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The role of the Coulomb interaction in isospin-violating direct nuclear reactions
Nuclear Physics A344 (1980) 151Not to be reproduced
by photoprint
115; @
North-Holland Publishing Co., Amsterdam
or microfilm without written permission from the publisher
THE ROLE OF THE COULOMB INTERACTION IN ISOSPIN-VIOLATING DIRECT NUCLEAR REACTIONS H. LENSKE
+ and
Znstitutftir Kernphysik, Kernforschungsanlage Received
G. BAUR JtXch, D-51 70 Jti’lich, West Germany
17 January
1980
Abstract: We investigate possible sources of isospin mixing in direct nuclear reactions. We find that there are mainly two effects which enter coherently into the direct reaction amplitude : “induced isospin mixing” caused by the Coulomb interaction between the target and the projectile, and “wavefunction effects” of the target nuclear states. These arise mainly through the difference of corresponding neutron and proton single-particle wave functions. These differences are especially noticeable just in the surface region where direct reactions take place. As illustrative examples we perform mode! calculations for the pick-up reactions ‘%(d, 3H)‘zC*(1+, 15.11 MeV) and ‘%(d, 3He)‘ZB,., (l+) and the inelastic scattering “C(d, d’)“C* leading to the l+ states at 12.71 MeV and 15.11 MeV. These model calculations are compared with the corresponding experimental data.
1. Introduction The concept of isospin in nuclear physics is directly connected with the very fundamental question whether the nuclear interactions are charge independent or not. Therefore, isospin violations have been studied extensively from the very beginning of nuclear physics. In ref. ‘) one can find an extensive review on former work on isospin violating effects in nuclear systems. However, it is well known that the nuclear hamiltonian does not commute with the operator of the total isospin, at least because of the presence of electro-magnetic interactions and the static Coulomb field which contain both charge-symmetric and charge-dependent pieces “). Therefore, it is necessary first to correct theoretically for the electromagnetic contributions to isospin-violating processes before one can draw conclusions about short-ranged nuclear charge-dependent interactions. The difficulties connected with this procedure can be seen most clearly in the analysis of lowenergy NN scattering ‘). Low-energy NN scattering seems to indicate a short-range charge symmetry breaking component in the free NN interaction of the order of 2 % of the average NN potential ‘). There is strong evidence that this charge-dependent NN interaction is caused by indirect electromagnetic effects like the mass splitting of the exchange bosons 2-4), + Present
address:
Center for Nuclear
Studies,
University 151
of Texas, Austin,
Texas 78712, USA.
152
N. Lenske, G. Baur / Coulomb interaction
and no new type of strong interaction is needed to explain the charge dependence of the free NN interaction, This result shows clearly how important a role the electroma~eti~ interactions play in ~osp~-violat~g processes. Although they are weak compared with the average nuclear interaction, it is not, in this case, justified to neglect electromagnetic processes of higher order. A great amount of work has been done on isospin-violating effects in more complex nuclear systems, experimentally and theoretically. Charge-dependent effects in ground-state properties and excited states of nuclei as well as in nuclear reactions were observed l3 3)_Whereas in the past several authors have claimed the need of a charge symmetry breaking interaction ‘, 6, in order to explain the oberved degree of isospin violation, more recent investigations stress the importance of the static Coulomb field for isospin violations in nuclei ‘- ‘). For example, Lawson ‘) has shown for lp shell nuclei that the differences of the proton and neutron single-particle wave functions can give rise to large isospin mixing effects even if the residual interaction is taken to be charge independent. This is entirely an effect of higher order in the Coulomb interaction and is comparable with the well-known Thomas-Ehrman shift lo) in the binding energy of the proton-rich members of nuclear isospin multiplets. In both cases a first-order perturbative treatment of Coulomb effects is not appropriate. The inte~retation of isospin mixing in nuclear many-body systems is in general much more complicated than in free NN scattering. Because one must necessarily employ a model, one has to check thoroughly to what degree the approximations influence the theoretical determination of isospin mixing. Furthermore, the need to use effective operators when nuclear models are employed may introduce further un~ertainti~ into the theoretical investigations. [For example, the calculated magnetic y-transition rates in nuclei may change appreciably if one changes from the bare electromagnetic operators to effective operators, which include important exchange effects and many-body renormalizations “)I Therefore, theoretical studies of isospin-violating processes in nuclei will in general determine the effective degree of isospin mixing within the model. The studies of Lawson ‘), Sato and Zamick ‘) and Shlomo and Waft 7, show that it may be hazardous to conclude from a model calculation the existence of a charge-dependent nuclear interaction. In the field of nuclear reactions, isospin violations have been studied in compound nucleus reactions I’- 14)(CNR) and for the excitation of isobar analog resonances “) (IAR). In CNR and IAR it is usual to subdivide the total isospin mixing into the “internal mixing”, due to isospin mixing within the space of ~ompo~d states, and the “external mixing” of the compound states, which is caused by the asymmetric coupling of the compound states to channel states differing in the charge of the projectile or ejectile, i.e. external mixing results from the reaction mechanism and the Coulomb interaction. Calculations show 16) that external mixing gives, at least in light nuclei, the main con~ibution to the observed total isospin mixing in CNR. Obviously, in CNR one studies the isospin impurities in the projectile plus target system. On the other hand, direct reactions have been used to measure the isospin
H. Lenske, G. Baur / Coulomb interaction
153
mixing of individual target states, mainly in light nuclei 17-20). Two kinds of experiments have been performed: (i) Certain pairs of isospin allowed nuclear reactions - like (d, 3H), (d, 3He) leading to the states of an isobaric multiplet - are observed. Isospin conservation predicts a definite value for the relative yield. In this class of experiments the transition operator is of mixed isospin. (ii) The intensities of an isospin-allowed and an isospin-forbidden excitation of the same J as seen for example in the inelastic scattering of deuterons or a-particles are compared. Here, the transition operator possesses a definite value of isospin. Therefore, if one assumes that charge-dependent projectile-target interactions and higherorder processes do not contribute to the excitation, the isospin-forbidden cross section is solely determined by the isospin impurity of the nuclear state in question. At this point we can already see the problem qualitatively: the theory of direct nuclear reactions -which is anyway never considered to be more exact than say to 20 % is usually most appropriate and accurate for rather strong transitions (e.g. good s.p. states in transfer reactions, strongly excited collective states in inelastic scattering). But if one studies weak transitions or deviations of the order of 1% from a predicted value, there is a great sensitivity to otherwise negligible details of the nuclear wave function and of the reaction mechanism. We address here ourselves to the question : Is there an unambiguous way to extract the degree of isospin mixing of a nuclear state which is excited in a direct nuclear reaction? This problem is investigated step by step in sect. 2. Firstly, we study the role of the residual projectile-target Coulomb interaction in isospin-violating direct reactions. This leads to the model of induced isospin mixing. It is shown that the residual projectile-target Coulomb interaction gives a contribution to the isospin-violating T-matrix which is comparable with the effect of the Coulomb force within the target nucleus. So it is in general not justified to neglect the charge-dependent parts of the residual projectile-target interaction in isospin-violating direct reactions even if the Coulomb interaction cannot contribute to the direct excitation of the final state. Secondly, the contributions of multistep processes are discussed for the general case that the nuclear states are of mixed isospin. It is shown that the interplay of the target isospin mixing and of the induced isospin mixing will in general lead to important interfering contributions to the isospin violating cross section. The ideas of sect. 2 are applied to the representative example 12C(d, d’)12C* leading to the 1 + states at E, = 12.71 MeV (mainly T = 0) and E, = 15.11 MeV (mainly T = 1). We have chosen this example because most of the previous experimental 16,18, and theoretical ‘-‘) work has been concentrated on these two levels. Our results, however, are of a general nature and can be applied - mutatis mutandis - to any kind of direct reactions. In sect. 3 we discuss the serious shortcomings of the conventional perturbation treatment of charge-dependent interactions when it is applied for nuclear structure
154
H. Lenske,
G. Baur / Coulomb interaction
and direct reaction studies of isospin-violating processes. As an example we study 13C(d, t)l’C*(l+, 15.11 MeV) and 13C(d, 3He)12B(1+, 0.0 MeV). The results for 12C(d, d’)12C* leading to the two l+ states are given in sect. 4. Nuclear wave-function effects will be studied for this reaction as well as the important role of induced isospin mixing effects in isospin-forbidden inelastic scattering. Finally, in sect. 5, a summary is given.
2. Induced isospin mixing effects in direct nuclear reactions The hamiltonian of the projectile-target
system,
H = H,+H,+H,+
T/p*,
(1)
consists of the internal hamiltonians HP and H, for the projectile and the target nucleus, the hamiltonian H, of the relative motion, which is thought to be given by the optical model, and a residual projectile-target interaction VP,. Alternatively, this hamiltonian can be split in an isospin-invariant piece H(O) and an isospin-violating (i.e. charge-dependent) piece H,,,, :
H = H"'+H H,,,_ contains charge-dependent
(2)
c.d:
interactions internal to the reaction partners as well
as those between them
H c.d.
=
%.
+
K:d.
+
v,‘d4.
(3)
For the sake of definiteness let us consider the case of an isospin-forbidden inelastic reaction. Let us furthermore assume that Vcy$cannot contribute to the direct excitation of the final state like it is the case for a O+ -+ l+ spin-flip transition. During the scattering process the projectile is assumed to remain in its ground The action of Ktd. within the target nustate of pure isospin, i.e. we can neglect VcTd,. cleus leads to isospin impure eigenstates 4,,(Y). In the index n, all quantum numbers other than J” are contained which are necessary to specify the state. The main contribution to T/$(and also to Kyd, and Vcy$)is given by isovector components. Then, for a nucleus with N neutrons and 2 protons, the states 4,(P) can be decomposed quite generally into 4”(Y) = 1J”n; To) + IJ”n; To + l),
(4)
where To = +(N-- Z). Often, the much stronger assumption is made that it is sufficient to treat Vctd,in first-order perturbation theory. 4,(Y) is then decomposed into states of good isospin 4JJ”, T),
H. Lenske, G. Baur / Coulomb interaction
155
and T’ may equal T [dynamical distortion ‘“)I_ If only one term of the sum with T’ # T is important, one ends up with the commonly used “two-state mixing’: model. The errors, which are possibly introduced into nuclear structure models and, more important, into nuclear reaction studies when one uses the perturbation method, are discussed further in sects. 3 and 4. But for the purpose of this section it will be sufficient to use the ansatz eq. (5) for the eigenstates +,(J”). What is the role of Vii for isospin-forbidden inelastic scattering? Although contributions of Vz$ to isospin-violating reaction processes are usually neglected, there are no general arguments for doing so. It appears very much on the same footing as VC$..V$ includes additional isospin mixing effects into the projectile-target system because by its action the target (or projectile) can be polarized into states with T # To. ‘* leads to a non-vanishing isospinEven if the target states are of pure isospin, V,.,. violating T-matrix element
In eq. (6) a typical T-matrix element due to Vp”hp. is given in second-order DWBA. The summation (or integration for continuous quantum numbers) is extended over intermediate channels with T # To. The distorted waves with wave vector k in the various channels are denoted by xl*) and ji(+) is the conjugate solution to x(+j41). Em is the energy of relative motion in channel m and EC+) = lim (E-t iy), E the energy in the incident channel. I$? is the isospin conserving part of VpA. It can be seen from eq. (6) that the isospin violation is mainly determined by the amplitude
(7) Induced isospin mixing appears in the A+ P system in complete analogy to the isospin mixing within the A-system alone [see eq. (5)]. It can be expected that qTd will be energy dependent because at different incident energies there can be quite different intermediate channels m of importance. Next let us consider the general case that the target states are of mixed isospin. According to eq. (5) the final state (mainly T # To) can be excited in a one-step process,
156
H. Lenske,
G. Baur / Coulomb
interaction
but the induced isospin mixing effects are also present. The total amplitude is given by the coherent sum T.fl = T?‘+T?‘d fl fl Therefore, the cross section for isospin-forbidden the interference of TrF and Tryd:
(9)
.
inelastic scattering is determined by
(10) If the target states are of mixed isospin, additional terms appear in cd. A typical T-matrix element for the excitation through a single intermediate channel m is, in second-order DWBA, given by
In the first step the T,-component of &(J’) is excited, in the second step the scattering goes from the T # Tocomponent of 4,(J) to the T-component of the final state. All the above T-matrix elements can be expected to be of the same order of magnitude. Typically they are given by the product of a Coulomb matrix element and a nuclear interaction matrix element. There are further contributions to ed which are of higher order in the chargedependent interactions Ktd. and l$y$. It is seen from eq. (10) that not only the magnitude of the total T,i”d is important but also the relative phase angle of Cd to qy. The induced mixing contributions may interfere destructively with the direct excitation resulting in a reduction of the isospin-forbidden inelastic cross section. If in an analysis Ted is neglected at all, then one would be led to the conclusion that the final state is of rather pure isospin. In the commonly used “two-state mixing model’ +1(J) = &(J,
T)-B&U,
4o(J) = c&G, To)+ &(J,
T,),
(12)
T)>
where in first order in T/td,,
(13)
H. Lenske, G. Raw / Coulomb ~t~ra~tio~
157
it is assumed that Tfpd can be neglected. Then the ratio R =
$
2
k(9) =
0 ;
(14)
would allow a direct measure of the isospin mixing amplitude of the target states. k(9) corrects for the differences in Q-values (see sect. 4). We note the following points: (i) If this model is reasonable, then R has to be independent of the reaction kinematics. (ii} The model depends seriously on the ass~ption that Coulomb effects within the target can be described in first-order perturbation theory. Thus the different radial dependence of proton and neutron s.p. wave functions is neglected. This point will be discussed in detail in sect. 3. (iii) Commonly one concludes from a non-vanishing R on a charge-dependent nuclear interaction. But one should notice that within this model in Vctd.necessarily the Coulomb interaction is contained.
3. Coulomb effects in the target wave function and their consequences for structure and direct reaction studies of i~pin-~olating prowsses The aim of this section is to show that the commonly used first-order perturbative treatment of charge-dependent interactions within the nucleus can lead to serious errors when single-particle Coulomb effects are included in this procedure. We want to stress the point that at least the monopole part of the target Coulomb interaction should be treated exactly in studies of isospin-violating processes ‘-‘). Let us set up a simple model. The hamiltonian of the nucleus A,
HA= %.p.+ b2,
05)
may consist of a s.p. h~ilton~ H_ (in general different for protons and neutrons, also in the short-range nuclear part - see below) and a residual (isosp~-conse~ing) two-body interaction
where V, and V, may depend not only on the relative coordinate but also on spin and tensor operators. If the hamiltonian is diagonalized in the 2-dimensional ph-space M$%),
&?(v)) with (17)
158
H. Lenske, G. Baur / Coulomb interaction
and CI= 7~(proton), v (neutron), the eigenstates are given by &(JM) = X,i&%J)
- Z,i&%4
&l(JM) = Z ,l&?(v) + Xpl&%). The coefficients are determined
(18) (19)
by
Z
-= ph xph
$
@ifi=&4,),
(20)
ZY
together with the normalization
condition
z,2,+Xih = 1.
(21)
In eq. (20) we have introduced the quantities A,, = s;h - s;h + v,, - v,,,
(22)
I/,p = <~;~(~)Iv,,I4;~(P)>~
(23)
and sEh = E~-$,(E~ < 0) are the ph energies. If A,, = 0. then X,, = Z,, = ,,/$ and the states & and & are of pure isospin T = 1 and T = 0. From eq. (20) it follows that the s.p. Coulomb effects introduce in two different ways isospin impurities in nuclear states: (i) through the differences in the proton and neutron ph energies, but also (ii) through the different values of the proton ph and neutron ph matrix elements of the originally isospin conserving residual interaction which are due to the differences in the radial behaviour of proton and neutron s.p. wave functions. One should notice that the second contribution is necessarily missing if one starts from a set of isospin pure basis states (which imply that proton and neutron wave functions are identical) and treats the Coulomb interaction - which is contained in l$$, - in first-order perturbation theory. This wave-function effect was studied first by Auerbach et al. “). Sato and Zamick *) have investigated the problem using HF wave functions. Lawson 7, considered the wave function effect for Woods-Saxon eigenfunctions within shell-model calculations. A strong sensitivity of the effect on the shape of the sp. potential and on the binding energy was observed. However, in refs. 7. *) a residual interaction of simple structure has been used. The analysis has been redone for the case of the already mentioned two lowest 1+ states in “CatE, = 12.71 MeV(“T= O”)andE, = 15.11 MeV(“T= 1”). Intable 1 theph matrix elements are listed as they are used in a RPA model calculation 22). The ph interaction includes, beside &function and density-dependent b-function contributions, also finite-range terms due to n- and p-meson exchange. The s.p. wave functions are generated in a Woods-Saxon potential which gives the correct binding energies 23). The numbers of table 1 show the importance of the s.p. wave function effect on the matrix elements of the residual interaction.
H. Lenske,
G. Burr
/ Coulomb
TABLE
159
interaction
1
Proton-proton, neuron-proton and neutron-neuron ph matrix elements of the type < lp,,r, lp$ /V/j& ‘) and j&t are given in the first coiumn; Woods-Saxon s.p. wave functions are used, Y contains densitydependent zero-range and finite-range contributions including tensor interactions “1 A""
._I
JpJh
CkeVl __-.-
_._---lP,,*lP$ 2P,,,lP,i: 2P,,*lP$ lf,,,lP$ ld,&,t, ZS,,,lS,:
5.8362 2.8508 2.7376 -0.2679 -0.1830 - 1.9554
- 1.0047 -0.3846 0.0969 2.4397 -2.3884 0.9940
5.6324 2.8157 2.7529 -0.2545 -0.1651 - 1.8781
203.8 35.1 - 15.3 - 13.4 - 17.9 - 77.3
- 1.0047 -0.3882 0.0728 2.4215 - 2.3197 0.8326
0.0 4.4 24.1 18.2 -68.7 161.4
12.05 34.00 35.80 36.60 41.80 37.15
12.86 34.00 35.30 36.30 40.64 35.95
810 0.0 -500 -300 -1160 -1300
The main content of the perturbation method, eq. (5), is to neglect the contributions of the ph matrix elements due to the wave function effect and to approximate the ph energy difference of the neutron and the proton ph pair in first-order perturbation theory in the Coulomb interaction [see, for example, the analysis of Shlomo and Wagner, ref. ‘)I. In terms of our amplitudes X, and Zph, this means
In the last step, the ph energy difference in the denom~ator has been neglected and in the two-state-model one has E, -E, = 21/,,. The Coulomb interaction is denoted by V,. It has to be realized from eqs. (20) and (24) that the first-order perturbation method is simply wrong when it is applied for the study of isospin mixing in nuclear levels. At least the monopole (single-particle) part of the Coulomb interaction has to be treated exactly. From the very concept the perturbation method is not able to give a correct description of the effect one is looking for. What are the consequences of the nuclear wave-function effect for isospin-violating direct nuclear reactions? There are mainly two important points: (i) The Coulomb intera~ion within the target will in general destroy the symmetry between neutron and proton ph amplitudes in a much more complicated way than the perturbation treatment suggests (see table 2). (ii) Direct nuclear reactions are mainly sensitive to the behaviour of the nuclear transition density in the surface and the asymptotic region. But in these regions of space the differences of the proton and neutron s.p. wave functions may be very pronounced. Both effects are due to the same reason and cannot be separated in general. (Within the ph model they enter multiplicatively.)
H. Lenske, G. Baur / Coulomb interaction
160
TABLE 2
RPA amp!itudes
and neutron configurations for the 1 + states at E, = 12.71 MeV and E, = 15.11 MeV in ‘% according to ref. 22)
of proton
l’C*(l”, h
1P 3/z
Is l/Z
proton
P
1P 112 2P 3/z 2P l/Z lf,,, 2s 1/z Id 312
12.7 MeV(“T
= 0”))
l*C*(l+,
neutron
15.11 MeV(“T
proton
= 1”)j neutron
x
Y
x
Y
x
Y
x
Y
-0.6163 0.0501 -0.0549 0.0987 0.0260 0.1133
-0.1392 0.0237 -0.0123 0.0364 -0.0147 0.0822
-0.7953 0.0672 0.0640 0.0833 0.0394 0.0966
-0.1749 0.0284 -0.0200 0.0383 -0.0052 0.0770
0.7856 -0.0812 -0.0616 0.0523 -0.0505 0.0364
0.1148 -0.0198 0.0218 - 0.0098 -0.0112 0.0046
-0.6086 0.0676 0.0459 - 0.0776 0.0444 -0.0598
-0.0848 0.0155 -0.0185 0.0004 0.0155 -0.0223
Therefore, if the perturbation method, as it stands, is used in direct reaction studies of isospin mixing, a serious error is introduced: one tries to describe a radially dependent effect in terms of a single multiplicative factor (apart from further errors which may enter when the induced mixing contributions are neglected). The perturbation method may be more reasonable when a “volume-like” process is involved as in y-transitions of low multipolarity. As an example of the wave-function effects in direct reactions let us consider the transfer reactions 13C(d, 3H)‘2C*(l+, 15.11 MeV) and r3C(d, 3He)‘2B,,,,(1+, T= 1) [ref. “)I. The final states belong to the same isobaric triplet. The “C-state is the T3 = 0 member while the “B._, forms the T3 = + 1 member. (We use the convention t, = +i for neutrons and t, = -3 for protons.) Correlations in the ground states of ’ “C(i-, T = T3 = i) and “B(1 +, T = T3 = 1) have been neglected. Therefore, 13C is assumed to be given by a lp+ neutron coupled to an inert “C core while r2B is chosen to be a lp, neutron particle, lp, proton hole state with respect to “C. Then, the first reaction proceeds via the pick-up of a lpi neutron while in the second a lp, proton is picked up. Within this model zero-range DWBA calculations have been performed. Non-locality effects were included in the usual manner 24). We do not want to discuss here the difficult theoretical problem of how to determine the correct transfer from factor [see, for example, ref. 25)], but we want to demonstrate how important it is to treat Coulomb effects within the target correctly in direct reaction studies of isospin mixing. In fig. 1 the lp, proton and neutron form factors are shown as calculated in a Woods-Saxon potential with standard geometry (R = 1.20 A*, a = 0.65) and a spin-orbit potential 25 times the Thomas spin-orbit term. The eigenvalues are taken to be equal to the binding energies of lp, protons and neutrons in “C (E, = - 17.16 MeV, .s, = - 14.80 MeV, ref. 23)). The depth of the potentials required for the proton and neutron is Vl = -61.93 MeV and Vi = -61.20 MeV.
Lenske,G. Baur 1 Coulomb interaction
H.
I
I
IO
20
,I
$
I
LO
I61
I
I
1
1
I
50
60
70
80
9.0
I 100 R (fm)
Fig. 1. The lp,,, proton and neutron wave functions in 13C as calculated in a Woods-Saxon well (see text). The binding energy is denoted by E.
It is seen from fig. 1 that the proton form factor shows an enhancement over the neutron one beyond the nuclear surface, while in the interior they are almost equal. In order to show the slectivity of direct reactions on the asymptotic part of the form factor, calculations have been done with a lower radial cutoff, i.e. the form factor is set equal 0 from the origin up to a certain R for the calculation of the DWBA T-matrix elements. In fig. 2 the ratio of the cutoff cross section to the full DWBA cross section at the first rn~irn~ for (d, 3H) and (d, 3He) is shown. It is clearly seen that the angular distributions in both cases are determined by more than 70% by the asymptotic region R 2 5.0 fin where the wave-function,di~eren~~ are large.
162
H. Lenske, G. Baur 1 Coulomb interaction
~:(mb/sterodl
i
12
11
10
09
O?
07 I
I
I
I
10
20
30
LO
I
III 50
60
RCUT-OF~
hl
I
I
I
I
10
20
30
LO
II 50
1 60
70
II 80
%.I
Fig. 2. The ratio of the cutoff cross section to the full DWBA cross section for 13C(d, t)“C* and 13C(d 3He)‘ZB, s. as a function of the cutoff radius’ R euloPf.The ratio is calculated at the first maximum of the angular distribution at 9 C.rn = 120.
Fig. 3. DWBA angular distribution fit to the experimental data of Lind et al. I’). The spectroscopic factors are denoted by S, for 13C(d, t) “C* and by S, for 13C(d, 3He)‘ZB,,,,. For &‘a two values are given. The one in parentheses corresponds to the case that the proton transfer form factor equals the neutron one. Non-locality corrections are included. Standard optical potentials were used [d: ref. 36), set 2, 3H/3He: ref. 37)] including 1 . s coupling.
In fig. 3 DWBA calculations are compared with the experimental data of Lindt et al. 17).
In a model calculation the s.p. Coulomb potential for the proton form factor in 13C(d, 3He)12B,.,, has been neglected, i.e. an isospin pure basis has been assumed. order to fit the experimental data, a spectroscopic factor is required which is greater by 27 % compared with the case the Coulomb potential is included. According to an ansatz of Lindt et al. l’) the ratio of the spectroscopic amplitudes for 13C(d, 3H)‘2C*(1+, 15.11 MeV) and 13C(d, 3He)12B,,,, enters as an important quantity into the analysis of isospin violations in transfer reactions. It is stated that the 12C 1 + states are expanded in basis states of pure isospin. On the other hand, the well depth method is used to calculate the form factor. But in doing this the energy shift of - 1.74 MeV of the “B_ relative to ’ “C( 1+, I 5.11 MeV) which is due to chargedependent interactions is automatically included in the calculation of the proton
H. Lenske, G. Bow / Coulomb interaction
163
transfer form factor, i.e. a basis of mixed isospin is used. Apart from the above discussed deficiencies of the perturbation method the ansatz is in contradiction to the assumption that the spectroscopic factor for “C(d, 3He)12B,.,. equals the spectroscopic factor for populating the T= 1 component of the 15.11 MeV state in 12C in 13C(d, 3H)12C*.
4. The interplay of induced isospin mixing and target isospin mixing in isospin violating inelastic scattering: application to “C(d, d’)“C* (l+, 15.11 MeV) The isospin mixing of the li states in “C at E, = 12.71 MeV (mostly T= 0) and E, = 15.11 MeV (mostly T = 1) was studied in the past by different means. The
two-state mixing model together with the perturbative method as described in sect. 2 has been used to interpret the experimentally observed deviations from strict isospin conservation. Adelberger et al.’ ‘) have determined from y-decay experiments an isospin mixing amplitude 6 = 0.046 +O.OlZ correspond~g to a charge dependent matrix element of (V$J = 110+30 keV within the two-state-model. In inelastic electron scattering a value of (V$J = 129.6k2.4 keV has been found (6 = 0.054+ 0.001) 2”). It has to tie mentioned that these numbers in addition depend seriously on the value of the theoretical isoscalar Ml matrix element of Cohen and Kurath26,28). Direct reaction studies of the isospin ~puriti~ in these two levels have been performed, for example, by Braithwaite et al. 20), Lindt et ~2.‘~) and recently also by Cossairt and May ‘*) in (d, d’) [refs. 17,20)] and in(d, 3H) [refs. 17P18~20)].In allcases the magnitude of ( l$:d.) overshoots the value derived out of the y-decay work. The deviations are especially large for (d, d’) reactions 17,20). As an application of our general method we have chosen the isospin violating inelastic scattering of deuterons leading to the 15.11 MeV state in “C . The actual calculations have been performed within the method of coupled channels for nuclear reactions 27). The total wave function of the projectile-plus-target system is projected onto a subset of states in which the deuteron remains in its ground state. Only a limited set of target states is considered. By means of the functions
IJ&f,, Y> =
CY&)JG @Ix dmJ,,,P
(26)
where &(r’) and #H[) denote the internal wave functions of projectile and target in the state y, l&J@ is a spherical harmonic depending on the orientation of the relative coordinate R (R = R/R) of projectile and target, a system of coupfed equations for the wave functions of relative motion is derived 28):
(27)
164
H. Lenske, G. Baur 1 Coulomb interaction
Here the diagonal operator
h,,(R) =
g2- q - 2
( Ulj,(R) -t V,(R))
(28)
contains an optical potential U,,,(R) and the single-particle Coulomb potential I/,(R). q, is the reduced mass in channel y. In eqs. (25) and (26) a shorthand notation has been used for the coupling of spherical tensor operators CT, x
I;C2ILM = c (Ilm,l,m*ILM)?;1,,7E2,2. m1mz
(29)
The coupling potentials “), I$J = (J,M,, YII$:’ f IGW,M,,
r’>,
(30)
have been calculated in the microscopic model for the projectile and the target. They are mainly given by folding the residual NN interaction VP, = c &&, &, R) with the ground-state density of the deuteron and the transition density of the target excitation 29, 30). The isospin conserving nuclear part of V,, is taken to be of the form V,, = v,,+V,,o.a’+l/,,z.2’+I:,a.o’z,2’+(~~/,,+~~’,,z.z’)S,,.
(31)
Here, S,, = (a * +)(a’* P)-+ * CT’is the usual tensor operator. The primed operators act on the projectile system. The folding of V,, by a T = 0 projectile density gives an effective interaction
where V denotes the folded potential. For the calculations the conservative assumption is made that the Coulomb interaction is the only charge-dependent interaction between projectile and target. The two-body Coulomb interaction (r = rl -rJ,
v,(r) = %i [T”‘+7p)+
7’3’23
(33)
can be split into an isoscalar part,
T(O) = 1+&z
29
(34)
the 3-components of an isovector operator,
T3'l'= T;+T~ 23
(35)
Ti2' = zfz;-+q.z2’
(36)
and an isotensor operator 3),
H. Lenske, G. Baur 1 Coulomb interaction
165
If the residual projectile target Coulomb interaction iS folded with the ground-state density of a T = 0 projectile, the isotensor part vanishes. Therefore, VJ$ is assumed to be given by the isovector part of the residual projectile-target Coulomb interaction for the case of a T = 0 projectile. In the model calculations the free NN T-matrix of Eikemeyer and Hackenbroich 31) has been taken for V,, in the form as given by Amos et al. 32). Whereas the Wigner part FoOof the effective deuteron target nucleon interaction is well known 33), there is only little information on the spin-dependent parts 00, and I& which are especially important for unnatural parity excitations. In a study of 12C(d, d’)l’C*(l+, 12.71 MeV) - which will be published separately it has been found that the tensor interaction together with non-central components out of the deuteron D-state give an important contribution to spin-flip excitations 2g, 30) in inelastic deuteron scattering. In the model calculations reported here these effects have been included. The ground-state wave function of the deuteron is approximated by a Os-harmonic oscillator function for the 3S, state and a Od-harmonic oscillator function for the 3D, component with an oscillator constant b = 2.05 fm. The D-state probability is taken to be pn = 0.04 [ref. ““)I. Calculations have been performed using two different lp-lh RPA models for the “C states. In the RPA calculations of Gillet and Vinh Mau 35) charge-dependent effects within the nucleus have been neglected. As s.p. wave functions harmonic oscillator functions were taken. These RPA wave functions have been used as the isospin pure basis states in a perturbative two-state mixing ansatz for the two 1’ states t. The mixing amplitude 6 was chosen according to the results of Adelberger et al. 21). Then, the Q-value corrected ratio, do’5.11 R = k(s) do12.71 = a2, (37) is simply given by the square of the mixing amplitude (6 = 0.046). The function k(9) is defined by k(9) = doo(12.71)/doo(15.11).
(38)
Here, do,@,) is the model differential cross section calculated with the form factor for the inelastic excitation of the 1 ‘, T = 0 model state at E, = 12.71 MeV and E, = 15.11 MeV. Clearly, if induced isospin mixing effects are important, the simple relation between the two cross sections is destroyed. R will, in general, become a function of the incident energy of the deuteron and of the scattering angle. In fig. 4, R(9) is shown as it results from a model calculation in which, apart from + The aim of the following model calculations is to test the two-state mixing model with respect to its application in isospin-forbidden inelastic scattering. For this purpose we desist from the errors which the model introduces in structure calculations because they mainly concern the interpretation of the mixing amplitude.
166
H. Lemke,
C. Baur 1 Coulomb
- R (a)
‘2C(d.d’) ‘*C* (I+.1511 Me’.‘)
I
fxlo-31
k
I
I
I,
interaction I
I
I
I.
I
-
$=280MeV ---WITHOUT ASC -WITH ASC
35-
r) I I ,,-... I
‘._/
I1
( , , ,
/
I
L_f
/
, , ,
, ml
50 3
/:
’
-
;
,’
( ,
I 150
CM
Fig. 4. R(9) as a function of the scattering angle 8_,. as it comes out when the RPA wave functions of Cillet and Vinh Mau are used. The asymptotic Coulomb coupling correction is denoted by ASC. For details see text. If induced isospin mixing were neglected, R(9) was a constant denoted here by 6*.
target mixing, induced mixing effects are also incorporated. The couplings between the two 1’ channels as well as the coupling of both to GDR states at E, = 17.6 MeV and E, = 20.9 MeV are included. Within these model calculations the GDR states can only be excited by Coulomb excitation. The average optical-potential set of Hinterberger et al. 36) has been used including a spin-orbit potential [set 2 of ref. ““)I. Theory and experiment I’) differ by roughly an order of magnitude. [The experimental values l’) for R(9) are collected in table 3.1 The model calculations shown in fig. 4 show that the interference of the induced mixing contributions and the target mixing contributions in the Fmatrix are important. It is not justified to neglect the induced isospin mixing effects. However, the question arises what are the important intermediate channels. In table 4, results for R(9) are shown for different coupling schemes at a variety of scattering angles (Ed = 28.0 MeV). It is seen that within these model calculations almost all intermediate channels give the same contribution. The individual contributions interfere preferentially destructively so that the net result does not differ very much from a single contribution. It seems to us that there are no general arguments for the selection of intermediate channels. Clearly, this makes it hard to calculate the induced mixing contribution quantitatively. Special attention has been paid to the long-range coupling due to the Coulomb interaction. The ~~ptoti~ coupling of the 15.11 MeV state to the GDR states due to the slow decrease of the Coulomb part in the coupling potential was treated analytic~ly within a separation method 2Q,37). As is also shown in fig. 4, in these model calcula-
H. Lenske, G. Baur / Coulomb interaction
167
TABLE 3 Experimentally derived value for R(9), eq. (37) Ed = 24.1 MeV ________~
-~
R,(g) (x 10-z)
9 21.7
1.269 *0.114 0.837 + 0.069 0.643 kO.077
34.2 45.6
-__--R(9) (x10-Z) 1.989 kO.179 1.537 kO.127 1.308 kO.157
9 25.8 38.4 50.9 63.0 89.3
Ed = 26.2 MeV
____
______
R,(9) (x 10-Z) 1.098 kO.074 0.871 kO.112 1.107 +0.088 0.954 kO.066 1.218 *0.140
W) (x 10-Z) 1.816 kO.122 1.450 kO.186 1.880 kO.149 1.627 F0.113 1.396 kO.160
9 19.2 25.5 34.3 44.3
Ed = 27.5 MeV R,(9) (x 10-z) 1.167 FO.072 1.164 kO.052 1.153 kO.068 0.985 kO.069
R(9) (x 10-z) 1.807 kO.111 1.928 kO.086 1.838 kO.115 1.491 kO.105
R, is just the ratio of the experimental cross sections whereas R is the Q-value corrected value, eq. (37).
tions the asymptotic coupling gives an important contribution to the cross section leading to the 15.11 MeV state. [If we were to try to fit the experimental magnitude we would need a charge-dependent matrix element of roughly 330 keV in accordance with the results of ref. “) where the data were analyzed within a model of this kind.] Apart from Coulomb excitation, the inelastically scattered deuteron acts like an isoscalar operator on the target nucleus. Now, it is well known that in N = 2 nuclei the proton and neutron components of the ground-state transition densities of T = 1 states are on the average of the same magnitude, but they carry a relative minus sign. On the other hand, if a transition is due to an isoscalar external operator, the sum TABLE4 Induced isospin mixing contributions due to different coupling schemes at different scattering angles 9 R(9) 9=S” (x lo-s)
Coupling scheme ____ O’++l’(T=
1)
R(9) R(9) R(9) R(9) W) W) R(9) 9= 12” 9=30” 9=44” 9=92°9=120”9=130” 9=150” (x lo-s) (x 10-a) (x 10-a) (x 10-3) (x 10-3) (x 10-3) (x 10-j)
2.204
2.100
2.173
2.174
2.194
2.211
2.238
2.188
0+~1+(7-=0)++1+(7-=
1)
2.565
2.588
2.404
2.378
2.542
2.496
2.675
2.714
O’++l’(T=
l)ul,(T=
1)
2.414
2.396
2.210
2.219
2.261
2.263
2.318
2.209
o+++l+(T=O)u1+(T= ttl,(T= l)ul;(T=
1) 1)
2.402
2.521
2.606
2.562
2.8i6
2.272
3.038
3.198
o+ t--) 2f(T = 0) +-+l.+(T = 0) ++ 1 +(T = 1)
3.435
3.320
2.408
2.415
2.607
2.622
2.650
3.383
168
H. Lenske, G. Baur 1 Coulomb interaction
of proton and neutron transition densities enters into the matrix elements, i.e. the excitation of an isovector (T = 1) state by an isoscalar field is mainly determined by the difference of the appropriate proton and neutron transition densities. Therefore, from the structure point of view, isospin-forbidden inelastic deuteron scattering tests the difference of the proton and neutron transition densities. If isospin would be conserved, the isoscalar transition density had to vanish for the excitation of an isovector state. In figs. 5 and 6 the proton and neutron components of the ground-state transition density for O+ -+ (l+, 15.11 MeV) in “C are shown together with the isoscalar density as they come out of a lp-1 h RPA calculation using realistic s.p. wave functions 22). In fig. 5 the L= 0, S = 1 transition density is shown, in fig. 6 the L= 2, S = 1 transition density can be seen for the O+ + 1 + spin-flip transition to the 15.11 MeV state in “C. In both cases it is seen that the proton and neutron transition densities behave quite differently. Because of configuration mixing effects these differences are much
RPA -TRANSITION DENS11 Is-1* 15 11Me” PROTDN --- NEVTRON -rscsC*Lb& L:o, s=1
10
20
30
LO
50
60
70 R[fml
Fig. 5. Proton,
neutron, and isoscalar transition density L = 0, S = 1 for O+ -+ l+ (15.11 MeV) in “C as they result from the RPA calculations of Speth et al. ‘*).
169
H. Lenske, G. Bmr 1 Coulomb ~tera~tio~ 7-i7, S,,, IF31f ti31
RPA-TRANSITION DENSITY Ofi E,:15 I, Me”
t
“C
/ I
10
I
20
I
t
I
I
I
3.0
LO
50
60
70
t
Rffml
Fig. 6. The L = 2, S = 1 transition densities for O+ I, l+ (15.11 MeV) in “% [ref. 22)].
more pronounced than in the case of the pick-up form factor discussed in the previous section. It is also seen from figs. 5 and 6 that the isoscalar densities (i.e. the difference of the proton and the neutron densities) cannot be derived by the simple first-order perturbation ansatz. In table 2 the RPA amplitudes 21) for the 12.71 MeV state and the 15.11 MeV state are given, The s.p. Coulomb effects destroy the symmetry of the proton and the neutron ph states in a much more complicated way than one would expect from a fnstorder treatment. The angular distributions for “C(d, d’)12C*(l+, 12.71 MeV) and W(d, d’)12C* (1 +, 15.11 MeV) can be seen in figs. 7 and 8 for several incident energies of the deuteron. The experimental data are those of Lind et al. “). Except for the case E, = 28.0 MeV, there are only few experimental points. Therefore, a detailed comparison of theory and experiment is not possible. However, in all cases the magnitude of the angular distribution& given correctly in the model calculations. Also the L,==2 shape of the experimental results is reproduced showing the importance of the tensor interaction and of non-central contributions out of the
H. Lenske, G. Baur / Coulomb interaction
n.
I’ 50
7 ‘I
1 100
‘I
II 150
‘CM
Fig. 7. Angular distributions for ‘W(d, d’)‘*C* (1 +, 12.71 MeV). Calculations without induced mixing are denoted by “direct”, those which include induced mixing are denoted by ‘W’. The RPA wave functions of ref. “) are used. Experimental points are taken from Lind et al. I’).
E,=2L,lMeV
1
’ QCM
Fig. 8. Same as fig. 7, but for ‘*C(d, d’)‘*C* (I+, 15.11 MeV) (isospin forbidden).
deuteron D-state for deuteron induced spin-flip transitions. A very striking feature shows up for E, = 28.0 MeV. While the experimental angular distribution leading to the 12.71 MeV state is rather flat, the corresponding one for the excitation of the 15.11 MeV state seems to decrease faster with increasing scattering angle. Like those of the other incident energies, the shapes of the theoretical angular distributions for the two transitions are quite similar. One may speculate that the enhancement of the “T = 0” cross section at backward angles is due to compound processes, which may give a larger contribution to “C(d, d’)l’C*(l+, 12.71 MeV) than to l’C(d, d’)“C*(l+, 15.11 MeV) because the “l 2C + d” compound systems preferentially occupy “T = O”states. Clearly, additional experimental information is needed in order to study this effect in detail. Finally, in fig. 9, the experimental and theoretical results for R(9), eq. (37), are compared for the different incident energies of the deuteron. These results demonstrate in a very clear way the errors which are introduced when R(9) is interpreted within the perturbation model discussed in sect. 2.
H. Lenske, G. Baur / Coulomb interaction 50
100
I’
5.0
3 ’ ,I’
150%M
“I
50
I/
RIBI k-m-2f
100
1% ‘I
150 %M
I II
t “II
‘ZC(d,~~‘2C*~l~,15~lMe#
- 5.0
Ed -28.OMeV -cc - -
4.0
171
DIRECT
- L.0
3.0
- 3.0
2.0
- 2.0
-1.0
1.0
t
5.0 -
R@’ Ixlo-21
12C(d,d~)12CX~?~15.11MeV~
__
Ed =26 2MeV
Ixlo-21
f
i
1ZCfd,d’~‘2C *(I:
15.11 MeW
- 5.0
Ed’2LlMeV -cc
-cc - -
R(9)
DIRECT EXP.
- f I
I
t
t
DIRECT
d.0
EXP.
I
I
t
3.0 -
- 3.0
2.0 -
- 2.0
1.0 -
- 1.0
~
Fig. 9. R(9) as defined in eq. (37). The RPA wave functions of Speth et al. 22) are used. The interference of the induced mixing contributions and the direct component is clearly seen. Experimental points are taken from ref. “). It is seen that at Ed = 57 MeV (no experiment available) the induced mixing effects are less important at least within the model calculations.
172
H. Lenske,
G. Baur / Coulomb
interaction
From figs. 8 and 9 it is seen that the magnitude of isospin-forbidden inelastic cross section is mainly determined by the isospin mixing in the nuclear wave function. The induced isospin mixing contributions to the T-matrix are mainly responsible for the angular dependence and energy dependence of R(9). They have to be included into a theoretical study of isospin-violating direct reactions. In these model calculations the channel coupling of the two 1’ channels gave the most important single contribution to the induced isospin mixing. It is also seen from fig. 9 that it can be expected that the induced mixing contributions are reduced at higher incident energies. Therefore, in order to study the isospin structure of the nuclear wave function, the experiment should be repeated at higher deuteron energies. At all incident energies the theoretical results for R(9) are larger than experiment. Also the angular dependence is not reproduced in detail. The remaining discrepancies may be partly due to the nuclear structure model which has been used for the calculation of the transition density (a spherical ground state is assumed for “C, but it is well known that 12C is oblate in the ground state 40). The theoretical y-width for O+ + 1 + (12.71) is too low [rtheor = 0.01 eV, reXP = 0.35 + O.O5eVforO+ + I+ (15.1 l), [ref. 21)], whereas it overshoots the experimental width Kheor = 58.27 eV, reXP = 37.0+ 1.1 eV, ref. “‘)I. On the other hand, it may be that there are important channel couplings other than those we included in the calculations. As already mentioned, it is hard to check this point because one can only consider a rather limited number of channels in the calculation. For example, in our calculations contributions to the induced isospin mixing due to the inelastic break-up of the deuteron and its subsequent recombination have been neglected.
5. Summary In sect. 2, the serious shortcomings of the conventional treatment of isospin-violating direct reactions were discussed. First, the conventional theoretical ansatz neglects the induced isospin mixing contributions which are due to channel-coupling effects caused in part by charge-dependent components in the projectile target nucleon residual interaction - like the well-known Coulomb interaction - and in part by isospin impurities in the wave functions of the intermediate state. The induced isospin mixing due to the projectile-target interaction would lead to a non-vanishing isospin-forbidden inelastic cross section even if the nuclear states are of pure isospin. It was shown that the induced mixing contributions and the direct excitation contribution due to an isospin impurity in the final nuclear state wave function enter coherently into the T-matrix element of an isospin-violating direct reaction. Therefore, induced isospin mixing effects have to be included into a theoretical study of isospinviolating direct reactions. Secondly, the conventional method assumes for the treatment of all charge depen-
H. Lenske, G. Baur / Coulombinteraction
173
dent interactions internal to the nucleus the validity of first-order perturbation theory even for the long-ranged Coulomb interaction. In sect. 3, this point has been discussed in detail. It was shown that this ansatz is simply wrong when it is applied in nuclear structure models because important isospin-violat~g contributions are neglected from the very beginning. The first-order perturbative method neglects the spatial differences of proton and neutron s.p. wave functions. In a simple lp-lh model it was shown that within this model isospinviolating contributions due to the differences of the proton and neutron ph matrix elements of the residual interaction are missing. Although these mixing effects are caused by the well known Coulomb interaction, they would be interpreted within this model as a hint for a new type of charge-dependent interaction. This wave-function effect has been verified by comparing proton and neutron ph-matrix elements calculated with realistic s.p. wave functions. Further, in sect. 3, we have shown that the perturbation method introduces serious errors into direct reaction studies of isospin mixing in nuclear states. For direct nuclear reactions mainly the surface and asymptotic region of the nuclear wave functions are of importance. But in this region the proton and neutron wave functions may be quite different especially for light target nuclei. In general, the different radial behaviour of the s.p. wave functions cannot be expressed by a single constant as is done in the perturbation method. Model calculations were performed for 13C!(d, t)“C*(l+, 15.1l.MeV) and 13C(d 3He)‘2B (I+, 0.0 MeV). In sect. 4 the interplay of induced isospin mixing and the target isospin mixing were studied for “C(d, d’)12C*(1 ‘, 15.11 MeV). We have applied our general method to this specific example because it is an experimentally very well studied isospin-forbidden direct reaction. Using the RPA wave functions of Gillet and Vinh Mau 35) and Speth et dz2), the perturbation method and a realistic treatment of the nuclear isospin mixing were compared. The model calculations of sect. 4 have shown that the perturbation method fails to explain the experiment, whereas good agreement was achieved between theory and experiment when realistic sp. wave functions were used and the interference of target and induced isospin mixing in the T-matrix is included into the calculations. It was pointed out that isosp~-forbidden direct reactions should be studied at higher incident energies (say 80 MeV or higher) in order to reduce the induced mixing effects. Iachello and Singh 37) have proposed that intermediate pick-up (d-3He-d and d-3H-d) or stripping (d-p-d and d-n-d) processes may be responsible for the observed isospin violation in (d, d‘) reactions. The calculations of Izumoto 38) show that these contributions are mainly due to the threshold behaviour of the intermediate channels. With increasing incident energy of the deuteron the pick-up stripping contributions decrease rapidly so that they will give only a minor contribution for the deuteron energies considered here. On the other hand, Izumoto 38) neglected the s.p. wavefunction effect.
174
H. Lenske,
G. Baur / Coulomb
interaction
Finally, we want to remark that there seems to be at the present stage no need to introduce some kind of new charge dependent nuclear interaction in order to explain isospin violations in direct nuclear reactions. We are indebted to Prof. Dr. J. Speth and Dr. J. Wambach for making available the RPA calculations and a lot of useful discussions on the RPA theory.
References 1) 2) 3) 4)
5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)
26) 27) 28) 29) 30) 31)
D. H. Wilkinson, ed., Isospin in nuclear physics (North-Holland, Amsterdam, 1969) E. M. Henley, Charge dependence and charge symmetry of nuclear forces, ref. i), p. 17 ff. R. J. Blin-Stoyle, Fundamental interactions and the nucleus (North-Holland, Amsterdam, 1973) P. Signell, Adv. Nucl. Phys. A141 (1969) 225; Riazuddin, Nucl. Phys. 7 (1958) 217; M. J. St. Stevens, Phys. Len. 19 (1965) 499; H. Goldberg, Nuov. Cim. 40 (1965) 243; R. H. Da!itz and F. von Hippel, Phys. Lett. 10 (1964) 155; J. S. Leuing and Y. Nogami, Nucl. Phys. B7 (1968) 527 R. J. Blin-Stoyle and J. Le Tourneux, Phys. Rev. 123 (1961) 627 S. Shlomo and D. 0. Riska, Nucl. Phys. A254 (1975) 281 S. Shlomo and G. J. Wagner, Z. Phys. A285 (1978) 283 H. Sato and L. Zamick, Phys. L&t. 70B (1977) 285 R. D. Lawson, Phys. Lett. 78B (1978) 371 J. B. Ehrman, Phys. Rev. 81 (1951) 412; R. G. Thomas, Phys. Rev. 81 (1951) 148; 88 (1957) 1109 M. Chemtob and M. Rho, Phys. Lett. 298 (1969) 540; Nucl. Phys. Al63 (1971) 1 N. T. Porie, J. C. Paar, T. Wiley and C. R. Lux, Phys. Rev. C9 (1974) 2171; C. R. Lux, N. T. Porie and S. M. Grimes, Phys. Rev. Cl5 (1977) 1308 H. L. Harney, H. A. Weidenmiiller and A. Richter, Phys. Rev. Cl6 (1977) 1774; H. A. Weidenmtiller, A. Richter and H. L. Harney, Phys. Rev. Cl8 (1978) 1953 D. Robson, Phys. Rev. 137 (1965) 835 D. Robson, Theory of nucleon induced analogue resonances, ref. ‘), p. 461 I. Rotter, J. Phys. G5 (1979) 75; 271 J. M. Lmd, G. T. Garvey and R. E. Tribble, Nucl. Phys. A276 (1977) 25 J. D. Cossairt and D. P. May, Nucl. Phys. A319 (1979) 182 G. J. Wagner, K. T. Kniipfle, G. Mairle, P. Doll and H. Hafner, Phys. Rev. Cl6 (1977) 1271 W. J. Braithwaite, J. E. Bussoletti, F. E. Cecil and G. T. Garvey, Phys. Rev. Lett. 29 (1972) 376 E. G. Adelberger, R. E. Marrs, K. A. Snover and J. E. Bussoletti, Phys. Lett. 62B (1976) 29; Phys. Rev. Cl5 (1977) 484 J. Speth, J. Wambach and V. Klemt, private communication J. Sanchez-Dehesa, doctoral thesis, University of Bonn, Jiil-Report 1425, Jtilich 1977 N. Austern, Direct nuclear reaction theories (Wiley, New York, 1969) T. Berggren, Nucl. Phys. 72 (1965) 337; W. T. Pinkston and G. R. Satchler, Nucl. Phys. 72 (1965) 641; A. Prakash and N. Austern, Ann. of Phys. 51 (1969) 418 S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1 T. Tamura, Rev. Mod. Phys. 37 (1965) 679 V. A. Madsen, Charge exchange and inelastic scattering, in Nuclear reactions and spectroscopy, ed. J. Cerny (Academic Press, New York, 1975) H. Lenske, doctoral thesis, University of Bonn, Jill-Spec-42, Jiilich 1979 H. Lenske, to be published H. Eikemeyer and H. H. Hackenbroich, Nucl. Phys. Al69 (1971) 407
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115
32) K. A. Amos, H. V. von Geramb, R. Sprickmann, J. Arvieux, M. Buenerd and G. Perrin, Phys. Lett. 52B (1974) 138 33) F. Hinterberger, G. Mairle, U. Schmidt-Rohr, G. J. Wagner and P. Turek, Nucl. Phys. AllS( 1968) 570 34) D. W. Sprung, Few body problems in nuclear and particle physics, ed. R. J. Slobodrian (Les Presses de l’universite Laval, Quebec, 1975) p. 475 35) V. Gillet and N. Vinh Mau, Nucl. Phys. 54 (1964) 321 36) F. Hinterberger, G. Mairle, U. Schmidt-Rohr and G. J. Wagner, Nucl. Phys. All1 (1968) 265 37) F. Iachello and P. D. Singh, Phys. Lett. 4uB (1974) 81 38) T. Izumoto, Nucl. Phys. A275 (1977) 342; and private communication 39) C. M. Perey and F. G. Perey, Atomic Data and Nucl. Data Tables 17 (1976) 40) W. H. Bassichis, A. K. Kerman and J. P. Svenne, Phys. Rev. 160 (1967) 746 41) D. Robson, Phys. Rev. C7 (1973) 1
by photoprint
115; @
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THE ROLE OF THE COULOMB INTERACTION IN ISOSPIN-VIOLATING DIRECT NUCLEAR REACTIONS H. LENSKE
+ and
Znstitutftir Kernphysik, Kernforschungsanlage Received
G. BAUR JtXch, D-51 70 Jti’lich, West Germany
17 January
1980
Abstract: We investigate possible sources of isospin mixing in direct nuclear reactions. We find that there are mainly two effects which enter coherently into the direct reaction amplitude : “induced isospin mixing” caused by the Coulomb interaction between the target and the projectile, and “wavefunction effects” of the target nuclear states. These arise mainly through the difference of corresponding neutron and proton single-particle wave functions. These differences are especially noticeable just in the surface region where direct reactions take place. As illustrative examples we perform mode! calculations for the pick-up reactions ‘%(d, 3H)‘zC*(1+, 15.11 MeV) and ‘%(d, 3He)‘ZB,., (l+) and the inelastic scattering “C(d, d’)“C* leading to the l+ states at 12.71 MeV and 15.11 MeV. These model calculations are compared with the corresponding experimental data.
1. Introduction The concept of isospin in nuclear physics is directly connected with the very fundamental question whether the nuclear interactions are charge independent or not. Therefore, isospin violations have been studied extensively from the very beginning of nuclear physics. In ref. ‘) one can find an extensive review on former work on isospin violating effects in nuclear systems. However, it is well known that the nuclear hamiltonian does not commute with the operator of the total isospin, at least because of the presence of electro-magnetic interactions and the static Coulomb field which contain both charge-symmetric and charge-dependent pieces “). Therefore, it is necessary first to correct theoretically for the electromagnetic contributions to isospin-violating processes before one can draw conclusions about short-ranged nuclear charge-dependent interactions. The difficulties connected with this procedure can be seen most clearly in the analysis of lowenergy NN scattering ‘). Low-energy NN scattering seems to indicate a short-range charge symmetry breaking component in the free NN interaction of the order of 2 % of the average NN potential ‘). There is strong evidence that this charge-dependent NN interaction is caused by indirect electromagnetic effects like the mass splitting of the exchange bosons 2-4), + Present
address:
Center for Nuclear
Studies,
University 151
of Texas, Austin,
Texas 78712, USA.
152
N. Lenske, G. Baur / Coulomb interaction
and no new type of strong interaction is needed to explain the charge dependence of the free NN interaction, This result shows clearly how important a role the electroma~eti~ interactions play in ~osp~-violat~g processes. Although they are weak compared with the average nuclear interaction, it is not, in this case, justified to neglect electromagnetic processes of higher order. A great amount of work has been done on isospin-violating effects in more complex nuclear systems, experimentally and theoretically. Charge-dependent effects in ground-state properties and excited states of nuclei as well as in nuclear reactions were observed l3 3)_Whereas in the past several authors have claimed the need of a charge symmetry breaking interaction ‘, 6, in order to explain the oberved degree of isospin violation, more recent investigations stress the importance of the static Coulomb field for isospin violations in nuclei ‘- ‘). For example, Lawson ‘) has shown for lp shell nuclei that the differences of the proton and neutron single-particle wave functions can give rise to large isospin mixing effects even if the residual interaction is taken to be charge independent. This is entirely an effect of higher order in the Coulomb interaction and is comparable with the well-known Thomas-Ehrman shift lo) in the binding energy of the proton-rich members of nuclear isospin multiplets. In both cases a first-order perturbative treatment of Coulomb effects is not appropriate. The inte~retation of isospin mixing in nuclear many-body systems is in general much more complicated than in free NN scattering. Because one must necessarily employ a model, one has to check thoroughly to what degree the approximations influence the theoretical determination of isospin mixing. Furthermore, the need to use effective operators when nuclear models are employed may introduce further un~ertainti~ into the theoretical investigations. [For example, the calculated magnetic y-transition rates in nuclei may change appreciably if one changes from the bare electromagnetic operators to effective operators, which include important exchange effects and many-body renormalizations “)I Therefore, theoretical studies of isospin-violating processes in nuclei will in general determine the effective degree of isospin mixing within the model. The studies of Lawson ‘), Sato and Zamick ‘) and Shlomo and Waft 7, show that it may be hazardous to conclude from a model calculation the existence of a charge-dependent nuclear interaction. In the field of nuclear reactions, isospin violations have been studied in compound nucleus reactions I’- 14)(CNR) and for the excitation of isobar analog resonances “) (IAR). In CNR and IAR it is usual to subdivide the total isospin mixing into the “internal mixing”, due to isospin mixing within the space of ~ompo~d states, and the “external mixing” of the compound states, which is caused by the asymmetric coupling of the compound states to channel states differing in the charge of the projectile or ejectile, i.e. external mixing results from the reaction mechanism and the Coulomb interaction. Calculations show 16) that external mixing gives, at least in light nuclei, the main con~ibution to the observed total isospin mixing in CNR. Obviously, in CNR one studies the isospin impurities in the projectile plus target system. On the other hand, direct reactions have been used to measure the isospin
H. Lenske, G. Baur / Coulomb interaction
153
mixing of individual target states, mainly in light nuclei 17-20). Two kinds of experiments have been performed: (i) Certain pairs of isospin allowed nuclear reactions - like (d, 3H), (d, 3He) leading to the states of an isobaric multiplet - are observed. Isospin conservation predicts a definite value for the relative yield. In this class of experiments the transition operator is of mixed isospin. (ii) The intensities of an isospin-allowed and an isospin-forbidden excitation of the same J as seen for example in the inelastic scattering of deuterons or a-particles are compared. Here, the transition operator possesses a definite value of isospin. Therefore, if one assumes that charge-dependent projectile-target interactions and higherorder processes do not contribute to the excitation, the isospin-forbidden cross section is solely determined by the isospin impurity of the nuclear state in question. At this point we can already see the problem qualitatively: the theory of direct nuclear reactions -which is anyway never considered to be more exact than say to 20 % is usually most appropriate and accurate for rather strong transitions (e.g. good s.p. states in transfer reactions, strongly excited collective states in inelastic scattering). But if one studies weak transitions or deviations of the order of 1% from a predicted value, there is a great sensitivity to otherwise negligible details of the nuclear wave function and of the reaction mechanism. We address here ourselves to the question : Is there an unambiguous way to extract the degree of isospin mixing of a nuclear state which is excited in a direct nuclear reaction? This problem is investigated step by step in sect. 2. Firstly, we study the role of the residual projectile-target Coulomb interaction in isospin-violating direct reactions. This leads to the model of induced isospin mixing. It is shown that the residual projectile-target Coulomb interaction gives a contribution to the isospin-violating T-matrix which is comparable with the effect of the Coulomb force within the target nucleus. So it is in general not justified to neglect the charge-dependent parts of the residual projectile-target interaction in isospin-violating direct reactions even if the Coulomb interaction cannot contribute to the direct excitation of the final state. Secondly, the contributions of multistep processes are discussed for the general case that the nuclear states are of mixed isospin. It is shown that the interplay of the target isospin mixing and of the induced isospin mixing will in general lead to important interfering contributions to the isospin violating cross section. The ideas of sect. 2 are applied to the representative example 12C(d, d’)12C* leading to the 1 + states at E, = 12.71 MeV (mainly T = 0) and E, = 15.11 MeV (mainly T = 1). We have chosen this example because most of the previous experimental 16,18, and theoretical ‘-‘) work has been concentrated on these two levels. Our results, however, are of a general nature and can be applied - mutatis mutandis - to any kind of direct reactions. In sect. 3 we discuss the serious shortcomings of the conventional perturbation treatment of charge-dependent interactions when it is applied for nuclear structure
154
H. Lenske,
G. Baur / Coulomb interaction
and direct reaction studies of isospin-violating processes. As an example we study 13C(d, t)l’C*(l+, 15.11 MeV) and 13C(d, 3He)12B(1+, 0.0 MeV). The results for 12C(d, d’)12C* leading to the two l+ states are given in sect. 4. Nuclear wave-function effects will be studied for this reaction as well as the important role of induced isospin mixing effects in isospin-forbidden inelastic scattering. Finally, in sect. 5, a summary is given.
2. Induced isospin mixing effects in direct nuclear reactions The hamiltonian of the projectile-target
system,
H = H,+H,+H,+
T/p*,
(1)
consists of the internal hamiltonians HP and H, for the projectile and the target nucleus, the hamiltonian H, of the relative motion, which is thought to be given by the optical model, and a residual projectile-target interaction VP,. Alternatively, this hamiltonian can be split in an isospin-invariant piece H(O) and an isospin-violating (i.e. charge-dependent) piece H,,,, :
H = H"'+H H,,,_ contains charge-dependent
(2)
c.d:
interactions internal to the reaction partners as well
as those between them
H c.d.
=
%.
+
K:d.
+
v,‘d4.
(3)
For the sake of definiteness let us consider the case of an isospin-forbidden inelastic reaction. Let us furthermore assume that Vcy$cannot contribute to the direct excitation of the final state like it is the case for a O+ -+ l+ spin-flip transition. During the scattering process the projectile is assumed to remain in its ground The action of Ktd. within the target nustate of pure isospin, i.e. we can neglect VcTd,. cleus leads to isospin impure eigenstates 4,,(Y). In the index n, all quantum numbers other than J” are contained which are necessary to specify the state. The main contribution to T/$(and also to Kyd, and Vcy$)is given by isovector components. Then, for a nucleus with N neutrons and 2 protons, the states 4,(P) can be decomposed quite generally into 4”(Y) = 1J”n; To) + IJ”n; To + l),
(4)
where To = +(N-- Z). Often, the much stronger assumption is made that it is sufficient to treat Vctd,in first-order perturbation theory. 4,(Y) is then decomposed into states of good isospin 4JJ”, T),
H. Lenske, G. Baur / Coulomb interaction
155
and T’ may equal T [dynamical distortion ‘“)I_ If only one term of the sum with T’ # T is important, one ends up with the commonly used “two-state mixing’: model. The errors, which are possibly introduced into nuclear structure models and, more important, into nuclear reaction studies when one uses the perturbation method, are discussed further in sects. 3 and 4. But for the purpose of this section it will be sufficient to use the ansatz eq. (5) for the eigenstates +,(J”). What is the role of Vii for isospin-forbidden inelastic scattering? Although contributions of Vz$ to isospin-violating reaction processes are usually neglected, there are no general arguments for doing so. It appears very much on the same footing as VC$..V$ includes additional isospin mixing effects into the projectile-target system because by its action the target (or projectile) can be polarized into states with T # To. ‘* leads to a non-vanishing isospinEven if the target states are of pure isospin, V,.,. violating T-matrix element
In eq. (6) a typical T-matrix element due to Vp”hp. is given in second-order DWBA. The summation (or integration for continuous quantum numbers) is extended over intermediate channels with T # To. The distorted waves with wave vector k in the various channels are denoted by xl*) and ji(+) is the conjugate solution to x(+j41). Em is the energy of relative motion in channel m and EC+) = lim (E-t iy), E the energy in the incident channel. I$? is the isospin conserving part of VpA. It can be seen from eq. (6) that the isospin violation is mainly determined by the amplitude
(7) Induced isospin mixing appears in the A+ P system in complete analogy to the isospin mixing within the A-system alone [see eq. (5)]. It can be expected that qTd will be energy dependent because at different incident energies there can be quite different intermediate channels m of importance. Next let us consider the general case that the target states are of mixed isospin. According to eq. (5) the final state (mainly T # To) can be excited in a one-step process,
156
H. Lenske,
G. Baur / Coulomb
interaction
but the induced isospin mixing effects are also present. The total amplitude is given by the coherent sum T.fl = T?‘+T?‘d fl fl Therefore, the cross section for isospin-forbidden the interference of TrF and Tryd:
(9)
.
inelastic scattering is determined by
(10) If the target states are of mixed isospin, additional terms appear in cd. A typical T-matrix element for the excitation through a single intermediate channel m is, in second-order DWBA, given by
In the first step the T,-component of &(J’) is excited, in the second step the scattering goes from the T # Tocomponent of 4,(J) to the T-component of the final state. All the above T-matrix elements can be expected to be of the same order of magnitude. Typically they are given by the product of a Coulomb matrix element and a nuclear interaction matrix element. There are further contributions to ed which are of higher order in the chargedependent interactions Ktd. and l$y$. It is seen from eq. (10) that not only the magnitude of the total T,i”d is important but also the relative phase angle of Cd to qy. The induced mixing contributions may interfere destructively with the direct excitation resulting in a reduction of the isospin-forbidden inelastic cross section. If in an analysis Ted is neglected at all, then one would be led to the conclusion that the final state is of rather pure isospin. In the commonly used “two-state mixing model’ +1(J) = &(J,
T)-B&U,
4o(J) = c&G, To)+ &(J,
T,),
(12)
T)>
where in first order in T/td,,
(13)
H. Lenske, G. Raw / Coulomb ~t~ra~tio~
157
it is assumed that Tfpd can be neglected. Then the ratio R =
$
2
k(9) =
0 ;
(14)
would allow a direct measure of the isospin mixing amplitude of the target states. k(9) corrects for the differences in Q-values (see sect. 4). We note the following points: (i) If this model is reasonable, then R has to be independent of the reaction kinematics. (ii} The model depends seriously on the ass~ption that Coulomb effects within the target can be described in first-order perturbation theory. Thus the different radial dependence of proton and neutron s.p. wave functions is neglected. This point will be discussed in detail in sect. 3. (iii) Commonly one concludes from a non-vanishing R on a charge-dependent nuclear interaction. But one should notice that within this model in Vctd.necessarily the Coulomb interaction is contained.
3. Coulomb effects in the target wave function and their consequences for structure and direct reaction studies of i~pin-~olating prowsses The aim of this section is to show that the commonly used first-order perturbative treatment of charge-dependent interactions within the nucleus can lead to serious errors when single-particle Coulomb effects are included in this procedure. We want to stress the point that at least the monopole part of the target Coulomb interaction should be treated exactly in studies of isospin-violating processes ‘-‘). Let us set up a simple model. The hamiltonian of the nucleus A,
HA= %.p.+ b2,
05)
may consist of a s.p. h~ilton~ H_ (in general different for protons and neutrons, also in the short-range nuclear part - see below) and a residual (isosp~-conse~ing) two-body interaction
where V, and V, may depend not only on the relative coordinate but also on spin and tensor operators. If the hamiltonian is diagonalized in the 2-dimensional ph-space M$%),
&?(v)) with (17)
158
H. Lenske, G. Baur / Coulomb interaction
and CI= 7~(proton), v (neutron), the eigenstates are given by &(JM) = X,i&%J)
- Z,i&%4
&l(JM) = Z ,l&?(v) + Xpl&%). The coefficients are determined
(18) (19)
by
Z
-= ph xph
$
@ifi=&4,),
(20)
ZY
together with the normalization
condition
z,2,+Xih = 1.
(21)
In eq. (20) we have introduced the quantities A,, = s;h - s;h + v,, - v,,,
(22)
I/,p = <~;~(~)Iv,,I4;~(P)>~
(23)
and sEh = E~-$,(E~ < 0) are the ph energies. If A,, = 0. then X,, = Z,, = ,,/$ and the states & and & are of pure isospin T = 1 and T = 0. From eq. (20) it follows that the s.p. Coulomb effects introduce in two different ways isospin impurities in nuclear states: (i) through the differences in the proton and neutron ph energies, but also (ii) through the different values of the proton ph and neutron ph matrix elements of the originally isospin conserving residual interaction which are due to the differences in the radial behaviour of proton and neutron s.p. wave functions. One should notice that the second contribution is necessarily missing if one starts from a set of isospin pure basis states (which imply that proton and neutron wave functions are identical) and treats the Coulomb interaction - which is contained in l$$, - in first-order perturbation theory. This wave-function effect was studied first by Auerbach et al. “). Sato and Zamick *) have investigated the problem using HF wave functions. Lawson 7, considered the wave function effect for Woods-Saxon eigenfunctions within shell-model calculations. A strong sensitivity of the effect on the shape of the sp. potential and on the binding energy was observed. However, in refs. 7. *) a residual interaction of simple structure has been used. The analysis has been redone for the case of the already mentioned two lowest 1+ states in “CatE, = 12.71 MeV(“T= O”)andE, = 15.11 MeV(“T= 1”). Intable 1 theph matrix elements are listed as they are used in a RPA model calculation 22). The ph interaction includes, beside &function and density-dependent b-function contributions, also finite-range terms due to n- and p-meson exchange. The s.p. wave functions are generated in a Woods-Saxon potential which gives the correct binding energies 23). The numbers of table 1 show the importance of the s.p. wave function effect on the matrix elements of the residual interaction.
H. Lenske,
G. Burr
/ Coulomb
TABLE
159
interaction
1
Proton-proton, neuron-proton and neutron-neuron ph matrix elements of the type < lp,,r, lp$ /V/j& ‘) and j&t are given in the first coiumn; Woods-Saxon s.p. wave functions are used, Y contains densitydependent zero-range and finite-range contributions including tensor interactions “1 A""
._I
JpJh
CkeVl __-.-
_._---lP,,*lP$ 2P,,,lP,i: 2P,,*lP$ lf,,,lP$ ld,&,t, ZS,,,lS,:
5.8362 2.8508 2.7376 -0.2679 -0.1830 - 1.9554
- 1.0047 -0.3846 0.0969 2.4397 -2.3884 0.9940
5.6324 2.8157 2.7529 -0.2545 -0.1651 - 1.8781
203.8 35.1 - 15.3 - 13.4 - 17.9 - 77.3
- 1.0047 -0.3882 0.0728 2.4215 - 2.3197 0.8326
0.0 4.4 24.1 18.2 -68.7 161.4
12.05 34.00 35.80 36.60 41.80 37.15
12.86 34.00 35.30 36.30 40.64 35.95
810 0.0 -500 -300 -1160 -1300
The main content of the perturbation method, eq. (5), is to neglect the contributions of the ph matrix elements due to the wave function effect and to approximate the ph energy difference of the neutron and the proton ph pair in first-order perturbation theory in the Coulomb interaction [see, for example, the analysis of Shlomo and Wagner, ref. ‘)I. In terms of our amplitudes X, and Zph, this means
In the last step, the ph energy difference in the denom~ator has been neglected and in the two-state-model one has E, -E, = 21/,,. The Coulomb interaction is denoted by V,. It has to be realized from eqs. (20) and (24) that the first-order perturbation method is simply wrong when it is applied for the study of isospin mixing in nuclear levels. At least the monopole (single-particle) part of the Coulomb interaction has to be treated exactly. From the very concept the perturbation method is not able to give a correct description of the effect one is looking for. What are the consequences of the nuclear wave-function effect for isospin-violating direct nuclear reactions? There are mainly two important points: (i) The Coulomb intera~ion within the target will in general destroy the symmetry between neutron and proton ph amplitudes in a much more complicated way than the perturbation treatment suggests (see table 2). (ii) Direct nuclear reactions are mainly sensitive to the behaviour of the nuclear transition density in the surface and the asymptotic region. But in these regions of space the differences of the proton and neutron s.p. wave functions may be very pronounced. Both effects are due to the same reason and cannot be separated in general. (Within the ph model they enter multiplicatively.)
H. Lenske, G. Baur / Coulomb interaction
160
TABLE 2
RPA amp!itudes
and neutron configurations for the 1 + states at E, = 12.71 MeV and E, = 15.11 MeV in ‘% according to ref. 22)
of proton
l’C*(l”, h
1P 3/z
Is l/Z
proton
P
1P 112 2P 3/z 2P l/Z lf,,, 2s 1/z Id 312
12.7 MeV(“T
= 0”))
l*C*(l+,
neutron
15.11 MeV(“T
proton
= 1”)j neutron
x
Y
x
Y
x
Y
x
Y
-0.6163 0.0501 -0.0549 0.0987 0.0260 0.1133
-0.1392 0.0237 -0.0123 0.0364 -0.0147 0.0822
-0.7953 0.0672 0.0640 0.0833 0.0394 0.0966
-0.1749 0.0284 -0.0200 0.0383 -0.0052 0.0770
0.7856 -0.0812 -0.0616 0.0523 -0.0505 0.0364
0.1148 -0.0198 0.0218 - 0.0098 -0.0112 0.0046
-0.6086 0.0676 0.0459 - 0.0776 0.0444 -0.0598
-0.0848 0.0155 -0.0185 0.0004 0.0155 -0.0223
Therefore, if the perturbation method, as it stands, is used in direct reaction studies of isospin mixing, a serious error is introduced: one tries to describe a radially dependent effect in terms of a single multiplicative factor (apart from further errors which may enter when the induced mixing contributions are neglected). The perturbation method may be more reasonable when a “volume-like” process is involved as in y-transitions of low multipolarity. As an example of the wave-function effects in direct reactions let us consider the transfer reactions 13C(d, 3H)‘2C*(l+, 15.11 MeV) and r3C(d, 3He)‘2B,,,,(1+, T= 1) [ref. “)I. The final states belong to the same isobaric triplet. The “C-state is the T3 = 0 member while the “B._, forms the T3 = + 1 member. (We use the convention t, = +i for neutrons and t, = -3 for protons.) Correlations in the ground states of ’ “C(i-, T = T3 = i) and “B(1 +, T = T3 = 1) have been neglected. Therefore, 13C is assumed to be given by a lp+ neutron coupled to an inert “C core while r2B is chosen to be a lp, neutron particle, lp, proton hole state with respect to “C. Then, the first reaction proceeds via the pick-up of a lpi neutron while in the second a lp, proton is picked up. Within this model zero-range DWBA calculations have been performed. Non-locality effects were included in the usual manner 24). We do not want to discuss here the difficult theoretical problem of how to determine the correct transfer from factor [see, for example, ref. 25)], but we want to demonstrate how important it is to treat Coulomb effects within the target correctly in direct reaction studies of isospin mixing. In fig. 1 the lp, proton and neutron form factors are shown as calculated in a Woods-Saxon potential with standard geometry (R = 1.20 A*, a = 0.65) and a spin-orbit potential 25 times the Thomas spin-orbit term. The eigenvalues are taken to be equal to the binding energies of lp, protons and neutrons in “C (E, = - 17.16 MeV, .s, = - 14.80 MeV, ref. 23)). The depth of the potentials required for the proton and neutron is Vl = -61.93 MeV and Vi = -61.20 MeV.
Lenske,G. Baur 1 Coulomb interaction
H.
I
I
IO
20
,I
$
I
LO
I61
I
I
1
1
I
50
60
70
80
9.0
I 100 R (fm)
Fig. 1. The lp,,, proton and neutron wave functions in 13C as calculated in a Woods-Saxon well (see text). The binding energy is denoted by E.
It is seen from fig. 1 that the proton form factor shows an enhancement over the neutron one beyond the nuclear surface, while in the interior they are almost equal. In order to show the slectivity of direct reactions on the asymptotic part of the form factor, calculations have been done with a lower radial cutoff, i.e. the form factor is set equal 0 from the origin up to a certain R for the calculation of the DWBA T-matrix elements. In fig. 2 the ratio of the cutoff cross section to the full DWBA cross section at the first rn~irn~ for (d, 3H) and (d, 3He) is shown. It is clearly seen that the angular distributions in both cases are determined by more than 70% by the asymptotic region R 2 5.0 fin where the wave-function,di~eren~~ are large.
162
H. Lenske, G. Baur 1 Coulomb interaction
~:(mb/sterodl
i
12
11
10
09
O?
07 I
I
I
I
10
20
30
LO
I
III 50
60
RCUT-OF~
hl
I
I
I
I
10
20
30
LO
II 50
1 60
70
II 80
%.I
Fig. 2. The ratio of the cutoff cross section to the full DWBA cross section for 13C(d, t)“C* and 13C(d 3He)‘ZB, s. as a function of the cutoff radius’ R euloPf.The ratio is calculated at the first maximum of the angular distribution at 9 C.rn = 120.
Fig. 3. DWBA angular distribution fit to the experimental data of Lind et al. I’). The spectroscopic factors are denoted by S, for 13C(d, t) “C* and by S, for 13C(d, 3He)‘ZB,,,,. For &‘a two values are given. The one in parentheses corresponds to the case that the proton transfer form factor equals the neutron one. Non-locality corrections are included. Standard optical potentials were used [d: ref. 36), set 2, 3H/3He: ref. 37)] including 1 . s coupling.
In fig. 3 DWBA calculations are compared with the experimental data of Lindt et al. 17).
In a model calculation the s.p. Coulomb potential for the proton form factor in 13C(d, 3He)12B,.,, has been neglected, i.e. an isospin pure basis has been assumed. order to fit the experimental data, a spectroscopic factor is required which is greater by 27 % compared with the case the Coulomb potential is included. According to an ansatz of Lindt et al. l’) the ratio of the spectroscopic amplitudes for 13C(d, 3H)‘2C*(1+, 15.11 MeV) and 13C(d, 3He)12B,,,, enters as an important quantity into the analysis of isospin violations in transfer reactions. It is stated that the 12C 1 + states are expanded in basis states of pure isospin. On the other hand, the well depth method is used to calculate the form factor. But in doing this the energy shift of - 1.74 MeV of the “B_ relative to ’ “C( 1+, I 5.11 MeV) which is due to chargedependent interactions is automatically included in the calculation of the proton
H. Lenske, G. Bow / Coulomb interaction
163
transfer form factor, i.e. a basis of mixed isospin is used. Apart from the above discussed deficiencies of the perturbation method the ansatz is in contradiction to the assumption that the spectroscopic factor for “C(d, 3He)12B,.,. equals the spectroscopic factor for populating the T= 1 component of the 15.11 MeV state in 12C in 13C(d, 3H)12C*.
4. The interplay of induced isospin mixing and target isospin mixing in isospin violating inelastic scattering: application to “C(d, d’)“C* (l+, 15.11 MeV) The isospin mixing of the li states in “C at E, = 12.71 MeV (mostly T= 0) and E, = 15.11 MeV (mostly T = 1) was studied in the past by different means. The
two-state mixing model together with the perturbative method as described in sect. 2 has been used to interpret the experimentally observed deviations from strict isospin conservation. Adelberger et al.’ ‘) have determined from y-decay experiments an isospin mixing amplitude 6 = 0.046 +O.OlZ correspond~g to a charge dependent matrix element of (V$J = 110+30 keV within the two-state-model. In inelastic electron scattering a value of (V$J = 129.6k2.4 keV has been found (6 = 0.054+ 0.001) 2”). It has to tie mentioned that these numbers in addition depend seriously on the value of the theoretical isoscalar Ml matrix element of Cohen and Kurath26,28). Direct reaction studies of the isospin ~puriti~ in these two levels have been performed, for example, by Braithwaite et al. 20), Lindt et ~2.‘~) and recently also by Cossairt and May ‘*) in (d, d’) [refs. 17,20)] and in(d, 3H) [refs. 17P18~20)].In allcases the magnitude of ( l$:d.) overshoots the value derived out of the y-decay work. The deviations are especially large for (d, d’) reactions 17,20). As an application of our general method we have chosen the isospin violating inelastic scattering of deuterons leading to the 15.11 MeV state in “C . The actual calculations have been performed within the method of coupled channels for nuclear reactions 27). The total wave function of the projectile-plus-target system is projected onto a subset of states in which the deuteron remains in its ground state. Only a limited set of target states is considered. By means of the functions
IJ&f,, Y> =
CY&)JG @Ix dmJ,,,P
(26)
where &(r’) and #H[) denote the internal wave functions of projectile and target in the state y, l&J@ is a spherical harmonic depending on the orientation of the relative coordinate R (R = R/R) of projectile and target, a system of coupfed equations for the wave functions of relative motion is derived 28):
(27)
164
H. Lenske, G. Baur 1 Coulomb interaction
Here the diagonal operator
h,,(R) =
g2- q - 2
( Ulj,(R) -t V,(R))
(28)
contains an optical potential U,,,(R) and the single-particle Coulomb potential I/,(R). q, is the reduced mass in channel y. In eqs. (25) and (26) a shorthand notation has been used for the coupling of spherical tensor operators CT, x
I;C2ILM = c (Ilm,l,m*ILM)?;1,,7E2,2. m1mz
(29)
The coupling potentials “), I$J = (J,M,, YII$:’ f IGW,M,,
r’>,
(30)
have been calculated in the microscopic model for the projectile and the target. They are mainly given by folding the residual NN interaction VP, = c &&, &, R) with the ground-state density of the deuteron and the transition density of the target excitation 29, 30). The isospin conserving nuclear part of V,, is taken to be of the form V,, = v,,+V,,o.a’+l/,,z.2’+I:,a.o’z,2’+(~~/,,+~~’,,z.z’)S,,.
(31)
Here, S,, = (a * +)(a’* P)-+ * CT’is the usual tensor operator. The primed operators act on the projectile system. The folding of V,, by a T = 0 projectile density gives an effective interaction
where V denotes the folded potential. For the calculations the conservative assumption is made that the Coulomb interaction is the only charge-dependent interaction between projectile and target. The two-body Coulomb interaction (r = rl -rJ,
v,(r) = %i [T”‘+7p)+
7’3’23
(33)
can be split into an isoscalar part,
T(O) = 1+&z
29
(34)
the 3-components of an isovector operator,
T3'l'= T;+T~ 23
(35)
Ti2' = zfz;-+q.z2’
(36)
and an isotensor operator 3),
H. Lenske, G. Baur 1 Coulomb interaction
165
If the residual projectile target Coulomb interaction iS folded with the ground-state density of a T = 0 projectile, the isotensor part vanishes. Therefore, VJ$ is assumed to be given by the isovector part of the residual projectile-target Coulomb interaction for the case of a T = 0 projectile. In the model calculations the free NN T-matrix of Eikemeyer and Hackenbroich 31) has been taken for V,, in the form as given by Amos et al. 32). Whereas the Wigner part FoOof the effective deuteron target nucleon interaction is well known 33), there is only little information on the spin-dependent parts 00, and I& which are especially important for unnatural parity excitations. In a study of 12C(d, d’)l’C*(l+, 12.71 MeV) - which will be published separately it has been found that the tensor interaction together with non-central components out of the deuteron D-state give an important contribution to spin-flip excitations 2g, 30) in inelastic deuteron scattering. In the model calculations reported here these effects have been included. The ground-state wave function of the deuteron is approximated by a Os-harmonic oscillator function for the 3S, state and a Od-harmonic oscillator function for the 3D, component with an oscillator constant b = 2.05 fm. The D-state probability is taken to be pn = 0.04 [ref. ““)I. Calculations have been performed using two different lp-lh RPA models for the “C states. In the RPA calculations of Gillet and Vinh Mau 35) charge-dependent effects within the nucleus have been neglected. As s.p. wave functions harmonic oscillator functions were taken. These RPA wave functions have been used as the isospin pure basis states in a perturbative two-state mixing ansatz for the two 1’ states t. The mixing amplitude 6 was chosen according to the results of Adelberger et al. 21). Then, the Q-value corrected ratio, do’5.11 R = k(s) do12.71 = a2, (37) is simply given by the square of the mixing amplitude (6 = 0.046). The function k(9) is defined by k(9) = doo(12.71)/doo(15.11).
(38)
Here, do,@,) is the model differential cross section calculated with the form factor for the inelastic excitation of the 1 ‘, T = 0 model state at E, = 12.71 MeV and E, = 15.11 MeV. Clearly, if induced isospin mixing effects are important, the simple relation between the two cross sections is destroyed. R will, in general, become a function of the incident energy of the deuteron and of the scattering angle. In fig. 4, R(9) is shown as it results from a model calculation in which, apart from + The aim of the following model calculations is to test the two-state mixing model with respect to its application in isospin-forbidden inelastic scattering. For this purpose we desist from the errors which the model introduces in structure calculations because they mainly concern the interpretation of the mixing amplitude.
166
H. Lemke,
C. Baur 1 Coulomb
- R (a)
‘2C(d.d’) ‘*C* (I+.1511 Me’.‘)
I
fxlo-31
k
I
I
I,
interaction I
I
I
I.
I
-
$=280MeV ---WITHOUT ASC -WITH ASC
35-
r) I I ,,-... I
‘._/
I1
( , , ,
/
I
L_f
/
, , ,
, ml
50 3
/:
’
-
;
,’
( ,
I 150
CM
Fig. 4. R(9) as a function of the scattering angle 8_,. as it comes out when the RPA wave functions of Cillet and Vinh Mau are used. The asymptotic Coulomb coupling correction is denoted by ASC. For details see text. If induced isospin mixing were neglected, R(9) was a constant denoted here by 6*.
target mixing, induced mixing effects are also incorporated. The couplings between the two 1’ channels as well as the coupling of both to GDR states at E, = 17.6 MeV and E, = 20.9 MeV are included. Within these model calculations the GDR states can only be excited by Coulomb excitation. The average optical-potential set of Hinterberger et al. 36) has been used including a spin-orbit potential [set 2 of ref. ““)I. Theory and experiment I’) differ by roughly an order of magnitude. [The experimental values l’) for R(9) are collected in table 3.1 The model calculations shown in fig. 4 show that the interference of the induced mixing contributions and the target mixing contributions in the Fmatrix are important. It is not justified to neglect the induced isospin mixing effects. However, the question arises what are the important intermediate channels. In table 4, results for R(9) are shown for different coupling schemes at a variety of scattering angles (Ed = 28.0 MeV). It is seen that within these model calculations almost all intermediate channels give the same contribution. The individual contributions interfere preferentially destructively so that the net result does not differ very much from a single contribution. It seems to us that there are no general arguments for the selection of intermediate channels. Clearly, this makes it hard to calculate the induced mixing contribution quantitatively. Special attention has been paid to the long-range coupling due to the Coulomb interaction. The ~~ptoti~ coupling of the 15.11 MeV state to the GDR states due to the slow decrease of the Coulomb part in the coupling potential was treated analytic~ly within a separation method 2Q,37). As is also shown in fig. 4, in these model calcula-
H. Lenske, G. Baur / Coulomb interaction
167
TABLE 3 Experimentally derived value for R(9), eq. (37) Ed = 24.1 MeV ________~
-~
R,(g) (x 10-z)
9 21.7
1.269 *0.114 0.837 + 0.069 0.643 kO.077
34.2 45.6
-__--R(9) (x10-Z) 1.989 kO.179 1.537 kO.127 1.308 kO.157
9 25.8 38.4 50.9 63.0 89.3
Ed = 26.2 MeV
____
______
R,(9) (x 10-Z) 1.098 kO.074 0.871 kO.112 1.107 +0.088 0.954 kO.066 1.218 *0.140
W) (x 10-Z) 1.816 kO.122 1.450 kO.186 1.880 kO.149 1.627 F0.113 1.396 kO.160
9 19.2 25.5 34.3 44.3
Ed = 27.5 MeV R,(9) (x 10-z) 1.167 FO.072 1.164 kO.052 1.153 kO.068 0.985 kO.069
R(9) (x 10-z) 1.807 kO.111 1.928 kO.086 1.838 kO.115 1.491 kO.105
R, is just the ratio of the experimental cross sections whereas R is the Q-value corrected value, eq. (37).
tions the asymptotic coupling gives an important contribution to the cross section leading to the 15.11 MeV state. [If we were to try to fit the experimental magnitude we would need a charge-dependent matrix element of roughly 330 keV in accordance with the results of ref. “) where the data were analyzed within a model of this kind.] Apart from Coulomb excitation, the inelastically scattered deuteron acts like an isoscalar operator on the target nucleus. Now, it is well known that in N = 2 nuclei the proton and neutron components of the ground-state transition densities of T = 1 states are on the average of the same magnitude, but they carry a relative minus sign. On the other hand, if a transition is due to an isoscalar external operator, the sum TABLE4 Induced isospin mixing contributions due to different coupling schemes at different scattering angles 9 R(9) 9=S” (x lo-s)
Coupling scheme ____ O’++l’(T=
1)
R(9) R(9) R(9) R(9) W) W) R(9) 9= 12” 9=30” 9=44” 9=92°9=120”9=130” 9=150” (x lo-s) (x 10-a) (x 10-a) (x 10-3) (x 10-3) (x 10-3) (x 10-j)
2.204
2.100
2.173
2.174
2.194
2.211
2.238
2.188
0+~1+(7-=0)++1+(7-=
1)
2.565
2.588
2.404
2.378
2.542
2.496
2.675
2.714
O’++l’(T=
l)ul,(T=
1)
2.414
2.396
2.210
2.219
2.261
2.263
2.318
2.209
o+++l+(T=O)u1+(T= ttl,(T= l)ul;(T=
1) 1)
2.402
2.521
2.606
2.562
2.8i6
2.272
3.038
3.198
o+ t--) 2f(T = 0) +-+l.+(T = 0) ++ 1 +(T = 1)
3.435
3.320
2.408
2.415
2.607
2.622
2.650
3.383
168
H. Lenske, G. Baur 1 Coulomb interaction
of proton and neutron transition densities enters into the matrix elements, i.e. the excitation of an isovector (T = 1) state by an isoscalar field is mainly determined by the difference of the appropriate proton and neutron transition densities. Therefore, from the structure point of view, isospin-forbidden inelastic deuteron scattering tests the difference of the proton and neutron transition densities. If isospin would be conserved, the isoscalar transition density had to vanish for the excitation of an isovector state. In figs. 5 and 6 the proton and neutron components of the ground-state transition density for O+ -+ (l+, 15.11 MeV) in “C are shown together with the isoscalar density as they come out of a lp-1 h RPA calculation using realistic s.p. wave functions 22). In fig. 5 the L= 0, S = 1 transition density is shown, in fig. 6 the L= 2, S = 1 transition density can be seen for the O+ + 1 + spin-flip transition to the 15.11 MeV state in “C. In both cases it is seen that the proton and neutron transition densities behave quite differently. Because of configuration mixing effects these differences are much
RPA -TRANSITION DENS11 Is-1* 15 11Me” PROTDN --- NEVTRON -rscsC*Lb& L:o, s=1
10
20
30
LO
50
60
70 R[fml
Fig. 5. Proton,
neutron, and isoscalar transition density L = 0, S = 1 for O+ -+ l+ (15.11 MeV) in “C as they result from the RPA calculations of Speth et al. ‘*).
169
H. Lenske, G. Bmr 1 Coulomb ~tera~tio~ 7-i7, S,,, IF31f ti31
RPA-TRANSITION DENSITY Ofi E,:15 I, Me”
t
“C
/ I
10
I
20
I
t
I
I
I
3.0
LO
50
60
70
t
Rffml
Fig. 6. The L = 2, S = 1 transition densities for O+ I, l+ (15.11 MeV) in “% [ref. 22)].
more pronounced than in the case of the pick-up form factor discussed in the previous section. It is also seen from figs. 5 and 6 that the isoscalar densities (i.e. the difference of the proton and the neutron densities) cannot be derived by the simple first-order perturbation ansatz. In table 2 the RPA amplitudes 21) for the 12.71 MeV state and the 15.11 MeV state are given, The s.p. Coulomb effects destroy the symmetry of the proton and the neutron ph states in a much more complicated way than one would expect from a fnstorder treatment. The angular distributions for “C(d, d’)12C*(l+, 12.71 MeV) and W(d, d’)12C* (1 +, 15.11 MeV) can be seen in figs. 7 and 8 for several incident energies of the deuteron. The experimental data are those of Lind et al. “). Except for the case E, = 28.0 MeV, there are only few experimental points. Therefore, a detailed comparison of theory and experiment is not possible. However, in all cases the magnitude of the angular distribution& given correctly in the model calculations. Also the L,==2 shape of the experimental results is reproduced showing the importance of the tensor interaction and of non-central contributions out of the
H. Lenske, G. Baur / Coulomb interaction
n.
I’ 50
7 ‘I
1 100
‘I
II 150
‘CM
Fig. 7. Angular distributions for ‘W(d, d’)‘*C* (1 +, 12.71 MeV). Calculations without induced mixing are denoted by “direct”, those which include induced mixing are denoted by ‘W’. The RPA wave functions of ref. “) are used. Experimental points are taken from Lind et al. I’).
E,=2L,lMeV
1
’ QCM
Fig. 8. Same as fig. 7, but for ‘*C(d, d’)‘*C* (I+, 15.11 MeV) (isospin forbidden).
deuteron D-state for deuteron induced spin-flip transitions. A very striking feature shows up for E, = 28.0 MeV. While the experimental angular distribution leading to the 12.71 MeV state is rather flat, the corresponding one for the excitation of the 15.11 MeV state seems to decrease faster with increasing scattering angle. Like those of the other incident energies, the shapes of the theoretical angular distributions for the two transitions are quite similar. One may speculate that the enhancement of the “T = 0” cross section at backward angles is due to compound processes, which may give a larger contribution to “C(d, d’)l’C*(l+, 12.71 MeV) than to l’C(d, d’)“C*(l+, 15.11 MeV) because the “l 2C + d” compound systems preferentially occupy “T = O”states. Clearly, additional experimental information is needed in order to study this effect in detail. Finally, in fig. 9, the experimental and theoretical results for R(9), eq. (37), are compared for the different incident energies of the deuteron. These results demonstrate in a very clear way the errors which are introduced when R(9) is interpreted within the perturbation model discussed in sect. 2.
H. Lenske, G. Baur / Coulomb interaction 50
100
I’
5.0
3 ’ ,I’
150%M
“I
50
I/
RIBI k-m-2f
100
1% ‘I
150 %M
I II
t “II
‘ZC(d,~~‘2C*~l~,15~lMe#
- 5.0
Ed -28.OMeV -cc - -
4.0
171
DIRECT
- L.0
3.0
- 3.0
2.0
- 2.0
-1.0
1.0
t
5.0 -
R@’ Ixlo-21
12C(d,d~)12CX~?~15.11MeV~
__
Ed =26 2MeV
Ixlo-21
f
i
1ZCfd,d’~‘2C *(I:
15.11 MeW
- 5.0
Ed’2LlMeV -cc
-cc - -
R(9)
DIRECT EXP.
- f I
I
t
t
DIRECT
d.0
EXP.
I
I
t
3.0 -
- 3.0
2.0 -
- 2.0
1.0 -
- 1.0
~
Fig. 9. R(9) as defined in eq. (37). The RPA wave functions of Speth et al. 22) are used. The interference of the induced mixing contributions and the direct component is clearly seen. Experimental points are taken from ref. “). It is seen that at Ed = 57 MeV (no experiment available) the induced mixing effects are less important at least within the model calculations.
172
H. Lenske,
G. Baur / Coulomb
interaction
From figs. 8 and 9 it is seen that the magnitude of isospin-forbidden inelastic cross section is mainly determined by the isospin mixing in the nuclear wave function. The induced isospin mixing contributions to the T-matrix are mainly responsible for the angular dependence and energy dependence of R(9). They have to be included into a theoretical study of isospin-violating direct reactions. In these model calculations the channel coupling of the two 1’ channels gave the most important single contribution to the induced isospin mixing. It is also seen from fig. 9 that it can be expected that the induced mixing contributions are reduced at higher incident energies. Therefore, in order to study the isospin structure of the nuclear wave function, the experiment should be repeated at higher deuteron energies. At all incident energies the theoretical results for R(9) are larger than experiment. Also the angular dependence is not reproduced in detail. The remaining discrepancies may be partly due to the nuclear structure model which has been used for the calculation of the transition density (a spherical ground state is assumed for “C, but it is well known that 12C is oblate in the ground state 40). The theoretical y-width for O+ + 1 + (12.71) is too low [rtheor = 0.01 eV, reXP = 0.35 + O.O5eVforO+ + I+ (15.1 l), [ref. 21)], whereas it overshoots the experimental width Kheor = 58.27 eV, reXP = 37.0+ 1.1 eV, ref. “‘)I. On the other hand, it may be that there are important channel couplings other than those we included in the calculations. As already mentioned, it is hard to check this point because one can only consider a rather limited number of channels in the calculation. For example, in our calculations contributions to the induced isospin mixing due to the inelastic break-up of the deuteron and its subsequent recombination have been neglected.
5. Summary In sect. 2, the serious shortcomings of the conventional treatment of isospin-violating direct reactions were discussed. First, the conventional theoretical ansatz neglects the induced isospin mixing contributions which are due to channel-coupling effects caused in part by charge-dependent components in the projectile target nucleon residual interaction - like the well-known Coulomb interaction - and in part by isospin impurities in the wave functions of the intermediate state. The induced isospin mixing due to the projectile-target interaction would lead to a non-vanishing isospin-forbidden inelastic cross section even if the nuclear states are of pure isospin. It was shown that the induced mixing contributions and the direct excitation contribution due to an isospin impurity in the final nuclear state wave function enter coherently into the T-matrix element of an isospin-violating direct reaction. Therefore, induced isospin mixing effects have to be included into a theoretical study of isospinviolating direct reactions. Secondly, the conventional method assumes for the treatment of all charge depen-
H. Lenske, G. Baur / Coulombinteraction
173
dent interactions internal to the nucleus the validity of first-order perturbation theory even for the long-ranged Coulomb interaction. In sect. 3, this point has been discussed in detail. It was shown that this ansatz is simply wrong when it is applied in nuclear structure models because important isospin-violat~g contributions are neglected from the very beginning. The first-order perturbative method neglects the spatial differences of proton and neutron s.p. wave functions. In a simple lp-lh model it was shown that within this model isospinviolating contributions due to the differences of the proton and neutron ph matrix elements of the residual interaction are missing. Although these mixing effects are caused by the well known Coulomb interaction, they would be interpreted within this model as a hint for a new type of charge-dependent interaction. This wave-function effect has been verified by comparing proton and neutron ph-matrix elements calculated with realistic s.p. wave functions. Further, in sect. 3, we have shown that the perturbation method introduces serious errors into direct reaction studies of isospin mixing in nuclear states. For direct nuclear reactions mainly the surface and asymptotic region of the nuclear wave functions are of importance. But in this region the proton and neutron wave functions may be quite different especially for light target nuclei. In general, the different radial behaviour of the s.p. wave functions cannot be expressed by a single constant as is done in the perturbation method. Model calculations were performed for 13C!(d, t)“C*(l+, 15.1l.MeV) and 13C(d 3He)‘2B (I+, 0.0 MeV). In sect. 4 the interplay of induced isospin mixing and the target isospin mixing were studied for “C(d, d’)12C*(1 ‘, 15.11 MeV). We have applied our general method to this specific example because it is an experimentally very well studied isospin-forbidden direct reaction. Using the RPA wave functions of Gillet and Vinh Mau 35) and Speth et dz2), the perturbation method and a realistic treatment of the nuclear isospin mixing were compared. The model calculations of sect. 4 have shown that the perturbation method fails to explain the experiment, whereas good agreement was achieved between theory and experiment when realistic sp. wave functions were used and the interference of target and induced isospin mixing in the T-matrix is included into the calculations. It was pointed out that isosp~-forbidden direct reactions should be studied at higher incident energies (say 80 MeV or higher) in order to reduce the induced mixing effects. Iachello and Singh 37) have proposed that intermediate pick-up (d-3He-d and d-3H-d) or stripping (d-p-d and d-n-d) processes may be responsible for the observed isospin violation in (d, d‘) reactions. The calculations of Izumoto 38) show that these contributions are mainly due to the threshold behaviour of the intermediate channels. With increasing incident energy of the deuteron the pick-up stripping contributions decrease rapidly so that they will give only a minor contribution for the deuteron energies considered here. On the other hand, Izumoto 38) neglected the s.p. wavefunction effect.
174
H. Lenske,
G. Baur / Coulomb
interaction
Finally, we want to remark that there seems to be at the present stage no need to introduce some kind of new charge dependent nuclear interaction in order to explain isospin violations in direct nuclear reactions. We are indebted to Prof. Dr. J. Speth and Dr. J. Wambach for making available the RPA calculations and a lot of useful discussions on the RPA theory.
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