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Eikonal approximation for forward and backward scattering
7 August 1972
PHYSICS LETTERS
Volume 40B, number 5
EIKONAL APPROXIMATION
FOR FORWARD AND BACKWARD SCATTERING J.V. NOBLE
Physrcs Department,
Unrverslty of Virgmicl, Charlottesville,
Vugmia 22901,
USA
Received 31 May 1972 Elastic scattermg m the presence of direct plus space-exchange potent& leads to differential cross-sections which are both forward- and backward-peaked. By expandmg about the forward and backward directions simultaneously, one can derive exactly soluble coupled integral equations for the forward- and backward-scattering amplitudes. Scattering cross-sections at medium and high energy usually are strongly forward peaked. In certain instances, owing to the presence of important exchange mechanisms, a prominent beackward peak may also be observed. The object of this note is to generalize the erkonal approximation to the case where both direct and space-exchange potentials are present, m order to provide an approximatron vahd at both forward and backward angles. To clarify the method, I first review the derivation of the eikonal approximation for direct potentials using the methods of Sugar and Blankenbecler [ 11. This provides the framework for the closed-form eikonal approximation solutron of the coupled mtegral equations describing simultaneous forward and backward scattering. Similar coupled equations for coupled-channel and/or spin-dependent problems have been discussed previously by Feshbach [2]. Moreover, a colleague [3] assures me that the central idea of this paper is well-known, but neither he or I have been able to find literature references to bolster this contention. Thus the motivation for this note is at least partly to transfer the ideas contained herein from the realm of folklore to that of literature. Sugar-Blankenbeckel derivation of forward eikonal approximation. Consider elastic scattering from a local (direct) potential: The scattering matrix satisfies the Lrppmann-Schwinger equation (m = fi2 = 1) -
vd(k -k jqK”’k’k2) I
T(k’;
k,
k2)
=
pd(k’_k)
+sdk”
If
i(k2 tie -k”2) where
Sugar and Blankenbecler [l] have shown that the classical eikonal approximation can be derived directly from eq. (1) via the following sequence of steps: i) Expand R” about the forward direction and neglect terms in the free-particle Green’s function quadratic in q” = k” _ k; ii) Write the resulting Green’s function, iii) Fourier transform
[k - q”-k]-’ as -iI
the resulting equation;
flk +qyk; k2)= J$&exp(-iq.r)
t(r)
O”dr exp [i?-(ie - q”-k)];
that is, letting kp=q + k,
(2)
we have
t(Y) = ~,&)[l
-iTdrt(y-nr)] 0
(3)
where v = fiklm; 52.5
I August 1912
PHYSICS LETTERS
Volume 40B, number 5 iv) Write f(r) = V(Y)@(Y),so that
~~)=I-iidd7(r-~)Q(T-V~). 0 Differentiate d$ -=dz
(4)
eq. (4) along the z-direction
(lettmg i 3 6)); this grves
-i O” u Sdru.~[Va(~-ur)0(~-v?)]=~~dr~[Vd(l--u7)9(~-U~)] 0 0
=~V&)$(r).
(9
v) thus @)=exp
[G /
d~‘l/~(Jb*+z’~) zexp{ix(r)} -
(6)
1
-00
and
- exp(-iq*r) lrIk + q, k k2)= Jc2;?
v&j exp{ ix(r)}
,
(7)
and if we assume 4.2 = 0, we find
(7’) with
s@)=G
j%z -00
Simultaneous (tlVlr’)=
and
V&/m)
q,=kme.
forward and backward erkonal approximatron.
V,(Y)G(Y-i)+
with the potential
Let us now suppose the potential
V,,(r)6(rtr’);
is of the form (9)
(9) the Lippmann-Schwinger
i’@‘, k; k*) = &(k’ - k) t Fex(k’ + k) + !$k”
eq. (1) becomes [ Vd(k’- k”) t i?‘,,(k’ t k”)] T(k”, k; k*) (10)
$(k* + IE- k"*)
Clearly the scattering amplitude defined by (10) can possess both a forward-peaked and a backward-peaked part, so that a straightforward expansion about the forward direction is obviated at the outset. To proceed, we must use a trick: Let us split the amplitude T into its forward-peaked and backward-peaked parts. Then T = Tf t T,,, and in operator language, eq. (10) splits into the coupled pair Tf = Vd + V&--K)-’
Tf + V&E-K)-1
Tb = Vex+ I/,,@‘-K)-lTft
V&-K)-’
T,,
(I14
Tb.
(1 lb)
In eq. (1 la), we expand k’ about k: k’ = k + q, and in the intermediate states mvolvmg Tf, we expand k” about k, whereas in those involving T,,, we expand k” about -k. This procedure yields the approximate equations vd(q-q”) TAk t q, k; k2) = f&q) +sdq”
Tf(k+q”,k,k2) ---+
E -q”.k
s
dq”
vex(q +q”) Tb(-k + q”, k; k2) 1e tq”.k
(124
Volume
408,
number
PHYSICS
4
LETTERS
24 July
1972
and T&k
+ 4; k, k2) = v,,(q)
Upon Founer
+ Jdq”
transformmg
Vex(q+q”)Tf(k+q’:R;k2) If -$k
+
s
v&q-q”) Tb(-k+q”, dq” ----g$--k-~-.
k;k2) (1 W
the above, we find
(Isa)
fb(r)=
-lTdifd-r-w)] -i v&)Tdrt,(r+vr) .
v,,(r)[f
0
(13b)
0
Letting tf(Y> = or(r), tb(-?)
= a2(@, we have (suppose vd and v,, are spherically symmetric)
the matrix equation
(14)
Welet
(I::)=
, so that
(
(15) and so, if z = u as before, d$l/dz
= (-l/b)(vd@l
+ vex@l) >
Definmg x(z) = (- l/k)Ji d@,/dz = (-i/u) Elimmatmg, k2 d2@,
--_-
vzx dz2
dz’ vd(@
Vex@‘2,
d@,/k = (--1/u)(~&+J~ + v&2) +zf2) and @, = Gh exp {-ix(z)}, d@,/dz = (-l/u)
(16)
.
we obtam the equations
VexQ, .
(17)
say, Q2 we have (u s k) k2 dVex d’~ ----++~=o. v,‘;, dz dz
The integrating
(18)
factor 2 d@, /dz allows us to write
;($($+(@l)2) =o
(19)
ex
(d@&hQ2 = (v;Jk2)(w2 where w IS a constant
- @;)
of mtegratron.
Letting aI = w$, we get the solutions
IL(r) = sin X(r), cos X(Y) where h(r) = (l/k)J-
dz’ Ve,(\lb2 +z’2). Since at z = --oo we have
(
zi = i , and smce x(-m) ) 0
= 0, we fiid 527
Volume
408, number
4
PHYSICS
LETTERS
24 July
1972
(20)
so that exp {-i(k’-
__ T&k: k k2) = s (2;j
k).r} exp{ix(r)} [v,(r)
cod(r)
-1 ?&(r) sinh(~)]
(21)
and Tb(k: k k2) = J(2;p
exp(+i(k’+
k).r} exp{ix(r)}
If. as in eq. (7’) we neglect the longitudinal * db b
0 (W2
m dbb
J,(b(K’
cash(r)-iVe(r)
momentum-transfer,
sinX(r)] .
(22)
we obtam
[cos~(b)exp{ix(b))-11,
T&k:k; k2) = -ik s -~,(b(k’-k),)
T,,(k: k; k2) = k j- __
[I/,,(r)
t k),) exp{ q(b))
(23)
sin h(b)
(24)
0 (W2
x(b)=;
where
jiz -00
V,(@+z2),
Conclusions. Eqs. (23) and (24) are the generalizations of the standard eikonal formula, eq.(7’). We note that although (k’ - k)L = (k’ + k), = k&f?, the two eqs. (23) and (24) are not expected to have a Common regron of validity and thus should not be combined to produce the patently incorrect result (for V, and Vex real) T=
-ik$
* dbb
---Jo(bksinO)
[exp{i(x+A)}
-11.
(25)
0 (2r92 Eq. (25) is clearly false since m several kinds of exchange scattering we expect h = -x so that eq. (25) would vanish identically. Applications of eqs. (23) and (24) to charge-exchange and (with some drastic assumptions) to heavy-particle exchange processes will be made in a subsequent publication. I am grateful to Dr. P.M. F&bane
and Dr. J.S. Trefrl for helpful discussions.
References [l] R L. Sugar and R. Blankenbecler, Phys. Rev. 183 (1969) 1387. [ 2) H. Feshbach, in Interaction of high-energy particles with nuclei, T.E.O. Erlcson (Academic Press, New York, 1967). [3] P.M. Fishbane, private communication
528
PIOC Intern.
School
of Physics
“Enrico
Fermi”
no. 38, ed.
PHYSICS LETTERS
Volume 40B, number 5
EIKONAL APPROXIMATION
FOR FORWARD AND BACKWARD SCATTERING J.V. NOBLE
Physrcs Department,
Unrverslty of Virgmicl, Charlottesville,
Vugmia 22901,
USA
Received 31 May 1972 Elastic scattermg m the presence of direct plus space-exchange potent& leads to differential cross-sections which are both forward- and backward-peaked. By expandmg about the forward and backward directions simultaneously, one can derive exactly soluble coupled integral equations for the forward- and backward-scattering amplitudes. Scattering cross-sections at medium and high energy usually are strongly forward peaked. In certain instances, owing to the presence of important exchange mechanisms, a prominent beackward peak may also be observed. The object of this note is to generalize the erkonal approximation to the case where both direct and space-exchange potentials are present, m order to provide an approximatron vahd at both forward and backward angles. To clarify the method, I first review the derivation of the eikonal approximation for direct potentials using the methods of Sugar and Blankenbecler [ 11. This provides the framework for the closed-form eikonal approximation solutron of the coupled mtegral equations describing simultaneous forward and backward scattering. Similar coupled equations for coupled-channel and/or spin-dependent problems have been discussed previously by Feshbach [2]. Moreover, a colleague [3] assures me that the central idea of this paper is well-known, but neither he or I have been able to find literature references to bolster this contention. Thus the motivation for this note is at least partly to transfer the ideas contained herein from the realm of folklore to that of literature. Sugar-Blankenbeckel derivation of forward eikonal approximation. Consider elastic scattering from a local (direct) potential: The scattering matrix satisfies the Lrppmann-Schwinger equation (m = fi2 = 1) -
vd(k -k jqK”’k’k2) I
T(k’;
k,
k2)
=
pd(k’_k)
+sdk”
If
i(k2 tie -k”2) where
Sugar and Blankenbecler [l] have shown that the classical eikonal approximation can be derived directly from eq. (1) via the following sequence of steps: i) Expand R” about the forward direction and neglect terms in the free-particle Green’s function quadratic in q” = k” _ k; ii) Write the resulting Green’s function, iii) Fourier transform
[k - q”-k]-’ as -iI
the resulting equation;
flk +qyk; k2)= J$&exp(-iq.r)
t(r)
O”dr exp [i?-(ie - q”-k)];
that is, letting kp=q + k,
(2)
we have
t(Y) = ~,&)[l
-iTdrt(y-nr)] 0
(3)
where v = fiklm; 52.5
I August 1912
PHYSICS LETTERS
Volume 40B, number 5 iv) Write f(r) = V(Y)@(Y),so that
~~)=I-iidd7(r-~)Q(T-V~). 0 Differentiate d$ -=dz
(4)
eq. (4) along the z-direction
(lettmg i 3 6)); this grves
-i O” u Sdru.~[Va(~-ur)0(~-v?)]=~~dr~[Vd(l--u7)9(~-U~)] 0 0
=~V&)$(r).
(9
v) thus @)=exp
[G /
d~‘l/~(Jb*+z’~) zexp{ix(r)} -
(6)
1
-00
and
- exp(-iq*r) lrIk + q, k k2)= Jc2;?
v&j exp{ ix(r)}
,
(7)
and if we assume 4.2 = 0, we find
(7’) with
s@)=G
j%z -00
Simultaneous (tlVlr’)=
and
V&/m)
q,=kme.
forward and backward erkonal approximatron.
V,(Y)G(Y-i)+
with the potential
Let us now suppose the potential
V,,(r)6(rtr’);
is of the form (9)
(9) the Lippmann-Schwinger
i’@‘, k; k*) = &(k’ - k) t Fex(k’ + k) + !$k”
eq. (1) becomes [ Vd(k’- k”) t i?‘,,(k’ t k”)] T(k”, k; k*) (10)
$(k* + IE- k"*)
Clearly the scattering amplitude defined by (10) can possess both a forward-peaked and a backward-peaked part, so that a straightforward expansion about the forward direction is obviated at the outset. To proceed, we must use a trick: Let us split the amplitude T into its forward-peaked and backward-peaked parts. Then T = Tf t T,,, and in operator language, eq. (10) splits into the coupled pair Tf = Vd + V&--K)-’
Tf + V&E-K)-1
Tb = Vex+ I/,,@‘-K)-lTft
V&-K)-’
T,,
(I14
Tb.
(1 lb)
In eq. (1 la), we expand k’ about k: k’ = k + q, and in the intermediate states mvolvmg Tf, we expand k” about k, whereas in those involving T,,, we expand k” about -k. This procedure yields the approximate equations vd(q-q”) TAk t q, k; k2) = f&q) +sdq”
Tf(k+q”,k,k2) ---+
E -q”.k
s
dq”
vex(q +q”) Tb(-k + q”, k; k2) 1e tq”.k
(124
Volume
408,
number
PHYSICS
4
LETTERS
24 July
1972
and T&k
+ 4; k, k2) = v,,(q)
Upon Founer
+ Jdq”
transformmg
Vex(q+q”)Tf(k+q’:R;k2) If -$k
+
s
v&q-q”) Tb(-k+q”, dq” ----g$--k-~-.
k;k2) (1 W
the above, we find
(Isa)
fb(r)=
-lTdifd-r-w)] -i v&)Tdrt,(r+vr) .
v,,(r)[f
0
(13b)
0
Letting tf(Y> = or(r), tb(-?)
= a2(@, we have (suppose vd and v,, are spherically symmetric)
the matrix equation
(14)
Welet
(I::)=
, so that
(
(15) and so, if z = u as before, d$l/dz
= (-l/b)(vd@l
+ vex@l) >
Definmg x(z) = (- l/k)Ji d@,/dz = (-i/u) Elimmatmg, k2 d2@,
--_-
vzx dz2
dz’ vd(@
Vex@‘2,
d@,/k = (--1/u)(~&+J~ + v&2) +zf2) and @, = Gh exp {-ix(z)}, d@,/dz = (-l/u)
(16)
.
we obtam the equations
VexQ, .
(17)
say, Q2 we have (u s k) k2 dVex d’~ ----++~=o. v,‘;, dz dz
The integrating
(18)
factor 2 d@, /dz allows us to write
;($($+(@l)2) =o
(19)
ex
(d@&hQ2 = (v;Jk2)(w2 where w IS a constant
- @;)
of mtegratron.
Letting aI = w$, we get the solutions
IL(r) = sin X(r), cos X(Y) where h(r) = (l/k)J-
dz’ Ve,(\lb2 +z’2). Since at z = --oo we have
(
zi = i , and smce x(-m) ) 0
= 0, we fiid 527
Volume
408, number
4
PHYSICS
LETTERS
24 July
1972
(20)
so that exp {-i(k’-
__ T&k: k k2) = s (2;j
k).r} exp{ix(r)} [v,(r)
cod(r)
-1 ?&(r) sinh(~)]
(21)
and Tb(k: k k2) = J(2;p
exp(+i(k’+
k).r} exp{ix(r)}
If. as in eq. (7’) we neglect the longitudinal * db b
0 (W2
m dbb
J,(b(K’
cash(r)-iVe(r)
momentum-transfer,
sinX(r)] .
(22)
we obtam
[cos~(b)exp{ix(b))-11,
T&k:k; k2) = -ik s -~,(b(k’-k),)
T,,(k: k; k2) = k j- __
[I/,,(r)
t k),) exp{ q(b))
(23)
sin h(b)
(24)
0 (W2
x(b)=;
where
jiz -00
V,(@+z2),
Conclusions. Eqs. (23) and (24) are the generalizations of the standard eikonal formula, eq.(7’). We note that although (k’ - k)L = (k’ + k), = k&f?, the two eqs. (23) and (24) are not expected to have a Common regron of validity and thus should not be combined to produce the patently incorrect result (for V, and Vex real) T=
-ik$
* dbb
---Jo(bksinO)
[exp{i(x+A)}
-11.
(25)
0 (2r92 Eq. (25) is clearly false since m several kinds of exchange scattering we expect h = -x so that eq. (25) would vanish identically. Applications of eqs. (23) and (24) to charge-exchange and (with some drastic assumptions) to heavy-particle exchange processes will be made in a subsequent publication. I am grateful to Dr. P.M. F&bane
and Dr. J.S. Trefrl for helpful discussions.
References [l] R L. Sugar and R. Blankenbecler, Phys. Rev. 183 (1969) 1387. [ 2) H. Feshbach, in Interaction of high-energy particles with nuclei, T.E.O. Erlcson (Academic Press, New York, 1967). [3] P.M. Fishbane, private communication
528
PIOC Intern.
School
of Physics
“Enrico
Fermi”
no. 38, ed.