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Theory of nuclear reactions
-~
Nuclear Physics 50 (1964) l m 1 6 ; (~) North-Holla~ui Publishing Co., Amsterdam Not
to be repx'oducedby photoprint of micxofilm without written permission from the publisher
T H E O R Y O F NUCLEAR REACTIONS
IV. Coulomb lnteractiom in the ~
and Penetration Factors
J. HUMBLET
University of Liege t, Belgium Received 16 May 1963 Abstract: The expansion o f the collision matrix as the sum o f a background term and a series o f resonant terms Oven in part I is extended to include the Coulomb interactions in the ehfmneIn without CUt Off at large distances. The penetration factors introduced are independent o f the channel radii.
1. Introdee~m When the incoming and outgoing radial wave functions associated with a pure Coulomb interaction are considered as functions o f the wave number k, it is well known that they have an essential singularity at the origin k -- 0 of the complex k-plane. The corresponding singularities o f the collision matrix are usually avoided by cutting off the Coulomb interaction at some large distance. Near the thresholds, the analytical behaviour of the collision matrix is then the same as that obtained for neutron channels, and the effect of the penetration o f the Coulomb barrier at very small energies no longer appears explicitly. In the first paper 1) of this series x-s), the element Uo,o o f the collision matrix corresponding to a nuclear reaction c' -~ c appears as a "background" term Qc'~ plus a series o f resonant terms. To establish such an expansion, the Coulomb interaction o f the two fragments in any channel c was neglected when their distance re was larger than be(~> ac). Under such conditions, near the thresholds o f channels c and c', which we now assume not to be neutron channels, the resonant terms are not decreasing exponentially with the energy. Nevertheless, as the sum o f Q¢,o and the resonant terms must exhibit such an exponential decrease, we are forced to conclude that, at least near the thresholds, the background term Q~,o plays an important part in the collision matrix. Since, in practice, very little can be said so far about the structure of the Qo,~ function, this is certainly a drawback of the theory, at least ff one is concerned with the practical fitting of experimental data at low energies. It is the purpose o f the present paper to show that it is possible to avoid the cut-off of the Coulomb field at large distance and, at the same time, to introduce explicitly an exponential energy dependence of the resonant terms near the thresholds. The background term is modified accordingly and in the applications treated up to now 4.), it turned out that it could be completely neglected. t Institut de Math6matique, 15 Avenue des Tilleuls, Liege. I January 1964.
2
J. HUMBLEr
Because of the essential singularity of the Coulomb wave functions in the complex k-plane, it has been necessary to revise here some arguments of part I regarding the definition and properties of the collision matrix. The Coulomb wave functions and their main properties are given in sect. 2. The corresponding expression for the collision matrix and some of its general properties, such as unitarity and symmetry, are established in sect. 3. Its proposed expansion is given in sect. 4; each resonant term is as in part I independent of the channel radii ac, but in contrast to part I, energy-dependent penetration factors are introduced. The resulting expression for the differential cross section is written down in sect. 5. In appendix 1, another expansion of the collision matrix is given, in which our definition of the resonant states is used together with conventional a-dependent penetration factors; this is done mainly to facilitatecomparisons with the R-matrix theory. Appendix 2 contains some of the mathematical arguments which are necessary to justify the expansion given in sect. 4. Preliminary reports on the results given in the present paper were already given last year to the Padua and Brussels conferences s). 2. Coulomb Wave F~c/iom When the Coulomb wave function is analysed in partial waves, its radial factors satisfy equations of the form
(dz~ +k 2- l(l+l)-'~)y,=O,rz
(2.1)
where
tl = ZI Z2eaMh -2.
(2.2)
Two solutions of this equation are the well known regular function Fz and irregular function Gl, whose asymptotic behaviour for large r is Fi "~ sin(kr-~ log(2kr)-½1~+~z)
(larg k[ < ½~),
(2.3a)
G~ ~ cos(kr-~ log(2kr)--½1~+al)
([arg kl < ½~).
(2.3b)
where • and at are defined by = _q
e2~, , = F ( l + 1 + i~)
(2.4)
k' The quantitiesFz, G~, • and as are realwhen k isitselfreal. The first solution F~ is proportional to the regular Whittaker function .At' = .A~.i +½
(2ikr), since Fj ----½P-tl!(-i)z+1 ~ . i+,(2ikr),
(2.5)
1 8~ = - e-~'=[F(l + 1 + i~,)F(l + 1 - i a ) ] * . l!
(2.6)
where
11q~oaY o v h'UCLI~R REACTIONS (IV)
3
The function .,dr is defined according to Buchholtz e), i.e. ~1/ : M,,.,+~(2ikr)/F(21+ 2) for a fixed r; ~Ig is an integral function of k. In terms of well known notations 7, s), we also have et
=
(21+ I)!!C~
=
2,~," 7 i
(
12+ ~t2) . . . (1 +,,2) e 2 " - lJ
-~. 1
= 1 [(/2 + ,'2)... (1 +,'z)]½C o;
(2.7a)
(2.7b)
lI the latter quantity has been normalized in order that et : 1 for k : oo or for a neutron channel ( ~ / : 0). Two other solutions of eq. (2.1) are conveniently defined as
(2.8a)
It(r, k) ---- (G~-iF,)e "~ = i Z e ~ W _ , O~(r, k) :
" - - - " ' : ( - i)'e*'W+, (~,+ zF,)c
(2.8b)
where W_ --- W~,.z+~(e~t~2kr),
W+ ffi W_u.i+~(e-~½~2kr)
(2.9)
are irregular Whittaker functions. Considered as functions of k, they have k ffi 0 and k ffi oo as essential singularities, but they have no other singularity in the complex k-plane. They are multivalued, according to the relations e) W,., ~+~(e't~2kre '2") = W.=,~+½(e'°2kr) + 2imt .~v~,.l+½(2ikr),
r(-l-i,')
W_,,. ~++(e-'°2kre '2"1) : W_u z++(e-'+12kr) +2inn .Ar_~,.
I++(-2ikr),
r(-
l+
i,')
(2.10a)
(2.10b)
where n is an integer and ~ l _ , . , ~ + t ( - 2 i k r ) f f i ( - 1 ) z+t . ~ , , t + t ( 2 1 k r ) . Together with the regular function ~tr, the functions W± satisfy the relation 6, 7)
..4f--
eu
r(l+l-i,')
W+
eZ(a + u)
r(l+l+i,')
IV_
(2.11)
and the Wronskian relations 6, i x) W(.~d', W±) : - 2ik
C ti,a(l ± 0
r(l+ I+ i,')'
W(W+, W_) -- - 2 i k e - n .
(2.12,) (2.12b)
4
J. RUMBI..IgT
According to eq. (2.8), the relations (2.11) and (2.12) may be rewritten as - 2iFt = e - ~ I t
-et~zOt,
(2.13a) (2.14)
W(O t , It) = - 2 i k ;
W(Ft, Or) = - R e - " ,
we also have (2.13b)
2Gi = e-i"Ii+e~'*Oz.
For large values of r, the asymptotic behaviour of W± is such that e) Il " e x p [ - i ( k r - u l o g ( 2 k r ) - ½ g l ) ]
( - 2 g < argk < g),
(2.15a)
Ot " e x p [ + i ( k r - u l o g ( 2 k r ) - ~ n l ) ]
( - g < argk < 2g)
(2.15b)
and accordingly they correspond to incoming and outgoing waves respectively. Later we shall also need the behaviour of .~' and IV+ for very small k. We have )
= ½ F(I + I + ia) W+
(any argk),
\t//
/ \+i~/t
[-/-)xg,t+,C=)[l+OCk)],
targkZ- i
n-0,
C2,6 ) (216b)
where x -- x/8"-~-;r = x / ~
(2.17)
and 0 are positive quantities, the latter being an arbitrarily small constant. Because of the definition (2.8) of It and Or, we have
It(r,ke-t*)= (-1)'O,(r,k ~ - = ,
(2.18a)
Or(r, Re+t') = (-1)'It(r, k)e -~'.
(2.18b)
Further, if k* is properly defined as [klexp(-i arg k), we have Or(r, k)*
=
ite*"/k'W~,/k, t+~(etO2k*r)
= It(r, k*) = (-1)tOt(r, k*e'~)¢"~*,
(2.19a)
It(r, k)* = O,(r, k*) -- ( - 1 ) t l ~ r , k*e-t')¢ 'a*.
(2.19b)
In an open channel, k is a real and positive quantity and we have in partion!ar Or(r, k)* = ll(r, k),
It(r, k)* = O~(r, k)
(open).
(2.20)
* Our definition of the modified Bessel function Ka~+I is that of Watson I). Eq. (2.16a) is easily derived from Buohholtz [ref. 6), eq. (16), p. 97l; eq. (2.16b) agrees with formulae given by Hull and Breit [ref. v), p. 421-432] and by Erd~lyi [ref. xo), eqs. (16) and (18) of section 6.9]. Eq. (2.16a) does not agree with the formulae given by Slater [ref. xl), eqs. (4.4.20) and (4.4.21)], but the latter are marred by a misprint: J and I have to be interchanged in the right.hand sides; a similar correction has to be introduced in his equations corresponding to our eq. (2.16b).
wmonY
or suc~
~cnom
Ov)
5
In a closed channel, arg k = ½n and
Ol(r, k)* -- (-1)'Oz(r, k)e -'~ l,(r, k)*
=
(closed),
(--1)Ill(r, ke-Z'~e -~"
(closed);
(2.21a)
(2.21b)
the last relation, which corresponds to a change of Riemann sheet, will in fact not be needed in what follows. From now on, we shall make I s and Ot single-valued functions of k by introducing a cut in the complex k-plane along the negative imaginary axis. Then -½n < arg k < ½7r
(2.22a)
~ - ~ < arg (k*e~=) < ½~.
(2.22b)
and also
3. Dellaition and Properties of the Collision Matrix Among the various solutions ofeq. (2.1), Fi, O~, .~', W+, W_, the only two which (1) are finearly independent for any definite value o f k oe 0 (cf. eq. (2.12)), (2) have no pole in the complex k-plane for a fixed value o f r (cf. eqs. (2.5), (2.6) and (2.13)) and (3) do not vanish identically (for any r) at some point of the k-plane, are the irregular Whittaker functions W+ and W_ (cf. eq. (2.12b)). Since Ii and Ol differ from the latter functions by factors which have neither zeros nor poles, we may always write the radial wave function in channel c as
uo(ro, k,) -~ xoO¢+ yol¢;
(3.1)
the amplitudes x,, y~ are then uniquely defined in the complex k©-plane for ko 4= 0, provided that the appropriate cuts have been made in this plane. Under such conditions, we are justified in defining a collision matrix by
go, uo,o =
m
(3.2)
where, as in part I, the upper index (c) refers to the fact that c is the only entrance channel present in the total wave function ~'. Because of the relation (2.14) we still have explicitly,as in part I,
U~.© = W(O(~¢')'Ic')/k~" W(#(o ' Oc)/k,
(3.3)
W e again assume that the radialfactors 0(o, 0(~)have the same analyticalproperties as the regular Whittaker functions .#f~, ~fo,.... , because these radial factors bclong to a wave function ~,
t The notation (I.8.2b) refers to eq. (8.2b) of part I.
6
$. HUMBLET
which in fact vanish for re : 0, re : 0 , . . . respectively. More precisely, the only singularities assumed for O~°), ~o,~) in the complex ko-plane as well as in the complex d-plane are branch points at the channel openings (ks = O, k e : 0 . . . . ). The resonant states are again defined as complex poles of Uec, i.e. by the boundary condition
W(¢~~), O~)
= 0.
(3.4)
Because of the relations (I.8.12) and (2.19a), it is evident that to each pole k~, = K~,-i~¢, (for which larg k~,[ < ½~) there corresponds another one, namely k*,d" = - ~ c ~ - i ~ , , the argument of which falls within the limits prescribed by eq. (2.22a). To prove that in any upper k-plane, poles, if any, m a y lie only on the imaginary axis, we may proceed as follows. Assuming that ~,~,< 0, let us consider eq. (I.9.12):
iF.
f
1~.12do~ =
'* (~o,* ~ o' , - oo,~0,)
(3.5)
If the integration, rather than being limited, to the interior region o~, is extended to the whole configuration space, the integral will remain finite, because, in a channel c, ~/'~ behaves like Oc,(r~) = u~(r~, k~,), which is itself proportional to exp(/k~,r~) = exp(-Ire, Ire). For the same reason, each term of the fight-hand side vanishes at infinity, and we are left with
r, afl~V.12doJ = ~
O,
(3.6)
i.e. with F, = 0, or ,o. = 0.
(3.7)
Equation (I.9.15) remains valid, provided that No, is now defined by eq. (I.9.14), but it seems difficult in the present case to give an explicit form for No, corresponding to eq. (I.1.43). The unitarity and symmetry properties of the collision matrix are also readily extended to the present situation. To prove the former of these properties, we have only to note that, whereas eq. (L7.10a) remains unchanged, eq. (I.7.10b), for a closed channel c- should be replaced by (the channel q being open) ¢,~.'* = ( - Vo~Oo)* = -(-1)'°V*~qOo(ko):"-
(closed).
(3.8)
The rest of the proof then proceeds as in ref. 1). In the proof of the symmetry property, eq. (I.7.15) should be replaced by ~"-"' = Z
p+
q'-,(a_,_.L,-v_,_.o_,)Z ~-' v_,_qo_, rp p - rp
(3.9)
THEORY OF N U C I ~
REACTIONS (IV)
and hence t (-- 1)/'t+MqKIiv(-q) = Z tp__£[t~mlOp(kp)_ Uv.qlp(kp)] p+ l'p
_ ~ tp__pU~qOp(kp)(_l)ipe_up.
(3.10)
p- rp
Here again we have used eqs. (2.20), (2.21a), but not (2.21b), and the rest o f the p r o o f also remains unchanged. 4. E x p a ~ o n of the Collision Matrix The differential cross section is not directly proportional to JUt,c]2, but rather to the squared modulus of Tc,c = Uo,~-6c,¢e 2'~°.
(4.1)
Using the relations (1.6.11) and (2.13a), one has
W(O(?), Fc,)/kc, W(+~°, O~e":)lk: l(,c'+-.) r ~+x W(OtJ),Fc'e71k7 v-l)
Tc.c = _2ie,(..,+..) • = - 2ze,. s, e
k~. k~
,-:,
Oosoe
k,)
(4.2)
(4.3)
In the numerator of the last factor, according to eq. (2.5), we have F~,82 lk~.r- 1 = ½1'!(ikc,)-v- 1 ~ , ° , r +,(2ike re);
(4.4)
this is an integral function of k~., whose asymptotic behaviour for small k©, is, according to eq. (2.16a), given by tt
F~,C, lk~,V- ~ = ¼r !n~,r- lxc, I2,,+ a(x~,)[1 +O(k~,)].
(4.5)
Furthermore, in the denominator of the last factor of eq. (4.3), according to eq. (2.8b), we have
Oos, e"°k~ = ~1 (-iko)'F(l+
1 + iGq)W+ ;
(4.6)
this quantity has singularities corresponding to the poles of the F function, namely kc =
ino l+l+n
(n = 0, 1, 2 . . . . ),
(4.7)
and to the essential singularity o f F and W+ at ko = 0. Nevertheless, for small k, and
--½n+O ~_ arg ko ~_ ½n-O, factor ~p]r~is missing in the last term.
(4.8a)
t In ~!. (1.7.16) a tt In eqs. (4.5) and (4.8b), the quantities x e, and xe are of course defined according to eq. (2.17); no confusion can arise with the symbol occurring in eq. (3.1).
8
J. HUM2LEr
where 0 is an arbitrarily small positive constant, eq. (2.16b) leads to the simple asymptotic behaviour O, se'k
ffi
x° K2,÷ i(xo)[1 + o(ko)].
(4.8b)
In particular, we see that the quantity (4.6) remains finite for /co --, 0, provided that the condition (4.8a) is satisfied. The situation is now very similar to the one we had in part I. Again we have the choice between two types of expansion corresponding to the application of the theorem (A2.5) of appendix 2 either to the last factor ofeq. (4.3), or to the product of this factor and kl©kl¢'.. Before writing down these expansions, however, we have to draw the cuts we must introduce in the complex plane of the chosen variable. In the physical ko-plane 1),
-ke-~
-/~ -at,-&, J~mk/~/,,, h: /~
**
.eko
Fig. 1. The physical ke-plane.
kit
:
kc
~-
kd
ke
~.
t
Re~
Fig. 2. The physical ~f-plane.
they should be drawn as shown in fig. 1. The "T-shaped" cut between - K ~ and + / ~ and along the negative imaginary axis makes Os a single-valued function of ks and ko as well; at the same time, it leaves us with a single branch of Ueo at the opening of channels, such as a or b, for which K.. b < Ko. The branch points at the openings ofchannels d, e . . . . with Kd, o.... > K"s make it necessary to draw symmetrical t cuts fromKd, s" to + ~ and --Kd, , .... to - - ~ . In practice, however, we have only to consider the expansion of the collision matrix in the 8-plane. If, for convenience, the internal energies of the fragments are referred to those of channel 0, i.e. if we take Eso = 0, the appropriate 8-plane is simply the t The symmetry is necessary to ensure eq. (1.8.9). In part I, we h a d made a different, but equivalent choice o f cuts joining the points --/Ca and +Ka, and - - K e a n d -t-Ke, . . . . The p r e e m p t i o n we are adopting n o w would only be imperative i f there were no upper b o u n d to the branch-points, in order to secure communication between the lower and the upper part o f the k0-plane.
TSBORY
oF I~'UCLS~.R~JSACTIONS(IV)
9
conformal mapping o f the right half of the physical ko-plane of fig. 1; it is shown in
fig. 2. As in part I, let us simply add an index n to all the quantities which have to be evaluated at the resonance ~',. The residue of the last factor of Tc,~ in eq. (4.3) is computed as in part I. The only difference worth mentioning is the fact that in the evaluation of the numerator at the resonance, the first relation (2.13) introduces an extra factor e x p ( - i ~ , , ) in eq. (L10.9), in whichj~, must be replaced by -2/Fo,. For the collision matrix elements normalized to unit flux, we are now justified in writing, instead of eq. CI.10.15),
~-o,o ---- qE~,~--~c,ce ~i~© ----
k c, k¢ 8c,~c ~
~,©~o)
t~
(
'°"
(4.9a)
'°
= k,/-#~-#~,ko[8o,8oei('°'+'°~o,c(~') 8o e,,,o,_~o, +~', ) Go,. Go. e,,,o_~o.+~o.)l T-z •
(4.9b)
In these equations, the functions Qo,~ and ~c'~ are of course completely different from those noted with the same symbols in eqs. (1.11.15); ~rc,¢©, are real energydependent phases when the energy is real; ~r©., ~c,. are complex constants, while ~ . , ~,., ~c., ~¢,. are real constant phases; G~, G©,. are positive constants; the ~ and O are defined by ~ '
( @~" ~
#~, = arg \8c.v~Oo, / , ~ , = ~ ¢ , - a r g k~,.
(4.11a)
(4.11b)
5. The Differential Cross ~ t i o n The factor e x p [ - i ( ~ c , . + # c . ) ] in the expansions (4.8) seems at first sight to introduce an unexpected complication in the practical computation of cross sections, since ¢©~ and ¢o'. are not real quantities. However, the occurrence of these factors is only the result of the definition adopted for Ii and O z for the purpose of analytical arguments, and we may now go over to a more practical notation. The definitions (2.8) were in fact chosen because they made I l and Ol free of poles I, As an exception to our usual notation, Ge. is not related to the value at the resonance of the function (2.12b): it keeps its meaning given by eq. (I.12.13a).
10
j. HUmlLWr
and zeros independently of r. Under such conditions, the amplitudes Xo and yo were properly defined in the complex k©-plane. Now that the expansions (4.9) have been established, we may define new ingoing and outgoing wave functions by
I ~ " .ffi G t - iFt -- ~tde-t~',
(5.1a)
O~ c" = Gt+ iFt - O~lde+l~'z,
(5.1b)
and, for real energies, new amplitudes Xc and Yo by
u©(G, k¢) = XcO'~" + Y~I'~c',
(5.2)
leading to a new collision matrix element Uo".'o"=
X~?) = U¢ce°'d-,(-c'+,c). yotc)
(5.3)
Dropping the indices "new" everywhere, let us rewrite the eqs. (4.9) with these new meanings of the symbols, at the same time introducing the partial width Fo, defined in terms of the new O , by eq. (1.10.16). When the new collision matrix (5.3) is normalized to unit flux, its expansions read To, o -- c?l¢o-6o,o = k~;+*k'o+*8¢soQ¢o(d')
]/ko, ko =
eo
x/kek.8¢8o~.o,o(8)-i~ q, V k o ' k o
co'
go
e g©.
e,&.r~,.,r~, e ~:°'. "
x ¢ , x . leo,,118.1
(5.4a)
(5.4b)
~'-¢,
Comparing these expressions with the corresponding ones (I. 10.22 a, b) of part I, we see that, if the Coulomb field is not cut off at large distances, the resonance terms of the collision matrix keep the same form, provided that the partial widths be now interpreted as functions of the energy, according to the substitution Fo, --*/~,(ko) -
no
2r."
(5.5)
8on
It is clear that, since 8o and ~ , are independent of ao the partial widths 1", are, just as the former Fo,, a-independent in the sense of part HI. Moreover, at the resonance, i.e. for dr ffi d'°, we still have E Fo,(kc,) = E ¢
r . = r,.
(5.6)
G
For practical applications, it is useful to introduce explicitly into eqs. (5.4a) and (5.4b) penetration of the respective forms
~21 Po = eG ~co, I0
Pc
= s c2,
(5.7)
THEORY OF NUCLEAR R E A C T I O N S (IV)
1I
which are directly suggested by the expression for the resonance terms and are aindependent. One thus obtains the final expressions
P~,Pc e,~c,"I~,.F~.
- i ~'. q.
!%,. ~c. --CtR P~ P ~ell
e i¢~-
(5.8a)
g - - 8n
In the first cases so far treated 4), it turns out that, when these penetration factors are used in the analysis of experimental data, good fits can be achieved with negligible contributions from the background terms Q¢.~, ~¢,~. Of course, one may expect that in other cases slowly varying background terms may have to be taken into account. The final expression for the differential cross section may just be taken over from Lane and Thomas' paper 12)t. Adopting their definition of Ce(Oa) and o~, ~, namely
Ce(O~) :
(4n)- ~ ~/~cosec 2 (½01) exp I'- 2i ~ log sin (½0~)], k~ k~ a)iz = a a ~ - a i o =- a ~ - a o
(for r / : ~/~),
(5.9) (5.10)
the differential cross section doa,. e's'¢ may be written as do~., ~'s'v, = IAe,,'v',a~(g]e')12dae• ,
(5.11)
with, in our (new) notation, 7~#
Ai.,.,,.,~, = ~ [C,,,(eE.)&o,.,,. ~ + i~J21+
1 e'(""r +"'O T,o,o,.,o,,. ,,,,,o Yr'02,.)]. (5.12)
Either of the expansions (5.8) of cFe,,,r¢,, ' ~ , o may be introduced in the amplitude
(5.]2). The author would like to thank Professors W. A. Fowler and L. Rosenfeld and Mr. C. Mahaux for stimulating discussions. A grant from the Institut Interuniversitaire des Sciences Nucl6aires is gratefully acknowledged. Appendix 1 ANOTHER
EXPANSION OF THE COLLISION MATRIX
We have insisted elsewhere 3) on the necessity of expanding the collision matrix in such a way that each term be a-independent, by which we mean that the separation t Note that
~,L.,.-~h.c,c : where (o~a is defined by eq. (5.10).
~ !de-~('i'° + "io)
12
J.m.r~t,wr
in background and resonance terms, the position and total width of each resonant state, and all partial widths should be independent of the channel radii. This point of view does not exclude the explicit introduction of such radii at some stage of the interpretation of experimental data, e.g. in evaluating the penetration of the Coulomb barrier with reference to some distance of the order of magnitude of the nuclear radius. For different reasons, however, this may lead to very uncertain results. Firstly, the conventional penetration factor [cf. eq. (AI.1) below] is only qualitatively related to the penetration of the actual Coulomb barrier is); in particular, its strong dependence on the channel radius does not reflect the fact that the surface on the nucleus is not sharply defined. Secondly, it is evident that, independently of the way it is expanded, the exact expression for the collision matrix remains a-independent; if therefore, the single terms of a certain expansion are not a-independent and only two or three such terms are retained, as is done in practice, the result may strongly depend on the channel radii. Under such conditions, as already pointed out by Mc Dermott ta), one must realize that the values of the channel radii giving the best fit to the experimental data will not have any necessary connexion with the nuclear radii: they will unavoidably have to be chosen primarily to compensate the neglect of the further terms of the collision matrix. With this warning, and only in order to facilitate comparison between our results and those of the R-matrix theory, we shall briefly derive, within the frame of the complex eigenvalue theory, an expansion of the collision matrix utilizing the same a-dependent penetration factors as those adopted in the R-matrix theory, namely (cf. ref. ,2))
P c f I m ( acO")
k~ao
= acImL© =
, o . ,,° =..
=
(Ff + Gf),o =.°
k~a°.
(A1.1)
(Io O°),. =..
However, in contrast to the latmr theory, we shall maintain our definition of the positions and total widths of the resonance levels. We consider the collision matrix defined by eq. (3.2) when the definitions (2.8) of I t and O~ are adopted, or equivalently the collision matrix (5.3) with the corresponding definition (5.1) of I t and O~. In the R-matrix theory, the collision matrix may in particular be written, in matrix notation, as 12) U = fI[I+2i/ft(I-RL°)-xRP~la
(A1.2)
where t~ is a diagonal matrix defined by
f~c'~ = 6,,,a, ffi tic,,(IdOc)t = 5,,,e"°;
(A1.3)
q). is real for real k c. In the R-matrix formalism only the quantity (1-RL°)-IR
(A1.4)
THEORYO~ NUCL~S~RSAC'nONS(Iv)
13
is expanded. Now, in our notation, taking into account the fact that W(@~*,), Oc,) - 0 for c' # c, it is easy to see that we have
....
V Mo
=
o,
oo,
4;:'
(ALSa)
Mo, ~ ? ) ' - L , ~ ? ) O, '
which may be written, in terms of the quantities (AI.1), (A1.3)
,,.,--,,..
M° L
,,:,_,
~ M c , a e a. ¢~.c), _ L. 0~¢)
since Q~p~ _ ~/k-~-o Oc
(real k~).
(AI.6)
The factor in eXl.(AI.5b) corresponding to the quantity (AIM) is thus indentifie~las • M©
~(c?)
V~4o7~.,.o~(:)'-L°~?" whose expansion is immediately written down as
-½i.~©,©-½~ Oc,nO¢.ei~©,.Ge.G~ e~o.. x[a.,a© Z--Z.
(A1.7)
Hence, we obtain the following type of expansion:
-,,
oo, oo.
,_,.
A1.8)
In this expression, F~,, ~ , have the same definition and the same property of aindependence as before. However, the background term -~c~ is different from those in our expansions (5.4): it is not independent of the channel radii; likewise, each resonance term of eq. (A1.8) is a-dependent, but, in contrast with the R-matrix, our method gives this dependence explicitly in terms of the known functions f~c, Po and Oc, of the channel radius ao. The actual derivation of the a-independent partial widths from experimental data by means of this formula (A1.8) is more difficult than by means of the expansions (5.4), because we have here to compute O~,, i.e. values of the function Oo for complex values k , of the wave number kc. For the very simple problem of the one-channel elastic scattering case, we could 14) compute explicitly the background term .~o,©, but in the general case, it is probably not easier to discuss the functions .~,~ than the background functions occurring in
eqs. (5.4). Preliminary results is) in the fitting of experimental data using eq. (A1.8) show that the penetration factor (A1.1) is not better than those given by eqs. (5.7).
14
J. HUMnLrr
Appendix 2 THE EXPANSION THEOREM The well known Mittag-Leffler expansion le) being concerned with single-valued functions of a complex variable, the expansion o f the collision matrix is not a straightforward application of this theorem. It is rather an extension of it and it is useful to give the main arguments leading to its justification. Let f ( k ) be a function defined in a complex k-plane in which a series o f cuts have been introduced to make it single-valued. Outside the cuts, let its only singularities be an infinite number of isolated poles kl, k2 . . . . . with residues Pl, P2 . . . . . Let us assume that 0 < Ikll < Ik21 _~ . . . (A2.1) and that the series nil
(A2.2)
IP,lk, I
Rqc
x
Fig. 3. The complex k-plane with its poles × and a contour C . (for m ----4) with indentations of df~'erent types (C. ~ ABCDEFOHIJKLA-J-PQRSP). is convergent. Under such conditions, it is easy to verify that the series
,~k-k. P"
(A2.3)
is also convergent for any finite value of k uniformly distant from the poles. Let now C,, be a closed contour drawn in such a way that the poles k 1, k2 . . . . k,, are interior to it and that all the other singular points are exterior to it. Then, according to the theorem of residues,
1 i" jc..--,k-k;
p. - k-k."
(,,2.4)
TtW.ORY oF NUCLg~ REACTIONS(rv)
15
Hence, if we let m go to infinity,
f(k) = Q(k)+ ~ft P" =
k-k.
(A2.5)
where Q(k) = lim f f(k') dk'. =-,= J c , k ' - k It is important to recall that, as a corollary to the fundamental Cauchy theorem, the theorem of residues used to derive eq. (A2.4) does not require 17) that each point of the contour be a point of holomorphy off(k). It is suJ~cient t h a t f ( k ) be continuous on the contour (3,; then, for any point k' on C=, we have lira L f ( k ) - f ( k ' ) ] = 0.
(A2.6)
when k is an interior point tending to k'. Accordingly, it is often possible to extend indentations just to the singular points (like ABC and DEF in fig. 3), rather than turning around them at a non-vanishing distance 0ike G H I and J K L in fig. 3). This remark finds application in sect. 4. There, we had two possible choices for the cut issuing from the threshold of channel c: (1) to draw a circle of finite radius around the essential singularity and extend it to infinity by a cut which could be infinitely narrow (like GHI), or (2) to expand the product o f f ( k ) and a factor g(k) chosen in such a way that with a cut extending just to the singular point (like DEF), the product f(k)g(k) is continuous on (3, and has inside exactly the same singular points (poles) as f(k). In sect. 4, we had to choose the latter alternative for physical reasons: we wanted an expression for T¢,= in which the energy dependence explicitly appeared in each term of eq. (A2.5) for real energies just above the thresholds. Two possible forms for g(k) were 8~,te~le-it~°'+~°)k~'k~Z-t and e=,8=e-l(~='+~°)k~"1. (A2.7) It will be noticed that the point k© = 0 is an essential singularity of
l/[r(l+
1 + i,to/ko) W+ ]
as well as of l/F/+, in spite of the fact that te foxmer quantity is O(k l) when arg k= statistics eq. (4.8a). Refermces 1) 2) 3) 4)
J. Humblet and L. Rosenfeld, Nuclear Physics 26 (1961) 529 L. Ro~nfeld, Nuclear Physics 26 (1961) 594 J. Humblet, Nuclear Physics 31 (1962) 544 C. Mahaux, Pruc. Brmmels Conf. on Nuclear Physics, Sept. 1962 (I.I.S.N., Bruxelles, 1962); Bull. Soc. Se. Li6ge 32 (1963) 62, 70 and 240 5) J. Humblet, Proc. Int. Symposium on Direct Interactions and Nuclear Reaction Mechanisms, Padua, Sept. 1962 (Gordon and Breach, New York, 1963); Proc. Brussels Conf. on Nuclear Physics, Sept. 1962 (I.I.S.N., Bruxelles, 1962)
16
J. h-ut~tzr
6) H. Buchholz, Die Konfluente Hypergeometrische Funktion (Springer, Berlin, 1953) 7) M. H. Hull and G. Breit, Handbuch der Physik, 41/1 (Springer, Berlin, 1959) 8) A. Messiah, Quantum mechanics, vol. I (North-Holland Publ. Co., Amsterdam, 1961) appendixB 9) G. N. Watson, Theory of Bessel functions, 2nd ed. (Cambridge University Press, Cambridge, 1952) 10) A. Erd61yi et al., Higher transcendental functions, vol. I (McGraw-Hill, New York, 1953) 11) L.J. Slater, Confluent hypergeometri¢ functions (Cambridge University Press, Cambridge, 1960) 12) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 13) L. C. McDermott, Thesis, Columbia University, New York 1959 (AEC document CU-187, Washington D.C.) 14) J. Humblet, Comptes Rendns Acad. So. 231 (1950) 1436 15) J. P. Jenkenne, private communication (1963) 16) A. Hurwitz and R. Courant, Funktionentheorie (Springer, Berlin, 1922); C. Carath6odory, Funktionentheorie (Birkhathter, Basel, 1950) 17) E. Goursat, Cours d'analyse mathC~atique, 5th ed. (Ganthier-Villars, Paris, 1933) footnote p. 74 18) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952)
Nuclear Physics 50 (1964) l m 1 6 ; (~) North-Holla~ui Publishing Co., Amsterdam Not
to be repx'oducedby photoprint of micxofilm without written permission from the publisher
T H E O R Y O F NUCLEAR REACTIONS
IV. Coulomb lnteractiom in the ~
and Penetration Factors
J. HUMBLET
University of Liege t, Belgium Received 16 May 1963 Abstract: The expansion o f the collision matrix as the sum o f a background term and a series o f resonant terms Oven in part I is extended to include the Coulomb interactions in the ehfmneIn without CUt Off at large distances. The penetration factors introduced are independent o f the channel radii.
1. Introdee~m When the incoming and outgoing radial wave functions associated with a pure Coulomb interaction are considered as functions o f the wave number k, it is well known that they have an essential singularity at the origin k -- 0 of the complex k-plane. The corresponding singularities o f the collision matrix are usually avoided by cutting off the Coulomb interaction at some large distance. Near the thresholds, the analytical behaviour of the collision matrix is then the same as that obtained for neutron channels, and the effect of the penetration o f the Coulomb barrier at very small energies no longer appears explicitly. In the first paper 1) of this series x-s), the element Uo,o o f the collision matrix corresponding to a nuclear reaction c' -~ c appears as a "background" term Qc'~ plus a series o f resonant terms. To establish such an expansion, the Coulomb interaction o f the two fragments in any channel c was neglected when their distance re was larger than be(~> ac). Under such conditions, near the thresholds o f channels c and c', which we now assume not to be neutron channels, the resonant terms are not decreasing exponentially with the energy. Nevertheless, as the sum o f Q¢,o and the resonant terms must exhibit such an exponential decrease, we are forced to conclude that, at least near the thresholds, the background term Q~,o plays an important part in the collision matrix. Since, in practice, very little can be said so far about the structure of the Qo,~ function, this is certainly a drawback of the theory, at least ff one is concerned with the practical fitting of experimental data at low energies. It is the purpose o f the present paper to show that it is possible to avoid the cut-off of the Coulomb field at large distance and, at the same time, to introduce explicitly an exponential energy dependence of the resonant terms near the thresholds. The background term is modified accordingly and in the applications treated up to now 4.), it turned out that it could be completely neglected. t Institut de Math6matique, 15 Avenue des Tilleuls, Liege. I January 1964.
2
J. HUMBLEr
Because of the essential singularity of the Coulomb wave functions in the complex k-plane, it has been necessary to revise here some arguments of part I regarding the definition and properties of the collision matrix. The Coulomb wave functions and their main properties are given in sect. 2. The corresponding expression for the collision matrix and some of its general properties, such as unitarity and symmetry, are established in sect. 3. Its proposed expansion is given in sect. 4; each resonant term is as in part I independent of the channel radii ac, but in contrast to part I, energy-dependent penetration factors are introduced. The resulting expression for the differential cross section is written down in sect. 5. In appendix 1, another expansion of the collision matrix is given, in which our definition of the resonant states is used together with conventional a-dependent penetration factors; this is done mainly to facilitatecomparisons with the R-matrix theory. Appendix 2 contains some of the mathematical arguments which are necessary to justify the expansion given in sect. 4. Preliminary reports on the results given in the present paper were already given last year to the Padua and Brussels conferences s). 2. Coulomb Wave F~c/iom When the Coulomb wave function is analysed in partial waves, its radial factors satisfy equations of the form
(dz~ +k 2- l(l+l)-'~)y,=O,rz
(2.1)
where
tl = ZI Z2eaMh -2.
(2.2)
Two solutions of this equation are the well known regular function Fz and irregular function Gl, whose asymptotic behaviour for large r is Fi "~ sin(kr-~ log(2kr)-½1~+~z)
(larg k[ < ½~),
(2.3a)
G~ ~ cos(kr-~ log(2kr)--½1~+al)
([arg kl < ½~).
(2.3b)
where • and at are defined by = _q
e2~, , = F ( l + 1 + i~)
(2.4)
k' The quantitiesFz, G~, • and as are realwhen k isitselfreal. The first solution F~ is proportional to the regular Whittaker function .At' = .A~.i +½
(2ikr), since Fj ----½P-tl!(-i)z+1 ~ . i+,(2ikr),
(2.5)
1 8~ = - e-~'=[F(l + 1 + i~,)F(l + 1 - i a ) ] * . l!
(2.6)
where
11q~oaY o v h'UCLI~R REACTIONS (IV)
3
The function .,dr is defined according to Buchholtz e), i.e. ~1/ : M,,.,+~(2ikr)/F(21+ 2) for a fixed r; ~Ig is an integral function of k. In terms of well known notations 7, s), we also have et
=
(21+ I)!!C~
=
2,~," 7 i
(
12+ ~t2) . . . (1 +,,2) e 2 " - lJ
-~. 1
= 1 [(/2 + ,'2)... (1 +,'z)]½C o;
(2.7a)
(2.7b)
lI the latter quantity has been normalized in order that et : 1 for k : oo or for a neutron channel ( ~ / : 0). Two other solutions of eq. (2.1) are conveniently defined as
(2.8a)
It(r, k) ---- (G~-iF,)e "~ = i Z e ~ W _ , O~(r, k) :
" - - - " ' : ( - i)'e*'W+, (~,+ zF,)c
(2.8b)
where W_ --- W~,.z+~(e~t~2kr),
W+ ffi W_u.i+~(e-~½~2kr)
(2.9)
are irregular Whittaker functions. Considered as functions of k, they have k ffi 0 and k ffi oo as essential singularities, but they have no other singularity in the complex k-plane. They are multivalued, according to the relations e) W,., ~+~(e't~2kre '2") = W.=,~+½(e'°2kr) + 2imt .~v~,.l+½(2ikr),
r(-l-i,')
W_,,. ~++(e-'°2kre '2"1) : W_u z++(e-'+12kr) +2inn .Ar_~,.
I++(-2ikr),
r(-
l+
i,')
(2.10a)
(2.10b)
where n is an integer and ~ l _ , . , ~ + t ( - 2 i k r ) f f i ( - 1 ) z+t . ~ , , t + t ( 2 1 k r ) . Together with the regular function ~tr, the functions W± satisfy the relation 6, 7)
..4f--
eu
r(l+l-i,')
W+
eZ(a + u)
r(l+l+i,')
IV_
(2.11)
and the Wronskian relations 6, i x) W(.~d', W±) : - 2ik
C ti,a(l ± 0
r(l+ I+ i,')'
W(W+, W_) -- - 2 i k e - n .
(2.12,) (2.12b)
4
J. RUMBI..IgT
According to eq. (2.8), the relations (2.11) and (2.12) may be rewritten as - 2iFt = e - ~ I t
-et~zOt,
(2.13a) (2.14)
W(O t , It) = - 2 i k ;
W(Ft, Or) = - R e - " ,
we also have (2.13b)
2Gi = e-i"Ii+e~'*Oz.
For large values of r, the asymptotic behaviour of W± is such that e) Il " e x p [ - i ( k r - u l o g ( 2 k r ) - ½ g l ) ]
( - 2 g < argk < g),
(2.15a)
Ot " e x p [ + i ( k r - u l o g ( 2 k r ) - ~ n l ) ]
( - g < argk < 2g)
(2.15b)
and accordingly they correspond to incoming and outgoing waves respectively. Later we shall also need the behaviour of .~' and IV+ for very small k. We have )
= ½ F(I + I + ia) W+
(any argk),
\t//
/ \+i~/t
[-/-)xg,t+,C=)[l+OCk)],
targkZ- i
n-0,
C2,6 ) (216b)
where x -- x/8"-~-;r = x / ~
(2.17)
and 0 are positive quantities, the latter being an arbitrarily small constant. Because of the definition (2.8) of It and Or, we have
It(r,ke-t*)= (-1)'O,(r,k ~ - = ,
(2.18a)
Or(r, Re+t') = (-1)'It(r, k)e -~'.
(2.18b)
Further, if k* is properly defined as [klexp(-i arg k), we have Or(r, k)*
=
ite*"/k'W~,/k, t+~(etO2k*r)
= It(r, k*) = (-1)tOt(r, k*e'~)¢"~*,
(2.19a)
It(r, k)* = O,(r, k*) -- ( - 1 ) t l ~ r , k*e-t')¢ 'a*.
(2.19b)
In an open channel, k is a real and positive quantity and we have in partion!ar Or(r, k)* = ll(r, k),
It(r, k)* = O~(r, k)
(open).
(2.20)
* Our definition of the modified Bessel function Ka~+I is that of Watson I). Eq. (2.16a) is easily derived from Buohholtz [ref. 6), eq. (16), p. 97l; eq. (2.16b) agrees with formulae given by Hull and Breit [ref. v), p. 421-432] and by Erd~lyi [ref. xo), eqs. (16) and (18) of section 6.9]. Eq. (2.16a) does not agree with the formulae given by Slater [ref. xl), eqs. (4.4.20) and (4.4.21)], but the latter are marred by a misprint: J and I have to be interchanged in the right.hand sides; a similar correction has to be introduced in his equations corresponding to our eq. (2.16b).
wmonY
or suc~
~cnom
Ov)
5
In a closed channel, arg k = ½n and
Ol(r, k)* -- (-1)'Oz(r, k)e -'~ l,(r, k)*
=
(closed),
(--1)Ill(r, ke-Z'~e -~"
(closed);
(2.21a)
(2.21b)
the last relation, which corresponds to a change of Riemann sheet, will in fact not be needed in what follows. From now on, we shall make I s and Ot single-valued functions of k by introducing a cut in the complex k-plane along the negative imaginary axis. Then -½n < arg k < ½7r
(2.22a)
~ - ~ < arg (k*e~=) < ½~.
(2.22b)
and also
3. Dellaition and Properties of the Collision Matrix Among the various solutions ofeq. (2.1), Fi, O~, .~', W+, W_, the only two which (1) are finearly independent for any definite value o f k oe 0 (cf. eq. (2.12)), (2) have no pole in the complex k-plane for a fixed value o f r (cf. eqs. (2.5), (2.6) and (2.13)) and (3) do not vanish identically (for any r) at some point of the k-plane, are the irregular Whittaker functions W+ and W_ (cf. eq. (2.12b)). Since Ii and Ol differ from the latter functions by factors which have neither zeros nor poles, we may always write the radial wave function in channel c as
uo(ro, k,) -~ xoO¢+ yol¢;
(3.1)
the amplitudes x,, y~ are then uniquely defined in the complex k©-plane for ko 4= 0, provided that the appropriate cuts have been made in this plane. Under such conditions, we are justified in defining a collision matrix by
go, uo,o =
m
(3.2)
where, as in part I, the upper index (c) refers to the fact that c is the only entrance channel present in the total wave function ~'. Because of the relation (2.14) we still have explicitly,as in part I,
U~.© = W(O(~¢')'Ic')/k~" W(#(o ' Oc)/k,
(3.3)
W e again assume that the radialfactors 0(o, 0(~)have the same analyticalproperties as the regular Whittaker functions .#f~, ~fo,.... , because these radial factors bclong to a wave function ~,
t The notation (I.8.2b) refers to eq. (8.2b) of part I.
6
$. HUMBLET
which in fact vanish for re : 0, re : 0 , . . . respectively. More precisely, the only singularities assumed for O~°), ~o,~) in the complex ko-plane as well as in the complex d-plane are branch points at the channel openings (ks = O, k e : 0 . . . . ). The resonant states are again defined as complex poles of Uec, i.e. by the boundary condition
W(¢~~), O~)
= 0.
(3.4)
Because of the relations (I.8.12) and (2.19a), it is evident that to each pole k~, = K~,-i~¢, (for which larg k~,[ < ½~) there corresponds another one, namely k*,d" = - ~ c ~ - i ~ , , the argument of which falls within the limits prescribed by eq. (2.22a). To prove that in any upper k-plane, poles, if any, m a y lie only on the imaginary axis, we may proceed as follows. Assuming that ~,~,< 0, let us consider eq. (I.9.12):
iF.
f
1~.12do~ =
'* (~o,* ~ o' , - oo,~0,)
(3.5)
If the integration, rather than being limited, to the interior region o~, is extended to the whole configuration space, the integral will remain finite, because, in a channel c, ~/'~ behaves like Oc,(r~) = u~(r~, k~,), which is itself proportional to exp(/k~,r~) = exp(-Ire, Ire). For the same reason, each term of the fight-hand side vanishes at infinity, and we are left with
r, afl~V.12doJ = ~
O,
(3.6)
i.e. with F, = 0, or ,o. = 0.
(3.7)
Equation (I.9.15) remains valid, provided that No, is now defined by eq. (I.9.14), but it seems difficult in the present case to give an explicit form for No, corresponding to eq. (I.1.43). The unitarity and symmetry properties of the collision matrix are also readily extended to the present situation. To prove the former of these properties, we have only to note that, whereas eq. (L7.10a) remains unchanged, eq. (I.7.10b), for a closed channel c- should be replaced by (the channel q being open) ¢,~.'* = ( - Vo~Oo)* = -(-1)'°V*~qOo(ko):"-
(closed).
(3.8)
The rest of the proof then proceeds as in ref. 1). In the proof of the symmetry property, eq. (I.7.15) should be replaced by ~"-"' = Z
p+
q'-,(a_,_.L,-v_,_.o_,)Z ~-' v_,_qo_, rp p - rp
(3.9)
THEORY OF N U C I ~
REACTIONS (IV)
and hence t (-- 1)/'t+MqKIiv(-q) = Z tp__£[t~mlOp(kp)_ Uv.qlp(kp)] p+ l'p
_ ~ tp__pU~qOp(kp)(_l)ipe_up.
(3.10)
p- rp
Here again we have used eqs. (2.20), (2.21a), but not (2.21b), and the rest o f the p r o o f also remains unchanged. 4. E x p a ~ o n of the Collision Matrix The differential cross section is not directly proportional to JUt,c]2, but rather to the squared modulus of Tc,c = Uo,~-6c,¢e 2'~°.
(4.1)
Using the relations (1.6.11) and (2.13a), one has
W(O(?), Fc,)/kc, W(+~°, O~e":)lk: l(,c'+-.) r ~+x W(OtJ),Fc'e71k7 v-l)
Tc.c = _2ie,(..,+..) • = - 2ze,. s, e
k~. k~
,-:,
Oosoe
k,)
(4.2)
(4.3)
In the numerator of the last factor, according to eq. (2.5), we have F~,82 lk~.r- 1 = ½1'!(ikc,)-v- 1 ~ , ° , r +,(2ike re);
(4.4)
this is an integral function of k~., whose asymptotic behaviour for small k©, is, according to eq. (2.16a), given by tt
F~,C, lk~,V- ~ = ¼r !n~,r- lxc, I2,,+ a(x~,)[1 +O(k~,)].
(4.5)
Furthermore, in the denominator of the last factor of eq. (4.3), according to eq. (2.8b), we have
Oos, e"°k~ = ~1 (-iko)'F(l+
1 + iGq)W+ ;
(4.6)
this quantity has singularities corresponding to the poles of the F function, namely kc =
ino l+l+n
(n = 0, 1, 2 . . . . ),
(4.7)
and to the essential singularity o f F and W+ at ko = 0. Nevertheless, for small k, and
--½n+O ~_ arg ko ~_ ½n-O, factor ~p]r~is missing in the last term.
(4.8a)
t In ~!. (1.7.16) a tt In eqs. (4.5) and (4.8b), the quantities x e, and xe are of course defined according to eq. (2.17); no confusion can arise with the symbol occurring in eq. (3.1).
8
J. HUM2LEr
where 0 is an arbitrarily small positive constant, eq. (2.16b) leads to the simple asymptotic behaviour O, se'k
ffi
x° K2,÷ i(xo)[1 + o(ko)].
(4.8b)
In particular, we see that the quantity (4.6) remains finite for /co --, 0, provided that the condition (4.8a) is satisfied. The situation is now very similar to the one we had in part I. Again we have the choice between two types of expansion corresponding to the application of the theorem (A2.5) of appendix 2 either to the last factor ofeq. (4.3), or to the product of this factor and kl©kl¢'.. Before writing down these expansions, however, we have to draw the cuts we must introduce in the complex plane of the chosen variable. In the physical ko-plane 1),
-ke-~
-/~ -at,-&, J~mk/~/,,, h: /~
**
.eko
Fig. 1. The physical ke-plane.
kit
:
kc
~-
kd
ke
~.
t
Re~
Fig. 2. The physical ~f-plane.
they should be drawn as shown in fig. 1. The "T-shaped" cut between - K ~ and + / ~ and along the negative imaginary axis makes Os a single-valued function of ks and ko as well; at the same time, it leaves us with a single branch of Ueo at the opening of channels, such as a or b, for which K.. b < Ko. The branch points at the openings ofchannels d, e . . . . with Kd, o.... > K"s make it necessary to draw symmetrical t cuts fromKd, s" to + ~ and --Kd, , .... to - - ~ . In practice, however, we have only to consider the expansion of the collision matrix in the 8-plane. If, for convenience, the internal energies of the fragments are referred to those of channel 0, i.e. if we take Eso = 0, the appropriate 8-plane is simply the t The symmetry is necessary to ensure eq. (1.8.9). In part I, we h a d made a different, but equivalent choice o f cuts joining the points --/Ca and +Ka, and - - K e a n d -t-Ke, . . . . The p r e e m p t i o n we are adopting n o w would only be imperative i f there were no upper b o u n d to the branch-points, in order to secure communication between the lower and the upper part o f the k0-plane.
TSBORY
oF I~'UCLS~.R~JSACTIONS(IV)
9
conformal mapping o f the right half of the physical ko-plane of fig. 1; it is shown in
fig. 2. As in part I, let us simply add an index n to all the quantities which have to be evaluated at the resonance ~',. The residue of the last factor of Tc,~ in eq. (4.3) is computed as in part I. The only difference worth mentioning is the fact that in the evaluation of the numerator at the resonance, the first relation (2.13) introduces an extra factor e x p ( - i ~ , , ) in eq. (L10.9), in whichj~, must be replaced by -2/Fo,. For the collision matrix elements normalized to unit flux, we are now justified in writing, instead of eq. CI.10.15),
~-o,o ---- qE~,~--~c,ce ~i~© ----
k c, k¢ 8c,~c ~
~,©~o)
t~
(
'°"
(4.9a)
'°
= k,/-#~-#~,ko[8o,8oei('°'+'°~o,c(~') 8o e,,,o,_~o, +~', ) Go,. Go. e,,,o_~o.+~o.)l T-z •
(4.9b)
In these equations, the functions Qo,~ and ~c'~ are of course completely different from those noted with the same symbols in eqs. (1.11.15); ~rc,¢©, are real energydependent phases when the energy is real; ~r©., ~c,. are complex constants, while ~ . , ~,., ~c., ~¢,. are real constant phases; G~, G©,. are positive constants; the ~ and O are defined by ~ '
( @~" ~
#~, = arg \8c.v~Oo, / , ~ , = ~ ¢ , - a r g k~,.
(4.11a)
(4.11b)
5. The Differential Cross ~ t i o n The factor e x p [ - i ( ~ c , . + # c . ) ] in the expansions (4.8) seems at first sight to introduce an unexpected complication in the practical computation of cross sections, since ¢©~ and ¢o'. are not real quantities. However, the occurrence of these factors is only the result of the definition adopted for Ii and O z for the purpose of analytical arguments, and we may now go over to a more practical notation. The definitions (2.8) were in fact chosen because they made I l and Ol free of poles I, As an exception to our usual notation, Ge. is not related to the value at the resonance of the function (2.12b): it keeps its meaning given by eq. (I.12.13a).
10
j. HUmlLWr
and zeros independently of r. Under such conditions, the amplitudes Xo and yo were properly defined in the complex k©-plane. Now that the expansions (4.9) have been established, we may define new ingoing and outgoing wave functions by
I ~ " .ffi G t - iFt -- ~tde-t~',
(5.1a)
O~ c" = Gt+ iFt - O~lde+l~'z,
(5.1b)
and, for real energies, new amplitudes Xc and Yo by
u©(G, k¢) = XcO'~" + Y~I'~c',
(5.2)
leading to a new collision matrix element Uo".'o"=
X~?) = U¢ce°'d-,(-c'+,c). yotc)
(5.3)
Dropping the indices "new" everywhere, let us rewrite the eqs. (4.9) with these new meanings of the symbols, at the same time introducing the partial width Fo, defined in terms of the new O , by eq. (1.10.16). When the new collision matrix (5.3) is normalized to unit flux, its expansions read To, o -- c?l¢o-6o,o = k~;+*k'o+*8¢soQ¢o(d')
]/ko, ko =
eo
x/kek.8¢8o~.o,o(8)-i~ q, V k o ' k o
co'
go
e g©.
e,&.r~,.,r~, e ~:°'. "
x ¢ , x . leo,,118.1
(5.4a)
(5.4b)
~'-¢,
Comparing these expressions with the corresponding ones (I. 10.22 a, b) of part I, we see that, if the Coulomb field is not cut off at large distances, the resonance terms of the collision matrix keep the same form, provided that the partial widths be now interpreted as functions of the energy, according to the substitution Fo, --*/~,(ko) -
no
2r."
(5.5)
8on
It is clear that, since 8o and ~ , are independent of ao the partial widths 1", are, just as the former Fo,, a-independent in the sense of part HI. Moreover, at the resonance, i.e. for dr ffi d'°, we still have E Fo,(kc,) = E ¢
r . = r,.
(5.6)
G
For practical applications, it is useful to introduce explicitly into eqs. (5.4a) and (5.4b) penetration of the respective forms
~21 Po = eG ~co, I0
Pc
= s c2,
(5.7)
THEORY OF NUCLEAR R E A C T I O N S (IV)
1I
which are directly suggested by the expression for the resonance terms and are aindependent. One thus obtains the final expressions
P~,Pc e,~c,"I~,.F~.
- i ~'. q.
!%,. ~c. --CtR P~ P ~ell
e i¢~-
(5.8a)
g - - 8n
In the first cases so far treated 4), it turns out that, when these penetration factors are used in the analysis of experimental data, good fits can be achieved with negligible contributions from the background terms Q¢.~, ~¢,~. Of course, one may expect that in other cases slowly varying background terms may have to be taken into account. The final expression for the differential cross section may just be taken over from Lane and Thomas' paper 12)t. Adopting their definition of Ce(Oa) and o~, ~, namely
Ce(O~) :
(4n)- ~ ~/~cosec 2 (½01) exp I'- 2i ~ log sin (½0~)], k~ k~ a)iz = a a ~ - a i o =- a ~ - a o
(for r / : ~/~),
(5.9) (5.10)
the differential cross section doa,. e's'¢ may be written as do~., ~'s'v, = IAe,,'v',a~(g]e')12dae• ,
(5.11)
with, in our (new) notation, 7~#
Ai.,.,,.,~, = ~ [C,,,(eE.)&o,.,,. ~ + i~J21+
1 e'(""r +"'O T,o,o,.,o,,. ,,,,,o Yr'02,.)]. (5.12)
Either of the expansions (5.8) of cFe,,,r¢,, ' ~ , o may be introduced in the amplitude
(5.]2). The author would like to thank Professors W. A. Fowler and L. Rosenfeld and Mr. C. Mahaux for stimulating discussions. A grant from the Institut Interuniversitaire des Sciences Nucl6aires is gratefully acknowledged. Appendix 1 ANOTHER
EXPANSION OF THE COLLISION MATRIX
We have insisted elsewhere 3) on the necessity of expanding the collision matrix in such a way that each term be a-independent, by which we mean that the separation t Note that
~,L.,.-~h.c,c : where (o~a is defined by eq. (5.10).
~ !de-~('i'° + "io)
12
J.m.r~t,wr
in background and resonance terms, the position and total width of each resonant state, and all partial widths should be independent of the channel radii. This point of view does not exclude the explicit introduction of such radii at some stage of the interpretation of experimental data, e.g. in evaluating the penetration of the Coulomb barrier with reference to some distance of the order of magnitude of the nuclear radius. For different reasons, however, this may lead to very uncertain results. Firstly, the conventional penetration factor [cf. eq. (AI.1) below] is only qualitatively related to the penetration of the actual Coulomb barrier is); in particular, its strong dependence on the channel radius does not reflect the fact that the surface on the nucleus is not sharply defined. Secondly, it is evident that, independently of the way it is expanded, the exact expression for the collision matrix remains a-independent; if therefore, the single terms of a certain expansion are not a-independent and only two or three such terms are retained, as is done in practice, the result may strongly depend on the channel radii. Under such conditions, as already pointed out by Mc Dermott ta), one must realize that the values of the channel radii giving the best fit to the experimental data will not have any necessary connexion with the nuclear radii: they will unavoidably have to be chosen primarily to compensate the neglect of the further terms of the collision matrix. With this warning, and only in order to facilitate comparison between our results and those of the R-matrix theory, we shall briefly derive, within the frame of the complex eigenvalue theory, an expansion of the collision matrix utilizing the same a-dependent penetration factors as those adopted in the R-matrix theory, namely (cf. ref. ,2))
P c f I m ( acO")
k~ao
= acImL© =
, o . ,,° =..
=
(Ff + Gf),o =.°
k~a°.
(A1.1)
(Io O°),. =..
However, in contrast to the latmr theory, we shall maintain our definition of the positions and total widths of the resonance levels. We consider the collision matrix defined by eq. (3.2) when the definitions (2.8) of I t and O~ are adopted, or equivalently the collision matrix (5.3) with the corresponding definition (5.1) of I t and O~. In the R-matrix theory, the collision matrix may in particular be written, in matrix notation, as 12) U = fI[I+2i/ft(I-RL°)-xRP~la
(A1.2)
where t~ is a diagonal matrix defined by
f~c'~ = 6,,,a, ffi tic,,(IdOc)t = 5,,,e"°;
(A1.3)
q). is real for real k c. In the R-matrix formalism only the quantity (1-RL°)-IR
(A1.4)
THEORYO~ NUCL~S~RSAC'nONS(Iv)
13
is expanded. Now, in our notation, taking into account the fact that W(@~*,), Oc,) - 0 for c' # c, it is easy to see that we have
....
V Mo
=
o,
oo,
4;:'
(ALSa)
Mo, ~ ? ) ' - L , ~ ? ) O, '
which may be written, in terms of the quantities (AI.1), (A1.3)
,,.,--,,..
M° L
,,:,_,
~ M c , a e a. ¢~.c), _ L. 0~¢)
since Q~p~ _ ~/k-~-o Oc
(real k~).
(AI.6)
The factor in eXl.(AI.5b) corresponding to the quantity (AIM) is thus indentifie~las • M©
~(c?)
V~4o7~.,.o~(:)'-L°~?" whose expansion is immediately written down as
-½i.~©,©-½~ Oc,nO¢.ei~©,.Ge.G~ e~o.. x[a.,a© Z--Z.
(A1.7)
Hence, we obtain the following type of expansion:
-,,
oo, oo.
,_,.
A1.8)
In this expression, F~,, ~ , have the same definition and the same property of aindependence as before. However, the background term -~c~ is different from those in our expansions (5.4): it is not independent of the channel radii; likewise, each resonance term of eq. (A1.8) is a-dependent, but, in contrast with the R-matrix, our method gives this dependence explicitly in terms of the known functions f~c, Po and Oc, of the channel radius ao. The actual derivation of the a-independent partial widths from experimental data by means of this formula (A1.8) is more difficult than by means of the expansions (5.4), because we have here to compute O~,, i.e. values of the function Oo for complex values k , of the wave number kc. For the very simple problem of the one-channel elastic scattering case, we could 14) compute explicitly the background term .~o,©, but in the general case, it is probably not easier to discuss the functions .~,~ than the background functions occurring in
eqs. (5.4). Preliminary results is) in the fitting of experimental data using eq. (A1.8) show that the penetration factor (A1.1) is not better than those given by eqs. (5.7).
14
J. HUMnLrr
Appendix 2 THE EXPANSION THEOREM The well known Mittag-Leffler expansion le) being concerned with single-valued functions of a complex variable, the expansion o f the collision matrix is not a straightforward application of this theorem. It is rather an extension of it and it is useful to give the main arguments leading to its justification. Let f ( k ) be a function defined in a complex k-plane in which a series o f cuts have been introduced to make it single-valued. Outside the cuts, let its only singularities be an infinite number of isolated poles kl, k2 . . . . . with residues Pl, P2 . . . . . Let us assume that 0 < Ikll < Ik21 _~ . . . (A2.1) and that the series nil
(A2.2)
IP,lk, I
Rqc
x
Fig. 3. The complex k-plane with its poles × and a contour C . (for m ----4) with indentations of df~'erent types (C. ~ ABCDEFOHIJKLA-J-PQRSP). is convergent. Under such conditions, it is easy to verify that the series
,~k-k. P"
(A2.3)
is also convergent for any finite value of k uniformly distant from the poles. Let now C,, be a closed contour drawn in such a way that the poles k 1, k2 . . . . k,, are interior to it and that all the other singular points are exterior to it. Then, according to the theorem of residues,
1 i" jc..--,k-k;
p. - k-k."
(,,2.4)
TtW.ORY oF NUCLg~ REACTIONS(rv)
15
Hence, if we let m go to infinity,
f(k) = Q(k)+ ~ft P" =
k-k.
(A2.5)
where Q(k) = lim f f(k') dk'. =-,= J c , k ' - k It is important to recall that, as a corollary to the fundamental Cauchy theorem, the theorem of residues used to derive eq. (A2.4) does not require 17) that each point of the contour be a point of holomorphy off(k). It is suJ~cient t h a t f ( k ) be continuous on the contour (3,; then, for any point k' on C=, we have lira L f ( k ) - f ( k ' ) ] = 0.
(A2.6)
when k is an interior point tending to k'. Accordingly, it is often possible to extend indentations just to the singular points (like ABC and DEF in fig. 3), rather than turning around them at a non-vanishing distance 0ike G H I and J K L in fig. 3). This remark finds application in sect. 4. There, we had two possible choices for the cut issuing from the threshold of channel c: (1) to draw a circle of finite radius around the essential singularity and extend it to infinity by a cut which could be infinitely narrow (like GHI), or (2) to expand the product o f f ( k ) and a factor g(k) chosen in such a way that with a cut extending just to the singular point (like DEF), the product f(k)g(k) is continuous on (3, and has inside exactly the same singular points (poles) as f(k). In sect. 4, we had to choose the latter alternative for physical reasons: we wanted an expression for T¢,= in which the energy dependence explicitly appeared in each term of eq. (A2.5) for real energies just above the thresholds. Two possible forms for g(k) were 8~,te~le-it~°'+~°)k~'k~Z-t and e=,8=e-l(~='+~°)k~"1. (A2.7) It will be noticed that the point k© = 0 is an essential singularity of
l/[r(l+
1 + i,to/ko) W+ ]
as well as of l/F/+, in spite of the fact that te foxmer quantity is O(k l) when arg k= statistics eq. (4.8a). Refermces 1) 2) 3) 4)
J. Humblet and L. Rosenfeld, Nuclear Physics 26 (1961) 529 L. Ro~nfeld, Nuclear Physics 26 (1961) 594 J. Humblet, Nuclear Physics 31 (1962) 544 C. Mahaux, Pruc. Brmmels Conf. on Nuclear Physics, Sept. 1962 (I.I.S.N., Bruxelles, 1962); Bull. Soc. Se. Li6ge 32 (1963) 62, 70 and 240 5) J. Humblet, Proc. Int. Symposium on Direct Interactions and Nuclear Reaction Mechanisms, Padua, Sept. 1962 (Gordon and Breach, New York, 1963); Proc. Brussels Conf. on Nuclear Physics, Sept. 1962 (I.I.S.N., Bruxelles, 1962)
16
J. h-ut~tzr
6) H. Buchholz, Die Konfluente Hypergeometrische Funktion (Springer, Berlin, 1953) 7) M. H. Hull and G. Breit, Handbuch der Physik, 41/1 (Springer, Berlin, 1959) 8) A. Messiah, Quantum mechanics, vol. I (North-Holland Publ. Co., Amsterdam, 1961) appendixB 9) G. N. Watson, Theory of Bessel functions, 2nd ed. (Cambridge University Press, Cambridge, 1952) 10) A. Erd61yi et al., Higher transcendental functions, vol. I (McGraw-Hill, New York, 1953) 11) L.J. Slater, Confluent hypergeometri¢ functions (Cambridge University Press, Cambridge, 1960) 12) A. M. Lane and R. G. Thomas, Revs. Mod. Phys. 30 (1958) 257 13) L. C. McDermott, Thesis, Columbia University, New York 1959 (AEC document CU-187, Washington D.C.) 14) J. Humblet, Comptes Rendns Acad. So. 231 (1950) 1436 15) J. P. Jenkenne, private communication (1963) 16) A. Hurwitz and R. Courant, Funktionentheorie (Springer, Berlin, 1922); C. Carath6odory, Funktionentheorie (Birkhathter, Basel, 1950) 17) E. Goursat, Cours d'analyse mathC~atique, 5th ed. (Ganthier-Villars, Paris, 1933) footnote p. 74 18) J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952)