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Corrections to the Alder-Winther theory of Coulomb excitation
Nuclear Physics ONorth-I-Iolland
A448 (1986) 333-364 Publishing Company
CORRECTIONS TO THE ALDER-WINTHER THEORY OF COULOMB EXCITATION F.D. DOS AIDOS*, Department
C.V. SUKUMAR
and D.M. BRINK
of Theoretic& Physics, 1 Keble Rd., Oxford OXI 3NP, UK Received 31 August 1984 (Revised 1 July 1985)
Abstract: An expansion scheme is developed for studying corrections to the Alder-Winther theory of Coulomb excitation. The zeroth-order term in the expansion of the excitation cross section is identical to the expression provided by the Alder-Winther theory and there are several kinds of first-order corrections. All of these can be calculated by making simple changes in existing computer programs for the Alder-Winther theory. These corrections can be interpreted as a change in the deflection function for the relative motion and a change in the excitation probabilities due to the deviations of the relative motion from a Rutherford orbit. Some of the terms that describe the change in the excitation probabilities correspond to an energy symmetrisation and others can be interpreted as an angular momentum symmetrisation. Numerical comparisons with results of full quanta1 coupled-channels calculations are presented.
1. Introduction Coulomb excitation is an important tool for studying electromagnetic properties of nuclear levels. When heavy ions are used as projectiles multiple transitions occur and it is possible to excite rotational bands up to high spins. A coupled-channels calculation for such a multistep process can be made only for not too heavy ions. Rose1 et al. I) have developed a coupled-channels code AROSA for this purpose, but its use is restricted to cases where the Sommerfeld parameter n is not too large (n G 30) and where only a few states in a rotational band are excited (J 6 10). A full coupled-channels calculation by conventional methods is not feasible for very heavy ions because of the large number of channels involved. This situation may change with the development of new more powerful methods [Tolsma’), Rhoades-Brown et al. 3)]. The semi-classical method for calculating multiple Coulomb excitation developed by Alder and Winther4) assumes that the relative motion of target and projectile is described by a Rutherford orbit. This defines a time-dependent potential which causes the excitation. This procedure has been implemented by de Boer and Winther in the computer code COULEX 5). The time-dependent Schrodinger equation for the *Permanent
address:
Departamento
de Fisica, Universidade 333
de Coimbra,
3000 Coimbra,
Portugal
F. D. dos Aidos et al. / Corrections
334
internal
excitation
is solved numerically,
and yields an excitation
probability
P,,( 13,)
for exciting a final internal state 1-1from an initial state v when the relative motion follows a Rutherford orbit with scattering angle 0,. The differential cross section is given by
where dun/d&! is the Rutherford cross section. It is found that the results are improved by the use of “energy-symmetrised” orbits 5). This energy symmetrisation consists of three steps: (a) substituting in expression (1) doJdO(B,) by
where u, and u,, are the relative velocities for the entrance and exit channels; (b) substituting in the evaluation of P,,(8,) the energy of the relative motion in all matrix elements of the coupling potential (xlVlS) by an average of the energies of the relative motion for the channels x and {; (c) including a factor u,/uV in the expression for the excitation cross section. With this procedure the excitation cross section will be given by
In ref. 5, the energy symmetrisation is defined in a slightly different way because of a change of variable in the time-dependent Schrodinger equation. Either symmetrisation method provides similar results for the cross section. It has been difficult to assess the accuracy of the Alder-Winther method because no systematic procedure for improving it and for calculating corrections has been available. De Boer et al. 6, have tried to improve the method by taking into account the influence
of the excitation
process
on the relative
motion
through
an angular
momentum symmetrisation of the orbit of relative motion. A refinement of this method was developed by de Boer and Dannhauser’). They assumed that the relative motion changes from one Rutherford orbit to another of different energy and angular momentum several times during the scattering. Both of these methods provide similar results but do not give a significant improvement on the results obtained by COULEX. It is likely that they include some but not all of the relevant corrections. Sukumar and Brink8) have developed an expansion for calculating inelastic scattering cross sections based on the path integral method. This method will be applied to the multiple Coulomb excitation problem in the present paper. In order to understand the nature of the expansion we consider the excitation of rotational states in a deformed target nucleus with charge Z,e and intrinsic quadrupole moment Q by a structureless projectile with charge Z,e. The reduced mass is m and
the incident characteristic
relative velocity is u. The Coulomb collision time rc are given by Z, Z,e 2
UC=-
mu2
length parameter
2aC rC = -
u
’
uc and a
’
and the ma~mum strengths of the central (monopole) and the quadrupole parts of the Coulomb interaction are proportional to Z,Z,e*
ho=77
C
There are several dimensionless parameters which characterise the scattering [Guidry et al. 9)]. One is the Sommerfeld parameter
Another is an interaction strength
Qzle*
4 _ rcF2
4Aua:
’
which gives an indication of the number of quanta excited. For backward scattering from a rotor in the sudden appro~mation q = $A,,, where L,, is the ma~mum rotational angular momentum in units of 87which can be excited classically. A third dimensionless constant is the adiabaticity parameter r,AE t=
21i
AE =nz,
where AE is the excitation energy of the first excited state. A Coulomb excitation cross section is a function of all of these parameters. The method of Sukumar and Brinks) is an asymptotic expansion in powers of l/n for fixed q and E where the leading term corresponds to the Alder-Winther theory. In this paper we calculate the l/n corrections to this theory. There are several kinds of corrections of order l/n. Quantum corrections to the relative motion do not involve the quadrupole interaction. They contribute a phase *) which does not affect the cross section and we do not consider them here. A second kind of correction gives a m~ifi~ation of the scattering angle due to the effect of the quadrupole potential. The relative magnitude of this effect is proportional to 9, &!$L;. EO
Another modification of the scattering angle is associated with the energy transfer This effect is proportional to
AE.
5 -_AE 2v,, - n *
F. D. dos A ides et al. / Corrections
336
These modifications of the trajectory of the relative projectile will in turn change the excitation amplitudes.
motion of the target and A measure of this effect is
given by q!+ EO
n'
AE q2V,,
_!6 n ’
All of these effects are of order l/n for fixed q and E and are included in the correction terms to the Alder-Winther theory which are derived in this paper. When developing semi-classical theories of Coulomb excitation it is important to recognise that there are several dimensionless parameters in the theory. This means that a number of different expansion schemes is possible. One alternative scheme is the classical limit S-matrix (CLSM). Guidry et al. 9, point out that this is an expansion in powers of l/n for exact classical dynamics. In other words it is an expansion in powers of l/n for fixed A and is therefore different from the expansion used here. The present paper is organised as follows. The results from ref. *) are collected in sect. 2. In sect. 3 the formulae for the correction terms are written in a coordinate system which is convenient for studying the scattering problem and the expressions are simplified. Expressions for the corrections to the cross section are given in sect. 4 and an interpretation for these corrections is discussed in sect. 5. Numerical results are presented in sect. 6. An alternative method for calculating the first-order corrections in the expansion method of ref. ‘) has been developed by dos Aidos and Brink lo). It is simpler than the present method but it is restricted to the sudden limit where all the excitation energies
are set equal to zero.
2. Statement of the problem The theory
developed
in ref. *) applies H=p2/2m+ +,5)=H&)+V(r,5),
to a scattering V,(r)+h(r,E),
problem
with hamiltonian (4) (5)
where r is the relative coordinate of the target and projectile, p is the momentum conjugate to r, and the 5 are internal coordinates. The first two terms on the right-hand side of (4) describe the relative motion, H,( 5) is the internal hamiltonian of the target, and V(r, 6) is the coupling which produces excitation. In the present paper we are concerned with an application of the theory developed in ref.8) to Coulomb excitation. We assume that the incident energy is well below the top of the Coulomb barrier so that the interactions V, and V have no nuclear contribution. We take V,(r) to be the point Coulomb potential
331
F. D. dos Aidos et al. / Corrections
and
V(r, 5) the non-central
scribes rotational
part
of the electromagnetic
states of a deformed 4,(E)
interaction.
target and V is a quadrupole =
If H,(t) interaction,
J*/W
dethen (7)
where Q is the target quadrupole moment and 0 is the angle between r and the symmetry axis of the target. The results obtained in sects. 2-5 of this paper do not depend on the particular forms (7) and (8) for HO and V. The formalism is also independent of the form of V,(r) as long as it is a central potential with a monotonic deflection function. In sects. 2-5 we shall always give the results for this general form of V,(r) unless otherwise stated. Ref. *) gives an approximate formula for the propagator from an initial relative momentum pO and internal state v of the target at t, to a final relative position r, and internal state p at t,, &,(r,L
T,,[r,(t)l+aqy}.
Pota) = K,(rltl,NO){
In eq. (9) K, is the semi-classical limit of the propagator motion in the potential V,(r). The matrix element Tpy is
describing
(9 the relative
T,“MGl= hwo(h~hJlv) where the unitary
operator
U, is a solution
of the equation
iAt$=h(r,(t), with initial
condition
path r,(t), which final position r,. The relation the propagator
()U,
U,( t, to) = 1 when t = t,. The operator
is a classical
trajectory
(10)
in V,(r)
01) U, is a functional
with initial
momentum
between the propagator and the cross section is discussed is given by eq. (9) the corresponding cross section is
of the pO and
in ref. ‘l). If
in which U, and uP are the relative velocities for the entrance and exit channels. The factor UJU, occurs in (12) because the inelastic cross section is a ratio of fluxes in the exit and entrance channels. duJdfi(8,) is the classical cross section for elastic scattering by the potential V,(r). If V,(r) is the Coulomb potential (eq. (6)) then duJdG(8,) is the Rutherford cross section and the lowest-order term for the excitation cross section in eq. (12) is equal to the result obtained in the Alder-Winther theory with unsymmetrised orbits (eq. (1)).
F. D. dos A idos et al. / Corrections
338
The correction &$ ref. 8, and is
to the transition amplitude is given by eqs. (22) and (38) of
6qy = TIP,,+ T;ly,
(13) (14)
115) where
bw= f-&k f&4 >
(df)l= w%(b d.
The aim of this paper is to calculate the first-order corrections Tiand T’ to T.
3. Simplification of the transition amplitude In this section we simplify expressions (14) and (15) for the correction to the transition amplitude. These depend on the orbit p,(t) and in (14) and (15) this orbit is specified by its initial momentum p. and its final position rl. As the independent variables are p,, and r, it is a simplification to write
where we have used the convention of summing over repeated indices. Then p. and r1 are treated consistently as independent variables in the partial derivatives. Each classical orbit rc( t) in V,(r) is specified by six parameters. In (14) and (15) these are the components of p. and rt, but other choices are possible and also more convenient. We denote a convenient set of parameters by C,, (Y= 1,. . . ,6, and choose them to be E,, r, 0,, (Y,/3, y, where E, is the energy of the orbit, r is the time at which the projectile reaches the point of closest approach, 0, is related to the scattering angle 8, by 6, = $(r - 8,) and (Y,/3, y are three angles defining the orientation of the orbit. We evaluate (14) and (15) for a standard orbit defined in fig. 1. The standard orbit r,(t) has r = 0 and a = /3 = y = 0. An orbit with orientation (Y,/3, y is obtained from the standard orbit by making a rotation through (Yin the
339
F.D. dos A ides et al. / Corrections
Fig. 1. Sketch
range (0,27r)
of the standard orbit: x is antiparallel to L, z is the symmetry axis of the orbit, y-component of the velocity is always greater than or equal to zero.
about
finally a rotation It is convenient
the x-axis, then a rotation
p in (-
ia, $7) about
and the
the y-axis and
through y in (0,2a) about the z-axis. to introduce the following notation for the derivatives
appearing
in eqs. (14) and (15):
Then
D*(f)
and
using the summation
D( f, g) can be expressed
convention
in terms of D*( C,) and
for the indices
(II and /L
D(C,, CP) by
F. D. dos A ides et al. / Corrections
340
With this notation
eqs. (14) and (15) can be written
as
(20)
The functions D2( C,) and D(C,, Cp) depend on the geometry of the orbits. They are calculated for a standard orbit in appendix A for a general central potential V,(r) and in the case where V,(r) is the point Coulomb potential (6). Only two of the functions D2(C,) are non-zero. For a Coulomb potential they reduce to D2(~)=
cot e,
1
+,
D’@o)=
0
2~
=
2t?n
(23)
>
where L = hn tan 0, is the angular momentum of the orbit. The results for D( C,, Cs) are given in table 1. Note that only the symmetric part of D(C,, Cs) contributes to T, in (20) while only the antisymmetric part contributes to (21). Hence the sums over (Yand p in (20) and (21) each contains only a small number of terms. The evolution operator Uo[tl, to, r,(t)] is a functional of the orbit rc( t). This orbit is specified by the six parameters C, which we take to be E,, 0,, 7, (Y,p, y. Only two, E, and So, of these six parameters affect the shape of the orbit. The dependence of U, on the other parameters can be found from symmetry arguments. The standard orbit r,(t) in fig. 1 has its time origin at the time of closest approach. A general orbit r,(t) can be obtained from rS(t) by a rotation R and a time translation T, r,(t) Here R is the rotation
specified
= RTr,(t)
.
by the angles (Y, /3, y, and T is defined
by
Tr,( t) = r,( t - T).
TABLE 1
D( C,, Cfl) for a general potential (appendix A)
0
0
0
0
0
0
0 0 0 0
A -B B 0 0
B C
B C
0 0 0 D -E
0 0 0 E F
-1
-C
-C 0 0
0 0
ED. dos Aides et al. / Corrections
341
In this equation ,r = t, - t,,
t, =
r’dr/L Jml
(24)
t, is the time taken to move from r,,, to ri ml is the distance of closest approach, along the orbit r,(t), and r is the time corresponding to the point of closest approach r,. The evolution operator Uo[tl, t,, r,(t)] is obtained from the hamiltonian h(r,( t), E). Clearly h will remain unchanged if we apply an arbitrary rotation to both the coordinates of relative motion r,(t) and the coordinates describing the orientation of the target. Thus rotating the trajectory by R is equivalent to rotating the intrinsic state of the target by R-‘. A similar argument can be used for T to give U&,0&)]
= U,[t,t’,RTr,(t)] =RU,,[t-qt’-~,r,(t)]R-‘.
The operator
(25)
R( a, /I, y) is given by R = exp( - iyJ,/Fz)exp(
- i&/A)exp(
- iaJ,/A)
,
(26)
where J,, Jy, J, are the angular momentum operators of the intrinsic state. Eqs. (25) and (26) lead to a number of simple results. In eq. (10) the influence the potential V is negligible at to and t,. Provided Q-is not too large
T,,]r,(t)]
Uo[to,t,--,r,(t)]
=exp(-i&r/A),
Uo[t,-T,ti,rs(t)]
=exp(i%/tZ),
=exp{i(~~--~)~/~}(~lRUg[t~,t~,rs(t)]R~’I~).
The T-dependence is isolated in the exponential to r can be calculated. The first derivative is
aL - ar
iAE
AT,,,
factor and derivatives
of
(27) with respect
(28)
where AE = E, - Ed. The excitation energy is - AE and the change in the energy of relative motion is AE. Derivatives of U, along the standard orbit (7 = 0, (Y= /I = y = 0) can be obtained from (25) and (26)
Derivatives
~U&,t~)=
~{h(t)L/,[t,t’,r,(t)l-L:[t,t’,r,(t)]h(t’)}, (29)
&v,(W)=
;[v,[t.t’,r,(t)],
Jx].
of V can be found in a similar
(30)
way, (31)
There
are expressions
for derivatives
with respect
to p and y in terms of commuta-
342
F. D. dos Aides et al. / Corrections
tom with J,, and J, which are similar to (30) and (31). When t and P are outside the interaction region h(t) = h(f) = Ho and (29) is equivalent to (28). These results can be used to simplify eqs. (20) and (21) for TI and T’. The results hold for any choice of the central potential Y&r). We write (20) as a sum of five terms, Tl,,=A,+A,+A3+A4+A5,
(32) (33)
(34) (35)
(36) A,=
-48
I
+D(LY)-$ Tpv.
(37)
The term A, can be evaluated by using (28) and noting that 11(7, E&I = - 1, (38) By using (28) and (A.22) A, can be written as (39) for the case where V,(r) is a general central potential and as AZ= - k5.Y 0
if it is a Coulomb potential. By using (A.23) and (A.28) A, can be written as A,=-aifi
(40)
Eq. (36) can be simplified by using (28) and (A-26) and A, becomes
(41) A 4 only gives corrections to the phase of the propagator and does not affect the cross
343
F. D. dos A idos et al. f Cwrections
section. Hence we do not need to evaluate it. The first term of A, is divergent as r1 increases as discussed in ref. *). Terms of this type should be summed to all orders in l/n but only modify the overall phase. The remaining term A, can be written in terms of commutators by using the equations for the derivatives of U, with respect to (Y,j3, y (eq. (30), etc.), A5=&(
-~~(~,~)(~I~U~,J,II~)A~+~(~,~)(ILI[[L~O,J~I,J~~~~) Iv) 1-~(Y~Ykmv_?1~41
+NPJ%PI[[I/,YJ,]?~]
lb)) * (42)
In the next section we show that A, does not cont~bute to the cross section although it does affect the polarisation. To simplify eq. (21) we use the formula
&Uo(t,t’)=-$dt”U&,t”)~ which can be derived by perturbation (22) as
av( P) dc
vow,
t’>
(43)
theory. This shows us to rewrite UC,, in eq.
(45) Using these formulae and eqs. (29) and (30) for the derivatives aU,/aC, write T’ as T,: = t, -t t,,
we can (46)
in which t, =
(PI v,lv) 3
tb=(~IUbtv),
u, = -;~t’dtUo(t~,t)h’(t)Uo(t,t,), 20
(47) (48) (4%
We have used the notation (A, B} = AB + BA for the anticommutator of the operators A and B. In (51) U, = Uo{tl, to). To illustrate this reduction we consider
F. D. dos Aidos et al. / Corrections
344
one matrix
element
in (21) and simplify
it by using (45) and (29)
(52) The second term in (52) contributes to the first term in eq. (50) for h’(t). The first term in (52) can be simplified by noting that h(t,) = Ho because t, is outside the interaction region. Using (43) with C = E, it reduces to -iAEJpI
au,Iv)
a~,
JT = -ihe,*
(53)
and contributes to the first term in (51) for U,. The other matrix elements in (21) can be calculated in a similar way. The contribution of t, to Tp’,,corresponds to the first-order perturbation of the transition amplitude by a perturbing potential h’(t). t, can be expressed in terms of the matrix elements T,,[r,( t)] or their derivatives with respect to E, and (3,. 4. Calculation of the cross section The results obtained in sect. 3 can be used to find expressions for the first-order corrections to the excitation cross section. For the case of rotational excitation the appropriate states are (PI = with energies the excitation summed
(JW 9
I’) = IJiM,)
E and .si, respectively. The differential cross section corresponding to of the target from level Ji to level J is given by (cf. eq. (12) where we
over all initial
and final M-values) (54)
where
u. and u1 are the relative velocities POJ(EO? 80) = c
in the entrance
and exit channels,
ItJM(~o? 00) I23
(55)
MM,
and is the transition amplitude evaluated along the standard orbit with energy E, and scattering angle 0, = s - 26,. In (54) duJd0 is the classical cross section for scattering by V,(r). If V,(r) is the monopole Coulomb potential then duJdS2 is the Rutherford cross section. AP, is given by AP,=
c 2Re[t,*,(A,+A,+A,+A,+A,+t,+t,)], MM,
where the A- and t-terms have been defined
in the previous
(57) section.
F. D. dos A ideset al. / Corrections
The contribution
of A, to the cross section
345
can be found
from eqs. (38) (54) and
(57) and we obtain P
-=Al POf From
If V,(r)
1 -aP,,
MM,
POJ
Coulomb
potential
This term is equal to the difference Rutherford cross sections
2
(58)
’
of A,
(39) we find for the contribution
is the monopole
AE
aE,
=G
then
PA2 -=--. POJ
AE
between
the symmetrised
(60)
EO
and the unsymmetrised
(61) (cf. eq. (2)). Usually the Coulomb excitation excitation probabilities PJ defined by
43
If V,(r)
is the monopole
the term A,
in eq. (40) to the cross section is
--L!Im
P,, - POJ2
(62)
of PJ using this formalism,
In the calculation
of A, defined
The contribution
values for the
J
da=U,7 as well as cross sections. should be left out.
codes provide
da~ym’,
u1
dUJ
computer
C MM
Coulomb
-!J-$+t&
tTM .[
potential,
P
0
i
(63)
then
a2 c0s2eo-+ as;
-=A3 POJ
1I tJM .
0
i1 t,,
cOteo$ 0
.
(64)
The derivatives of tJ,,,, in this term can be evaluated by numerical differentiation. A, is given by eq. (41). It will contribute to the phase of the transition amplitude but will not affect the cross sections. The contribution of A, to the cross section contains terms of the form zM { (JMIJk”uOJ,“IJiMi)(JiMil
QtlJM)
- (JMI
u~IJiM~)(J~M~IJ,“u~I;I,“IJM)}
3 (65)
where k stands
for x, y, .z and n, m can take the values 0,1,2. It is obvious
that the
F. D. dos A idos et al. / Corrections
346
commutator
Jk,
C
IJA~)(J~~I
M
I
= 0.
(66)
Therefore the contribution from A 5 vanishes. The argument over A4 and Mi, hence A, can affect polarisations.
depends
upon summing
The contribution from t, defined in eq. (48) consists of a sum of four terms. Using eqs. (A.25), (A.27) (A.28) and (A.30) they can be written as (67)
(68) C (JMI(au,/aeo).oJ,+J,(av,/aeo).oIJiMi)t~M
P P OJ
-= b3
MM,
-
>
P OJ
(69)
C (JMI[UO,Jy]Jz+Jz[Uo, Jy]lJiM,>t?~
P P OJ
-=b4
_ 2-1,
MMl
(70)
P OJ
The first term can be added to PAI defined ‘,I
+
in eq. (58) to yield
‘Al =- 1
POJ
aPOJ POJaE,&i.
(71)
If the energy &i of the initial state IJiMi) is zero the contributions from A, and tbl to the cross section cancel. The contribution from the three remaining terms of t, (expressions (68) to (70)) can be calculated by using the expressions (A.27) (A.28) and (A.30) as well as the commutator (66). For the case where the target is initially in a level with zero spin and zero excitation energy ( Ji = 0, q = 0) and V,(r) is the monopole Coulomb potential we obtain
+_d”)tJ,-,)t,*,~
P b4 P OJ
-=
---
I
P oJ
cot
4n
(73)
8, ;(j+(“)fJA4+l
+d”)tJM-,h,
(74)
341
F. D. dos Aides et al. / Corrections
where j+(M)=JJ(J+l)-M(M+l))
(75)
j_(M)=JJ(J+l)-M(M-1).
(76)
The derivatives in (72) and (73) can again be obtained by numerical differentiation. The term t, in eq. (47) corresponds to a first-order correction to the transition amplitude by a perturbing potential h’(t) given in eq. (50). It can be evaluated by modifying the interaction term in a Coulomb excitation code like COULEX, I’+,(t),
t) + V(‘&),
6) + ‘CA’(t).
Here K is a parameter and we need the first-order correction in K. This can be obtained by running the modified code for several values of K and evaluating the derivative of the amplitude at K = 0 numerically. The matrix elements of h’(t) required for the modification can be calculated from (50). Matrix elements of h(t) already exist in the code and matrix elements of J,, J, and J, are standard. The new matrix elements are those of hV’(t)/aE, and W(t)/M, on a standard Rutherford orbit. They are easy to calculate from formulae for V(t) given in ref. 4).
5. Interpretation of the correction terms In this expression theory of approach
section we (1) for the Coulomb the relative
discuss the interpretation of the correction terms (57) to the excitation cross section of a level J given by the Alder-Winther excitation with unsymmetrised orbits. In the Alder-Winther motion is described classically while the excitation process is
treated quantum-mechanically. Furthermore, this method assumes that the relative motion follows a Rutherford orbit, unperturbed by the coupling with the intrinsic degrees of freedom. The correction terms that were obtained in this paper describe the deviations
of the relative
motion
from a Rutherford
orbit due to the excitation
process and the influence of those deviations on the intrinsic motion. If we wish to follow the same philosophy as in the Alder-Winther theory, we should try to describe the correction terms as a modification of the orbit of relative motion yielding an excitation cross section
(77) where da,/dG is the classical cross section corresponding to the modified orbit rl(t) and P,[r,(t)] is given by expressions (55) and (56) where now r,(t) must be replaced by the modified orbit rl(t). An attempt to build a theory as outlined in eq. (77) was made by Pechukas12). The method that has emerged has the disadvantage of not preserving the superposition principle and is different from the method that we use in this paper. A comparison between the two methods can be found in ref. 13).
348
F. D. dos A idos et al. / Corrections
The formalism
that we use avoids the problems
at the cost of not allowing difficulties the variables
inherent
for such a straightforward
in the interpretation
arise because
that can be used to describe
to the Pechukas
interpretation.
of the noncommutativity
the intrinsic
method
Some of the of some of
state of the target as shall be
seen later. For the case where the target is excited from the initial level Ji = 0 with &i= 0, we find that the first-order correction to the excitation cross section of a level J can be interpreted as a change in the classical deflection function for the relative motion and a change in the coupling potential that causes the excitation. The change in the deflection function is due to an additional time-dependent potential V’( r, t) which is a Pechukas-like average of the coupling potential V( r, 5) suitably changed to account for the sum over the initial and final M-values, C CJMI U3Ct1Y‘jVCr? 04Ct2 V’(r,t)=Re
MM,
tO)IJiMi)fJ*M
(78)
P OJ
The change in the potential causing the excitation consists of three kinds of terms that must be added to the coupling potential V(r,( t), 5). Some of these terms describe the change in the relative kinetic energy due to V( r, 5); others correspond to an energy symmetrisation procedure similar to the one used by Alder and Winther; the remaining terms apply a similar method to the angular momentum and correspond to a symmetrisation of its modulus and z-component. If the initial intrinsic angular momentum Ji and energy &i are not zero, then there are extra terms which account for the change of energy and angular momentum transfer when a level J is excited. Our aim is then to write the excitation cross section of a level J as (79) where‘daJdS2 is the classical cross section corresponding to the modified deflection function and P,, is evaluated in a similar way to PO, but where an extra potential is added to I’( rs( t), 5). A Pi, is the additional term that accounts for the initial energy and angular momentum of the target. We shall proceed in two steps. First we evaluate the effect of the perturbing potential V’(r, t) defined in eq. (78) on the classical cross section for the relative motion and obtain an expression for da,/dL?. This expression will contain some of the correction terms (57). Then we interpret the remaining corrections in terms of P,, + A Pi,. The deflection function for classical scattering from the central potential V,(r) is (80)
4 = %,(b? GJ) assumed to be a monotonic function relation we obtain the function
of the impact
b = b&‘,,
E,).
parameter
b. Inverting
this
(81)
F. D. dos Aidos et ul. / Corrections
349
If the deflection function is changed by the coupling potential, then a particle that starts with energy E, and impact parameter b will be deflected by the angle &@, E,) = @,a@, E;,) +A&,
(82)
where A& describes the effect of the extra potential V’(r, 1). Inverting this relation we obtain, to first order b&i,
&) = b&r
-A&
&,I
where b,(@,, Eo) is the inverse deflection function for the potential V,(r) defined in (81) and to first order it is irrelevant whether the derivative is taken at 0, = OS.or 0, = 8,,. Expression (83) is the inverse deflection function for the perturbed case. Writing it in terms of the initial angular momentum Li of the orbit we get &(e,,, J$,) = L($,,
-&) +AL,
(84)
where
aL AL= - ae,, &A%. i
1
The classical cross section can be written as (87) where the perturbed deflection applicable
derivative is taken at constant initial momentum po. Although the potential &(P) + V’(r, t) is not spherically symmetric it leads to a function that is independent of the azimuthal angle and eq. (87) is to this case. Using expression (84) in eq. (87) we obtain, to first order,
where duJdS2 is the classical cross section corresponding to the unperturbed deflection function (80) and AL is given by eq. (86). An expression for A@, was derived in appendix B. Using eqs. (86) (B.lO) and (B.16) we find de,=
(~)f_~+(~)E,$-6e,
(89)
F. D. dos Aides et ul. / Corrections
350
with
6E,,86 and SL given by 6E,,=AE=q-E,
(91)
c ( atJM/aeO) Eof;M
MM,
POf
1
(92)
(93) The three terms in eqs. (89) and (90) describe the effects due to energy loss, angular momentum transfer and change of the effective spherically symmetric field. It is convenient to express the derivative in eq. (88) in terms of the parameters E,,7,(Y, p,y,8,defined in sect. 3. Using
aAL r=,
o
C=ff,P,y,r,
we obtain
(94) the derivative being taken for standard orbits (at constant E, and (Y= /3 = y = r = 0). Now we substitute expression (90) in (94) and evaluate the contribution of each one of the terms (91) (92) and (93) to the first-order corrections to the unperturbed cross section. The contribution from (91) is
(95) These terms are equal to part of the correction Pa2/PoJto the excitation cross section found in the previous section (eq. (59)). The remaining term of Pd2/PoJ is, to first order, equivalent to including a factor uo/ul in the excitation cross section that cancels the factor ul/uo in (54) [ref. ‘)I. The term (92) gives a contribution
(96)
to the cross section. (63).
The first term in (96) is equal to the correction
Pd3/PoI in eq.
351
F. D. dos A idos et ul. / Corrections
The contribution
from (93) is found
to be
where we have used the relation
which can be proved by using the commutator
(66). The first two terms of (97) have
a similar structure to P&P,, and P,,/P,, defined in eqs. (69) and (70). P,,( E,, 0,) is the excitation probability when the relative motion follows an unperturbed orbit rS( t) with energy E, and scattering angle 8, = v - 20,. The terms in (96) and (97) that depend on the derivative aP,,J,/&?o correspond to changing the scattering angle of the orbit along which we evaluate PoJ. In order to obtain a change de, given by eq. (89) we add and subtract the term
to duJdS2 in eq. (94). Using expressions (94)-(97) for AP,
t,,,,,( esot E,) state
as well as the results obtained
we can write the excitation
IJM)
is the transition corresponding
amplitud’e
from the initial
state
corrections
IJiMi)
section (54) as
to the final
to the hamiltonian
f&l(~) + J+&>, E)+ h’(t) orbit with energy where r,(t) is the unperturbed h’(t) is defined in eq. (50). BtiJM is given by StiJM =
in the previous
cross section with first-order
7
E, and scattering
angle
t9,, and
(9%
F. D. dos Aides et al. / Corrections
352
where we have used eq. (B.ll). Eq. (98) has the same form as eq. (79). We have thus shown that some of the correction terms account for a change in the relative motion due to the extra potential
V’(r, t) defined
in eq. (78).
In order to interpret the remaining terms we first look at the case where ei = Ji = 0, for which 8tiJ,,, vanishes. From eq. (98) we see that the effect of the change in relative motion is not taken into account by changing the trajectory in eqs. (55) and (56) as would be the case in the Pechukas treatment. On the contrary, it is necessary to subtract the effect caused by the perturbing potential V’(r, t) on the orbit with energy E, and scattering angle and evaluate t,,, along the unperturbed scattering angle 19~~.The effect of the deviation of the orbit on the excitation of the target is described by the additional term h’(t) in the hamiltonian (cf. eq. (50)). In order to understand h’(t)
the effect of this extra potential
we write the matrix
element
of
as
where we have used one of the relations
(B.ll)
and
The first term in expression (100) describes the decrease in relative kinetic energy due to the additional coupling potential V(r, t) which is neglected when the relative motion is described by the unperturbed orbit r,(t). The second term is similar to the Alder-Winther energy-symmetrisation procedure but corresponds to keeping the angular momentum fixed instead of the scattering angle. This can be seen by adding this term to the matrix element of the coupling potential V(t, E,) = V(r,(t, E,), E) to obtain
” (xl v(t>4,) - a~,
q,[,
=(x,V(t,
E,++.
If we had chosen the basis states lx) and 15) to be eigenstates terms in eq. (100) would have had the form
001) of J, the next two
(102) The angular momentum of the relative motion L points towards the negative side of the x-axis and this term corresponds to a symmetrisation of the modulus of L in a similar way to the energy symmetrisation in eq. (101). The final term in eq. (100) has a similar meaning. It corresponds to a symmetrisation of the z-component of the
F. D. dos A idos et a/.
angular excitation
momentum
and
describes
the effect
Corrections
353
of the tilting
of the target. In this last term the tilting
of the orbit
is done through
This term comes from eqs. (20) and (21) where the derivatives
a change
on the in /3.
are taken with respect
to p and y. It can easily be shown by reversing the roles of j3 and y on the subsequent treatment leading to expression (100) that the last term of this expression can also be written as
where again we used eqs. (B.ll). This shows that the effect of the tilting of the orbit can equally well be described by a symmetrisation of L, due to a change of fi as by a symmetrisation of Ly due to a change of y. However, as the basis states Ix) and I{) cannot be simultaneous eigenstates of J, and J, or of JY and J,, these terms cannot be incorporated in V(r,(t), 5) through a change in the angular momentum of the orbit r,(t) as was done for the energy in eq. (101). If the initial level does not have zero spin and energy then 8tiJM, which was defined in eq. (99), does not vanish. The first term of 8tiJM corresponds to substituting the excitation energies ex and eS in the matrix element of h’(t) (eq. (100)) by ex - .si and sS - .si. This is in agreement with the energy-symmetrisation procedure as now the difference between the energy of relative motion in channel x and in the entrance channel is - (ex - ei) instead of - eX. If the initial state is an eigenstate IJiM,,) of J, then the second term has a similar interpretation. It corresponds to substituting in eq. (102) MXx and MXs by MXX- MXi and MXr - MXi. The same applies for the last term of StiJM if the initial state is an eigenstate of J,. It should be noted that the formulae presented in this section were not used in practical calculations. They are useful only insofar as they provide some insight into the interpretation of the corrections.
6. Numerical Three
sets of numerical
results
tests of the theory developed
in sects. 3 and 4 have been
made. The first was designed to test the correctness of the formulae and the reliability of the numerical methods, while the others test the accuracy of the expansion up to first order in l/n used in this paper. The calculations corresponding to the first two sets of results were made for the scattering of 160 by “‘Sm at E,, = 109.2 MeV. The target nucleus is assumed to be in its ground state J = M = 0 and the excited states are those of an axially symmetric rotor with an infinite moment of inertia. This is the sudden case with all excitation energies Ed = 0. The values of the relevant parameters are n =
29.90,
5=0,
q =
2.990 >
A = q/n
=
0.100.
The third set of results corresponds to the non-sudden case where the excitation energies of the target are finite. The calculations were made for the scattering of ‘Be
354 by
F. D. dos A ides et al. / Corrections
lS2Sm at E,,
= 30 MeV. The excitation
energies
.sJ of 15*Sm were taken to be
Q=OM~V,
~~ =
0.122
MeV,
Em= 0.361 MeV,
.s6= 0.705 MeV,
Ed =
1.122 MeV,
q0 = 1.539 MeV.
The fact that these energies do not correspond exactly to a rotational spectrum is of no consequences. As we have already mentioned the formulae obtained in sect. 4 are valid for any form of the intrinsic hamiltonian He([). The relevant parameters for this case are equal to n = 20.17,
5 = 0.043)
q = 1.212,
A = 0.060.
These values were chosen because the parameters A and [ are large enough for there to be significant corrections to the Alder-Winther theory and because some coupled-channels results calculated with the code AROSA were available. The tables show the values for excitation probabilities PJ and deviations 8,. The excitation probabilities PJ are defined by da 3 dafYrn) ‘J, d0 - ua d0 J
(103)
TABLE 2
Zero- and first-order excitation probabilities J 180~/178~
1500
Pib)
PI’)
Pid)
0 2 4 6 8 10
8.320 3.138 4.174 1.552 2.738 2.852
-2 -1 -1 -1 -2 -3
8.318 3.141 4.173 1.551 2.734 2.847
-2 -1 -1 -1 -2 -3
6.334 3.475 4.213 1.426 2.292 2.168
-2 -1 -1 -1 -2 -3
6.333 3.477 4.213 1.425 2.289 2.163
-2 -1 -1 -1 -2 -3
0
8.196 3.819 3.935 1.223 1.850 1.671
-2 -1 -1 -1 -2 -3
8.196 3.819 3.935 1.224 1.851 1.671
-2 -1 -1 -1 -2 -3
6.544 4.141 3.918 1.118 1.551 1.280
-2 - 1 -1 -1 -2 -3
6.545 4.141 3.918 1.118 1.552 1.281
-2 -1 -1 -1 -2 -3
1.316 5.300 2.807 5.238 5.023 2.951
-1 -1 -1 -2 -3 -4
1.316 5.300 2.807 5.240 5.025 2.952
-1 -1 -1 -2 -3 -4
1.266 5.488 2.725 4.762 4.270 2.340
-1 - 1 - 1 -2 -3 -4
1.265 5.488 2.725 4.763 4.272 2.341
-1 -1 -1 -2 -3 -4
2 4 6 8 10 1200
Pea)
0
2 4 6 8 10
“) PO is the lowest-order excitation probability from ref. I’). ‘) Ph is the lowest-order excitation probability from COULEX. ‘) PI is the first-order excitation probability from ref. lo). d, P; is the first-order excitation probability from eqs. (54) and (103). P{ cannot be calculated for 8, = 180° and so we used b’, = 178O in P,‘, Pi.
F.D. dos Aides et ul. / Corrections
355
where du,/dti is the excitation cross section calculated from COULEX, from eq. (54) or from AROSA, and da, @Ym)/ds2is the energy-symmetrised Rutherford cross section defined in eq. (2). The S, are the deviations of the excitation probabilities from the AROSA results PAJ and are defined by S,=lOO
4 - PA.7
P
004)
AJ
We note that the expressions for the corrections to the cross section obtained in this paper are not applicable for backward scattering. For this case we can either evaluate the analytic limit of those ex$ressions when 6, + 180° or make the calculation for a value of the scattering angle close to 180°. We follow the second method and use the value 0, = 178O. In the sudden case the corrections contained in eq. (54) can be calculated by an analytical method which is different from the one used in this paper [dos Aidos and Brink”)]. Note that for the sudden case du~Y*~/d~ = do,/dS2 and ul/uO= 1. Table 2 compares results obtained by using the numerical methods of sect. 4 with TABLE 3
Comparison
0s 180”/178”
1500
120°
of first-order
corrections
with AROSA “): sudden case
.J
AROSA
&lb)
6,‘)
0 2 4 6 8 10
6.508 3.460 4.204 1.429 2.283 2.%
-2 - 1 -1 - 1 -2 -3
28 -9 -0.7 9 19 36
-3
0 2 4 6 8 10
6.720 4.135 3.916 1.120 1.547 1.364
-2 -1 -1 - 1 -2 -3
22 -8 0.5 9 19 34
-3
0 2 4 6 8 10
1.280 5.443 2.715 4.742 4.206 2.349
-1 ‘--1 -1 -2 - 3 -4
3
-1
-3 3 10 19 32
0.5 0.2 0.3 -0.3 2
0.2 0.04 -0.2 -0.1 2
0.8 0.4 0.4 1 4
“) AROSA is the exact excitation probability. h, So is the deviation of the COULFX results for the excitation probability from the AROSA results in percent. ‘) 6, is the deviation of the results for the excitation probability, eqs. (54) and (103). from the AROSA results in percent. 6, cannot be calculated for 8, = lSO” and so we used @,= 178O.
356
F. D. dos Aides et al. / Corrections
those from the method of ref. lo). In this table PO is the zero-order value of PJ from ref. lo) while PO’is obtained from COULEX. The results should be identical and they are except for a small difference in the last figure. The columns P, and Pi show the corresponding zero- plus first-order values of PJ (P, from ref. lo) and P; from eq. (54)). Again the results should be identical because they correspond to different ways of calculating the same correction. Again the numbers are the same except for a small difference in the last figure. This is a check of the numerical methods used to evaluate the formulae of sect. 4 of this paper. The difference between P, and PO is the correction to the Alder-Winther theory calculated in this paper. In some cases it is more than 20% of the zero-order value for the parameters used in this case. Table 3 compares the values of PJ obtained from the method developed in the present paper with those calculated with the coupled-channels code AROSA for the same values of the parameters as in table 2. In these calculations only levels up to J = 10 were included in order to have the same cut-off as the one used in AROSA.
TABLET Comparison
of first-order
corrections
with AROSAa):
non-sudden
case
180°/1780
0 2 4 6 8 10
6.098 3.578 3.197 1.037 1.574 1.399
-1 -1 -2 -3 -5 -7
-0.9 0.2 8 25 59 124
-0.6 0.2 5 15 32 59
0.2 -0.2 - 0.3 0.9 -8 -33
150”
0 2 4 6 8 10
6.457 3.282 2.584 7.525 1.048 8.721
- 1 -1 -2 -4 -5 -8
-0.8 0.5 8 25 59 123
-0.5 0.4 6 15 32 59
0.1 -0.2 - 0.1 - 0.06 -4 -24
1200
0 2 4 6 8 10
7.423 2.444 1.304 2.706 2.819 1.810
- 1 - 1 -2 -4 -6 -8
-0.5 0.9 8 24 57 119
-0.3 0.7 5 15 32 59
0.07 -0.2 0.07 0.6 -2 -18
“) AROSA is the exact excitation probability. ‘) S, is the deviation of the COULRX results for the excitation probability with unsymmetrised orbits from the AROSA results in percent. ‘) S, is the deviation of the COULEX results for the excitation probability with energy-symmetrised orbits from the AROSA results in percent. d, 6, is the deviation of the results for the excitation probability, eqs. (54) and (103) from the AROSA results in percent. 6, cannot be calculated for 6’, = 180° and so we used 0, = 178’.
F. D. dos Aidos et al. / Corrections
We show in this table the deviations (eq. (104)) 6, for method developed in this paper. The agreement between (54) and the AROSA coupled-channels results is good. the AROSA values is 4% in the case of the J = 10 state
351
COULEX and 6, for the the results provided by eq. The largest deviation from at 0, = 120”. In most cases
the agreement is better than 1%. The results provided by the Alder-Winther theory are less accurate in almost all cases. The average error is about 10% and in several cases it is more than 30%. Table 4 compares the values of the excitation probabilities PJ for a case where the excitation energies are not all equal to zero and where the corrections in E/n and [q/n are important. In this table we show the deviations 6, for COULEX without the energy-symmetrisation procedure, 8, for COULEX with the energy-symmetrisation procedure and 6, for the results provided by eq. (54). In general the energy symmetrisation on its own corresponds to an improvement over the unsymmetrised results, especially for the higher states. The results S, that correspond to the method developed in this paper show a more drastic improvement. For the states up to J = 6 the deviation 8, is always under 1%. For the higher states the deviations 8, are still smaller than S, or 8, although the agreement is not as good as for the other states. The reason for such large deviations may be the smallness of the values involved that prevents rigorous calculations due to the limited precision of the computers, or it may indicate that the energy of the state is becoming too high and the higher-order terms in t/n can no longer be neglected. These comparisons suggest that eq. (54) gives a significant improvement over cross sections calculated by the Alder-Winther method. The authors would like to thank Prof. J. de Boer for his interest and for providing some results calculated from the coupled-channels programme AROSA. C.V.S. would like to acknowledge support from the SERC and the Hahn-Meitner Institute. F.D.d.A. would like to acknowledge support from Comissao Permanente INVOTAN and from the Danish
Ministry
of Education.
Appendix A The purpose
of this appendix
is to derive expressions
for the quantities
(A.1)
defined
in eqs.
(16) and
(17) where
f and
g are
the
orbit
parameters
Cj =
358 {
F. D. dos Aidos ef al. / Correciions
E,, 7, So, a, j?, y
}.
In these expressions
Aij is the inverse
‘ii = arli
aPoj
of the matrix
(A.3)
9
where S,,(r,t,, pot,) is the classical action for an orbit with initial momentum p,, and final position ri. With this definition the expressions (A.l) and (A.2) are independent of the coordinate system used to specify r,, po. It is convenient to use polar coordinates for both vectors, r, = (r,, 0, @) 7 The action
S, can be expressed
Po=(PoJM.
simply in polar coordinates
64.4) as
where L is the angular momentum, r,,, the distance of closest approach, initial position at time t,. p, is the radial momentum p,(r) and 6, is related As a first step defining the slope by relating the
= {2m(E,-
Y(r))
-
L2/r2}l'*,
and r, the
(A.6)
to the scattering angle 19,by 8, = $(v - 8,). in evaluating (A.l) and (A.2) we express the angles e,, (Y,j?, y and orientation of an orbit in terms of 8, c$, 5, 4. This can be done incident direction (5, +!J) and the outgoing direction (8, +) to
t$,, (Y,/3, y. We have c~s e = cos( e, -
&OS
p,
(A-7)
sin~cos~=cos(f3,-cu)sinj3cosy--sin(8,-cu)siny, cos l= sin
useful
relation
- c~~(
e, + +OS
(A.@
p ,
(A.9)
-cos(e,+ar)sinpcosy-sin(8,+cu)siny.
(A.lO)
is
cos(?r-2e,)=cos8cos~+sinesin~cos(~-~).
(A.ll)
We need derivatives of r3,, (Y,fl, y with respect to 6, {, $, 4 for a standard orbit (fig. 1). A standard orbit has (Y= p = y = 0 and 8 = e,, {= P - 8,, 4 = + = $T. These can be obtained by differentiating (A.7)-(A.11). The non-vanishing first derivatives are
aeo
ae,_l -7, a
al
ap
q=-s=*tanea,
aa
aa
-...-=_L
ae
-=_1 ad
23
au-
a+-q=f.
2,
ay
x=-t,
(A.12)
F. D. dos A idos et al. / Corrections
359
We also need the mixed second derivatives
with f = e,, a, p, y. Only one of these is non-zero,
a=e, = ftane,.
(A.13)
a+a+
The non-zero
matrix
elements
of Sij (A.3) in polar coordinates
for a standard
orbit
are
(s'P s,, S*$’ SBP ,s
OP
%
se+
s,,
s,,,
=
i
s,,
0
0
s2,
s,,
0
0
0
s3,
I
The matrix elements can be calculated from (A.5) by using (A.12) and (A.13). It can be shown from (A.5) that ( aS/&90)p = 2L. Hence s,, =
S,, = P/G,)?
s,, = - f aL/ae, In (A.14)
o is the asymptotic
t, -+ 00. For a Rutherford
orbit
velocity
aLlap
S,, = fL tan
, and
u(ti)
e, .
is the velocity
L = n tan e,, where n is the Sommerfeld
,
S,, = -qant?,
(A.14) at t,;
S,, -+ 1 as
parameter
c02e,,
s,, = - hn/2
S,, = +An tan28,.
The non-zero
matrix
elements
Ai, = &VP, A,,=~/s,,=
and
(A.15)
of the inverse
matrix
A,, = - S2,/S2,
A,, are = - 2( aeO/aPo)
L2
A,, = 2/L tan e, .
-2wom&,,
(~.16)
The remaining ingredients needed for evaluating (A.l) and (A.2) are the expressions for E, and r in terms of (r, 8, c#J),(p, l, 4). E, is the total energy and r is the time at which the particle reaches the point of closest approach. They are given by & =po2/2m
+ KMtcJ)
(A.17)
3
7 = t, - t,, ) where t,, is the time to pass from the distance ts1 =
It is a function the dependence
(A.18) of closest approach
r,
to ri,
s
rldr/F. rrn
of ri, E, and t9,. The dependence of r on 8, +, {,$ is contained on 6,. Neglecting the term V,(r,( to)) which goes to zero
in as
F. D. dos Aidm et ul. / Corrections
360
1, + - co, the only non-zero first derivatives of E, and r are ~E~/~P~ = 0, a7/&,
+wh
at~~/aE~)~~,
aqae = - aqac = - e( at,,/ae,) E,,
= -l/U(t,),
a27 dr,ap,=zqp
= -4
1 a?,,
a% m=--
i
4 a@
-
’
a2T
at,,
= - +tan9,m.
aQa+
0
(A.19) Terms of order l/r, have been neglected except in the case of u(tJ in &r/&-r. This term is multiplied by rt in a correction to the phase of the propagator. The derivatives of t,, can be evaluated easily by using the equations of a Rutherford orbit. They are
at~~/ae~= - (~~/v)t~e~ f a%,/ae,” = - ( aC/u)sec2fJo, atsl/aEo = - &
( rl 4 3a,ln(2r,cos
$/u,)
- 5ac).
In the general case the derivatives of t,, with respect to 8, can be written in terms of derivatives of L by using the identity ( at,,/ae,)
E, = ( aL/aE,)
(A.20)
e,.
To prove this we define s(r,,
&,, 0 = j%,(r) G?
dr
where p, is the radial momentum (eq. (A.@). It is easy to see that t,, =
(as/aq
L,
do = - ( as/aL)
Using (A.21) and a well-known transformation
Eo.
(A.21)
of partial derivatives,
Collecting these results we can evaluate D2(C,) and D(C,, C,), where C, are the orbit parameters E,, r, @,,(Y,j3, y. The non-zero results for D2(C,) are
1
(A.22)
=2E,
(A.23)
D*(e,) = 112~ for a general central potential. For a Coulomb potential they reduce to 02(T) = l/E,)
02(eo) = (i/2An)cot
e,.
(A.24)
361
F. D. dos Aides et 01. / Corrections
The expressions for the non-zero D(C,, CO) in the notation of table 1 are D(r,Ea)=
(A-25)
-1,
(A.26)
= - i( ~~O/dE,)L= c=D@,,~,)=
-L>(~,~)=D(~~,c+=
= +( i)eojaL)Eo=
(A.27)
-(1/4EO)sin8,cos8,, -D@,e,)
(A.28)
(1/2nh)cos2eo,
D=D(/3,/3)=
-tan8,/2L=
E=D(~,Y)=
-~(~,j3>=
F=D(y,y)=
1/2Ltan&=
(A.29)
-l/2&, -1/2~=
-~i/2~~)cote~,
(i/2nfi)c0t2eo.
(A.30) (A.31)
Here two expressions are given for each term. The first corresponds to a general central potential and the second to a Coulomb potential. The function A = D( 7,7) is not needed for the calculations in this paper. it contributes to corrections to the phase of the scattering amplitude but not to corrections to the cross section.
Appendix B
In this appendix we find an expression for the change in the scattering angle f?, caused by the perturbing potential V’(r, t) given by eq. (78). The unperturbed potential I$(P) is assumed to have spherical symmetry and a monotonic deflection function. The unperturbed orbit rO(t), when specified on the standard axes defined in fig. 1, lies on the yz plane. The orbit rl(t) that corresponds to scattering off the potential G(r)
+ I+,
t)
(B-1)
with the same initial conditions as r,(t) will not necessarily lie on the same plane. Its scattering direction will be (8, + be, $r + A+), where de and Acp describe the effect of the extra coupling potential V’(r, t). To first order its scattering angle will be given by e,, = e,, + de,, de,= -de,
03.2) (B.3)
where 0, is the scattering angle of the unperturbed orbit. A@can be evaluated by using standard classical perturbation theory methods. We choose a set of six parameters C,( r, p) that are constants of the unperturbed motion C,(rO(t),pO(t))
= const
(B.4)
F. D. dos Aides er al. / Corrections
362
and use them to specify the orbit r,,(t) = ro( C,, t). When C, are evaluated orbit
rr( t) they will no longer be constant
but will vary according
along the
to the equation
14)
(B.5) where { C,, Cs>PB are the Poisson brackets
of C, and Cp. To first order we can substitute all the parameters C, on the right-hand side of eq. (B.5) by their unperturbed constant values. The first-order change on the value of C, after the scattering due to the additional potential V’( r, t) can be obtained by integrating eq.
(B.5)
where C,, and CpO are the initial values of the parameters C, and CD. I is the integral of the perturbation along the unperturbed orbit. Any six independent constants of the unperturbed motion can be chosen for C,. The Poisson brackets in eq. (B.6) are easily evaluated if the parameters C, are taken to be the position r, 0, + and conjugate momenta p,, pe, p+, of the unperturbed orbit at a specified time t,. As we are interested in obtaining the direction of scattering we choose t, to be very large. Eq. (B.6) for C, = O(t,) becomes
03.8) In (B.8) constant,
the derivative the scattering
should be taken while keeping the energy of the orbit direction fixed and the orbit r,(t) in the yz plane. It is
convenient to express the derivative in terms of the parameters E,, e,, 7, (Y,p, y that were defined in sect. 3. Expressions that relate these parameters to r, 8, +,, p,, pe, pg can be easily obtained and we find, for t, large enough,
(B.9) 0,( L, E,) is related
to the deflection
function
of the unperturbed
e&c, 4,) = t( -77- e,,(L, 4,)). Other
useful relations
motion
by (B.10)
are (B.ll)
F. D. dos A idos et al. / Corrections
363
which can be obtained by noting that a=y=Q*sin/3=L,/L, -L,/L.
a=p=O*siny= Using (B.7) and (B.9) in (B.8) we obtain
(B.12) where, using the expression (78) for V’(r, t), (B.13)
(B.14) c
(atJM/a@o)E"GM
MM, POJ
(B.15)
From eq. (B.3) we can write the first-order correction to the scattering angle as (B.I~) where we have used relation (10). AS, is a sum of three terms. The first term describes the effect of the energy loss on the scattering direction. The second describes the effect of angular momentum transfer. In (B.14) the projectors on levels J and Ji can be expressed in terms of eigenstates IJ, MX) of J, instead of J, and we obtain SL=
_
t:
(“xi-Mx)ltJMx12
MxM.d
p
(B.17) OJ
6L
is, up to a sign, the change of polarisation along the x-axis. The negative sign is due to the fact that in the standard axes the relative angular momentum points towards the negative side of the x-axis. The third term of (B.16) is independent both of the energy and of the angular momentum transfer. It describes the effect of the change of the effective spherically-symmetric field due to the perturbation. References 1) F. RKsel, J.X. Saladin and K. Alder, Comp. Phys. Comm. 8 (1974) 35 2) 3) 4) 5) 6)
L.D. Tolsma, Phys. Rev. CZO (1979) 592 M.J. Rhoades-Brown, M.H. Macfarlane and S.C. Pieper, Phys. Rev. C21 (1980) 2417, 2436 K. Alder and A. Winther, Coulomb excitation (Academic, New York, 1966) A. Winther and J. de Boer, Caltech technical report (Nov. 1965), reprinted in ref.4) J. de Boer, G. Dannhguser, H. Massmann, F. Riisel and A. Winther, J. of Phys. G3 (1977) 889
364
F. D. dos Aidos et al. / Corrections
7) J. de Boer and G. Dannhluser, Nuclear physics ed. C.H. Dasso, R.A. Broglia and A. Winther (North-Holland, Amsterdam, 1982) p. 451 8) C.V. Sukumar and D.M. Brink, Nucl. Phys. A404 (1983) 121 9) M.W. Guidry, R. Donangelo, J.O. Rasmussen and J.P. Boisson, Nucl. Phys. A295 (1978) 482 10) F.D. dos Aidos and D.M. Brink, to be published 11) F.D. dos Aidos and D.M. Brink, J. of Phys. Gll (1985) 249 12) P. Pechukas, Phys. Rev. 181 (1969) 174 13) C.V. Sukumar, J. of Phys. GlO (1984) 81 14) H. Goldstein, Classical mechanics, 2nd ed., (Addison-Wesley, Reading, Mass., 1980)
A448 (1986) 333-364 Publishing Company
CORRECTIONS TO THE ALDER-WINTHER THEORY OF COULOMB EXCITATION F.D. DOS AIDOS*, Department
C.V. SUKUMAR
and D.M. BRINK
of Theoretic& Physics, 1 Keble Rd., Oxford OXI 3NP, UK Received 31 August 1984 (Revised 1 July 1985)
Abstract: An expansion scheme is developed for studying corrections to the Alder-Winther theory of Coulomb excitation. The zeroth-order term in the expansion of the excitation cross section is identical to the expression provided by the Alder-Winther theory and there are several kinds of first-order corrections. All of these can be calculated by making simple changes in existing computer programs for the Alder-Winther theory. These corrections can be interpreted as a change in the deflection function for the relative motion and a change in the excitation probabilities due to the deviations of the relative motion from a Rutherford orbit. Some of the terms that describe the change in the excitation probabilities correspond to an energy symmetrisation and others can be interpreted as an angular momentum symmetrisation. Numerical comparisons with results of full quanta1 coupled-channels calculations are presented.
1. Introduction Coulomb excitation is an important tool for studying electromagnetic properties of nuclear levels. When heavy ions are used as projectiles multiple transitions occur and it is possible to excite rotational bands up to high spins. A coupled-channels calculation for such a multistep process can be made only for not too heavy ions. Rose1 et al. I) have developed a coupled-channels code AROSA for this purpose, but its use is restricted to cases where the Sommerfeld parameter n is not too large (n G 30) and where only a few states in a rotational band are excited (J 6 10). A full coupled-channels calculation by conventional methods is not feasible for very heavy ions because of the large number of channels involved. This situation may change with the development of new more powerful methods [Tolsma’), Rhoades-Brown et al. 3)]. The semi-classical method for calculating multiple Coulomb excitation developed by Alder and Winther4) assumes that the relative motion of target and projectile is described by a Rutherford orbit. This defines a time-dependent potential which causes the excitation. This procedure has been implemented by de Boer and Winther in the computer code COULEX 5). The time-dependent Schrodinger equation for the *Permanent
address:
Departamento
de Fisica, Universidade 333
de Coimbra,
3000 Coimbra,
Portugal
F. D. dos Aidos et al. / Corrections
334
internal
excitation
is solved numerically,
and yields an excitation
probability
P,,( 13,)
for exciting a final internal state 1-1from an initial state v when the relative motion follows a Rutherford orbit with scattering angle 0,. The differential cross section is given by
where dun/d&! is the Rutherford cross section. It is found that the results are improved by the use of “energy-symmetrised” orbits 5). This energy symmetrisation consists of three steps: (a) substituting in expression (1) doJdO(B,) by
where u, and u,, are the relative velocities for the entrance and exit channels; (b) substituting in the evaluation of P,,(8,) the energy of the relative motion in all matrix elements of the coupling potential (xlVlS) by an average of the energies of the relative motion for the channels x and {; (c) including a factor u,/uV in the expression for the excitation cross section. With this procedure the excitation cross section will be given by
In ref. 5, the energy symmetrisation is defined in a slightly different way because of a change of variable in the time-dependent Schrodinger equation. Either symmetrisation method provides similar results for the cross section. It has been difficult to assess the accuracy of the Alder-Winther method because no systematic procedure for improving it and for calculating corrections has been available. De Boer et al. 6, have tried to improve the method by taking into account the influence
of the excitation
process
on the relative
motion
through
an angular
momentum symmetrisation of the orbit of relative motion. A refinement of this method was developed by de Boer and Dannhauser’). They assumed that the relative motion changes from one Rutherford orbit to another of different energy and angular momentum several times during the scattering. Both of these methods provide similar results but do not give a significant improvement on the results obtained by COULEX. It is likely that they include some but not all of the relevant corrections. Sukumar and Brink8) have developed an expansion for calculating inelastic scattering cross sections based on the path integral method. This method will be applied to the multiple Coulomb excitation problem in the present paper. In order to understand the nature of the expansion we consider the excitation of rotational states in a deformed target nucleus with charge Z,e and intrinsic quadrupole moment Q by a structureless projectile with charge Z,e. The reduced mass is m and
the incident characteristic
relative velocity is u. The Coulomb collision time rc are given by Z, Z,e 2
UC=-
mu2
length parameter
2aC rC = -
u
’
uc and a
’
and the ma~mum strengths of the central (monopole) and the quadrupole parts of the Coulomb interaction are proportional to Z,Z,e*
ho=77
C
There are several dimensionless parameters which characterise the scattering [Guidry et al. 9)]. One is the Sommerfeld parameter
Another is an interaction strength
Qzle*
4 _ rcF2
4Aua:
’
which gives an indication of the number of quanta excited. For backward scattering from a rotor in the sudden appro~mation q = $A,,, where L,, is the ma~mum rotational angular momentum in units of 87which can be excited classically. A third dimensionless constant is the adiabaticity parameter r,AE t=
21i
AE =nz,
where AE is the excitation energy of the first excited state. A Coulomb excitation cross section is a function of all of these parameters. The method of Sukumar and Brinks) is an asymptotic expansion in powers of l/n for fixed q and E where the leading term corresponds to the Alder-Winther theory. In this paper we calculate the l/n corrections to this theory. There are several kinds of corrections of order l/n. Quantum corrections to the relative motion do not involve the quadrupole interaction. They contribute a phase *) which does not affect the cross section and we do not consider them here. A second kind of correction gives a m~ifi~ation of the scattering angle due to the effect of the quadrupole potential. The relative magnitude of this effect is proportional to 9, &!$L;. EO
Another modification of the scattering angle is associated with the energy transfer This effect is proportional to
AE.
5 -_AE 2v,, - n *
F. D. dos A ides et al. / Corrections
336
These modifications of the trajectory of the relative projectile will in turn change the excitation amplitudes.
motion of the target and A measure of this effect is
given by q!+ EO
n'
AE q2V,,
_!6 n ’
All of these effects are of order l/n for fixed q and E and are included in the correction terms to the Alder-Winther theory which are derived in this paper. When developing semi-classical theories of Coulomb excitation it is important to recognise that there are several dimensionless parameters in the theory. This means that a number of different expansion schemes is possible. One alternative scheme is the classical limit S-matrix (CLSM). Guidry et al. 9, point out that this is an expansion in powers of l/n for exact classical dynamics. In other words it is an expansion in powers of l/n for fixed A and is therefore different from the expansion used here. The present paper is organised as follows. The results from ref. *) are collected in sect. 2. In sect. 3 the formulae for the correction terms are written in a coordinate system which is convenient for studying the scattering problem and the expressions are simplified. Expressions for the corrections to the cross section are given in sect. 4 and an interpretation for these corrections is discussed in sect. 5. Numerical results are presented in sect. 6. An alternative method for calculating the first-order corrections in the expansion method of ref. ‘) has been developed by dos Aidos and Brink lo). It is simpler than the present method but it is restricted to the sudden limit where all the excitation energies
are set equal to zero.
2. Statement of the problem The theory
developed
in ref. *) applies H=p2/2m+ +,5)=H&)+V(r,5),
to a scattering V,(r)+h(r,E),
problem
with hamiltonian (4) (5)
where r is the relative coordinate of the target and projectile, p is the momentum conjugate to r, and the 5 are internal coordinates. The first two terms on the right-hand side of (4) describe the relative motion, H,( 5) is the internal hamiltonian of the target, and V(r, 6) is the coupling which produces excitation. In the present paper we are concerned with an application of the theory developed in ref.8) to Coulomb excitation. We assume that the incident energy is well below the top of the Coulomb barrier so that the interactions V, and V have no nuclear contribution. We take V,(r) to be the point Coulomb potential
331
F. D. dos Aidos et al. / Corrections
and
V(r, 5) the non-central
scribes rotational
part
of the electromagnetic
states of a deformed 4,(E)
interaction.
target and V is a quadrupole =
If H,(t) interaction,
J*/W
dethen (7)
where Q is the target quadrupole moment and 0 is the angle between r and the symmetry axis of the target. The results obtained in sects. 2-5 of this paper do not depend on the particular forms (7) and (8) for HO and V. The formalism is also independent of the form of V,(r) as long as it is a central potential with a monotonic deflection function. In sects. 2-5 we shall always give the results for this general form of V,(r) unless otherwise stated. Ref. *) gives an approximate formula for the propagator from an initial relative momentum pO and internal state v of the target at t, to a final relative position r, and internal state p at t,, &,(r,L
T,,[r,(t)l+aqy}.
Pota) = K,(rltl,NO){
In eq. (9) K, is the semi-classical limit of the propagator motion in the potential V,(r). The matrix element Tpy is
describing
(9 the relative
T,“MGl= hwo(h~hJlv) where the unitary
operator
U, is a solution
of the equation
iAt$=h(r,(t), with initial
condition
path r,(t), which final position r,. The relation the propagator
()U,
U,( t, to) = 1 when t = t,. The operator
is a classical
trajectory
(10)
in V,(r)
01) U, is a functional
with initial
momentum
between the propagator and the cross section is discussed is given by eq. (9) the corresponding cross section is
of the pO and
in ref. ‘l). If
in which U, and uP are the relative velocities for the entrance and exit channels. The factor UJU, occurs in (12) because the inelastic cross section is a ratio of fluxes in the exit and entrance channels. duJdfi(8,) is the classical cross section for elastic scattering by the potential V,(r). If V,(r) is the Coulomb potential (eq. (6)) then duJdG(8,) is the Rutherford cross section and the lowest-order term for the excitation cross section in eq. (12) is equal to the result obtained in the Alder-Winther theory with unsymmetrised orbits (eq. (1)).
F. D. dos A idos et al. / Corrections
338
The correction &$ ref. 8, and is
to the transition amplitude is given by eqs. (22) and (38) of
6qy = TIP,,+ T;ly,
(13) (14)
115) where
bw= f-&k f&4 >
(df)l= w%(b d.
The aim of this paper is to calculate the first-order corrections Tiand T’ to T.
3. Simplification of the transition amplitude In this section we simplify expressions (14) and (15) for the correction to the transition amplitude. These depend on the orbit p,(t) and in (14) and (15) this orbit is specified by its initial momentum p. and its final position rl. As the independent variables are p,, and r, it is a simplification to write
where we have used the convention of summing over repeated indices. Then p. and r1 are treated consistently as independent variables in the partial derivatives. Each classical orbit rc( t) in V,(r) is specified by six parameters. In (14) and (15) these are the components of p. and rt, but other choices are possible and also more convenient. We denote a convenient set of parameters by C,, (Y= 1,. . . ,6, and choose them to be E,, r, 0,, (Y,/3, y, where E, is the energy of the orbit, r is the time at which the projectile reaches the point of closest approach, 0, is related to the scattering angle 8, by 6, = $(r - 8,) and (Y,/3, y are three angles defining the orientation of the orbit. We evaluate (14) and (15) for a standard orbit defined in fig. 1. The standard orbit r,(t) has r = 0 and a = /3 = y = 0. An orbit with orientation (Y,/3, y is obtained from the standard orbit by making a rotation through (Yin the
339
F.D. dos A ides et al. / Corrections
Fig. 1. Sketch
range (0,27r)
of the standard orbit: x is antiparallel to L, z is the symmetry axis of the orbit, y-component of the velocity is always greater than or equal to zero.
about
finally a rotation It is convenient
the x-axis, then a rotation
p in (-
ia, $7) about
and the
the y-axis and
through y in (0,2a) about the z-axis. to introduce the following notation for the derivatives
appearing
in eqs. (14) and (15):
Then
D*(f)
and
using the summation
D( f, g) can be expressed
convention
in terms of D*( C,) and
for the indices
(II and /L
D(C,, CP) by
F. D. dos A ides et al. / Corrections
340
With this notation
eqs. (14) and (15) can be written
as
(20)
The functions D2( C,) and D(C,, Cp) depend on the geometry of the orbits. They are calculated for a standard orbit in appendix A for a general central potential V,(r) and in the case where V,(r) is the point Coulomb potential (6). Only two of the functions D2(C,) are non-zero. For a Coulomb potential they reduce to D2(~)=
cot e,
1
+,
D’@o)=
0
2~
=
2t?n
(23)
>
where L = hn tan 0, is the angular momentum of the orbit. The results for D( C,, Cs) are given in table 1. Note that only the symmetric part of D(C,, Cs) contributes to T, in (20) while only the antisymmetric part contributes to (21). Hence the sums over (Yand p in (20) and (21) each contains only a small number of terms. The evolution operator Uo[tl, to, r,(t)] is a functional of the orbit rc( t). This orbit is specified by the six parameters C, which we take to be E,, 0,, 7, (Y,p, y. Only two, E, and So, of these six parameters affect the shape of the orbit. The dependence of U, on the other parameters can be found from symmetry arguments. The standard orbit r,(t) in fig. 1 has its time origin at the time of closest approach. A general orbit r,(t) can be obtained from rS(t) by a rotation R and a time translation T, r,(t) Here R is the rotation
specified
= RTr,(t)
.
by the angles (Y, /3, y, and T is defined
by
Tr,( t) = r,( t - T).
TABLE 1
D( C,, Cfl) for a general potential (appendix A)
0
0
0
0
0
0
0 0 0 0
A -B B 0 0
B C
B C
0 0 0 D -E
0 0 0 E F
-1
-C
-C 0 0
0 0
ED. dos Aides et al. / Corrections
341
In this equation ,r = t, - t,,
t, =
r’dr/L Jml
(24)
t, is the time taken to move from r,,, to ri ml is the distance of closest approach, along the orbit r,(t), and r is the time corresponding to the point of closest approach r,. The evolution operator Uo[tl, t,, r,(t)] is obtained from the hamiltonian h(r,( t), E). Clearly h will remain unchanged if we apply an arbitrary rotation to both the coordinates of relative motion r,(t) and the coordinates describing the orientation of the target. Thus rotating the trajectory by R is equivalent to rotating the intrinsic state of the target by R-‘. A similar argument can be used for T to give U&,0&)]
= U,[t,t’,RTr,(t)] =RU,,[t-qt’-~,r,(t)]R-‘.
The operator
(25)
R( a, /I, y) is given by R = exp( - iyJ,/Fz)exp(
- i&/A)exp(
- iaJ,/A)
,
(26)
where J,, Jy, J, are the angular momentum operators of the intrinsic state. Eqs. (25) and (26) lead to a number of simple results. In eq. (10) the influence the potential V is negligible at to and t,. Provided Q-is not too large
T,,]r,(t)]
Uo[to,t,--,r,(t)]
=exp(-i&r/A),
Uo[t,-T,ti,rs(t)]
=exp(i%/tZ),
=exp{i(~~--~)~/~}(~lRUg[t~,t~,rs(t)]R~’I~).
The T-dependence is isolated in the exponential to r can be calculated. The first derivative is
aL - ar
iAE
AT,,,
factor and derivatives
of
(27) with respect
(28)
where AE = E, - Ed. The excitation energy is - AE and the change in the energy of relative motion is AE. Derivatives of U, along the standard orbit (7 = 0, (Y= /I = y = 0) can be obtained from (25) and (26)
Derivatives
~U&,t~)=
~{h(t)L/,[t,t’,r,(t)l-L:[t,t’,r,(t)]h(t’)}, (29)
&v,(W)=
;[v,[t.t’,r,(t)],
Jx].
of V can be found in a similar
(30)
way, (31)
There
are expressions
for derivatives
with respect
to p and y in terms of commuta-
342
F. D. dos Aides et al. / Corrections
tom with J,, and J, which are similar to (30) and (31). When t and P are outside the interaction region h(t) = h(f) = Ho and (29) is equivalent to (28). These results can be used to simplify eqs. (20) and (21) for TI and T’. The results hold for any choice of the central potential Y&r). We write (20) as a sum of five terms, Tl,,=A,+A,+A3+A4+A5,
(32) (33)
(34) (35)
(36) A,=
-48
I
+D(LY)-$ Tpv.
(37)
The term A, can be evaluated by using (28) and noting that 11(7, E&I = - 1, (38) By using (28) and (A.22) A, can be written as (39) for the case where V,(r) is a general central potential and as AZ= - k5.Y 0
if it is a Coulomb potential. By using (A.23) and (A.28) A, can be written as A,=-aifi
(40)
Eq. (36) can be simplified by using (28) and (A-26) and A, becomes
(41) A 4 only gives corrections to the phase of the propagator and does not affect the cross
343
F. D. dos A idos et al. f Cwrections
section. Hence we do not need to evaluate it. The first term of A, is divergent as r1 increases as discussed in ref. *). Terms of this type should be summed to all orders in l/n but only modify the overall phase. The remaining term A, can be written in terms of commutators by using the equations for the derivatives of U, with respect to (Y,j3, y (eq. (30), etc.), A5=&(
-~~(~,~)(~I~U~,J,II~)A~+~(~,~)(ILI[[L~O,J~I,J~~~~) Iv) 1-~(Y~Ykmv_?1~41
+NPJ%PI[[I/,YJ,]?~]
lb)) * (42)
In the next section we show that A, does not cont~bute to the cross section although it does affect the polarisation. To simplify eq. (21) we use the formula
&Uo(t,t’)=-$dt”U&,t”)~ which can be derived by perturbation (22) as
av( P) dc
vow,
t’>
(43)
theory. This shows us to rewrite UC,, in eq.
(45) Using these formulae and eqs. (29) and (30) for the derivatives aU,/aC, write T’ as T,: = t, -t t,,
we can (46)
in which t, =
(PI v,lv) 3
tb=(~IUbtv),
u, = -;~t’dtUo(t~,t)h’(t)Uo(t,t,), 20
(47) (48) (4%
We have used the notation (A, B} = AB + BA for the anticommutator of the operators A and B. In (51) U, = Uo{tl, to). To illustrate this reduction we consider
F. D. dos Aidos et al. / Corrections
344
one matrix
element
in (21) and simplify
it by using (45) and (29)
(52) The second term in (52) contributes to the first term in eq. (50) for h’(t). The first term in (52) can be simplified by noting that h(t,) = Ho because t, is outside the interaction region. Using (43) with C = E, it reduces to -iAEJpI
au,Iv)
a~,
JT = -ihe,*
(53)
and contributes to the first term in (51) for U,. The other matrix elements in (21) can be calculated in a similar way. The contribution of t, to Tp’,,corresponds to the first-order perturbation of the transition amplitude by a perturbing potential h’(t). t, can be expressed in terms of the matrix elements T,,[r,( t)] or their derivatives with respect to E, and (3,. 4. Calculation of the cross section The results obtained in sect. 3 can be used to find expressions for the first-order corrections to the excitation cross section. For the case of rotational excitation the appropriate states are (PI = with energies the excitation summed
(JW 9
I’) = IJiM,)
E and .si, respectively. The differential cross section corresponding to of the target from level Ji to level J is given by (cf. eq. (12) where we
over all initial
and final M-values) (54)
where
u. and u1 are the relative velocities POJ(EO? 80) = c
in the entrance
and exit channels,
ItJM(~o? 00) I23
(55)
MM,
and is the transition amplitude evaluated along the standard orbit with energy E, and scattering angle 0, = s - 26,. In (54) duJd0 is the classical cross section for scattering by V,(r). If V,(r) is the monopole Coulomb potential then duJdS2 is the Rutherford cross section. AP, is given by AP,=
c 2Re[t,*,(A,+A,+A,+A,+A,+t,+t,)], MM,
where the A- and t-terms have been defined
in the previous
(57) section.
F. D. dos A ideset al. / Corrections
The contribution
of A, to the cross section
345
can be found
from eqs. (38) (54) and
(57) and we obtain P
-=Al POf From
If V,(r)
1 -aP,,
MM,
POJ
Coulomb
potential
This term is equal to the difference Rutherford cross sections
2
(58)
’
of A,
(39) we find for the contribution
is the monopole
AE
aE,
=G
then
PA2 -=--. POJ
AE
between
the symmetrised
(60)
EO
and the unsymmetrised
(61) (cf. eq. (2)). Usually the Coulomb excitation excitation probabilities PJ defined by
43
If V,(r)
is the monopole
the term A,
in eq. (40) to the cross section is
--L!Im
P,, - POJ2
(62)
of PJ using this formalism,
In the calculation
of A, defined
The contribution
values for the
J
da=U,7 as well as cross sections. should be left out.
codes provide
da~ym’,
u1
dUJ
computer
C MM
Coulomb
-!J-$+t&
tTM .[
potential,
P
0
i
(63)
then
a2 c0s2eo-+ as;
-=A3 POJ
1I tJM .
0
i1 t,,
cOteo$ 0
.
(64)
The derivatives of tJ,,,, in this term can be evaluated by numerical differentiation. A, is given by eq. (41). It will contribute to the phase of the transition amplitude but will not affect the cross sections. The contribution of A, to the cross section contains terms of the form zM { (JMIJk”uOJ,“IJiMi)(JiMil
QtlJM)
- (JMI
u~IJiM~)(J~M~IJ,“u~I;I,“IJM)}
3 (65)
where k stands
for x, y, .z and n, m can take the values 0,1,2. It is obvious
that the
F. D. dos A idos et al. / Corrections
346
commutator
Jk,
C
IJA~)(J~~I
M
I
= 0.
(66)
Therefore the contribution from A 5 vanishes. The argument over A4 and Mi, hence A, can affect polarisations.
depends
upon summing
The contribution from t, defined in eq. (48) consists of a sum of four terms. Using eqs. (A.25), (A.27) (A.28) and (A.30) they can be written as (67)
(68) C (JMI(au,/aeo).oJ,+J,(av,/aeo).oIJiMi)t~M
P P OJ
-= b3
MM,
-
>
P OJ
(69)
C (JMI[UO,Jy]Jz+Jz[Uo, Jy]lJiM,>t?~
P P OJ
-=b4
_ 2-1,
MMl
(70)
P OJ
The first term can be added to PAI defined ‘,I
+
in eq. (58) to yield
‘Al =- 1
POJ
aPOJ POJaE,&i.
(71)
If the energy &i of the initial state IJiMi) is zero the contributions from A, and tbl to the cross section cancel. The contribution from the three remaining terms of t, (expressions (68) to (70)) can be calculated by using the expressions (A.27) (A.28) and (A.30) as well as the commutator (66). For the case where the target is initially in a level with zero spin and zero excitation energy ( Ji = 0, q = 0) and V,(r) is the monopole Coulomb potential we obtain
+_d”)tJ,-,)t,*,~
P b4 P OJ
-=
---
I
P oJ
cot
4n
(73)
8, ;(j+(“)fJA4+l
+d”)tJM-,h,
(74)
341
F. D. dos Aides et al. / Corrections
where j+(M)=JJ(J+l)-M(M+l))
(75)
j_(M)=JJ(J+l)-M(M-1).
(76)
The derivatives in (72) and (73) can again be obtained by numerical differentiation. The term t, in eq. (47) corresponds to a first-order correction to the transition amplitude by a perturbing potential h’(t) given in eq. (50). It can be evaluated by modifying the interaction term in a Coulomb excitation code like COULEX, I’+,(t),
t) + V(‘&),
6) + ‘CA’(t).
Here K is a parameter and we need the first-order correction in K. This can be obtained by running the modified code for several values of K and evaluating the derivative of the amplitude at K = 0 numerically. The matrix elements of h’(t) required for the modification can be calculated from (50). Matrix elements of h(t) already exist in the code and matrix elements of J,, J, and J, are standard. The new matrix elements are those of hV’(t)/aE, and W(t)/M, on a standard Rutherford orbit. They are easy to calculate from formulae for V(t) given in ref. 4).
5. Interpretation of the correction terms In this expression theory of approach
section we (1) for the Coulomb the relative
discuss the interpretation of the correction terms (57) to the excitation cross section of a level J given by the Alder-Winther excitation with unsymmetrised orbits. In the Alder-Winther motion is described classically while the excitation process is
treated quantum-mechanically. Furthermore, this method assumes that the relative motion follows a Rutherford orbit, unperturbed by the coupling with the intrinsic degrees of freedom. The correction terms that were obtained in this paper describe the deviations
of the relative
motion
from a Rutherford
orbit due to the excitation
process and the influence of those deviations on the intrinsic motion. If we wish to follow the same philosophy as in the Alder-Winther theory, we should try to describe the correction terms as a modification of the orbit of relative motion yielding an excitation cross section
(77) where da,/dG is the classical cross section corresponding to the modified orbit rl(t) and P,[r,(t)] is given by expressions (55) and (56) where now r,(t) must be replaced by the modified orbit rl(t). An attempt to build a theory as outlined in eq. (77) was made by Pechukas12). The method that has emerged has the disadvantage of not preserving the superposition principle and is different from the method that we use in this paper. A comparison between the two methods can be found in ref. 13).
348
F. D. dos A idos et al. / Corrections
The formalism
that we use avoids the problems
at the cost of not allowing difficulties the variables
inherent
for such a straightforward
in the interpretation
arise because
that can be used to describe
to the Pechukas
interpretation.
of the noncommutativity
the intrinsic
method
Some of the of some of
state of the target as shall be
seen later. For the case where the target is excited from the initial level Ji = 0 with &i= 0, we find that the first-order correction to the excitation cross section of a level J can be interpreted as a change in the classical deflection function for the relative motion and a change in the coupling potential that causes the excitation. The change in the deflection function is due to an additional time-dependent potential V’( r, t) which is a Pechukas-like average of the coupling potential V( r, 5) suitably changed to account for the sum over the initial and final M-values, C CJMI U3Ct1Y‘jVCr? 04Ct2 V’(r,t)=Re
MM,
tO)IJiMi)fJ*M
(78)
P OJ
The change in the potential causing the excitation consists of three kinds of terms that must be added to the coupling potential V(r,( t), 5). Some of these terms describe the change in the relative kinetic energy due to V( r, 5); others correspond to an energy symmetrisation procedure similar to the one used by Alder and Winther; the remaining terms apply a similar method to the angular momentum and correspond to a symmetrisation of its modulus and z-component. If the initial intrinsic angular momentum Ji and energy &i are not zero, then there are extra terms which account for the change of energy and angular momentum transfer when a level J is excited. Our aim is then to write the excitation cross section of a level J as (79) where‘daJdS2 is the classical cross section corresponding to the modified deflection function and P,, is evaluated in a similar way to PO, but where an extra potential is added to I’( rs( t), 5). A Pi, is the additional term that accounts for the initial energy and angular momentum of the target. We shall proceed in two steps. First we evaluate the effect of the perturbing potential V’(r, t) defined in eq. (78) on the classical cross section for the relative motion and obtain an expression for da,/dL?. This expression will contain some of the correction terms (57). Then we interpret the remaining corrections in terms of P,, + A Pi,. The deflection function for classical scattering from the central potential V,(r) is (80)
4 = %,(b? GJ) assumed to be a monotonic function relation we obtain the function
of the impact
b = b&‘,,
E,).
parameter
b. Inverting
this
(81)
F. D. dos Aidos et ul. / Corrections
349
If the deflection function is changed by the coupling potential, then a particle that starts with energy E, and impact parameter b will be deflected by the angle &@, E,) = @,a@, E;,) +A&,
(82)
where A& describes the effect of the extra potential V’(r, 1). Inverting this relation we obtain, to first order b&i,
&) = b&r
-A&
&,I
where b,(@,, Eo) is the inverse deflection function for the potential V,(r) defined in (81) and to first order it is irrelevant whether the derivative is taken at 0, = OS.or 0, = 8,,. Expression (83) is the inverse deflection function for the perturbed case. Writing it in terms of the initial angular momentum Li of the orbit we get &(e,,, J$,) = L($,,
-&) +AL,
(84)
where
aL AL= - ae,, &A%. i
1
The classical cross section can be written as (87) where the perturbed deflection applicable
derivative is taken at constant initial momentum po. Although the potential &(P) + V’(r, t) is not spherically symmetric it leads to a function that is independent of the azimuthal angle and eq. (87) is to this case. Using expression (84) in eq. (87) we obtain, to first order,
where duJdS2 is the classical cross section corresponding to the unperturbed deflection function (80) and AL is given by eq. (86). An expression for A@, was derived in appendix B. Using eqs. (86) (B.lO) and (B.16) we find de,=
(~)f_~+(~)E,$-6e,
(89)
F. D. dos Aides et ul. / Corrections
350
with
6E,,86 and SL given by 6E,,=AE=q-E,
(91)
c ( atJM/aeO) Eof;M
MM,
POf
1
(92)
(93) The three terms in eqs. (89) and (90) describe the effects due to energy loss, angular momentum transfer and change of the effective spherically symmetric field. It is convenient to express the derivative in eq. (88) in terms of the parameters E,,7,(Y, p,y,8,defined in sect. 3. Using
aAL r=,
o
C=ff,P,y,r,
we obtain
(94) the derivative being taken for standard orbits (at constant E, and (Y= /3 = y = r = 0). Now we substitute expression (90) in (94) and evaluate the contribution of each one of the terms (91) (92) and (93) to the first-order corrections to the unperturbed cross section. The contribution from (91) is
(95) These terms are equal to part of the correction Pa2/PoJto the excitation cross section found in the previous section (eq. (59)). The remaining term of Pd2/PoJ is, to first order, equivalent to including a factor uo/ul in the excitation cross section that cancels the factor ul/uo in (54) [ref. ‘)I. The term (92) gives a contribution
(96)
to the cross section. (63).
The first term in (96) is equal to the correction
Pd3/PoI in eq.
351
F. D. dos A idos et ul. / Corrections
The contribution
from (93) is found
to be
where we have used the relation
which can be proved by using the commutator
(66). The first two terms of (97) have
a similar structure to P&P,, and P,,/P,, defined in eqs. (69) and (70). P,,( E,, 0,) is the excitation probability when the relative motion follows an unperturbed orbit rS( t) with energy E, and scattering angle 8, = v - 20,. The terms in (96) and (97) that depend on the derivative aP,,J,/&?o correspond to changing the scattering angle of the orbit along which we evaluate PoJ. In order to obtain a change de, given by eq. (89) we add and subtract the term
to duJdS2 in eq. (94). Using expressions (94)-(97) for AP,
t,,,,,( esot E,) state
as well as the results obtained
we can write the excitation
IJM)
is the transition corresponding
amplitud’e
from the initial
state
corrections
IJiMi)
section (54) as
to the final
to the hamiltonian
f&l(~) + J+&>, E)+ h’(t) orbit with energy where r,(t) is the unperturbed h’(t) is defined in eq. (50). BtiJM is given by StiJM =
in the previous
cross section with first-order
7
E, and scattering
angle
t9,, and
(9%
F. D. dos Aides et al. / Corrections
352
where we have used eq. (B.ll). Eq. (98) has the same form as eq. (79). We have thus shown that some of the correction terms account for a change in the relative motion due to the extra potential
V’(r, t) defined
in eq. (78).
In order to interpret the remaining terms we first look at the case where ei = Ji = 0, for which 8tiJ,,, vanishes. From eq. (98) we see that the effect of the change in relative motion is not taken into account by changing the trajectory in eqs. (55) and (56) as would be the case in the Pechukas treatment. On the contrary, it is necessary to subtract the effect caused by the perturbing potential V’(r, t) on the orbit with energy E, and scattering angle and evaluate t,,, along the unperturbed scattering angle 19~~.The effect of the deviation of the orbit on the excitation of the target is described by the additional term h’(t) in the hamiltonian (cf. eq. (50)). In order to understand h’(t)
the effect of this extra potential
we write the matrix
element
of
as
where we have used one of the relations
(B.ll)
and
The first term in expression (100) describes the decrease in relative kinetic energy due to the additional coupling potential V(r, t) which is neglected when the relative motion is described by the unperturbed orbit r,(t). The second term is similar to the Alder-Winther energy-symmetrisation procedure but corresponds to keeping the angular momentum fixed instead of the scattering angle. This can be seen by adding this term to the matrix element of the coupling potential V(t, E,) = V(r,(t, E,), E) to obtain
” (xl v(t>4,) - a~,
q,[,
=(x,V(t,
E,++.
If we had chosen the basis states lx) and 15) to be eigenstates terms in eq. (100) would have had the form
001) of J, the next two
(102) The angular momentum of the relative motion L points towards the negative side of the x-axis and this term corresponds to a symmetrisation of the modulus of L in a similar way to the energy symmetrisation in eq. (101). The final term in eq. (100) has a similar meaning. It corresponds to a symmetrisation of the z-component of the
F. D. dos A idos et a/.
angular excitation
momentum
and
describes
the effect
Corrections
353
of the tilting
of the target. In this last term the tilting
of the orbit
is done through
This term comes from eqs. (20) and (21) where the derivatives
a change
on the in /3.
are taken with respect
to p and y. It can easily be shown by reversing the roles of j3 and y on the subsequent treatment leading to expression (100) that the last term of this expression can also be written as
where again we used eqs. (B.ll). This shows that the effect of the tilting of the orbit can equally well be described by a symmetrisation of L, due to a change of fi as by a symmetrisation of Ly due to a change of y. However, as the basis states Ix) and I{) cannot be simultaneous eigenstates of J, and J, or of JY and J,, these terms cannot be incorporated in V(r,(t), 5) through a change in the angular momentum of the orbit r,(t) as was done for the energy in eq. (101). If the initial level does not have zero spin and energy then 8tiJM, which was defined in eq. (99), does not vanish. The first term of 8tiJM corresponds to substituting the excitation energies ex and eS in the matrix element of h’(t) (eq. (100)) by ex - .si and sS - .si. This is in agreement with the energy-symmetrisation procedure as now the difference between the energy of relative motion in channel x and in the entrance channel is - (ex - ei) instead of - eX. If the initial state is an eigenstate IJiM,,) of J, then the second term has a similar interpretation. It corresponds to substituting in eq. (102) MXx and MXs by MXX- MXi and MXr - MXi. The same applies for the last term of StiJM if the initial state is an eigenstate of J,. It should be noted that the formulae presented in this section were not used in practical calculations. They are useful only insofar as they provide some insight into the interpretation of the corrections.
6. Numerical Three
sets of numerical
results
tests of the theory developed
in sects. 3 and 4 have been
made. The first was designed to test the correctness of the formulae and the reliability of the numerical methods, while the others test the accuracy of the expansion up to first order in l/n used in this paper. The calculations corresponding to the first two sets of results were made for the scattering of 160 by “‘Sm at E,, = 109.2 MeV. The target nucleus is assumed to be in its ground state J = M = 0 and the excited states are those of an axially symmetric rotor with an infinite moment of inertia. This is the sudden case with all excitation energies Ed = 0. The values of the relevant parameters are n =
29.90,
5=0,
q =
2.990 >
A = q/n
=
0.100.
The third set of results corresponds to the non-sudden case where the excitation energies of the target are finite. The calculations were made for the scattering of ‘Be
354 by
F. D. dos A ides et al. / Corrections
lS2Sm at E,,
= 30 MeV. The excitation
energies
.sJ of 15*Sm were taken to be
Q=OM~V,
~~ =
0.122
MeV,
Em= 0.361 MeV,
.s6= 0.705 MeV,
Ed =
1.122 MeV,
q0 = 1.539 MeV.
The fact that these energies do not correspond exactly to a rotational spectrum is of no consequences. As we have already mentioned the formulae obtained in sect. 4 are valid for any form of the intrinsic hamiltonian He([). The relevant parameters for this case are equal to n = 20.17,
5 = 0.043)
q = 1.212,
A = 0.060.
These values were chosen because the parameters A and [ are large enough for there to be significant corrections to the Alder-Winther theory and because some coupled-channels results calculated with the code AROSA were available. The tables show the values for excitation probabilities PJ and deviations 8,. The excitation probabilities PJ are defined by da 3 dafYrn) ‘J, d0 - ua d0 J
(103)
TABLE 2
Zero- and first-order excitation probabilities J 180~/178~
1500
Pib)
PI’)
Pid)
0 2 4 6 8 10
8.320 3.138 4.174 1.552 2.738 2.852
-2 -1 -1 -1 -2 -3
8.318 3.141 4.173 1.551 2.734 2.847
-2 -1 -1 -1 -2 -3
6.334 3.475 4.213 1.426 2.292 2.168
-2 -1 -1 -1 -2 -3
6.333 3.477 4.213 1.425 2.289 2.163
-2 -1 -1 -1 -2 -3
0
8.196 3.819 3.935 1.223 1.850 1.671
-2 -1 -1 -1 -2 -3
8.196 3.819 3.935 1.224 1.851 1.671
-2 -1 -1 -1 -2 -3
6.544 4.141 3.918 1.118 1.551 1.280
-2 - 1 -1 -1 -2 -3
6.545 4.141 3.918 1.118 1.552 1.281
-2 -1 -1 -1 -2 -3
1.316 5.300 2.807 5.238 5.023 2.951
-1 -1 -1 -2 -3 -4
1.316 5.300 2.807 5.240 5.025 2.952
-1 -1 -1 -2 -3 -4
1.266 5.488 2.725 4.762 4.270 2.340
-1 - 1 - 1 -2 -3 -4
1.265 5.488 2.725 4.763 4.272 2.341
-1 -1 -1 -2 -3 -4
2 4 6 8 10 1200
Pea)
0
2 4 6 8 10
“) PO is the lowest-order excitation probability from ref. I’). ‘) Ph is the lowest-order excitation probability from COULEX. ‘) PI is the first-order excitation probability from ref. lo). d, P; is the first-order excitation probability from eqs. (54) and (103). P{ cannot be calculated for 8, = 180° and so we used b’, = 178O in P,‘, Pi.
F.D. dos Aides et ul. / Corrections
355
where du,/dti is the excitation cross section calculated from COULEX, from eq. (54) or from AROSA, and da, @Ym)/ds2is the energy-symmetrised Rutherford cross section defined in eq. (2). The S, are the deviations of the excitation probabilities from the AROSA results PAJ and are defined by S,=lOO
4 - PA.7
P
004)
AJ
We note that the expressions for the corrections to the cross section obtained in this paper are not applicable for backward scattering. For this case we can either evaluate the analytic limit of those ex$ressions when 6, + 180° or make the calculation for a value of the scattering angle close to 180°. We follow the second method and use the value 0, = 178O. In the sudden case the corrections contained in eq. (54) can be calculated by an analytical method which is different from the one used in this paper [dos Aidos and Brink”)]. Note that for the sudden case du~Y*~/d~ = do,/dS2 and ul/uO= 1. Table 2 compares results obtained by using the numerical methods of sect. 4 with TABLE 3
Comparison
0s 180”/178”
1500
120°
of first-order
corrections
with AROSA “): sudden case
.J
AROSA
&lb)
6,‘)
0 2 4 6 8 10
6.508 3.460 4.204 1.429 2.283 2.%
-2 - 1 -1 - 1 -2 -3
28 -9 -0.7 9 19 36
-3
0 2 4 6 8 10
6.720 4.135 3.916 1.120 1.547 1.364
-2 -1 -1 - 1 -2 -3
22 -8 0.5 9 19 34
-3
0 2 4 6 8 10
1.280 5.443 2.715 4.742 4.206 2.349
-1 ‘--1 -1 -2 - 3 -4
3
-1
-3 3 10 19 32
0.5 0.2 0.3 -0.3 2
0.2 0.04 -0.2 -0.1 2
0.8 0.4 0.4 1 4
“) AROSA is the exact excitation probability. h, So is the deviation of the COULFX results for the excitation probability from the AROSA results in percent. ‘) 6, is the deviation of the results for the excitation probability, eqs. (54) and (103). from the AROSA results in percent. 6, cannot be calculated for 8, = lSO” and so we used @,= 178O.
356
F. D. dos Aides et al. / Corrections
those from the method of ref. lo). In this table PO is the zero-order value of PJ from ref. lo) while PO’is obtained from COULEX. The results should be identical and they are except for a small difference in the last figure. The columns P, and Pi show the corresponding zero- plus first-order values of PJ (P, from ref. lo) and P; from eq. (54)). Again the results should be identical because they correspond to different ways of calculating the same correction. Again the numbers are the same except for a small difference in the last figure. This is a check of the numerical methods used to evaluate the formulae of sect. 4 of this paper. The difference between P, and PO is the correction to the Alder-Winther theory calculated in this paper. In some cases it is more than 20% of the zero-order value for the parameters used in this case. Table 3 compares the values of PJ obtained from the method developed in the present paper with those calculated with the coupled-channels code AROSA for the same values of the parameters as in table 2. In these calculations only levels up to J = 10 were included in order to have the same cut-off as the one used in AROSA.
TABLET Comparison
of first-order
corrections
with AROSAa):
non-sudden
case
180°/1780
0 2 4 6 8 10
6.098 3.578 3.197 1.037 1.574 1.399
-1 -1 -2 -3 -5 -7
-0.9 0.2 8 25 59 124
-0.6 0.2 5 15 32 59
0.2 -0.2 - 0.3 0.9 -8 -33
150”
0 2 4 6 8 10
6.457 3.282 2.584 7.525 1.048 8.721
- 1 -1 -2 -4 -5 -8
-0.8 0.5 8 25 59 123
-0.5 0.4 6 15 32 59
0.1 -0.2 - 0.1 - 0.06 -4 -24
1200
0 2 4 6 8 10
7.423 2.444 1.304 2.706 2.819 1.810
- 1 - 1 -2 -4 -6 -8
-0.5 0.9 8 24 57 119
-0.3 0.7 5 15 32 59
0.07 -0.2 0.07 0.6 -2 -18
“) AROSA is the exact excitation probability. ‘) S, is the deviation of the COULRX results for the excitation probability with unsymmetrised orbits from the AROSA results in percent. ‘) S, is the deviation of the COULEX results for the excitation probability with energy-symmetrised orbits from the AROSA results in percent. d, 6, is the deviation of the results for the excitation probability, eqs. (54) and (103) from the AROSA results in percent. 6, cannot be calculated for 6’, = 180° and so we used 0, = 178’.
F. D. dos Aidos et al. / Corrections
We show in this table the deviations (eq. (104)) 6, for method developed in this paper. The agreement between (54) and the AROSA coupled-channels results is good. the AROSA values is 4% in the case of the J = 10 state
351
COULEX and 6, for the the results provided by eq. The largest deviation from at 0, = 120”. In most cases
the agreement is better than 1%. The results provided by the Alder-Winther theory are less accurate in almost all cases. The average error is about 10% and in several cases it is more than 30%. Table 4 compares the values of the excitation probabilities PJ for a case where the excitation energies are not all equal to zero and where the corrections in E/n and [q/n are important. In this table we show the deviations 6, for COULEX without the energy-symmetrisation procedure, 8, for COULEX with the energy-symmetrisation procedure and 6, for the results provided by eq. (54). In general the energy symmetrisation on its own corresponds to an improvement over the unsymmetrised results, especially for the higher states. The results S, that correspond to the method developed in this paper show a more drastic improvement. For the states up to J = 6 the deviation 8, is always under 1%. For the higher states the deviations 8, are still smaller than S, or 8, although the agreement is not as good as for the other states. The reason for such large deviations may be the smallness of the values involved that prevents rigorous calculations due to the limited precision of the computers, or it may indicate that the energy of the state is becoming too high and the higher-order terms in t/n can no longer be neglected. These comparisons suggest that eq. (54) gives a significant improvement over cross sections calculated by the Alder-Winther method. The authors would like to thank Prof. J. de Boer for his interest and for providing some results calculated from the coupled-channels programme AROSA. C.V.S. would like to acknowledge support from the SERC and the Hahn-Meitner Institute. F.D.d.A. would like to acknowledge support from Comissao Permanente INVOTAN and from the Danish
Ministry
of Education.
Appendix A The purpose
of this appendix
is to derive expressions
for the quantities
(A.1)
defined
in eqs.
(16) and
(17) where
f and
g are
the
orbit
parameters
Cj =
358 {
F. D. dos Aidos ef al. / Correciions
E,, 7, So, a, j?, y
}.
In these expressions
Aij is the inverse
‘ii = arli
aPoj
of the matrix
(A.3)
9
where S,,(r,t,, pot,) is the classical action for an orbit with initial momentum p,, and final position ri. With this definition the expressions (A.l) and (A.2) are independent of the coordinate system used to specify r,, po. It is convenient to use polar coordinates for both vectors, r, = (r,, 0, @) 7 The action
S, can be expressed
Po=(PoJM.
simply in polar coordinates
64.4) as
where L is the angular momentum, r,,, the distance of closest approach, initial position at time t,. p, is the radial momentum p,(r) and 6, is related As a first step defining the slope by relating the
= {2m(E,-
Y(r))
-
L2/r2}l'*,
and r, the
(A.6)
to the scattering angle 19,by 8, = $(v - 8,). in evaluating (A.l) and (A.2) we express the angles e,, (Y,j?, y and orientation of an orbit in terms of 8, c$, 5, 4. This can be done incident direction (5, +!J) and the outgoing direction (8, +) to
t$,, (Y,/3, y. We have c~s e = cos( e, -
&OS
p,
(A-7)
sin~cos~=cos(f3,-cu)sinj3cosy--sin(8,-cu)siny, cos l= sin
useful
relation
- c~~(
e, + +OS
(A.@
p ,
(A.9)
-cos(e,+ar)sinpcosy-sin(8,+cu)siny.
(A.lO)
is
cos(?r-2e,)=cos8cos~+sinesin~cos(~-~).
(A.ll)
We need derivatives of r3,, (Y,fl, y with respect to 6, {, $, 4 for a standard orbit (fig. 1). A standard orbit has (Y= p = y = 0 and 8 = e,, {= P - 8,, 4 = + = $T. These can be obtained by differentiating (A.7)-(A.11). The non-vanishing first derivatives are
aeo
ae,_l -7, a
al
ap
q=-s=*tanea,
aa
aa
-...-=_L
ae
-=_1 ad
23
au-
a+-q=f.
2,
ay
x=-t,
(A.12)
F. D. dos A idos et al. / Corrections
359
We also need the mixed second derivatives
with f = e,, a, p, y. Only one of these is non-zero,
a=e, = ftane,.
(A.13)
a+a+
The non-zero
matrix
elements
of Sij (A.3) in polar coordinates
for a standard
orbit
are
(s'P s,, S*$’ SBP ,s
OP
%
se+
s,,
s,,,
=
i
s,,
0
0
s2,
s,,
0
0
0
s3,
I
The matrix elements can be calculated from (A.5) by using (A.12) and (A.13). It can be shown from (A.5) that ( aS/&90)p = 2L. Hence s,, =
S,, = P/G,)?
s,, = - f aL/ae, In (A.14)
o is the asymptotic
t, -+ 00. For a Rutherford
orbit
velocity
aLlap
S,, = fL tan
, and
u(ti)
e, .
is the velocity
L = n tan e,, where n is the Sommerfeld
,
S,, = -qant?,
(A.14) at t,;
S,, -+ 1 as
parameter
c02e,,
s,, = - hn/2
S,, = +An tan28,.
The non-zero
matrix
elements
Ai, = &VP, A,,=~/s,,=
and
(A.15)
of the inverse
matrix
A,, = - S2,/S2,
A,, are = - 2( aeO/aPo)
L2
A,, = 2/L tan e, .
-2wom&,,
(~.16)
The remaining ingredients needed for evaluating (A.l) and (A.2) are the expressions for E, and r in terms of (r, 8, c#J),(p, l, 4). E, is the total energy and r is the time at which the particle reaches the point of closest approach. They are given by & =po2/2m
+ KMtcJ)
(A.17)
3
7 = t, - t,, ) where t,, is the time to pass from the distance ts1 =
It is a function the dependence
(A.18) of closest approach
r,
to ri,
s
rldr/F. rrn
of ri, E, and t9,. The dependence of r on 8, +, {,$ is contained on 6,. Neglecting the term V,(r,( to)) which goes to zero
in as
F. D. dos Aidm et ul. / Corrections
360
1, + - co, the only non-zero first derivatives of E, and r are ~E~/~P~ = 0, a7/&,
+wh
at~~/aE~)~~,
aqae = - aqac = - e( at,,/ae,) E,,
= -l/U(t,),
a27 dr,ap,=zqp
= -4
1 a?,,
a% m=--
i
4 a@
-
’
a2T
at,,
= - +tan9,m.
aQa+
0
(A.19) Terms of order l/r, have been neglected except in the case of u(tJ in &r/&-r. This term is multiplied by rt in a correction to the phase of the propagator. The derivatives of t,, can be evaluated easily by using the equations of a Rutherford orbit. They are
at~~/ae~= - (~~/v)t~e~ f a%,/ae,” = - ( aC/u)sec2fJo, atsl/aEo = - &
( rl 4 3a,ln(2r,cos
$/u,)
- 5ac).
In the general case the derivatives of t,, with respect to 8, can be written in terms of derivatives of L by using the identity ( at,,/ae,)
E, = ( aL/aE,)
(A.20)
e,.
To prove this we define s(r,,
&,, 0 = j%,(r) G?
dr
where p, is the radial momentum (eq. (A.@). It is easy to see that t,, =
(as/aq
L,
do = - ( as/aL)
Using (A.21) and a well-known transformation
Eo.
(A.21)
of partial derivatives,
Collecting these results we can evaluate D2(C,) and D(C,, C,), where C, are the orbit parameters E,, r, @,,(Y,j3, y. The non-zero results for D2(C,) are
1
(A.22)
=2E,
(A.23)
D*(e,) = 112~ for a general central potential. For a Coulomb potential they reduce to 02(T) = l/E,)
02(eo) = (i/2An)cot
e,.
(A.24)
361
F. D. dos Aides et 01. / Corrections
The expressions for the non-zero D(C,, CO) in the notation of table 1 are D(r,Ea)=
(A-25)
-1,
(A.26)
= - i( ~~O/dE,)L= c=D@,,~,)=
-L>(~,~)=D(~~,c+=
= +( i)eojaL)Eo=
(A.27)
-(1/4EO)sin8,cos8,, -D@,e,)
(A.28)
(1/2nh)cos2eo,
D=D(/3,/3)=
-tan8,/2L=
E=D(~,Y)=
-~(~,j3>=
F=D(y,y)=
1/2Ltan&=
(A.29)
-l/2&, -1/2~=
-~i/2~~)cote~,
(i/2nfi)c0t2eo.
(A.30) (A.31)
Here two expressions are given for each term. The first corresponds to a general central potential and the second to a Coulomb potential. The function A = D( 7,7) is not needed for the calculations in this paper. it contributes to corrections to the phase of the scattering amplitude but not to corrections to the cross section.
Appendix B
In this appendix we find an expression for the change in the scattering angle f?, caused by the perturbing potential V’(r, t) given by eq. (78). The unperturbed potential I$(P) is assumed to have spherical symmetry and a monotonic deflection function. The unperturbed orbit rO(t), when specified on the standard axes defined in fig. 1, lies on the yz plane. The orbit rl(t) that corresponds to scattering off the potential G(r)
+ I+,
t)
(B-1)
with the same initial conditions as r,(t) will not necessarily lie on the same plane. Its scattering direction will be (8, + be, $r + A+), where de and Acp describe the effect of the extra coupling potential V’(r, t). To first order its scattering angle will be given by e,, = e,, + de,, de,= -de,
03.2) (B.3)
where 0, is the scattering angle of the unperturbed orbit. A@can be evaluated by using standard classical perturbation theory methods. We choose a set of six parameters C,( r, p) that are constants of the unperturbed motion C,(rO(t),pO(t))
= const
(B.4)
F. D. dos Aides er al. / Corrections
362
and use them to specify the orbit r,,(t) = ro( C,, t). When C, are evaluated orbit
rr( t) they will no longer be constant
but will vary according
along the
to the equation
14)
(B.5) where { C,, Cs>PB are the Poisson brackets
of C, and Cp. To first order we can substitute all the parameters C, on the right-hand side of eq. (B.5) by their unperturbed constant values. The first-order change on the value of C, after the scattering due to the additional potential V’( r, t) can be obtained by integrating eq.
(B.5)
where C,, and CpO are the initial values of the parameters C, and CD. I is the integral of the perturbation along the unperturbed orbit. Any six independent constants of the unperturbed motion can be chosen for C,. The Poisson brackets in eq. (B.6) are easily evaluated if the parameters C, are taken to be the position r, 0, + and conjugate momenta p,, pe, p+, of the unperturbed orbit at a specified time t,. As we are interested in obtaining the direction of scattering we choose t, to be very large. Eq. (B.6) for C, = O(t,) becomes
03.8) In (B.8) constant,
the derivative the scattering
should be taken while keeping the energy of the orbit direction fixed and the orbit r,(t) in the yz plane. It is
convenient to express the derivative in terms of the parameters E,, e,, 7, (Y,p, y that were defined in sect. 3. Expressions that relate these parameters to r, 8, +,, p,, pe, pg can be easily obtained and we find, for t, large enough,
(B.9) 0,( L, E,) is related
to the deflection
function
of the unperturbed
e&c, 4,) = t( -77- e,,(L, 4,)). Other
useful relations
motion
by (B.10)
are (B.ll)
F. D. dos A idos et al. / Corrections
363
which can be obtained by noting that a=y=Q*sin/3=L,/L, -L,/L.
a=p=O*siny= Using (B.7) and (B.9) in (B.8) we obtain
(B.12) where, using the expression (78) for V’(r, t), (B.13)
(B.14) c
(atJM/a@o)E"GM
MM, POJ
(B.15)
From eq. (B.3) we can write the first-order correction to the scattering angle as (B.I~) where we have used relation (10). AS, is a sum of three terms. The first term describes the effect of the energy loss on the scattering direction. The second describes the effect of angular momentum transfer. In (B.14) the projectors on levels J and Ji can be expressed in terms of eigenstates IJ, MX) of J, instead of J, and we obtain SL=
_
t:
(“xi-Mx)ltJMx12
MxM.d
p
(B.17) OJ
6L
is, up to a sign, the change of polarisation along the x-axis. The negative sign is due to the fact that in the standard axes the relative angular momentum points towards the negative side of the x-axis. The third term of (B.16) is independent both of the energy and of the angular momentum transfer. It describes the effect of the change of the effective spherically-symmetric field due to the perturbation. References 1) F. RKsel, J.X. Saladin and K. Alder, Comp. Phys. Comm. 8 (1974) 35 2) 3) 4) 5) 6)
L.D. Tolsma, Phys. Rev. CZO (1979) 592 M.J. Rhoades-Brown, M.H. Macfarlane and S.C. Pieper, Phys. Rev. C21 (1980) 2417, 2436 K. Alder and A. Winther, Coulomb excitation (Academic, New York, 1966) A. Winther and J. de Boer, Caltech technical report (Nov. 1965), reprinted in ref.4) J. de Boer, G. Dannhguser, H. Massmann, F. Riisel and A. Winther, J. of Phys. G3 (1977) 889
364
F. D. dos Aidos et al. / Corrections
7) J. de Boer and G. Dannhluser, Nuclear physics ed. C.H. Dasso, R.A. Broglia and A. Winther (North-Holland, Amsterdam, 1982) p. 451 8) C.V. Sukumar and D.M. Brink, Nucl. Phys. A404 (1983) 121 9) M.W. Guidry, R. Donangelo, J.O. Rasmussen and J.P. Boisson, Nucl. Phys. A295 (1978) 482 10) F.D. dos Aidos and D.M. Brink, to be published 11) F.D. dos Aidos and D.M. Brink, J. of Phys. Gll (1985) 249 12) P. Pechukas, Phys. Rev. 181 (1969) 174 13) C.V. Sukumar, J. of Phys. GlO (1984) 81 14) H. Goldstein, Classical mechanics, 2nd ed., (Addison-Wesley, Reading, Mass., 1980)