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Stationary state currents in nuclear reactions: Rotational coupling in alpha-particle scattering
Nuclear Physics A483 (1988) 173-194 North-Holland, Amsterdam
STATIONARY STATE CURRENTS IN NUCLEAR REACTIONS: Rotational coupling in alpha-particle scattering R.S. MACKINTOSH, A.A. IOANNIDES and S.G. COOPER
Physics Department, The Open University, Milton Keynes MK7 6AA, UK Received 19 January 1988 (Revised 26 February 1988) Abstract: The graphical presentation of stationary-state probability currents occurring in nuclear reactions is described, and some results are presented for an alpha-scattering example involving channel coupling. We find that a natural way of expressing channel coupling effects is through the introduction of non-spherical potentials ('O-potentials") which precisely reproduce the elasticscattering channel wavefunction. The relationship to more conventional coupling-induced polarization potentials is discussed and arguments presented for an "anti-Percy" effect related to channel coupling non-locality.
1. Introduction More than two decades ago, McCarthy and others 1) and subsequently Amos 2), presented graphical representations of the stationary-state wavefunctions for spinless particles scattering from optical-model potentials. In particular, they presented figures showing the behaviour of the probability current density and its divergence. These pioneering calculations were particularly motivated by the need to accredit various approximations for the wavefunction, the intention being to allow computationally simpler calculations of observables. This motivation is no longer generally relevant owing to subsequent great increases of computing power, but there are nevertheless now reasons to exploit this computing power to examine these quantities once more. There have been many serious attempts to calculate nucleon-nucleus and nucleusnucleus potentials. An important question which inevitably arises is the nature of the polarization potential which is due to the coupling to inelastic and reaction channels. Sometimes it is attractive and sometimes it is repulsive; in fact is usually both at different radii. Although some systematics have been noted 3) and explanations put forward in particular cases e.g. refs. 4,5), there is no generally agreed understanding of the underlying physics of these polarization potentials. Moreover, they embody features absent from most local-density models: probable underlying angular-momentum dependence and possible strong non-local effects. It is in the hope that it will eventually throw light on these questions that we have chosen to establish means of studying the various quantities related to probability current. 0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) June 1988
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R.S. Mackintosh et aL/ Stationary state currents in nuclear reactions
The emphasis of our work is quite different from that of McCarthy and of Amos. Since we study the currents connected with the elastic channels of coupled-channel calculations, we cannot take their short cut to the calculation of the divergence of the probability current, namely, multiplying the absorptive potential by IqJ[2. We calculate V - j directly. Dividing by I,I 2 then gives an angle-dependent function which we refer to for convenience as the imaginary part of a " 0 - p o t e n t i a l " , indicating that it is calculated directly from 0- We can also calculate a real term which added to it gives a complex non-spherically symmetric "0-potential" which, when inserted into Schrrdinger's equation, exactly reproduces 0(r). Because of its non-spherical nature, this 0-potential needs some care in its interpretation. At the same time, we shall argue that it is a rich source of understanding concerning the polarization effects due to coupled channels. Many of the properties of the local polarization potentials as calculated by more conventional means can be related to it. In fact, we find that these 0-potentials turn out to be much more informative than the divergence of the current since the behaviour of this latter quantity is so much dominated by local variations in 1012. For heavy ions in particular, 101 = can vary over many orders of magnitude in interesting regions of the nucleus. This paper contains our formulation of the problem as applied to spinless particles, followed by detailed results for one particular case chosen to exemplify the information concerning polarization potentials. The case studied here involves the polarization effects due to rotational model coupling for alpha particles scattering from 2°Ne at 104 MeV. We then present an interpretation of the "0-potentials" and some assessment of their significance. This discussion will be continued in subsequent papers in which we examine the possibility of exploiting 0-potentials as an alternative representation o f / - d e p e n d e n c e effects, and discuss the particular question of polarization potentials in sub-barrier rotational excitation of heavy ions. 2.
Formulation
We begin with the expression for the probability current j in terms of the stationary state wavefunction 0(r):
j(r) = h Im [0*V~b]. /z
(1)
The quantity V • j is of great interest because of its immediate connection with the absorption of flux by the non-hermitian part of the single-particle potential. Given a n / - i n d e p e n d e n t , local potential, V(r)+ iW(r), it is clear that: h2 -o*v2 -- 1 12[(V(r) - E) + iW(r)]. (2) 2/* Taking the imaginary part and using
V . j(r) = h Im [ 0 * V i 0 ] /x
(3)
R.S. Mackintosh et aL / Stationary state currents in nuclear reactions
175
we get the well known equation by which McCarthy calculated V • j: V-/(r)=
W(r)tO(r)[ 2.
(4)
Moreover, taking the real part of eq. (2), we get an equation which will be of considerable importance in what follows: Re (0*V2~) _ 2 (V(r) - E)[0[ 2 .
(5)
11
For reasons to be described, it will be appropriate to evaluate the divergence of the current from 0 using eq. (3) instead of eq. (4); the algorithm for this at once gives us the left-hand side of the last equation too, since the real and imaginary parts of the same quantity are involved. We now describe how we exploited these last three equations. Eqs. (4) and (5) as they stand apply only to local and/-independent potentials. Our intention was not to study the properties of 0 for such potentials. We therefore did not use eq. (4) to calculate the divergence of the probability current, but used eq. (3) directly from 0. We then applied eq. (2), taking the real and imaginary parts, to verify our program by precisely reproducing the real and imaginary parts of local, /-independent optical potentials. Thus, we construct V. j and [0[2 directly, with no restriction to/-independent local potentials, by evaluating the real and imaginary parts of 0*V20. We may now consider eqs. (5) and (4) to define functions V(r) and W(r). Note that these will now in general depend upon angle. We refer to these functions as the real and imaginary parts of a "0-potential" which when inserted into Schr6dinger's equation would give 0(r). We adopt this terminology for convenience, and to distinguish them from conventional potentials; all terms in the hamiltonian of the projectile plus target-nucleus system must be spherically symmetric*. These nonspherical functions must therefore be a local equivalent representation of a scalar term in the hamiltonian composed of the momentum operator - i h V and the gradient of the nuclear density (i.e., in the radial direction). It is the fact that the beam direction defines a preferred momentum direction which leads to the non-spherical effective potentials. We shall make very extensive use of these 0-potentials in discussions of polarization effects due to channel coupling. 2.t. M E T H O D O F C A L C U L A T I O N
We assume that the stationary state wavefunction has been generated by an optical model or coupled channel code; we write down the following form to establish the notation:
O(,) =IE u~(r) Y,o(~), r i
(6)
* The exception being for an odd nucleus of spin > ½, with a deformed intrinsic state, which can be aligned in an external magnetic field.
176
where
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
u~(r) has the asymptotic form: il+t
ut(r) ~ T k eW'(47r(21 + 1))'/2(It - &Ol)
(7)
in which/i and 05 are the incoming and outgoing Coulomb radial wavefunctions. It follows from eq. (1) and the expression
- Lr Or2
that the divergence of the probability current may be expressed as:
V "j(,) = ~ ~ ZL(r) YLo(~) ,
(9)
where r(21+ 1)(2/,+ 1)]
&(r)=E L 4 7 ( ~ )
5/2 trL 2 [ l'(l'+l) u,(r)ur(r) ] (Cooo) Im u*(r)u~'(r) r2
.1
(lO)
in which C represents a Clebsch-Gordan coefficient.
2.2. EVALUATION O F T H E C U R R E N T VECTOR
In order to express the current vector in a natural way, we shall depart from the usual implicit convention of using the z-axis for the incident direction, and, confining ourselves to the scattering plane, we shall use the x-axis in the horizontal direction for this purpose. Then the y-axis is vertical. Theta remains the angle from the incident direction. With these conventions, we can express the x and y components of the current j as follows: h
Yao(O,0),
(11)
(Ya(r)) Ya,(O, O)
(12)
jx = 7 - - Z Im (--wa(r)) A m ,x
h
Jy = Z Im 2m A
in which ~ and Y depend upon the first derivatives of the radial wavefunctions. These are incorporated in the following functions f and g:
r '+' 1"7\dr-r/ ~(r)=-L2-~-~.l g,(r>=
[ t ]1/2[ d
(13) r
'
I+1\ u,(r) ,
(14)
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
177
which occur in the following expressions for ~ and Y:
F ul(r)__fr(r) ut(r)gr(r) Hn,x] Ya (r) = ~ L r 2 Gu,x 4 r2
(15)
1
[ ut(r)__fv(r) Su,a -~ ut(r)gr(r) Tu,,x] ~a(r) =~ L r2 r2 "
(16)
The coefficients G,/4, S, and T are as follows:
F(21+1)(21'+3)ql/2r~,'+ll ,'r~tl'+,~r~,,'+la G,,,~=L ~ ~ j .-., -,o,--oo o.--o, , , h,,,,x
S/,'x
=F(21+1)(21'-1)]'/2c,,'-11
rcar-tAc,,r-,~
(17)
o. o,
(18)
= F(21+ 1)(21'+ 3)]ll/2rr+llr,p,r+lX~ 2 L 4--'~--~ _] ,--o oo,,--oo o, ,
(19)
u
j
.-,
-,o.
oo
[(2•+ 1)(21'- 1)1'/2,..,,,_,1,,,,,._,tr_,~,, 2 T,,,~=L 4 - - ~ j ,--o oo,,-.oo o , .
(20)
2.3. ALTERNATIVE APPROACHES The expressions given above are precise, computationally straightforward, and the multipole expansion coefficients such as Zt(r) may be stored for any form of graphical processing. Nevertheless, an alternative approach is simply to evaluate $(r) on a grid and evaluate the various gradients by straightforward numerical differentiation. This is facilitated for spinless particles by the fact that symmetry dictates that the derivatives normal to the scattering plane are zero. There is a particular advantage to us in the 'square-mesh' approach in that it requires minimal adaptation of existing software developed 6) for the graphical representation of biomagnetic fields. Because of the relatively coarse mesh, all quantities involving second derivatives, such as the ~b-potentials, were determined by calculating V25 at all points on the mesh by using eqs. (6) and (8). The algebraic approach will give immediate results in the absence of such software and we use it to calibrate the accuracy of the mesh approach. In this paper, we use the mesh approach to give graphical representations conveying the global properties, and the algebraic approach to give simple graphs showing the precise effects of channel coupling on various quantities evaluated along the zero impact parameter line. We emphasize that this division of labour is historical rather than essential, and both methods give exactly the same results.
3. Rotational coupling of alpha particles We have studied the wavefunctions corresponding to the inelastic scattering of alpha particles from 2°Ne treated as a good rotor. The 2 + and 4 + states are included
178
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
with realistic values of the deformation parameters fiE and 134. In some calculations the 4 + state was omitted without change of parameters. The interest, of course, lies in the characteristic channel coupling effects, so we compare our results throughout with " D W " , or "uncoupled" wavefunctions obtained by switching off the backward coupling in the coupled channel code 7) from which we extract the wavefunctions. A bonus coming from a calculation with "uncoupled" wavefunctions is that it enables us to verify that we do indeed recover the real and imaginary potentials by dividing the real and imaginary parts of ff*V2ff by 1012. In this way we have checked the program to high accuracy. Channel-coupling O-potentials. We first present various quantities calculated by the algebraic approach, plotted along a line of zero impact parameter, i.e. through the nuclear centre in the direction of the incident momentum. Generally, we compare the quantities as calculated in three ways: (a) when the coupling is switched off, (b) when coupling to the 2 + state is included, and (c) when coupling to both the 2 + and 4 + states is included. The potential used in the coupled channel, CC, calculations is fixed at that which is appropriate for the full coupling of case (c). This will sometimes be referred to as the "bare" poten,'!al in the discussion below. Fig. la shows II]/I2 for these cases with the no-coupling case a solid line, the 2 + case long dashes, and the three state case short dashes. The roughly exponential fall-off through the nucleus is followed by a focusing effect in which the focus actually lies outside the nucleus on the far side. The halfway point for the real potential is at about 3.75 fm. The focusing effect found by McCarthy at much lower energies generally led to a focus within the nucleus. Of more interest is the very large effect that the channel coupling has had on the wavefunction on the far side of the nucleus compared to the rather modest reduction for r < +3.5 fm. Later, we shall refer back to this figure in order to compare the results with 1012 from inverted and empirical elastic scattering potentials. It turns out that the variation in magnitude of V • j along the same central line is so dominated by the behaviour of I~/I2 that it is much more informative to divide out by the latter quantity. This just gives the imaginary 0-potential, and in fact most of our attention will fall upon both 0-potentials. Concerning the divergence itself, we simply mention that in the present case, unlike some to be discussed, V . j nowhere becomes positive so that there are no absolutely emissive regions generated by coupling in this particular case. Nevertheless, there are emissive regions in the polarization 0-potential, i.e. the difference between the extracted ~b-potential and the spherical bare potential, as will be demonstrated. As we have mentioned, the 0-potentials provide a good check on our programs since these must be the " b a r e " potentials in the no-coupling cases. In fig. lb, the solid line is the real potential calculated using eq. (5) for the no-coupling case. The bare potential is reproduced to three or four significant figures. The Coulomb dominated regions beyond 6 fm are obvious. The long dashed curve in this figure is the 0-potential for the case with one excited state and the short dashed curve
2,00 1,75
nepl rot02 rot024
~ .....
1.50 1.25 1.00 0.75 0,50 0,25 I
-5
I
0
5
10
0
5
l0
0 -10 -20 -30 V(re~) MeV
-40 -50 -60 -70 -80 ,
!
-5
i/
I
V(imag) MeV -15
-20
(c) -5
0 x/fro
5
10
Fig. 1. (a) Potentials for 104 MeV alpha particles scattering from 20 Ne, 1~12 for the elastic channel plotted along the zero impact parameter axis through the nucleus with the flux incident from the left and with x = 0 located at the nuclear centre. The solid line is for no inelastic coupling, the long dashes are for rotational model coupling to the 2 ÷ state, and the short dashes correspond to the further inclusion of the 4 + state in the calculation. (b) As for part (a), but showing the real potentials as calculated from the wavefunctions using the method described in the text. The repulsive Coulomb term dominates beyond 6 fln. (c) As for part (b), but showing the imaginary potentials. In this and subsequent figures, the quantities notated as V(real) and V(imag) are the V and W of eqs. (2)-(5).
180
R.S. M a c k i n t o s h et al. / Stationary state currents in nuclear reactions
results w h e n the 4 + state is i n c l u d e d . T h e effect is n o t n e g l i g i b l e at the n u c l e a r surface, a n d r e p r e s e n t s a h i g h l y a s y m m e t r i c a l effect, the p o t e n t i a l n o l o n g e r b e i n g s p h e r i c a l l y s y m m e t r i c . I n fig. 2a we s h o w the differences b e t w e e n the d a s h e d curves a n d t h e s o l i d curve o f fig. l b . T h e s e curves r e p r e s e n t the p o l a r i z a t i o n effect d u e to the c h a n n e l c o u p l i n g , a n d e x h i b i t r e p u l s i o n o n the " i l l u m i n a t e d " side o f the n u c l e u s a n d in the far p a r t o f t h e s h a d o w side, a n d a t t r a c t i o n f u r t h e r in on the s h a d o w side. T h e c o n s e q u e n c e s o f this b e h a v i o u r will be e x a m i n e d below. T h e i m a g i n a r y p o l a r i z a t i o n p o t e n t i a l is also a s y m m e t r i c . Figs. l c a n d 2b are the c o u n t e r p a r t s for the i m a g i n a r y c o m p o n e n t s o f figs. l b a n d 2a. W h i l e the i m a g i n a r y p o t e n t i a l s in fig. l c n e v e r b e c o m e positive (emissive), the p o l a r i z a t i o n p o t e n t i a l s o f fig. 2b d o b e c o m e emissive in the tail b e y o n d + 5 fm. E v i d e n t l y , flux is b e i n g fed b a c k into the elastic
AV(real) MeV -1 -2 -3
rot02 m _ _ rot024 (a) I
t
-5
i
0
5
I0
0 -2
AV(imag) MeV
-4
A
t
-6 -8
(b) I
-5
i
i
0 x/fro
I
5
I
10
Fig. 2. (a) As for figure lb, but showing the polarization potentials calculated from the curves of fig. lb by subtracting from the potentials calculated with coupling included, the no-coupling potential. In this figure, the solid lines are the polarization potentials for 2+ state only, and the dashed lines are for both the 2+ and 4+ states. (b) The imaginary polarization potentials as in fig. lc calculated by subtraction as in fig. 2a for the real part.
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
181
channel here, a clear example of a coupled-channel non-locality. Flux generation as a non-local effect has been discussed by Austern 8). One feature of the 0-potentials is exactly as expected: the polarization potentials, real and imaginary, are peaked just in the region of the inelastic scattering form factors, something which is less obvious in the case of spherical S-matrix equivalent polarization potentials for a reason which we will attempt to make intelligible later. We have shown only the behaviour along the zero impact parameter line. In figs. 3-6 we present figures conveying the spatial behaviour of these quantities and also j(r) itself. The upper and lower parts of fig. 3 represent 1012 for the no-coupling and "2 + only" cases. Note that the plotting program has changed the scale by a factor of 1.7, thus de-emphasizing the reduction in the focusing effect brought about by the coupIing. Fig. 4 throws some light on this focusing: the current j ( r ) is directed inwards to become strong just along the axis on the unilluminated side of the nucleus. The left-hand side of fig. 5 shows, somewhat squashed by the plotting routines, the spherical real no-coupling potential. The non-spherical real 0-potential due to 2 + coupling can be compared with it in the right-hand side of fig. 5. Fig. 6 shows how nonspherical the imaginary 0-potential is in the coupled case. The scale in these figures can be deduced from fig. 1 which shows the behaviour of these quantities along the y = 0, zero impact parameter, line. Comparison with empirical optical potential. The optical potential appropriate for single-channel scattering is quite different from that appropriate to coupled channel calculations, and the difference between the potentials required in the two cases has for a long time been taken as a rough way of getting the polarization potential due to channel coupling 9,1o). In fig. 7a we compare [0[ 2 for a spherical optical potential fitted to forward-angle elastic scattering 11) and compare it to the same quantity calculated for the elasticchannel wavefunction from a coupled channel calculation including both the 2 + and 4 + states with a potential adjusted to fit the elastic and inelastic scattering as used for all the cases above. Although the magnitude on the far side of the nucleus is approximately correct, the phase of the evident interference feature is wrong. Indeed, comparing with fig. la, although the focussing effect has been correctly suppressed, [0[ 2 elsewhere is hardly closer to the CC result than the no-coupling results shown in fig. la. We conclude that much that is interesting about polarization potentials might be missed from the comparison of the two roughly fitted empirical potentials. In fig. 7b we present two imaginary potentials which ostensibly should be the same. The solid line is Rebel's empirical imaginary potential, and the dashed line is that shown in short dashes in fig. lc. This is the imaginary part of a potential which we know would reproduce in a single-channel calculation the elastic-channel wavefunctions from the 0 ÷ 2 ÷ 4 ÷ coupled channel calculation for which the parameters have also been adjusted to "fit" the (largely forward angle) data. Comparing with spherical local equivalent potentials. The polarization effects due to channel coupling can also be quantified by applying S-matrix to potential
Fig. 3. Upper: Histogram representation of I~[ 2 for the elastic channel on the scattering plane for the no-coupling case. The incident b e a m direction is from the u p p e r left. The absorption o f the beam is d e a r l y visible as is the focus; in spite o f the focus, the obvious effect off the beam axis is the creation o f a diffractive shadow on the "'dark" side. Lower: As for the u p p e r part, but for the case in which rotational coupling to the 2 + state is included. Note that the scale is renormalized by a factor o f 1.7, as can be seen from the unperturbed region on the large y edge. The features at the corners are artefacts of the plotting routine.
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
183
10
8-
6
m
~
4-
2•
~.
N.
/.
.p
y/fro 0
-2
-4
p
-6
-8
-10
....
I .... -8
I .... -6
I .... -4
I . . . . . . . . -2 0 x/fm
I .... 2
I .... 4
I .... 6
I .... 8
10
Fig. 4. T h e p r o b a b i l i t y f l u x j o n the scattering p l a n e f o r the n o - c o u p l i n g c a s e . T h e a r r o w size is g e n e r a l l y p r o p o r t i o n a l to IJl e x c e p t f o r p l o t t i n g r o u t i n e artefacts at the e x t r e m e l e f t - h a n d e d g e .
inversion techniques, see refs. 5,12,13), to the elastic channel S-matrix elements and subtracting the "bare" potential from the coupled-channel calculation. This gives a spherically symmetric, local and /-independent representation of the channelcoupling effects which can be compared with potentials found by phenomenological fits to data. However, this potential will not be expected to give the same wavefunction in the nuclear interior as the 0-potentials described above, although the wavefunction must be identical in the asymptotic region. In fig. 8 we exhibit 1012 for the coupled-channel calculation (solid line, identical to the short-dashed curve of fig. la). Indeed, this quantity is indistinguishable from that obtained by inversion (dashed curve) in the external region. What is very surprising is that the dashed curve falls below the solid curve in the nuclear interior. This is evidently an "anti-Percy effect". The wavefunction containing the non-local effects due to channel coupling is larger in the nuclear interior than that for the local potential giving the same asymptotic wavefunction, i.e. S-matrix elements [cf. refs. 14.15)where the Percy effect is discussed]. In the context of nucleon-nucleus interactions, Mahaux and
184
R.S. M a c k i n t o s h et al. / S t a t i o n a r y state currents in nuclear reactions
0
0
7
-7
~ ~ -~ .~.
~.~ ~ ~.~ ~ .~
•-
.~
0
?
g~
~.~ I
I
I
.
I
0
g
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
_=
0 e'~
0
?
0
~A
~o
u~
_o e~
~*
¢-4
I
I
I
o 0
~r
185
186
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
1.4
- -
-
-
dw rebel rot024
-
1.2 1.0 0.8 0.6 0.4 0.2 -5
0
5
10
-5
-10 V/MeV
_••I
-15
.I
-20
\I
(b) iW
-5
0
I
5
10 x/fm Fig. 7. (a) Comparing 1012 for the elastic-channel wavefunction for two different calculations which nominally fit the elastic-scattering cross section and hence the asymptotic wavefunction. The solid line is for the one-channel optical-model fit and the dashed line represents the coupled-channel calculation including 2+ and 4+ states with appropriately-refitted potential parameters. (b) The solid line is the imaginary potential corresponding to the solid line of part (a) and the dashed line is the projection on the x-axis of the imaginary part of the non-spherical 0-potential corresponding to the dashed line of part (a). Sartor have recently f o u n d that the radial w a v e f u n c t i o n s of b o u n d n e u t r o n s in Pb are i m p r o v e d by the u n j u s t i f i a b l e o m i s s i o n o f the Perey factor, see p. 269 o f ref. t6). The existence o f a n effect, p r e s u m a b l y d u e to the n o n - l o c a l effect of c o u p l i n g , which is the o p p o s i t e sense to the Perey effect, i m m e d i a t e l y suggests that such effects as we have f o u n d s h o u l d also be explored in the context of n u c l e o n - n u c l e u s interactions. It is very instructive to c o m p a r e the local p o t e n t i a l o b t a i n e d b y i n v e r s i o n with the n o n - s p h e r i c a l p o t e n t i a l which c o r r e s p o n d s to the e l a s t i c - c h a n n e l C C w a v e f u n c tion. First we study the i n v e r t e d potential. I n fig. 9, the solid curve is the real " b a r e " c o u p l e d - c h a n n e l p o t e n t i a l with the C o u l o m b part removed. The d a s h e d line is the
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
1.4I
187
rot/cc rot/inv.
------
1.2 1.0
P~l 2
0.8 0.6 0.4 0.2 -5
0 xffm
5
10
Fig. 8. Comparing [ol2 for the rotational coupling case (4+ state included, solid line), with that calculated using the spherical S-matrix equivalent potential which was found by S-matrix to potential inversion.
bare f cc/inv. Z -20 -40 V/MeV -60 -80 I
I
I
I
2
4
6
8
10
r/fm Fig. 9. Two potentials showing the polarization effect on the real potential due to rotational coupling, as represented by spherical potentials. The solid line is the bare potential in the coupled channel calculation, and the dashed line is the potential found by inversion from the S for the elastic channel given by the coupled-channel calculation. p o t e n t i a l o b t a i n e d b y a p p l y i n g S - m a t r i x to p o t e n t i a l i n v e r s i o n to the e l a s t i c - c h a n n e l S - m a t r i x e l e m e n t s f r o m t h e c o u p l e d - c h a n n e l c a l c u l a t i o n in w h i c h b o t h the 2 + a n d 4 + states o f 2°Ne were i n c l u d e d . F r o m fig. 10, w h e r e the difference b e t w e e n these curves is p l o t t e d , it is c l e a r t h a t t h e p o l a r i z a t i o n effect is g e n e r a l l y r e p u l s i v e e x c e p t n e a r 3.3 fm w h e r e it is attractive. I n fig. 11 we p r e s e n t the i m a g i n a r y c o u n t e r p a r t to t h o s e in fig. 9. W e shall n o w c o m p a r e these p o l a r i z a t i o n p o t e n t i a l s o b t a i n e d b y i n v e r s i o n with t h e ~0-potentials. I n fig. 12 the solid line is the b a r e i m a g i n a r y p o t e n t i a l , the l o n g
188
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
15.0 12.5 10.0 7.5 AV/MeV 5.0 2.5 0.0 -2.5 i
2
i
i
4
i
10 r/fm Fig. 10. The difference between the two curves in fig. 9. This is the polarization potential generated by channel coupling and is repulsive almost everywhere.
bare -5 -10
6
8
f
'._ ~~/llff/
V(imag) MeV -15
\,J
-20 I
i
i
I
2
4
6
8
10 r/fm Fig. 11. The imaginary potentials corresponding to the real potentials shown in fig. 9. The extra absorption peaked near the surface is expected; the decrease in absorption for smaller radii is less so. d a s h e d line is the p r e v i o u s l y given ~O-potential a n d the s h o r t - d a s h e d curve is the i n v e r t e d p o t e n t i a l s h o w n d a s h e d in fig. 11. T h e s h a r p central d i p visible in the latter figure has b e e n lost d u e to the c o a r s e r r e s o l u t i o n n e c e s s a r y for e v a l u a t i n g the qJ-potentials (which c a n n o t in a n y case s h o w a d i s c o n t i n u i t y at the n u c l e a r centre.) I n fig. 12, the s p h e r i c a l l y s y m m e t r i c i n v e r t e d p o t e n t i a l m a k e s the a s y m m e t r y o f t h e ~O-potential m o r e o b v i o u s . It is fairly clear h o w the s h a r p o s c i l l a t i o n s o f the i n v e r t e d p o t e n t i a l a r e n e c e s s a r y for the s p h e r i c a l p o t e n t i a l if it is to give the s a m e a s y m p t o t i c b e h a v i o u r as the n o n - s p h e r i c a l ~0-potential. This p o i n t is c l e a r e r f r o m the two p o l a r i z a t i o n p o t e n t i a l s c o m p a r e d as in fig. 13. T h e qJ-potential s h o w n in the s o l i d line a p p e a r s to force the d a s h e d line u p n e a r +5 fm, b u t b e i n g s p h e r i c a l this t a k e s
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
~no
189
[oupling 2+,4 + ~-pot. ~'
--
~ - 2 +, 4 + invert.fl
-5
II
It
-10 V(imag)
.//I
MeV -15 -20
-5
0 x/fm
5
10
Fig. 12. Imaginary potentials along the zero impact parameter axis. Solid line: "bare" potential (spherical); long dashes: 0-potential (non-spherical); short dashes: inverted potential (spherical). The two spherical potentials are as in fig. 11, except that the programs used to produce this figure could not handle the cusp very close to r = 0 in the inverted potential. 10 8
/
6
\
/ /
4 AV(real) MeV
f "\
2 0 -2
2+4 + 2 +, 4 + invert
-4 I
i
]
J
I
-5
0 5 10 x/fm Fig. 13. Real potentials along the zero impact parameter axis but with the 'bare' potential subtracted. The solid line is the non-spherical 0-potential and the dashed line is the spherical potential obtained by inverting the S-matrices. it t o o f a r n e a r --5 fm, so it m u s t d r o p s h a r p l y for Ir1<4.5 fro. This is also n e c e s s i t a t e d b y the n e g a t i v e e x c u r s i o n o f the 0 - p o t e n t i a l n e a r +3.5 fro. But since [0[ 2 is m u c h larger at the negative x, lit, side o f the n u c l e u s the negative p o t e n t i a l o p e r a t i v e at - 3 fm m u s t be c o m p e n s a t e d by the large r e p u l s i v e f e a t u r e at the n u c l e a r centre. O f course, such a r g u m e n t s c a n o n l y b e t a k e n as c o n c l u s i v e if t h e y a p p l y u n i f o r m l y to m a n y cases as t h e y are s t u d i e d . The r a t i o o f V • j as c a l c u l a t e d from 0 to t h a t f r o m i n v e r s i o n m i g h t b e closer to u n i t y t h a n the ratio o f the i m a g i n a r y p o t e n t i a l s o w i n g to the fact that ~/, is different
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R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
in the two cases. In fact the ratio is closer to unity for r = - 4 fm along the zero impact parameter line, but further from unity near r = +4 fm. S u m m a r y o f O-potential properties. Without doubt, the 0-potentials are the quantities related to probability current and its divergence which most dramatically reveal the effects of channel coupling upon the interaction between nuclei. The perturbations in the 0-potentials induced by coupling, which may be called the polarization 0-potentials, are largest in magnitude quite close to the radius where the inelasticcoupling form factors peak. This might seem inevitable, but we have seen that it is far less completely true for the local potential found by inversion. Other striking features of the 0-potential are: (i) The 0-potential is not spherically symmetric. (ii) The imaginary part of the 0-potential exhibits a small region on the "far" side of the nucleus where it is actually less than the bare potential. The first of these features is most naturally thought of as arising from an effective k. r dependence in the polarization potential. The second feature, which is a particular aspect of the first, might most naturally be ascribed to the underlying non-locality of the coupling effect. This is because the difference between the 0-potential and the bare potential in this region represents a generative or emissive polarization potential. Flux is being restored to the beam by the coupling here, a characteristic non-local effect 8). We give a more extended discussion of this in the next section. The difference between the 0-real potential and the bare real potential is comparable to the imaginary polarization potential, but the magnitude of this difference is much smaller than the bare real potential itself. In fig. 2a, therefore, we only plotted the difference, i.e. the real polarization potential. Like the imaginary polarization 0-potential, this is peaked at the maxima of the inelastic scattering form factors, but the degree of k. r dependence is very marked. In fact, at the front of the nucleus it is basically repulsive, and at the far side of the nucleus, basically attractive. This correlates with the polarization potential obtained by S-matrix inversion,* where there is surface repulsion (the near-side repulsion in fig. 2a extends to a greater radius than the attrractive part on the far side; there is actually repulsion for r > 4.75 fm on the far side), and attraction further in. The interior oscillations on the inverted, spherical,/-independent potential are the result of attempting to express this k. r dependence.
4. Interpreting non-spherical ~-potentials Symmetry dictates that a local interaction potential between a projectile and a spherical nucleus must be spherical (also for a spin-zero deformed nucleus in the * As described above, but for further discussion of what can be learned from this reaction using inversion, see ref. 17).
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
191
laboratory, non-intrinsic, frame). Nevertheless, it is quite obvious that there is a way in which the interaction potential can depend upon angle if the incoming momentum vector is allowed to define a direction. This is because there is a very natural way in which the interaction can depend upon the local momentum vector k. It is quite plausible that, for example, the local g-matrix could depend upon whether the nuclear nucleon interacted with would recoil into a nuclear region of positive or negative density gradient. This could plausibly lead to a (k. r) 2 term in the effective interaction, which would imply a term involving (k x r) 2, i.e. the angular momentum squared. This was one motivation for studying possible phenomenological evidence 18-20) for such /-dependence; the other was the emerging importance of coupled inelastic and reaction channels, which have a manifestly angularmomentum dependent effect 21). It is /-dependence of this kind which should be amenable to study through the evaluation of the various quantities described above, the 0-potentials in particular. It should therefore be no surprise that the 0-potentials presented in the last section are indeed non-spherical. We do not yet have a "dictionary" which will enable us to read the physics behind the details of the extracted potentials. However, it does appear that the calculated divergence and 0-potentials exhibit features that are most naturally interpreted as the signature of the underlying non-locality of the polarization potential. The difference between the extracted 0-potential and the bare potential of the coupled-channel calculation is a local representation of the couplinggenerated polarization potential. The imaginary part of this potential consistently exhibits positive regions, i.e. regions where the polarization potential is effectively generative, or emissive. Regions of emissivity are a likely characteristic of non-local potentials. It is, of course, the case that if the angle-dependent 0-potentials are inserted into a one-channel Schroedinger equation, that they would exactly reproduce the elastic channel component of the coupled channel wave function. They may be compared to the trivially-equivalent local potentials introduced for radial wavefunctions by Austern, ref. s). This property suggests that they may have an application in DWBA calculations where it is desirable to have elastic-channel wavefunctions which embody all the exchange and channel-coupling effects. 4.1. I M P L I C A T I O N S F O R S P H E R I C A L INVERSIONS
Our previous studies of the polarization potentials induced by channel coupling have exploited the new S-matrix to potential inversion technique which has been described in refs. 4.5,13). Such inversions were compared with the 0-potentials in sect. 3 above. The highly non-spherical nature of these 0-potentials, which we have associated with underlying /-dependence, constitutes a probable explanation of a phenomenon which invariably appears when polarization potentials are being studied by inversion. It is that the inversions achieved when the S-matrix elements
192
R.S. M a c k i n t o s h et al. / Stationary state currents in nuclear reactions
are related to channel coupling are n e v e r as good as those found with model calculations. The goodness of any inversion is quantified 4.5,13) by means of a "phase shift distance", tr. When model inversions are performed in which S-matrix elements corresponding to known potentials are processed through the inversion procedure, the potentials are regained with precision. The corresponding values of ~r are always from one to several orders of magnitude lower than those obtained when S-matrix elements from coupled-channel calculations or k n o w n / - d e p e n d e n t potential are so processed. Almost any potential composed of Woods-Saxon-like components can be found within the space spanned by our "inversion basis". On the other hand, the polarization potentials cannot be perfectly described within the same space. These difficulties, relating to the determination by inversion of spherical polarization potentials, are probably associated with the underlying l-dependency/non-sphericity of the polarization contribution. They appear to indicate that the spherical equivalent potentials of non-spherical potentials may indeed have components which are too rapidly varying to be completely within the space spanned by the normal choice of basis function set. We note that within the context of electron scattering, there are upper limits on the effective wave number of Fourier components in the nuclear charge density. These are imposed by limits on the physical momenta present. Such limits might be expected to apply to local nuclear potentials, except that it is far from clear how angular features of "reasonable" wavelength translate into radial features of a spherical potential. The difficulty of achieving satisfactory spherical potentials becomes acute in certain cases of heavy-ion scattering near the fusion barrier. This matter will be discussed in a future paper on the subject of the rotation-induced polarization potential in heavy-ion scatttering near the Coulomb barrier. 5. Summary and outline of future work
The purpose of this paper has been to establish that the study of the wavefunction, and such characteristics as the @-potentials we have defined, constitutes both a readily feasible and a potentially fruitful approach to learning more of what goes on in direct nuclear reactions, and in particular the nature of the polarization potentials generated by channel coupling. Because I 12 can vary very rapidly through the nucleus, it turns out that the behaviour of V • j becomes more transparent if it is divided by I 12 This leads to the imaginary ~b-potential. We also introduce the analogous real @-potential. The overall complex @-potential appears to be a useful means of expressing, in local /-independent form, the effects of channel coupling, non-locality, or /-dependence. In sect. 3 we summarized the properties of the ~b-potentials arising from a particular kind of channel coupling, that due to rotational states. The particular coupling we have studied here does not begin to exhaust the contribution of channel coupling to elastic scattering. For rotational coupling, we
R.S. Mackintosh et aL / Stationary state currents in nuclear reactions
193
see that the coupling effects when represented as a local potential depart markedly from sphericity. It cannot be assumed that the cumulative effect of an expanded set of coupled channels would be a restoration of spherical symmetry. Until such a restoration of symmetry is shown explicitly to be the case, we must conclude that one of the main results of this work is a clear demonstration of the breakdown of the local-density approximation which may be said to omit many essential features of direct reaction effects. According to the local-density approximation, the optical potential must be spherical, even for the one-channel potential for a 0 ÷ "deformed" nucleus. Because of the strength of channel coupling, we conclude that the problem of establishing the optical model or internuclear potential cannot be isolated from the complexities of direct reaction processes. One line of exploration suggested by this work, is the further study of what we have called an "anti-Perey effect", in the context of nucleon-nucleus interactions. If it occurs for nucleons, it could be of relevance to resolving an anomaly discovered by Mahaux and Sartor relating to the normalization of the tails of bound neutron wavefunctions, see p. 269 of ref. 16). More generally, the procedures described here show promise of throwing light on channel coupling non-locality as distinct from the exchange non-locality. This question seems recently to have been largely ignored apart from the work of Rawitscher, see for example ref. 22). We mention the small generative region near x = 6 fm in fig. 2 as a signature for such non-local effects, and it would be interesting to attempt a more direct link with Rawitscher's calculations. There is a clear scope for a much more extensive application of the methods introduced here. In particular, it appears that the coupling of reaction channels has a more powerful effect than inelastic channels, and it is plausible that this might be connected with the strikingly large generative effects which have been indicated by phenomenology 23). These reactions and the general questions they raise as to where and why the polarization potential is attractive and repulsive hold promise as a fruitful area for study. We are grateful to SERC for grant number N G 17179 in support of Dr. Cooper.
References 1) 2) 3) 4) 5) 6)
I.E. McCarthy, Nucl. Phys. 10 (1959) 583; Nucl. Phys. 11 (1959) 574 K.A. Amos, Nucl. Phys. 77 (1966) 225 R.S. Mackintosh, Nucl. Phys. A230 (1974) 195 A.A. Ioannides and R.S. Mackintosh, Phys. Lett. B161 (1985) 43 A.A. Ioannides and R.S. Mackintosh, Nucl. Phys. A467 (1987) 482 A.A. Ioannides, in Inverse problems: an interdisciplinary study, ed. P.C. Sabatier (Academic Press, London, 1987) 7) R.S. Mackintosh, Nucl. Phys. A210 (1973) 245 8) N. Austern, Phys. Rev. 137 (1965) B752 9) N.K. Glendenning, D.L. Hendde and O.N. Jarvis, Phys. Lett. B26 (1968) 131
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10) R.S. Mackintosh, Nucl. Phys. A164 (1971) 398 11) H. Rebel, G.W. Schweimer, G. Schatz, J. Specht, G. Mauser, D. Habs and H. Klewe-Nebenius, Nucl. Phys. A182 (1972) 145 12) R.S. Mackintosh and A.M. Kobos, Phys. Lett. Bl16 (1982) 95 13) A.A. Ioannides and R.S. Mackintosh, Nucl. Phys. A438 (1985) 354 14) F.G. Perey and B. Buck, Nucl. Phys. 32 (1962) 353 15) F.G. Percy, in Direct interactions and nuclear reaction mechanisms, ed. E. Clemental and C. Villi (Gordon and Breach, New York, 1963) p. 125 16) C. Mahaux and R. Sartor, Nucl. Phys. A475 (1987) 247 17) R.S. Mackintosh and A.A. Ioannides, in Advanced methods in the evaluation of nuclear scattering data (Springer, Berlin, 1985) 18) R.S. Mackintosh and L.A. Cordero, Phys. Lett. B68 (1977) 213 19) A.M. Kobos and R.S. Mackintosh, J. of Phys. G5 (1979) 97 20) A.M. Kobos and R.S. Mackintosh, Acta Phys. Polon. B12 (1981) 1029 21) H. Feshbach, Ann of Phys. 5 (1958) 357 22) G.H. Rawitscher, Nucl. Phys. A475 (1987) 519 23) R.S. Mackintosh, S.G. Cooper and A.A. Ioannides, Nucl. Phys. A476 (1988) 287
STATIONARY STATE CURRENTS IN NUCLEAR REACTIONS: Rotational coupling in alpha-particle scattering R.S. MACKINTOSH, A.A. IOANNIDES and S.G. COOPER
Physics Department, The Open University, Milton Keynes MK7 6AA, UK Received 19 January 1988 (Revised 26 February 1988) Abstract: The graphical presentation of stationary-state probability currents occurring in nuclear reactions is described, and some results are presented for an alpha-scattering example involving channel coupling. We find that a natural way of expressing channel coupling effects is through the introduction of non-spherical potentials ('O-potentials") which precisely reproduce the elasticscattering channel wavefunction. The relationship to more conventional coupling-induced polarization potentials is discussed and arguments presented for an "anti-Percy" effect related to channel coupling non-locality.
1. Introduction More than two decades ago, McCarthy and others 1) and subsequently Amos 2), presented graphical representations of the stationary-state wavefunctions for spinless particles scattering from optical-model potentials. In particular, they presented figures showing the behaviour of the probability current density and its divergence. These pioneering calculations were particularly motivated by the need to accredit various approximations for the wavefunction, the intention being to allow computationally simpler calculations of observables. This motivation is no longer generally relevant owing to subsequent great increases of computing power, but there are nevertheless now reasons to exploit this computing power to examine these quantities once more. There have been many serious attempts to calculate nucleon-nucleus and nucleusnucleus potentials. An important question which inevitably arises is the nature of the polarization potential which is due to the coupling to inelastic and reaction channels. Sometimes it is attractive and sometimes it is repulsive; in fact is usually both at different radii. Although some systematics have been noted 3) and explanations put forward in particular cases e.g. refs. 4,5), there is no generally agreed understanding of the underlying physics of these polarization potentials. Moreover, they embody features absent from most local-density models: probable underlying angular-momentum dependence and possible strong non-local effects. It is in the hope that it will eventually throw light on these questions that we have chosen to establish means of studying the various quantities related to probability current. 0375-9474/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) June 1988
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R.S. Mackintosh et aL/ Stationary state currents in nuclear reactions
The emphasis of our work is quite different from that of McCarthy and of Amos. Since we study the currents connected with the elastic channels of coupled-channel calculations, we cannot take their short cut to the calculation of the divergence of the probability current, namely, multiplying the absorptive potential by IqJ[2. We calculate V - j directly. Dividing by I,I 2 then gives an angle-dependent function which we refer to for convenience as the imaginary part of a " 0 - p o t e n t i a l " , indicating that it is calculated directly from 0- We can also calculate a real term which added to it gives a complex non-spherically symmetric "0-potential" which, when inserted into Schrrdinger's equation, exactly reproduces 0(r). Because of its non-spherical nature, this 0-potential needs some care in its interpretation. At the same time, we shall argue that it is a rich source of understanding concerning the polarization effects due to coupled channels. Many of the properties of the local polarization potentials as calculated by more conventional means can be related to it. In fact, we find that these 0-potentials turn out to be much more informative than the divergence of the current since the behaviour of this latter quantity is so much dominated by local variations in 1012. For heavy ions in particular, 101 = can vary over many orders of magnitude in interesting regions of the nucleus. This paper contains our formulation of the problem as applied to spinless particles, followed by detailed results for one particular case chosen to exemplify the information concerning polarization potentials. The case studied here involves the polarization effects due to rotational model coupling for alpha particles scattering from 2°Ne at 104 MeV. We then present an interpretation of the "0-potentials" and some assessment of their significance. This discussion will be continued in subsequent papers in which we examine the possibility of exploiting 0-potentials as an alternative representation o f / - d e p e n d e n c e effects, and discuss the particular question of polarization potentials in sub-barrier rotational excitation of heavy ions. 2.
Formulation
We begin with the expression for the probability current j in terms of the stationary state wavefunction 0(r):
j(r) = h Im [0*V~b]. /z
(1)
The quantity V • j is of great interest because of its immediate connection with the absorption of flux by the non-hermitian part of the single-particle potential. Given a n / - i n d e p e n d e n t , local potential, V(r)+ iW(r), it is clear that: h2 -o*v2 -- 1 12[(V(r) - E) + iW(r)]. (2) 2/* Taking the imaginary part and using
V . j(r) = h Im [ 0 * V i 0 ] /x
(3)
R.S. Mackintosh et aL / Stationary state currents in nuclear reactions
175
we get the well known equation by which McCarthy calculated V • j: V-/(r)=
W(r)tO(r)[ 2.
(4)
Moreover, taking the real part of eq. (2), we get an equation which will be of considerable importance in what follows: Re (0*V2~) _ 2 (V(r) - E)[0[ 2 .
(5)
11
For reasons to be described, it will be appropriate to evaluate the divergence of the current from 0 using eq. (3) instead of eq. (4); the algorithm for this at once gives us the left-hand side of the last equation too, since the real and imaginary parts of the same quantity are involved. We now describe how we exploited these last three equations. Eqs. (4) and (5) as they stand apply only to local and/-independent potentials. Our intention was not to study the properties of 0 for such potentials. We therefore did not use eq. (4) to calculate the divergence of the probability current, but used eq. (3) directly from 0. We then applied eq. (2), taking the real and imaginary parts, to verify our program by precisely reproducing the real and imaginary parts of local, /-independent optical potentials. Thus, we construct V. j and [0[2 directly, with no restriction to/-independent local potentials, by evaluating the real and imaginary parts of 0*V20. We may now consider eqs. (5) and (4) to define functions V(r) and W(r). Note that these will now in general depend upon angle. We refer to these functions as the real and imaginary parts of a "0-potential" which when inserted into Schr6dinger's equation would give 0(r). We adopt this terminology for convenience, and to distinguish them from conventional potentials; all terms in the hamiltonian of the projectile plus target-nucleus system must be spherically symmetric*. These nonspherical functions must therefore be a local equivalent representation of a scalar term in the hamiltonian composed of the momentum operator - i h V and the gradient of the nuclear density (i.e., in the radial direction). It is the fact that the beam direction defines a preferred momentum direction which leads to the non-spherical effective potentials. We shall make very extensive use of these 0-potentials in discussions of polarization effects due to channel coupling. 2.t. M E T H O D O F C A L C U L A T I O N
We assume that the stationary state wavefunction has been generated by an optical model or coupled channel code; we write down the following form to establish the notation:
O(,) =IE u~(r) Y,o(~), r i
(6)
* The exception being for an odd nucleus of spin > ½, with a deformed intrinsic state, which can be aligned in an external magnetic field.
176
where
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
u~(r) has the asymptotic form: il+t
ut(r) ~ T k eW'(47r(21 + 1))'/2(It - &Ol)
(7)
in which/i and 05 are the incoming and outgoing Coulomb radial wavefunctions. It follows from eq. (1) and the expression
- Lr Or2
that the divergence of the probability current may be expressed as:
V "j(,) = ~ ~ ZL(r) YLo(~) ,
(9)
where r(21+ 1)(2/,+ 1)]
&(r)=E L 4 7 ( ~ )
5/2 trL 2 [ l'(l'+l) u,(r)ur(r) ] (Cooo) Im u*(r)u~'(r) r2
.1
(lO)
in which C represents a Clebsch-Gordan coefficient.
2.2. EVALUATION O F T H E C U R R E N T VECTOR
In order to express the current vector in a natural way, we shall depart from the usual implicit convention of using the z-axis for the incident direction, and, confining ourselves to the scattering plane, we shall use the x-axis in the horizontal direction for this purpose. Then the y-axis is vertical. Theta remains the angle from the incident direction. With these conventions, we can express the x and y components of the current j as follows: h
Yao(O,0),
(11)
(Ya(r)) Ya,(O, O)
(12)
jx = 7 - - Z Im (--wa(r)) A m ,x
h
Jy = Z Im 2m A
in which ~ and Y depend upon the first derivatives of the radial wavefunctions. These are incorporated in the following functions f and g:
r '+' 1"7\dr-r/ ~(r)=-L2-~-~.l g,(r>=
[ t ]1/2[ d
(13) r
'
I+1\ u,(r) ,
(14)
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
177
which occur in the following expressions for ~ and Y:
F ul(r)__fr(r) ut(r)gr(r) Hn,x] Ya (r) = ~ L r 2 Gu,x 4 r2
(15)
1
[ ut(r)__fv(r) Su,a -~ ut(r)gr(r) Tu,,x] ~a(r) =~ L r2 r2 "
(16)
The coefficients G,/4, S, and T are as follows:
F(21+1)(21'+3)ql/2r~,'+ll ,'r~tl'+,~r~,,'+la G,,,~=L ~ ~ j .-., -,o,--oo o.--o, , , h,,,,x
S/,'x
=F(21+1)(21'-1)]'/2c,,'-11
rcar-tAc,,r-,~
(17)
o. o,
(18)
= F(21+ 1)(21'+ 3)]ll/2rr+llr,p,r+lX~ 2 L 4--'~--~ _] ,--o oo,,--oo o, ,
(19)
u
j
.-,
-,o.
oo
[(2•+ 1)(21'- 1)1'/2,..,,,_,1,,,,,._,tr_,~,, 2 T,,,~=L 4 - - ~ j ,--o oo,,-.oo o , .
(20)
2.3. ALTERNATIVE APPROACHES The expressions given above are precise, computationally straightforward, and the multipole expansion coefficients such as Zt(r) may be stored for any form of graphical processing. Nevertheless, an alternative approach is simply to evaluate $(r) on a grid and evaluate the various gradients by straightforward numerical differentiation. This is facilitated for spinless particles by the fact that symmetry dictates that the derivatives normal to the scattering plane are zero. There is a particular advantage to us in the 'square-mesh' approach in that it requires minimal adaptation of existing software developed 6) for the graphical representation of biomagnetic fields. Because of the relatively coarse mesh, all quantities involving second derivatives, such as the ~b-potentials, were determined by calculating V25 at all points on the mesh by using eqs. (6) and (8). The algebraic approach will give immediate results in the absence of such software and we use it to calibrate the accuracy of the mesh approach. In this paper, we use the mesh approach to give graphical representations conveying the global properties, and the algebraic approach to give simple graphs showing the precise effects of channel coupling on various quantities evaluated along the zero impact parameter line. We emphasize that this division of labour is historical rather than essential, and both methods give exactly the same results.
3. Rotational coupling of alpha particles We have studied the wavefunctions corresponding to the inelastic scattering of alpha particles from 2°Ne treated as a good rotor. The 2 + and 4 + states are included
178
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
with realistic values of the deformation parameters fiE and 134. In some calculations the 4 + state was omitted without change of parameters. The interest, of course, lies in the characteristic channel coupling effects, so we compare our results throughout with " D W " , or "uncoupled" wavefunctions obtained by switching off the backward coupling in the coupled channel code 7) from which we extract the wavefunctions. A bonus coming from a calculation with "uncoupled" wavefunctions is that it enables us to verify that we do indeed recover the real and imaginary potentials by dividing the real and imaginary parts of ff*V2ff by 1012. In this way we have checked the program to high accuracy. Channel-coupling O-potentials. We first present various quantities calculated by the algebraic approach, plotted along a line of zero impact parameter, i.e. through the nuclear centre in the direction of the incident momentum. Generally, we compare the quantities as calculated in three ways: (a) when the coupling is switched off, (b) when coupling to the 2 + state is included, and (c) when coupling to both the 2 + and 4 + states is included. The potential used in the coupled channel, CC, calculations is fixed at that which is appropriate for the full coupling of case (c). This will sometimes be referred to as the "bare" poten,'!al in the discussion below. Fig. la shows II]/I2 for these cases with the no-coupling case a solid line, the 2 + case long dashes, and the three state case short dashes. The roughly exponential fall-off through the nucleus is followed by a focusing effect in which the focus actually lies outside the nucleus on the far side. The halfway point for the real potential is at about 3.75 fm. The focusing effect found by McCarthy at much lower energies generally led to a focus within the nucleus. Of more interest is the very large effect that the channel coupling has had on the wavefunction on the far side of the nucleus compared to the rather modest reduction for r < +3.5 fm. Later, we shall refer back to this figure in order to compare the results with 1012 from inverted and empirical elastic scattering potentials. It turns out that the variation in magnitude of V • j along the same central line is so dominated by the behaviour of I~/I2 that it is much more informative to divide out by the latter quantity. This just gives the imaginary 0-potential, and in fact most of our attention will fall upon both 0-potentials. Concerning the divergence itself, we simply mention that in the present case, unlike some to be discussed, V . j nowhere becomes positive so that there are no absolutely emissive regions generated by coupling in this particular case. Nevertheless, there are emissive regions in the polarization 0-potential, i.e. the difference between the extracted ~b-potential and the spherical bare potential, as will be demonstrated. As we have mentioned, the 0-potentials provide a good check on our programs since these must be the " b a r e " potentials in the no-coupling cases. In fig. lb, the solid line is the real potential calculated using eq. (5) for the no-coupling case. The bare potential is reproduced to three or four significant figures. The Coulomb dominated regions beyond 6 fm are obvious. The long dashed curve in this figure is the 0-potential for the case with one excited state and the short dashed curve
2,00 1,75
nepl rot02 rot024
~ .....
1.50 1.25 1.00 0.75 0,50 0,25 I
-5
I
0
5
10
0
5
l0
0 -10 -20 -30 V(re~) MeV
-40 -50 -60 -70 -80 ,
!
-5
i/
I
V(imag) MeV -15
-20
(c) -5
0 x/fro
5
10
Fig. 1. (a) Potentials for 104 MeV alpha particles scattering from 20 Ne, 1~12 for the elastic channel plotted along the zero impact parameter axis through the nucleus with the flux incident from the left and with x = 0 located at the nuclear centre. The solid line is for no inelastic coupling, the long dashes are for rotational model coupling to the 2 ÷ state, and the short dashes correspond to the further inclusion of the 4 + state in the calculation. (b) As for part (a), but showing the real potentials as calculated from the wavefunctions using the method described in the text. The repulsive Coulomb term dominates beyond 6 fln. (c) As for part (b), but showing the imaginary potentials. In this and subsequent figures, the quantities notated as V(real) and V(imag) are the V and W of eqs. (2)-(5).
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R.S. M a c k i n t o s h et al. / Stationary state currents in nuclear reactions
results w h e n the 4 + state is i n c l u d e d . T h e effect is n o t n e g l i g i b l e at the n u c l e a r surface, a n d r e p r e s e n t s a h i g h l y a s y m m e t r i c a l effect, the p o t e n t i a l n o l o n g e r b e i n g s p h e r i c a l l y s y m m e t r i c . I n fig. 2a we s h o w the differences b e t w e e n the d a s h e d curves a n d t h e s o l i d curve o f fig. l b . T h e s e curves r e p r e s e n t the p o l a r i z a t i o n effect d u e to the c h a n n e l c o u p l i n g , a n d e x h i b i t r e p u l s i o n o n the " i l l u m i n a t e d " side o f the n u c l e u s a n d in the far p a r t o f t h e s h a d o w side, a n d a t t r a c t i o n f u r t h e r in on the s h a d o w side. T h e c o n s e q u e n c e s o f this b e h a v i o u r will be e x a m i n e d below. T h e i m a g i n a r y p o l a r i z a t i o n p o t e n t i a l is also a s y m m e t r i c . Figs. l c a n d 2b are the c o u n t e r p a r t s for the i m a g i n a r y c o m p o n e n t s o f figs. l b a n d 2a. W h i l e the i m a g i n a r y p o t e n t i a l s in fig. l c n e v e r b e c o m e positive (emissive), the p o l a r i z a t i o n p o t e n t i a l s o f fig. 2b d o b e c o m e emissive in the tail b e y o n d + 5 fm. E v i d e n t l y , flux is b e i n g fed b a c k into the elastic
AV(real) MeV -1 -2 -3
rot02 m _ _ rot024 (a) I
t
-5
i
0
5
I0
0 -2
AV(imag) MeV
-4
A
t
-6 -8
(b) I
-5
i
i
0 x/fro
I
5
I
10
Fig. 2. (a) As for figure lb, but showing the polarization potentials calculated from the curves of fig. lb by subtracting from the potentials calculated with coupling included, the no-coupling potential. In this figure, the solid lines are the polarization potentials for 2+ state only, and the dashed lines are for both the 2+ and 4+ states. (b) The imaginary polarization potentials as in fig. lc calculated by subtraction as in fig. 2a for the real part.
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
181
channel here, a clear example of a coupled-channel non-locality. Flux generation as a non-local effect has been discussed by Austern 8). One feature of the 0-potentials is exactly as expected: the polarization potentials, real and imaginary, are peaked just in the region of the inelastic scattering form factors, something which is less obvious in the case of spherical S-matrix equivalent polarization potentials for a reason which we will attempt to make intelligible later. We have shown only the behaviour along the zero impact parameter line. In figs. 3-6 we present figures conveying the spatial behaviour of these quantities and also j(r) itself. The upper and lower parts of fig. 3 represent 1012 for the no-coupling and "2 + only" cases. Note that the plotting program has changed the scale by a factor of 1.7, thus de-emphasizing the reduction in the focusing effect brought about by the coupIing. Fig. 4 throws some light on this focusing: the current j ( r ) is directed inwards to become strong just along the axis on the unilluminated side of the nucleus. The left-hand side of fig. 5 shows, somewhat squashed by the plotting routines, the spherical real no-coupling potential. The non-spherical real 0-potential due to 2 + coupling can be compared with it in the right-hand side of fig. 5. Fig. 6 shows how nonspherical the imaginary 0-potential is in the coupled case. The scale in these figures can be deduced from fig. 1 which shows the behaviour of these quantities along the y = 0, zero impact parameter, line. Comparison with empirical optical potential. The optical potential appropriate for single-channel scattering is quite different from that appropriate to coupled channel calculations, and the difference between the potentials required in the two cases has for a long time been taken as a rough way of getting the polarization potential due to channel coupling 9,1o). In fig. 7a we compare [0[ 2 for a spherical optical potential fitted to forward-angle elastic scattering 11) and compare it to the same quantity calculated for the elasticchannel wavefunction from a coupled channel calculation including both the 2 + and 4 + states with a potential adjusted to fit the elastic and inelastic scattering as used for all the cases above. Although the magnitude on the far side of the nucleus is approximately correct, the phase of the evident interference feature is wrong. Indeed, comparing with fig. la, although the focussing effect has been correctly suppressed, [0[ 2 elsewhere is hardly closer to the CC result than the no-coupling results shown in fig. la. We conclude that much that is interesting about polarization potentials might be missed from the comparison of the two roughly fitted empirical potentials. In fig. 7b we present two imaginary potentials which ostensibly should be the same. The solid line is Rebel's empirical imaginary potential, and the dashed line is that shown in short dashes in fig. lc. This is the imaginary part of a potential which we know would reproduce in a single-channel calculation the elastic-channel wavefunctions from the 0 ÷ 2 ÷ 4 ÷ coupled channel calculation for which the parameters have also been adjusted to "fit" the (largely forward angle) data. Comparing with spherical local equivalent potentials. The polarization effects due to channel coupling can also be quantified by applying S-matrix to potential
Fig. 3. Upper: Histogram representation of I~[ 2 for the elastic channel on the scattering plane for the no-coupling case. The incident b e a m direction is from the u p p e r left. The absorption o f the beam is d e a r l y visible as is the focus; in spite o f the focus, the obvious effect off the beam axis is the creation o f a diffractive shadow on the "'dark" side. Lower: As for the u p p e r part, but for the case in which rotational coupling to the 2 + state is included. Note that the scale is renormalized by a factor o f 1.7, as can be seen from the unperturbed region on the large y edge. The features at the corners are artefacts of the plotting routine.
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
183
10
8-
6
m
~
4-
2•
~.
N.
/.
.p
y/fro 0
-2
-4
p
-6
-8
-10
....
I .... -8
I .... -6
I .... -4
I . . . . . . . . -2 0 x/fm
I .... 2
I .... 4
I .... 6
I .... 8
10
Fig. 4. T h e p r o b a b i l i t y f l u x j o n the scattering p l a n e f o r the n o - c o u p l i n g c a s e . T h e a r r o w size is g e n e r a l l y p r o p o r t i o n a l to IJl e x c e p t f o r p l o t t i n g r o u t i n e artefacts at the e x t r e m e l e f t - h a n d e d g e .
inversion techniques, see refs. 5,12,13), to the elastic channel S-matrix elements and subtracting the "bare" potential from the coupled-channel calculation. This gives a spherically symmetric, local and /-independent representation of the channelcoupling effects which can be compared with potentials found by phenomenological fits to data. However, this potential will not be expected to give the same wavefunction in the nuclear interior as the 0-potentials described above, although the wavefunction must be identical in the asymptotic region. In fig. 8 we exhibit 1012 for the coupled-channel calculation (solid line, identical to the short-dashed curve of fig. la). Indeed, this quantity is indistinguishable from that obtained by inversion (dashed curve) in the external region. What is very surprising is that the dashed curve falls below the solid curve in the nuclear interior. This is evidently an "anti-Percy effect". The wavefunction containing the non-local effects due to channel coupling is larger in the nuclear interior than that for the local potential giving the same asymptotic wavefunction, i.e. S-matrix elements [cf. refs. 14.15)where the Percy effect is discussed]. In the context of nucleon-nucleus interactions, Mahaux and
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R.S. M a c k i n t o s h et al. / S t a t i o n a r y state currents in nuclear reactions
0
0
7
-7
~ ~ -~ .~.
~.~ ~ ~.~ ~ .~
•-
.~
0
?
g~
~.~ I
I
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.
I
0
g
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
_=
0 e'~
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u~
_o e~
~*
¢-4
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I
I
o 0
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185
186
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
1.4
- -
-
-
dw rebel rot024
-
1.2 1.0 0.8 0.6 0.4 0.2 -5
0
5
10
-5
-10 V/MeV
_••I
-15
.I
-20
\I
(b) iW
-5
0
I
5
10 x/fm Fig. 7. (a) Comparing 1012 for the elastic-channel wavefunction for two different calculations which nominally fit the elastic-scattering cross section and hence the asymptotic wavefunction. The solid line is for the one-channel optical-model fit and the dashed line represents the coupled-channel calculation including 2+ and 4+ states with appropriately-refitted potential parameters. (b) The solid line is the imaginary potential corresponding to the solid line of part (a) and the dashed line is the projection on the x-axis of the imaginary part of the non-spherical 0-potential corresponding to the dashed line of part (a). Sartor have recently f o u n d that the radial w a v e f u n c t i o n s of b o u n d n e u t r o n s in Pb are i m p r o v e d by the u n j u s t i f i a b l e o m i s s i o n o f the Perey factor, see p. 269 o f ref. t6). The existence o f a n effect, p r e s u m a b l y d u e to the n o n - l o c a l effect of c o u p l i n g , which is the o p p o s i t e sense to the Perey effect, i m m e d i a t e l y suggests that such effects as we have f o u n d s h o u l d also be explored in the context of n u c l e o n - n u c l e u s interactions. It is very instructive to c o m p a r e the local p o t e n t i a l o b t a i n e d b y i n v e r s i o n with the n o n - s p h e r i c a l p o t e n t i a l which c o r r e s p o n d s to the e l a s t i c - c h a n n e l C C w a v e f u n c tion. First we study the i n v e r t e d potential. I n fig. 9, the solid curve is the real " b a r e " c o u p l e d - c h a n n e l p o t e n t i a l with the C o u l o m b part removed. The d a s h e d line is the
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
1.4I
187
rot/cc rot/inv.
------
1.2 1.0
P~l 2
0.8 0.6 0.4 0.2 -5
0 xffm
5
10
Fig. 8. Comparing [ol2 for the rotational coupling case (4+ state included, solid line), with that calculated using the spherical S-matrix equivalent potential which was found by S-matrix to potential inversion.
bare f cc/inv. Z -20 -40 V/MeV -60 -80 I
I
I
I
2
4
6
8
10
r/fm Fig. 9. Two potentials showing the polarization effect on the real potential due to rotational coupling, as represented by spherical potentials. The solid line is the bare potential in the coupled channel calculation, and the dashed line is the potential found by inversion from the S for the elastic channel given by the coupled-channel calculation. p o t e n t i a l o b t a i n e d b y a p p l y i n g S - m a t r i x to p o t e n t i a l i n v e r s i o n to the e l a s t i c - c h a n n e l S - m a t r i x e l e m e n t s f r o m t h e c o u p l e d - c h a n n e l c a l c u l a t i o n in w h i c h b o t h the 2 + a n d 4 + states o f 2°Ne were i n c l u d e d . F r o m fig. 10, w h e r e the difference b e t w e e n these curves is p l o t t e d , it is c l e a r t h a t t h e p o l a r i z a t i o n effect is g e n e r a l l y r e p u l s i v e e x c e p t n e a r 3.3 fm w h e r e it is attractive. I n fig. 11 we p r e s e n t the i m a g i n a r y c o u n t e r p a r t to t h o s e in fig. 9. W e shall n o w c o m p a r e these p o l a r i z a t i o n p o t e n t i a l s o b t a i n e d b y i n v e r s i o n with t h e ~0-potentials. I n fig. 12 the solid line is the b a r e i m a g i n a r y p o t e n t i a l , the l o n g
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R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
15.0 12.5 10.0 7.5 AV/MeV 5.0 2.5 0.0 -2.5 i
2
i
i
4
i
10 r/fm Fig. 10. The difference between the two curves in fig. 9. This is the polarization potential generated by channel coupling and is repulsive almost everywhere.
bare -5 -10
6
8
f
'._ ~~/llff/
V(imag) MeV -15
\,J
-20 I
i
i
I
2
4
6
8
10 r/fm Fig. 11. The imaginary potentials corresponding to the real potentials shown in fig. 9. The extra absorption peaked near the surface is expected; the decrease in absorption for smaller radii is less so. d a s h e d line is the p r e v i o u s l y given ~O-potential a n d the s h o r t - d a s h e d curve is the i n v e r t e d p o t e n t i a l s h o w n d a s h e d in fig. 11. T h e s h a r p central d i p visible in the latter figure has b e e n lost d u e to the c o a r s e r r e s o l u t i o n n e c e s s a r y for e v a l u a t i n g the qJ-potentials (which c a n n o t in a n y case s h o w a d i s c o n t i n u i t y at the n u c l e a r centre.) I n fig. 12, the s p h e r i c a l l y s y m m e t r i c i n v e r t e d p o t e n t i a l m a k e s the a s y m m e t r y o f t h e ~O-potential m o r e o b v i o u s . It is fairly clear h o w the s h a r p o s c i l l a t i o n s o f the i n v e r t e d p o t e n t i a l a r e n e c e s s a r y for the s p h e r i c a l p o t e n t i a l if it is to give the s a m e a s y m p t o t i c b e h a v i o u r as the n o n - s p h e r i c a l ~0-potential. This p o i n t is c l e a r e r f r o m the two p o l a r i z a t i o n p o t e n t i a l s c o m p a r e d as in fig. 13. T h e qJ-potential s h o w n in the s o l i d line a p p e a r s to force the d a s h e d line u p n e a r +5 fm, b u t b e i n g s p h e r i c a l this t a k e s
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
~no
189
[oupling 2+,4 + ~-pot. ~'
--
~ - 2 +, 4 + invert.fl
-5
II
It
-10 V(imag)
.//I
MeV -15 -20
-5
0 x/fm
5
10
Fig. 12. Imaginary potentials along the zero impact parameter axis. Solid line: "bare" potential (spherical); long dashes: 0-potential (non-spherical); short dashes: inverted potential (spherical). The two spherical potentials are as in fig. 11, except that the programs used to produce this figure could not handle the cusp very close to r = 0 in the inverted potential. 10 8
/
6
\
/ /
4 AV(real) MeV
f "\
2 0 -2
2+4 + 2 +, 4 + invert
-4 I
i
]
J
I
-5
0 5 10 x/fm Fig. 13. Real potentials along the zero impact parameter axis but with the 'bare' potential subtracted. The solid line is the non-spherical 0-potential and the dashed line is the spherical potential obtained by inverting the S-matrices. it t o o f a r n e a r --5 fm, so it m u s t d r o p s h a r p l y for Ir1<4.5 fro. This is also n e c e s s i t a t e d b y the n e g a t i v e e x c u r s i o n o f the 0 - p o t e n t i a l n e a r +3.5 fro. But since [0[ 2 is m u c h larger at the negative x, lit, side o f the n u c l e u s the negative p o t e n t i a l o p e r a t i v e at - 3 fm m u s t be c o m p e n s a t e d by the large r e p u l s i v e f e a t u r e at the n u c l e a r centre. O f course, such a r g u m e n t s c a n o n l y b e t a k e n as c o n c l u s i v e if t h e y a p p l y u n i f o r m l y to m a n y cases as t h e y are s t u d i e d . The r a t i o o f V • j as c a l c u l a t e d from 0 to t h a t f r o m i n v e r s i o n m i g h t b e closer to u n i t y t h a n the ratio o f the i m a g i n a r y p o t e n t i a l s o w i n g to the fact that ~/, is different
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R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
in the two cases. In fact the ratio is closer to unity for r = - 4 fm along the zero impact parameter line, but further from unity near r = +4 fm. S u m m a r y o f O-potential properties. Without doubt, the 0-potentials are the quantities related to probability current and its divergence which most dramatically reveal the effects of channel coupling upon the interaction between nuclei. The perturbations in the 0-potentials induced by coupling, which may be called the polarization 0-potentials, are largest in magnitude quite close to the radius where the inelasticcoupling form factors peak. This might seem inevitable, but we have seen that it is far less completely true for the local potential found by inversion. Other striking features of the 0-potential are: (i) The 0-potential is not spherically symmetric. (ii) The imaginary part of the 0-potential exhibits a small region on the "far" side of the nucleus where it is actually less than the bare potential. The first of these features is most naturally thought of as arising from an effective k. r dependence in the polarization potential. The second feature, which is a particular aspect of the first, might most naturally be ascribed to the underlying non-locality of the coupling effect. This is because the difference between the 0-potential and the bare potential in this region represents a generative or emissive polarization potential. Flux is being restored to the beam by the coupling here, a characteristic non-local effect 8). We give a more extended discussion of this in the next section. The difference between the 0-real potential and the bare real potential is comparable to the imaginary polarization potential, but the magnitude of this difference is much smaller than the bare real potential itself. In fig. 2a, therefore, we only plotted the difference, i.e. the real polarization potential. Like the imaginary polarization 0-potential, this is peaked at the maxima of the inelastic scattering form factors, but the degree of k. r dependence is very marked. In fact, at the front of the nucleus it is basically repulsive, and at the far side of the nucleus, basically attractive. This correlates with the polarization potential obtained by S-matrix inversion,* where there is surface repulsion (the near-side repulsion in fig. 2a extends to a greater radius than the attrractive part on the far side; there is actually repulsion for r > 4.75 fm on the far side), and attraction further in. The interior oscillations on the inverted, spherical,/-independent potential are the result of attempting to express this k. r dependence.
4. Interpreting non-spherical ~-potentials Symmetry dictates that a local interaction potential between a projectile and a spherical nucleus must be spherical (also for a spin-zero deformed nucleus in the * As described above, but for further discussion of what can be learned from this reaction using inversion, see ref. 17).
R.S. Mackintosh et al. / Stationary state currents in nuclear reactions
191
laboratory, non-intrinsic, frame). Nevertheless, it is quite obvious that there is a way in which the interaction potential can depend upon angle if the incoming momentum vector is allowed to define a direction. This is because there is a very natural way in which the interaction can depend upon the local momentum vector k. It is quite plausible that, for example, the local g-matrix could depend upon whether the nuclear nucleon interacted with would recoil into a nuclear region of positive or negative density gradient. This could plausibly lead to a (k. r) 2 term in the effective interaction, which would imply a term involving (k x r) 2, i.e. the angular momentum squared. This was one motivation for studying possible phenomenological evidence 18-20) for such /-dependence; the other was the emerging importance of coupled inelastic and reaction channels, which have a manifestly angularmomentum dependent effect 21). It is /-dependence of this kind which should be amenable to study through the evaluation of the various quantities described above, the 0-potentials in particular. It should therefore be no surprise that the 0-potentials presented in the last section are indeed non-spherical. We do not yet have a "dictionary" which will enable us to read the physics behind the details of the extracted potentials. However, it does appear that the calculated divergence and 0-potentials exhibit features that are most naturally interpreted as the signature of the underlying non-locality of the polarization potential. The difference between the extracted 0-potential and the bare potential of the coupled-channel calculation is a local representation of the couplinggenerated polarization potential. The imaginary part of this potential consistently exhibits positive regions, i.e. regions where the polarization potential is effectively generative, or emissive. Regions of emissivity are a likely characteristic of non-local potentials. It is, of course, the case that if the angle-dependent 0-potentials are inserted into a one-channel Schroedinger equation, that they would exactly reproduce the elastic channel component of the coupled channel wave function. They may be compared to the trivially-equivalent local potentials introduced for radial wavefunctions by Austern, ref. s). This property suggests that they may have an application in DWBA calculations where it is desirable to have elastic-channel wavefunctions which embody all the exchange and channel-coupling effects. 4.1. I M P L I C A T I O N S F O R S P H E R I C A L INVERSIONS
Our previous studies of the polarization potentials induced by channel coupling have exploited the new S-matrix to potential inversion technique which has been described in refs. 4.5,13). Such inversions were compared with the 0-potentials in sect. 3 above. The highly non-spherical nature of these 0-potentials, which we have associated with underlying /-dependence, constitutes a probable explanation of a phenomenon which invariably appears when polarization potentials are being studied by inversion. It is that the inversions achieved when the S-matrix elements
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R.S. M a c k i n t o s h et al. / Stationary state currents in nuclear reactions
are related to channel coupling are n e v e r as good as those found with model calculations. The goodness of any inversion is quantified 4.5,13) by means of a "phase shift distance", tr. When model inversions are performed in which S-matrix elements corresponding to known potentials are processed through the inversion procedure, the potentials are regained with precision. The corresponding values of ~r are always from one to several orders of magnitude lower than those obtained when S-matrix elements from coupled-channel calculations or k n o w n / - d e p e n d e n t potential are so processed. Almost any potential composed of Woods-Saxon-like components can be found within the space spanned by our "inversion basis". On the other hand, the polarization potentials cannot be perfectly described within the same space. These difficulties, relating to the determination by inversion of spherical polarization potentials, are probably associated with the underlying l-dependency/non-sphericity of the polarization contribution. They appear to indicate that the spherical equivalent potentials of non-spherical potentials may indeed have components which are too rapidly varying to be completely within the space spanned by the normal choice of basis function set. We note that within the context of electron scattering, there are upper limits on the effective wave number of Fourier components in the nuclear charge density. These are imposed by limits on the physical momenta present. Such limits might be expected to apply to local nuclear potentials, except that it is far from clear how angular features of "reasonable" wavelength translate into radial features of a spherical potential. The difficulty of achieving satisfactory spherical potentials becomes acute in certain cases of heavy-ion scattering near the fusion barrier. This matter will be discussed in a future paper on the subject of the rotation-induced polarization potential in heavy-ion scatttering near the Coulomb barrier. 5. Summary and outline of future work
The purpose of this paper has been to establish that the study of the wavefunction, and such characteristics as the @-potentials we have defined, constitutes both a readily feasible and a potentially fruitful approach to learning more of what goes on in direct nuclear reactions, and in particular the nature of the polarization potentials generated by channel coupling. Because I 12 can vary very rapidly through the nucleus, it turns out that the behaviour of V • j becomes more transparent if it is divided by I 12 This leads to the imaginary ~b-potential. We also introduce the analogous real @-potential. The overall complex @-potential appears to be a useful means of expressing, in local /-independent form, the effects of channel coupling, non-locality, or /-dependence. In sect. 3 we summarized the properties of the ~b-potentials arising from a particular kind of channel coupling, that due to rotational states. The particular coupling we have studied here does not begin to exhaust the contribution of channel coupling to elastic scattering. For rotational coupling, we
R.S. Mackintosh et aL / Stationary state currents in nuclear reactions
193
see that the coupling effects when represented as a local potential depart markedly from sphericity. It cannot be assumed that the cumulative effect of an expanded set of coupled channels would be a restoration of spherical symmetry. Until such a restoration of symmetry is shown explicitly to be the case, we must conclude that one of the main results of this work is a clear demonstration of the breakdown of the local-density approximation which may be said to omit many essential features of direct reaction effects. According to the local-density approximation, the optical potential must be spherical, even for the one-channel potential for a 0 ÷ "deformed" nucleus. Because of the strength of channel coupling, we conclude that the problem of establishing the optical model or internuclear potential cannot be isolated from the complexities of direct reaction processes. One line of exploration suggested by this work, is the further study of what we have called an "anti-Perey effect", in the context of nucleon-nucleus interactions. If it occurs for nucleons, it could be of relevance to resolving an anomaly discovered by Mahaux and Sartor relating to the normalization of the tails of bound neutron wavefunctions, see p. 269 of ref. 16). More generally, the procedures described here show promise of throwing light on channel coupling non-locality as distinct from the exchange non-locality. This question seems recently to have been largely ignored apart from the work of Rawitscher, see for example ref. 22). We mention the small generative region near x = 6 fm in fig. 2 as a signature for such non-local effects, and it would be interesting to attempt a more direct link with Rawitscher's calculations. There is a clear scope for a much more extensive application of the methods introduced here. In particular, it appears that the coupling of reaction channels has a more powerful effect than inelastic channels, and it is plausible that this might be connected with the strikingly large generative effects which have been indicated by phenomenology 23). These reactions and the general questions they raise as to where and why the polarization potential is attractive and repulsive hold promise as a fruitful area for study. We are grateful to SERC for grant number N G 17179 in support of Dr. Cooper.
References 1) 2) 3) 4) 5) 6)
I.E. McCarthy, Nucl. Phys. 10 (1959) 583; Nucl. Phys. 11 (1959) 574 K.A. Amos, Nucl. Phys. 77 (1966) 225 R.S. Mackintosh, Nucl. Phys. A230 (1974) 195 A.A. Ioannides and R.S. Mackintosh, Phys. Lett. B161 (1985) 43 A.A. Ioannides and R.S. Mackintosh, Nucl. Phys. A467 (1987) 482 A.A. Ioannides, in Inverse problems: an interdisciplinary study, ed. P.C. Sabatier (Academic Press, London, 1987) 7) R.S. Mackintosh, Nucl. Phys. A210 (1973) 245 8) N. Austern, Phys. Rev. 137 (1965) B752 9) N.K. Glendenning, D.L. Hendde and O.N. Jarvis, Phys. Lett. B26 (1968) 131
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