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Coulomb scattering of dirac particles
ANN;\LS
OY
27, 338-361
I’ILYSIC’H:
Coulomb
(1964)
Scattering
of Dirac
Particles*
The continuum Coulomb wave functiun in the Johnson and Deck forn--a. nlatrix operator acting on a plane wave spinor-is discussed. (;enerul relat,il,ns for the elastic diRerrnt,ial cross section, the asymnretry parameters, and the precession and nutation of polarization are developed. A11 expansion of the asymptotic wave function corrert t,o third order in X = LYZ for all Born pnraneters Y = aZ/@ is ~~l)t:tined. This expansion involves a two paranleter function T(o, V) which in turn may he expanded for small Y to give results previous11 uht.xined 1)~ Johnson, Weber, and Mullin. On the other hand, a large Y asymptotic expansion of 7'(0, V) valid for 1 2av ! > 1 permits a discussion of the nonrelativist,ic limit. Both the small and large apprnsinrations suhstsnt,ially agree for 1 v 1 1. 1;. Consequently, these t,wo approximations may be used to span the whule range of Born parameters. The asymptotic Born parameter approximation predicts markedly different scattering behavior for electrons and positrons. 9 striking feature is that functions pertaining t,o electron srattering are oscillatory in the scattering angle 8 while t,he positron functions arc not. In particular, the oscillations in the nsymmetr)paranreter for electron scatt,ering obtained numerically 1)~ Sherman are cont,ained in the analytical approximate form. Also. the approach to the Rutherford cross section is different t,hun that quot,ed 1)s illott and Massey.
A.
l’KELIMIS;\ItT
kmmss
The Coulomb scattering wave function for a I)irac electron has been given exactly by ;1Zott (1, $2) in terms of an infinite series of angular momentum eigenfunctions, constructed so that at large distances the wave function behaves like a dist#orted plane wave plus an outgoing spherical wave. The scat,tering wave function depends upon the Coulomb interaction by means of two parameters, namely, the potential strength parameter X = aZ and the dynamical Born paramet’er Y = aZl?,lp, where E and p are the energy and momentum of the incident particle. lcor certain ranges of X and V, the infinite Mott series has been summed numerically (3-6) for large distances I’. These numerical sums also depend on the * Work
was performed
in the Ames
Laboratory 33x
of the U. S. Atomic
Energy
Commission.
s:attering angle so it is easily seen that a full multiparameter table of numerical values of the wave function is not practicable. It is desirable therefore to develop apptwimations involving a finite number of l~nown tabulated fuuctiolw Approximations of the infinit,e .\Iott, series that have been developed may be characterized as expansions of the exact wave function in certain orders of the parameters x and V. liar esample, t,he Born approximation involves an erpansioll in powers of x and @ = p ‘I!: ‘v 1. The Sommerfeld-Maue (7, 8) approximation is the exact’ wave function expanded to first order in h mit#h v being considered as an independent parameter. 12 modification (9--l 1 ) of the Sommcrfeltl-~laue approsimat,ion, which is equivalent to the third order 13orn approximation, consists of the exact wave function expanded to t’hird order in x where agaiu v is considered a function of X and @ = p’l? ‘v 1. This is a good approsimation even for large Z as long as v is small. :\ll of these approsimatiolls alp limited by the condition that v must be small. Recently, Johnson and Deck ( 12) have reorganized the exact scattering wave function in the convenient form of a matrix operator acting 011 a plane wave spinor which corresponds to the asymptotic incident wave. 111the present paper, some of the consequences of this form are investigated. These include the formulation of t’he scattering problem, relationships among the scatt’ering functions, and the effect of scattering on the polarization. Starting with the exact wave function in the .Johnson-Deck form, an approximation ia developed based on an expansion in powers of x through t.hird order for arbitrary -\~alues of V. This approximation involves a two parameter functiou T(O, v) which is given by an infinite sum represent’ation or alternatively by an integral represeiit~ation. Icor small L’, a Taylor’s expansion allows the sum to be performed in terms of known functions. This small v expansion leads to results which are iti complete agreement, with .Johnson et ul. (9). l”or 12av 1 > 1, an asymptotic expansion of 7’( 8, V) obtained from the integral representation provides an approximation fol T(0, V) in terms of kuo~n funct,iotrs. The small v and large v expansions give results in substantial agreement for 1 v 1 2 1,;. C’onsequently, these approsimations may be used to span t’he whole range of Horn parameters. The asymptotic approsimation for large 1u 1 is used to evaluate the e1ast.k Coulomb scattering cross section and the polarization functions. l;or the case of clcctron scattering ( attractive potential), oscillations as a function of 0 are predicted in the different’ial cross section and the asymmetry functiw. In part,icular, the oscillat.ions in the asymmetry paramet,er S obtained in the numerical calculations of Sherman (6) are contained in the analytical form presented in this paper. Also, for large 1v j which corresponds to the nonrelativistic limit, it is found that, the correction to the Rutherford differential cross section is different frnm that, c+mted by Alott and Alassey i 13).
340 B. (fi
FR.4IKIN,
WEBER,
AA-D
H.-\MMEH
particle
in a Coulomb
NOTATIOS
The Hamiltonian = 777 = c = 1)
for a Dirac
H = --ia.V
field is taken
to he
+ fl - X,/r,
where x = (YZ, O( being the fine structure constant and Z being the (shielded) atomic number. For h > 0, the potential is attractive. The symbols (Y, /3, and d represent the usual Dirac matrices.’ The symbols A* and A+ denote the complex and Hermitian conjugate of any matrix 9, respectively. The caret fi indicates a unit vector, and the vector cross product is given typically by the notation a A b. Quantities which describe the scattering process are: the asymptotic momentum p = [E” + I]““; the scattering angle 0 given by cos 0 = $. f; the Born parameter v = kE/p; the distorted plane wave phase factor 6, = -v In{ pr( 1 cos 0)) ; the distorted outgoing spherical wave phase factor 6, = a/2 2 argr(1 + iv) + v ln(pl.(l - cos 0)). In Section III, the notation x = -22ip~, y = [I? - X2]“‘, a = y - iv, and b = 27 + 1 is employed. II.
A.
CROSS
POTENTIAL
SCATTERING
FORhllILATIOX
SECTION
The Johnson-Deck
(1.2) form for the exact wave function
is given by
\k(E, I,) = D(,E, ;, $1) C( s1 , $1) where the operator D(E,
(2.1)
D is defined by ;, $1) = G + iUlla~ ($1 -
P) + iL$, f, i.6.
(2.2)
Here, $I is a unit vector in the direction of the incident plane wave, U(sl , ljl) is a free particle plane wave spinor for positive energy E and arbitrary polarization direction given by the unit vector s1 , and G, M, L are functions independent of s1 , which are given explicitly in Eqs. (3.1 ))(3.3). The plane wave spinor T:(sl , ljl) satisfies the equation [o(~l)~s,]c~T(sl)
$1)
where the three vector polarization O(fA) The asymptotic
form
1 See, for example,
ref.
=
operator
= ?d + d.$l(l
of the Johnson-Deck
14.
t7cs1,
(2.3)
$11,
(16) O(&)
is given by (2.4)
- S)$l. wave
function,
required
for a
discussion
of elastic scattering
is
‘li(lC:, r) = ei(pl~r+*p)l~(sl,
&) + r-1ei(P’+6s’Ds(jr
$,)CY(s,,
PI).
(2.5)
Here D,( ;, $1) = G,? + iXdl.&~ ($1 -
P) + iL,$, A F.d,
( 2.6)
where G, , A/, , and L, are obtained from the asymptotic form of G’, M, and L, and the energy dependence of 11, has been suppressed. 12syn~ptotically, the outgoing scattered wave of momentum p$3= pF
( 2.7 )
itself must be proportional to a plane wave spinor (of positive energy E) in a definite polarization state. Consequently, D.&r?, , $,)G’(sl,
(2.8)
$5) = c(O)U(sn, $2)
where c(B) is the proportionality constant., and ~12is a definite polarization direction related to s1that will be discussedin Section 11,B. If the scattered wave is analyzed not for polarization direction ~~1, but is analyzed for arbitrary direction of polarization S? , then the component of the scattered wave relevant to the scattering processis (‘(ss ) $2 ; Sl ) @l)U(S? ) $2)
(2.9)
where (‘(s, , $2 ; s1, $1) = 7’+( s:!, &)D,C& , Ij,)l:(sl
, $1).
(2.10)
l’or the scattering from initial state characterized by s1, ljl , to the final state characterized by sg, $2 , it follows that the differential cross section is given by du(s2,
fi2 ; sl,
13,)idti = 1(I(,s~, fh ; s1 , fll) j’.
(2.11)
The derivation uses the fact that the matrix element of t)he current operator a.$ is given by I:+@., $)a.$l:(s, $) = p/E. (2.12) The matrix element C’ of Eq. ( 2.11) may be expressedsimply in terms of the functions appearing in the asymptotic Johnson-Deck form. If one defines free particle Hamiltonians H, by H, = ‘y.pn + P;
11= 1, 2,
(2.13)
and usesthe identities cr.(pl - ~2) = HI - Hz ip, A p2.d = ~1.~2 + P;
+ II,/3 - 1 - H-H,,
(2.1.5)
then it is found that C(S,, j% ; sl, Ij,) = I’+(+, Here A1 and AZ are functions
lj2)[ill
of 0, (fll.&
$,j.
(2.16)
= cos e), given by
Al = G, + 2L,[c020,‘2 A, = QpEL,
+ pA2]C:(sl,
-
(Mp)2] (2.17)
.
The square of the matrix element involved in the differential cross section is evaluated by inserting appropriate projection operators and using standard trace techniques. The result, in a form closely related to that given by Tolhoek (l/S’), is 2d~(s~, & ; sl, &)/dQ
= ([I + (n.sl)(n.sz)]lI
- [(n.sI)
+ (n.s2)]I,
+ (n.s2 A ~1113 + [(n A sd. (n A s2)lLl.
(2.18)
Here n, the normal direction to the scattering plane, is given by n = (~2 A &):/I p? A pl I
(2.19)
and the four I functions, which are real functions of 8 but independent of polarization directions, are given by the formulas: I1 = [I -
(p!f~)‘)“sin’8/2]~4~ill* + [l - (p~E’)“co&9/2]dsA2* + (Z,!E)Re Al&*;
I2 = [( p/E’)kn
0]Im AISZ*;
(2.20) (2.21)
Z3 = f~~(p/E’)‘sin Oi[l + (E - l)(E
+ 1)-l cos e]il,A,*
- [I - (E - ‘l)(E
+ l)~‘cos e].4aAz*
- [2(E - l)(E
(2.22)
+ I)-‘cos 8]Red1,4?*1;
I4 = (E-‘sin% + cos ff[cos 8 + (p/li:)“sin’?0/2] IS1i41* + (K’sin’B + cos 8[cos19- (p,:~)‘cos’ej2])dpA~*
(2.23)
+ [2r(:-’ + (h’ - l)“E-‘sinSB]ReAl.4?*. These four functions are not independent but are related through the equation II? = 122+ Is2 + IJ’),
(2.24)
which may be verified by direct substitution. The functions r(0) given by Eqs. (3.20)-(X23) may be determined directly from scattering experiments. For a single scattering with an initially unpolarized beam, the differential cross section dar/dQ is obtained from Eq. (2.18) by summing over final polarization directions and averaging over the initial polari-
COULOMB
zation directions.
SChTTERTX(:
OF
Dllt.i(’
PARTICLES
3-13
This procedure yields dar/dQ = II(e).
(2.25)
If there is a second scattering (shielded from the direct incident beam) a long distance 1.~2from the first scatterer, located at a polar direction 0 with respect to the incident beam direction fil , then the beam incident on the second scatterer is a plane wave ( i.~2)-‘eiB”” Ds(& , &) l’($, , sl). Here, the prime signifies distance measured with respect to the second scatterer. Substituting this incident wave into Eq. (2.~3) and foIlowixig the same steps leading to ICq. (2.12), it is found that the differential cross section for scattering from initial state characterized by s1 , 13~, to a final state, analyzed after the second scattering, characterized by s1 , 13~, is given by
where C’Cs:! , lj3 ; $2 ; s1 , $1) = L”( $4 ) $a)L&i$a ) fi?)l)&&
) $,)U(Sl
) $1).
(2.27)
Inserting a sum over the complete set of states I -, (~12 , $3) CTi+(~s12, $2) between the txo 1),$‘s, where 6 sunis over f and i sums over positive and negative energy spinors, it is found t.hat the sum reduces to one term with the result C(s3 ) $5 ; $2 ; Sl ) $1) = cI(s.3 , p, ; Sl? ) ~~)C(S12 ) p2 ; Sl ) $1). The C’s on the right are defined by Eq. (2.10). one obtains rJuis:i , /A ; $2 ; SI , 1;1)-=
1 __ (l.lZ)')
&
ddSa,&
Consecluently,
;s12,@2)
using Eq. (2.12)
dds1n,fi2;
dfl'
(2.28)
s1,131) dll
("',9) .d.d
,
The differential cross section rL~~~/dti’ for double scattering from an initially unpolarized beam is obtained by averaging over initial polarization directions ( +sl) and summing over final polarization directions ( &se). The calculations are simplified by using the fact (proven in Section 11,H) that if s1 is chosen as n = (p2 A pl)/j p2 A pI 1, then sle is also n. Substituting I2q. (2.18) into l31. (2.39) and performing the sums and averaging, one obtains the simple result (r12)2d~II fdd = 1l(O)11(O’)
+ (n.n’)1,(0)1,(0’),
( 2.30 )
where cos0 = pl.& , cos 0’ = &.j&, n’ = (p3 A p?),:l pa A p:! 1. It follows 1\Iott’s (1) definition of the asymmetry, 6(0, O’), is given by s(e, and the asymmetry
e’)
parameter
=
(2.31)
r,(e)r,(e’)ll,(e)r,(e’),
o(e) defined
that
by Sherman
(6)
is the angular
344
FRhL)KIN,
WEBER,
AND
HAMMER
function s(e)
(2.32)
= r,(e)lllcs).
Double scattering may also be analyzed with the formalism of density matrices, as is done, for example, by Fano (17) and Rose (18), which yields identical results. Similarly, information regarding both I,(e) and 14( 0) may be obtained from a triple scattering experiment. If an initially unpolarized beam propagating in the $1 direction is scattered through consecutive intermediate direct’ions $2 and $3 into a final direction fi4 , and the final beam is not analyzed for polarization, then the differential cross section for this process is given by duIII/dd
= ~~~12~2~)-2~l~e12)rl~e2,)rl~e,~)
x [l + (n43-n32j6(ea3, ez2) + jnx2~n21P(832, e2d
(2.33)
+ (.n43.n32)(n21.n32)~(Bra , e2d + (n132.n43A n21)s(e43, e21)Ia~e32)/L0b) + (n32 A n43>. (n32 A n21)6(e43, e2,)r,(e,,)!)rl(e,,)1. Here, cos eij = $i.$j, nZi = (pi A pj)/( pi A pj /, and rij is the distance tween the ith and jth scatterers. B.
I-'oL~~RI~AT~o~
OF
THE
SC.~TTERED
be-
WA4w
Since a wave scattered in a given direction is asymptotically proportional to a plane wave state, it must be in a state of definite polarization (1.5). This polarization direction si2of the scattered wave is some function of s1, the polarization direction of the incident wave. Since there is no component of the scattered wave in the -sri direction, it is clear that for fixed & , Ij, , and s1, the maximum cross section a~( SP, $2 ; sl , $l)idCi occurs when s2 = sr?. Expressing s1 and s2 in terms of the basis vectors $1 , n A $1 , and n (where n is given by Ey. (2.19)) so that s1 = (sin e1 cos &)$A + (sin e1 sin &)n
A fil + (cos el)n, (2.31) s2 = (sin e2 cos (b?)& + (sin e2 sin +,)n A @I+ (cos e,)n, one finds by maximizing the cross section of Eel. (2.18) that s12(ep , 42) is determined by the relations sin(+l - &) = I,( Ily - 122)--li’, cos(& - 42) = I,( 11’ - 1y2, and tan(eJ2)
= [(I1 + IZ)l(I1
- 1?)]“” tan(e1/2).
(2.36)
C’OULOMR
SChTTERIKG
OF
Z)THAC
:i?l:j
PARTICLES
These ecluations explicitly indicate that, in general, both the azimuthal and polar angles of the polarization direction change in the scattering process. Thus, the polarization direction precesses and nutates about the normal to the plane of scattering from the original direction s1 to the final direction s12 . Ls’or t’lle special case of & = 0 (or 81 = a), one sees from Eq. (2.36) that 82 = 0 ( or 82 = s). Thus, if the incident polarization direction is parallel (antiparallel) t,o the normal to pla.ne of scattering, the polarization direction of the scattered ware is also parallel (antiparallel) to the normal. Also, it follo\j-s from Et{. (2.136) that if the asymmetry parameter s = I?.lI, is equal to zero, then 0? eclualx & ( no nutation j. The maximum differential cross section occurring when s2 = sir is found to be du( sy? , fi? ; Sl ) $,)ldQ
= II - Iz cos 8,
( 2.37 )
and, as expected, (IIcJ(-sI? , p2 ; sI , rjl).;rlQ = 0. III. A.
EX.\C’T
THE
kPKESSIONS
JOHSXON-DKK ;\SI)
WAVIS
FI3CTIOS
bL.\TIOlUS
The functions (;, 111, and I, occurring in Eq. (2.2) have been explicitly t,ermined by .Johnson and Deck ( 12) and are given by the infinite series:
de-
(8.1 )
In the preceding formulas, N(r,
k) = [r(u)!r(b)](,-l)“e’“‘~,
(3.4)
the symbol Pi represents the Legendre polynomial of order Ii whose argument is cos 8, the prime indicates a derivative with respect to the argument, and ,Fl denotes the confluent hypergeometric function. The functions G and ); are related to 91 by the first order partial differential ecluations : (; = ,-yI,
=
( 1 - cos 19)x8 /dx + sin2 t!M,‘d( cos 0) - 2]eY"d/,
-ee-X!‘[xt3:‘ax
+
(1
-
cm
O)d:d(cos
B)]eX’JI.
(X5) (3.6)
This can be verified by performing relations : (cl/&)
91(a, b; x) = z-‘(1 Pk’ + P:-l
= --/;(I
the indicated
operations
- b)[,Fl(CX, b; 2) - co9 e)-‘(&
-
with
IK(U, b -
the aid of t,he 1; r,],
P-1).
(Xi) ( 3.8)
A derivation of Eqs. (X5) and (X.6) will be given in a future paper. In terms of the parabolic coordinates (1 = ip1*(1 + cos l9), the differential
relations given by Ills.
<2 = ipr( 1 - cos B),
(3.9)
(3.5) and (3.6) become
c; = [& - 2 - “&3/d&J]L~Z,
(3.10)
I, = ((1 + (?)[f~~ -
(3.1 I )
d/d&]M.
.Johnsonand Deck have shown that if one sets A’ equal to zero, so that y ecluals A, the corresponding zero order approximation ~11,~ obtained from Eq. ( 3.1 ) can be summed to obtain Jf” = - ;,ir(]
- gpp+f~‘~2
,F,( 1 + iv, 2; &).
(3.12)
This approximation leads to the well known Sommerfeld-Rlaue approximation.
In terms of an expansion in X2,the function JI is
It has been demonstrated by Mott (19) that the asymptotic expansion of a sum of an infinite number of terms of the type occurring in Eq. ( :3.IS) is equivalent to the sum of the asymptotic expansions of each term, provided the resulting series is summed as the limit of a power series on its circle of convergence. Interpreting the sum this way, the asymptotic form of the X2 contribut’ion to 11 may be obtained from a knowledge of the asymptotic form of R. This is obtained from Eq. (X.1) by using the asymptotic expansion of $I( a, 0, x) fol large X, and the Taylor’s expansion y = k - h’,l(Y~) + 0(X’). The result is
+ **Cl< + 1 + iv) - i!(2p,.)
(3.11)
+ O(?))
- .i( _1)‘;(~p~)-3-‘“e-“~‘tl
+ ()(),7’) ),
where q1 is the derivative of the logarithm of the gamma function. The term
COULOMB
SCATTERIh-G
OF
DIEtA<’
X47
PARTICLES
in Eq. (3.14) proportional to e-l” is zero to order t,-4 when summed with the Legendre polynomials, so it is clear that asymptotically the X’ correction to NO contributes only to the outgoing scattered wave. The asymptotic form of A/,, (Es. 3.12) involves both a plane wave part and an outgoing scattered wave. The differential relations of G and IJ to d/ may now be used to obtain the asymptotic form of the wave function itself. This wave function is correct through order h” and has the form given in Eqs. (2.5) and (2.6). In this approximation the functions G,- , 11, , and I,, are: G,< = (4~ sin” O/2)-‘{ ;I/, = (4~ sin’ O/Z)-‘(1
--2iv( 1 + X’Y’(O, v)) - X”(,sin O)r’(O,
(3.15)
v)),
(3.16)
+ X’7’(,6, Y)),
L, = (4~ siri” 0.i2)-1X’( tan O/2)2+( 0 v). The prime signifies the derivative y(e, v) = (zivB( -iv,
(X.17)
with respect to 0 and 1 + 2iv)}-‘e-‘i”‘”
(SinS12’,\‘(e,v),
(3.18)
where (,X19) Here B represents the beta function. In addition to the series representation of the two parameter function S(e, v), it is also possible to obtain an integral representation. The term involving R alone can be written in terms of the Eulerian integral representation for the beta function (20). Also, the terms involving the product of B and the ql’s can be written in terms of an integral representation of a similar st,ructure ( 21). The result is
The infinite sum can be performed Legendre polynomials” so that
with
the aid of the generating
I - .r- [I +2 - 2~cos e]'!'} ,q(,g,v)=s0I.r-‘y-l(1 - - s)Yiy{ ~1 + .c? - 2.~ cos ep
(2x
-
function
Ins)
d.l-
for
(,X21 )
The integration canrlot be performed in terms of known functions. However, for large / v 1 an asymptotic expansion can be obtained from the integral representation, whereas for small V, a result in terms of known functions can be cow veniently obtained from the infinite series form of S(8, Y). ? See ref. 20, p.
154.
IV. POTENTIAL
SCATTERING:
RESULTS
In the preceding section, the approximate wave function correct through order X” for all Y has been obtained in terms of the two parameter functions T(o, V) and T’( 8, v). By means of appropriate substitutions, the potential scattering functions I(@) of Section II can now be expresseddirectly in terms of T(e, V) and T’(@, v). The results are: I1,!R = (1 - /? sill’?O/2)] 1 + x”T* 1’ + Xp(sin 0) Im[T’( 1 + X’T*)] + i’p’( sir? e/2)1 T” /, 12,1R= -2x0( E’))‘(sin’ O/2) Re[T’( 1 + x”T*)],
(4.1) (4.2)
/cll’R = $@” sin @[I + (I!: - l)(,E + 1))’ cos 0111 + X’T” 1’ + 2Ap(sin’O/2)[1 + (E - l)(E)-’
cos 01Im[T’(l
+ A’T*)]
(1.3)
+ x’p”( sill” e,/2)sin el T’ I’, I1”R = 11 - $[sin’O&
+ lj2(f!: -
I)(&
+ I)-‘sin”
+ xp sin O[l - 2(& - l)(E)-‘sin’O/2] + i2p?(sin’ e/2)
cos
O])] 1 + X’?T* 1’ Im[T’(I
+ X’T*)]
(4.4)
81T’ /‘,
where p = p/E = x/v,
(45)
R = [ui(2p sin’ O/2)]‘.
(4.6)
and
In the limit of small momentum, R becomes the Rutherford differential cross section. The function T(0, V) given by Eq. ( 3.18) is the approximation to zero order in A” of that function which would give an exact solution. Therefore, to obtain the I(e) functions to a consistent order in X2,the terms quadratic in the T’s should be neglected because these are part of the higher order corrections. However, if the wave function itself is approximated very closely by its expansion to third order in X, it may be useful to retain these terms. The preceding results reduce to those obtained by using the Sommerfeld-Maue approximation if one setsboth T and T’ equal to zero. In this case,one finds that the differential cross section I,/R is equal to 1 - /3” sin ’ O/2, and I,,/R (and consequently the asymmetry parameter S) is equal to zero. Thus, an interesting feature of the Sommerfeld-Maue approximation, which differs from the exact physical case, is that the polarization direction of the incident particle processes
C'OT'LOMB
SCATTERING
OF
1)IRAC
P.-\RTICLES
without nutation and ( 2.X) ).
about an axis normal to the scattering
B.
I'A~~AMETER
Sh.-1Lr,
Ron
349
plane (see Eqx. (2.35)
It is possible to obtain an expression in terms of 1aow1~ functions for the angular function IT’(e, U) in the case of small Born parameter V. Using a Taylor’s expansion in Eq. (3.18) and (3.19), it is found that through order v T(O, v) = -$
[f’1.~cos~)
/.=I
- PL,(COSe)]
X / 1 - 2iv(r +
- liisinBl%)(i7r,/k
- 2iik + 2Tk*l(k)
Vi/i)-'[T
+ 2i/<*z(k)
where \IlY is the polygamma function *z(z) = ~PF~(z)/&, roni constant y = -*,( 1 ) = 0.5772 . . . . Using the Legendre polynomial identit’y i I + cos 8)[f%‘(cos
8) -
fl:-,( cos O)] = x[f’p(:cos
one finds t’hat T’(@, v) through T’(e,
V) = !+
cot e,tzg
( 4.7 )
+ l;!k’) -
2i*l(/i)]]
and the Euler-Alas&e-
e) + /),,--1( COB O)],
(4.8)
order v may be expressed as [f’,,(cos
0) -
P-rjcos
e)](&,‘k
+
1 !I,,‘)
k=l +
l$
tan
ei'2
Ag
[PA-ccO~
e)
+
e)]
f'kpl(~~~
(4.9)
X 11 + v(k-I[*
- “i,;k
2zi(y
+
111
sine”Z)(kr
+ 27rlPP~(l~) + 2ik\kr(k)
+ 1,/l<)
- mQ(k)]}.
The divergent sums in the preceding equation are t#o be interpreted as the limits of the convergent sums on their circle of convergence. If X and v are both small and of the same order (i.e., v may be considered not as an independent parameter but equal to A/@ where 0 is of order L), then a con sistent approximation requires T(e, V) to zero order in v and T’(o, V) to first order in Y. These sums may be performed using the expressions given by .Johnson et al. (9), which lead to the results lie T(e, V) = a’;1 3 -
12C?( 1 - 2)
+ O(V),
II11 T(6, v) = T ln( 1 + .r) + O(V),
(4.10 (4.11
He T’( 8, U) = - ( tan 0/Y) ln s + &r(sin
0)-‘{ (1 + r) ln( 1 + .r) - .r In 4 - 2 111x)
(4.12
350
FRADKIN, +
WEBER,
AND
HAMMER
O(4,
Im I”l.8, v) = r,i( 1 + r))‘7r
cos e/2
+ v(sin 6)-‘((1 -(a’/6)(1
+ z)&(l
- 2.c)(l
-
s’)
-
4xCz(l
- r) + 2.r%%\
-
s)
(4.13)
+ O(;‘).
In the preceding equations, x equals sin B,l2 and Ed represents Euler’s dilogarithm.3 The 1(e) functions which follow from these expressions with the aid of Eqs. (4.1)-C 4.4) are in complete agreement with Johnson et al. (9). C.
Bores
LARGE
PARAMETER
When the Born parameter is large, 0 = x /v is small. Consequently, in the expressions for 1(e) given by Eqs. (d.l)-(4.4), it is consistent with maintaining the approximation to order X’ to simplify the exact dynamical factors multiplying the functions of T and T’. The resulting expressions are:
(I1 - R)/R
= (I, - R).IR = X2(2( Re T) + v-’ sin 0( Im T’)
IJR
= -2sX”(Ev)-‘(
I,!R
=
-
V? sir? e/2),
sin’ 0j2) Re T’,
(4.15)
sin’ 612) Im T’ + ;,
h’v-‘{2(
(4.14)
(4.16)
The functions T and T’ to be used in these equations, which are appropriate for large Born parameters, may be approximated by means of asymptotic expansions. When ( v j ---) 00, an asymptotic expansion of S(6, V) can be obtained from the integral representation given in Eq. (3.21). The methods and results of this expansion are presented in the following paper (23). Also derived there is the asymptotic expansion for B( -iv, 1 + 2iv). Using - these results and the definition of T(e, v) given by Eq. (3.18), one obtains for the case v -+ - 00
T(e,v)
,w~{c+
$[(se$i)c
- L?tan’i]
..
4-!)sec’~)c+2(tad~)(“+!Isec’~)+
(4.17)
where c = (7r - 0) tan e/2. For
the case
v
-+ + 00, one obtains T(e,
3 For
definition
(4.18)
v) = TI(@, v) + Tz(e, v) + Tde, v>
and tabulation,
see, for
example,
ref.
22.
(4.19)
whelp
1 1 (1+b> it? v5+ 1_[,'+4see? s'y'? ---[L+-0 + xiu -. 1 - J- ;+(l+o)v; 2i 2V[ 89
16
2
av
(3
2)
&)I
Ii ei”Hh-”(a)I , r/7,(0, vj = -(A/v) e”“‘, C-M) + ...
and n = v In csc 0,‘2, b = {se2 O/2 -
(4.23) 3; -
Ij(ln
csc” O/2)).
(1.34)
Here H”’ and H’-” indicate the Hake1 function of the first and second kind, respectively. These asymptotic expansions are valid for t#he whole range of 6, and as shown ill the following papel , give good results for finite v as long as 127w
> 1.
For large positive v, the somewhat formidable expression for ?‘(O, V) may he simplified if one restricts consideration to a = v In csc 8/2 > 1 (forward angles). Then, the Hake1 function occurring in Eel. (4.20) may be expanded asymptotically
so that
Consequently, for large positive expansion of 7’(0, v) is
v such that v ln csc e/2 > I, the asymptotic
where i; = - (a + 19) tan O/2.
(4.28)
The asymptotic expansion of d7’(0, v)/83 may he obtained by differentiating the relevant expansions of 7’( 0, V) since it can he shown that dT(f?, V) ia0 possesses an asymptotic expansion. The functions (I1 - Iz) ‘R, Ir/R, and Ix/R given by Eqs. (1.1-k)-(4.16) are plotted (with appropriate scale factors) in Icigs. 1 to 6. For negative v (repulsive potential) the asymptotic expansion of 7’ given by Eq. (-1.17) and its derivative have been used, and for positive v (attractive potential) the asymptotic expansion of 7’ given by Eqs. (-2.19.-C&.22) and its derivatives have been employed. A. POSITIVE: BORN PAKAMETER
The Born parameter is positive for attractive potential scattering which occurs, for example, when an electron is scattered by a positive point charge. h’rom Figs. 1 to 3 it is immediately apparent that oscillations occur in all three functions, (II - R)!‘R, It/R, and IJR. These functions oscillate with a very
a \ z t!
1.2
I-
0.8 0.4 i
u
“14
‘A
-0.4 2.4-
u=3
-
2.0
1.6 -
1.2
0.0 -
0.4
o-0.4”
I
-I
2.OC
8 (degrees) FIG. 1. The gh frequency
expression behavior
X-‘(Ii - RI/IL st small angles
as a function not plotted.) 352
of 0 fur positive
Y. (Small
amplitude
0.06
-
0.06
-
0.04
-
0.02
I Il.30
O-
-0.02
-
-0.04
-
-0.06
-
O.lO-
v.,o
0.,60.06
-
0.04
-
-0.04
-
-0.00
-
U=5
K
0.2
-
0.1
-
-0.1
I
0.6
I/=
FI :G. 2. The expression high frequency behavior
OS
=--
0.4
-
SIMPLE
I FORMULA
- !/2EX-2(12/X) at small angles
as a function not plotted.) 354
of 0 for positive
Y. (Small
amplitude,
0.‘6Y*30 0.12
-
0.06
-
0.04
O-
-0.06
-
-0.06
I
1
y. ,
FIG. 3. The expression h-z(Ij/R) as a function frequency behavior at small angles nclt plotted.) 355
of B for
positive
Y. (Small
amplitude,
higt
-\ -l.O0
.--I
I
30
60
I so
I
I
120
150
180
8 (degrees) FIG.
-1. The
enpressiox~
v’A-‘(I,
-
/Z)/I/
as a funct~ion
of 0 for negative
Y.
8 (degrees) FIG.
5. The
expression
-?/,Y”EX-*(Z~/R)
as a function
of 0 for negative
Y
high frequency, low amplitude behavior at extreme forward angles (not plotted), and as 0 increases, the period and amplitude of the oscillations increase. These oscillations may be seenexplicitly in the analytical form of T(O, V) for the condition IJ ln csc O/2 > 1 (Eq. (4.27)). The oscillations result from the term involving exp(2iu In csc 8/Z), which shall henceforth be described as the os-
0.24
0.16 c s N ‘X cd
In
P
0. I 2
0.06
0 0 Frc:.
6. The
30
60
expression
Y~X-~(I;~/~~)
120 as a frmc~tion
150
of 0 for negative
160 v’.
cillatol~y part of 7’( 0, v). This telm adecluatrly gives the ZCJNM, maxima, and minima of the oscillations. Oscillations in the asymmetry parameter s (which is proportional to I2 :K) welp obtained by Shelman ( fi) in his exact nunic~kal calculations for the cases in which v is of the order of me or gwabeJ*. Shwman’s values and the values obt’ained from the analytical form S = -?A”(
[
It? 7”
in which 7’ is given by Eqs. (4. I!))-i4.2%), is plott,ed in E’ig. 7. The analytical applnsimation gives good agwment with Sherman’s data. The cluantitative discrepancy is of order h2 which is the error expected from the approximation of the exact ?’ function. The error arising from the asymptotic expansion, which applies whenever 12~ 1 > 1, is seen t,o be of sccondal:v importance even when v = 0.474. The asymmetry parameter obtained with the use of t,hc small v expansion ( 9) for the case of v = 0.474 is also plotted in lcig. 7. Hew, there arc no oscillations (for angles glratcr than about 5”) and both t’hc kwgc v and the small v appwsi-
I?I(:. 7. The ,zpproximation for the asymmetry function S compared with Sherman’s vn111es. (8rmll amplitude, high frequency hehnvior :tt. srnnll an,qles not plotted.)
mations are in fairly good agreement with Sherman’s numerical calculations. Thus, one sees that as long as X” is small, good approximations are available for the whole range of T: for v 5 3:.I one may use the smaH v expansion and for Y Y > ,.d I4 one may use the asymptotic large Y expansion. A simple formula for the asymmetry parameter w may he obtained from the leading tcrnl of the oscillatory pal% of T(0, V) given in Eel. (4.27). To the first order in l/v and A”, one obtains D N_ f2,iR
‘v
-27rX”(Ev)P’(sin
8,‘2)(1
-
cos O/2)
sin[&
hi csc 0:2].
(5.2)
This formula is expected to apply for large v only in the region v ln csc O/2 > 1 which roughly includes the whole angular region between 0 = 0 and the 0 adjacent to x which is associated with the penultimate zero of the asymmetry function. However, as illustrated by the dashed lines for Y eclual to 1 and 3 in Fig. 2, the simple formula even roughly describes the behavior ill the region of the last maximum of the asymmetry function although v In csc O/2 is greater than 1 here and the approximation is not expected to apply at all. The function (II - R),IR is the fractional difference between the actual and the Rutherford differential cross sections, as related to the Rutherford value. From Fig. X, it is seen that for large Y, this function is oscillatory and terminates at o = g with a nonzero value. Using a small argument expansion of the Ha&e1 functions in the relevant expressions for 7’ and ?“, it may be shown that the value at 0 = T is given by
(I1 - R)/R
= (3’j-?:J + ~T[(>$T/v)%‘?
+ v/8 - ;.;28) - ~1).
(5.3)
FIG. 8. The approximation for the correction to the Kutherford difl’erential u%+(J, - I?)/&, compared with Sherman’s values. (Small amplitude, high havior at small angles not. plotted.)
CROSS section, freclueucy he-
As a function of u, this terminal value changes sign from negative t,o positive at v = 0.626 and reaches a maximum value of (2.61)X” near v = 2.4. Figure 8 illust,rates the agreement between Sherman’s value for (I1 - R) ‘R and that obtained from the large v asymptotic expansion involving Eq. (4.14) in cow junction with Eqs. (4.18)P(4.22). The error flags on Sherman’s data for the case Y = 0.4T4 stems from the fact that he gives IJR to only three significant figures. It’ is seen that the agreement is quite good, and here also the numerical discrepartcy may be attributed primarily to the high value of A’. In particular, the positive values of ( I1 - R)/R as 0 approaches rr are predicted, which is also in agreement with the Doggett and Spencer (5) calculation for V’S of order 1 or greater. l:or v Iii csc 0 ; 2 >> 1, the functions (II - R)/R, /2,/R, and I:(/R may be approximated using only the nonoscillatory part of ?‘(0, v) given by Eq. (4.27). Only the nonoscillatory part of ?‘( 8, v) is of experimental significance, since in ally measurement over Ae, the oscillatory part averages to zero.
The Horn parameter is negative for repulsive potential scatt.ering which occurs, for example, when a positron is scattered by a positive point charge. It is evident from Figs. 4, 5, and 6 that there are no oscillations in the functions (I1 - R)/R, IS/R, and t,/R. (An important physical distinction between attractive and repulsive potential scattering is that physical bound states exist only for attractive potential scattering. Thus, the oscillations occurring for attractive potential scattering might be attributed to resonances reflecting t,he
hound state structure.) In these figures, the scale fact)ors have been chosen SO that the plotted functions are universal functions of V. The value of ( I1 - R ) /R at 0 = = is - %:#. kigure 1 also illustrates the furlction -sin” .0,‘2 which Mott and Alassey ( 1~) claim is the appropriate value of a-‘(Il - R)/IZ for all values of p as long as X is small compared with unity. Their conclusion was reached on the basis of a small A expansion of the exact cross section in which h and @ were considered independent parameters. The results displayed here in lcigs. 1 and I are in disagreement with their assertion. Instead of a common curve for +X and -A (attractive and repulsive potential), these results exhibit quite different character for these two cases. The l\tott and L\Iassey argument derives from the formula (I, -
R)!R
= -p”
sin’B/L’
+ ahfl(sin
@2)( 1 - sin 0,‘2) + 0( A’)
which is valid for small /\ and small v. The second term may he neglected only when 1 X / < /3, which is no longer true when j v / > 1. Of course, the above formula is not valid for this case and thus one cannot extrapolate to the non relativistic limit on the basis of it. Instead, one must use the small X, large v expansion in order to discuss the large v case, and this in effect introduces additional terms of order fi:‘. It is to be noted that as a function of 1v (, the functions (I, - R)iR = (I, - R),;R, 12/R, and I,!R go to zero much faster in the nonrelativistic limit for negative v than the corresponding functions do for positive V. Since these functions all vanish in this limit, Eqs. (235) and (2.X) imply that in a single scattering at nonrelativistic energies, the polarization direction does not change but remains fixed in space.
The authors of the functions RECEIVED:
wish to thank Mr. W. J. Highy for performing displayed graphically in this paper.
the
computer
calculations
October 22, 19G:i REFERESCES
1. S. F. Mow, Ptwc. &I!/. Sot. (k&on.) A124, 425 (1929). 2. NC‘. F. Mow, Pwc. Roy. Sot. (London) A136, 129 (1932). 3. J. H. HARTLETT AND R. E. WATSON, Proc. .-IPL. .-lead. .4t+s Sci. 74, 53 (1940). 4. W. A. MCKINLEY AND H. FESHBACH, Phys. Rev. 74, 1759 (1948). 5. J. A. DOGGETT ANI) L. V. SPENCER, Phys. Rev. 103, 1597 (1956). 6. S. SHERMAS, Phys. Rev. 103, 1601 (1956). 7. W. H. FURRY, Phys. Ret). 46, 391 (193-l). 8. A. SOMMERFELD AND A. W. MADE, dnn. Physik22,629 (1935). 9. W. R. JOHNSON, T. A. WERER. AND C.J.MI-LLIN, Phys. Kent. 121,933 (1960). 10. I'. G. GORSHKOV, Zh. Eksperim. i Teor. Fir. 41, 977 (19Gl), English transl.: Soviet --JETP 14, 694 (1962).
Phys.
11. H. NAGEI,, ZQl. l’ek. Ho&ml. Hnntll. So. 157 (1960). 12. W. I?. JOHNWK ANU It. I'. DECK. J. M&h. Ph,qs. 3, 319 (1962). is. N. I'.b~W'rhNI~ Ii. 8.W.i%4hSE'i. “The Theory of Atomic C’ollisions,” Pntl ed., p. X0. (kford Cniv. Press, London, I~~ngland, 1949. r/i. I:. H. (:om, Ii’o. Xotl. Phys. 27, 187 11!)55). 15. D. 31. FR.~I)KIA ANIJ II. H. C;oou, JR., Hrz~. Moth. Ph!/s. 33, 343 (l!Cl). f/j. II. :\. '~l~l,llOEK, ft'crl. .Ifot/. Ph.//s. 28, 277 (1951;). 1 Y. IT. FANO, /:rr. Jfrrtl. P/u/S. 29, 79 (1957). 18. nr. I<;. I:osE. “Relativistic &:lectron Theory.” Wiley, Sew \iork, 1961. 19. S. I;. Mrrr, Proc. ZZu!/. Sot. (Lonrh) A118, 512 (1928). in. “Higher Tr:~nscrndental E’unctions,” l~atenmn 1\lanuscCpt Project, edited by A. ICrdClyi, 1.01. 1, p. 9. Mc(:r:tw-Hill, Kew York, 1!)53. 21. “‘l’ahlr of Integral Transfornls,” Hatenxm Mitnuwript Project, edited by A. Erdfilyi, \‘ol. 1, p. 314. ~Ic(Grsw--Hill, l;ew IFork, 1041. 3”. I,. I.WVIN, “Dilogarithms and Associated Functions.” Macdonald, London, 1958. 33. 1’. 9. WEBEW, II. hr. FRADKIS. ASI) C. L. HAMMER, .lnn. Php. (4.Y.) 27, 36’2-376 (1964.
OY
27, 338-361
I’ILYSIC’H:
Coulomb
(1964)
Scattering
of Dirac
Particles*
The continuum Coulomb wave functiun in the Johnson and Deck forn--a. nlatrix operator acting on a plane wave spinor-is discussed. (;enerul relat,il,ns for the elastic diRerrnt,ial cross section, the asymnretry parameters, and the precession and nutation of polarization are developed. A11 expansion of the asymptotic wave function corrert t,o third order in X = LYZ for all Born pnraneters Y = aZ/@ is ~~l)t:tined. This expansion involves a two paranleter function T(o, V) which in turn may he expanded for small Y to give results previous11 uht.xined 1)~ Johnson, Weber, and Mullin. On the other hand, a large Y asymptotic expansion of 7'(0, V) valid for 1 2av ! > 1 permits a discussion of the nonrelativist,ic limit. Both the small and large apprnsinrations suhstsnt,ially agree for 1 v 1 1. 1;. Consequently, these t,wo approximations may be used to span the whule range of Born parameters. The asymptotic Born parameter approximation predicts markedly different scattering behavior for electrons and positrons. 9 striking feature is that functions pertaining t,o electron srattering are oscillatory in the scattering angle 8 while t,he positron functions arc not. In particular, the oscillations in the nsymmetr)paranreter for electron scatt,ering obtained numerically 1)~ Sherman are cont,ained in the analytical approximate form. Also. the approach to the Rutherford cross section is different t,hun that quot,ed 1)s illott and Massey.
A.
l’KELIMIS;\ItT
kmmss
The Coulomb scattering wave function for a I)irac electron has been given exactly by ;1Zott (1, $2) in terms of an infinite series of angular momentum eigenfunctions, constructed so that at large distances the wave function behaves like a dist#orted plane wave plus an outgoing spherical wave. The scat,tering wave function depends upon the Coulomb interaction by means of two parameters, namely, the potential strength parameter X = aZ and the dynamical Born paramet’er Y = aZl?,lp, where E and p are the energy and momentum of the incident particle. lcor certain ranges of X and V, the infinite Mott series has been summed numerically (3-6) for large distances I’. These numerical sums also depend on the * Work
was performed
in the Ames
Laboratory 33x
of the U. S. Atomic
Energy
Commission.
s:attering angle so it is easily seen that a full multiparameter table of numerical values of the wave function is not practicable. It is desirable therefore to develop apptwimations involving a finite number of l~nown tabulated fuuctiolw Approximations of the infinit,e .\Iott, series that have been developed may be characterized as expansions of the exact wave function in certain orders of the parameters x and V. liar esample, t,he Born approximation involves an erpansioll in powers of x and @ = p ‘I!: ‘v 1. The Sommerfeld-Maue (7, 8) approximation is the exact’ wave function expanded to first order in h mit#h v being considered as an independent parameter. 12 modification (9--l 1 ) of the Sommcrfeltl-~laue approsimat,ion, which is equivalent to the third order 13orn approximation, consists of the exact wave function expanded to t’hird order in x where agaiu v is considered a function of X and @ = p’l? ‘v 1. This is a good approsimation even for large Z as long as v is small. :\ll of these approsimatiolls alp limited by the condition that v must be small. Recently, Johnson and Deck ( 12) have reorganized the exact scattering wave function in the convenient form of a matrix operator acting 011 a plane wave spinor which corresponds to the asymptotic incident wave. 111the present paper, some of the consequences of this form are investigated. These include the formulation of t’he scattering problem, relationships among the scatt’ering functions, and the effect of scattering on the polarization. Starting with the exact wave function in the .Johnson-Deck form, an approximation ia developed based on an expansion in powers of x through t.hird order for arbitrary -\~alues of V. This approximation involves a two parameter functiou T(O, v) which is given by an infinite sum represent’ation or alternatively by an integral represeiit~ation. Icor small L’, a Taylor’s expansion allows the sum to be performed in terms of known functions. This small v expansion leads to results which are iti complete agreement, with .Johnson et ul. (9). l”or 12av 1 > 1, an asymptotic expansion of 7’( 8, V) obtained from the integral representation provides an approximation fol T(0, V) in terms of kuo~n funct,iotrs. The small v and large v expansions give results in substantial agreement for 1 v 1 2 1,;. C’onsequently, these approsimations may be used to span t’he whole range of Horn parameters. The asymptotic approsimation for large 1u 1 is used to evaluate the e1ast.k Coulomb scattering cross section and the polarization functions. l;or the case of clcctron scattering ( attractive potential), oscillations as a function of 0 are predicted in the different’ial cross section and the asymmetry functiw. In part,icular, the oscillat.ions in the asymmetry paramet,er S obtained in the numerical calculations of Sherman (6) are contained in the analytical form presented in this paper. Also, for large 1v j which corresponds to the nonrelativistic limit, it is found that, the correction to the Rutherford differential cross section is different frnm that, c+mted by Alott and Alassey i 13).
340 B. (fi
FR.4IKIN,
WEBER,
AA-D
H.-\MMEH
particle
in a Coulomb
NOTATIOS
The Hamiltonian = 777 = c = 1)
for a Dirac
H = --ia.V
field is taken
to he
+ fl - X,/r,
where x = (YZ, O( being the fine structure constant and Z being the (shielded) atomic number. For h > 0, the potential is attractive. The symbols (Y, /3, and d represent the usual Dirac matrices.’ The symbols A* and A+ denote the complex and Hermitian conjugate of any matrix 9, respectively. The caret fi indicates a unit vector, and the vector cross product is given typically by the notation a A b. Quantities which describe the scattering process are: the asymptotic momentum p = [E” + I]““; the scattering angle 0 given by cos 0 = $. f; the Born parameter v = kE/p; the distorted plane wave phase factor 6, = -v In{ pr( 1 cos 0)) ; the distorted outgoing spherical wave phase factor 6, = a/2 2 argr(1 + iv) + v ln(pl.(l - cos 0)). In Section III, the notation x = -22ip~, y = [I? - X2]“‘, a = y - iv, and b = 27 + 1 is employed. II.
A.
CROSS
POTENTIAL
SCATTERING
FORhllILATIOX
SECTION
The Johnson-Deck
(1.2) form for the exact wave function
is given by
\k(E, I,) = D(,E, ;, $1) C( s1 , $1) where the operator D(E,
(2.1)
D is defined by ;, $1) = G + iUlla~ ($1 -
P) + iL$, f, i.6.
(2.2)
Here, $I is a unit vector in the direction of the incident plane wave, U(sl , ljl) is a free particle plane wave spinor for positive energy E and arbitrary polarization direction given by the unit vector s1 , and G, M, L are functions independent of s1 , which are given explicitly in Eqs. (3.1 ))(3.3). The plane wave spinor T:(sl , ljl) satisfies the equation [o(~l)~s,]c~T(sl)
$1)
where the three vector polarization O(fA) The asymptotic
form
1 See, for example,
ref.
=
operator
= ?d + d.$l(l
of the Johnson-Deck
14.
t7cs1,
(2.3)
$11,
(16) O(&)
is given by (2.4)
- S)$l. wave
function,
required
for a
discussion
of elastic scattering
is
‘li(lC:, r) = ei(pl~r+*p)l~(sl,
&) + r-1ei(P’+6s’Ds(jr
$,)CY(s,,
PI).
(2.5)
Here D,( ;, $1) = G,? + iXdl.&~ ($1 -
P) + iL,$, A F.d,
( 2.6)
where G, , A/, , and L, are obtained from the asymptotic form of G’, M, and L, and the energy dependence of 11, has been suppressed. 12syn~ptotically, the outgoing scattered wave of momentum p$3= pF
( 2.7 )
itself must be proportional to a plane wave spinor (of positive energy E) in a definite polarization state. Consequently, D.&r?, , $,)G’(sl,
(2.8)
$5) = c(O)U(sn, $2)
where c(B) is the proportionality constant., and ~12is a definite polarization direction related to s1that will be discussedin Section 11,B. If the scattered wave is analyzed not for polarization direction ~~1, but is analyzed for arbitrary direction of polarization S? , then the component of the scattered wave relevant to the scattering processis (‘(ss ) $2 ; Sl ) @l)U(S? ) $2)
(2.9)
where (‘(s, , $2 ; s1, $1) = 7’+( s:!, &)D,C& , Ij,)l:(sl
, $1).
(2.10)
l’or the scattering from initial state characterized by s1, ljl , to the final state characterized by sg, $2 , it follows that the differential cross section is given by du(s2,
fi2 ; sl,
13,)idti = 1(I(,s~, fh ; s1 , fll) j’.
(2.11)
The derivation uses the fact that the matrix element of t)he current operator a.$ is given by I:+@., $)a.$l:(s, $) = p/E. (2.12) The matrix element C’ of Eq. ( 2.11) may be expressedsimply in terms of the functions appearing in the asymptotic Johnson-Deck form. If one defines free particle Hamiltonians H, by H, = ‘y.pn + P;
11= 1, 2,
(2.13)
and usesthe identities cr.(pl - ~2) = HI - Hz ip, A p2.d = ~1.~2 + P;
+ II,/3 - 1 - H-H,,
(2.1.5)
then it is found that C(S,, j% ; sl, Ij,) = I’+(+, Here A1 and AZ are functions
lj2)[ill
of 0, (fll.&
$,j.
(2.16)
= cos e), given by
Al = G, + 2L,[c020,‘2 A, = QpEL,
+ pA2]C:(sl,
-
(Mp)2] (2.17)
.
The square of the matrix element involved in the differential cross section is evaluated by inserting appropriate projection operators and using standard trace techniques. The result, in a form closely related to that given by Tolhoek (l/S’), is 2d~(s~, & ; sl, &)/dQ
= ([I + (n.sl)(n.sz)]lI
- [(n.sI)
+ (n.s2)]I,
+ (n.s2 A ~1113 + [(n A sd. (n A s2)lLl.
(2.18)
Here n, the normal direction to the scattering plane, is given by n = (~2 A &):/I p? A pl I
(2.19)
and the four I functions, which are real functions of 8 but independent of polarization directions, are given by the formulas: I1 = [I -
(p!f~)‘)“sin’8/2]~4~ill* + [l - (p~E’)“co&9/2]dsA2* + (Z,!E)Re Al&*;
I2 = [( p/E’)kn
0]Im AISZ*;
(2.20) (2.21)
Z3 = f~~(p/E’)‘sin Oi[l + (E - l)(E
+ 1)-l cos e]il,A,*
- [I - (E - ‘l)(E
+ l)~‘cos e].4aAz*
- [2(E - l)(E
(2.22)
+ I)-‘cos 8]Red1,4?*1;
I4 = (E-‘sin% + cos ff[cos 8 + (p/li:)“sin’?0/2] IS1i41* + (K’sin’B + cos 8[cos19- (p,:~)‘cos’ej2])dpA~*
(2.23)
+ [2r(:-’ + (h’ - l)“E-‘sinSB]ReAl.4?*. These four functions are not independent but are related through the equation II? = 122+ Is2 + IJ’),
(2.24)
which may be verified by direct substitution. The functions r(0) given by Eqs. (3.20)-(X23) may be determined directly from scattering experiments. For a single scattering with an initially unpolarized beam, the differential cross section dar/dQ is obtained from Eq. (2.18) by summing over final polarization directions and averaging over the initial polari-
COULOMB
zation directions.
SChTTERTX(:
OF
Dllt.i(’
PARTICLES
3-13
This procedure yields dar/dQ = II(e).
(2.25)
If there is a second scattering (shielded from the direct incident beam) a long distance 1.~2from the first scatterer, located at a polar direction 0 with respect to the incident beam direction fil , then the beam incident on the second scatterer is a plane wave ( i.~2)-‘eiB”” Ds(& , &) l’($, , sl). Here, the prime signifies distance measured with respect to the second scatterer. Substituting this incident wave into Eq. (2.~3) and foIlowixig the same steps leading to ICq. (2.12), it is found that the differential cross section for scattering from initial state characterized by s1 , 13~, to a final state, analyzed after the second scattering, characterized by s1 , 13~, is given by
where C’Cs:! , lj3 ; $2 ; s1 , $1) = L”( $4 ) $a)L&i$a ) fi?)l)&&
) $,)U(Sl
) $1).
(2.27)
Inserting a sum over the complete set of states I -, (~12 , $3) CTi+(~s12, $2) between the txo 1),$‘s, where 6 sunis over f and i sums over positive and negative energy spinors, it is found t.hat the sum reduces to one term with the result C(s3 ) $5 ; $2 ; Sl ) $1) = cI(s.3 , p, ; Sl? ) ~~)C(S12 ) p2 ; Sl ) $1). The C’s on the right are defined by Eq. (2.10). one obtains rJuis:i , /A ; $2 ; SI , 1;1)-=
1 __ (l.lZ)')
&
ddSa,&
Consecluently,
;s12,@2)
using Eq. (2.12)
dds1n,fi2;
dfl'
(2.28)
s1,131) dll
("',9) .d.d
,
The differential cross section rL~~~/dti’ for double scattering from an initially unpolarized beam is obtained by averaging over initial polarization directions ( +sl) and summing over final polarization directions ( &se). The calculations are simplified by using the fact (proven in Section 11,H) that if s1 is chosen as n = (p2 A pl)/j p2 A pI 1, then sle is also n. Substituting I2q. (2.18) into l31. (2.39) and performing the sums and averaging, one obtains the simple result (r12)2d~II fdd = 1l(O)11(O’)
+ (n.n’)1,(0)1,(0’),
( 2.30 )
where cos0 = pl.& , cos 0’ = &.j&, n’ = (p3 A p?),:l pa A p:! 1. It follows 1\Iott’s (1) definition of the asymmetry, 6(0, O’), is given by s(e, and the asymmetry
e’)
parameter
=
(2.31)
r,(e)r,(e’)ll,(e)r,(e’),
o(e) defined
that
by Sherman
(6)
is the angular
344
FRhL)KIN,
WEBER,
AND
HAMMER
function s(e)
(2.32)
= r,(e)lllcs).
Double scattering may also be analyzed with the formalism of density matrices, as is done, for example, by Fano (17) and Rose (18), which yields identical results. Similarly, information regarding both I,(e) and 14( 0) may be obtained from a triple scattering experiment. If an initially unpolarized beam propagating in the $1 direction is scattered through consecutive intermediate direct’ions $2 and $3 into a final direction fi4 , and the final beam is not analyzed for polarization, then the differential cross section for this process is given by duIII/dd
= ~~~12~2~)-2~l~e12)rl~e2,)rl~e,~)
x [l + (n43-n32j6(ea3, ez2) + jnx2~n21P(832, e2d
(2.33)
+ (.n43.n32)(n21.n32)~(Bra , e2d + (n132.n43A n21)s(e43, e21)Ia~e32)/L0b) + (n32 A n43>. (n32 A n21)6(e43, e2,)r,(e,,)!)rl(e,,)1. Here, cos eij = $i.$j, nZi = (pi A pj)/( pi A pj /, and rij is the distance tween the ith and jth scatterers. B.
I-'oL~~RI~AT~o~
OF
THE
SC.~TTERED
be-
WA4w
Since a wave scattered in a given direction is asymptotically proportional to a plane wave state, it must be in a state of definite polarization (1.5). This polarization direction si2of the scattered wave is some function of s1, the polarization direction of the incident wave. Since there is no component of the scattered wave in the -sri direction, it is clear that for fixed & , Ij, , and s1, the maximum cross section a~( SP, $2 ; sl , $l)idCi occurs when s2 = sr?. Expressing s1 and s2 in terms of the basis vectors $1 , n A $1 , and n (where n is given by Ey. (2.19)) so that s1 = (sin e1 cos &)$A + (sin e1 sin &)n
A fil + (cos el)n, (2.31) s2 = (sin e2 cos (b?)& + (sin e2 sin +,)n A @I+ (cos e,)n, one finds by maximizing the cross section of Eel. (2.18) that s12(ep , 42) is determined by the relations sin(+l - &) = I,( Ily - 122)--li’, cos(& - 42) = I,( 11’ - 1y2, and tan(eJ2)
= [(I1 + IZ)l(I1
- 1?)]“” tan(e1/2).
(2.36)
C’OULOMR
SChTTERIKG
OF
Z)THAC
:i?l:j
PARTICLES
These ecluations explicitly indicate that, in general, both the azimuthal and polar angles of the polarization direction change in the scattering process. Thus, the polarization direction precesses and nutates about the normal to the plane of scattering from the original direction s1 to the final direction s12 . Ls’or t’lle special case of & = 0 (or 81 = a), one sees from Eq. (2.36) that 82 = 0 ( or 82 = s). Thus, if the incident polarization direction is parallel (antiparallel) t,o the normal to pla.ne of scattering, the polarization direction of the scattered ware is also parallel (antiparallel) to the normal. Also, it follo\j-s from Et{. (2.136) that if the asymmetry parameter s = I?.lI, is equal to zero, then 0? eclualx & ( no nutation j. The maximum differential cross section occurring when s2 = sir is found to be du( sy? , fi? ; Sl ) $,)ldQ
= II - Iz cos 8,
( 2.37 )
and, as expected, (IIcJ(-sI? , p2 ; sI , rjl).;rlQ = 0. III. A.
EX.\C’T
THE
kPKESSIONS
JOHSXON-DKK ;\SI)
WAVIS
FI3CTIOS
bL.\TIOlUS
The functions (;, 111, and I, occurring in Eq. (2.2) have been explicitly t,ermined by .Johnson and Deck ( 12) and are given by the infinite series:
de-
(8.1 )
In the preceding formulas, N(r,
k) = [r(u)!r(b)](,-l)“e’“‘~,
(3.4)
the symbol Pi represents the Legendre polynomial of order Ii whose argument is cos 8, the prime indicates a derivative with respect to the argument, and ,Fl denotes the confluent hypergeometric function. The functions G and ); are related to 91 by the first order partial differential ecluations : (; = ,-yI,
=
( 1 - cos 19)x8 /dx + sin2 t!M,‘d( cos 0) - 2]eY"d/,
-ee-X!‘[xt3:‘ax
+
(1
-
cm
O)d:d(cos
B)]eX’JI.
(X5) (3.6)
This can be verified by performing relations : (cl/&)
91(a, b; x) = z-‘(1 Pk’ + P:-l
= --/;(I
the indicated
operations
- b)[,Fl(CX, b; 2) - co9 e)-‘(&
-
with
IK(U, b -
the aid of t,he 1; r,],
P-1).
(Xi) ( 3.8)
A derivation of Eqs. (X5) and (X.6) will be given in a future paper. In terms of the parabolic coordinates (1 = ip1*(1 + cos l9), the differential
relations given by Ills.
<2 = ipr( 1 - cos B),
(3.9)
(3.5) and (3.6) become
c; = [& - 2 - “&3/d&J]L~Z,
(3.10)
I, = ((1 + (?)[f~~ -
(3.1 I )
d/d&]M.
.Johnsonand Deck have shown that if one sets A’ equal to zero, so that y ecluals A, the corresponding zero order approximation ~11,~ obtained from Eq. ( 3.1 ) can be summed to obtain Jf” = - ;,ir(]
- gpp+f~‘~2
,F,( 1 + iv, 2; &).
(3.12)
This approximation leads to the well known Sommerfeld-Rlaue approximation.
In terms of an expansion in X2,the function JI is
It has been demonstrated by Mott (19) that the asymptotic expansion of a sum of an infinite number of terms of the type occurring in Eq. ( :3.IS) is equivalent to the sum of the asymptotic expansions of each term, provided the resulting series is summed as the limit of a power series on its circle of convergence. Interpreting the sum this way, the asymptotic form of the X2 contribut’ion to 11 may be obtained from a knowledge of the asymptotic form of R. This is obtained from Eq. (X.1) by using the asymptotic expansion of $I( a, 0, x) fol large X, and the Taylor’s expansion y = k - h’,l(Y~) + 0(X’). The result is
+ **Cl< + 1 + iv) - i!(2p,.)
(3.11)
+ O(?))
- .i( _1)‘;(~p~)-3-‘“e-“~‘tl
+ ()(),7’) ),
where q1 is the derivative of the logarithm of the gamma function. The term
COULOMB
SCATTERIh-G
OF
DIEtA<’
X47
PARTICLES
in Eq. (3.14) proportional to e-l” is zero to order t,-4 when summed with the Legendre polynomials, so it is clear that asymptotically the X’ correction to NO contributes only to the outgoing scattered wave. The asymptotic form of A/,, (Es. 3.12) involves both a plane wave part and an outgoing scattered wave. The differential relations of G and IJ to d/ may now be used to obtain the asymptotic form of the wave function itself. This wave function is correct through order h” and has the form given in Eqs. (2.5) and (2.6). In this approximation the functions G,- , 11, , and I,, are: G,< = (4~ sin” O/2)-‘{ ;I/, = (4~ sin’ O/Z)-‘(1
--2iv( 1 + X’Y’(O, v)) - X”(,sin O)r’(O,
(3.15)
v)),
(3.16)
+ X’7’(,6, Y)),
L, = (4~ siri” 0.i2)-1X’( tan O/2)2+( 0 v). The prime signifies the derivative y(e, v) = (zivB( -iv,
(X.17)
with respect to 0 and 1 + 2iv)}-‘e-‘i”‘”
(SinS12’,\‘(e,v),
(3.18)
where (,X19) Here B represents the beta function. In addition to the series representation of the two parameter function S(e, v), it is also possible to obtain an integral representation. The term involving R alone can be written in terms of the Eulerian integral representation for the beta function (20). Also, the terms involving the product of B and the ql’s can be written in terms of an integral representation of a similar st,ructure ( 21). The result is
The infinite sum can be performed Legendre polynomials” so that
with
the aid of the generating
I - .r- [I +2 - 2~cos e]'!'} ,q(,g,v)=s0I.r-‘y-l(1 - - s)Yiy{ ~1 + .c? - 2.~ cos ep
(2x
-
function
Ins)
d.l-
for
(,X21 )
The integration canrlot be performed in terms of known functions. However, for large / v 1 an asymptotic expansion can be obtained from the integral representation, whereas for small V, a result in terms of known functions can be cow veniently obtained from the infinite series form of S(8, Y). ? See ref. 20, p.
154.
IV. POTENTIAL
SCATTERING:
RESULTS
In the preceding section, the approximate wave function correct through order X” for all Y has been obtained in terms of the two parameter functions T(o, V) and T’( 8, v). By means of appropriate substitutions, the potential scattering functions I(@) of Section II can now be expresseddirectly in terms of T(e, V) and T’(@, v). The results are: I1,!R = (1 - /? sill’?O/2)] 1 + x”T* 1’ + Xp(sin 0) Im[T’( 1 + X’T*)] + i’p’( sir? e/2)1 T” /, 12,1R= -2x0( E’))‘(sin’ O/2) Re[T’( 1 + x”T*)],
(4.1) (4.2)
/cll’R = $@” sin @[I + (I!: - l)(,E + 1))’ cos 0111 + X’T” 1’ + 2Ap(sin’O/2)[1 + (E - l)(E)-’
cos 01Im[T’(l
+ A’T*)]
(1.3)
+ x’p”( sill” e,/2)sin el T’ I’, I1”R = 11 - $[sin’O&
+ lj2(f!: -
I)(&
+ I)-‘sin”
+ xp sin O[l - 2(& - l)(E)-‘sin’O/2] + i2p?(sin’ e/2)
cos
O])] 1 + X’?T* 1’ Im[T’(I
+ X’T*)]
(4.4)
81T’ /‘,
where p = p/E = x/v,
(45)
R = [ui(2p sin’ O/2)]‘.
(4.6)
and
In the limit of small momentum, R becomes the Rutherford differential cross section. The function T(0, V) given by Eq. ( 3.18) is the approximation to zero order in A” of that function which would give an exact solution. Therefore, to obtain the I(e) functions to a consistent order in X2,the terms quadratic in the T’s should be neglected because these are part of the higher order corrections. However, if the wave function itself is approximated very closely by its expansion to third order in X, it may be useful to retain these terms. The preceding results reduce to those obtained by using the Sommerfeld-Maue approximation if one setsboth T and T’ equal to zero. In this case,one finds that the differential cross section I,/R is equal to 1 - /3” sin ’ O/2, and I,,/R (and consequently the asymmetry parameter S) is equal to zero. Thus, an interesting feature of the Sommerfeld-Maue approximation, which differs from the exact physical case, is that the polarization direction of the incident particle processes
C'OT'LOMB
SCATTERING
OF
1)IRAC
P.-\RTICLES
without nutation and ( 2.X) ).
about an axis normal to the scattering
B.
I'A~~AMETER
Sh.-1Lr,
Ron
349
plane (see Eqx. (2.35)
It is possible to obtain an expression in terms of 1aow1~ functions for the angular function IT’(e, U) in the case of small Born parameter V. Using a Taylor’s expansion in Eq. (3.18) and (3.19), it is found that through order v T(O, v) = -$
[f’1.~cos~)
/.=I
- PL,(COSe)]
X / 1 - 2iv(r +
- liisinBl%)(i7r,/k
- 2iik + 2Tk*l(k)
Vi/i)-'[T
+ 2i/<*z(k)
where \IlY is the polygamma function *z(z) = ~PF~(z)/&, roni constant y = -*,( 1 ) = 0.5772 . . . . Using the Legendre polynomial identit’y i I + cos 8)[f%‘(cos
8) -
fl:-,( cos O)] = x[f’p(:cos
one finds t’hat T’(@, v) through T’(e,
V) = !+
cot e,tzg
( 4.7 )
+ l;!k’) -
2i*l(/i)]]
and the Euler-Alas&e-
e) + /),,--1( COB O)],
(4.8)
order v may be expressed as [f’,,(cos
0) -
P-rjcos
e)](&,‘k
+
1 !I,,‘)
k=l +
l$
tan
ei'2
Ag
[PA-ccO~
e)
+
e)]
f'kpl(~~~
(4.9)
X 11 + v(k-I[*
- “i,;k
2zi(y
+
111
sine”Z)(kr
+ 27rlPP~(l~) + 2ik\kr(k)
+ 1,/l<)
- mQ(k)]}.
The divergent sums in the preceding equation are t#o be interpreted as the limits of the convergent sums on their circle of convergence. If X and v are both small and of the same order (i.e., v may be considered not as an independent parameter but equal to A/@ where 0 is of order L), then a con sistent approximation requires T(e, V) to zero order in v and T’(o, V) to first order in Y. These sums may be performed using the expressions given by .Johnson et al. (9), which lead to the results lie T(e, V) = a’;1 3 -
12C?( 1 - 2)
+ O(V),
II11 T(6, v) = T ln( 1 + .r) + O(V),
(4.10 (4.11
He T’( 8, U) = - ( tan 0/Y) ln s + &r(sin
0)-‘{ (1 + r) ln( 1 + .r) - .r In 4 - 2 111x)
(4.12
350
FRADKIN, +
WEBER,
AND
HAMMER
O(4,
Im I”l.8, v) = r,i( 1 + r))‘7r
cos e/2
+ v(sin 6)-‘((1 -(a’/6)(1
+ z)&(l
- 2.c)(l
-
s’)
-
4xCz(l
- r) + 2.r%%\
-
s)
(4.13)
+ O(;‘).
In the preceding equations, x equals sin B,l2 and Ed represents Euler’s dilogarithm.3 The 1(e) functions which follow from these expressions with the aid of Eqs. (4.1)-C 4.4) are in complete agreement with Johnson et al. (9). C.
Bores
LARGE
PARAMETER
When the Born parameter is large, 0 = x /v is small. Consequently, in the expressions for 1(e) given by Eqs. (d.l)-(4.4), it is consistent with maintaining the approximation to order X’ to simplify the exact dynamical factors multiplying the functions of T and T’. The resulting expressions are:
(I1 - R)/R
= (I, - R).IR = X2(2( Re T) + v-’ sin 0( Im T’)
IJR
= -2sX”(Ev)-‘(
I,!R
=
-
V? sir? e/2),
sin’ 0j2) Re T’,
(4.15)
sin’ 612) Im T’ + ;,
h’v-‘{2(
(4.14)
(4.16)
The functions T and T’ to be used in these equations, which are appropriate for large Born parameters, may be approximated by means of asymptotic expansions. When ( v j ---) 00, an asymptotic expansion of S(6, V) can be obtained from the integral representation given in Eq. (3.21). The methods and results of this expansion are presented in the following paper (23). Also derived there is the asymptotic expansion for B( -iv, 1 + 2iv). Using - these results and the definition of T(e, v) given by Eq. (3.18), one obtains for the case v -+ - 00
T(e,v)
,w~{c+
$[(se$i)c
- L?tan’i]
..
4-!)sec’~)c+2(tad~)(“+!Isec’~)+
(4.17)
where c = (7r - 0) tan e/2. For
the case
v
-+ + 00, one obtains T(e,
3 For
definition
(4.18)
v) = TI(@, v) + Tz(e, v) + Tde, v>
and tabulation,
see, for
example,
ref.
22.
(4.19)
whelp
1 1 (1+b> it? v5+ 1_[,'+4see? s'y'? ---[L+-0 + xiu -. 1 - J- ;+(l+o)v; 2i 2V[ 89
16
2
av
(3
2)
&)I
Ii ei”Hh-”(a)I , r/7,(0, vj = -(A/v) e”“‘, C-M) + ...
and n = v In csc 0,‘2, b = {se2 O/2 -
(4.23) 3; -
Ij(ln
csc” O/2)).
(1.34)
Here H”’ and H’-” indicate the Hake1 function of the first and second kind, respectively. These asymptotic expansions are valid for t#he whole range of 6, and as shown ill the following papel , give good results for finite v as long as 127w
> 1.
For large positive v, the somewhat formidable expression for ?‘(O, V) may he simplified if one restricts consideration to a = v In csc 8/2 > 1 (forward angles). Then, the Hake1 function occurring in Eel. (4.20) may be expanded asymptotically
so that
Consequently, for large positive expansion of 7’(0, v) is
v such that v ln csc e/2 > I, the asymptotic
where i; = - (a + 19) tan O/2.
(4.28)
The asymptotic expansion of d7’(0, v)/83 may he obtained by differentiating the relevant expansions of 7’( 0, V) since it can he shown that dT(f?, V) ia0 possesses an asymptotic expansion. The functions (I1 - Iz) ‘R, Ir/R, and Ix/R given by Eqs. (1.1-k)-(4.16) are plotted (with appropriate scale factors) in Icigs. 1 to 6. For negative v (repulsive potential) the asymptotic expansion of 7’ given by Eq. (-1.17) and its derivative have been used, and for positive v (attractive potential) the asymptotic expansion of 7’ given by Eqs. (-2.19.-C&.22) and its derivatives have been employed. A. POSITIVE: BORN PAKAMETER
The Born parameter is positive for attractive potential scattering which occurs, for example, when an electron is scattered by a positive point charge. h’rom Figs. 1 to 3 it is immediately apparent that oscillations occur in all three functions, (II - R)!‘R, It/R, and IJR. These functions oscillate with a very
a \ z t!
1.2
I-
0.8 0.4 i
u
“14
‘A
-0.4 2.4-
u=3
-
2.0
1.6 -
1.2
0.0 -
0.4
o-0.4”
I
-I
2.OC
8 (degrees) FIG. 1. The gh frequency
expression behavior
X-‘(Ii - RI/IL st small angles
as a function not plotted.) 352
of 0 fur positive
Y. (Small
amplitude
0.06
-
0.06
-
0.04
-
0.02
I Il.30
O-
-0.02
-
-0.04
-
-0.06
-
O.lO-
v.,o
0.,60.06
-
0.04
-
-0.04
-
-0.00
-
U=5
K
0.2
-
0.1
-
-0.1
I
0.6
I/=
FI :G. 2. The expression high frequency behavior
OS
=--
0.4
-
SIMPLE
I FORMULA
- !/2EX-2(12/X) at small angles
as a function not plotted.) 354
of 0 for positive
Y. (Small
amplitude,
0.‘6Y*30 0.12
-
0.06
-
0.04
O-
-0.06
-
-0.06
I
1
y. ,
FIG. 3. The expression h-z(Ij/R) as a function frequency behavior at small angles nclt plotted.) 355
of B for
positive
Y. (Small
amplitude,
higt
-\ -l.O0
.--I
I
30
60
I so
I
I
120
150
180
8 (degrees) FIG.
-1. The
enpressiox~
v’A-‘(I,
-
/Z)/I/
as a funct~ion
of 0 for negative
Y.
8 (degrees) FIG.
5. The
expression
-?/,Y”EX-*(Z~/R)
as a function
of 0 for negative
Y
high frequency, low amplitude behavior at extreme forward angles (not plotted), and as 0 increases, the period and amplitude of the oscillations increase. These oscillations may be seenexplicitly in the analytical form of T(O, V) for the condition IJ ln csc O/2 > 1 (Eq. (4.27)). The oscillations result from the term involving exp(2iu In csc 8/Z), which shall henceforth be described as the os-
0.24
0.16 c s N ‘X cd
In
P
0. I 2
0.06
0 0 Frc:.
6. The
30
60
expression
Y~X-~(I;~/~~)
120 as a frmc~tion
150
of 0 for negative
160 v’.
cillatol~y part of 7’( 0, v). This telm adecluatrly gives the ZCJNM, maxima, and minima of the oscillations. Oscillations in the asymmetry parameter s (which is proportional to I2 :K) welp obtained by Shelman ( fi) in his exact nunic~kal calculations for the cases in which v is of the order of me or gwabeJ*. Shwman’s values and the values obt’ained from the analytical form S = -?A”(
[
It? 7”
in which 7’ is given by Eqs. (4. I!))-i4.2%), is plott,ed in E’ig. 7. The analytical applnsimation gives good agwment with Sherman’s data. The cluantitative discrepancy is of order h2 which is the error expected from the approximation of the exact ?’ function. The error arising from the asymptotic expansion, which applies whenever 12~ 1 > 1, is seen t,o be of sccondal:v importance even when v = 0.474. The asymmetry parameter obtained with the use of t,hc small v expansion ( 9) for the case of v = 0.474 is also plotted in lcig. 7. Hew, there arc no oscillations (for angles glratcr than about 5”) and both t’hc kwgc v and the small v appwsi-
I?I(:. 7. The ,zpproximation for the asymmetry function S compared with Sherman’s vn111es. (8rmll amplitude, high frequency hehnvior :tt. srnnll an,qles not plotted.)
mations are in fairly good agreement with Sherman’s numerical calculations. Thus, one sees that as long as X” is small, good approximations are available for the whole range of T: for v 5 3:.I one may use the smaH v expansion and for Y Y > ,.d I4 one may use the asymptotic large Y expansion. A simple formula for the asymmetry parameter w may he obtained from the leading tcrnl of the oscillatory pal% of T(0, V) given in Eel. (4.27). To the first order in l/v and A”, one obtains D N_ f2,iR
‘v
-27rX”(Ev)P’(sin
8,‘2)(1
-
cos O/2)
sin[&
hi csc 0:2].
(5.2)
This formula is expected to apply for large v only in the region v ln csc O/2 > 1 which roughly includes the whole angular region between 0 = 0 and the 0 adjacent to x which is associated with the penultimate zero of the asymmetry function. However, as illustrated by the dashed lines for Y eclual to 1 and 3 in Fig. 2, the simple formula even roughly describes the behavior ill the region of the last maximum of the asymmetry function although v In csc O/2 is greater than 1 here and the approximation is not expected to apply at all. The function (II - R),IR is the fractional difference between the actual and the Rutherford differential cross sections, as related to the Rutherford value. From Fig. X, it is seen that for large Y, this function is oscillatory and terminates at o = g with a nonzero value. Using a small argument expansion of the Ha&e1 functions in the relevant expressions for 7’ and ?“, it may be shown that the value at 0 = T is given by
(I1 - R)/R
= (3’j-?:J + ~T[(>$T/v)%‘?
+ v/8 - ;.;28) - ~1).
(5.3)
FIG. 8. The approximation for the correction to the Kutherford difl’erential u%+(J, - I?)/&, compared with Sherman’s values. (Small amplitude, high havior at small angles not. plotted.)
CROSS section, freclueucy he-
As a function of u, this terminal value changes sign from negative t,o positive at v = 0.626 and reaches a maximum value of (2.61)X” near v = 2.4. Figure 8 illust,rates the agreement between Sherman’s value for (I1 - R) ‘R and that obtained from the large v asymptotic expansion involving Eq. (4.14) in cow junction with Eqs. (4.18)P(4.22). The error flags on Sherman’s data for the case Y = 0.4T4 stems from the fact that he gives IJR to only three significant figures. It’ is seen that the agreement is quite good, and here also the numerical discrepartcy may be attributed primarily to the high value of A’. In particular, the positive values of ( I1 - R)/R as 0 approaches rr are predicted, which is also in agreement with the Doggett and Spencer (5) calculation for V’S of order 1 or greater. l:or v Iii csc 0 ; 2 >> 1, the functions (II - R)/R, /2,/R, and I:(/R may be approximated using only the nonoscillatory part of ?‘(0, v) given by Eq. (4.27). Only the nonoscillatory part of ?‘( 8, v) is of experimental significance, since in ally measurement over Ae, the oscillatory part averages to zero.
The Horn parameter is negative for repulsive potential scatt.ering which occurs, for example, when a positron is scattered by a positive point charge. It is evident from Figs. 4, 5, and 6 that there are no oscillations in the functions (I1 - R)/R, IS/R, and t,/R. (An important physical distinction between attractive and repulsive potential scattering is that physical bound states exist only for attractive potential scattering. Thus, the oscillations occurring for attractive potential scattering might be attributed to resonances reflecting t,he
hound state structure.) In these figures, the scale fact)ors have been chosen SO that the plotted functions are universal functions of V. The value of ( I1 - R ) /R at 0 = = is - %:#. kigure 1 also illustrates the furlction -sin” .0,‘2 which Mott and Alassey ( 1~) claim is the appropriate value of a-‘(Il - R)/IZ for all values of p as long as X is small compared with unity. Their conclusion was reached on the basis of a small A expansion of the exact cross section in which h and @ were considered independent parameters. The results displayed here in lcigs. 1 and I are in disagreement with their assertion. Instead of a common curve for +X and -A (attractive and repulsive potential), these results exhibit quite different character for these two cases. The l\tott and L\Iassey argument derives from the formula (I, -
R)!R
= -p”
sin’B/L’
+ ahfl(sin
@2)( 1 - sin 0,‘2) + 0( A’)
which is valid for small /\ and small v. The second term may he neglected only when 1 X / < /3, which is no longer true when j v / > 1. Of course, the above formula is not valid for this case and thus one cannot extrapolate to the non relativistic limit on the basis of it. Instead, one must use the small X, large v expansion in order to discuss the large v case, and this in effect introduces additional terms of order fi:‘. It is to be noted that as a function of 1v (, the functions (I, - R)iR = (I, - R),;R, 12/R, and I,!R go to zero much faster in the nonrelativistic limit for negative v than the corresponding functions do for positive V. Since these functions all vanish in this limit, Eqs. (235) and (2.X) imply that in a single scattering at nonrelativistic energies, the polarization direction does not change but remains fixed in space.
The authors of the functions RECEIVED:
wish to thank Mr. W. J. Highy for performing displayed graphically in this paper.
the
computer
calculations
October 22, 19G:i REFERESCES
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11. H. NAGEI,, ZQl. l’ek. Ho&ml. Hnntll. So. 157 (1960). 12. W. I?. JOHNWK ANU It. I'. DECK. J. M&h. Ph,qs. 3, 319 (1962). is. N. I'.b~W'rhNI~ Ii. 8.W.i%4hSE'i. “The Theory of Atomic C’ollisions,” Pntl ed., p. X0. (kford Cniv. Press, London, I~~ngland, 1949. r/i. I:. H. (:om, Ii’o. Xotl. Phys. 27, 187 11!)55). 15. D. 31. FR.~I)KIA ANIJ II. H. C;oou, JR., Hrz~. Moth. Ph!/s. 33, 343 (l!Cl). f/j. II. :\. '~l~l,llOEK, ft'crl. .Ifot/. Ph.//s. 28, 277 (1951;). 1 Y. IT. FANO, /:rr. Jfrrtl. P/u/S. 29, 79 (1957). 18. nr. I<;. I:osE. “Relativistic &:lectron Theory.” Wiley, Sew \iork, 1961. 19. S. I;. Mrrr, Proc. ZZu!/. Sot. (Lonrh) A118, 512 (1928). in. “Higher Tr:~nscrndental E’unctions,” l~atenmn 1\lanuscCpt Project, edited by A. ICrdClyi, 1.01. 1, p. 9. Mc(:r:tw-Hill, Kew York, 1!)53. 21. “‘l’ahlr of Integral Transfornls,” Hatenxm Mitnuwript Project, edited by A. Erdfilyi, \‘ol. 1, p. 314. ~Ic(Grsw--Hill, l;ew IFork, 1041. 3”. I,. I.WVIN, “Dilogarithms and Associated Functions.” Macdonald, London, 1958. 33. 1’. 9. WEBEW, II. hr. FRADKIS. ASI) C. L. HAMMER, .lnn. Php. (4.Y.) 27, 36’2-376 (1964.