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Elastic Scattering of high Energy Electrons by Nuclei.

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360
Annalen der Physik
*
7. Folgc
*
Band 16, Heft 7-8
*
1965
Elastic Scattering of high Energy Electrons by Nuclei
By SAILAJANANDA
MUKHERJEEand S. DATTAMAJUMDAR
With 2 Figures
Abstract
DIRAC’S
equation for the scattering of high energy electrons by a modified coulomb potential is solved approximately after converting i t into .a second order equation and a closed
expression for the scattering amplitude is obtained. The expression can be used t o obtain information about the charge-distribution in the nuclei by analysing t h e experimental d a t a
on electron scattering. To test t h e usefulness of t h e approach some preliminary calculations
were done on Au and C nuclei, and t h e results are found t o be satisfactory.
1. Introduction
For more than a decade HOFSTADTER
and his coworkers have been carrying
on experiments on the elastic scattering of electrons to determine the chargdistribution in the nuclei. A fairly complete bibliography of this series of investigation can be obtained in the review article by HOFSTADTER
[l].Some attempts
have also been made t o correlate the experimental data with the solution of
DIRAC’S
equation. Of all the methods so far suggested the only reliable one is, however, the phase shift method of YENNIEet al. [S]. The method is exact but involves a lot of numerical calculation. The approximate methods are generally of
limited accuracy. For high atomic numbers, the results of the first BORN
approximation (briefly, PRA) differ widely from those of trhephase shift analysis. The
defects of the FBA are well-discussed in the literature and are brought t o clear
relief in the case of a uniform or “smoothed uniform” charge distribution. The
F B A in this case yields diffraction zeros which occur neither experimentally nor
in the exact solution of DIRAC’Sequation. SCHIFF[3] tried t o improve the approximation by summing the infinite Born series by the stationary phase method.
Inspite of the elegance of the approach t,he formula arrived a t is not likely t o
possess the accuracy needed for an analysis of the experimental data.
The difficulty in obtaining an accurate approximate formula stems from the
fact that for electron energies of a few hundred Mev the scattering cross-sect,ion
drops by a factor of
as the scattering angle increases from 30’ t o 90”.
It seems difficult to achieve this degree of accuracy by any simple approximation method. A simple approximate formula for the scattering cross-section may
still serve a useful purpose by making possible a rough estimate of the charge distribution and thus acting as a guide t o the accurate phase-shift calculations a t
S. MUKHERJEEarid S.D. MAJTMDAR:Elastic Scattering of high Energy Electrons 362
high energies. In t,he present paper we believe t,o have partly attained t,his goal
by deriving a closed expression for the scattering cross-section which gives better
agreement, with the exact, calculation and is free from the inherent defects of hhe
FBA. A satisfact,ory feature of t,he present t,reatment is that the scattering amplitude is complex (while in the FBA it is always real) and gives diffractmionminima in place of diffraction zeros. I n the determinabion of the nuclear charge distribution we have made a departure from t,he usua.1 procedure. Instead of t,rying
standard models with adjustable parameters we have adopted an analytical
method of building up bhe charge distxihution from the scat,tering dat.a. The
charge distribution, assumed spherical, is represenbed analytically by an esponential times a polynomial in r . The integration difficult,ies are, thereby. largely
removed and a lot of heavy calculation is avoided.
For light nuclei t,he present method yields result,s similar t o those obtained
with the FBA but is still free from the defect,s of t,he latter. The method is specially designed for the actual case of interest, heavy nuclei and high energies of the
incident electron: and essentially consists in solving t.he iteratled DIRACequation
[eqn. ( 2 ) below] by an approximatre procedure. If t,he term on the right hand side
of equation ( 2 ) is put, equal t,o zero we get, a SCHR~DINGER
t,ype equation (3).This
equation can be solved exactly in the rase of a pure coulomb field if t,he Ti2 t,erm
is neglected. When the DE BROGLIE
wave 1engt)hof t'he incident electmn becomes
comparable with t h r nuc1ea.r dimensions, the modification of the coulomb pot.enequation
t8ialinside the nucleus must be taken int,o account. The SCHRODINGER
(3a) then becomes complicated and can be so!ved only approximately. However,
once this is done an approximate solut,ion of DIRAC'Sequation for the problem
can be easily obt8ainetl.By studying the asymptot.ic behavior of this solution it is
found that at)high elect>ronenergies the S C H R ~ D I N Gand
E RD1~.4c-cross-sections
calculated on the basis of eqn. ( 2 ) and (3) respect,ively differ merely by afactor
cos2t9/2,where t9 is t,he scatt,ering angle. For the derivation of the scattering formula a t high energies it, is t,herefore sufficient t.0 consider, instead of the two linked DIRACequations, a single equat,iori of t,hc type ( 3 ) .
I n the case of a point nucleus t,he procedure outlined above leads to t*heSOMMERFELD-MAUE [4] wave function which has been exDensively used for deriving
improved cross-sections for bremsst,rahlung [5]>pair-production and the photoelectric effect [ti]. The scattering cross-section calculated from it is, however, just
what is obt,a,inedfrom the FBA. It is not difficult to understand why no improvement is obtained. I n tlie case of a point charge the t,erm on the right hand side
of eqn. (3) and the Ti2 term on the left hand side should not be treabed as small.
Owing to the singular nature of the potent,ial a t the origin these terms are always
important however large the incident energy may be. Neglect of these terms
means the loss of a substantial contribution to the scattered part. I n fact, the
scattered amplitude is roughly 0 (I/&)
the incident amplitude so that a t high energies any approximate solution is likely t o yield poor scattering results if due
aktention is not paid t o the singular terms in the potential. The nucleus with its
extended charge dist,ribution, however, presents an ideal case for the application
of this method. The potential in this case is finitre a t the origin and nowhere becomes too large. The t8ermsin question may now be considered small over the
entire doma.in, and the approximation, though not good enough for the purpose
for which it' was originally int,ended, ma.y be profit.ably used t o explore t,he det.ails of the charge distribution.
362
Annalen der Physik
*
7. Bolge
*
Band 16, Heft 7-8
*
1965
The scheme of presentation is as follows: I n 8 2 and 8 3 on outline of the
theory is given, and t.he relevant formulae are listed in 8 4. I n § 5 some results
are presented and compared with the experimental and other theoretical results.
For lack of computational facilities we could not carry the numerical calculations
far and had t o stop a t the point where we could convince the reader of the efficacy of the method. Further calculations are in progress and will be reported as
soon as they are finished.
2. DIRAC’Sequation at high energies and the scattering cross-section
DIRAC’Sequation for an electron in an electrostatic field may be written as
(&+ V - p + i a - v ) y = o
(1)
where a and j3 are the DIRACmatrices. Equation (1)is written in natural relativistic units with ti = na = c = 1. Distances are then measured in units of lilmc,
energy in mc2 and momentum in mc ; V is the potential energy. Mult,iplying (1)by
tmheoperator ( E
v
p - ia V)we have
L y E ( 6 2 p2
2&V 7 2 ) y = ia . ( 0 V ) y
(2)
where c2 = p2
1,p being the momentum.
Equation (2) may be solved approximately by treating the term on the right
hand side as a small correction. Dropping this term altogether we have
-+ +
+
+ +
+
Lyo = 0
(3)
Differentiation of this equation gives
LVyo
so t h a t for
+ 2(& + 7) ( 0 V ) y o
=
0
i
( 4)
y1= AYo --a-vYo
the function
w
= vl0
+ y1 satisfies the equation
+
L y = ia. ( V V ) (1 ?‘/&)yo.
This is sufficiently close t o equation (2) and yo + y1 may by taken t o be a good
approximation t o the solution of the latter for V << E . The constant matrix A
occurring in eqn. (4) is arbitrary and should be chosen so as t>oremove from y1
any term proportional t o yo,the solution of the homogeneous equation.
The case of a point-charge was investigated by XOMMERFELD and MAUE.They
neglected the V2 term in (2) and determined yo and y1 by successive approximation. These functions are given by
yg= N e i p T U ( p P
) ( i a & / p 1,
, ipr
-
i p .r)
(‘;I
and
yp = NetPr(- i / 2 e ) [ a . VF ( i a & / p1,
, i p r - i p . r ) ]U ( p )
(7)
where F ( i a e ! p , 1, i p r - ip . r ) is the confluent hypergeometric function, U ( p )
is the DIRACplane wave spinor and N is the normalisation constant. The constant a is given by a = Ze2/lic.
The function I& in (6) is the exact solution of the equation
for V =
+ + 2eV)yo
(02 p 2
0
and is important for the subsequent development.
4
=
13a)
S . MUKHERJEEand S. D. MAJVMDAR:Elastic Scattering of high Energy Electrons
363
We now assume that i t is possible t o write. in the case of a general potential,
yo = U(p)eiP'f(r)
(8)
where f(r)is a funct,ion of tht: space coordinates. Differentiating eqn. (8) we get,
v y 0 = U ( p ) e i p r [ i P f ( r+)W ( r ) l
so that put>tingA
=
we have
- a:;-*,
-E
y1 = - i / 2 & [a. ~ f ( r ) ] e i U
p .( ~
p)
and the complete wave-function
YJ = yo
+ y1 =
[ f ( r )- ( i / " )
. ~ f ( r )U]( p ) .
(9)
I n the asymptotic region the wave-funchion (9) permit,s further simplificat.ion. I n
order that yo should give a spherically outrgoing part, f ( r ) must contain t.erms
which beoomt asymptotically
eiP.'
1
.
- e7pr
,Y
.
~
113 r
r
@(O,y)
where @ is a function of t h e angular variables 0, y onljr. In the case of a coulomb
field there is a n additional t,erm in the exponent, which makes a logarithmic contribution t.0 the phase, thus modifying tthe waves cven a t great, distances. Hence
if asymptotically
1 . , .
(y~o)scatt
+ e z p r t n l l l o ~ l r @ (0. Y ) U ( P )
-8
we h a w
I
(Yl)scatt + - e
r
.i ~ -r,7 ~. n l n 2 2
~7.
')&
( P r^ - P)@ ( 6 (PI 7
I (PI
3
(10)
where r is the unit vector in tshedirectlion of r . Thc terms in (10) contributed by
t,he logarithmic phase term on differentiation vanish asymptotically while the
term containing
(6,q)is negligible compared t o those retained which are proportional t o p @ .
The scattered part o f thr solntion is thus givcn by
1 .
(v)scatt + - e t P r -i- ialn2pr
r
[I
(?IF - - P ) l @ U ( P I .
(11)
v@
+ 2.
This formula holds in the general case. In the case of n point charge the asymptotie behaviour of y: is known and the formula is deduced more simply as follows:
-4s r -+ 00 t,hc scattered part, of f ( r )can he writ.t,ent,hrough terms of order l / r as
when terms of order L
a r e neglected. When t h e w are substituted in cqn. (9) thc
TZ
scattered part of y: takes exactly the form ( 11).
Before we pass on t o a discussion of the equation (3), i t is perhaps desirable
t o obtain an expression for the scattered current. The plane wave DIRACspinor
sat,isfies the equation
(J . P PI "(PI = c o - ( P )
(12)
+
364
Annalrn drr Physik
*
7. Folge
*
Band 16, Heft 7-8
*
1965
so that the positive energy solution may be written as
where u is an arbitrary two component spinor. The scattered current is given b y
J = e y,aatt a yscntt
= e I@lZ
+ g.( P i .
U" (PI11
- PI1 f3 [1
+ 5 (Pi-P)l
*
U(p).
We note that
(a.B)a=B-i(Bxa)
&(a * B ) = B
+ i ( B x a)
and
(a.B ) &(or. B ) = 2 B ( n . B ) - B2a
where B is a vector which commutes with the a*.Using these formulae a n d normalising the incident current t o unit flux, we have
J = e l @ I 2 ( p / &- p3/&3sin28/2)2
where 6 is the angle of scattering. This reduces t o
J
=
e l@lz cos2 612. i
(14)
when the rest mass is neglected in comparison with the Kinetic energy. The factor cos2 612 is also obtained in BORN'S
approximation by trace calculation.
3. Determination of !Po
The integral equation equivalent t o cqn. ( 3 a) is
(15)
To solve this we substituh tfhe function (6) for yo in the right hand side arid
the replace the kernel by its approximate form for r
r'. The function (6) is a
good choice since the field outside the nuclear charge cloud is still coulombic and
t,he function (6) is a n exact solution of (15)for a pure coulomb field. The function
deviates little from a plane wave near the origin and tends t o a plane wave for
vanishing at,omic number. It accounts properly for the modifications in phase
and amplitude caused by the long range coulomb field and is really a much bett8erfunction t o start with.
The scattered part of yo now becomes, after omission of the primes
where q = p - pF is the momentum transfer. An analytical evaluation of t h e
integral (16) requires that V ( r ) be given as a simple function of r. Most of the
standard models for the nuclear charge-distributions, viz. FERMI,
Gaussian, harmonic-shell etc. do not fulfil this requirement; the potential in such cases are
usually evaluated numerically. We, therefore, do not consider these models a s
such in the present theory and instead represent t,he potential b y an expression
S . MUKHERJEE:and S. I).~IIAJVNDAR:Elastic Scattering of high Energy Electrons 365
of t,he type
where thc pariLmet,erx x , . n , . 6 , ctc. are conncctccl by t h r rclation.
which ensurcs charge normalisation ancl thtb rclatioii x u , = - 1 which leads t o
the cancellat,ion of the singularity at t~hcorigin. The potential ( I i )can br obtained from a charge distribution
The knowledge obtained so far of t,he nuclear c h a r g ~distribut'ion indicates t h a t
t,he expansion (19)can be uscd with atfvantagc. For accuracy a few trrinu involving relatively high powers of r might be necessary but, t,heir inclusion does not,
require any special t,reat,ment. For t,he light, nuclei ha,rmonic:shell t,ype of distribution gives t h e best, f i t wit,h the rxperimentd data. Some five or six parameters
are found tlo be sufficient, for reproducing t,he dist,ribut,ionaccurately. The niodified exponent,ial modrl h a s already been tried by EHRENBERG
and ot,hers [7] for
016 and by FREGEAU
[PI for el2.The inadequacy of t,his model. as noticed by
hhem seems to bc due t o the restricted typc of the rxpression they used for the
charge distribution. For heavier nuclei FERMI
clist,ribut,ion is the accepted model.
An accurate representsatmion
of the plateau in this ca,se by an expansion of the t8ype
(19) may require a gootl number of para.met,ers.If fewer parameters are taken the
distribution curve may not be smooth t,hough all it,s important feat.ures will still
be revealed. The number of paramet,ers in t,his met'hod, however, presents no
serious difficulty. On the contrary, it is bet,ter t'o st.art8with an adequate number
of them, so as t,o obt'ain as detailed a knowledge of t.he distribution as is permit'ted by the rxperiniental data.
The expression ( 17) for t,he potjentlia.lhas many advant,ages. For an appropriate choice of tJhe , x i ' s . a11 tfhr integrals iii ( l t i ) can be el-aluated once for all. A
comparison with the cxperirnental data t8hengives a set, of linear equat,ions for
the determinabion of t,hr paramet,ers a j , b i etc. This enables us t o see cleaily t'he
of the charge distributJion
dependence of the scat,t,eringresults on toheparan~rt~ers
and also makcs it possible t o explore models which do not fall i1it.o t8hest,andard
categories.
4. The scattering amplitude
All the integrals occurring in t,he relat8ion(10) can be derived from
1, can be cvaluat(ec1by the method of rrsidues (see ,4ppendix) ancl i s found t o
have the value
('21)
366
Arinalen der Physik
*
7. Folge
*
*
Band 16, Heft 7-8
1965
The value of the integral:
is obtained by differentiating 1, n-times with respect to -A :
('22)
The integral with the pure coulomb term V ( r ) =
aneeds
separate considera-
tions. It may be evaluated by allowing 2 t o approach zero in the relat.ion (21).
This gives
e- nu12
I , =eiulnbin28/2
7 4ialn2pa
P
and
where we have substituted
DI =
and N =r(l- ia)enu12 thus normalising
the incident beam t o unit flux. It is interesting t o see that but for the spinor
part and a numerical constant l/r(l ; a ) this is identical with the non-relavistic solution for a point charge. Also for low atomic numbers one gets back Mottscattering results as expected. As DI -+00 the factor eialn2pa contributes an indeterminate phase and corresponds t o the logarithmic phase-factor eialn2pr of
the coulomb field. I n the case of a point charge this phase factor drops out of the
scatt,ering cross-section and causes no difficulty. This does not happen when the
+
potential contains additional short-range terms falling faster than
, and t o
determine @ (0) in the latter case we proceed as follows:
We note that all distributions are practically of finite extent. It is, therefore,
possible t o speak of a cut-off radius R of the chargedistribution, so t h a t the field
is purely coulombic for r > R and includes additional short-range terms for
r < R. The scattered wave may be obtained by matching the external and the
internal solutions a t r = R. Taking the asymptotic from of the wave-functions
for r = R and fitting the spherically outgoing parts across the boundary, we
have
fC(e)$ e i p R + i a l n 2 ~ B + f,(o), 1 eapR = @ ( ( j ~ e i p R + i a l % Z ~ B
R
(24)
where f c ( 0 ) and f , ( O ) are the amplitudes for the coulomb and the short-range
terms and @ ( O ) is the resultant amplitude. Equation (24) gives
@ ( e ) = f c ( 0 ) + f,(@
(23)
The charge-distribution can also be terminated effectively by the introduction of
a screening factor e-rl.8 where 01, i s sufficiently large. Examination of the relation (23) reveals t h a t whenever the condition a s p
1 is satisfied, the equation
(25) can be rewritten a s
@ ( O ) = f c ( 0 ) f , ( 0 ) e--iulnzpns.
(26)
If the nuclear charge-distribution is sufficiently spread out or if the electronenergy is sufficiently high, DI,may be put equal to olg, where 0 1 ~is the greatest of
ecialn-p R .
>
+
367
S. MUHHERJEEand S. D. MAJUMDAR:
Elastic Scattering of high Energy Electrons
>
the parameters a, of the charge distribution. But in the cases where asp 1 is
not satisfied, the screening factor a,is t o be chosen and it is better to treat i t a s
a free parameter t o be determined by comparison with the experimental data.
However, a,will be very close t o aswith x , = agas the lower limit. This complication is a consequence of the long range nature of the coulomb field.
5. Numerical results
Some simple calculation were carried out, on Gold and Carbon nuclei to test
the usefulness of the method and t,o ascertain, the order of approximation
involved. The results are givm below.
G o 1d : For simplicity, a pure exponent>ialdistribution
-
("7)
,o (1.) = p O e - 7 / a with x p = 9 1
was chosen. The electron energy was taken t80be 150 MeV. The scattering crosssection in this case can be written in the closcd form
G(e) = l a 2
cos2
o/;
(28)
where
1
30"
40d 50"
I
L
60" 70" 80"
Scattering angle in degrees
Scattering angle i7 degrees
Fig. 2
Fig. 1
Fig. 1. Cross-sect,ions at 150 Mev for scattering by gold for the exponential distribution with
"1)
= . 91
Fig. 2. Cross-sections for 420 Mev electrons scattered from C12. The solid curve is obtained
by the present method forthedistribution (30).The dotted curve gives the BORN'S
approximation results. The experimental points are taken from reference [lo]
368
Annalen der Physik
*
7. Folge
*
Band lG, Heft 7-8
*
19G5
The results obtained with a g p = 1.07 were compared with the phase-shift calet al. 191. The agreement may be considered as satisfa,ctory
culations of TENNIE
though the phase-shift curve is a bit steeper (Fig. 1).
C a r b o n : We chose, for simplicity, a charge-distribution
e ( r ) = * 2 6 0 0 e - e r -k e-1.25r (--.2600
-7233r - -1S1Sr2)
(30)
+
which starts with a zero at, the origin. The scattering cross-section for 420 MeV.
elect,rons obtained by the present method (with a, =.9) as wellas by BORN'Smethod are presented in Fig. 2. The agreement with the experimental results of
H O F S T A D T ESOBOTTKA
R ~ ~ ~ [ l o ] is reasonablygoodbut leaves scope for further improvement. This is likely since the charge distribution (30) deviates slightly from
the harmonic shell distribution determined for Carbon. Inclusion of a term proportional t o r in the first part of (30) is expected to give a much more satisfactory
agreement with the experimental data.
6. Appendix
An outline of the method for evaluating I , is given below. This follows closely
t,he method developed by NORDSIECK
[ll]for evaluating a class of integrals connected with the problems of pair production and bremsstrahlung.
The confluent hypergeometric function in I, is represented as
t8heclosed contour encircles the points 0 and 1 once anticlockwise. At the point
where the contour cuts the real axis t o the right, of 1, arg. t and arg. (t - 1)are
both zero.
On substituting this in I,, we may interchange the order of space- and contour-integrations and carry out the space-integration first provided the contour
can be chosen to satisfy everywhere
--I - 2 p I m ( t ) < 0.
(A 2)
The space-integral part can easily be evaluated for real t and the results can
be continued analytically for complex t , whenever the condition (A 2 ) holds. After
all this is done, we have
The int,egrand has thus a simple pole a t
Since
2pIm(to)=
-
+
+
-I P2P2 12& < - 1
IP *412 A2P2
unlessqllp when there is no scattering, we see that to must lie outside the contour.
The integral can now be evaluated by expanding the contour t o infinity and considering only the contribution from the pole a t to. This gives
4n (to)'a - 1 (to - 1)- ia (2.pq
+ 2ilg1-l
S. A~UKHERJEE and S. D. MAJUMDAR:Elastic Scattering of high Energy Electrons
It may also he shown that for A sinall hut positivc, arg. to
+ ()(A). b o that,
-
369
arg. (to - 1) =
7c
References
Rev. Nucl. Sci. i (1957) 231.
[l] HOFSTADTER. R.,
r 2 i 'I-ENNIE. D. K . . RAVENHALL. D. G. and WILSON, K.s.,
Phhvs. RW. 9.5 (1964) 5no.
i3j SCHIFF,L. I., Phys. Rev. 108 (1956) 443.
r-il SOMMERFELD.
A. and MAUE.A. W.. Ann. Phvsik 92 (1935) 6299; M. E. ROSE. Relativistic Electron Theory (John Wiley & Sons, Inc.). p. 237.
[5] BETHE,
H. A. and M ~ Y I M O NL., C., Phys. Rev. 93 (1954) 768.
[GI BANERJEE,
H.. Nuovo Cimento, Vol. 10, No: 5, (1958) 863.
[7] EHRENBERG, H. F., and others - Phys. Rev. 113. (1959) 6(X.
[8] F R E G E A
J.~H, . , Phys. Rev. 104 (1956) 28.5
[9] See reference 2, pp 505
[lo] See referenw 1, pp. '244.
[ll] SORDSIEC'K.-1..Phyh. Rev. 93 (1954) 783.
L
A
C a l c u t t a , Department of Physics. Universiby College of Science.
K h a r a g p u r . Department. of Physics. Indian Institute of Technology.
h i tler Hedaktion eingegangen a m 13.J u l i 1965.
'25 Ann. I'hysik.
7. Folge. Rd. l ( i
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