements with electrons of extreme relati-
vistic energy. The effects of the Coulomb field on the electron
are treated exactly for both initial and final states, but the
energy of the initial state is taken to be very large compared
to the electron rest energy, Relativistic effects are treated
rigorously. Screening is neglected. The differential and total
cross-sections are obtained numerically, using lead for the
target, for three values of the final electron momentum* namely
0, -g-mc» and mc. It is found that the angular spread of the X-
ray beam is of the order of mc2/El» where Ei is the bombarding
energy, and that the bremsstrahlung intensity has a non-zero
value at the upper limit of the spectrum.
The author is deeply indebted to Professor A. T. Nordsieck
for much valuable guidance and criticism, end for his active
interest in the mathematical difficulties which arose during
the course of the study. He also wishes to thank Dr. M. S.
Watanabe for many stimulating discussions during the final
stages of the work.
INTRODUCTION
1.1 History and Discussion of the Problem
The theoretical determination of the differential and total
cross-sections for the production of bremsstrahlung by the decel
eration of electrons in the Coulomb field of an atomic nucleus
has been carried out satisfactorily only in certain cases." Lue
case in which the bonbarding electrons have non-relativistic
energy has been treated exhaustively. For relativistic bom
barding energies, however, no complete study has been made. The
problem has been solved using the Born approximation for both
the incoming and outgoing electron states.'" This approximation
is probably satisfactory for nuclei of small atomic number, but
is unreliable for nuclei of large atomic number.
Although there exists an accurate solution of the Dirac
equation in a Coulomb field, no solution in closed form has been
obtained which behaves asymptotically like a plane wave at in
finity. Such behavior is necessary for this application, and
the lack of a solution with this asymptotic behavior is the
chief obstacle in the path of an accurate treatment of the pro
blem. In order for such a solution in closed form to be possible,
Dirac's equation in a Coulomb field must be separable in rec
tangular coordinates, or possibly in some other coordinate
system in which one set of coordinate surfaces approximates a
iHeitler, "The Quantum Theory of Radiation," (2nd Edit.; Oxford.
The University Press, 1947) pp. I6lff
Loc. cit.
set of parallel planes at infinity. No coordinate system of this
type has yet been found in which the Dirac equation in a Coulomb
field is separable. The only known closed solution of the Dirac
equation in a Coulomb field is an expression in terms of spherical
harmonics. " Solutions in non-closed form are available,^ usually
in the form of a series of Dirac-Gordon functions. When a solution
of this type is applied to the present problem, however, it is
found that the resulting series expression for the cross-section
either fails to converge or converges so slowly that computations
are impractical.
Recently an approximate solution of the type we need has be-
3
come available. This is an improvement on the Born approxi
mation, expressed in the form of a series of which only the first
two terms are obtained. We find that the contributions from the
second term to the bremsstrahlung cross-section calculated here
are negligible, and it therefore seems reasonable to assume that
the series given is suitable, and in fact very rapidly convergent
when used for this particular purpose. This solution is, however,,
only valid for extreme relativistic energies. This solution
has been applied to the case where the electron has extreme
relativistic energy both before and after the collision,^" and
therefore the shape of the tip of the spectrum, which is of
]W. Gordon, z.f. Phys. M> H (1928); h. Bethe, Hdb. d. Phys. ZJk
(1), 311 ff; J. H. Bartlett and T. A. Welton, P.P.. j£, 281 (1941)
%. H. Furry, P, R. 46, 391 (1934)
•*L. Bess, P.P.. 77, 550 (1950)
4-Loc. Cit.
3
interest for experimental purposes, is yet to be calculated.
The present paper contains a calculation of the cross-
section for extreme relativistic bombarding energy in the case
where the outgoing electron has non-relativistic or moderately
relativistic energy. To this end, the Dirac-Gordon wave function
mentioned above has been used for the outgoing electron, and the
calculations have been carried out for a final S-state and also a
final P-state. It is to be expected that the states of higher
angular momentum will give negligible contributions at the very
tip of the spectrum, (see sec. 4,2 for further discussion of this
point.) For the bombarding electron, the Bess wave function has
been used,
1,2 The Method of Attack. The problem is formulated as a first-
order perturbation problem, since the Coulomb field is already
included in the initial and final wave functions, and the only
perturbing influence is the free radiation field. Our goal is
therefore easily presented, but the achievement thereof leads us
by a tortuous path. The scattering matrix element is written
down in equation 10. Since both the initial and final wave func
tions are quite complicated, the integral in (10) cannot be direct
ly evaluated. We therefore express both the initial and final
wave functions in terms of contour integrals, and then perform the
space integration first. In the process of the work, it becomes
clear that a certain four-parameter set of expressions, written
out in equation IS, will be very important for our work, and a
whole section (sec. 3.2) is devoted to expressing the cross-section
in terms of the members of this set.of quantities, with the purpose
of determining which values of the parameters we will need. In
section 3.3 the general member of this set is reduced to a form
convenient for numerical evaluation.
Early in the work we anticipate the result that the major
contribution to the cross-section will come from a cone whose
angle is of the same order of magnitude as mc2/E-]_, and the
detailed calculations are carried out in this region only. Con
tributions from outside this cone are evaluated roughly as
described in sections 3.4 and 4.1.
5
II FORMULATION OF THE PROBLEM
2.1 The Incident Beam. The Bess wave function may be written in
the following form:
Here and in subsequent work we will use the subscript l(one) to
designate quantities referring to the incoming electron, and 2
for a similar purpose with reference to the outgoing electron.
In all cases p (with the appropriate subscript) will represent
the momentum of the electron divided by H, and e will represent
the energy of the electron divided by lie. ex is the Dirac matrix
vector, asZe2/fiv = (Ze2/ftc) (e/p), u1 is the Dirac free-particle
spinor, and 3F-1 denotes the confluent hypergeometric function.
We shall use the following integral representation for the
confluent hypergeometric functiom
The contour may go to infinity in any direction such that
</?(zt) < 0, the equal sign being allowed if and only if (R{/$) > 1.
The initial wave-function (eq. 1) then takes the form:
v A v
I * ft -*
We shall use the direction of px as the polar axis in subsequent
v/ork, and Q in eq. 3 is the polar angle between ? and p The
1#
normalization factor is given by
where L is the side of the normalization box.
