Download - Nuclear Stripping Reactions
NUCLEAR STRIPPING REACTIONS1,2 By NORMAN K. GLENDENNING
Lawrence Radiation Laboratory, University of California, Berkeley, California
CONTENTS
1. INTRODUCTION...................................................... 191 2. SINGLE-NuCLEON STRIPPING REACTIONS. • . • . . . . .. . . . . . . . . . . . . . ... . . . .. 193
2.1 General form of the cross section.... .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193 2.2 A ntisymmetrization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 196 2.3 Plane-wave calculation-Butler formula. . . . . . . . . . . . . . . . . . . . . . . . . .. 197 2.4 Distorted-wave method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 206 2.5 Distorting potential in stripping reactions. . . . . . . . . . . . . . . . . . . . . . . . .. 212 2.6 Some examples of distorted-wave calculations.... . . . . . . . . . . . . . . . . . .. 215 2.7 Nuclear structure and spectroscopic factors. . . . . . . . . . . . . . . . . . . . . . . .. 216 2.8 Polarization and angular correlation ............................. , 2 2 9 2.9 Rearrangement stripping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 3 7
3. SINGLE-NuCLEON TRANSFER REACTIONS OTHER THAN (d, P). . . . . . . . . . . .. . 239 4. Two-NUCLEON TRANSFER REACTIONS................................... 2 40
4.1 General features. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 240 4.2 Interpretation of some experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 50
ApPENDIX. . . • • . • • • • • • . . • • . • • • • • • • . . . . . • • • • . . • . . • . . . . . • • . • • . • • • • . • . . • . 253
1. INTRODUCTION The transmutation of elements by bombardment with slow deuterons,
followed by the emission of protons, and the interpretation of the reaction as a stripping process date from 1935. Oppenheimer & Phillips (138) explained the experimental results of Lawrence, McMillan & Thornton (113)
in terms of the penetration of the neutron into the target nucleus while the proton remained outside, beyond the Coulomb barrier. In these experiments the cross section was found to increase less rapidly over the energy region measured (up to 3.6 MeV) than was anticipated from the Gamow penetration factor for transmission of a charged particle through the Coulomb barrier of the nucleus. The physical picture treated shows a deuteron moving in the field of the nucleus, being stretched by the action of the Coulomb repulsion on the proton, while the neutron attracted by the interaction with the nucleus is captured. Because the deuteron binding energy is small, it is easily polarized, and the reaction can occur without the proton having to enter the Coulomb barrier. Hence the variation of the cross section with energy is quite different from that of reactions in which a charged particle must actually enter the nucleus.
1 The survey of literature for this review was concluded in early 1963. • Written while the author was a visitor during 1962-63 at Centre d'Etudes
Nucleaires, Saclay, and Institut du Radium, Orsay, France. The author expresses his appreciation to Professors C. Block and M. Jean for the opportunity of making that sojourn.
191
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192 GLENDENNING
However, the importance of stripping reactions relevant to the study of nuclear states was not rf'cognized until at higher energies the angular distributions of resolved groups of protons corresponding to particular energy levels of the residual nucleus were measured (30, 93) . These angular distributions exhibit a pronounced structure at forward angles, which Butler recognized as implying the importance of high angular momenta (14, 31, 32, 33) . Since high angular momenta must correspond to large impact parameters, he concluded that the reactions proceed, at least in part, by a stripping process in which one of the particles of the deuteron is absorbed into the nucleus, while the other merely carries off the balance of energy and momentum. Moreover, since the reaction connects the ground state of the target to a specific final state, the stripped particle can have carried into the nucleus only such angular momentum and parity as are consistent with their conservation. Their subtraction from the incident wave is reflected in the angular distribution of the outgoing protons.
That a connection exists between the angular distribution of the outgoing proton and the angular momentum of the state into which the neutron is stripped can be understood in the light of the following classical argument (36, 100). Let the momenta of the incident deuteron, the ·outgoing proton, and the stripped neutron be likd, likp, and liq, respectively. Energy conservation specifies the magnitude of kp, and momentum conservation requires that q=kd-kp' If the neutron is stripped into a state of orbital angular momentum lli and if we suppose that the deuteron is broken up at the point R, then 1 equals R Xq. Consequently
Ika-k,.IR�l 1. For 1 ¢O this inequality can be satisfied only for scattering angles greater than some minimum value which depends on 1. In any case, because of the small deuteron binding, the deuteron is broken up preferentially with the proton moving in the forward direction.
For single-nucleon transfer reactions the angular momentum 1 of the bound state, which according to the above argument can be deduced by a measurement of the angular distribution, is uniquely connected with the parity of the final nuclear state and determines within limits its spin, if the initial state is known. This establishes the importance of stripping reactions and their inverse, pickup reactions, as a means of studying the properties of nuclear energy levels, as emphasized by Butler, and· as confirmed subsequently by the abundance of information emanating from stripping experiments.
The stripping reaction is discussed in this review from the point of view of its usefulness as a probe of the structure of nuclear states. The foundations of the theory are not examined. However, some heuristic derivations of important results are given.
In Section 2, the (d, p) reaction is discussed in considerable detail because it is the most thoroughly studied reaction and it serves to illustrate many
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NUCLEAR STRIPPING REACTIONS 193
points relevant to other direct reactions. This reaction by its nature populates and has been used to identify single-particle states in the final nucleus [cf. (41,42,43, 128, 163)]. The inverse reaction (p, d) can be used to study the hole-states [d. (41, 73)]; and these reactions, together with several other single-nucleon transfer reactions, are discussed briefly in Section 3. In Section 4, reactions involving the transfer of two nucleons are considered. These reactions make levels of two-particle or hole excitation accessible for study [d. ( 10, 79)].
The literature on stripping reactions is vast. I have not seen, let alone read, all that has been published on the subject. Undoubtedly much interesting work has been overlooked, and to the reader thus cheated I make my apologies. Other reviews which impinge upon the subject have been written by Huby ( 100), Horowitz (98), Butler (36) , Breit (27), Banerjee ( 1 1), French (61), Macfarlane & French ( 117), Tobocman (178), and Austern (8) .
2. SINGLE-NuCLEON STRIPPING REACTIONS 2.1 General form of the cross section.-The angular distribution of out
going particles from the (d, p) stripping reaction was first obtained by Butler (31, 32, 33, 36). His method involves matching, at the nuclear boundary, the wave function for the system in the interior and exterior regions of the nucleus, found after certain simplifying assumptions are made. However, the method is lengthy and involved. Other authors soon gave alternative treatments of the reaction, which under the appropriate assumptions gave the same or similar results (7, 34, 40, 44, 45, 57, 58, 62, 64, 65, 74, 75, 94,95,99,173,174,184,185).
In this section we shall develop the general form of the differential cross section for single-particle stripping reactions and discuss the spectroscopic significance of the results. In subsequent sections the several current forms of the theory will be discussed together with its application to the analysis of experiments.
The exact transition matrix element for the (d, p) reaction can be written (d. 64)
2.
where\)i d<+> is the exact-state vector for the system and has outgoing spherical waves of deuterons at infinity; tPP is the wave function for the final system comprising proton and residual nucleus (A + 1) with no interaction between these two parts; Vpn is the proton-neutron interaction and VpA is the protontarget interaction. Since the exact solution qr d is not available, an approximate expression for T has to be found (see Appendix) :
T(d. p),...., (xp(-) I V pn + (1 - P) VpAP I xP» 3.
where P is an operator which projects onto the ground state of the target nucleus.
The wave functions appearing in Equation 3 may be written in more
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194
detail as
where
GLENDENNING
Xd(+) = !/td(+)(k.J, R),/>d(r)X1I'd(dn, dp)'lrJ.M.(A) XpH = !/tp(-)(kr" rp')Xl/f'1'(dp)'lrJ,M/(A, rA, dn)
r = r" - 7", 2R = r,,+ r"
4.
5
Here CPd is the internal wave function of the deuteron, Xl is its spin function [we consider only the S state since the Sand D states contribute incoherently, and the D state admixture in the deuteron is less than 10 percent (45));
�Ji is the target wave function; Xl/2 is the proton spin function and �J/ is the wave function for the final nucleus. The functions �i+) and �p(-) are elastic scattering wave functions to be computed by use of a one-body interaction between the deuteron and target, and proton and target. The (+) and (-) refer to their asymptotic behavior: �i+) has outgoing spherical waves and 1/11,1(-) incoming spherical waves. The coordinate 7r.'
6. is the coordinate of the proton referred to the center of mass of the final nucleus.
Because of the physical picture we have of the stripping process, we expand the wave function for the final nuclear state on a basis exhibiting the target plus the stripped nucleon.
7.
where.p is a wave function constructed by vector coupling the extra nucleon in the spin-orbit state CPli to a target wave function with angular momentum Jc. [Aside from the ground state Jc=Ji, the wave functions for the core nucleons do not necessarily correspond to unique states of the target nucleus (117).]
if>(Jc1I)J!M, = L (JeM.,jmj I J,M!) 'lrJeMc(A)cf>I/"f(r", ,dn) M.mj , 8 .
where the bracket is a Clebsch-Gordan coefficient. Other representations than the spin orbit could have been used, such as the L-S or channel-spin representations. They are all connected by a unitary transformation, and the choice in any situation could be governed by requiring'that Equation 7 have a minimum of terms.
The expansion coefficients {3 represent the degree to which the final state has the configurations indicated by .p, and they are directly related to the reduced widths of the state.
.
If we insert the above expressions into Equation 3, it is apparent that the Vpn term is nonvanishing only if the final state contains;components corresponding to the core (comprising the target nucleons), being in its ground state. In contrast, the term in VpA is nonvanishing only if t�e final state does contain components corresponding to core excitations. In the literature scant -
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NUCLEAR STRIPPING REACTIONS 195
attention has been paid to the second term. It is often neglected for incorrect reasons. However, it may be intrinsically smaller than the first term, becoming important only when the final state is almost purely a core excited state. The success of the current treatment of stripping suggests that this is so. We shall therefore devote most attention to what we call simple stripping, corresponding to the first term in Equation 3, and reserve for Section 2.9 a brief discussion of the rearrangement stripping process. Accordingly we drop the index on (J;z(JcJ/) referring to the core, and obtain for the simple stripping amplitude
TCd.,,) = L (J;M"jmj I J,M,)(lm!tlm. 1 jmj) il"'j"'l"'.
X (!p", !p.llpah/21 + 1 (jjlBI ml
where we have used for the spin-orbit functions
<!>z;"'i(r", d,,) = L (lml, lm. l jmj)il<!>,ml(r")x1I2,,..(d,,) ml"'.
",,-'(T .. ) = u,(r,,)y,-,(r,,)
9.
10.
The quantity Br, which is the amplitude for the absorption of a neutron with quantum numbers (1m), contains the dependence on the scattering angle and is
Blm(k.J, kp) = i-'(21 + 1)-112 f .p"C-)*(kp, T"')</>,"'*(T,,)V,,,,(r)
X .pd(+)(k.J, R),Mr)dr"dr" 11.
The differential cross section is found from Equation 9 by averaging (2'l1/n) I Tit over the spin directions in the initial state, and summing over the final spin directions, multiplying by the density-of-states factor and dividing by the incident flux:
d · · k 1 . -..:!... _ ma m" 2 '" I T 12 dO -
(21rn2)2 kd 3(2J, + 1) M,fi;dI'p Cd.,,) 12.
where md*, mp* are the reduced masses and kd, kp are the wave numbers for relative motion in the initial and final states, respectively:
kpl = 2m,,*E,,/nl, kdl = 2md*Ed/nl
Ep = Ed + Q 55 Ed + B,. - Bd 13.
where Q is the usual symbol for the increase in kinetic energy, Bn and Bd are respectively the binding energy of the neutron in the final nucleus and of the deuteron , and Ep and Ed are center-of-mass energies.
The magnetic sums in Equation 12 can be explicitly executed yielding
�_� md*mp* k" 2J,+I", .-IBml' 14 dO -
2 (21rn2)2 kd 2J, + 1 ft;: {J" , •
The amplitude Br depends, in general, very sensitively on the value of l. To this fact the stripping reaction owes its value as a source of spectroscopic information (31, 32, 33). For the transition matrix, Equation 9, exists only
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1 96 GLENDENNING
for values of 1 and j for which
J, = J. + j = J. + I + t 7rJ = 7rH-l1
15. 16.
are satisfied. Here 11"; is the parity of the initial state. If, because of the sensitivity of the angular dependence of I Brl1, one can identify which 1 (or 1's) are involved, the parity of the residual level is given uniquely, while the spin is specified, at least within limits (see, however, Sec. 2.9). Also, since the spin-orbit partners are considerably separated in energy, reference to the shell-model level ordering often reduces any ambiguity (47, 120).
We should remark that the above discussion and Equation 14 refer to the simple stripping process and not to rearrangement stripping. In the latter, more general selection rules apply which will, be discussed in Section 2.9.
2.2 Antisymmetrization.-So far we have treated the nucleons in the incident deuteron as though they were distinguishable from the others. The effect of antisymmetrizing the total wave function is to introduce exchange integrals. However, the exchange integrals, because they involve the overlap between bound and free states, will be smaller than the direct term. The only evaluation of such effects has been done in the plane-wave approximation and agrees with our assertion (51, 59). We shall therefore ignore them in our considerations of the simple stripping reactions (see, however, Sec. 2.9).
There is, however, an important way in which the equivalence of nucleons makes itself felt. This is through statistical factors, which of course are of no significance for the angular distribution but must be included if the theory is used to extract reduced widths from the experimental data (61). These factors can be easily constructed. Since we shall be interested later in reactions such as (He3, d), we shall consider the reaction A (a, a -1)A + 1 where A is the atomic mass of the target and a the atomic mass of the projectile. In the initial state we want to construct an antisymmetrical wave function from the product of the target A and projectile a wave functions. There will be
, = (A + a) =
(A + a) I N. a AlaI 17.
such product terms in the antisymmetrical function corresponding to the different ways in which the particles can be distributed' between the two groups. Similarly, for the final-state wave function
(A + a) (A + a)1 N,= a-I = (A+l)!(a-l)!
18 .
Schematically, therefore, the matrix element looks like
T(A, a -+ A + 1, a-I) = (N,N.)-1I2 X f <1,[,pA+l(l, 2, ... A + 1),pa-l(A + 2· • • A + a)]* 19.
X V a. [,yA(I , 2, . . • A),ya(A + I . • . A + a)]
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NUCLEAR STRIPPING REACTIONS 197 where at and a .. are antisymmetrization operators. The N; terms in the initial wave function will each connect with a terms in the final wave function to form the direct in tegrals we are in teres ted in. Accordingly there are Ni eq ual integrals which, together with the normalizing factors, give us va(A + 1) as the factor by which the integral evaluated with nonsymmetrized wave functions should be multiplied. Moreover, in the reduction of the integral to the form we considered earlier, the overlap between ,pa-l and ,pu will contribute a factor
(td I til) (tpn,,, !1n, I tim,,) 20.
where ({ I ) is a fractional parentage coefficient for the isospin. For a = 2, 3, or 4 this product is ± vl/2. Similarly the overlap between ,pA+l and ,pA will contribute.
21.
being the Clebsch-Gordan coefficient for the isospin, where mt is the z component for the outgoing particle (-t for protons). The fractional parentage coefficient from the overlap we absorb into the definition of fl.
With these results we now have for the differential cross section
22.
where
S(l) = � S(jl) 23.
and
S1I2(jl) = (A + 1) 1/2�;1
= (A + 1)1/2 f if?(J.jI)J/�J/J(A + 1) 24.
The symbol d(A+l) denotes integration over the A+l nucleons in the final nucleus. Here S is called the spectroscopic factor or relative reduced width. We shall discuss it at greater length in Section 2.7.
2.3 Plane-wave calculation-Butler formula.-The general form of the differential cross section for single-particle stripping has been derived above in Born approximation. There remains now the explicit evaluation of the amplitudes Br. There are two current methods employed in their evaluation, known commonly as the plane-wave and distorted-wave calculation. In the latter, the scattering of the incident deuteron and outgoing proton by the nucleus is taken into account. In this section we shall discuss the first method. It leads us to the Butler formula for the angular distribution which has been used extensively in the literature as a means of extracting spectroscopic information from stripping reactions [see for example (117) and references contained therein).
To obtain the wave functions describing the relative motion between deuteron and target nucleus, and proton and final nucleus, we assume that
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198 GLENDENNING the interaction between the two parts of the system can be neglected. They then satisfy the field-free Schroedinger equation and are therefore the plane waves:
25.
In this approximation we are able to obtain a closed foim expression for Br. We must anticipate of course that the result will have only a limited range of applicability.
By changing the variables of integration in Equation 11 to T and Tn we obtain
where
B,m = ;-·'(21 + 1)-112 J e-' K.rVnl'(r)q,d(r)dr
X f u,(rn)Y,"'*(Tft)e,q'tndrft
q = k.J - (MT/MF)kp is the momentum carried into the nucleus by the neutron, and
K = kp - (!)k.J
26.
27.
28.
is the momentum transferred to the .proton by its interaction with the neutron.
The first integral can be simplified by using the Schroedinger equation for the deuteron:
' .
(_�V2 + Vnp(r) + Ed) q,a(r) = 0 2m*
29.
where m* is the nucleon reduced mass in the deuteron and Bd is the binding energy. Define a by
then
where
(b)2 = 2m*Bd
G(K) = f e-iK.rVnp(r)q,a(r)dr
III = - - (K2 + a?:}P(K)
2m"
P(K) = f e-iK·rq,d(r)dr
30.
31.
32.
The integral P(K) is (211')3/2 times the probability amplitude that the momentum K is to be found in the deuteron. Not unexpectedly,· this factor damps the cross section at large angles since the deuteron does not contain large momentum components in the abundance of the smaIl ones: The quantity P(K) is a smoothly varying function of angle and dependsJor its speCific form on the choice of wave function for the deuteron. For several choices it
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is shown below: (a)1/2 e-.. r
<pa(r) = - -, 2... r
(8 ... )1/2 1 P(K) = a3 1 + (K/a)2
P(K) = (8 ... / C �'a') ') 1/2 1 + (�Ja)' 1 + (�/"')'
199
3 3.
34.
[See also Banerjee (11).] Corresponding to the deuteron binding energy 2.226 MeV, a=0.2317 p-l while the parameter a' is usually taken to be a' �7a. We may note that the wave function Equation 33 corresponds to a zero-range force and that the corresponding function G(K) is a constant.
The second integral in Equation 26 is the one which depends on the orbital angular momentum transferred in the reaction, and whose variation with scattering angle is sensitive to 1. To evaluate it we use the expansion of a plane wave
e'q.r = 4,.- f iLjL(qr) L YJ!f*(q) YLMG) L_O M 35.
The orthogonality of the spherical harmonics selects the L = 1 term. Then
f ul(r)Y,"'*(r)eiq.rdr = 4,.-iIYlm*(q) fo"u1(r)jl(qT)r2dr 36.
Some early work approximated this integral by its value at r�RN, near the nuclear surface, yielding the angular dependence j,(qR) (15). We shall, however, introduce the so-calJed cutoff approximation by neglecting the contribution to the integral from the inner region r <R?, RN. The cutoff approximation is certainly better founded than the surface approximation. In fact, it simulates in a rough way the absorption of the deuteron into a compound nucleus state and is therefore also better than a complete evaluation of the integral. To best see this, one would make the partial-wave expansion Equation 35 separately for deuteron and proton waves (Eq. 25). Then for 1=0, as an example, one has a sum of integrals
fo "jL(kpr) Uo (r)jL(kar)T2dr 37. Now the partial wave L corresponds to an impact parameter L/ka. To simulate the absorption of particles penetrating the nuclear interior, we might therefore set all integrals with L <Lc equal to zero, where Lc is chosen so that Lc/k�RN. But it is just these integrals (whose integrands are plotted in Fig. 1) which receive the largest contribution from the interior. Therefore, neglecting the contribution to the integral, Equation 36, from the interior, simulates roughly the absorption of the lower partial waves. Of course it is essential that a sufficient number of partial waves be involved for these considerations to be valid.
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200 GLENDENNING I
I'
ro • I 3
7 A -t VJ 1\1f\ '\ K / 1\
K 1/ / Y X 'X 'v � lL v / \ j.y \-/\ /' \
\" \ '! U '/ >Z / y� /\ /' \ \ \ '\ ..-/ , '-
Z 1 4 5 6 7 8 9 10 11 12 13 14 15 r(1 O·U ems)
FIG. 1. The integrand of Eq. 37 for various values of the angular momentum of the incident deuteron partial waves. Note that the principal contribution for the low partial waves comes from the interior region. In this example, 1==0, Ed=8 MeV, Q=2 MeV, and R=5 F. [From Butler (36).]
We can evaluate the cutoff integral by introducing the Schroedinger equations ( d2 1(1 + 1) ) . -- + --- - q2 rjl(qr) = 0
dr2 r2 ( d2 1(1 + 1) ) -- + --- + t2 rUI(r) = 0, dr2 r2
38.
39.
where h2t2/2M*=Bn is the binding energy of the neutron in the residual nucleus, and M* is the reduced mass of the neutron-target system. We note that Uz satisfies the Bessel equation only for r �RN where RN is so large that the nuclear potential is zero. After an obvious manipulation
(q2 + t2) r "'jz(qr)ul(r)r2dr = - r"' i. [(rul) .!:... (rjl) - (Tjl) .!:... (rul)] dT JR JR dr dr dr [ d . jl(qR) d .]
= R2UI(R) -jl(qR) - -- -- hl(ztR) dR hl(itR) dR
40.
where we have used the fact that the Hankel function of the first kind hl satisfies the Equation 39 and has the bound-state boundary condition that it vanishes at infinity (162, p. 79).
Gathering together the above results, we have:
h2 Elm = - 41r(21 + 1)-1/2_ ylm*(q)Rul(R)P(K)WI(q, R) 41.
2M*
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where
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[d. NqR) d . ] WI(q, R) = R dR JI(qR) - h!(itR) dR h!(ztR) and we have used energy conservation to show that
K2 + ",' 1 MF mp 1 q2 + t2 m* = M/ md m* = M*
201
42.
43.
We may now introduce this value of Br into Equation 22. Before doing so, however, we generalize the results somewhat so that they will also be valid for single-nucleon stripping from projectiles heavier than the deuteron. In the notation of Section 2.2 the reduced masses are
A + 1 m *-+-- (a - I)M P A + a A M*-+ -- M A + 1
where M is the nucleon mass. Then we find
where
82(/) = S(/)802(/) 802(1) = iR3UI2(R)
44.
46.
47.
which are respectively the reduced width of the level for angular momentum l, and the single-particle reduced width. In general, P(K) is to be interpreted as the Fourier transform of the wave function for the nuclide a with respect to the relative motion between the stripped nucleon, and a - 1, and
a-I K = kl - -- k. a 48.
The expression for the differential cross section, Equation 45, has the dependence on the scattering angle first found by Butler. Both the diffractionlike pattern, which it usually yields, and the magnitude are strong functions of the orbital angular momentum l of the state into which the neutron is captured, as Figure 2 and Table I show. The strong preference for smalll's shown in the table led Bethe & Butler (14) to point out that small admixtures in the bound-state wave function of angular momentum smaller than the dominant one can be detected easily in stripping reactions.
There appears in the Butler theory a radius parameter R (Eq. 42), whose value affects the positions of the maxima and minima in the angular distributions. One customarily seeks a value of R lying reasonably close to the nuclear radius R�1.3 Alia such that for some value of I, the theoretical and experimental main stripping peaks coincide. The radius so determined is
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-.: o-
s:: ::» >.. f -
:a .. c( ..... s:: o
4= u ell ." III 1/1 o ..
U
'0 0--. s;; � .. J! -Q
'\,/ V\�
,
I
/ "
/ o
GLENDENNING
1�1 /" r\ fo-\. 3 \� 0 X \ �
I \ / \ \ , I 1/ \ '\ 1\/
<
'/ I� 1 \ \
V\ / \ I \ 1\ \ , \ I � \f �\ \ \
V I' ) ' " ." � ",\ " --- ......... ��
20 40 60 80 Angle of Scattering (Degrees)
FIG. 2. Angular dependence of the Butler formula for several values of 1, with parameters Ed=8.8 MeV, Q=3.2 MeV, R=5 F. Relative heights. of the first peaks are not shown in the figure, but are listed in Table II. [From Butler (36). ]
Height
TABLE I RELATIVE HEIGHTS OF THE FIRST PEAKS IN THE BUTLER
THEORY FOR THE REACTION OF FIGURE 2
o 1 2
0.223 0.065
3
0.022
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NUCLEAR STRIPPING REACTIONS 203
called the stripping radius. (Of course it is necessary that R be equal to or greater than RN for the use of Hankel functions in Equation 42 to be correct.) In general one considers the agreement between the Butler theory and experiment to be acceptable if, when the radius R is chosen as above, the width of the main stripping peak and the position, but not the magnitude of the second peak are approximately reproduced. Data with which the Butler theory is in acceptable agreement are illustrated for several reactions in Figures 3 and 4 (140, 189).
The stripping radii found empirically are roughly given (8, 117, 145) by the formula
R = (4.37 + O.042A)F
R ... (1.7 + 1.22A 1/8)F
for light nuclei and bombarding energies above the Coulomb barrier. Recent data indicate that R is energy dependent; decreasing as the bombarding energy increases (77, 125, 167). It appears that for bombarding energies greater than 1 5 MeV on light and intermediate nuclei, the Butler theory gives less satisfactory agreement than for lower energies. At the higher energies the second peak of the Butler curve falls at too large an angle (see Fig. 10). Evidently distortion effects from the interaction of the incident and outgoing particles with the nucleus are no longer adequately simulated by the cutoff. In fact the cutoff simulates principally the absorption, but not the elastic scattering in the entrance and exit channels. The latter becomes relatively more important at higher energies where the mean free pat1{in the nuclear tail becomes longer.
•
e" GROUND STATE O·9.24Mev
• • •
••• •
. . . ..... -
°0 40 10 120 110
°0 40 80 120 160
CUTU-Of-MASS ANGLE IN DEGREES
FIG. 3. Angular distribution for BIO(d, p)Bll ground-state reaction is for incident energy Ed=10 MeV, Q=9.24 MeV. The stripping radius is R=S.2 F and l=1. [From Zeidman�et al. (189).1 The second part of the figure refers to the anomalous first excited state referred to in 2.9.
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204 GLENDENNING
5�----------------------�------�
4 '-... en
"-D E
-
z 3 0 -t-o W
PI
.I , 0
0 t CI) CJ) CJ) 2 0
0 0 a:: 0 -I o <t .... Z w a:: w u. L&.. -C
o
•
•
0
Zr90 (d, p) Zr"
Q=5.02MeV - BUTLER CURVE
l=2 ro =-6.5 x 10-15 em
o
0 0 0 0 0 0 0 0
0 0 0
90 120
BLAB (deg)
0
FIG. 4. Angular distribution for the Zr90(d, p)Zr91 ground-state reaction for incident deuterons of energy Ed=10.85 MeV, Q=4.91 MeV. Str.ipping radius of the Butler curve is R=6.S F and 1=2. [From Preston et al. (140).]
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NUCLEAR STRIPPING REACTIONS 205
The low energy limit on the valid region of the Butler theory is obviously connected with the Coulomb barrier. As Butler points out, the classical deflection angle in the Coulomb field for the stripping process is small at bombarding energies larger than the barrier height (36).
It has been observed, however, that for (d, p) reactions leading to levels for which the neutron is loosely bound (i.e., small Q = Bn- Bd), the Butler theory reproduces the observed angular distributions even at very low energies (166, 167, 183) (see Fig. 5). Wilkinson suggests two reasons for this:
(a) The wave function for the loosely bound neutron extends out some considerable distance from the main distribution of matter in the nucleus, so that the neutron can be stripped at a large distance where the distortion of the free-particle wave functions from plane waves is slight. [The stripping radius found in (166, 167) is �6.S Fat Ea= 1.5 MeV while it is R=4.2 F at Ea=14.4 MeV (l1S).1
(b) At low energy and low Q, the outgoing momentum of the proton will be close to one half of the momentum carried by the incident deuteron. No demand is made for momentum from the internal motion of the deuteron, therefore, and the neutron and proton can be far apart (cr1 = 4.32 F) at the time of stripping, again a situation for which distortion is small. The condition that the proton momentum be one half the deuteron momentum is fulfilled when E�-tQ and, as Warburton & Chase (180) point out, corresponds to the condition for closest approach to the stripping pole where, Amado (4) argues, the distortion effects should be at their minimum. To test these conjectures, Gibbs & Tobocman (66) have done a series of calculations in which the Q of the reaction was varied and in which the interactions in the incident and exit channels are taken into account. They found that these interactions were most important in producing deviations from the Butler theory just where the above argument suggests they should be least important. The apparent success of the Butler theory for these low energies, they claim, is due merely to the very simple shape of the angular distribution. However, Wilkinson has pointed out that the distorted-wave calculations (see next section) have always employed a zero-range force between neutron and proton to reduce the integrations (141) . Therefore the second effect depending on the separation of neutron and proton is not taken into account at all . Indeed, Oppenheimer & Phillips (138) found that the deuteron's large size is important at these low energies.
Generally speaking, the Butler theory does not yield a particularly detailed account of the angular distributions. However, in spite of this it gives reliable spectroscopic assignments for levels in light nuclei within a certain energy range, when handled in the fashion we have discussed. Its fruitfulness as measured by the wealth of spectroscopic information derived from its use cannot be chronicled. We merely cite several references. In the review by Macfarlane & French (117) , reduced widths (discussed in Sec. 2.7) for levels of light nuclei are tabulated and discussed from the point of view of their content of nuclear structure information, and a compilation of energy levels was
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206
C Q :0; Q
40
:030 � on e ,520 -• o l:Z 10 b
1.0
GLENDENNING
.NGUL.AR DISTRIBUTIONS
Li' (d.p) Lie _I': oJ.'.'. U1.LICIIOfI
'"TI. "cY. lit, IIIU"
�o 40 60 80 100 .20 '40 160"" 180
L4
CENTER-OF�MASS PROTON ANGLE
1 10
4' f:lCCITATION
CURVE INTERACTION
AADIUS
1.8 2.2 2.6 1.0 1.4 I.' 2.2 DEUTERON ENERGY IN MeV
z .•
FIG. �. Angular distribution for Li'(cl, p)Li8 reaction for bombarding "energies between 1.2 and 1.9 MeV. The reaction Q= -0.192 MeV. Soli9 lines are Butler curves for l = 1 corresponding to stripping radii shown in figure. Excitation cUrve is also shown. [From SeIlschop (167).]
made by Ajzenberg-Selove & Lauritsen (1) and Endt & Van der Leun (49) for which spectroscopic information from stripping reactions is given.
2.4 Distorted-wave method.-To obtain a detailed agreement between theory and experiment for stripping reactions involving iight and intermediate nuclei, and to extract spectroscopic information in the heavy-
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NUCLEAR STRIPPING REACTIONS 207 element region where the Coulomb force at the bombarding energies usually available is too important to be ignored, it is necessary to take proper account of the interactions in the entrance and exit channels. That is to say, the scattering and partial absorption of the deuteron before the actual stripping event, and of the proton after this event, can have an important effect on the differential cross section. We saw that the cutoff procedure used in the plane-wave treatment simulates distortion effects due to absorption of the lower partial waves. However, the optical-model analysis of elastic scattering of nucleons from nuclei indicates that the nucleus is not a perfect absorber (53, 54, 67). The incident particles may interact without being absorbed. Moreover, the elastic cross section is usually much larger than the stripping cross section. Thus, although the lower partial waves may still be strongly absorbed, the higher ones, which are of importance for stripping, may be distorted from the plane-wave form.
The distortion of the wave functions is caused by two interactions, the long-range repulsive Coulomb force and the short-range attractive nuclear force. At energies sufficiently below the top of the Coulomb barrier, the deuteron and proton will not come close enough to the nucleus to be much affected by the nuclear force. In this case, most readily attainable with heavy targets, the distortion can be treated as purely Coulomb (12, 17, 114, 171). (vVe avoid the misnomer "Coulomb stripping," which implies a conceptually dilIerent process.) The angular distributions are backward peaked, and devoid of structure (50, 169). Above the limit for pure Coulomb distortion the nuclear force becomes important also (176). However, for light and intermediate nuclei at energies in the vicinity of the top of the Coulomb barrier, it is apparently possible for the two interactions, one attractive, the other repulsive, to approximately cancel each other for the partial waves of most importance in stripping. In such cases the Butler theory gives a good account of the angular distribution (but not the magnitude of the cross section). This cancellation is illustrated in calculations by Tobocman (175) for Ca44(d p) and is shown in Figure 6. The Coulomb barrier is about E�1.08 ZZ'/Al/3=6.1 MeV, and the bombarding energy is Ed=7.01 MeV. Indeed, there will always be a partial cancellation of the effects of the two interactions, but in heavier nuclei it is not so exact that the angular distribution has a Butler-like form. The appropriateness of the various calculations is roughly summarized in Figure 7 where we plot the boundaries of the several types of calculations in the energy-mass plane. The boundaries are Q dependent: that is, the larger the Q of the reaction, the more energetic is the outgoing proton. Thus, for a bombarding energy at which the angular distribution of the protons is of the pure Coulomb type for an excited-state transition, the more energetic protons from the ground state may penetrate sufficiently into the interior region to feel the nuclear force. Their angular distribution may therefore deviate from the pure Coulomb type. This phenomena can be seen in the data of Stokes, for example (169).
The distorted-wave method for taking into account the scattering of the
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. _. Ct> a:
o T i!
•
•
•
•
AII'tlW911W
• I
I
0 • -.., c g ::! Q)
a
s
FIG. 6. Cross section and polarization for Ca44(d, p)Ca45 calculated for Ed=7.01 MeV, Q=3.3 MeV, 1= 1, and R=6 F. (a) Butler theory, (b) Coulomb distortion with cutoff, (c) Coulomb and nuclear distortion with cutoff, (d) Coulomb and nuclear
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.. .
o
without cutoff. Experimental points are normalized to calculations since the magnitude of cross section is not measured. Optical-model parameters are listed in· the original paper. [From Tobocman (175).]
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210 20
IS
16
14
Ed 12
(AElWl 10
8
6
4
2
00
GLENDENNING
Nuclear (+ Coulomb)
Butler
40 80
. \OUlomb bOr�!�� •••
• ••• ••• • •
.' .' .' .' .' .' .' .... Nuclear • •• •
• ,.;. Coulomb ..........
. .. .'
Pure Coulomb
120 160 A
200
FIG. 7. The energy-mass plane is divided into various regions according to what treatment need be accorded the nuclear and Coulomb distortions. Boundaries are only roughly definable and are somewhat Q dependent as discussed in the text.
incident and outgoing particles by the nuclear field was first used for the scattering of electrons by Mott & Massey (127). A preliminary investigation of the role played by distortion in stripping reactions was carried out by Horowitz & Messiah (95) who replaced the plane-wave function for the proton with that of a proton scattered by a hard sphere. Tobocman (174, 177) formulated the complete calculation and, in a series, of papers, investigated the distortion effects over a wide range of energies and atomic masses (66, 175, 176, 177).
A formal derivation of the distorted-wave method is given in the Appendix. In Section 2.1 we saw that the differential cross section depends on a set of amplitudes Br for the capture of a neutron with quantum numbers (lm) (Eq. 11). These amplitudes are matrix elements containing the wave functions for the free parti<;les. By neglecting their interaction with the nucleus, we obtain the plane-wave or Butler theory of Section 2.2. Here we want to include these effects. In this case the wave fu�ctions if;p and if;d satisfy Schroedinger equations containing the interaction of these particles
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NUCLEAR STRIPPING REACTIONS 211 with the nucleus, and they are therefore distorted from the field-free plane
wave solutions. Assuming that the interactions are central, the Schroedinger equation separates in spherical coordinates. Therefore, we expanded the wave functions (analogous to Eq. 35).
tfa<+) = h L i).!A(k,d?) YIf(R) YIf*(kd) N<
tf,,(-) = 4". L .jXf)"(k,,r,,') YIf(;,,)Y1f*(£,,) N<
49.
and the radial functions satisfy � d2 1(1 + 1) 2!Jj ( zZe.) � -- + --- + - Vj(r) + - - kjt r/l{kfT) = 0
dr' r2 Ii' r 50.
with /J.j, kj the mass and wave number in the channel j = d or p. Here Vi is the nuclear interaction of particle j with the nucleus while zZe2/r is the Coulomb interaction. Solutions to this equation are obtained by numerical integration, but we shall not discuss the execution of this purely technical
problem. Upon substitution of the above expansion into the expression for Br
CEq. 11), it is at once apparent that the integrations are much more difficult than in the plane-wave case. In the latter, the fact that Rand Tp' are contained in the exponentials of the plane waves meant that we could easily transform to new variables of integration Tn and T on which the bound-state and deuteron wave functions depend, respectively. Here no such easy transformation is possible and all published results based on the distorted-wave treatment of stripping employ the zero-range approximation. Calculations for a finite-range force have been reported to be in progress (76). We introduce a zero-range force defined by
Ii" Vnp(r)4>d(r) = - (8,..,,)112 -- aCT) 51.
2m*
where oCr) is the Dirac delta function and a and m* were defined in Section 2.3. (The wave function Eq. 33 is the corresponding deuteron wave function.) We are now able to transform the integration in Equation 11 to new coordinates Tp' and R on which the distorted waves depend. The volume element and ij function tr ans form t o
52.
th en
53.
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212 where
GLENDENNING
J " (Mlkp ) R)./z). = 0 1>.' My r uz(r)!>.(kdf)r2dr 54.
In the plane-wave treatment, the only effect of the finite range of the force was to introduce a monotonically decreasing function of angle G(K) (Eq. 3 1) . Under the above assumption of a zero-range force, this becomes simply a constant
Go = - (81rOl) 1/2/N2m* 55.
The damping of the cross section at large angles due to the distribution of momentum in the deuteron should be present. If this were the only effect of the finite range, we could correct our result for Br by multiplying it by the factor
G(K) K2 + Ol2 � = (&cOl)1/2 P(K) 56.
Undoubtedly there will be other effects on the angular distribution, particularly at energies near or below the Coulomb barrier, as discussed near the end of Section 2.3. For higher energies we hope that the success of the distorted-wave calculations in reproducing in detail the experimental angular distributions means that the finite-rangc effects are of minor importance.
2 .5 Distorting potential in stripping reactions.-The optical model has proved to be a successful way of describing the elastic scattering of light nuclides from complex nuclei (53, 54, 67). Accordingly, in performing a distorted-wave calculation, one usually adopts the optical. potential as a representation of the interaction in the entrance and exit channels (Eq. 50) . Generally one employs the optical-model parameters which best represent the elastic scattering at the energies relevant to the reactions. Although elastic scattering data are not always available at the required energies, the slow variation with energy of the optical model allows one to interpolate. In this way no new free parameters are introduced, in principle. However, there are often several sets of optical-model parameters that yield an equally satisfactory account of any given set of elastic scattering data. The reason for this is that the elastic scattering determines the phase shifts at infinity that the scattered waves have experienced. But these are not sufficient to determine the potential uniquely. On the other hand, the reactions are sensitive to the wave functions in the vicinity of the nucleus. This is just the region where wave functions belonging to different potentials that have the same phase shifts will themselves be different. Accordingly reactions impose additional constraints on the optical-model parameters.
Several groups of workers have illustrated the sensitivity of stripping reactions to potentials which are equivalent in the sense that they yield the same elastic scattering (5, 85) . This can be seen in Figure 8.
The role that the various optical-model parameters play in determining the shape of angular distributions has been investigated by Tobocman &
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Z5
20
VI !:: z ::;) >- IS � • Ill:: t- t CD Ill:: •
10 Q) b
, I I
5 I I I I I
, ,
0 20
NUCLEAR STRIPPING REACTIONS
Se16 ( d . p ) Se77 GROUND STATE
Ed - 7. S MeV . Q- S. 1 9 MeV I PLANE WAVE
II D.W.B.A., MEASURED PARAMETERS m D. W.B.A., BEST FIT
n
I
213
.,� .,-:-re ...... .. � - - - - - -
4 0 60 80 100 120 140 160 180
ANGLE (c,. '""�) DEGREES FIG. 8. The distorted-wave calculations were obtained with different deuteron
optical potentials which give essentially the same elastic scattering. (II) V = 75.8 MeV, W=16 MeV, fo = 1 .48, a = .55 F; ( I I I ) V=65 MeV, W= 16 MeV, Yo = 1.35 F,
a = .6 F. [From Hinds et al. (85).1
Gibbs ( 176) for the reaction Ca4°(d, p) Ca41 at Ed = 4. 13 MeV, Q = 4. 19 MeV. In Figure 9 a comparison is made between the three calculations : Butler, the distorted wave, and the distorted wave with cutoff (that is, the radial integrations are not carried throughout the nuclear volume as in Sec. 2 .3) . In this calculation we see that the formation of the second maximum is due p ri ncipally to contributions from t he nuclear i nterior. A strong second
maxi m um i s not uncommon a t energies below t he Coulomb barrier b ut not
so low as to fall within the pure Coulomb region [see, for example, Pb207 and Ti48 in (175)]. Since the second peak is caused mainly by contributions from the nuclear interior, it is very sensitive to changes in the optical potentials,
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214 GLENDENNING
•
" O../'!" '.tJJl . ..... . . . ..... .
....... .......... _--_ ...... _ -
.» : -U, - I I . So " . .... r , t -1.1, -" • •• " . .. �
FIG. 9. Differential cross section and polarization for Ca'�(d, p)Ca41, Ed =4.13 MeV, Q=4.12 MeV, and 1= 1 . Solid curve is distorted-wave calc:ulation, dashed curve is same except with cutoff, and dash-dot curve is the Butler curve. [From Tobocman & Gibbs (176) . ] '
and p ar tic ul arly to th e im aginary p art. Th e im agin ary p ar t, of co ur se, govern s th e m agnitud e of th e inter ior con tr ib ution s inc e it de term ine s ho w
m uch of th e flux is absorbed in to o th er re ac tion s. Reac tions in th is ener gy re gio ,a (s ee Fig. 7) prov ide , therefor e, a me an s of study in g th e op tic al -mo del
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NUCLEAR STRIPPING REACTIONS 2 1 5
parameters. The above results should not be construed a s invalidating our earlier arguments for introducing a cutoff in the plane-wave calculation to simulate absorption of the lower partial waves. In the above reaction the incident momentum is very low kpO.5 F-l. The total number of partial waves that can contribute is bounded by 2kRN which for this case is about five. In the above low energy reaction, only a very few partial waves can interact with the nucleus ( :::;; 5) ; and if we use a crude device, the cutoff, to simulate the absorption of the lower ones, it not unexpectedly will yield poorer results than if many partial waves are involved, which is the case in the region of applicability of the Butler theory.
2.6 Some examples of distorted-wave calculations.-The distorted-wave method usually gives a satisfactory account of the angular distributions in stripping reactions and we shall illustrate this by a few examples drawn from the considerable number of calculations which have been done.
We have already seen in Figure 3 a comparison of the Butler theory with the data for the BlO(d, p) B10 ground-state reaction at Ea = 10 MeV. At this
. energy the agreement is satisfactory; but at higher energies, as shown in Figure 10, a considerable adjustment in stripping radius from R = 5.2 F to
" I I
I
o 20
FIG. 10. Angular distribution of BIO(d, P)Bll at several energies. Compare with Fig. 3 at lower energy. The curves are from distorted-wave calculations except for the one marked 'Butler.' Optical-model parameters are given in original paper. [From ZeidIIlan�et al. ( 190.)]
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GLENDENNING 216
��������������-P�
Zr�d.p) Ed · 10.85 MeV
4
2
o 160 180
FIG. 1 1 . Distorted-wave calculation for Zr90(d, p)Zr91 ; compare with Fig. 4. Opticalmodel parameters are listed in original paper. [From Smith & !vash (168). ]
3.5 F has to be made to retain agreement with the principal peak. But now the second peak of the theory falls at an experimental minimum. The distorted-wave calculation, however, is in very good agreement with the experiments (190) . The optical-model parameters for the calculations illustrated were allowed to vary linearly with the energy in a fashion expected from experience.
In Figure 4 we saw a comparison of the Butler theory with the Zr9°(d, p)Zr91 reaction. A distorted-wave calculation for this reaction by Smith & Ivash is shown in Figure 1 1 . These authors have published a number of calculations in the mass region A �59 ( 168) .
For the heavy elements we show the Pb206 Cd, p) Pb207 results at an energy near the Coulomb barrier in Figure 12. Miller et al . have compiled data for this reaction for incident energies ranging from 8.3 to 15 MeV, and report distorted-wave calculations which generally confirm the usefulness of these experiments at the higher energies (124) . At lower energies the Coulomb force causes the angular distributions to become backward peaked and they lose their structure (12, 17 , 1 14, 1 7 1) . This is illustrated in Figure 13 by Erskine et al. (SO) whose experiments we shall discuss in Section 2.7 .
2 . 7 Nuclear structure and spectroscopic factors.-Because stripping reactions like all direct processes are characterized by the fact that only several nucleons are actively involved in the reaction, overlap integrals involving the passive nucleons must always enter in the expression for the cross section. Such an overlap measures the degree to which the passive nucleons occupy the same configuration in the initial and final states. This overlap for stripping, called the spectroscopic factor (1 17), is denoted by S and was
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NUCLEAR STRIPPING REACTIONS 217 introduced in Section 2.2.
S'I2(lj) = (A + 1) 1/2 I i11fJ,iIlJI''IfJ,d(A + 1) 57.
Here .p is a wave function constructed by the vector coupling to spin J, of a particle in the state lj to the target wave function of spin J,.
If the radial cutoff approximation is made, a second factor, the singleparticle reduced width
58.
also appears as a multiplying factor. We recall that in the cutoff approximation, the radial function of the bound neutron, uz, is required only at the one point R, where it normalizes the Hankel function to which it is proportional outside the nucleus. The cross section, in this approximation, is proportional to the reduced width
.. lit ... ,Q e 3
'0 ... b . '0
I'P ,------,------,...----,-----,.------,-----, 0'8 0-6
0'2
0·1 0'08
0'06
0-04 Pb206 (d p) Pb207
Q = 4'51 MeV Ex ': 0 P I1Z Ln = I
Ed = 14 '0 MeV
59.
O·020�-----=3;':::O:-----::6:':O:----...,9:-':O::----�12--0----:-:15:-:0:-----::!180 8 ( OeQrees)
FIG. 12. Angular distribution for Pb·06(d, p)pb,07 reaction for Ed = 14 MeV, Q = 4.51 MeV. Solid curve is distorted-wave calculation for 1 = 1 . (From Miller et al.
(124). ]
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218
c .2 � 1.0
o
GLENDENNING
ANGULAR DISTRIBUTION 8i 209 (d, p) Bi 210 RE ACTION
Ed • S.O MeV
• Q Q - 0.203 MeV, .! :2 0 6 Q 1:1 0. 397 MeV , '; = 2 • Q :; 1 .936 MeV, .1 = 4
/
180
FIG. 13. Angular distribution for Bi209(d, p)B210 reaction at Etl=8 MeV. The Coulomb barrier is about 15 MeV; consequently it is the dominating distorting interaction. Under such circumstances the angular distribution is backward peaked. Solid curves are distorted-wave calculations. [From Erskine et al. (50). ]
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NUCLEAR STRIPPING REACTIONS 2 1 9
which i s related to the width o f Lane & Thomas ( 1 12) through
3 h' 'YI = - __ 82 2 MR' 60 .
In the noncutoff theory of stripping, the single-particle reduced width does not appear, of course, because the wave function Uz is required at all points, not just those exterior to R. Therefore the non cutoff theory, insofar as the interior region is actually important, is capable of yielding more detailed information than (}02 contains. As we have already implied in our discussion at the end of Section 2, the interior region may be more or less important depending on where in the energy·mass plane of Figure 7 the experiment is done. Tobocman has illustrated how the angular distribution can be influenced by the shape of the wave function for the neutron under favorable circumstances [Fig. 13 of ( 1 75)].
The analysis of reduced widths falls into two parts. First, empirical values of the widths are determined by computing the other factors in the cross section and adj usting the factor S as required to obtain agreement with experiment. The first systematic attempt to measure spectroscopic factors was performed by Holt & Marsham ( 8 8-92) ; see also ( 2 5 , 1 17) . The second part of the analysis consists in attempting to interpret the significance of the spectroscopic factors for the nuclear wave functions. This may be done by computing them on the basis of a nuclear model and comparing the answers with the empirical results ( 6 , 41 , 61, 117 , 1 18, 144).
The extraction of reduced widths from experiments can be effected through use of either the Butler or distorted-wave theory. Until the recent accessibility of electronic computers, the Butler theory was used almost exclusively ( 1 17) . It is appealing because of its simplicity. However, it is inadequate in several ways as has already been discussed elsewhere ( 1 17) . First, it is limited to the l ight and intermediate elements at the energies currently available. Second, the Butler theory overestimates the cross section, sometimes by an order of magnitude, and therefore underestimates the reduced width (}2 ( 9 4, 1 72) . Figure 14 illustrates this point. Third, because of the approximations made to obtain a simple result, the cross section does not possess the correct dependence on energy, nuclear charge, and Q value; therefore corresponding artificial dependences are introduced into the widths. Accordingly, when one extracts the widths via the Butler theory one abandons the interpretation of 002 as a single-particle reduced width. Instead it is to be determined from experiment as a function of various parameters including its proper dependence on the single-particle quantum numbers, etc., and its artificial dependence on bombarding energy, etc. [See ( 1 1 7 ; Sec. 1 1 1-2) or ( 7 7).] By its artificial dependences it is supposed to compensate for the shortcomings of the theory. Having once established the relevant 002, possibly by extrapolation from neighboring nuclei, one can then deduce the spectroscopic factor S. Aside from the loss of information (the actual value of (02) , the philosophy is good. Indeed, as has been emphasized by Cohen ( 4 1 , 124)
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220
4
z
GLENDENNING
/\ I \ i \
0'6 , ... , ) 0" L" 2 Eo -I MeV
i \ i ,(NO DISTORTION; CuT OFF & F
I ' •
I ' . \ I ' j \ jNUCLEAR \
• \ DISTORTION tCOULOM8 DISTORTION ONLY I ' ......... ONLY .,,- - / I I \x/ --- '-., 1 / \ -- -
I / \ ' , I \ \ I " / \ COULOMB AND
, / � \ NUCLEAR DISTORTIONS /,1 � '\ " # ". ....... ---. -.- � -.--,
1..0 160 180
C£NTER-OF; MASS ANGL£ : DEGREES
FIG. 14. Calculations for OUCd, P)017, illustrating reduction of cross section by the distortion effects. In this case the Coulomb and nuclear distortions together retrieve the Butler shape but with a magnitude reduced by 5. [From Buck & Hodgsen, (29). ]
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NUCLEAR STRIPPING REACTIONS 2 2 1
from thc point o f view o f spectroscopy, i f one knew the differential cross section corresponding to all relevant angular momenta transfers as a function of bombarding energy, Q, Z, etc., one would not need a theory by which to calculate the amplitudes Br. Their squares would be experimentally available. So much data is not, however, available except in isolated regions.
Now that high speed computers have become available to a greater number of researchers, undoubtedly the extraction of spectroscopic factors will be effected ever increasingly through use of distorted-wave calculations. Even here there remain ambiguities, especially concerning the absolute values of the spectroscopic factors. The obvious sources of uncertainty are as follows :
(a) The optical-model parameters characterizing the distortion are not always unique. The stripping reaction may then serve to discriminate between several sets of parameters that fit the elastic scattering equally as well (recall Sec. 2 .5) . If, however, the stripping cross section is not measured to high precision, this discrimination may be not possible. The cross sections calculated with the several sets of parameters may differ by a factor of two or so (103) with a corresponding uncertainty in S.
(b) The wave functions U I for the captured neutron are not known. This together with S is an object of interest in the analysis. Even if the shell model of the nucleus can be used to suggest which orbit is in question, uncertainty in the wave function remains; what is the shape of the nuclear field and its radius, for example?
(c) Use of the zero-range force approximation underestimates the cross section (expecially at energies much below the Coulomb barrier) . One can estimate the effect on the distorted-wave calculation by noting that for plane waves the cross section would be underestimated by (see Sec. 2.3) :
",ma[t + (K/7a)2]2 which at the main peak is typically a 20-30 percent correction. If the prescription suggested in Section 2.4 for approximating the correction duc to finite-range force has been applied, this effect is already accounted for. However, at energies much below the Coulomb barrier such approximate ways of simulating finite-range effects probably are poor.
Scott has reported the extraction of reduced widths in the nickel isotopes by use of the distorted-wave method (164) . In Table II we show his results. For the Butler analysis, (2J,+ 1)02 is shown, which has to be divided by the appropriate value of the single-particle width 00' before a comparison can be made with the distorted-wave evaluation of (2J, + l)S. The relative values are roughly comparable, although here again variation of (M with radius R should be taken into account before the comparison can be taken seriously. The absolute values of (21/ + l)S are subject to the uncertainties discussed above. For example, Cohen reports values of the spectroscopic factor which differ by almost a factor of two from Scott's, presumably because different optical-model parameters and possibly radial functions UI, were used (43).
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222
Final nucleus
Ni69
Ni81
Ni69
NiBI
Ni69
Ni8I
Na24
GLEN DENNING
TABLE II
EXTRACTION OF REDUCED WIDTHS IN NICKEL ISOTOPES WITH DISTORTED-WAVE METHOD"
Butler Distorted-wave
Excitation Q l (2J,+1)0" (2J, + l)S (MeV) (MeV) R
Absolute Relative Absolute Relative
.0 6 . 779 1 6 .4 .035 100 1 . 61 100
.466 6.313 1 6 .4 .019 54 ' .79 49
.888 5 .899 1 6 .4 .0039 1 1 . 20 12
.0 5 . 598 1 5 . 7 .024 69 1 . 1 1 69
. 284 5 . 3 14 1 5 . 1 .019 54 .82 51 1.104 1 . 137 4 .45 1 4 . 8 .0073 21 .31 19 1 . 190 ;
.-5 .69 1 .09 0 4 .0 .0077 100 . 18 100
f 3 .07 2 .53 0 5 . 1 .0022 29 .052 28 4 .91 .69 0 3 .75 .0086 112 .25 133
. .341 6.438 3 6 .4 .023 100 2 .64 100
.068 5 .530 3 5 .7 .037 161 2 . 87 109
.0 4 .731 2 5 . 2 100 g 100 .472 4 . 259 2 5 . 2 51 . 5 57
a See Scott (165).
The interpretation of the spectroscopic factors in terms' of nuclear structure has been discussed by a number of authors on the basis of the several models of the nucleus : the shelI model (9, 20, 2 1 , 41 , 42, 48, 69, 61 , 1 10 , 1 1 1 , 1 17 , 1 18, 136, 137 , 144, 146, 153) , the rotational model ( 1 16, 154, 155, 161 , 186) , and the vibrational model (187). In particular, the work of Macfarlane & French ( 1 1 7) and of Lane ( 1 1 1) contains large sections devoted to the calculation of reduced widths from specific-model wave functions. In general such calculations involve recoupling of angular momenta and fractional parentage expansions and consequently are exercises in the techniques of Racah (46, 142, 143, 150). We shall be brief in our discussion since the subject has already been treated extensively.
We have chosen to discuss nuclear wave functions on the basis of j-j coupling but as already mentioned our results are general, being related to
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NUCLEAR STRIPPING REACTIONS 223
other coupling schemes by a unitary transformation . The spectroscopic factor for stripping into the j orbit is, according to Equation 24,
S(jl) = (A + 1) (P X j I mal = n;(P X j I D)2 61 .
where we have introduced a more schematic notation. Here A + 1 denotes the number of nucleons in the heavier nucleus, I D) denotes the wave function of the daughter nucleus (final nucleus in d, p) while I P Xj) is its parent, being formed by vector coupling the lighter nucleus (target for d, p) to the extra nucleon in the j orbit to give the spin of D. The subscript a denotes that I P) and I D) are antisymmetrized wave functions for A and A + 1 nucleons, respectively. The extra nucleon j is not i ncluded in this antisymmetrization, account of this neglect being taken by the factor (A + 1) as concerns the simple direct stripping (see Sec. 2.2) . The second overlap integral in Equation 61 is to be computed with wave functions that are antisymmetrized only with respect to nucleons in the same shells. The two overlap integrals are connected by the factor (n;/(A + 1))1/2 where nj is the number of nucleons of the heavier nucleus which occupy the j shell (61, 1 1 7) . Obviously i t i s more convenient t o work with the second overlap. In case the isotopic spin formalism is not used, A and nj refer to the number of particles of the type that is stripped.
We take from the literature (21 , 1 1 7) the following example for which the spectroscopic factor can be deduced from inspection of the wave function. The reaction is 0'7(d, p)0'8from the 5/2 +ground state to the 2 + excited state at 1 .98 MeV. The Butler theory was used to analyze the experiment, and it was concluded that the transition involved two fs, namely, 1 =0 and 2, and the ratio of the reduced widths was deduced as 82(2)/82(0)"-'4.4. I n the region of oxygen the Id5/2 and 2S1/2 levels lie close to each other. The authors postulated that the wave function for 018 is
i 018) = a I (<<6/22)2) + ·b I (<<6/2S1l2)2)
while that for 017 is simply I d6/2). The spectroscopic factors are :
therefore
S(2) = 2(0'7 X d6/2 1 0'8)2 = 2a2
S(O) = (0'7 X S1I2 1 0'8)2 = b'
82(2) =
2a2 802(2) 82(0) b2 802(0)
Macfarlane & French ( 1 1 7) estimate that
2 < 802(0) < 3 - 802(2) -
which yields for the wave function
0.43 � I bl � 0.36, 0.89 ::::; I a l ::::; 0.93
As another example, consider a simple situation : beyond the closed shells of the nucleus P, n neutrons occupy the state j and have total angular mo-
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224 GLENDENNING
mentum Jp. A nucleon is stripped (or picked up) from the same shell, the daughter nucleus being in the configuration (jn+l)J D. Then
S(lj) = (n + 1) «jn)�'Jp X j I (jn+!)aJ D)' 62 .
where a denotes any additional quantum numbers that may be required to form a complete set (for example, the seniority) . To evaluate the integral, we introduce the fractional parentage expansion
I (r+!)aJD) = L «jn)cx"J,jjJD I I (r+!)aJD) I (jn)cx"J X j) 63 . a" J
where ( I } ) are fractional parentage coefficients which have the orthonormality property
L «r)cx'J, j; JD I I (jn+!)aJD)! - 1 64. ,,'J
Then we obtain at once
S(jl) = (n + 1) «jn)cx'Jp, jj J D I } (jn+!)aJ D)' 65 . In the case of pickup reactions, JD is the target and we obtain a sum rule for the final states
L: S(jl) = n + 1 66. alJJ
This provides a check on the empirical reduced widths. The application of many types of sum rules to stripping reactions may be found in the review of Macfarlane & French ( 1 17) .
For the states of lowest seniority (even particles coupled to zero) , an explicit formula exists for the fractional parentage coefficients ( 165)
.n • • .n+! . ( 2j + 1 - n ) 1/2 «J )O,}iJ I } (J )J) = (n + 1)(2j + 1) , n = even « jn)j, j, ° I I (jn+l)O) = 1, n = odd
which yield at once
S ( + 1) 1 1 - n/(2j + 1), j n +-7 n = n + 1, n = even n = odd
67.
68.
If one takes into account the residual interaction between nucleons in the shell model, there will always be some configuration mixing in the wave functions. If one adopts the pairing force as an approximation to this residual interaction ( 13 , 108) , one can also obtain simple expressions for S which for even targets is proportional to the probability that the level j is unoccupied in the target, and for odd targets, that it is occupied in the final nucleus (41 , 187).
As a final example of the calculation of spectroscopic factors which involves the use of the Racah techniques, consider the C)35(d, p)C)36 groundstate transition analyzed by Okai & Sano (136, 137). For the target wave function they postulate
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NU CLEAR STRIPPING REACTIONS
I ClM: 3/2 +) = l ?rdm, "[(S1l22)0, (da/22)0]0: 3/2) 2
+ L O:J l ?rdw, "[S1I2, (d3/23)3/Z]J; 3/Z) J�I
and for the residual nucleus
I ClM; 2+) = l 'lrd'/2, "[(S1l.2)0, (d3i23)3/2]3/2; 2)
225
where 7f' and v indicate proton and neutron configurations, respectively, and the vector coupling notation used for the wave functions is obvious. The observed differential cross section, Figure 15, when analyzed with the Butler theory, requires an admixture of l = 0 and 2 contributions. The residual state is reached by l = 2 stripping from the first term in the Cpo wave function and by l = O from the second. We now derive the expression for the S connecting the two states :
I P) = I Jp', [(j,n,)h', (N,-')Jz']Jn'; Ip) I D) = I Ip, [U,n,)l" (j,n,)l.]ln; ID)
69.
Applying the fractional parentage expansion to (j.no) and using twice the recoupling coefficient
Ca, (be)B, JI (ab)A, e; J) = U(abJc; AB) = «2A + 1) (2B + l» '/'W(abJc; AB)
where W is a Racah coefficient (142, 143) , we find
I D) = L «Nz-')J.", h; J2 1 I CN,)Jz) U(J,Jz"Jnh; IJ2) U(JpIlDh; KJn) Jt1." IK I {Ip, [(N,)]" (j2n,-I)U'jIIK, j2; ID)
The wave functions on the right are in the form I P' Xi.) which is what we sought because the overlap 'can now be calculated immediately :
Thus we find
(P X j. , D) = aJpJp,aJ,J,'«N'-')J.',j2; J2 1 1 (ib)J.) U(J,Jz'Jnj.; In'J2) U(JpJn'Inj2; Ipln) 70 .
Sed) = 3«d3l22)0, d3l2; 3/2 1 I cd3l2')3/2)2U2(00 3/Z 3/2; 0 3/2)U2(0 3/2 3/2 2; 3/2 3/2) = 1/2
and
S(s) = 2 [ � aJ( -)JU(1/2 3/2 1/2 3/2; JO) U(J3/2 1/2 2; 3/2 3/2)] 1 _
= - (v'3 al - 0:.) 2 5 The authors estimate al and a. by perturbation theory to be al = 0.061
and a. = 0.389, using an energy difference sl/. -da/. = 0.841 MeV to obtain S (s) = 0.016.
To estimate the single-particle reduced widths Oo'(s) , 00' (d) , the authors use a square well ( 136) . Thus the reduced widths 02(S) , 02(d) are calculated
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226 GLENDENNING
"='· 20 'S :s
>. till-O)-Hi" -�2) w ell ... � . -..0 ... 15 ClI "-'
C 0 . -
� 1,,-2 en
� 10
Cd . -C " ...
� .5 a J \ / \-- //1-0
I '\ . ....
20 40 60 80 FIG. 15. Angular dhitributions for CI3G(d, P)Cl"'i, Ed =6.9 MeV, Q=6.3 MeV.
Butler curve is a combination of 1=0 and 2 with very small amplitude for 1=0. Radius is R = 5.5 F. [From Okai & Sano (136) . ]
and can be inserted :lnto Equation 4S to obtain the solid curve shown in Figure 15 . We see here an example of how a very small spectroscopic factor for the s state can nonetheless be detected in the presence of the d state because of the larger .intiinsic probability (measured by Br) of the former.
The interpretation given above for the CI35(d, p)C136 r:.eaction is not the only one possible. An alternative one has been suggested (121) based on the possibility that CIM if: a spheroidal nucleus. In this case the single-particle
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NUCLEAR STRIPPING REACTIONS 227
wave function has several values of particle angular momentum mixed into it, there being a sharing of the total angular momentum between the particle motion and the rotational motion of the nucleus (134). The cross section for stripping into such states therefore has contributions from the several l's that are mixed (154, 155, 161, 186).
We conclude this section with a discussion of the Bi209(d, p)B210 reaction and its interpretation. In the most recent, high resolution experiments, some 40 levels were observed in Bi210 up to an excitation of 3.2 MeV (50). Early low resolution experiments exhibited four groups of particles which were interpreted as stripping of the neutron into the g9/2, d6/2, g7/2, and d3/2 orbits (80). Better resolution experiments identified two additional groups attributed to iU/2 and S1/2 capture (87). Within each group there are many levels arising out of the different spin states of the nucleus that can be formed by the odd proton in the k9/2 level and the stripped neutron. The spectroscopic factor is unity for all levels, and the amplitude B,m is the same for all members of a given group aside from the different proton energies. Since the width of each group is typically .5 MeV and the bombarding energy is 10 MeV, the variation in proton energy across a group is not expected to affect Br. Hence each level in a group should be excited with an intensity proportional to 2J,+1 according to Equation 22. The shapes of the groups in the lower resolution experiments were interpreted on this basis (87), and results consistent with shell-model calculations (130) were obtained. The results of the high resolution experiments are shown in Figure 16. The group at lowest excitation contains ten levels (the level labeled 5 has such a width as to indicate that it contains two). They are attributed to neutron capture into the g9/2 level in agreement with the shell-model sequence of singleparticle states (47). The resulting proton-neutron configuration k9/2g9/2 is split into ten levels with spins from zero to nine. The spin assignments are made according to the relative intensities of the levels and the latter are compared with (2J, + 1) in Table I I I . The agreement is excellent.
The intensities of the various groups vary considerably. The two levels at Q ",, -0.2 MeV which are attributed to S1/2 capture are the strongest in the spectrum. The group at Q�+1.1 MeV, most of whose levels are attributed to ill/2 capture, are the weakest. (Note that the lines in the right half of Figure 16 are multiplied by 10.) This dependence on the orbital angular momentum is. of course. expected. The incident energy. 10 MeV, is far below the Coulomb barrier of 15 MeV.
-The classical trajectories under these cir
cumstances do not approach the nucleus sufficiently closely for the neutron to be captured, except for small impact parameters, and for these the deflection is very large. For large scattering angles the linear momentum transfer is large. �kp+k�1.5 F-l ; but because the impact parameter is small. small angular momentum transfer is highly favored. The angular distributions for three values
' of 1 are �hown in Figure 13. The characteristic dif
fraction-like pattern which allows an identification of the angular momentum for bombarding energies above the "pure Coulomb" region (see Fig. 7) is
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-c: o i :iI!�OO .. 8. ... c j e .W !IOOO .8 l g � 5CO C � .. is
o
27
26
I 38
31
39 40 35
33 28
29
ENERGY LEVELS IN Bi210 FROM
THE Bi 209 (d,p) Bi210 REACTION
Ed " S.O MeV e - 172.5 ° 5
18,19
2
8 7
20 22 6
4
2524 21 17 3 10 0
.IDo -Q� 0 '.0.50 .1.00 •• 1.50 '.2.00 '.zjo o Value (MeV)
� --. 3.0-' is . - . 2.0'--'-' . 1.5 ' 1.0 0.5 b
Extitction Enerqy (MeV) FIG. 16. Proton groups corresponding to levels in Bi210. Absolute differential cross sections measured at 172.5 deg and S.O-MeV
bombarding energy are indicated by height of lines. [From Erskine et al. (50). ]
N N 00
� Z t::l tr. Z Z Z Q
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Level
Q 1 2 3 4 5 6 7 8
NUCLEAR STRIPPING REACTIONS
TABLE III
SPINS OF B210 LEVELS OF h9/. gO/2 CONFIGURATION [After Erskine, Buechner & Enge (50)]
Excitation Spin
229
Experimental energy (MeV) JI
2J, + 1 relative intensity&
0 1 3 2 . 7 ± 0 . 3 0 . 047 0 1 0 . 7 ± 0 . 2 0 . 268 9 19 18 . 3 ± 0 . 6 0 . 320 2 5 4 . 3 ± 0 . 3 0 . 347 3 7 7 . 1 ± 0 . 4 0 .433 5+8 28 28 . 4 ± 0 . 8 0 . 501 4 9 10 . 0 ± 0 . 5 0 . 547 6 13 1 3 . 4 ± 0 . 5 0 . 581 7 15 15 . 2 ± 0 . 6
• Refers to differential cross section at 172.5 deg and 8.0-MeV bombarding energy.
here absent because of the dominating importance of the Coulomb barrier. In addition to the variation of intensity with 1, discussed above, the differential cross section also shows a weak angular dependence on 1: Erskine et al. were able to show that the width of the backward peak varies with 1, and in a way that can be reproduced by the distorted-wave calculations (SO) .
2.8 Polarization and angular correlation.-In earlier sections we saw that the angular distribution is characterized by the orbital angular momentum transfers 1 (which act incoherently when more than one contributes) but does not depend on which of the spin-orbit states j =1 ± ! is involved. The additional information is desirable. To obtain it, the angular correlation between the outgoing proton in Cd, p) and the de-excitation 'Y ray was suggested by several authors shortly after the appearance of Butler's papers (16, 63, 158) . Newns ( 1 3 1) suggested measuring the polarization of outgoing protons for the same reason, and gave a simple qualitative picture by which its origin can be understood. The polarization has subsequently been calculated by a number of authors (39, 70, 7 1 , 86, 96, 101, 147, 148, 1 60, 175 , 181 , 1 84) and there exist several reviews of the subject (19, 72) .
We shall briefly recall the classical picture of Newns, pointing out first that it assumes the absence of any spin-dependent interaction in either the incident or outgoing channels. In this case the possibility of polarization of the proton spin is due to the fact that it is correlated in the deuteron with the neutron spin through sp+S,, = Sd and through the neutron spin it is correlated to the neutron orbital angular momentum 1 because of the spin-orbit interaction. In the limit of j-j coupling, the single-particle states are characterized by j=l+s". We shall assume that the neutron is captured into a specific state j = l+! or I-i. Then any mechanism which tends to prefer a particular orientation of l with respect to the scattering plane will cause
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230 GLENDENNING
the outgoing protons to be polarized, while no polarization is possible if 1 = 0. Referring to Figure 1 7 where q = ka - kp is the linear momentum carried by the neutron, we note that the orbital angular momentum l = r X q of the neutron is oppositely directed in the two regions I and II .' Thus if more of the protons which reach the counting apparatus have come from one region than the other, they wiII be polarized. The sign of the polarization is determined by which spin-orbit state j = l ± ! is populated in the reaction, and which region contributes the greatest number of protons. If; for example, the protons are strongly absorbed as they traverse the nuclear interior, then the flux of protons reachi.ng the counting apparatus that were liberated in stripping events in region I will be smaller than the flux ,coming from II . The neutrons stripped in II have their orbital momentum directed into the plane of the figure and Bince Sd = 1, then the sign of the polarization is given by
1 +, j = l - l .
p = . 1 (strong proton absorption) '- , J = 1 + li" ,
where we take kd X kp as the direction of positive polarization. If the deuteron absorption is strongest, the opposite conclusion is reached. Classically the proton and deuteron absorption are additive in th.eir effects on the polarization. Quantum·mechanically this is not generally true, but can under appropriate circumstances be approximately correct as we shall see later (133). Moreover, the polarization may change sign as a function of scattering angle, though the quantum-mechanical calculations confirm the results of the classical argument iin the vicinity of the stripping peak (133) .
FIG. 17. Schematic di.,gram for classical polarization argument showing nucleus divided into two regions by a line passing through the center and parallel to q. Consequently, the stripped neutron whose momentum is q has angular momentum oppositely oriented in I and II.
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NUCLEAR STRIPPING REACTIONS 231 Once the relative importance of deuteron and proton distortion is estab
lished, the above model of Newns shows that a measurement of the sign of polarization near the stripping peak measures the spin-orbit state j into which the neutron has been captured, provided that only one such state is involved, and that the orbital angular momentum I has been determined, say, from the angular distribution. The experimental results (81) indicate that usually the sign of polarization is positive or negative according to whether j = 1 ± t. This corresponds to the dominance of the deuteron distortion. Indeed, Satchler has shown that the optical potential well depth is about twice as effective for the deuteron as for the proton in producing polarization, and that if V�2 Vd and kp = kd (where the V's are the distorting potentials) , they cancel each other : the polarization is zero (19, 135, 157) .
The magnitude of the polarization caused by such a mechanism as described above must be less than 33 percent (96, 131) . Indeed we can carry further the semiclassical treatment, deriving this limit, and showing the effect of a mixture of the spin-orbit douhlets in the final wave function. From the vector model of angular momentum we find that if j = 1 +Sn and I has the projection (ml) on the z axis, then the average projection of s" is
( ) _ ( ) 1(1 + 1) + s .. (S,. + 1) - j(j + 1) s" - m, . 21(1 + 1) 71 .
Similarly, i f Sd = S" + sp and s" has the projection (s,,) , then the average projection of sp is :
. ( ) _ ( ) s"(s,, + 1) + s"(s,, + 1) - Sd(Sd + 1) Sp - Sn
2s"(s,, + 1) 72 .
Combining these two equations we have for the polarization of the proton
p =
1 � (ml) , 3 1 + 1
1 (m,) -- -- , . 3 I
j = 1 + 1/2 73 .
j = 1 - 1/2
In case (m/) = 1 this gives us the limit I p i �l. If both spin-orbit partners contribute to the reaction with amplitudes i3+ and i3-, respectively, then 1 5 fJ+' fJ-' t (m,) P(O) =
311 + 1 - -1- 5 fJ+' + fJ-' 74.
a formula obtained quantum-mechanically by Horowitz & Messiah (96). The quantum-mechanical value for the average z projection of 1 is
75 .
since we have already learned that Br' is the amplitude for stripping of a neutron with the state 1m, (see Sec. 2.1) . We note the important point that, whereas the differential cross section is proportional to ((3+2+.B_2) ,.....S(j = 1+t) +S(j=l-t) and yields us the sum (d. Eq. 23), a measurement of the polari-
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232 GLENDENNING
zation yields us a second combination from which we can obtain the individual factors S( +), S( - ) .
I n practice, because of the large spin-orbit splittings, only one member of each spin-orbit doublet will contribute to a given stripping reaction. Nevertheless, the polarization always gives us a different combination of spectroscopic factors from that given by the differential cross section whenever more than one sp:in-orbit state is involved. We indicate briefly how the polarization is obtained from our previous results. The expression obtained for T(d.p) corresponds to the transition JiMifJ.r->J/M/fJ.p. The probability that the proton will ha.ve spin fJ.p is therefore proportional to
2:(I'P) = L I T(d.p)(M'J.ld � Mn£p) I I 76. M,M,l'd
and hence the polariza.tion is given by
P = (2:(+) - 2:(-» /(2:(+) + 2:(-» 77.
By employing the explicit expressions for Clebsch-Gordan coefficients having t as one of the angular momenta, the expression for P can be reduced to the simple form (96)
L (2j + I)-Ie -) i-l-l12S(jl) ( L m I Blm I ') 2 ;1 m P(O) = "3 L S(jl) I Blm I I
78. Ilm
from which Equation 74 follows immediately as a special case. Thus whereas the cross section depends on Li SUl), the polarization depends on Li (2j+ 1)-1( - )H-l/2SUl) . We note, however, that Equation 78 holds only in the absence of spin-dependent interactions in the distorting potentials. In this situation the polarization is zero for 1 =0. If spin-dependent interactions are present, the expression for T becomes
where
T(d.p)(Mi/>d -" MI/>p) = E (J;M;, jm;IJ,M,) jlmjm,m.Pop'Pd' (lml, 1/2m.Jjmi) (1/2J.1p', 1/2m.I IJ.1i) 79.
y(21 + l)fJjJhmtPI'P'IldIld' (ktJ, kp)
Blmr-'I'dPd' (kl, kp) = _ (4?r)3/2(87ra) 1I2 fl'* E i"-">" -IY.,/(kp) y>!,o(ka) 2m ""-'1'1"
[(2X' + 1)/(2X + l)]I /2 E (X'J.I', !J.lplJ'M/) JM.rmJ'MJ'm' (A'm', lJ.l/ I J'M/)(X'm', lmz l Xm) (X'O, 10 I XO) (XJ.I, IJ.ld l JMJ) (Xm, IJ.ld' 1 JMJ) f (Mlkp ) . !W' MF
r fll (r)!M (kar) r'dr
80.
Here j"J are the solutions to the radial Schroedinger equation which now contains a spin-dependent optical potential. No reduction of the polarization in this case to a simple form such as Equation 78 has been found.
The Butler theory, because it embraces no mechanism that favors one
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NUCLEAR STRIPPING REACTIONS 233
orientation of angular momentum over another, predicts zero polarization. (One can see that the Butler theory gives zero polarization because I Brl = I Bl-m l in this theory.) The polarization is caused in fact by the distortion and therefore, in addition to its spectroscopic interest, is a sensitive
probe of the optical-model parameters causing the distortion. Several investi
gations of this sensitivity have been published (133, 1 75 , 181) and some examples can be seen in Figure 6.
It has been discovered experimentally that the polarization in several reactions is much larger than the limit of 33 percent derived earlier (2, 3, 24, 82, 105). For the C12(d, p) C13 reaction, we show in Figure 18 a summary of polarization data compiled by Goldfarb (72) . In this reaction, polarizations as large as 60 percent are observed. Indeed, polarization corresponding to stripping of the neutron into an 1=0 orbit has been reported for several nuclei (81, 104), an example of which is shown in Figure 19. Both these ob-
I
I
I
i��iJ; 60 10 100 120 o .<60
8 cIII
0 .. · 0 5 MeY • 6 · ' • Ie 7 ' 1 • 0 8 · ' •
• 10' 0 H
6 to· 1 II • I I · ' II • 15· 0 II
FIG. 18. Protons polarization for C12(d, p)Cla measured at several bombarding energies. Note values in excess of 33% limit. [Compiled by Goldfarb (72). ]
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234 GLENDENNING
20
1 6 -0.4
12 -... In ...... .a E
8 -
bl 3 'a 'a
4
'e.m. FIG. 19. Polarization and angular distribution for Si27(d, p)Si28, Ed- IS MeV.
This is an 1 = 0 transitil)fi for which polarization can occur only if spin-dependent distortion affects either or both the proton and deuteron. Note that polarization changes sign near minima of angular disttibutions. [From Isoya & Marrone (104). ]
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NUCLEAR STRIPPING REACTIONS 235
servations require an additional polarizing mechanism besides the orbital angular momentum alignment, and they have been explained in terms of spin-orbit terms in the optical model (70, 147, 148) . Such an interaction is well known from the optical-model analysis of elastic sctattering of nucleons or deuterons, where its presence is required to provide the observed polarizations (23, 122). Although it is a small term and does not much affect the differential cross section, it can give rise to the large polarizations observed. A qualitative picture of the polarization in the presence of a spin-orbit interaction in the optical potential has been given by Butler (35) .
Some attempts have been made to find relationships between the angular distribution and polarization such as those that exist in the elastic scattering of nucleons from nuclei (149) . For the reactions in which l = O, a derivative rule has been suggested such thatP(O) oc djdO(da/dO) (19, 107) . The experimental results for Si28 (d, p)Si29 shown in Figure 19 do not confirm the rule except at the minima of the angular distribution ( 104) . Such a change in sign of the polarization near the minima of the angular distribution has been predicted for all stripping reactions, not just those for which 1 =0, and indeed it appears sometimes in the calculations ( 133) while sometimes it does not (175) . Evidently any connection of this type is rather sensitive to the various conditions of the reaction. The most likely circumstance for its existence is that in which the angular distribution has a well-developed Butler-like pattern. For this situation we may imagine the amplitude for the reaction divided into two types of terms. The first is proportional to the Butler amplitude. If it alone were present, no polarization would exist, as already pointed out. The other terms are those arising from distortion effects. If the angular distribution has a Butler-like pattern, evidently these effects are small. Such a division is particularly appropriate if, by virtue of strong absorption in the interior, the reaction is concentrated in the surface region. The polarization now contains two types of terms, the cross terms between Butler and each of the distortion terms, and the square of the distortion terms. In the present circumstance, the latter are small compared to the former. Since the Butler amplitude changes sign at the minima of the angular distribution, which are zeros in the Butler theory, the polarization also changes sign. However, since in the amplitude we can write the non-Butler part as a sum of parts arising from the various distortion effects, that is, the proton absorption and deuteron absorption, and spin-orbit distortion, each of these appears additively in the interference term of the polarization. This has already been remarked upon by Newns ( 133). (In addition there are terms in the amplitude whose presence cannot be attributed to only one distortion effect.)
Satchler has pointed out that information equivalent to that obtained in observing the polarization of outgoing protons from an unpolarized deuteron beam can be obtained by observing the left-right asymmetry in the angular distribution of protons when a polarized deuteron beam is used (156) . If all spin dependence in the distorting potentials is neglected, the following
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236 GLENDENN ING
relation holds
81 .
where (du/dQ)unpol is the angular distribution obtained when un polarized deuterons are used, E.nd (du/dO)pol is the angular distribution when the deuteron vector polarization is Pd. Since the proton polarization is perpendicular to the reaction plane, the maximum asymmetry is obtained when the deuteron polarization is perpendicular to the same plane. Whereas the proton polarization (because of the poor alignment provided by the small spin of the deuteron) cannot exceed 33 percent, we see that here a 100 percent effect is possible. The above simple connection becomes complicated, however, if spin-dependent interactions are present in the entrance or exit channels. It becomes dependent 011 the tensor polarization of the deuteron as well as on the kinematics of the reaction (70, 148) . Thirion, Beurtey and collaborators have commenced experiments with tensor-polarized deuterons at Saclay.
We discuss now very briefly the angular correlation between the outgoing proton, and the de-excitation "I ray that may follow if the stripping process has left the residual nucleus in a excited state. The correlation function in the plane-wave approximation is ( 16, 63, 97, 158)
W(k.J, kp, ky) = L: [(21 + 1) (21' + 1) ] 112PilPJ' I'BzBl'" lil'i' L: CLCL' L: 'TJx(jj'J,J)!x(LL'JjJ)Px(cos 0) LL' x
82 .
Here 8 is the angle between the proton and "I-ray direction and Bz is defined by
83 . and Equation 41 , CL are the amplitudes for the Lth pole radiation, fA and 1JA are geometric parameters for the 'Y and neutron radiations which are defined and tabulated elsewhere (18, 152). The sum on A is over even values satisfying
o � h � l + l', j + j', L + L', 2J 84. In general the correlation function depends on stilI another combination
of the overlaps parameters {3il and so may give information on the coupling scheme additional to what can be obtained from the differential cross section and polarization .
The correlation in the plane-wave approximation is predicted to have certain symmetries with respect to the recoil direction q. There is axial symmetry about q (i .e. , W is independent of cPy in Fig. 20) and forward-back symmetry through the plane to which q is normal (invariance to ()y->'1r- ()y) .
If, however, the distortion of the incident and outgoing particles is taken into account, these symmetries are modified or destroyed (101). The axial symmetry is reduced to reflection symmetry through the reaction plane (cPy -> -cPy) . However, if the sum on A in Equation 82 is limited to A = 0 and 2 , there remains an axis of symmetry 8, which is rotated away from q and lies
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NUCLEAR STRIPPING REACTIONS 237 in the reaction plane (101, 159) . About this axis the distortion introduces an anisotropy, but the forward"back symmetry through the plane to which it is normal remains.
It has been pointed out that the assumption of the stripping mechanism for (d, p) reactions implies certain connections between the cross section, polarization, and angular correlation which do not exist a priori (97, 101). [Specifically there are relations between the elements of the density matrix Pkq (101) .] This led to the suggestion that the form of the theory could be tested independently of such details as the distorting potentials and n-p interaction by exploiting these relationships. Some work along these lines has been done (109, 1 19, 159) ; however, this work assumed that there are no spin-dependent distortions present. Since polarizations much in excess of the limit allowed when they are absent have been observed, the significance of such analysis becomes obscured.
2.9 Rearrangement stripping.-The simple stripping process that we have been discussing so far is characterized by the fact that the configuration of the core, comprising the nucleons in the target, is unchanged in the final state. The final state may not be pure in this respect ; it can have components in which the core is rearranged. Such components can be excited by the interaction of the outgoing proton with the core nucleons. This process, which we call rearrangement stripping, is represented by the second term in Equation 2. The tacit assumption has been made that, when the final state of a nucleus has components of both types, the simple stripping amplitude domi-
(0)
.... - J
y (Ll
It
z
I ky I I
�� __ .;.I _ .. ___ ... Y
FIG. 20. (a) In (d, p, 'Y) reaction, a neutron n with angular momentumj is stripped by the target of spin Ji to form an excited state J. This decays by emission of an L-pole 'Y ray to final state of spin J,. (b) Coordinate system for the reaction showing the angles of various radiations.
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238 GLENDENNING
nates. The validity of such an assumption obviously depends on one or both of the following : (a) the rearrangement amplitude is intrinsically small ; (b) the components in the wave function corresponding to core rearrangement happen to be small. For most nuclear states at low excitation the second statement is often true. There are exceptions, the first e�cited state of Bll
being an interesting example. This state at 2.14 MeV is excited by the BIO(d, p) Bll reaction. The spins involved are J. = 3 + and J, = t - ( 182) . The ordinary selection rules in this case require that 1 = 3. Not only is this an unlikely assignment in view of the fact that the h/2 orbit should fall at a much higher excitation, but the angular distribution measured at Ed = 7.7 MeV suggests that 1 == 1 (52), which is in fact expected frqm the shell-model sequence. The angular distributions vary considerably at energies so high that Coulomb effects cannot be held responsible. The differential cross sections at 20° for the ground state (3/2 -) , the first excited state ( 1/2 -) , and a state at 9 .19 M eV (1/2 +) are excited in the ratios 5 : 1 : 23 when the statistical factor 2Jt+ 1 is removed (20) . The intensity of the state in question appears smaller than normal. It is significant that the same level is excited in the Be9(He8, p) B ll reaction and appears normal compared to neighboring states (84) . This would be consistent -with an interpretation of this level as having a proton excited relative to the ground state. It then could not be excited by the simple (d, p) stripping reaction, a view consistent with the anomalous results mentioned above.
Several mechanisms have been suggested which lead .'to a relaxation of the simple selection rules and thus allow the above transition to proceed by stripping of an 1 = 1 neutron. French & Evans suggested a process in which the proton in the deuteron is exchanged with one in the target (51 , 59) . The ejection of a P3/2 and capture of. a Pl/2 proton satisfies the ;known initial and final spins. A similar relaxation of the selection rules could follow if the departing proton exCited a proton from the PI/2 to the Pa/2 level. This has been referred to as spin-flip stripping in the literature (26, 182) but obviously is only a special event of the type that falls within our definition of rearrange-ment stripping. i
French estimates that the exchange stripping, which must always be present because of the equivalence principle, is smaller than -simple stripping by a factor of about te:n. No reliable estimates of the importance of rearrangement stripping have been made.
The selection rules for rearrangement stripping in which a core nucleon is excited from lij, to IdJ are
� = Ji + l + t +j, +j, fr,fr/ = (-) 1+1,+'l
There is now so much flexibility in satisfying these rules that spin and parity assignments will be difficult if not impossible to make •
. The angular distributions from rearrangement stripping are expected to differ from the usual simple stripping. In particular, the. structure will be
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NUCLEAR STRIPPING REACTIONS 239
less pronounced if the rearrangement amplitude alone is present. If the final state is such as to admit both amplitudes, they are coherent. If the rearrangement term has contributions from several l's, these are again coherent, in contrast with simple stripping.
Evidently no positive spectroscopic information can, in general, be extracted for states believed to have been populated substantially by the rearrangement process.
Examples, in addition to the BID(d, p)Bll reaction, in which simple stripping is not expected to contribute because the known spins would require stripping into an unlikely orbit, have been listed elsewhere (8, Sec. V. D) . The experiments of Middleton & Hinds on Mg24(d, p) Mg25 may exhibit a number of examples of rearrangement stripping (123). In their work at 10 MeV, some 44 levels were excited. These levels fall into two groups : those that are strongly excited and have stripping like angular distributions of which there are 16, and those that are weakly excited and whose angular distributions possess little structure.
We now mention briefly the so-called heavy-particle stripping process. This process was introduced to explain the backward peaks occasionally observed in stripping reactions (139) . In this process one envisages the target divided into a proton and the rest of the nucleus, which is called the heavy particle. The incident deuteron "strips" the heavy particle from the proton. Since in the center-of-mass system the target is moving backward, the proton is observed at a backward angle. While this is a valid physical picture, its importance has not so far been properly evaluated. The experiments which inspired the concept and from which the heavy-particle stripping amplitudes have been extracted were performed at very low energies. Under such circumstances, which we have discussed several times, the Coulomb repulsion is of great importance and cannot be neglected. However, the analysis of such experiments in terms of heavy-particle stripping has always done so (cL 55). At higher energies the angular distributions have normal stripping patterns (d. 37) .
3. SINGLE-NUCLEON TRANSFER REACTIONS OTHER THAN (d, p) I t is clear that much of the discussion of the (d, p) reactions contained
in the preceding sections can be carried over to other stripping reactions such as (d, n) or (He3, d) , or to pick up reactions such as (p, d) or (d, t) where one nucleon is transferred. Of course, the same type of information is obtained from all single-particle stripping reactions. Information of a new type, concerning the hole-states, is obtained from pickup reactions (41 , 73, 1 18, 128) . We shall not discuss these here, save to indicate briefly how the previous formulas may be adapted to them. For this purpose we re-Iabel the reaction as follows
85.
where the spins of the nucleus and light nuclide are indicated by J and I and their mass numbers by A and a, respectively. Define for transfer of a spin-t
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240 GLENDENNING
particle
86 .
Then for stripping reactions
--+ � =
2JA+I + 1 ka_1 (�) dn 2h + 1 ka dn IIi
87. while for pickup
<-d<T 2Ia + l ka ( dU ) dn = 2I a-I + 1 ka_1 dn If!
88.
The amplitude Br is calculated in a similar way for all reactions. The center-of-mass correction is ka-1 --+(Aj A + l)ka-l for the wave number in the distorted wave. The wave function for the nuclide a which is needed for the calculation of Br is subject to greater uncertainty in magnitude when a refers to a mass-three or -four particle since these wave functions are less well known than the deuteron function ( 1 18) . As a result, only relative values of the spectroscopic factors can be extracted. In ca\le the plane-wave theory is used, the general definitions of the momenta transfer q and K are
A q = Je,. - A + l k..-l,
a - l K = k..-l - -- III. a
4. Two-NuCLEON TRANSFER REACTIONS
89.
4. 1 Generalfeatures.-In this section, reactions in which two nucleons are transferred will be discussed. The obvious interest in these stems from the fact that nuclei can be produced that cannot otherwise be studied, because of the absence of suitable targets. Further, such reactions can reach states which are characterized by the fact that they have two particles or holes excited relative to the ground state, so that additional states not excited in single-nucleon transfer reactions can be investigated.
As was the case for single-nucleon transfer reactions, our interest is the spectroscopy and structure of the nuclear states ; therefore we shall discuss reactions in which the light nuclide has a mass number not greater than four. The reason for this restriction is that the interpretation in terms of the nuclear coupling scheme of the heavy nucleus is more direct : simple selection rules follow because of the simple structure of the light nuclides.
The spectrum of olltgoing particles from a two-nucleon transfer reaction shows that it is strongly selective in the levels that are excited. In 016 there are some 30 levels known up to an excitation of 1 7 MeV, yet less than a third of them are excited with an appreciable cross section in the N14(a, d)016
reaction (38) . For the levels that are excited the intensities vary widely, partly because of statistical factors. After such trivial factors have been re-
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N UCLEAR STRIPPING REACTIONS 241
moved, the residual fluctuations reflect the structure of the excited levels including the degree to which they are characterized by configurations consisting of an unexcited core (the target nucleus) plus two additional particles or holes, depending on whether the reaction is a stripping or a pickup process (38, 78, 106) .
The angular distributions of the outgoing particles corresponding to the formation of the strongly excited levels are peaked at a forward angle which, together with the high selectivity of the reactions, is characteristic of direct reactions (38, 78, 83, 106) .
In addition to the selectivity that we mentioned above, and that is analogous to the selectivity exhibited by single-nucleon transfer reactions, there may exist in two-nucleon transfer reactions an additional selectivity that depends upon the degree to which the transferred nucleons are correlated in the nu cleus. Such correlations can be imposed by the angular momentum coupling as well as by the internucleon force. To illustrate what is meant by a correlation imposed by angular momentum coupling, we may consider two particles of angular momentum j coupled to total angular momentum I. The classical orbits of such particles are coplanar if I is zero or has its maximum value 2j. Otherwise they move in planes that are tilted with respect to each other. Obviously their motion is spatially more highly correlated in the first two situations. Additional correlation can be induced by the attractive nuclear force. In shell-model language, this takes the form of configuration-mixed wave functions. If the correlation is one which has a large overlap with the correlation possessed by the two nucleons in the light nuclide, we may expect an unusually strong transition.
In this subsection we shall sketch the derivation of the differential cross section, leaving for the following part a discussion of some experimental results. We may write for the transition matrix [analogous to the (d, p) reaction]
90.
where V is the interaction between the outgoing deuteron and the final nucleus. As in Equation 2, V is made up of two parts : one is nondiagonal in the core nucleons and leads to rearrangement stripping; the other is the interaction between the two parts of the incident nuclide that are separated by the stripping reaction. The second part causes excitation of states in the final nucleus in which the core nucleons retain the same configuration they occupied in the target. This we shall again refer to as simple stripping and we focus attention on it. 'vVe shall for the sake of definiteness refer to (a, d) reactions, indicating at certain places how the results would be altered for other reactions such as (He3, p) or (He3, n) reactions.
The wave functions in Equation 90 may be written in more detail as
xi-) = I/IdH(ka, Ri)</>d(rd)x,l'd(dd)'PJ?JI(A + 2) Xa(+) = I/I,,(+)(ka, R,,)</>,,(T,,)xoO(db, dd)'PJiMi(A) 91 .
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242 GLENDENNING
Here we use the subscripts b and d to denote, respectively, the nucleons that have been stripped and that are contained in the outgoing deuteron. The notation for spins is schematic; i.e., dd stands for dn and dp while Ta denotes internal coordinates of the a particle. All parts of Xd(-) were defined in Section 2.1 except
92 . the center-of-mass coordinate of the outgoing deuteron referred to the center of mass of the residual nucleus with the center of the core defining the origin. In the wave function for the initial state, 1/Ia(+) is the elastic scattering wave function of the a particle, cf>a is its internal space wave function, and xoo is its spin function.
For the wave function of the final nucleus we introduce the expression analogous to Equatio:l 7
93.
where cf> is a wave function constructed by the vector coupling of the wave function 'l' Jc(A) for the core to wave functions for the stripped pair of nucleons. The basis we shall use is the L-S representation. The reason for this will be evident shortly. The symbols r and l' stand for all other quantum numbers required to define the basic states of core and stripped nucleons, respectively, including: the single-particle quantum numbers. Thus
where
ip(f'�LSJ)J;MI = L (JcMc, JMJI J,MI)>¥JcMc(A)if>�LS�J(7), Rb, db) 94. McMJ I
if>�LS�J = iL L (LML, SMsI JMJ)xsM.(db)if>�LML(�b' Rb) 95. MLMS
s the spin-orbit wave function for the stripped nucleons. (We have written the center-of-mass and relati�e coordinates as arguments although the usual shell-model wave function would refer to the single-particle coordinates. This may be regarded simply as a convenient notation to distinguish the four nucleons.)
.
Concerning the ex particle, we note that, except for sm411 admixtures induced by noncentraI forces, the space wave function corresponds to relative s-state motion among all the particles and is hence symmetric. The spin wave function must therefore be antisymmetric upon the interchange of the protons or neutrons. That is,
XOO( db, dd) = XOo( dp, dP')xoO( dn, dn ')
== ..!.. L (_ 1)8'+MS'Xs,-MS'(db)XS,M!' Cdd) 2 S'MS'
where S' is summed over 0 and 1.
96.
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NUCLEAR STRIPPING REACTIONS 243 Inserting the above expansions in Equation 90, we find
T(a.d) = .!.. L f3r.,LSJ L (J.M.,JMJ \ hM,) ('l1J.M'\ 'l'J.M,) L (LM, SMs \ JMJ) 2 r.,LSJ M"MJ MMS
L (-)S'+MS'(x,1Ad I XS,MS') (X�s \ XB,-MS'h/(2L + I)B.,LM 97 . SIMS'
where
B.,LM(ka, ka) = i-L(2L + 1)-1/2 J [.J-d(-) (kd, Ri)c/>dCrd)</>.,LM(rb, Rb) ]* 98 .
X V[.J-a(+)(ka, Ra)</>a(ra)]dr"dR"drddRd
We note that the overlap (\]!J.Mc l\]!J�i) =OJiJ.Orri states that the core nucleons retain their original configurations. The spin overlap requires that S = 1 i that is, only the triplet part of the wave function for the stripped nucleons can be excited.
For other two-nucleon transfer reactions the spin selection rule is different. Thus, for example, in He', because the spin function for the protons must be antisymmetric, they are in an s = o state. Therefore, in (He3, n) reactions, only singlet components of the final state are excited. On the other hand, in (He3, p) reactions, because the spin state of neutrons and either proton is mixed, both S = 0 and 1 components of the final states can be excited. In summary, the spin states in the final state that can be excited in the various two-nucleon transfer reactions are
S = 1 s = o S = 0, 1
for (a:, d) for (He", n), (t, p) for (He3, p), (t, n)
99.
We have till now distinguished between the nucleons in the a particle and those in the target. The equivalence principle introduces two modifications, similar to those discussed in Section 2 .2 . According to that discussion, we should mUltiply our T(ad) by a suitable statistical factor which in the notation of that section is je where
and
f =
.!.. (_a_, _ (A + 2) ')'/2 2 (a - 2) ' A ' 100.
C = (TiM T" TM T\ T,M T,) 101 . is a Clebsch-Gordan coefficient referring to the isotopic spin. For (a, d) reactions, the isotopic spin of the transferred pair is T = O so that only states in the final nucleus which have the same isotopic spin as the target can be excited. With these results
(_ ) l+lAd T(a.cI) = ---fC :E fJ.,LIJ(JoM., JMJ I J,M,)
2 .,LJMJ
:E (LML, 1 - I'dI JMJ)v'(2L + I)B.,LML ML 102.
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244 GLENDENNING
where we can now drop reference to r, understanding that {3 refers to the core ground state.
The cross section is now given by d", 1 ma *md* kd 2Jt + 1 " I " M 12 dO = "4 (2","'1,2)2 ka 2J; + 1 (fC)IL7tt 7 f3�LIJB�L 103 .
Again the different L, J, and M contribute incoherently. But the sum on 'Y introduces a coherent effect. This sum might refer, for example, to con· figuration mixing in the wave function. In the case of collective states this coherence can give ri:;e to an enhancement of the cross section relative to a pure state, as discussed by Yoshida ( 188) .
For the two· nucleon stripping reaction the cross section can be summa· rized by
where
b�11 li081' (a, d) 2S .0 = 080, (t, p) or (ReS, n)
Hoso +
os,), (t, n) or (Hea, p)
104.
105 .
The cross section for pickup reactions is similar except that the factor (2J,+ 1)/(2J; + 1) is replaced by (21,+ 1)/(21; + 1) where Ii and I, are the spins of the incident and outgoing nuclides. This factor has the value t for (d, a) and unity for the other reactions. Also the first and last indices on the isotopic spin factor C are interchanged.
The selection rule on the total orbital angular momentum L transferred in the reaction is
106.
The connection of L with the single-particle states into which the nucleons are stripped is
L + S = J = jn + jp 107. In general, several values of L for any given final state and particle configuration Un' jp) can contribute to the reaction.
The connection of L with the parity is not immediately obvious as it was in the case of Hingle-nucleon transfer reactions. There the angular momentum transferred in the reaction was that of a single particle. Here the parity change is obviously given by
108 .
However, we note tha,t the angular momentum L can be written as the sum of the relative, A, and center of mass, A, angular momenta of the stripped nucleons with (_)h+A = (_) !n+1p. Now in the a particle the neutron and proton have zero relative angular momentum, A = O. This has nonzero over-
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NUCLEAR STRIPPING REACTIONS 245 lap with the bound-state function in the final nucleus only for those components that also have A = O. Hence only the A = O, L = A components of the wave function are reached in the reaction. This establishes the parity selection rule (69) :
109.
In special cases additional selection rules hold. If both particles are stripped into the same spin-orbit state with the configuration (P)J, then the terms in the L-S expansion of this state satisfy
L + S + J == even 110.
Since for the P configuration the parity is unchanged, L must be even. For (a, d) reactions, because only S = 1 components can be excited, J must be odd (the same conclusion can be reached by considering the isotopic spin) . For a (He3, n) or (t, p) reaction, on the other hand, J must be even, while for (Hes, p) or (t, n) there is no such selection rule.
We shall illustrate with a few examples how the nuclear structure coefficients (3 can be found in some typical situations. First consider an eveneven target nucleus, and let the nucleons be stripped into spin-orbit states jn and jp. In this case their total spin is the spin of the final nucleus, and the form of the final states excited in simple stripping is
'l'J,(A + 2) = 'l'O(A)iJ>Cinip)J, 1 1 l .
We want to express this in terms of L-S wave functions and then compare it with Equations 93 and 94 to find (3. This transformation involves the 9 -j symbols (46)
where
[ 1'1 lp L
1/2 1/2 S
1/2 jn ] 11" 1/2 jp = [(2jn + 1) (2jp + 1) (2L + 1) (2S + 1)]1/2 1p
S h L
112 .
1/2 jn) 1/2 jp 113. S J,
and { } is the 9 -j symbol. We see at once that (3 is equal to this transformation coefficient X OJJr If the wave function is configuration mixed, then
1 14.
where, in this example, 'Y includes the single-particle quantum number j"jp (in this usage, j symbolizes nlj) . Then
115.
For an odd-odd target in which the two odd nucleons occupy jn'jP' and
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246 GLENDENNING
lie outside of closed subshells so that the target spin is Ji = in' + ip', then [i" i,,' ''' ] [ In I�LSJ = i", i/ '", I",
, J; ', L
1/2 in] 1/2 jp S J
116.
Here the final state is assumed to be described by the coupling scheme [U,j"')J,,, Upjp') Jp]J; .. If the final state is a sum over such wave function, say over JnJp, such sums are coherent and are symbolized by 'Y in Equation 103.
.
Finally, for an odd-proton nucleus in which the even 'core has spin zero, the odd proton is :in jP', and the final state has the coupling scheme [UP'jp)Jp, jn]J,
1/2 in] 1/2 i", S ,
117 .
Again, if the final nucleus contains several such components, corresponding to variousj,jpJp, such sums are the coherent 'Y sum in the cross section. (The connection between tr and the coefficients CL2 in (69) is CL2 = LJ (hsJ2.)
The explicit evaluation of the transfer amplitude BLM has been performed by several authors in the plane-wave approximation (22, 69, 129, 132, 151) . We shall not reiterate any of the details of that work. Depending on the approximations or the model that is adopted for the reaction, the angular distribution takes on several forms, all of which lead to similar results. The gross character of the angular distribution is given by
BLM � i-L(2L -I- 1)-1I2c5MO[(21" + 1) (21", + 1) ]1/2(1,.0, 11'0 I LO)G(K)iL(qR) 118.,
when the nucleons are stripped into orbitals In and lp. (The connection of ELM with F{ln1pL; M) , Equation 14 of (69) , is F=iLV(2L+ l) B.) Here q and K are defined in a manner analogous to Equation 89, and G(K) is a monotonically decreasing function of angle and depends on themomentum distribution in the alpha particle. The interesting angle dependence is contained in jL' There is a correction term to this result arising from the fact that the stripped nucleons do not adhere as a unit in the nucleus ( 132) ; it is, however, usually small. In place of the Bessel function one could as well use the wronskian Butler form of angle dependence (78) .
The distorted-wave calculation of the amplitude B'YLM is done in much the same way as for (d, p) reactions. Indeed the form of B'YLM is the same for all direct reactions. A few distorted-wave calculations have been reported. They are based on the approximation that the two nucleons are stripped as a lump. That is, reference to the single-particle orbits In, 1'1' into which the nucleons are stripped is suppressed. The reaction is characterized by the total angular momemum L that is transferred, which, as we have discussed, is also the center-of-mass angular momentum of the pair for those states (or components thereof) which can be excited. This allows one to introduce
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N UCLEAR STRIPPING REACTIONS 247
a wave function UL for the center-of-mass motion of the pair, which is analogous to the function Uz for the stripped neutron in (d,p) reactions. The angular distribution calculations for single and double stripping are identical in this approximation aside from the trivial changes in masses and charges. It will be recognized that the above description is the same as that one would apply for a cluster description of the final state. Therefore, if the independent motion of the particles in their shell-model states has only a small effect on the cross section, a cluster and shell-model description of the final state cannot be distinguished by the angular distribution.
The effect produced on the calculated angular distribution by allowing the nucleons to be stripped independently has not been investigated in distorted-wave approximation. In the plane-wave approximation the effect does not appear to be very important. To see more clearly how the independent motion enters into the calculation, we recall that B'YLM contains the wave function c/lLM of the stripped pair. In the usual shell-model description this represents the vector coupling of the single-particle angular momentum 1,. and Ip to L. Such a wave function can be transformed to the relative and center-of-mass coordinates of the pair.
</>(I,I,)L = I nlllnJ.; L) = :E (nX, NA; L l n1llnJ.; L) l nX, NA; L)
119 . nWA
where n ln2 are the principal quantum numbers for the single-particle states and n, N for the relative and center-of-mass motion with A, A the corresponding angular momenta. Explicit expressions for the transformation coefficients can be obtained only for harmonic oscillator wave functions (126, 170) and they are tabulated in (28). The summation in Equation 1 19 is restricted by
120.
We have already pointed out that the relative s-state motion in the incident nuclide overlaps only with the ).. =0 terms. Introducing the following wave function for the a particle,
N" 22) ' N" 2 2 0+ 0 '" = --- e-� ri; "" --- e-2� (,. +rd IP ) " (41r)3/. (41r)8/'
where p = Rd-Rb, we obtain
B,LM = i-L(2L + 1)-II2N" J "'d(Td)B-�2rlTd2dTd :E (no, NL; L I nlll, nol.; L)
.. N
121 .
J [I/ti-)(k.J, Rl)UNL(Rb)YLM(Rb)]*VI/t,,<+)(ka. R")e-4�'P'dR,,dRd 122.
Here UnO and UNL are harmonic oscillator functions (126). The last integral has exactly the same form as Equation 1 1 for (d, p) reactions. Now, however,
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248
Q) c: c: c ..r:: . () ... Q) Co
(fJ -c: ::::J 0 U
Q) c: c: c
..r:: () ... Q) Co (fJ -c: :l 0
u
GLENDENNING
1400 9.0
I ZOO
1 000
800 5.7+5·8
6 0 0
400
6.Zt6.4 ! l 4.9+5.1 1 3.95
1 2.31
1
10 20 30 40 50 60 70 80 90 100
Channel number
1 400 7.6 MeV
1 20 0 01 7
1 000
800
600 9.0 MeV NI�a.d ) 016 (8.88 MeV! 5.70+5.73
I 400 14.56 3.85
1 1 200 �
o L-__ L-__ L-__ � __ � __ ���-=����-L�� o 10 20 30 40 50 . 60 80 90 100
Channel num ber FIG. 21. Deuteron energy spectra at 15" corresponding to levels produced in N14,
0'°, 017, and p. by bombardment with 47-MeV ex particles (from left to right). Note that of the many levels known to exist in the energy range spanned, only a few are excited. [From Harvey et al. ( 79).]
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Qj c: 1000 c: 0 .c: 0 BOO ... Q) Q. 600 <I) -c: ::l 400 8
200
0 0
NUCLEAR STRIPPING REACTIONS
'0 I " .,
10
... ., Co
20
o
1 4. 7 !
30
1 1.0 B.9 7.06.1 ! ! ! !
40 50 60 70 Channel num ber
40 80 120 ·Channel nurn ber
FIG. 21-Continued
0 !
0 16
BO 90
F18� 1.1 MeV
F'· ground state
100
160 200
249
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250 GLENDENNING
B"LM is a sum of such integrals corresponding to different states of motion of the center of mass of the stripped pair ; i .e. , to different principal quantum numbers N. Aside from the structure factors fJ which depend on the nuclear coupling scheme, this is the way in which the single-particle character of the bound states expresses itself. The first two integrals are overlap integrals with respect to the pair that forms the outgoing deuteron and the relative motion of the pair of nucleons that is stripped, respectively.
The bracket ( ) is the amplitude of the relative s motion in the bound state of the pair (28). This factor tends to favor single-particle states for which L = O or 11+l2• Additional overlap with the motion in the a particle [e-2.rTb2] can be effected through configuration mixing. This is contained in the sum on 'Y.
4 . 2 Interpretation of some experiments.-We have seen that the twonucleon transfer reactions are expected to be very selective in the type of states that are excited. Strongly excited states will be those for which the core nucleons are unexcited. The state of relative motion of the stripped nucleons will be largely s-state, and the spin state will be predominantly the one indicated in Equation 99. Thus the mere fact that a level is strongly excited already indicates that its structure has some special characteristics. We consider as an example of these ideas the work of :Harvey et ai . , who studied the reaction:3 Cl2(a, d) NU, NU(a, d)016, NI6(a, d)017, and 016(01, d) p8 with 47-MeV a particles (79) . The spectrum of outgoing' deuterons is shown in Figure 2 1 . The striking thing about these spectra is that of all the le�els in the energy range covered, only a few are excited and, of these, several levels are much more intense than the others. The Q value corresponding to the most intense levels varies from one nucleus to another in a regular fashion, decreasing with increasing mass as can be seen' in Figure 2 2 . This behavior would follow if the nucleons were stripped into the same singleparticle states which are such that they are at high excitations in the light nucleus but, as the lower shells become filled, approach �the ground state in the heavier nuclei. The angular distributions are all similar; a typical one is shown in Figure 23. Harvey interprets these levels as having the structure [Ii, (d6/22hlI, : that is, the neutron and proton go into the d5/2 single-particle state with their spins coupled to 5, and this subunit forms, with the target whose spin is Ii, levels of spin I, = 5 - li to 5 + 1 •. The states that can be formed in this way are shown in Table IV. The basi� on which the identification of the d6/2 state as the one that is involved, aside from the fact that it gives a consistent interpretation of the data, is twofold. In Fl8 the shell model predicts that the last neutron and proton occupy the d6/2 state, and there is known to be a 5 + level at the energy observed in the experiment. Of the shell-model states available in this mass region, ,d5/2 is the only one which can give rise to a 5 + state in fi8, In addition, a shell-model calculation of the states of N14 predicts a level at about 9 MeV ju�t where the strong level is observed (1 79) . Furthermore, the overlap argument mentioned in Section 4. 1 suggests that the total orbital angular momeritum L should have
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FIG. 22. Reaction Q, corresponding to the most strongly excited levels of Fig. 2 1 as discussed in the text. [From Harvey et al. (79) . ]
7
6 0 0
- 5 "- 0 1/1 ......
.Q E 4 0
- •
c: 3 • 0 "0 • ...... b 2 "0
• 0 0
• 0 0
• 0 • • • •
O L-__ -L __ � ____ L-__ -L __ � ____ L_ __ � __ � ____ L_� o 1 0 20 30 40 50 100
Ang le ( deg , c . m. ) FIG. 23. Angular distributions of deuteron groups corresponding to the 7.6- and
9.0-MeV levels produced in NIG(l>, d)017 by 47-MeV l> particles. These are typical of all the strongly excited groups in Fig. 2 1 . [From Harvey et al. (79) . ]
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252
Reaction
CI2(a, d)N14 NI4(a, d)018 N16(a, d)017
018(a, d)F18
GLENDENNING
TABLE IV
J;
0+ 1 + 1/2 -0+
J,
5 + 4+, 5 +, 6 + 9/2 - , 1 1/2 -5 +
a minimum (0) or maximum (4) value. In a grazing collision on the surface, the latter is dynamically favored according to the following considerations :
Classically, angular momentum / r X q / is transferred in the reaction if it occurs at the point r. For the above reactions, q""1 F-l in the forward direction so that in a classical reaction with impact parameter �RN the angular momentum transferred is 3 to 4. [Of course this sort of argument should be supported by a quantum-mechanical calculation in which the particles interact with the nuclew; through optical potentials, as we have discussed for (d, p) reactions, for it depends upon the assumption that the particles are strongly absorbed if they must traverse the nuclear interior.] Now, if we expand the wave functions for the configuration (dS!22)J, J = 1 , 3, 5 in terms of the L-S wave functions, we find (see Eq . 1 12) :
I J = 5) = BG
. /108 . / 4 . / 63
I J = 3) = 11 175 3D - 11 175 3G + 11 175 'F
I J = 1 ) = . / ..2.. 3S - . / � 3D + . / � Ip 11 25 11 25 11 25
The only s tate with a large S = 1, L = 4 component (i.e., 3G) is the J = S. A rough calculation of the form of the cross section for these states yields (schematically)
CT. � 100 1 B. 1 2 CT 3 � 2 .3 1 B. I ' + 4. 5 1 B2 1 ' CTl � 1 . 2 I B2 1 2 + 4 .0 I Bo l '
According to our classical argument
I B. I ' > I B, I ' > I Bo l '
so that we are led to conclude that of the states of d5!22 configuration, J = 5 should be most stron gly populated.
The consistency of the above interpretation with other experimental results has also been di.5cussed by Harvey. The strongly excited levels in (a, d) which have two nucleons excited should not be strongly excited by inelastic scattering of nucleons or a particles because the internucleon force is a twobody force and the incident particle can therefore only excite one nucleon
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NUCLEAR STRIPPING REACTIONS 253
at a time (68) . I n N14(a, a') experiments, levels whose configurations are believed to correspond to single-particle excitations are indeed strongly populated while those corresponding to double excitation are not. The 9-MeV level made so strongly in the (0', d) reaction is not observably populated in the (0', a') reaction. Similar evidence on the two-particle excitation character of this level is provided because it is not populated in the OI6(d, a)N14 reactions.
Ball & Goodman have studied the (p, t) reaction at 22 MeV on nuclei near the neutron closed shell N = 50 (10) . This reaction can excite two neutron hole states. The single-particle states in this region are 199/2 which is closed at N = 50, the 2d5/2 which lies above, and the 2Pl/2 which lies below it. The energy spectrum of the tritons is shown in Figure 24. The groups at Q = - 12.8, - 1 1 .2 , and -8 MeV correspond, respectively, to pickup of neutrons from g2, dg, and d2 configurations, respectively. In Zr90 only the g2 group corresponding to the ground state is observed since the d orbit is empty. In 2r91 the d orbit has one neutron so that the ground-state transition now involves pickup of a d and g neutron, while the g2 group is present also, corresponding to an excited state. The nuclei 2r92 and Nb93 each have two d neutrons so that the d2 and g2 groups are both seen. The dg group does not appear, however. In these nuclei there are two d particles and, because of their interaction energy, the dg transition will be shifted and is therefore unresolved from the g2 group.
In Zr92(p, t) the d2 group is very broad because of the thick target used in the experiment; however, it probably hides other transitions involving the pickup of d2 neutrons. We know that Zr90 has two 0+ states whose wave functions are orthogonal mixtures of p2 and g2 protons (56) . Therefore, Zr92
can have these two proton states mixed into the ground-state wave function by the neutron-proton interaction. Thus the protons can be left in the excited 0+ state of Zr90 by the pickup of two neutrons from Zr92. In Zr90 these states are separated by 1. 75 MeV. In addition the 2r92 ground state could have also the component ['/I'(g2) ., v(d2) z]O and the pickup of the neutrons would excite the 2 + level in Zr90. This broadening is largely absent for Nb(p, t) although there may be a weak group at Q ,,-, - 10.5 MeV. The presence of the odd proton in this nucleus reduces the mixing in the wave functions.
The transition involving pickup from different orbits, the d and g, appears to be of lower intensity than for pickup from the same orbit. This is expected since for the latter the appropriate correlation discussed earlier can more easily be built up.
ApPENDIX Here we shall derive the approximation to the transition matrix intro
duced in Section 2 . 1 . Our starting point is the expression obtained by GellMann & Goldberger (64) which for (d, p) reactions is
T = <I/lp I Vnp + Vp I 'lrP» = ('lrp(-) I Vp + V" I I/ld) 123 .
Here VP and Vn denote the interaction o f the proton and neutron respectively
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254
-" .. u o Q. '" :0 � :; .. :i .. D '" .. :.... ., u -
i--
�
, I I I I
I J
. � 1 \
i I i _ I
I ,-r-I � I'� r�· 'J
I I\�
GLENDENNING
, I)
I
.-1 . . . / ...-.JW
•
I t.
1 11\ . �I
/i r �,'-O . ;r-\::I
,. I! ... 1\ Jl : f
: .. , / I , \ f'oo-O I y " V�� 1 � v
l \ VI . I . I
; , ! " =\ j � \0 I / \ I I V I 0
•
, I
I • ;1.(1)0"
, . �, , \ v \
.' i • Uzr92 (Po"
. --- Zr9' (p,f ,
,
., .
If z,tO (p,f'_
,
I , i
• • " '2 -I) CM.V )
" t, '1
FIG. 24. Triton groups corresponding to pickup of two neutrons from configura
tions d2(Q---- -8 MeV), dg(Q"-' - l 1 MeV), and g2(Q"-' - 13 MeV) from various tar-
gets. [From Ball & Goodman (10).] .
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NUCLEAR STRIPPING _REACTIONS
wi th the n ucleon s in the ta rge t
The Hamiltonian H for the system is written in two alternate forms
H = Hd + V" + V" = H" + V" + V",.
255
124.
where Ha is the Hami lto nian fo r the in te rnal s tructure of the nucleus and deuteron, and their relative motion, and a similar definition applies to H". The functions </>d and </>" are plane waves in the relative motion in the entrance and exit channels, respectively,
(HI' - E)cf>" = 0, 125 .
The exact-state vector for the system of A +2 nucleons is denoted by \[I (E - H)'Ir = 0 126.
The subscript d or p refers to a boundary condition at infinity. For example, \)fa means that there is a plane wave of deuterons. The integral equation which exhibits the boundary condition is
1 'Iri+) = cf>d + (VI' + V")'Iri+)
E(+) - Hd 121 . E(+) = E + iTj
where 1J 15 a small positive number. The (+) or ( -) refers to a boundary condition of outgoing or incoming spherical waves. A formal solution to Equation 127 i s
128.
We now introduce the identity
VI' = (Q + P)VP(Q + P)
= VQ� + Vp� + VQpI' + Vppl' 129 .
where Q+P= 1 and P is a projection operator onto the ground state. Introd uce also the solution to
(HI' + Vpp"P - E)xp = 0 130.
We recognize tha t Vpp1' is the first term of the optical potential. We can write \[11' in terms of this function
131.
where
132 .
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256 GLENDENNING
Then
T = (x,,(-) i V" + v" l <Pd) + (x,,(-) 1 v + v",, 1 '1'i+) - <Pd) = (x"<-) I v + V "" I '1'i+)} + (xp(-) 1 Vp + yn + VI''' - v i <Pd}
the hermiticity of the i::lOtentials being assumed. The last term is
(xp(-) 1 V ppP + V" - V" .. I <Pd} = (xp(-) 1 Hp + V ppP - Hel 1 <Pel) = «H" + V ppp)x,,(-) 1 <Pel} - (x,,(-) I Hel 1 <Pel) = (E - E) (x"<-) 1 <Pd) = 0
So
133 .
134.
Since Xp(-) satisfies a Schroedinger equation containing the first term of the optical potential, we shall calculate it from an optical potential. (This means that higher-order terms should be substracted from the operator in Eq. 134. We shall ignore them.)
We can introduce an optical potential Ucl for the deuteron and the corresponding optical-model function Xd,
(E - Hel - Uel)xel = 0
in terms of which \)icl<+> can be written 1
'1'a<+) =, xa<+) + (Vp + V" - Ud)xa<+) . E(+) - H Then in Born approximation
T� (xp(-) I Vpn + v I xa<+»
However, we note from Equation 132 that
v i xa<+)} = V QP"I xP» because QI Xd> =0. SO
T"-' (xp(-) I v"" + V QPP I xP» which is the result quoted in Section 2 . 1 .
135 .
136.
137.
138.
139.
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N UCLEAR STRIPPING REACTIONS 257
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