ANNALS
OF PHYSICS
148, 327-345 (!983)
Symmetries
in Inelastic
Proton-Nucleus
Scattering
G~~RAN FALDT The GustafWerner Institute. Bo.u 531. S-751 21 Uppsala, Sweden Received
Various discussed. amplitude. not equal.
symmetries They are In particular,
November
30. 1982
of inelastic proton-nucleus scattering within the Glauber model are obtained from symmetries of the corresponding proton-nucleon it is shown that the polarization and analyzing power are in general
1. INTR~DLJCTION Symmetries in inelastic proton-nucleus scattering have recently become a topic of much interest. This event was triggered by the work of Bleszynski et al. [ 11, which discussesconsequencesof parity and time reversal invariance in conjunction with the adiabatic approximation. In particular, it was claimed that time reversal invariance implies the equality of the polarization and analyzing power for the Ot + 1+ transition in carbon. This equality was later claimed to be proven also by Amado [2 ], using completely different methods. If true these results would have far-reaching consequences.They would mean that the adiabatic approximation in fact restricts the number of dynamical degrees of freedom. Such a consequenceis exceedingly unlikely and we shall demonstrate that already in second-order multiple scattering we do encounter terms violating the equality A = P. Nevertheless, the above-mentioned studies have inspired us to look more closely into the question of symmetries in nuclear scattering. Our investigation is performed within the Glauber model [3]. Starting from symmetry properties of the nucleonnucleon amplitude we derive the corresponding symmetry properties of the nucleonnucleus amplitude. The detailed derivations demonstrate that even in the adiabatic approximation we do not expect additional restrictions on the nuclear amplitude. Apart from the pedagogical value of such derivations, they serve to verify that the nuclear model amplitude does not violate elementary symmetry requirements. When limiting the number of invariant amplitudes in the nucleon-nucleon amplitude one does obtain additional symmetry relations for the nuclear amplitude. They are often present only when the nuclear states considered are simultaneous eigenstatesof the total nuclear orbital angular momentum L, and the total nuclear spin angular momentum S,. In many applications of interest this is indeed the case. Our definition of the Glauber amplitude follows that of 141.Other definitions have 327 0003-4916/83x$7.50 Copyright 0 1983 by Academic Press, Inc. All rights of reproduction in any form reserved.
328
C&RAN
FjiLDT
been proposed. They lead to contradictions with conventional DWBA and are unacceptable. We shall demonstrate how such models can give rise to additional selection rules, besides giving different numerical values for the amplitudes themselves. Finally, as a warning, we remark that the explicit structure of the symmetry relations derived in many cases depend on the choice of coordinate system. Also, we would like to emphasize that our investigation does not incorporate charge exchange or exchange terms due to the identity of the projectile and target nucleons.
2. THE NUCLEON-NUCLEON
AMPLITUDE
We first review the symmetries of the nucleon-nucleon amplitude. The most general rotationally invariant amplitude, conforming to parity and time reversal invariance, is
f(k’, k 01,d =fh
sol, 4
= A(q) + B(q) 01 . l-k, . ii + C(q)@, + 02) ’ ii +D(q)a,.ijo,.~+E(q)a,.Ko,.K
(1)
with K = +(k + k’),
(24
q = k - k’,
(2b)
ii=qxii.
(2c)
For the moment it is not necessary to specify whether this amplitude refers to the c.m. system or any other reference system. The actual values of the invariant amplitudes, though, depend on the reference system. The specific form of the symmetry relations to be discussed below depends on the choice of coordinate axes. We shall (except for Section 5) employ the coordinate system defined in Fig. 1 with axes (2, y, Z) = (ii, 4, ii). Invariance under parity implies f(K, 97 01, 02) =f(-K,
-q, 019 02).
(3)
Parity alone does not determine the structure of the amplitude J In addition to the invariant amplitudes of Eq. (l), bilinear combinations proportional to o1 . ija, . K and o1 . K o2 . q are also permitted. Invariance under time reversal implies that f(K, %~IV ‘Jz) = [f(--K, Q, -fJ:, -@)jT, where T denotes transposition
(4)
(with respect to both spin matrices) and the star
SYMMETRIES
IN
Impact
FIG.
1.
PROTON
329
SCATTERING
plane
Definition
of coordinate
axes.
complex conjugation. When the amplitude f is expressed as a linear form in CI, and uz, the time reversal condition reads
Time reversal alone does not determine the structure of the amplitude$ In addition to the invariant amplitudes of Eq. (I), linear terms proportional to o, . K and u2 . I? are also permitted. It is customary to decompose the time reversal operator into a product R = WT,; where To is complex conjugation and W a linear, unitary operator with properties wuw-’
=-u,
(6)
W]S,M)=(-l)~+“/S,-M).
(7)
The second relation is valid, not only for spin l/2 states, but for any spin state of total spin S and z-component M. We shall also consider another class of symmetries which we shall call reciprocity, in analogy with the symmetry discussed by Glauber [3]. There are three related versions of this symmetry. Reciprocity
I
f(‘G %u, 302)= u; ‘f(-Kv %UI, $) UK, U, = U&u, Reciprocity
+ uJ) = exp[iti
Pa) . +(a, + a,)].
WI
II fk
%u, 902) = u;‘f(K, UC7= u7Gh
-q, u,, uz) U,,
(94
+ 4).
(9b)
330 Reciprocity
&RAN
FiiLDT
III
Each of them leads to the same restriction on the nucleon-nucleon amplitude as parity. Therefore, it is natural to expect that also for the nucleon-nucleus amplitude they do lead to the same restriction. We have found them useful when used in conjunction with time reversal.
3. THE NUCLEAR AMPLITUDE We define the nuclear profile function as in Ref. 141. A typical Nth-order multiple scattering contribution is of the form (we neglect possible multiplicative factors)
where the Z-ordering
(or time ordering)
is defined by
The Z-ordering is necessary since the elementary amplitudes do not commute. This circumstance is due to the dependence on the projectile spin variable cr. The target spin variables commute. The above definition is valid only when K = t(k’ + k) = (0, 0, K) and K > 0. When K < 0, the Z-ordering must be replaced by an anti Z-ordering d({zi]), so that the amplitude of the first hit nucleon always stands furthest to the right, etc. Another important remark concerns the amplitudes& themselves. We observe that K has been chosen the same for all nucleon amplitudes. Only the momentum transfers pi are variable. They all lie in the impact plane, i.e., C . qi = 0. In order to avoid misunderstanding, we stress that along with qi also the vectors qi and iii = i& X K, which determine the spin directions in Eq. (1), vary independently for each nucleon. The nucleon amplitudes in Eq. (11) are in principle off-shell amplitudes. When the proton-nucleus amplitude is evaluated in the cm or Breit systems it is customary to employ for the invariant amplitudes A, B,..., E the corresponding nucleon-nucleon lab amplitudes. Our definition, Eq. (1 I), is not used by everybody. Some authors neglect the Zordering. Sometimes also the spin directions in the individual amplitudes fi are assumed the same, and determined by the external vectors ij and i?. In such
SYMMETRIES
331
IN PROTON SCATTERING
calculations the only Si dependenceresidesin the invariant amplitudes. Our definition is in agreement with conventional DWBA. As application of our methods we shall repeatedly consider excitation of vibrational states in spherical nuclei Ot + 2’, 3’,..., and the transition O+ + 1 + in carbon. For both types of transitions we decomposethe nuclear amplitude as follows, (k’;JMI.
Xlk;O)
= F,,,(q) + +b+G:,‘(q)
4. SYMMETRIES
+ o-G::,‘(q)1 + &%d.
OF THE NUCLEAR
(13)
AMPLITUDE
Here we shall demonstrate how symmetry relations for the nuclear amplitudes are derived from those of the nucleon-nucleon amplitude. The initial and final nuclear states are denoted IJi, Mi) and IJ,, M,), respectively. We start with parity. Insertion of the parity condition, Eq. (3) into Eq. (11) gives after a change of variables qi + -qi
=-i-‘({zi})(
fi dZqid t-q-
\“- qi) fi
i-l
,T,
{~(--K,qi,O,Oi)e-iq”“i).
(14)
i=l
Now, when K < 0, our prescription for the nuclear profile function demands that the Z-ordering be replaced by an anti Z-ordering. Since P((zi}) =.G((-z,}) we obtain T,(k’, k, a; (ri, q}) = T,(-k’,
-k. a; (-ci, cq}).
(15)
Then we derive the ensuing relations for the matrix elements.Introducing the parity operator ,TJ acting on the nuclear variables (pi, ai\ alone, we rewrite Eq. (15) as cv(k’,k,o;
(ri,oi})=
P;‘T,,(-k’,-k.o:
(ri,ui}).4,.
(16)
The nuclear states are eigenstatesof .fl, .f4 1.4M) = Pi,(-l)J
IJ. M),
(17)
where Pi, is the intrinsic parity. Consequently, the nuclear matrix elementssatisfy the relation (k’, m‘; JrM,I T(k mi; JiMi) = PfPi(-1)Jf7Ji(-k’,
m,;J,M,I
T/-k,
m,;JiMi),
(18)
where P, and Pi are the intrinsic parities of the final and initial states. This result is a well-known condition for invariance under parity. Our derivation demonstratesthat it
332
G6RAN
Fk+DT
is independent of the choice of coordinate axes and also that our model for the nuclear profile function defines a partiy invariant T-matrix. Equation (18) can be rewritten on a more useful form. To this end we rewrite the profile function on the right-hand side of Eq. (16). Performing a rotation through an angle II around the ii-axis we have TN(-k’,
-ky O; {ri, Oi}) = U,‘T,y(k’,
ky 0; {ri, Ui}) tJ,y
(194
U, = U,(J, + +a),
where J, = L, + S, is the identity is easily understood. and k. Its action is therefore numerical vectors in the Appendix A we get (k’,m,;J,M,I
(19b)
total angular momentum of the nuclear system. This U,, rotates all variables, except the numerical vectors k’ equivalent to keeping the variables fixed and rotating the opposite direction. Finally, exploiting Eq. (A.l) of
m,;J,M,)=P,P,(L',-m,;J,,--M,I
i’lk,
TIk-mi;Ji,-Mi).
(20)
This relation is valid only for our choice of coordinate axes. The same relation could have been obtained directly from Eq. (18) making use of the rotational invariance of the T-matrix. The corresponding operator U, would then have to include also the angular momentum 1 of the projectile. Next comes the more delicate time reversal condition. The nucleon-nucleon condition must be employed in the form (4), since the nuclear profile function is a multilinear function of cr. Inserting this condition into Eq. (11) gives
)( ,fi,&r,qi.-o*,[
* r iqf’si.
Oi>
’
I
(21)
A transposition changes the ordering of the amplitudes, turning a Z-ordering into an anti Z-ordering, i.e,, io [h(--K, Qj, -U*, --OF)’ eiqi’“i]
8((Zi})
=
[
d({Zi})
I”r {fi(-K,
qi, -o*,
-tJ:,eiq~si~]
The anti-ordering is appropriate for average momentum reversal condition becomes T,(k’,
k, o; {ri, ai}) = [I’,,,(-k,
We then work out the implications
?
(22)
i=l
-k’,
--K. Thus, the nuclear time
--CT*; {ri, -u:))]~.
(23)
for the matrix elements. First, we concentrate on
SYMMETRIES
IN
PROTON
333
SCATTERING
the spin degrees of freedom only. We denote a direct product of spin states by jm; {M}), where {M} stands for a specific set of target nucleon spin values. In this notation the matrix elements of Eq. (23) are (m,; {fif} I r,J$‘, k 0; {ri, ai}> (mi; {ai}) = (m,; {@,)I I’,,,(-k, -k’,
--a*; {ri, -a?])
Im,; {A?,}).
(24)
This relation is further rewritten employing the operator W = Wpr n Wi, where the action of the individual W-operators is as described in Eqs. (6) and (7). Consequently, WT,(-k, Combining
-k’,
--a*; {ri, -a)})
W-’ = T,(-k,
-k’, o; (ri, ui)).
(25)
this identity with the time reversal symmetry of the nuclear states ({rj, ai} IJ9 M) = V(J7 M)
fi (’ a.) (J, -Ml {rj, -fii)), Ij=lv2’ ’ 1
q(J, M) = (-l)J+M
Wd Wb)
we obtain for the nuclear T-matrix elements (k’, mf;JfMfl T/k, mi;JiMi) = (-l)J’+Mi-J’-M’+mi-m
f (-k, -m,; Ji, -Mi / T l-k’,
-m,; J,, -MF).
(27)
This is the standard time reversal condition. It is independent of the choice of quantization axis. As for parity we transform Eq. (27) into a more useful relation. A rotation through an angle 71around the q-axis brings (-k, -k’) into (k’, k). By a reasoning analogous to that employed in connection with Eq. (19) we rewrite the right-hand side of Eq. (25) as T&,(-k, -k’, a; {ri, ui}) = U;‘I’,(k’,
k, u; (ri, a,)) U,,
U, = U&J, + fu). The action of U, on the spin states is given by Eq. (A.2). We end up with (k’,m,;J,M,I
TIk,mi;JiMi)=(k’,m,;JiM,I
TIk,m,;J,M,).
(29)
This form of the time reversal condition is valid only for our choice of coordinate axes. We remark that, as expected, time reversal gives a relation between the matrix elements of the direct and time reversed processes. It does not give a restriction on the matrix elements in any one particular channel (except the elastic). Our derivations demonstrate, that even in the adiabatic approximation, time reversal does not in genera1 give any restriction apart from the obvious one, Eq. (29).
334
G6RAN
FALDT
Now we turn to invariance under reciprocity I. This symmetry is related Glauber’s reciprocity [3]. Since the restriction on the nucleon-nucleon amplitude to reciprocity is identical to that for parity, we expect the same to be true for nuclear amplitude. A direct demonstration is instructive and demonstrates methods to be applied below. Insertion of condition (8a) into Eq. (11) gives
to due the the
W) u, = U,(fa, + +J).
CT.’= 2 ui. i I
(job)
In the last step we employed that -K goes with anti-ordering and the fact that Z((Zi}) = .d((-zi}). P er f orming a rotation of the nuclear space coordinates around the z-axis through an angle rr we change (xi, yi) into (-xi. -yi). This operation is executed by the operator U,(L,4), i.e., &(k’,
k, a; (ri, oi}) = D;‘T,.(-k,
D, = exp[iti
-k’,
a; (-ri,
ai}) D,,
* (J., + io)]
@la) (3lb)
with J,4 = L + fo,, . The operator D, rotates all variables around the z-axis through an angle rr, except the numerical vectors (k’, k). Its action is therefore equivalent to keeping the variables fixed and rotating the numerical vectors by the same amount in the opposite direction. This rotation transforms (-k, -k’) into (-k’, -k) so that, finally, T,(k’,
k, O; (ri, Oi)).= l-,(-k’,
-k, 0; i-r;,
ai}).
(32)
This condition is identical to that for parity, Eq. (15). Thus, we have shown that Reciprocity I is equivalent to parity also for the nuclear amplitude. Identical results are valid for reciprocities II and III. We end the section with a remark on Glauber’s reciprocity theorem [ 3 1. In our language it reads T,(k’, k; {r(}) = T,,,(-k, -k’; {pi}). For spin-independent interactions this relation follows immediately from Eq. (30a), since in this case the nuclear profile function is independent of zi. With a small modification (the presence of the operator U,(ia,)) a similar theorem can be formulated in the absence of projectile spin dependence (e.g., pion scattering). There is no theorem in the general case.
SYMMETRIES INPROTON 5. PROPERTIES
SCATTERING
335
OF SPIN VARIABLES
In order to make every step clear we shall quickly review the basic properties of polarization variables. Since the relations to be derived are independent of the adiabatic approximation we shall use a slightly modified notation. The directions of the projectile momenta are denoted 1’ and i. It is always understood that the nucleon going with the “final” state Jr has a momentum of magnitude k’ and that with the “initial” state Ji a momentum of magnitude k. More important, we shall use as quantization z-axis the normal ti to the scattering plane. This choice does not affect the symmetry relations to be derived but simplifies their actual derivation. We first define (this is not the unpolarized cross section)
a(iv,, i.fi) = x ((irm,; J,h4,/T) im, ; JUAN,)/* m,M = 0(iJ,, i f.r,).
(33)
Rotation through an angle rr around the ii-axis changes (i’, 7) into (-i’, -7) but does not change the spin components. It follows, from the rotational invariance of T. that
u(I~, i.q = @.f,. -iq.
(34)
Parity invarians, Eq. (IS), leads to the same condition. Polarization and analyzing powers are defined by
0(iq, iq p(iq, i.q
with matrices p and a given by
(36) This choice of p and a is valid only when the quantization axis is along ii. Let us now look at the symmetries. Performing a rotation through an angle II around the ii-axis as in the derivation of Eq. (34) we f?nd
p(iq, i.q = fy-iq, -iJi). Again the sameresult follows from parity.
595/148/2
6
(37)
336
GZjRANFiiLDT
We then perform a rotation through an angle K around the q-axis (i.e., the x-axis). Then @‘, i) + (-i, -i’) and spin directions change according to Eq. (A. 1). We get
U(i’J‘,iJi)P(i’J,,iJi)= 1 1(-i,m,; J,M,)
T 1-i/m,; JiMi)]* p(-m,,
-m,).
(38)
m.M
Since’p(-m,,
-m,) = -p(m,, mi) we conclude that
P(Iv,,i.ri)= -q-i.rr, -iq). Next, we turn our attention to time reversal. Exploiting formulation, Eq. (27), we obtain
O(lvl,i.ri)fyiv,, i.ri)= v 1(-i,m,; JiMi
(39) the coordinate independent
1T /-il, m,; JcMi)lz p(-m,,
-mi).
(40)
lzf
But p(-m,,
-m,) = -a(mi,
m,)
SO
that
P&I,, iJi)= 4 (-Q, -iq) = A(iq ,iq,
(41)
where the last step exploited Eq. (39), or rather its identical relation for the analyzing power. Equation (41) expresses the well-known fact that the polarization in the direct channel is equal to minus the analyzing power in the time reversed process. We end this section by giving the explicit expressions for the polarization and the analyzing power appropriate for transitions described by the amplitude of Eq. (13). - - Remember, that this expression refers to our standard coordinate system (x, y, z) = (ii, q, I?). Denoting the unpolarized cross section by uO we have 2o,P(k’J,
k0) = c Sp[F$,a
. iiFJM]
M
= 2 Re c [F&,(GjL)
+ G$;)) + (G$&” - Gi;)*) Gyd],
(424
M
2o,A(k’J,
k0) = c Sp[Y&.FJ,a
. ii]
M
= 2 Re x [F&,(G:&) + G$,‘) - (G$;)’ - GJ,“) Gyd].
(42b)
M
Equality of A and P, A = P, demands
x (G;h’*_ G;,“) M
Gy; = 0.
(43)
SYMMETRIES
INPROTON
SCATTERING
331
6. APPLICATIONS We shall now illustrate with a number of examples how our methods can be used to derive restrictions on inelastic matrix elements. In many of the cases a restriction is obtained only when the nuclear states are simultaneous eigenstates of the total nuclear orbital angular momentum L, and the total nuclear spin angular momentum S, . Such nuclear states are denoted IJ, M) = I@, S), (M’, MS)) with M = M” + M’. We shall constantly illustrate with two particular cases of transitions. Case A. O+ 4 2’, 33,... with S, = Si = 0, i.e., excitation of vibrational spherical nuclei. The intrinsic parities are P, = Pi = 1.
states in
Case B. Ot -+ l+ transition with L,= Lj = 0. The final state is an unnatural parity state P, = -1, whereas Pi = + 1. In both cases the scattering amplitude &,,,, is decomposed according to Eq. (13). EXAMPLE 1. The nucleon-nucleon amplitude is invariant under parity. The corresponding nuclear symmetry relations are given by Eq. (20). Applied to our two cases they read,
Case A.
G(o) = -G(O)L. MT LM
Gi’,’ = G”’L.-M'
Case B. FIM=-Ft,-M, G(o) = G’O’ IM
1,
G(F) = -G(F) IM
I.-M’
My
2.
Consider a situation where the nuclear profile function, Eq. (1 1), is requires the nucleon amplitude to be independent of the projectile spin variables, except for the Eterm in Eq. (1). Such a situation would be typical of pion scattering. By assumption EXAMPLE
independent of the z-coordinate. For proton scattering this condition
T,(k’, k 0; {ri, of}) = rN(k’, k 0; {(Xi,Yi, -Zi), a,))* Rotation of the nuclear space coordinates around the z-axis through an angle rr gives r,(k’,
k 0; (ri, ai)> = U,-‘(L,)
. T,(k’, k, a; (-ri, q}) . U,(L,)
= PFPi(-l)J’-Ji
U,‘(L,)
. T,(k’, k, a; {ri, q}) . iIT,(
where in the last step we anticipated that the profile function is going to be sandwiched between nuclear states of definite parity. This relation differs from the
GtiRAN FiiLDT
338
parity relation. However, it is useful only when the nuclear states are eigenstatesof L,. We get Case A.
Case B. EXAMPLES
No restriction on the amplitudes. 3.
Consider the amplitude
.f(K,$0, ui) = A(4) + c(q) a . ii which is commonly used to describe excitation of natural parity states. It is invariant under parity and also under time reversal applied to the projectile variables alone. The time reversed condition for the nuclear profile function becomes (cf. Eq. (23)) T,y(k’, k, O; (ri}) = [I’,(-k,
v-k’, --o*; (ri})]‘“,
Proceeding as in Section 4 (time reversal transformation of nuclear states not necessary) we obtain
(k’>m,;JfMrI Tlk, mi;JiMi) = (-l)Ji-“i-JltM f(k’, mi; J,,
-M,l
T 1k, m,; Ji, -Mi).
The operator considered cannot induce the O+ + 1+ transition. For the natural parity transition Case A.
FL&,= (-l)L-s’f FL.-,w, G;% = (-l)L-”
GF’M,
Gli,’ = (-l)Lm.” G;,+‘,%,.
These selection rules as well as those of parity are in agreement with the symmetries of Ref. [4). EXAMPLE 4. Now consider a more sophisticated symmetry. Take time reversal for the projectile nucleon and reciprocity for the target nucleon. Thus,
f(K, 97UToi> = UJ ‘($ai)[f(-k,
+q, -a*, o~)]‘P ~,(+o,),
SYMMETRIES
IN
PROTON
339
SCATTERING
where the transposition acts only on the projectile nucleon variables. This symmetry allows all terms in Eq. (1), except the u . Koi . %term which is odd. In particular both the cr . ii and oi . ii terms are permitted. Again the symmetry leads to selection rules only when L, and S, are separately conserved. The condition on the nuclear profile function becomes
TN@‘, k 0; {ri,Of})= U,‘(S,)[TN(-k-k’, -u*; (ri,oi))]‘pU,(S,d) which leads to the symmetry for the matrix elements
(k’, m,; (LJf)(#M”)( = (-1)
Tlk mi; (LiSi)(M(Mf))
Lf+Sf-Li-Si+.~f(-,Zl~
X (k’, mi; (LfSf)(-M:,
-M:)(
TI k, m,; (Li Si)(-MI,
AM:)).
The applications: Case A. FL,, = (-l)L-”
FL,-,&,,
G:;, = (-l)‘,-“’
G&,.
G;p,: = (- 1)” -,M G;,:‘- ,,I, Together with parity invarians these selection rules imply F,,,,. Gjt,’ # 0 when L - A4 = even and Gy,), # 0 when L -M = odd. Hence, there is no interference between G$ and G:P,: and Eq. (43) tells us that A = P. Case B. F,,, = -F,,-,,,. G’o’ = -G(o) 1 XI
I.
G(i) =-G(T) I.,,
I, -
If.
21'
Together with parity these selection rules imply G’,:: = 0 and as a consequenceA = P. Exercise 1. Suppose that the nuclear profile function is invariant under the substitution xi --) -xi. Show, that for CaseA, this symmetry in combination with parity invariance leads to A = P. No additional information for Case B. With our choice of profile function this symmetry is realized only when all invariant amplitudes are zero except A(q). Do we in this case get an additional restriction on FL.,,?
Exercise 2. There is a symmetry similar to that of Example 4, which allows all terms except the (I . qai . S term. Determine it! Explain why this symmetry of the nucleon-nucleon amplitude does not give rise to a corresponding symmetry of the nucleon-nucleus amnlitude!
340
G6RAN
FtiLDT
7. WHYA#P We shall now prove by the example 0’ -+ 1 + that A #P also in the adiabatic approximation. To this end we consider an idealized situation where both initial and final states have orbital angular momentum L, = 0. The 1’ state could for example be a 2s,,, 0 (Is,,,))’ particle-hole state. Even though the strength of such a configuration is weak it offers a particularly transparent illustration of our tenet. In Appendix B the corresponding shell model result for Cl2 is given, considering the 1’ state as a lp,,, @ (lp,,,)-’ particle-hole state. We start from the symmetry considered in Example 4 of the previous section. The nuclear profile function is decomposed into a sum of two terms, one containing only even powers of u I Kai . K, the other only odd powers. The former is symmetric under the symmetry in question, the latter anti-symmetric. Each term is invariant under parity. Exploiting the symmetry relations of parity invariance example 1, we rewrite Eq. (43) as A -P The decomposition
= const . RejG(,t”G{i’
+ (Gf:’
- Cl;))
of the nuclear amplitudes is formally
F=f+f, where f, g are the symmetric rules of Example 4 we get Even
G:!“].
written
as
G=g+g,
(45)
terms and f, g the anti-symmetric.
f,,
(44)
= --&,,,,
g;;’ = -g\“,,
g I; = -g;o’,,
Applying
the selection
.
When combined with parity invariance these relations give f,, = 0,
giz = 0,
all M.
We conclude that if only even terms are considered A, = P,:
then from Eq. (44) we have
Odd f,, = f,-M, fl; = g:“‘,, Combined with parity invariance these relations give
f,, = 0,
gi$’ = 0
all M.
A further consequence is that when only odd terms are considered, then A, = P,.
SYMMETRIES
IN PROTON
SCATTERING
341
We conclude from this exercise that only interference terms between the even and odd amplitudes contribute to the difference A - P. In effect, Eq. (44) becomes A - P = const . Re(&‘*gj~’
+ g’$*( g!:’ - gl;‘)}.
In practice most calculations incorporate the dependence on the nuclear spin variable only in the linear approximation, i.e., in the excitation step. The projectile distortion, however, may depend on ,4(q) + C(q) o . ii in any order. We shall now prove that in this approximation, indeed A = P. We first consider g,(O) I . This term must, according to its definition, come from the linear o . K oi . C. However, oi . ii cannot change the M-value of the nuclear state, and therefore g:!’ = 0. We then consider gii’. Since a term of the form A + Co . ii cannot produce the desired nuclear transition, by definition only Co, . ii + Bo . iio, . ii + Do . qai . 4 can. However, all three terms change the nuclear M-value by one unit. Thus, we have (+) =o g10 As a result, in the linear approximation A = P. Now consider the general case. In the linear approximation g\:’ # 0 and g\{’ # 0. Thus, A #P if we can prove that either giy’ # 0 or g\i’ # 0. Remember that g involves only odd powers of a . Kai . % and g only even. Clearly, already in secondorder multiple scattering do we encounter prospective candidates, for g\i’
and for g’# E,u . Ku, . KC2u2. ii,.
It only remains to ascertain that they do not vanish by some kind of symmetry argument. If they would, it would be necessary to make the excitation mechanism more complex by introducing suitable powers of the non-commuting Cu . ii operators. We shall now demonstrate that indeed g\:’ # 0. For this purpose it is not necessary to treat the fully symmetrized nuclear problem. It is sufficient to consider the pair which is broken up in the excitation process. The initial nuclear spin state is 10,O) and the final 11, 1). The transition density must therefore be anti-symmetric p(l,2) = -~(2, 1). Furthermore, since the orbital angular momenta are zero p(r, , r2) = P(Jr, I,1 r2 I). Neglecting trivial multiplicative numerical factors, we have 819’ =
I
d3r, d3r,p(r,,
r2) d*q, d3q,6(q - q, - q2) eiql’sl+iqz~S2
X (1, 1 I E(qJ C(q2) uI * ka2 . n2 + C(qJ E(q2) u1 . n1u2 . k IO, 0)
(47)
with n = q x ic and a slight resealing of C(q). Observe that the directions of the
342
GijRAN
FALDT
normals ni vary when qi vary. After a number of simplifications we get g\:’ =
1
d3R d3r p(R, r) sin(iq
a(q, s) = - “-,f 2fi
d’Q sin(Q - sW($q
/qq,s)=-Ie- ~ {EC&
+
s)l,
i(Qs-pq) !’ d2QQe-i(vpy-'d +
(484
Q) C(h - Q)
Q) C(fs + Q)lv
- E(tq -
x
. S){a(q, s) -P(q,
and rearrangements
(48b) sin(Q
. s)
Q> C(fs - Q) + E(fq - Q) C
(48~)
Here we have introduced the notation p(r,, r2) = p(f(r, + rZ), r, - r2). We remark that when E and C have the same slopes then a(q, s) = 0. We also remark that a(q, s) and /3(q, s) must be anti-symmetric in s in order to give a non-vanishing contribution to g’ly. Thus, we have shown that A #P for the Ot + 1’ transition. It must be stressed, however, that the mechanism investigated is not necessarily the one responsible for the difference observed in the recent experiment by Cary et al. 151. We have only demonstrated that there is no theorem requiring A to be equal P in the adiabatic approximation.
8.
MISAPPLICATIONS
Some applications of the Glauber model employ nuclear profile functions different from the one we use, Eq. (11). The most commonly neglected feature is the zordering, which is equivalent to working with a symmetrized product of nucleon amplitudesfi. Such a choice defines a nuclear amplitude with the correct parity and time reversal symmetries, Eqs. (18) and (27). Furthermore, for the lowest-order terms in the projectile spin variables (i.e., up to linear) the model becomesidentical to ours. When higher-order terms are considered that is no longer true. This fact is easily understood since the nuclear profile function has an additional symmetry. It is invariant under the substitution zi + -zi, i.e., the selection rules of Example 2 apply to the complete nuclear amplitude. An alternative way of expressing the samething is to observe that time reversal can now be utilized also in the form of Eq. (5) having as consequencea restriction on the transition amplitude itself. Applied to the natural parity transitions the model predicts non-zero amplitudes only for L - M = even, in disagreement with Ref. [4] where Gp,),# 0 only when L -M = odd. A calculation with the restricted nucleon-nucleon amplitude A + Ca . ti would give G:~y,\= 0, all M, and equality of polarization and analyzing power. There are other ways in which profile functions different from ours can be
SYMMETRIES
IN PROTON SCATTERING
343
constructed. One method which has been used consists in having the q,-dependence of fi only in the arguments of the invariant amplitudes A, C,.... The vectors which determine the spin directions are assumedidentical to the external vectors S, ti and ii. Naturally, such a model is at variance with ours. Independent of the ordering problem, the profile function becomesinvariant under the substitution xi + -xi. The reader who has solved Exercise 1 knows that in Case A this symmetry leads to the equalities Gl”A = 0, all M, and A = P. The same result could be proved through Reciprocity III, since with the present construction also the nuclear profile function enjoys the symmetry of Eq. (10). Still another possibility is to employ time reversal as expressedby Eq. (5). The fact that different nuclear profile functions are sometimes used has been pointed out also by Harrington 161.
9. SUMMARY With the Glauber model as base, we have studied symmetry properties of inelastic nucleon-nucleus amplitudes, starting from the symmetries of the underlying nucleonnucleon amplitude. We have demonstrated that the nuclear amplitude satisfies the well-known parity and time reversal constraints. The proof, which is straightforward, confirms that the Glauber model observes fundamental invariance requirements. Time reversal relates the matrix elements of the direct processto those of the time reversed process. When the complete nucleon-nucleon amplitude is employed, the details of our proof reveals that the Glauber model, in conjunction with the adiabatic approximation, does not in general introduce any restrictions, apart from those of parity and time reversal. In particular, the equality of A and P is not universal. For the transition Ot + 1’ we show that already in second-order multiple scattering do we encounter terms violating this equality. When only part of the nucleon-nucleon amplitude is taken into account, additional restrictions on the nuclear matrix elements are frequently obtained. We have shown, with a number of examples,how such restrictions are derived. Often, they are realized only when the nuclear states are simultaneouseigenstatesof the total orbital angular momentum L, and the total spin angular momentum S,. Fortunately, this is the case in many practical applications, e.g., in the excitation of vibrationally excited states in spherical nuclei. Some investigations start from nuclear profile functions different from ours. Such alternatives do give rise to additional symmetries and in someof the models one can even prove rigorously that for selected transitions A = P. Naturally, the numerical values of the nonvanishing amplitudes also differ from ours. Our analysis has been performed within the Glauber model but most of our conclusions would remain valid in any equivalent model employing the adiabatic approximation.
344
GiiRAN
FALDT
A
APPENDIX
Here we collect some useful formulas concerning rotations of physical systems. Our choice of coordinate system is displayed in Fig. 1. The axes are chosen as (X, ji, Z) = (ii, S, c) and the vectors (k’, k) lie in the (y, z)-plane. There are three particular rotations employed. For simplicity, they are given below in the adiabatic approximation k = k’. Phases are defined following Brink and Satchler [ 71: I. (-k’, -k) -t (k’, k). This transformation is achieved by rotating the physical system through an angle rc around the x-axis (ii). We have einJx I -k; J, M) = einJ 1k; J, -AI),
(A.la)
eiqJx I -k’; J, M) = einJ 1k’; J, -M),
(A.lb)
where J, is the x-component of the total angular momentum. II. (-k’, -k) + (k, k’). This transformation is achieved by rotating the physical system through an angle n around the y-axis (ii). We have ei”Jv)-k;J,M)=(-l)J+Mlk’;J,-M),
(A.2a)
einJY( -k’; J, M) = (-l)-‘+M
(A.2b)
/ k; J, -44).
(k’, k) + (k, k’). This transformation is achieved by rotating the physical III. system through an angle 71around the z-axis (E). We have
einJz 1k; .I, M) = einM ) k’; .I, M),
(A.3a)
eizJz 1k’; J, M) = einM 1k; J, M).
(A.3b)
B
APPENDIX
The calculation of G’,:’ in Section 7 was more pedagogical than realistic. Here we give the complete double scattering contribution to G\:’ for the 0’ -+ 1’ (T = 0) transition in Cl*. We assume the 1’ state to be a pure p ,,* @ p,: particle-hole excitation. Denoting by I G) the ground state of i2C, with filled s,,~ and pJ12 shells, we can write
1171; T= 0) = I/@ ,z n w%aP(t, + v%:(P&
f>> a,(~(;.
- f>b,
>l
- 3))
I@.
(B-1)
The double scattering operator is d’q,
d*q, S(q -9,
- qJ eiq”r1+‘PZ.r2~(q,)f(q2).
P.2)
SYMMETRIES
INPROTON
A rather lengthy but uninteresting calculation
345
SCATTERING
gives the result
G!?(q) =?I,. Co bdbJ,(qb) joffl q,dq, C(q,) jam 42 dq,%) 1- soI S,,(q*)[J0(q,~)+ J*(q,b)l *J,(q,b) X
3
2fi
+ S,,(qJ
S,,(q*) [ (&
- &)
Jl(4l~)J&2~)
+ with
So,(q) =jom r*dru,W u,W.A (v),
(B.4a)
S,,(q) =p r*drIu,(r)12j2(q)
(B.4b)
and u,(r) and Up(r) the radial wavefunctions for the S- and P-states of “C. Our calculation assumes an isospin independent nucleon-nucleon amplitude. It is a trivial matter to include isospin dependence, as long as exchange effects are neglected. It is also trivial to expend Eq. (B.3) to include distortion in the spin-independent approximation.
ACKNOWLEDGMENTS I would like to thank Marek Bleszynski for illuminating discussions related to their work. I would also like to thank Ben Gibson and Terry Goldman for their kind hospitality at the Los Alamos Theoretical Division, where this work was carried out.
REFERENCES I. E. BLESZYNSKI, M. BLESZYNSKI, AND C. A. WH~T~EN JR., Phys. Rev. C 27 (1983), 902. R. D. AMADO, Phys. Rev. C 26 (1982), 270. R. J. GLAUBER, in “Lectures in Theoretical Physics” (W. E. Brittin ef al.. Eds.), Vol. I, Interscience. New York, 1959. 4. G. FXLDT AND P. OSLAND, Nucl. Phys. A 305 (1978), 509; G. F~~LDT AND A. INGEMARSSON,Nucl. 2. 3.
Phys. A 392 (1983),
249.
5. T. A. CAREY et al., Phys. Rev. Lett. 49 (1982) 266. 6. D. R. HARRINGTON, Rutgers University preprint, 1982. 7. D. M. BRINK AND G. R. SATCHLER, “Angular Momentum.” Oxford Univ. Press, Oxford, 1968.