Relativistic effects in nucleon -nucleon scattering

60 Pages·04.171 MB·English

by  Macy, Spencer

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HXLAZXVXrnC IFFSOfS
III NUCUftN-WGUBON SCATTSRllfC
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Bpmmr Macy
A Pl»®«rt&tIo*i Submitted to the
Graduate faculty In Partial fulfillment of
The Requirements for the Degree of
DOCTOR OF PHILOSOPHY
Major Subjects Theoretical Physics
Approved:
of wajor WSrK
/mm~ot Sajor:^partaiIt:
J r n/f ^04^L/7C&v}
lean or OrKuate OoSeg®
Iowa state College
I960
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UMI Number: DP12035
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TABLE OF CONTENTS
I. INTRODUCTION ......... 1
II. RELATIVXSTIC CORRECTIONS TO THE WAVS
EQUATION ................... . . . 5
A..Unite and notation . . . . . . . 5
B. The Potential Function ........ 9
0. The Wave Equation . . . . . . . 15
D. The Equation for the large
Component . . . . . . . . . . 15
III. THE SCATTERING CROSS SECTION . . . . . . 22
A. derivation of the Formulae ... 22
B. The Variational Principle for
the Phase Shifts ......... 55
IV. APPLICATION TO A SCATTERING FUQBLS!. * . 58
?. summary. . . . . . . . . . . . . . . . . 48
VI* ACKNOWLEDGMENTS. . ................ 50
VII. APPENDIX . . . . . . . . . . . . . . . . 51
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I, INTRODUCTION
The recent availability of high energy nucleon sources
has made It possible to obtain data on nucleon-nueleon scat­
tering in the 100-400 Mev range. A great number of experi-
mental papers* have appeared giving 'the differential and
total scattering cross sections for particular case® in
which, the scattered particle has an energy in this range.
The most striking feature of these experiments 1® the appar­
ent symmetry about 90° in the center -of mass system of the
angular distribution of the differential cross section for
neutron-proton scattering, this has led ierber^ to suggest
a type of exchange force which acts only in even states.
This type of exchange fore# lead# to considerable simpli­
fication of the present work and will be used In this paper.
Along with the experimental work have come theoretical
papers attempting to explain and interpret the experimental
results®. Most of these papers use the Sehrbdinger equation
^Soae recent papers are 0. Chamberlain and C. Wiegand,
Phys. Rev., |£, 81 (1950); Kelly, Leith, Segre, and Wiegand,
Phys. Rev., W, 96 (1950); Hadley, Kelly, Leith, Segre, and
fork, Phys. Rev., 75, 351 (1949); Sruekner, Hartsough,
Hayward,' and PowelTT Phys. Rev., JjJ»* 655 (1949).
%#« for example?. J. Ashkin and T. f. fu, Phye. Rev.,
73, 972 (1948).
%or a summary of recent paper® see: L. Rosenfeld,
Waolaar Forces {Intersclence Publishers, Inc., 1948),
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with a potential function chosen to fit the experimental
data, flit ranges and well depths of the potential functions
are derived f m the low energy data* If tensor forces are
agauaied, they are chosen to give the correct electric quad- .
impel#'moment.of the deuteroa. One of the most recent
attempts of this Mind is that of Christian and Hart4. they
w e the Serfeer potential with various combinations of central
and tensor for©#s. Using a radial, dependence of the Yukawa,
•xpo&antlol or square well type, they find a total cross
section at least 10$ larger than the experimental cross
section for an energy of about 80 lev. fhie indicate® that
if the phonowmological description of the neutron-proton
interaction in tews of a potential is to he applied to
scattering in the region of 100 lev, the potential should
he chosen so that the relativist!© corrections to he applied
would decrease the cross section by about 10$.
In order to deal with the relativist!© corrections,
there are two approach® that might he used, fhe first
approach, the most satisfactory theoretically, is to use
meson field theory. However, at the'present time there is
no, theory which account® for all the phenomena to b# explained,
fhe problem Is somewhat the same as that of selecting a po­
tential In the ordinary Sehrodlnger theory, in that there art
4a. s. Christian and 1. W, hart, fhya* lev., 77, 441 (1950).
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several types of meson fields to choose from. Marty® chose
a combination of psewfiosoalar and vector fields. With, this
combination he found too large a total cross section* and
the relativist!© terms increased it. jyuofon6 took pseudo-
scalar' coupling with exchange of tooth charged 'and' neutral
mesons to aoeount for the symmetry about' 90°. With this
choice he found the relativlstie terms decreased the cross
section toy 5-10$. ' Snyder' and Marshak7 made a calculation
using the Miller method® on the scalar and vector theories,
fhe correction to the scalar erosa section was small, while
the vector cross section was increased toy about 10$. fhus
It is seen that both the amount and direction of the correc­
tion depend upon the 'particular theory.
fhe second approach, the one used' in this paper, is
to apply the ordinary Dirac theory, choosing a potential
function to express the interaction between the particle®.
Thus the nucleon is treated as a Dirac particle with spin &.
fhe second order corrections' to the Dirac equation describing
the system can toe found and the result® used to find the
SC. Marty, nature JL§5, 361 (1950)1.
®P. 1. Almfm, Phys. lev. 75, 1773 (1949)1.
7H. Snyder and a. I. Marshak, Phys. lev, 72, 1253
(1§47)1, "
'8G. teller, Salt. f. Jhysik 70, 786 (1931).
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change in the cross section. Since the only truly .relatlv-
Istlo potential that eaa he chosen has .the S -function as
the "radial dependence9* a correction mmt he applied to
the laniltoaian. when a finite range Is.used. This has been
10
done for-certain potential* by Brett' * The analysis, of
this ■ additional torn will follow closely that.given by
Brett. :
9S. leaner, Belv* Phys. Acta 10* 4? (193?).
10£K Brett,■ Phys, lev. Zm 11936).
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^
■KW
II. RSLATIVISTXQ COHEECflOMS TO fllE WAVS I^UATIOU
A. Gait® and Rotation
The units and notation In this paper are■thoso■used
•j
In Rosenfeld . Mass and..momentum are expressed in energy
2
units so that M stands for M e , and p stands for p©» This
means that Planch*® -constant becomes fi » ho, and the orbital
angular momentum h • t x § will contain the velocity of
light c. Thee© mite are very convenient when we are dealing
with the Dirac equation,' because the velocity of light then
does not explicitly appear.
Rosenfeld introduces the concept of a ^dichotomic* ■
variable to treat the case in which w# must distinguish
between two possible states ©f a system. Two well-known
examples of such a variable are the ordinary spin and
the isotopie spin r . If we let s » §h<r, then the eigen­
values of <r are *1 and -1, corresponding to the two pos­
sible orientations of the spin. In the same way, the two
eigenvalues of fhe isotopic spin correspond to the two
states- of different charge.
We can think of the spin <r as being the projection of
a vector f on a unit vector I in a ©artesian coordinate
system. The components of 3r ©an be taken as 2x2 matrices.
^Rosenfeld, op. elt.
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