- Email: [email protected]
Threshold behavior of cross sections of charged particles
ANNALS
OF PHYSICS:
Threshold
7,
133-145
Behavior LUCIANO Indiana
(196%)
of Cross
Sections
FONDLY
AND ROGER
ITniversity,
Bloomington,
of Charged CT.
Particles*
NEWTON
Indiana
The energy dependence of scattering and reaction cross sections at the threshold of a new channel wit,h Coulomb forces is invest,igated. In the case of opposite charges an anomaly is found in which the average over the Coulomb bound state resonances below the threshold energy is discont,inuously different from the limit from above. The resolution and precision necessary, however, make it very difficult to measure the phenomenon.
The energy dependence of scattering and reaction cross sections near the threshold of a new channel has been investigated theoretically in a number of papers (l-7). It was found that when the newly opened processis a discrete and neutral one, i.e., in which each of the outgoing particles has a definite energy and there are no Coulomb forces between them, then certain other elastic and inelastic: cross sections have infinite energy derivatives at the new t,hreshold. The anomaly is usually known by the name “Wigner cusp,” although it need in fact not be a cusp but can be a “rounded step.” It has recently been pointed out (3, 6-7) that the effect could lend itself usefully to the experimental determination of scattering phase shifts as well as pnrit’ies and spins of the reaction participants. It, is known that no infinity in the energy derivative occurs if the two particles in t,he new channel are charged. There is, however: another type of anomaly which occurs in casethey are oppositely charged. It is t’his kind of threshold effect which we want to describe in this paper. As was pointed out in Ref. 7, the physical reason for the infinite derivat’ive of a cross section at the onset of another when the threshold energy is approached from above, is t.he well-known fact that the newly opened cross section starts with an infinite slope. It is the resulting sudden decrease of incident flux available to the old channels that makes itself felt in them. (The fact that the energy derivative of the old cross sections is infinite also when the threshold is approached from below is lesseasily understandable physically. Since that depends * Work supported in part by the National Science Foundation. t On leave of absence from Trieste University on a Fulbright traveling 133
grant.
on the imaginary part of the reaction amplitude, it is an essentially quantum mechanical interference phenomenon.) Nom it is well known that a reaction cross section for charged pnrticles does not start with an infinite slope at it,s threshold (4, 8); consequently other cross sections then have no infinite derivative there either. However, it is also known that if t,he two part,icles in the new channel have opposite charges, then the new cross section immediately starts with a nonzero value at its threshold (4). As a result of the concomitant sudden decrease in available flux, other cross sections should experience a discontinuity at the new threshold. In ot,her words, if the open channel X-matrix is discontinuously enlarged by rows and columns of nonzero values, then there must be compensating discontinuities in the old elements in order for t,he matrix to remain unitary. The details of the phenomenon, as we shall see, are somewhat more complicated than just indicated. The limit of an old S-matrix element as the threshold energy of a new channel with Coulomb attraction is approached from below does in fact not exist. The reason is the presence of oscillations representing resonances at each of the infinitely many Coulomb bound states. Since the nodes of these oscillations have the new t’hreshold as an accumulation point, they cannot be experimentally resolved and it is the aaerage which is observed. This average has a step-like discontinuity when compared to the limit when the threshold is approached from above. The oscillations around the average are large and narrow in one direction and smaller (about one half the step amplitude) and wider in the other. As interesting as the above described threshold effect is in principle, there is at present no hope of observing it experimentally. The width AR of the region around the threshold energy within which the essential deviat,ions owing to the charges of the outgoing particles take place is approximately equal to the depth of t’he Coulomb ground state between them. This energy is 27 ev multiplied by the channel’s reduced mass in Mev and by the square of t,he product of the two charges (in units of the electronic charge). In order to see the effect, the energy resolution in the incoming beam must be good compared t’o AE. It is clear that such a resolution is possible at present only in very low energy experiments such as ionic scatterings of electrons or photons. On the other hand, the relative size of the discontinuity, being a Coulomb effect, will be seen to be proportional to the fine structure constant o( (if the new channel’s charge is unity) and also, roughly, to the matrix element which leads from the incoming to the new channel. In the atomic case the latter again introduces a factor of LY. Therefore in all practical cases the effect is either too small or else requires too fine an energy resolution to be observable with present techniques. In case the two particles in the new channel are repulsively charged, the new cross section starts with all its derivatives equal to zero. Consequently it will
THRESHOLD
CROSS
SECTIOSS
135
make itself felt in the other cross sections only very gradually and no anomalies of any kind can be expected. However, the above argument about the width of the region around the threshold energy within which Coulomb effects are essent,ial applies here also (although there are, of course, no bound states). We can therefore conclude the following. Whenever a threshold anomaly in the form of a cusp or a “rounded step” is expected to be big enough to be experimentally observable, provided Coulomb effects are neglected, it will not be wiped out for practical purposes by the inclusion of the Coulomb field. In other words, if the Coulomb effects destroy the cusp or step, t,hen it was too small to be observable in the first place. The fine structure of the anomaly in the presence of Coulomb forces prevents the actual formation of an infinit,e derivative; instead it shows the details described above. But that fine structure is, practically, not observed. For present experimental purposes, when looking for or measuring the details of a threshold cusp or rounded step, one may therefore disregard entirely whether the particles are charged or neutral. Before proceeding to the details of the formulations, we may make another observation. The question arises if there is a similar anomaly at the threshold of a continuum channel, i.e., one in which there are more than two outgoing particles. Such a process can be either one of dissociation, e.g., ionization, or of particle production. The essential difference is that there is a continuous range of energies available to the individual particles. Let) us first look at the neutral case. The new differential cross section con-tains (pilaX - p’)l” as a factor (where p is the momentum of one of the particles and p%1,,is proportional Do the final energy). The quantity which exerts an influence on the old cross sections via removal of flux is, of course, the new cross section integrated over the energy range available to one of the particles. As we now approach the threshold from above, this integral must vanish since its range shrinks to naught. Moreover, its derivative with respect to the incoming chnmlel energy (i.e., with respect to pi,,) also vanishes, since in addition the integrand vanishes at, the limits of integration. In other words, the integrated new cross section &arts with zero value and slope. Hence it makes itself felt in ot’her channels only gradually and no anomaly will occur there. In casethere is a Coulomb repulsion between particles in t)he new channel, the above statement’ holds a fortiwi. If there is Coulomb &traction, t,hen the nonintegrated new cross section may have a nonzero value at, either or both ends of the energy range available. Hence the int,egrat’ed new cross section now has zero value but) an injnite derivative with respect to the entrance channel energy as t,he threshold is approached from above. Consequently the old cross sections will also have infinit)e energy derivativrs at the new threshold and a cusp or rounded
136
FONDA
AKD
NEWTON
step results. However, the previous arguments concerning the observability of such Coulomb effects apply here also. Therefore the anomaly is expected to be too small to be of experimental consequence. We can therefore conclude that at the threshold of a continuum channel other cross sect’ions do not exhibit any observable anomalies, regardless of whether the particles are charged or neutral. We now proceed to the details for t’he case of a discrete channel with charges. Section II contains a brief generalization of the formulation of Ref. 7 t’o the presence of Coulomb fields. Section III deals with the threshold behavior. There is an Appendix concerning some limiting properties of confluent hypergeometric functions needed for the t’hreshold behavior of Coulomb wave functions. II
We shall formulate the scattering and reaction amplitudes as in Ref. 7, but with the changes necessary if there are Coulomb fields present in some or all of the channel. It will be assumed that Coulomb-like tails exist in the diagonal potentials only. If the problem is one of internal excitation as in the illustrative example at the beginning of Ref. 7, then it is readily seen t’hat that is so. We write the amplitude mat’rix (9) e(k,k’)
= e,(k,k’)
+ 2?riY*(k)iLK-‘(3,
-
S)i?Y(k’),
(11
where we have suppressed all indices, in the notation of Ref. 7; k is the incoming momentum, k’ the outgoing momentum, K the diagonal matrix of the channel momenta, S is the unitary and symmetric scattering matrix, and the subscript ‘lc” refers to pure Coulomb values. 8, = >dNK? is the amplitude
csc’( ,4@) exp[ - iN log sin’( >$O) + i7r]
whose modulus squared is the Rutherford 8,
is the Coulomb
scattering
matrix,
=
cross section;
paL-4
with
uL = arg l?(L + 1 + iiy), N = AK-l, A = (e2/h2)M2&
= (aZ1Z2)Mc/h.
Notice that all these quantities are diagonal matrices referring to all channels and the elements of A can vary from channel to channel. For a neutral channel the corresponding elements of A and 8, vanish and the result goes over into that of Ref. 7. Notice also that we have multiplied the entire expression by the phase factor exp( -2iuo) in order to avoid unobservable oscillations at zero energy.
THRESHOLD
CROSS
137
SECTIOSS
The regular matrix solution G(K,r) is defined by the same boundary tion as in Ref. 7, while an irregular solution F(K,r) is defined by
condi-
By F( --K,r) we shall always mean the value obtained from F(K,r) by going from K bo -K around the origin in the negative (clockwise) sense. We then have for real K F( -K,r) The solutions tentials are
= F”(K,r).
in the absence of anything Fo(K,r) Go(K,r)
but the Coulomb
and centrifugal
= (iK)LIYiN*L+I,2(2iKT),
po(2)
= M~N,L+I,z (2iKr)/(ziK>“+‘(aL
+ l)!!,
(3)
where IV and M are the irregular and regular confluent hypergeometric functions in t,he notation of Whittaker and Watson (10). In t’he presence of the matrix potential V in addition to the (diagonal) Coulomb and centrifugal potentials, the solutions are F(K,r)
= Fo(K,r)
- Irn dr’F(K,r’)V(r’)$jO(K,r’,r) 7
(4)
G(K,r)
= Go(K,r)
+ [‘dr’G(K,r’)V(r’)so(K,r’,r),
(5)
and
where =: (2M/L!)r(L
+ 1 - iN)[Go(K,r)Fo(K,r’)
The S-matrix is obtained by expressing and r;l( - K,r). Since the Wronskian F(K,r) 7, is (F(K,r);
F( -K,r))
(6)
- Go(K$)Fo(K,r)].
G(K,r) as a linear combination of matrix, as defined by (3.11)) Ref.
= 2iK2L+11Pe--N~,
we find G(K,r)
== 6 (FT(-K)K--?L-leNrL![r(Z;
+ 1 + iN)]-lF(K,r)
- FT(K)K-2L-‘eN”L![r(L
+ 1 - iN)]-‘F(
-K,r)],
where F(K)
= @!)-‘I‘@
+ 1 - iN)M(F(K,r);
G(K,r).
(7)
138
FOSDA
The &matrix
AND
NEWTON
then is, in view of its symmetry,
,y = e--i’o~~~-1’zk,-L-1’zeN=‘2[r(~ + 1 - iN)]?L!F(K)P( x
r(~
Insertion of the integral equation for F(K,r) F(K)
= 1 + (2M/L!)
+
-K)
1 +
(L!)-’
iN)e-Na/2KL+li2n~l/Z~-iuo.
and G(K,r)
into F(K)
leads to
r (L + 1 - iN) ~-d~~~(~,~)~(rjC’.(K.,~).
(8)
This in turn yields the generalization of (5.1), Ref. 7, 8, - s = e-iyp+l’pyy~
+ 1 + iN) (L !)-I x mm(~!)-lr(~
+ 1 + iN)e-Nai2~l.+ll?e-iso,
(‘3)
where m = +&“”
s0
m drG:c(k’,r)V(r)G”(k’,r)F-‘(
-K)M1’2.
(10)
These equations contain all relevant statement’s about t’he behavior of an Smatrix element at its own threshold. The matrix functions G and Goare well behaved as any k tends to zero. They are so in the absenceof Coulomb forces, and their integral equation’ shows that for every fixed r, they are independent of the Coulomb tail. The matrix F-‘( -K) has in general a finite limit as any /i tends to zero from above, as we shall see. The entire low energy behavior of any channel is therefore contained in the diagonal outside factors in (9). ?;ow an individual element of that factor is (with 17and a being elements of N and -4)
z.
&ii
e- aYLz+1’2z.z/I!,
if
a > 0,
dG
1a I”+““( -i)‘/Z!,
if
a < 0,
if
u=O.
2+1/z
1 k:
9
(11)
These limits contain all the known low-energy statements about elastic, exoergic and endoergic cross sections near their threshold, for Coulomb repulsive, attractive and neutral channels. In particular notice that in t.he Coulomb attractive case all angular momenta start with nonzero values at zero energy. Since in practical casesa is usually small, however, the 1 = 0 term will dominate. ’ Go(K,r) can be defined by an integral equation like (5), with an inhomogeneity equal to Go in the absence of a Coulomb field, and B equal to the Coulomb potential. It is then clear that Go(K,r), as well as G(K,r), depends only on the potential between zero and T.
THRESHOLD
CROSS
SECTIOIG3
139
111 We now want to examine the energy dependence of an S-matrix element in the vicinity of the threshold of another. The anomalous behavior, if any, must then come from m in (9), since the out’side factors depend on the energies of their own channels only. Equation (10) shows that t’he anomalous Coulomb behavior we are looking for must come from the mat)rix F’( -K) on t)he right, since t’he funct,ions Ga and G are, for fixed r, independent of t,he Coulomb tail. If V(r) is well enough behaved at r --f 00, the integral is a smooth function of all Ii’s. We must therefore examine F( -h’) in the vicinity of k, = 0, for real positive k, and for positive imaginary I<, (it’ was shown in Ref. 7, that below threshold, k, must be taken t’o the positive imaginary axis). Equation (8) shows, by the same argument as above concerning bhe functions G(K,r) that the entire anomalous Coulomb behavior of F(K) must come from the diagonal left-hand factor r(L + 1 - iN)F,(K,r). Hence only the F,p(K) element comains an anomaly at the a-t,hreshold. We t.herefore need to look only at t,he function r(l + 1 - iq) . ja(x,r) in t,he neighborhood of k = 0 along t.he negative real and along the negative imaginary axes. We shall write I?(Z + 1 -
iq)fo(k,r)
= -(“ry+l
/ a IPz+le-ikrIl(/ a 1r,r]),
(12)
where I,(/
a j r,q)
= - (ik)‘(2r)-‘-’
1a 1--21p1e2krr(Z + 1 - i71)Wli~,~+~~~(2Z’X.~). (13)
If me use the integral representations for the irregular confluent hypergeometric function given by Whittaker and Watson (Ref. 10, p. X39), the function I,( 1a j r,q) is readily seen to he
where the contour C is given in Fig. 1; t,he upper sign in the integrand holds in the respulsive case, when a = ?iq > 0, and the lower in the attractive case, a < 0. The limiting behavior of 11 as k -+ 0 along the real and imaginary axes is determined in the Appendix. It is found that in the repulsive case Iyz + 1 -
iq)fe(xI,T)
- (2ry+l kY+l
j a ,22+1 s,m dxe-2~“l’“x2ze-“-~
(14)
for both kinds of limits, i.e., along the negative real axis as Ivell as along the negative imaginary axis. As a result there is no discontinuity, and, as expected, there is no anomaly in the Coulomb repulsive case.
140
tive
FONDA
FIG.
I. Path
C of integration
FIG. 7.
2. Distortion
NEWTON
in the complex
of the path
FIG.
AND
C in Fig.
3. Path
plane
1 for
D for
for the integral
the attractive
the
integral
11 , Eqs.
Coulomh
(139,
(A.1).
case and real posi-
in (A.3).
In the attractive case it is found that the limit as lz--, 0 along the real axis exists, but that along the negative imaginary axis it does not. If we define 4+-f(k) = ;i%P(k) then for 12on the positive imaginary A[r(Z
- f(k)
(15)
axis near k = 0,
+ 1 + iq>fo(-k,r)l
GZ l, zle::,
, ! a 121+1ga(O,r)
(16)
or, using (15) for k, 4 0, A[I’(L
+ 1 + iN)Fo(
-K,r)l
= s;;
where PC” is the projection operator use of (10) gives in the neighborhood
i*‘N’ 1A izL+l @X’GQ(K r) ,N , 7 >
onto the a-channel. Insertion below the a-threshold
k,=Q
,,z -
M1,2 I 9 y+l ire ___ (L!)2 Py3?73P2 sin ?r j N (
(16’)
in (8) and
irl‘v
AF(--K)
,
F( -K)]k,="+.
THRESHOLD
CROSS
141
SECTIONS
From t)his we obtain after some simplification
where on the right 3~ is meant as t’he limit as the a-threshold is approached from above. The corresponding expression for the t’hreshold behavior of the S-matrix is readily found, by use of (9), to bc: A(,9 -
&)
= (8 -
&)P’“‘[(-l)L”(l
- e-2*i’qa’) + P(“‘(X
Since we are interested
-
sop’]-‘P’“‘(S
in an element of the S-matrix ,iimO (S,), a’
-
SC).
(18)
other than LYand since
= (-l)‘,,
we get the final expression:
where S, = P’“‘S Pea’ is the submatrix of &’ referring to elastic scattering in the a-channel. The sum on o( and CY’ext)ends over the all possible independent variables such as orbital angular momentum and spin belonging to the a-channel. lcrom (1)) (IS’) and using rotational invariance [see (4), Ref. 7] we get the threshold energy dependence of t,he scattering amplitude: ABgjv,y,,,J(k’)
= --id; X
,,,,m~j,j,,
v’m
i’-z’~~-lY;~Y’(k’)
O,V)Czfjr(J,V;
Cl,j(J,V; X
{[S,”
-
V -
V’,V’)f$~j,~Q,
eir(L”-“‘~ai)]-l~a~,j,,a~,‘j,‘SJ,~,’j,r,y~,j,
(19) ,
where the z-axis has been t,aken in the direction of the incoming beam. Equation (19) immediately yields bhe corresponding result for the tot,al cross section via the optical theorem: ACTZY1 =- -2aRe
c l/q21 2Z’Z,Z,‘j,j,‘J
+ 1) (21’ + 1) iz-%~-”
All the S-matrix elements appearing on the right-hand side of (20) are meant as limits when the a-threshold is approached from above. Because of t#he pres-
142
FONDA
AND
NEWTON
ence of the oscillatory term in the curly bracket, the limit of the total cross section from below the a-threshold is not a well-defined quantity. As stated in Section I, the oscillations represent resonances at each of the infinitely many Coulomb bound states in the a-channel. They become faster and faster as / k, 1 approaches zero and have X-, = 0 as an accumulation point. It is then clear that experimentally no more than their average can be expected to be observable. The average of (20) is simply given by: total AW”,,
=
-
2n Re ,,,,,Ij,j,,J
Am
+ 1) (21’ + l)~z-z’4-2 (21)
X
Clj(J,Y; O,v)CI’j(J,v; O,~)S~lj,,l,i,~[X~!Jl--IJal,j,.~l,~j,,XJ,l,’j,’.~l’j.
We get a qualitative idea of the size of the maxima and minima of the oscillalations by taking the special case when S, is diagonal in 1 and the Coulomb field is weak. Then by (9) and (11)
and I Xpa[Xm - e--?‘q--1& since
As@ = O( / a yi2)
and
I&* = O( j (J,I) As, -
(-l)L
= O(l a [?l”+l).
In other words, the absolute maxima are high above t’he average (and narrow) while the absolute minima are only little below the average (and broad). It is easily seen that the minima are below the average by about half the size of the st,ep of the latter. The expression of the jump in the differential cross section will be more complicated. In that case one has to average t’he square of the modulus of the scattering amplitude. For practical purposes, however, the jump is usually small and comes almost entirely from the X-wave. It is then easy t’o obtain all relevant results from (19) in a straighforward approximation. If the incident beam is in the positive z-direction and P # 01 # y we have: A gfi aj,rjf(k’)
w - ?!?2j+1
Im C 0Bju,~~“iJve~~“,yj,u’(k’)8*gjv,rj)v)(k’) J
,
(22)
(23) where
is the inelastic cross section from the P-channel to the a-channel.
THRESHOLD
CROSS
SECTIONS
143
APPEXDIX
We are interested here in t,he low energy limits of irregular Coulomb wave functions or the limits of irregular confluent hypergeometric functions as the argument tends to zero while one of the indices tends to infinity. The function t,o be considered is given by ( 13) :
where L = 2 / a 1T > 0, 1is an integer, 7 = ) a l/l{, t’he contour C is given in Fig. 1, and the argument of ( -2) is to be zero when C cuts the negative real axis. The contour C is so chosen that t’he point x = &l/iv is outside it, We first take the repulsive case, i.e., when the upper sign in the integrand is taken, and f is either negabive real or posit.ive imaginary. In either case we can proceed as in Whittaker and Writ-son, pp. 244-245, and let, the circle shrink to naught. We then get
(A.2)
This is the limit as A+ 0 in the repulsive case, both for negative real and negat,ive imaginary k. Nom consider the attractive case. For real positive q we distort t*hecontour to that in Fig. 2 and then let the circle shrink to naught. We then obtain
(A.3)
where the path of integration, D, is shown in Fig. 3. For 7 on the negative imaginary axis we have
which is the sum of integrals over the contours Cl and Cz indicated in Fig. 4. In the first we may let 17 j -+ m and get sin 4 171I -
7r
0 I 1(0 3
e--in(l-19j)
191-m
- 1 1 dXe-=Z21e=1. 2?ri c
144
FONDA
APiD
XEWTON
FIG. 4. Split of contour C into CI and CZ for the attractive
case with negative imaginary
FIG. 5. Contour
Dl for the integral
$I in (A.4).
FIG. 6. Contour
D2 for the integral
$2 in (A.4).
v.
In the second we first shrink the circle to naught and then let 177I-+ co: sin 74 191 - I> I (2) 3 (eir(i-lTp _ e-ir(l-lel) ) ki 1 nIvl-00
s,, dze-fPz21ez-1 .
Consequently
I1 -+ cotdlrll III-J
- 031 - 32 7
(A.41
where
32 = ; s,, d.ze-fzz2’e=1. The paths of integration D1 and Dz are shown in Figs. 5 and 6. The limit does not exist in this case, because of the cotangent term.
THRESHOLD
CROSS
145
SECTIONS
Now with the definition AIL
=
lim 11 - II k-O+
we can write
for large q on the negative imaginary
by the use of (A.3). 31 is readily converted
Furthermore
axis
into Bessel’s integral
and one obtains
it is easily seen that
M,,J+1,2(2ikr)
iz
iz+’ (2r)“2k2+1
1a ~-z-“2(2z + l)!J*1+1(2(2
/ a ( p),
so that go(O,r) = ($#”
1a 1-z-1’21!J21+1(2(2 [ a 1rp2)
and e-iris1 AII
RECEIV.ED:
February
%
-
~
?!
sin 7rIpj I!
2-zrz-$o(0,r).
18, 1959 REFERENCES
1. E. P. WIGNER, Ph+s. Rev. 73. 1002 (1948). 8. R. H. CAPPS AND W. G. HOLLADAY, Phys. Rev. 99, 931 (1955), Appendix B. S. A. I. BAZ, Soviet Phys. JETP, 6, 709 (1958); J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 923 (1957). 4. G. B.REIT, Phys. Rev. 107, 1612 (1957). 5. R. G. NEWTON, Annals of Physics 4, 29 (1958). 6. R. Ii. ADAIR, Phys. Rev. 111, 632 (1958). 7. R. G. NEWTON, Phys. Rev. in press (1959). 8. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics.” Wiley, New York and London, 1952. 9. See, e.g., L. I. SCHIFF, “Quantum Mechanics,” 2nd ed., p. 120, Eq. (20.24), McGrawHill, New York, 1955. “A Course of Modern Analysis.” Macmillan, 10. l3. T. WHITTAKER AND G. N. WATSON, New York, 1948.
OF PHYSICS:
Threshold
7,
133-145
Behavior LUCIANO Indiana
(196%)
of Cross
Sections
FONDLY
AND ROGER
ITniversity,
Bloomington,
of Charged CT.
Particles*
NEWTON
Indiana
The energy dependence of scattering and reaction cross sections at the threshold of a new channel wit,h Coulomb forces is invest,igated. In the case of opposite charges an anomaly is found in which the average over the Coulomb bound state resonances below the threshold energy is discont,inuously different from the limit from above. The resolution and precision necessary, however, make it very difficult to measure the phenomenon.
The energy dependence of scattering and reaction cross sections near the threshold of a new channel has been investigated theoretically in a number of papers (l-7). It was found that when the newly opened processis a discrete and neutral one, i.e., in which each of the outgoing particles has a definite energy and there are no Coulomb forces between them, then certain other elastic and inelastic: cross sections have infinite energy derivatives at the new t,hreshold. The anomaly is usually known by the name “Wigner cusp,” although it need in fact not be a cusp but can be a “rounded step.” It has recently been pointed out (3, 6-7) that the effect could lend itself usefully to the experimental determination of scattering phase shifts as well as pnrit’ies and spins of the reaction participants. It, is known that no infinity in the energy derivative occurs if the two particles in t,he new channel are charged. There is, however: another type of anomaly which occurs in casethey are oppositely charged. It is t’his kind of threshold effect which we want to describe in this paper. As was pointed out in Ref. 7, the physical reason for the infinite derivat’ive of a cross section at the onset of another when the threshold energy is approached from above, is t.he well-known fact that the newly opened cross section starts with an infinite slope. It is the resulting sudden decrease of incident flux available to the old channels that makes itself felt in them. (The fact that the energy derivative of the old cross sections is infinite also when the threshold is approached from below is lesseasily understandable physically. Since that depends * Work supported in part by the National Science Foundation. t On leave of absence from Trieste University on a Fulbright traveling 133
grant.
on the imaginary part of the reaction amplitude, it is an essentially quantum mechanical interference phenomenon.) Nom it is well known that a reaction cross section for charged pnrticles does not start with an infinite slope at it,s threshold (4, 8); consequently other cross sections then have no infinite derivative there either. However, it is also known that if t,he two part,icles in the new channel have opposite charges, then the new cross section immediately starts with a nonzero value at its threshold (4). As a result of the concomitant sudden decrease in available flux, other cross sections should experience a discontinuity at the new threshold. In ot,her words, if the open channel X-matrix is discontinuously enlarged by rows and columns of nonzero values, then there must be compensating discontinuities in the old elements in order for t,he matrix to remain unitary. The details of the phenomenon, as we shall see, are somewhat more complicated than just indicated. The limit of an old S-matrix element as the threshold energy of a new channel with Coulomb attraction is approached from below does in fact not exist. The reason is the presence of oscillations representing resonances at each of the infinitely many Coulomb bound states. Since the nodes of these oscillations have the new t’hreshold as an accumulation point, they cannot be experimentally resolved and it is the aaerage which is observed. This average has a step-like discontinuity when compared to the limit when the threshold is approached from above. The oscillations around the average are large and narrow in one direction and smaller (about one half the step amplitude) and wider in the other. As interesting as the above described threshold effect is in principle, there is at present no hope of observing it experimentally. The width AR of the region around the threshold energy within which the essential deviat,ions owing to the charges of the outgoing particles take place is approximately equal to the depth of t’he Coulomb ground state between them. This energy is 27 ev multiplied by the channel’s reduced mass in Mev and by the square of t,he product of the two charges (in units of the electronic charge). In order to see the effect, the energy resolution in the incoming beam must be good compared t’o AE. It is clear that such a resolution is possible at present only in very low energy experiments such as ionic scatterings of electrons or photons. On the other hand, the relative size of the discontinuity, being a Coulomb effect, will be seen to be proportional to the fine structure constant o( (if the new channel’s charge is unity) and also, roughly, to the matrix element which leads from the incoming to the new channel. In the atomic case the latter again introduces a factor of LY. Therefore in all practical cases the effect is either too small or else requires too fine an energy resolution to be observable with present techniques. In case the two particles in the new channel are repulsively charged, the new cross section starts with all its derivatives equal to zero. Consequently it will
THRESHOLD
CROSS
SECTIOSS
135
make itself felt in the other cross sections only very gradually and no anomalies of any kind can be expected. However, the above argument about the width of the region around the threshold energy within which Coulomb effects are essent,ial applies here also (although there are, of course, no bound states). We can therefore conclude the following. Whenever a threshold anomaly in the form of a cusp or a “rounded step” is expected to be big enough to be experimentally observable, provided Coulomb effects are neglected, it will not be wiped out for practical purposes by the inclusion of the Coulomb field. In other words, if the Coulomb effects destroy the cusp or step, t,hen it was too small to be observable in the first place. The fine structure of the anomaly in the presence of Coulomb forces prevents the actual formation of an infinit,e derivative; instead it shows the details described above. But that fine structure is, practically, not observed. For present experimental purposes, when looking for or measuring the details of a threshold cusp or rounded step, one may therefore disregard entirely whether the particles are charged or neutral. Before proceeding to the details of the formulations, we may make another observation. The question arises if there is a similar anomaly at the threshold of a continuum channel, i.e., one in which there are more than two outgoing particles. Such a process can be either one of dissociation, e.g., ionization, or of particle production. The essential difference is that there is a continuous range of energies available to the individual particles. Let) us first look at the neutral case. The new differential cross section con-tains (pilaX - p’)l” as a factor (where p is the momentum of one of the particles and p%1,,is proportional Do the final energy). The quantity which exerts an influence on the old cross sections via removal of flux is, of course, the new cross section integrated over the energy range available to one of the particles. As we now approach the threshold from above, this integral must vanish since its range shrinks to naught. Moreover, its derivative with respect to the incoming chnmlel energy (i.e., with respect to pi,,) also vanishes, since in addition the integrand vanishes at, the limits of integration. In other words, the integrated new cross section &arts with zero value and slope. Hence it makes itself felt in ot’her channels only gradually and no anomaly will occur there. In casethere is a Coulomb repulsion between particles in t)he new channel, the above statement’ holds a fortiwi. If there is Coulomb &traction, t,hen the nonintegrated new cross section may have a nonzero value at, either or both ends of the energy range available. Hence the int,egrat’ed new cross section now has zero value but) an injnite derivative with respect to the entrance channel energy as t,he threshold is approached from above. Consequently the old cross sections will also have infinit)e energy derivativrs at the new threshold and a cusp or rounded
136
FONDA
AKD
NEWTON
step results. However, the previous arguments concerning the observability of such Coulomb effects apply here also. Therefore the anomaly is expected to be too small to be of experimental consequence. We can therefore conclude that at the threshold of a continuum channel other cross sect’ions do not exhibit any observable anomalies, regardless of whether the particles are charged or neutral. We now proceed to the details for t’he case of a discrete channel with charges. Section II contains a brief generalization of the formulation of Ref. 7 t’o the presence of Coulomb fields. Section III deals with the threshold behavior. There is an Appendix concerning some limiting properties of confluent hypergeometric functions needed for the t’hreshold behavior of Coulomb wave functions. II
We shall formulate the scattering and reaction amplitudes as in Ref. 7, but with the changes necessary if there are Coulomb fields present in some or all of the channel. It will be assumed that Coulomb-like tails exist in the diagonal potentials only. If the problem is one of internal excitation as in the illustrative example at the beginning of Ref. 7, then it is readily seen t’hat that is so. We write the amplitude mat’rix (9) e(k,k’)
= e,(k,k’)
+ 2?riY*(k)iLK-‘(3,
-
S)i?Y(k’),
(11
where we have suppressed all indices, in the notation of Ref. 7; k is the incoming momentum, k’ the outgoing momentum, K the diagonal matrix of the channel momenta, S is the unitary and symmetric scattering matrix, and the subscript ‘lc” refers to pure Coulomb values. 8, = >dNK? is the amplitude
csc’( ,4@) exp[ - iN log sin’( >$O) + i7r]
whose modulus squared is the Rutherford 8,
is the Coulomb
scattering
matrix,
=
cross section;
paL-4
with
uL = arg l?(L + 1 + iiy), N = AK-l, A = (e2/h2)M2&
= (aZ1Z2)Mc/h.
Notice that all these quantities are diagonal matrices referring to all channels and the elements of A can vary from channel to channel. For a neutral channel the corresponding elements of A and 8, vanish and the result goes over into that of Ref. 7. Notice also that we have multiplied the entire expression by the phase factor exp( -2iuo) in order to avoid unobservable oscillations at zero energy.
THRESHOLD
CROSS
137
SECTIOSS
The regular matrix solution G(K,r) is defined by the same boundary tion as in Ref. 7, while an irregular solution F(K,r) is defined by
condi-
By F( --K,r) we shall always mean the value obtained from F(K,r) by going from K bo -K around the origin in the negative (clockwise) sense. We then have for real K F( -K,r) The solutions tentials are
= F”(K,r).
in the absence of anything Fo(K,r) Go(K,r)
but the Coulomb
and centrifugal
= (iK)LIYiN*L+I,2(2iKT),
po(2)
= M~N,L+I,z (2iKr)/(ziK>“+‘(aL
+ l)!!,
(3)
where IV and M are the irregular and regular confluent hypergeometric functions in t,he notation of Whittaker and Watson (10). In t’he presence of the matrix potential V in addition to the (diagonal) Coulomb and centrifugal potentials, the solutions are F(K,r)
= Fo(K,r)
- Irn dr’F(K,r’)V(r’)$jO(K,r’,r) 7
(4)
G(K,r)
= Go(K,r)
+ [‘dr’G(K,r’)V(r’)so(K,r’,r),
(5)
and
where =: (2M/L!)r(L
+ 1 - iN)[Go(K,r)Fo(K,r’)
The S-matrix is obtained by expressing and r;l( - K,r). Since the Wronskian F(K,r) 7, is (F(K,r);
F( -K,r))
(6)
- Go(K$)Fo(K,r)].
G(K,r) as a linear combination of matrix, as defined by (3.11)) Ref.
= 2iK2L+11Pe--N~,
we find G(K,r)
== 6 (FT(-K)K--?L-leNrL![r(Z;
+ 1 + iN)]-lF(K,r)
- FT(K)K-2L-‘eN”L![r(L
+ 1 - iN)]-‘F(
-K,r)],
where F(K)
= @!)-‘I‘@
+ 1 - iN)M(F(K,r);
G(K,r).
(7)
138
FOSDA
The &matrix
AND
NEWTON
then is, in view of its symmetry,
,y = e--i’o~~~-1’zk,-L-1’zeN=‘2[r(~ + 1 - iN)]?L!F(K)P( x
r(~
Insertion of the integral equation for F(K,r) F(K)
= 1 + (2M/L!)
+
-K)
1 +
(L!)-’
iN)e-Na/2KL+li2n~l/Z~-iuo.
and G(K,r)
into F(K)
leads to
r (L + 1 - iN) ~-d~~~(~,~)~(rjC’.(K.,~).
(8)
This in turn yields the generalization of (5.1), Ref. 7, 8, - s = e-iyp+l’pyy~
+ 1 + iN) (L !)-I x mm(~!)-lr(~
+ 1 + iN)e-Nai2~l.+ll?e-iso,
(‘3)
where m = +&“”
s0
m drG:c(k’,r)V(r)G”(k’,r)F-‘(
-K)M1’2.
(10)
These equations contain all relevant statement’s about t’he behavior of an Smatrix element at its own threshold. The matrix functions G and Goare well behaved as any k tends to zero. They are so in the absenceof Coulomb forces, and their integral equation’ shows that for every fixed r, they are independent of the Coulomb tail. The matrix F-‘( -K) has in general a finite limit as any /i tends to zero from above, as we shall see. The entire low energy behavior of any channel is therefore contained in the diagonal outside factors in (9). ?;ow an individual element of that factor is (with 17and a being elements of N and -4)
z.
&ii
e- aYLz+1’2z.z/I!,
if
a > 0,
dG
1a I”+““( -i)‘/Z!,
if
a < 0,
if
u=O.
2+1/z
1 k:
9
(11)
These limits contain all the known low-energy statements about elastic, exoergic and endoergic cross sections near their threshold, for Coulomb repulsive, attractive and neutral channels. In particular notice that in t.he Coulomb attractive case all angular momenta start with nonzero values at zero energy. Since in practical casesa is usually small, however, the 1 = 0 term will dominate. ’ Go(K,r) can be defined by an integral equation like (5), with an inhomogeneity equal to Go in the absence of a Coulomb field, and B equal to the Coulomb potential. It is then clear that Go(K,r), as well as G(K,r), depends only on the potential between zero and T.
THRESHOLD
CROSS
SECTIOIG3
139
111 We now want to examine the energy dependence of an S-matrix element in the vicinity of the threshold of another. The anomalous behavior, if any, must then come from m in (9), since the out’side factors depend on the energies of their own channels only. Equation (10) shows that t’he anomalous Coulomb behavior we are looking for must come from the mat)rix F’( -K) on t)he right, since t’he funct,ions Ga and G are, for fixed r, independent of t,he Coulomb tail. If V(r) is well enough behaved at r --f 00, the integral is a smooth function of all Ii’s. We must therefore examine F( -h’) in the vicinity of k, = 0, for real positive k, and for positive imaginary I<, (it’ was shown in Ref. 7, that below threshold, k, must be taken t’o the positive imaginary axis). Equation (8) shows, by the same argument as above concerning bhe functions G(K,r) that the entire anomalous Coulomb behavior of F(K) must come from the diagonal left-hand factor r(L + 1 - iN)F,(K,r). Hence only the F,p(K) element comains an anomaly at the a-t,hreshold. We t.herefore need to look only at t,he function r(l + 1 - iq) . ja(x,r) in t,he neighborhood of k = 0 along t.he negative real and along the negative imaginary axes. We shall write I?(Z + 1 -
iq)fo(k,r)
= -(“ry+l
/ a IPz+le-ikrIl(/ a 1r,r]),
(12)
where I,(/
a j r,q)
= - (ik)‘(2r)-‘-’
1a 1--21p1e2krr(Z + 1 - i71)Wli~,~+~~~(2Z’X.~). (13)
If me use the integral representations for the irregular confluent hypergeometric function given by Whittaker and Watson (Ref. 10, p. X39), the function I,( 1a j r,q) is readily seen to he
where the contour C is given in Fig. 1; t,he upper sign in the integrand holds in the respulsive case, when a = ?iq > 0, and the lower in the attractive case, a < 0. The limiting behavior of 11 as k -+ 0 along the real and imaginary axes is determined in the Appendix. It is found that in the repulsive case Iyz + 1 -
iq)fe(xI,T)
- (2ry+l kY+l
j a ,22+1 s,m dxe-2~“l’“x2ze-“-~
(14)
for both kinds of limits, i.e., along the negative real axis as Ivell as along the negative imaginary axis. As a result there is no discontinuity, and, as expected, there is no anomaly in the Coulomb repulsive case.
140
tive
FONDA
FIG.
I. Path
C of integration
FIG. 7.
2. Distortion
NEWTON
in the complex
of the path
FIG.
AND
C in Fig.
3. Path
plane
1 for
D for
for the integral
the attractive
the
integral
11 , Eqs.
Coulomh
(139,
(A.1).
case and real posi-
in (A.3).
In the attractive case it is found that the limit as lz--, 0 along the real axis exists, but that along the negative imaginary axis it does not. If we define 4+-f(k) = ;i%P(k) then for 12on the positive imaginary A[r(Z
- f(k)
(15)
axis near k = 0,
+ 1 + iq>fo(-k,r)l
GZ l, zle::,
, ! a 121+1ga(O,r)
(16)
or, using (15) for k, 4 0, A[I’(L
+ 1 + iN)Fo(
-K,r)l
= s;;
where PC” is the projection operator use of (10) gives in the neighborhood
i*‘N’ 1A izL+l @X’GQ(K r) ,N , 7 >
onto the a-channel. Insertion below the a-threshold
k,=Q
,,z -
M1,2 I 9 y+l ire ___ (L!)2 Py3?73P2 sin ?r j N (
(16’)
in (8) and
irl‘v
AF(--K)
,
F( -K)]k,="+.
THRESHOLD
CROSS
141
SECTIONS
From t)his we obtain after some simplification
where on the right 3~ is meant as t’he limit as the a-threshold is approached from above. The corresponding expression for the t’hreshold behavior of the S-matrix is readily found, by use of (9), to bc: A(,9 -
&)
= (8 -
&)P’“‘[(-l)L”(l
- e-2*i’qa’) + P(“‘(X
Since we are interested
-
sop’]-‘P’“‘(S
in an element of the S-matrix ,iimO (S,), a’
-
SC).
(18)
other than LYand since
= (-l)‘,,
we get the final expression:
where S, = P’“‘S Pea’ is the submatrix of &’ referring to elastic scattering in the a-channel. The sum on o( and CY’ext)ends over the all possible independent variables such as orbital angular momentum and spin belonging to the a-channel. lcrom (1)) (IS’) and using rotational invariance [see (4), Ref. 7] we get the threshold energy dependence of t,he scattering amplitude: ABgjv,y,,,J(k’)
= --id; X
,,,,m~j,j,,
v’m
i’-z’~~-lY;~Y’(k’)
O,V)Czfjr(J,V;
Cl,j(J,V; X
{[S,”
-
V -
V’,V’)f$~j,~Q,
eir(L”-“‘~ai)]-l~a~,j,,a~,‘j,‘SJ,~,’j,r,y~,j,
(19) ,
where the z-axis has been t,aken in the direction of the incoming beam. Equation (19) immediately yields bhe corresponding result for the tot,al cross section via the optical theorem: ACTZY1 =- -2aRe
c l/q21 2Z’Z,Z,‘j,j,‘J
+ 1) (21’ + 1) iz-%~-”
All the S-matrix elements appearing on the right-hand side of (20) are meant as limits when the a-threshold is approached from above. Because of t#he pres-
142
FONDA
AND
NEWTON
ence of the oscillatory term in the curly bracket, the limit of the total cross section from below the a-threshold is not a well-defined quantity. As stated in Section I, the oscillations represent resonances at each of the infinitely many Coulomb bound states in the a-channel. They become faster and faster as / k, 1 approaches zero and have X-, = 0 as an accumulation point. It is then clear that experimentally no more than their average can be expected to be observable. The average of (20) is simply given by: total AW”,,
=
-
2n Re ,,,,,Ij,j,,J
Am
+ 1) (21’ + l)~z-z’4-2 (21)
X
Clj(J,Y; O,v)CI’j(J,v; O,~)S~lj,,l,i,~[X~!Jl--IJal,j,.~l,~j,,XJ,l,’j,’.~l’j.
We get a qualitative idea of the size of the maxima and minima of the oscillalations by taking the special case when S, is diagonal in 1 and the Coulomb field is weak. Then by (9) and (11)
and I Xpa[Xm - e--?‘q--1& since
As@ = O( / a yi2)
and
I&* = O( j (J,I) As, -
(-l)L
= O(l a [?l”+l).
In other words, the absolute maxima are high above t’he average (and narrow) while the absolute minima are only little below the average (and broad). It is easily seen that the minima are below the average by about half the size of the st,ep of the latter. The expression of the jump in the differential cross section will be more complicated. In that case one has to average t’he square of the modulus of the scattering amplitude. For practical purposes, however, the jump is usually small and comes almost entirely from the X-wave. It is then easy t’o obtain all relevant results from (19) in a straighforward approximation. If the incident beam is in the positive z-direction and P # 01 # y we have: A gfi aj,rjf(k’)
w - ?!?2j+1
Im C 0Bju,~~“iJve~~“,yj,u’(k’)8*gjv,rj)v)(k’) J
,
(22)
(23) where
is the inelastic cross section from the P-channel to the a-channel.
THRESHOLD
CROSS
SECTIONS
143
APPEXDIX
We are interested here in t,he low energy limits of irregular Coulomb wave functions or the limits of irregular confluent hypergeometric functions as the argument tends to zero while one of the indices tends to infinity. The function t,o be considered is given by ( 13) :
where L = 2 / a 1T > 0, 1is an integer, 7 = ) a l/l{, t’he contour C is given in Fig. 1, and the argument of ( -2) is to be zero when C cuts the negative real axis. The contour C is so chosen that t’he point x = &l/iv is outside it, We first take the repulsive case, i.e., when the upper sign in the integrand is taken, and f is either negabive real or posit.ive imaginary. In either case we can proceed as in Whittaker and Writ-son, pp. 244-245, and let, the circle shrink to naught. We then get
(A.2)
This is the limit as A+ 0 in the repulsive case, both for negative real and negat,ive imaginary k. Nom consider the attractive case. For real positive q we distort t*hecontour to that in Fig. 2 and then let the circle shrink to naught. We then obtain
(A.3)
where the path of integration, D, is shown in Fig. 3. For 7 on the negative imaginary axis we have
which is the sum of integrals over the contours Cl and Cz indicated in Fig. 4. In the first we may let 17 j -+ m and get sin 4 171I -
7r
0 I 1(0 3
e--in(l-19j)
191-m
- 1 1 dXe-=Z21e=1. 2?ri c
144
FONDA
APiD
XEWTON
FIG. 4. Split of contour C into CI and CZ for the attractive
case with negative imaginary
FIG. 5. Contour
Dl for the integral
$I in (A.4).
FIG. 6. Contour
D2 for the integral
$2 in (A.4).
v.
In the second we first shrink the circle to naught and then let 177I-+ co: sin 74 191 - I> I (2) 3 (eir(i-lTp _ e-ir(l-lel) ) ki 1 nIvl-00
s,, dze-fPz21ez-1 .
Consequently
I1 -+ cotdlrll III-J
- 031 - 32 7
(A.41
where
32 = ; s,, d.ze-fzz2’e=1. The paths of integration D1 and Dz are shown in Figs. 5 and 6. The limit does not exist in this case, because of the cotangent term.
THRESHOLD
CROSS
145
SECTIONS
Now with the definition AIL
=
lim 11 - II k-O+
we can write
for large q on the negative imaginary
by the use of (A.3). 31 is readily converted
Furthermore
axis
into Bessel’s integral
and one obtains
it is easily seen that
M,,J+1,2(2ikr)
iz
iz+’ (2r)“2k2+1
1a ~-z-“2(2z + l)!J*1+1(2(2
/ a ( p),
so that go(O,r) = ($#”
1a 1-z-1’21!J21+1(2(2 [ a 1rp2)
and e-iris1 AII
RECEIV.ED:
February
%
-
~
?!
sin 7rIpj I!
2-zrz-$o(0,r).
18, 1959 REFERENCES
1. E. P. WIGNER, Ph+s. Rev. 73. 1002 (1948). 8. R. H. CAPPS AND W. G. HOLLADAY, Phys. Rev. 99, 931 (1955), Appendix B. S. A. I. BAZ, Soviet Phys. JETP, 6, 709 (1958); J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 923 (1957). 4. G. B.REIT, Phys. Rev. 107, 1612 (1957). 5. R. G. NEWTON, Annals of Physics 4, 29 (1958). 6. R. Ii. ADAIR, Phys. Rev. 111, 632 (1958). 7. R. G. NEWTON, Phys. Rev. in press (1959). 8. J. M. BLATT AND V. F. WEISSKOPF, “Theoretical Nuclear Physics.” Wiley, New York and London, 1952. 9. See, e.g., L. I. SCHIFF, “Quantum Mechanics,” 2nd ed., p. 120, Eq. (20.24), McGrawHill, New York, 1955. “A Course of Modern Analysis.” Macmillan, 10. l3. T. WHITTAKER AND G. N. WATSON, New York, 1948.