A microscopic theory of inelastic scattering of nucleons by deformed nuclei (I)

i

2.F

[

Nuclear Physics A195 (1972) 415--448; (~) North-HollandPublishiny Co., Amsterdam

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Not to be reproduced by photoprint or microfilm without written permission from the publisher

A MICROSCOPIC

THEORY OF INELASTIC SCATTERING

O F N U C L E O N S BY D E F O R M E D N U C L E I (I) ALPAR SEVGEN

Physikalisches Institut der Unicersitiit, 78 Freibury im Breisyau, W. Germany t

Received 23 May 1972 Abstract: A microscopic theory of inelastic scattering of nucleons from rotational nuclei is formulat-

ed. Angular-momentum conservation and full antisymmetrization require a careful choice of the many-body basis states for the scattering problem. A very convenient set of basis states {ArEc, qs} turns out to be a non-orthogonal system. The advantages and properties of this representation are discussed and the scattering matrix is calculated. Numerical analysis shows that our formalism gives an accurate and practical description of the inelastic scattering by doubly even deformed nuclei and by odd-even rotational nuclei where Coriolis coupling effects are weak. The study of low-energy resonances via the example of the splitting of the swave neutron strength function gives us additional confidence in our approach so that in paper (II) we shall present some applications which we hope contribute to a better understanding of scattering by deformed nuclei.

I. Introduction

M i c r o s c o p i c theories o f nuclear reactions have the following aims: (i) T o u n d e r s t a n d the d y n a m i c a l c h a r a c t e r o f resonances. One hopes to calculate the p o s i t i o n s a n d widths o f resonances. Structural i n f o r m a t i o n o b t a i n e d f r o m the scattering d a t a can in this w a y be reliably c o m p a r e d with the m o d e l used for the target. Such an a p p r o a c h bears the l i m i t a t i o n o f the m o d e l used. In t h a t sense one can view this as a test o f the m o d e l H a m i l t o n i a n . (ii) P h e n o m e n o l o g i c a l descriptions o f nuclear reactions focus their a t t e n t i o n o n one o r a few d o m i n a n t aspects o f the c o m p l i c a t e d scattering m e c h a n i s m (e.g. direct or c o m p o u n d scattering) a n d thereby the d a t a are suitably a n d conveniently p a r a m etrized. T h e m i c r o s c o p i c a p p r o a c h a t t e m p t s to u n d e r s t a n d the f o r m s o f the p h e n o m enological interactions (shape a n d m a g n i t u d e o f e.g. the optical p o t e n t i a l ) a n d to o b t a i n these forms a n d p a r a m e t e r s as an " o u t p u t " f r o m a m o r e basic study. By " m i c r o s c o p i c a p p r o a c h " we m e a n t h a t a r o t a t i o n a l l y i n v a r i a n t H a m i l t o n i a n is e m p l o y e d , which d e p e n d s on the c o o r d i n a t e s a n d m o m e n t a o f all the nucleons, a n d t h a t the states describing the t o t a l system (A nucleons) are fully antisymmetric. O f course this a p p r o a c h is also p h e n o m e n o l o g i c a l as r e g a r d s m o r e m i c r o s c o p i c theories; h a d r o n s o t h e r t h a n nucleons are neglected a n d the i n t e r a c t i o n between the nucleons w h i c h p r o p a g a t e s with finite speed is replaced by a n i n s t a n t a n e o u s effective interact Based on a thesis submitted by the author to Yale University in partial fulfilment of the requirements for the degree of Doctor of Philosophy (1971). 415

416

A. SEVGEN

tion. However for our purposes [(i) and (ii)] there is little to be gained by such considerations, not to mention the tremendous complexity of such theories. The last decade has seen much activity in which microscopic approaches to nuclear reactions were studied specifically in the frame of the shell model. This topic is extensively studied and reviewed in ref. 1), henceforth denoted by MW. During recent years work has concentrated on the inclusion of states with two particles in the continuous spectrum of the shell-model Hamiltonian, on the application of the shellmodel approach to experimental data and on the incorporation of more sophisticated target-structure models (described in terms of the random phase approximation RPA). The RPA involves a correlated ground state. However, if the residual interaction V between the particles is strong enough to cause coherent excitations from the occupied orbitals then the RPA breaks down. In this case we have a highly collective deformed nuclear ground state. This paper is concerned specifically with the microscopic analysis of inelastic scattering of nucleons by deformed nuclei. The reasons for the relevance of this problem are: (a) There are very few nuclei whose states can be well described as a superposition of a few shell-model states, but the number of nuclei which are deformed is very large. (b) Some nuclei which are supposed to have closed shells (e.g. 160, 4°Ca) are known to have excited levels of deformed character. (c) There is also the "practical" interest of how far the microscopic approach can be pushed. This is because if the low-lying target states have a complicated structure, then in the formulation of the scattering problem one faces serious non-orthogonalities of the basis states. We would like to understand the importance of these non-orthogonalities for the case of scattering from deformed nuclei. (d) We shall see that some of the assumptions and techniques employed in the shellmodel approach do not hold for the rotational targets. This first paper deals mainly with the construction of the formalism. In the second paper we shall present some interesting applications. In sect. 2 the structure of the rotational target (which enters into the reaction formalism in an essential way) is reviewed. We shall consider the projected Hartree-Fock (PHF) description of the target, although there are also other descriptions 2, 3). The basis states {X~:, ~} for the scattering problem are chosen in sect. 3. The choice of channel states requires that particular attention be paid to the asymptotic conditions. Our basis states form a non-orthogonal set. In sect. 4 the non-orthogonalities are estimated numerically. The S-matrix is calculated in sect. 5 where the non-orthogonalities are taken into account. In sect. 6, low-energy resonances typical for our problem are considered and the splitting of the s-wave neutron strength function is discussed from a microscopic point of view. We present a summary and draw the conclusions in sect. 7. It ought to be mentioned that Afnan '~) has studied numerically, neutron scattering and proton capture by 19F using approximate wave functions of the deformed target

MICROSCOPIC THEORY

417

and the formulae developed in the shell-model approach. It is interesting to have such a numerical application for a specific case, nevertheless the problem has not been studied in its generality and from a completely microscopic point of view, since (i) Afnan mentions that in principle one should calculate the wave function of the projectile nucleon in a deformed potential. Let us note that as a channel-coupling interaction, the "deformed potential" is a phenomenological prescription which gives good fits to data but whose justification is needed from basic principles (as will be shown in paper (II)). (ii) Antisymmetrization is not fully taken into account. It is neglected in the channel states. The study of antisymmetrization requires detailed numerical investigation as will be seen in sect. 4. (iii) It is not clear why the formulae developed in the shell-model reaction theory should apply just as well without any modifications [see e.g.eqs. (5.2.2) and (5.2.3.)]. (iv) There are other very interesting topics in the theory that must be worked out, such as shape resonances, properties of the optical potential etc. Therefore before starting large computer calculations we must first understand well the structure of the formalism and how it should be applied. We shall see the need for carrying out the formalism on a specific model because the assumed properties of some components of the general theory [e.g. the hermiticity of the projection operators P, Q of Feshbach 5, 6); see appendix B] may not hold.

2. Description of the target structure

We assume that the deformed target nucleus can be described by projected HartreeFock (PHF) states. The usual procedure is that one looks for a determinantal intrinsic state iX) such that

6 ( X I H I X ) _ O.

(2.1)

The single-particle orbitals in X are varied with the constraint that they remain orthogonal in the variation process. If one lifts the requirement that the single-particle orbitals have good angular momenta one can further lower the energy. If axial symmetry is kept Xwill havea good z-projection, K. However since H commuteswith the total angular momentum Jr, the actual nuclear states have good angular momentum. Therefore states of good angular momentum (denoted by ~ ) must be projected out of ]ARK), and the projection operators that do this are denoted by P~K"It is usually assumed that the projected wave functions

Q~ -
(2.2)

are good approximations to the nuclear states. It is known that the PHF wave functions do not have enough correlations built into them. For a better description of the

418

A. S E V G E N

target states one can proceed in the following way: once the H F potential is determined, one can construct a set of determinants {Xt(, ~,,},vt( distinguishing between determinants with the same K, and diagonalise the nuclear Hamiltonian t within the space of functions {PUtKXt(, ~.,,). Then the target states are given by q)ut, where , Xt(, ~K" ~Put = ~ at(, vK Pu1( K, VK

(2.3)

However, even the P H F wave functions themselves (e.g. (2.2)) are already complicated, since the projection operator excites particles out of the occupied intrinsic states. Therefore, such single-particle states are in effect only partially occupied, i.e. there are fractional holes in the target so that for example in a (d, p) reaction, the neutron may drop into these fractional holes making the extraction of spectroscopic factors difficult. Therefore in view of the complicated structure of the states ~o~ (eq. (2.3)) we shall in the following confine ourselves to states Out (eq. (2.2)) as an approximate description of the target. 3. Choice of representation for the scattering problem Let the wave function 7~ describe the scattering problem. We want to expand this wave function in terms of channel states {X~} and the bound states embedded in the continuum (BSEC) {~/'} which are constructed in subsects. 3.1 and 3.2 respectively. c (b} do not form an orthogonal set. In subsect. 3.3 overlaps are The basis vectors {X E, calculated and in subsect. 3.4 the choice of {X~, ~b} is discussed. 3.1. T H E C H A N N E L

S T A T E S {XE¢}

A channel state X~ represents the situation in whichan asymptotically non-vanishing single-particle wave function is coupled to the target in its ground or one of its excited states. The state X~ is antisymmetric in all the nucleon coordinates:

The antisymmetrization in (3.1.1) is with respect to the Ath nucleon; the C-bracket is a Clebsch-Gordan coefficient. The target state f2ut is already antisymmetric in the ( A - 1) nucleon coordinates. The continuum orbital can be written as OCjlm(A) _ HIj(F A, ke) o~j,mj(~A), ra

(3.1.2)

where ~a is the unit vector in the direction of r a. Then X~ can be written in a more compact form X~ = ~¢a(u~j(r a, kc)q~c}, (3.1.3) t T h e s e t {PUtKXr,~K} in general are n o n - o r t h o g o n a l a n d m a y even be linearly dependent. O f course, before diagonalizing the nuclear H a m i l t o n i a n , o n e m u s t construct a linearly i n d e p e n d e n t set o f wave functions f r o m {PXuKXK,vx} (see e.g. subsect. 3.3.3).

MICROSCOPIC THEORY

419

where q~c, called the surface function, is defined by comparing eqs. (3.1.1)-(3.1.3). The state X~c without antisymmetrization of the continuum particle, denoted by Xn,-c satisfies the equation H o XE -~ = E . ~ (3.1.4) where n o ~ ntarget(1 . . . . . ( A - 1))-+ t A 71- U(YA) , (3.1.5)

E = ec+h2kZ~/ZMv,

(3.1.6)

where ec is the threshold energy of the channel; My is the nucleon mass. In eq. (3.1.5), ta is the kinetic-energy operator of the Ath nucleon; U(ra) is the single-particle potential in which ~O~m is calculated. We now discuss the choice of U(ra). One may at first think of constructing the channel states by using a "deformed potential" for the particle in the continuum. We shall show however that such states are not appropriate for our purposes: (a) One way to employ a deformed potential would be to use the Hamiltonian H o defined as HPo =- Htarget .+tA.+ U(p, R, o~), (3.1.7) where the deformed potential U(r, R, ~) represents the interaction between the projectile and the nucleus, R is the nuclear radius and e is an operator that creates collective excitations of the target and depends upon the deformation parameters of the target and its orientation. The potential U(r, R, ~) can be written as

U(r, R, ct) =- U(r, R+~) = Uo(r, R)+AU(r, R, ct),

(3.1.8)

where Uo is spherical. For rotational nuclei ~ and A U are given by

- R ~ fix Y°(OP),

(3.1.9)

AU = ~ ( - 1 ) " O"Uo ct", .=1 n! Or"

(3.1.10)

A

where fl~ is a deformation parameter, angle 0' is measured from the nuclear symmetry axis and we have made the usual assumption that

Uo(R, r) = Uo(r-R)

(3.1.11)

for commonly used potentials. This approach is phenomenological and it is one of the aims of this work to understand why this approach works so well from a more basic study. Hence, we cannot employ the potential U(r, R, ~) given by eq. (3.1.8) in a microscopic analysis of scattering. (b) If one uses the deformed HF potential to construct the continuum orbitals, then meaningful channel states cannot be constructed, though it is possible to construct many-body states with good angular momentum. A channel state of the form

IcE, JM, K) oc pSra~rt(E)lgo) ,

(3.1.12)

A. SEVGEN

4,20

where (r'[a~t(E)lO) is a deformed continuum orbital and X o is the intrinsic state of the doubly even target, is not acceptable for two reasons: (i) It is not possible to satisfy simultaneously the conditions that there is an incoming wave with quantum numbers l,j and that the target is in a state of good angular momentum. Instead, in IcE, JM, K) the incoming wave (l,j) is coupled to several target rotational states. (ii) Due to the splitting of the members of a rotational band, the state IcE, JM, K) does not have a well-defined energy. Therefore the threshold behaviour cannot be described correctly. We see that the objection to the state IcE, JM, K) comes about because of the asymptotic conditions. We shall see that the functions describing the bound state of J t # the A-nucleon system, lobs) oc PMKaK,~,,IXo) , where ( r ' [aK, vKI0) is a bound-state wave function in the deformed well with K denoting the intrinsic z-projection and vK distinguishing levels with the same projection, are acceptable. It is now clear that the potential U(rA) in eq. (3.1.5) has to be spherical. The question is what the best choice for U is. The potential U, defined such that the matrix element Mcc vanishes, where Mcc is given by

Mcc =- (X~I VIX~,), _~

(X

v(i, A ) - V(A)I(A)

c

Xb),

i

xE, = .,j(rA, k )q,c, --C

t

(3.l,13)

is a very convenient choice. The fully antisymmetric channel state X~ occurring in eq.(3.1.13) involves the doubly even rotational target in its J = 0 ground state and v(i, j) is the two-body interaction. If the target were a (true) closed-shell nucleus then with this definition U(r) would have been the spherical Hartree-Fock potential. Since our target is deformed, U(r) is not the (conventional) deformed Hartree-Fock field used to describe the ground state of the A-particle system, but is now the potential that the incoming projectile feels due to its interaction with the target in its physical ground state. In practice, satisfying the self-consistency criterion eq. (3.1.13) could prove to be a tedious task, in which case one may make a practical choice for the scattering potential U: (i) by choosing that real Woods-Saxon potential which is obtained from a phenomenological optical potential U(r, R, ~(/~)) [eq. (3.1.8)] appropriate for the target, by letting/3 equal zero, and taking the real term, i.e. U = Re [Uo(r, R)],

(3.1.14)

where U o is given by eq. (3-1-8); (ii) by interpolating the parameters of the potentials for neighboring spherical nuclei.

MICROSCOPIC THEORY

421

Since the parameters of U(r) depend upon the mass number smoothly, we expect the choices made for U in (i) and (ii) to agree reasonably well. 3.2. T H E B O U N D STATES E M B E D D E D

IN THE CONTINUUM

-- B S EC

Suppose the projectile nucleon interacts with the nucleons in the target and drops to a suitable single-particle bound level while exciting the target. For the description of such phenomena we use the BSEC, and it is convenient to make the following choice for them:

I~s,~s>sM oc P~s{a~,~[excitation]i'}s, vslXo> - P~sIS, Vs> -

(3.2.1)

PMslXs,

where [excitation] i' = (n-particle-n-hole) i', n = I, 2 . . . ( A - I ) , and a~,~ creates a particle bound in the deformed HF field above the doubly even core. The state X o is the even-A target intrinsic state. Particularly interesting are those BSEC ~N. s vN which have intrinsic states Xs,~, ~ such that [XN,~u> = a~,~NlXo>. (3.2.2) Such BSEC, which we denote by q~N,~,,s (in contrast to the ~pSs,~ of eq. (3.2.1)) involve the coupling of the single particle to the rotational states of the target, all orbitals asymptotically vanishing. This can be seen explicitly 7),

jr L 2 J + l

d

[J ~3 0

[J/IJ]q~'(A)f2~},

, (3.2.3)

,

N is the component with angular momentum j of the deformed level where goj. t" J 0 > . The BSEC cPN,,, g , (eq. (3.2.3.)) would be important in the rpN,,.,. = /(Norm.)-

~ 2(2I+1) j 2 J +C2l [

OI[JN] etN~ . . . . E a r ( A - 1),

(3.2.4)

where e~ is the excitation energy of the rotational level 12~, N~ ' ' ' is the probability that a particle in the deformed level ¢pu,,~, has the angular momenta./', l. The energy Env ( A - 1) is the H F energy of the ( A - 1)-particle system. When the target rotational * The a d i a b a t i c a p p r o x i m a t i o n to projection integrals is discussed in a p p e n d i x A.

422

A. SEVGEN

levels are degenerate, the difference A (eq. (3.2.4)) vanishes so that the BSEC ~ JN, VN then are not in general important for the description of scattering, except in the following two cases. (ii) Even if the rotational energies are not large the deformed H F potential of the target may have a bound level very close to the threshold so that the projectile nucleon with very low energy excites the nucleus to a collective excited state and itself drops to J v,, play an important role in the low-energy this weakly bound level. Then such cbN, scattering. (iii) I f a level is nearly bound in the Hartree-Fock potential of the target then we can account for this case in terms of BSEC 4~N. s v~,. This can be achieved for example in the following ways: (a) We deepen the H F well so as to bind this reasonance, and call this new level J 9N, v,,- Then we construct q0N. v,,. New channel states ~ nc can be constructed using the procedure outlined earlier in subsect. 3.1. The new spherical potential/~ is defined by the relation:

( ~ 1 E v(i, A)-/~(A)I(A)~2~,) = 0,

(3.2.5)

i

The difference ( U - U) is then treated as a perturbation. An objection to this method is that . ~ may not represent the target states very well, since in deepening the H F well we affect the occupied bound orbitals. Also if (13"- U) is large it cannot be treated as a perturbation. In this case it is more convenient to resort to another method: (b) Generalizing Wang and Shakin's s) procedure to deformed single-particle potentials, one can define a deformed resonance wave packet ~R(r) which is obtained from the resonant deformed continuum wave of energy ER:

49R(r) = CWk(ER, r) = 0

for

r < Re,

for

r > Re,

(3.2.6)

where C is a normalization constant and R c is the cut-off radius. The function qSkR(r) can be made orthogonal to the bound orbitals by subtracting the bound-state components

E

f?

gp~(r) = C' wk(Ea, r)-- ~ q~k,,.k Vk

dr q~k,~k(r')wk(ER, r')

]

.

(3.2.7)

We identify ~oN,v~ of the BSEC ~JN , 'VN with q~(r) as given in eq. (3.2.7). Deformed continuum orbitals which are orthogonal to q~(r) need not be constructed since anyway such orbitals are not used in our theory. One may think of constructing new J vN and are positive-energy solutions channel states :gE which are orthogonal to 45N, of the modified Hamiltonian ~4~o which is given by

~ o -- AHoA,

(3.2.8)

where H o is given by eq. (3.1.5) and A = 1 --I~N.~N)(q'N,~N[. J s

(3.2.9)

MICROSCOPIC THEORY

423

J Due to the structure of ~N, vN (eq. (3.2.3)), states Zr are of the form,

~ = Z g~(c', r)
(3.2.10)

c'

It is not desirable to construct the channel states X~, because they will further get mixed via the new residual interaction ( H - ~ o ) and it is also difficult to construct them especially if there are a few states of type ~JN , V N " Therefore we keep our old channel states X~ (eq. (3.1.1.)). J In all the cases (i)-(iii) considered above one must make sure that ~N,~N is linearly independent of the channel states {X~}. This means that the deformed single-particle level ~PN,vN (eq. (3.2.2)) must not be a linear combination of the scattering states of potential U (eq. (3.1.13)) only. Otherwise there is no point in constructing the state J (PN, vN "

The existence of BSEC of type ~JN, vN is peculiar to the deformed region. We shall see that they have large overlaps with the channel states {X~} because they have a similar structure. Although we assume that the projected states f2~ (eq. (2.2)) approximate the lowlying eigenstates of the target Hamiltonian, we do not expect ~bS,~s s (eq. (3.2.1)) to approximate the eigenstates of the full A-nucleon Hamiltonian. This is because rotational motion is mostly characteristic of the low-lying states whereas the excitation energy in the compound system is fairly high ( > 10 MeV), and the true eigenstates are rather complicated. In our model the compound states are linear combinations of the BSEC. Low-energy resonances in the channel subspace {X~:} also play a role in the description of compound-nucleus levels. In our problem such low-energy resonances require a careful study since we are using a mixed representation t. Therefore we delay the discussion of this topic to sect. 6. 3.3. OVERLAPS We have specified the basis states {X~, ~} in terms of which we want to expand the complete scattering wave function. However since we are taking into account the antisymmetrization fully and since we use a mixed representation, the basis states are not orthogonal. We would like to consider now this problem of non-orthogonality. 3.3.1. The overlap (X~[X~) between the channel states. Let us assume that t h e target is doubly even and has an intrinsic H F state with K = 0. For simplicity we shall do the calculation using the adiabatic approximation to the projection integrals. In appendix A it is shown that the adiabatic approximation to the projection integrals is equivalent to using target states of the following f o r m : ~

,I ,I . . . . . = (2•+ 1) ~ D~oXo(r

, r.4-1),

(3.3.1)

t We call the representation {Xr c, q)) mixed, because the continuum orbitals are constructed from the spherical potential U, whereas bound orbitals are obtained from the deformed HF potential of the target.

424

A. SEVGEN

where the primed coordinates refer to the intrinsic system IXo) =

1-I

akt, ~kl0).

(3.3.2)

(k, vk) ~ occupied

We can expand ak*,~k in terms of the operators that create the complete set of states in the spherical potential U (eq. 3.1.13)):

,;fo

at

k , vk ~ k * k, ~k = Z nn,.'09" -,.'rj" + n'l'S"

k, vk "i'k , dE tBrj, (E , )ars,(E ).

(3.3.3)

We make use of the anticommutator {a~j(E), a t , ~k)

= B,jk. vk(E)fk,k,,

(3.3.4)

and find c c" (XEIXE,)

=

6cc,6(E_E,)_

co"

(3.3.5)

REE,,

where the function R is given by

~, [(21+1)(21'+1)]~E C j I REE, = 2,/+ l ,. m 0 ×

0

Z

m). (3.3.6)

~'J,'"~k'~r~-'ec')B,~J~(E-3' a(k,~~'

(k, vk) ~occupied

To compare the importance of the distribution R with that of the 6-function we must integrate over both distributions. If a smooth f u n c t i o n f ( E ) is square integrable and normalised to unity, then we define F such that

F =-

i

If

•

f(E -e~,)[a(E - E )6~,~,,-Rww,]f( E - e~,,). !

.

f

i, e

c * c tt

Ip

(3.3.7)

,t

For e' = c" this expression reduces to the following form:

e = 1 - 2(2i + 1)

Z

J 2~(k, m) f~dE,B~k(E, o --vn)

e~,)f(E, e~,) 2,

m>O (k, vk) ~ occupied

(3.3.8) where the round brackets indicate a 3j symbol. Using the Schwarz inequality we find the lower bound of F, F > 1-Aa (3.3.9) where A 1 is given by A, --- 2 ( 2 I + 1 ) y °

0

dE'IB'~'""(E')] 2.

(3.3.10)

The levels (m, Vm) in eq. (3.3.10) are occupied orbitals of the target intrinsic state. The quantity A x measures the importance of the distribution R with respect to the 6-function and indicates the strength with which the continuum wave ~b~.t,,(r) (eq. (3.1.2)) is contained in the description of the rotational target state g2~. The magnitude

MICROSCOPIC THEORY

425

of A~ is restricted: 0 < A, __< I.

(3.3.11)

Numerical values for the important quantity A ~ will be given in subsect. 4.4. 3.3.2. The overlap between B S E C and channel states. If we calculate this overlap in the adiabatic approximation, using for BSEC 4,

¢ ~J , . , = [½(2J + 1)]~{D~N x ~ , ~ + ( - 0

J+s o M J, , _ ~ _ ~ , . . } ,

(3.3.12)

where XN,,~ is given by eq. (3.2.2) and (3.3.13)

7(_s. ,~ - exp ( - inJr)X N..... we find



= L 2--~1

J

c

BN'~NtE jl ~.

0

~

e~ c/

(3.3.14)

(3.3.15) The last equality defines the bracket. We also assume that the target intrinsic state X o, transforms under the time-reversal operator with a plus sign. The bracket satisfies the orthogonality relations, 0

The probability that the BSEC • JN , VN contains the many-body continuum states {X}} is given by A2 - (~,.~ll~x~ollel, v.>, (3.3.17) where l~xEc~is the unit operator for the channel subspace: --

l~x~c } =

zP

dE

!

C ' ¢ " ~" g c '

tt

c'

C'¢"

C"

(3.3.18)

dE [XE,>ge,E,,(X~,,I. ~c'"

We use the fact that the channel states {X~:} are not orthogonal to each other (subsect. 3.3.1). The quantities Ye'e'" c,c,, are the elements of the inverse of the metric tensor. We calculate A2 making use of eq. (3.3.15)t:

~=EE l'r'E rr" j'j"

0 x

0 !

t!

N, vN

P

c'c"

N, vN

tt

f ~ dE f ~ d E Brj, (E --ec,)gE,E,,B,,,j,,(E -e~,,) . ( ) 3.3.19 get

~ 8Ctt

• We exhaust the sum rule, eq. (3.3.18), only with those channel states that involve the target in its rotational states; other channel states, due to their complicated character, have vanishing overlap with ~ , v N in the adiabatic approximation.

426

A. SEVGEN

I f A a can be neglected then A 2 a s s u m e s the simple form: A2

/X~ ,| l ' j "*1 o

~

d E qiB lN' ' j ' v:ffE'~12 ~, ] l •

(3.3.20)

The quantity A 2 is also restricted to the range of values, 1 > d2 > 0.

(3.3.21)

I f A 2 is equal to unity then q /N, VN will be linearly dependent on (X~}. Numerical values of A2 will be given in sect. 4. Let us note that when we do not make the adiabatic approximation, the overlaps of channel states with complicated BSEC q0ss, ~ (eq. (3.2.1)) do not vanish. 3.3.3. Overlap (~r,~,,l~r,~,,, s J ~ o f the BSEC. These overlaps in general do not vanish for K -~ K ' (K = N or S ) and vK :/: (vK)' except for the adiabatic limit. These non-orthogonalities do not represent much of a problem since we can orthonormalize the set {q~s). One should be careful that all the wave functions are linearly independent. I f any two are proportional we take only one of them. This can be achieved by diagonalizing the sub-metric tensor #,p, ~ = (K, vr), and then discarding the vectors with eigenvalue zero. Let us denote this linearly independent orthonormal set by { ~ } . From here on, when we use projected wave functions our BSEC are the set {~s}. When for simplicity we use the adiabatic approximation, we continue to employ d ¢~K, VK"

3.4. DISCUSSION OF THE CHOICE {X~C, fib) We expand the full scattering wave function ~0"c( - ~ + ) which obeys the equation H~'*'

= E ~ '+'

(3.4.1)

in terms of the basis set {X~, ~b): tuck*, = Z b~(i)~b,+ --E i

dE ,a tc( c ,, E )' X ~¢", ,

(3.4.2)

c" ~ ec,

where c indicates the incoming channel and the plus sign indicates that there are outgoing waves in channels c' which can couple to c via the interaction. We have truncated the space of functions to at most one particle in the continuum. Besides the set o f functions {X~, ~} introduced above, one may consider other choices for the set into which tv~c*~ -E is expanded. Among these, we mention the following possibilities: (i) The spherical potential U ( r ) (eq. (3.1.5)) that we used to generate the continuum states possesses a complete set of single-particle states. For the basis states, one may therefore propose to use the functions defined such that,

where ~j,,t is either a bound or a continuum level of the potential U(r). When ~j,,~ is a continuum state then ~ is identical with X~:. However the set {~} is not convenient

MICROSCOPIC THEORY

427

to use for the BSEC, i.e. when ~kj,,t denotes a bound level in U(r). We denote such functions by ~b. The non-orthogonality of the functions Eb is rather considerable. Our BSEC • (eq. (3.2.1)) are orthogonal to each other in the adiabatic limit, i.e. for large deformations, whereas no such simplification can be made for the overlap (,Ebl2 b') except for the neglect of the antisymmetrization with the projectile which is in no way justified. Another objection to Eb is that these functions will be mixed strongly with each other so that the resonance energies are shifted quite far from the unperturbed energies t e b , where Ho(~ b) ---- eb(~b); (3.4.4) again the bar indicates that particle A is not antisymmetrized. The operator Ho is given in eq. (3.1.5) and e b = ec+E°nt, (3.4.5)

Eot being the binding energy of the level $~,,t in U(r). Therefore it is impractical to use {~b). (ii) The deformed H F potential provides us with another complete set of singleparticle states. For our BSEC {~) we use this set. However, as we discussed in subsect. 3.3.1, it is not possible to construct channel states that satisfy the asymptotic boundary condition using this representation. We see that for our present purposes, the set (X~, q~} is very convenient. Is it sufficiently complete? Suppose we generate the deformed HF potential by starting from the spherical potential U(r) and by turning on the deformation. Those bound states of U(r) which are pushed above threshold by this process are, from the construction of the set LX~, c q~) not available to a single particle. The lost strength is given by ~3,lj

~ (~1 2"~mJ ,,,~,,,,\ _ _1 1 - - 2j +1 ,,j t ~ n j I U n j l l ~ , / ,

(3.4.5)

where the sum over ~ goes over all the levels bound in the deformed potential, When the deformed H F potential is such that there is a level nearly bound or just bound, the low-energy scattering is affected, We see that if the level that is nearly bound has spherical bound levels as its components we cannot account for such a resonance. However the situation can easily be remedied by accounting for this J resonance in terms of the BSEC ~bu, v,, as described in subsect. 3.2.3. 4. Continuum admixtures 4.1. C O N T I N U U M

A D M I X T U R E S IN SPECTROSCOPIC F A C T O R S A N D T H E N O N -

ORTHOGONALITY OF BASIS STATES FOR DOUBLY EVEN AND ODD-EVEN ROTATIONAL TARGETS It is the purpose of this section to estimate the magnitudes of d 1 and d 2 defined in eqs. (3.3.10) and (3.3.19) respectively, for doubly even and odd-even rotational targets. We shall see that d 1 and A 2 depend on the mass number A. For the calculation t The reader may convince himself that this is the ease by reading sect. 6.

428

A. S E V G E N

of A1 and A 2 we need the continuum content of the deformed bound single-particle levels. In principle, the deformed bound levels are eigenstates of the non-local singleparticle HF Hamiltonian. However for the ease of numerical work, we calculate them in a local deformed potential. In subsect. 4.2, the continuum admixture to deformed levels is calculated in first-order perturbation theory. We study the continuum admixture as a function of binding energy or, equivalently, well depth. The continuum

~d~

0.01C.A.(l'#nlm)

0.001 __

..........

C.A. (d- 2so)

.... B=ol CA (g* Ido) :

/t : Ig . E2s(MeV}

o.oooi ~

iU

io . . . .

3"o

~.b

5b

o'o E~(M~v)

Fig. l. The d- (g-) wave continuum admixture to the level 12s) ( l i d ) ) as a function of the binding energy of the level 12s) ( l l d ) ) (eq. (4.2.5)), fl = 0.1. (Arrows indicate the configurations where levels (s'l'O) indicated by small circles become bound.) l.C .A.(l'-nlm)

O.

__ C.A,(s-Idol

....C.A.Id-Idol I~=0"I 0.01!

oo~

I,

~-~

-- - .......................

'ff--~o. ~

~,d

.,~ o'v

.,. ~'v

o~ oo

oo o'v

Fig. 2. The s- (d-) wave continuum admixture to the level I l d ) as a function of the binding energy Ela (eq. (4.2.5)), t5' = 0.1.

content of the levels is calculated exactly in subsect. 4.3, as a function of the deformation parameter. In subsect. 4.4 we present tables for A 1 andA2 for neutron scattering from 2°Ne, 24Mg and 22Ne and extend our discussion to odd-even rotational targets in subsect. 4.5. 4.2. P E R T U R B A T I O N C A L C U L A T I O N O F T H E C O N T I N U U M D E F O R M E D B O U N D LEVELS

ADMIXTURE

TO

For the unperturbed system we take a spherical square well, r = - I V o I O ( d - r),

(4.2.1)

MICROSCOPIC THEORY

429

where d is the nuclear radius, so that

A V = -IVolfidY°~(d- r)

(4.2.2)

is the perturbing term due to deformation. In the first-order perturbation theory, the continuum admixture (CA) of an l' wave to a bound level IE°lm) is given by CA(/'--+ nlm) =

f;

d~' I(E~'m(d)IAVIE°~=)I2

'

(4.2.3)

where ( r l E O > _ to~(h.t, r) yl ~

(4.2.4)

F

is a bound level of the unperturbed potential, and the continuum orbital is given by eq. (3.1.2). We use the notation, normalization and phase conventions of MW. Explicitly we have

CA(/'

--+ nlm) =

/~21Vo12d=

dO Yr* m Y~0 •,. m I-co,bt(~,,,d)32I(l ', nl), (4.2.5)

where

I(l', nl) = f o~d# u'3(e'' d) _ ( e ' - E°,) 2

d {gr(d, d, E ) - ~ [o9~1,(,~sr!/)]21 E ~ , " dE " E - Esr ~ =

(4.2.6)

The Green function for the square well evaluated at the surface is

g,(d, d, E) - 2rod jt(p')h~"(p) h 2 [ph~l)'(p)jt(p')-p~(p')h~l)(p)]'

(4.2.7)

wherej~ and h~~ are the spherical Bessel and Hankel functions, and

p2 = 2mEd2/h 2, p,2 = 2re(E+ [Vol)d2/h 2.

(4.2.8)

The primes on the functions in eq. (4.2.7) indicate derivatives with respect to the argument. The quantities CA(/'-+ nlm) as functions of the binding energy of the level IE°,m> are shown in figs. 1 and 2, for (d -+ 2s0), (s -+ ld0), (d -+ ld0) and (g -+ ld0). The deformation parameter fl is taken equal to 0.1. From the curves we see that as the potential depth 1Io increases, each time a new state is bound there is a discontinuity in the function CA (l' -+ nlm). We can understand this in the following way: let C~,~t and C~,~_.~be the continuum admixtures corresponding to the potential depth immediately before and after level l' with s' nodes becoming bound. Then we have

CI,-., = C~!-+z+las,,r-+~l2,

(4.2.9)

where lal 2 is the admixture of the newly bound state IE~r,,) to IE°l,,). We also note from the CA (l' -+ nlm) curves that as a new state IE,°,r,,) is about to be bound CA

430

A. S E V G E N

(l' --,. nlm) increases. That is, the effect of the 1' resonance is that the continuum wave is more and more localized near the potential well and it gets mixed with the level IE°~) with higher probability. 200 100

10 rl-

i

i +

0130

1

1

40

I

50

60

I

70

I

'

1

9o

_o

lo0 Vo (MeV)

Fig. 3. The function Rlao measures the importance of the continuum admixtures to level ( l d 0 ) as c o m p a r e d to bound-state admixtures (eq. (4.2.14)).

005

.

004

C£

1

// //

003 OO2

•

,/

l

i

]

OOl o

10

210

310

40

510

60

E2s(MeV)

Fig. 4. The function R2~o.

To understand the relative importance of the continuum and bound admixtures, let us normalize the perturbed wave function I~,,) to order /72. The state [{=) can be written 9) as:

Ig,,,) = IE,°t,,)+ ~

IE°) (E°IA't'-AVIg''>,

,.~<...)

(4.2.10)

E~ -E°~

where a.,m -

¢~.%lAVl~m>.

(4.2.11)

We denote by I~,.) the state normalized to order/32: I{,,> = NI{=>,

(4.2.12)

MICROSCOPIC THEORY

431

where N is the normalization coefficient and is given by N = I-I+/?2(B+C)] -~ ..~ 1-½/~2(B+C),

(4.2.13)

where B and C are the bound and continuum contributions. We define R n l m a s the ratio of the norms of continuum and bound states (except that of IE,°t,,)) in I~,,)

R,,t,,'

_

fl2N2C Fl2N2B

_

C

(4.2.14)

B "

The ratio Rnt,, , measures in the first-order perturbation theory the relative importance of continuum and bound admixtures to IE~tm) ; R,z,, is shown in figs. 3 and 4 for (ld0) and (2s0). F r o m these figures we see that the continuum admixtures are much more important for the ld level than for the 2s level. We can infer from R2s o (fig. 3)that the continuum in the perturbed level I~m) (eq. (4.2.12)) evolving from 12s) is negligible. This result will be of importance when we discuss the strength function in subsect. 6.2. For very small binding energies of the level Jl d ) , the rather large numerical values Of Rlao are also due to the fact that the s-continuum wave does not have any penetration factor and thereby gets mixed with the ld level even for zero energy, resulting in the large amount of continuum admixture in the perturbed level evolving from ld. We conclude that the continuum admixture to a given bound level [E~,,,) depends upon: (i) The number of nearby bound levels of the same parity that IE~tm) can mix with. It is seen from figs. 3 and 4 that each time a new level is bound R,~,, decreases i.e. the importance of the continuum in the description of the state I~,,) is reduced. (ii) The binding energy of IE°tm), which can be seen from the expression for l ( r , nl), eq. (4.2.6). (iii) Whether or not there are narrow resonances above threshold. F r o m the CA (l' --* nlm) curves we see that there is an increase in the amount of continuum admixture when the resonance is very close to threshold, i..e nearly bound. 4.3. EXACT CALCULATION OF THE CONTINUUM ADMIXTURE TO DEFORMED BOUND LEVELS We now investigate the continuum admixture as a function of the strength/3 of the deformation. Clearly the perturbation calculation is not at all suitable for this case, so that we resort to a numerical calculation. We assume that a deformed WoodsSaxon potential provides a good approximation to the H F field. The bound levels in this deformed optential are determined from a coupled-channel calculation a discussed by Rost t o),. The bound states tpk, vk in the deformedwell satisfythe equation,

(ho+ dV)tPk,~,, = Ek.v,, q~k,~k,

(4.3.1)

t Deformed levels have also been calculated exactly in ref. 1~) by expanding them in terms of Sturmian functions.

432

A. S E V G E N TABLE 1 C o n t i n u u m in t h e single-particle d e f o r m e d levels, C ~ p , u s i n g the potential V~

[k ~r, ~'k]

fl

0.1

[½+, 3]~

BE (MeV) J=½

5.7070 0.00091 0.00018 0.00027 0.00013

total

0.20

0.30

0.40

0.50

5.6326 0.00134 0.00070 0.00078 0.00130

5.5925 0.00173 0.00143 0.00117 0.00407

5.6272 0.00212 0.00227 0.00132 0.00854

5.6816 0.00262 0.00324 0.00125 0.01435

0.00149

0.00412

0.00840

0.01425

0.02146

7.5203 0.00040 0.00060

6.5443 0.00103 0.00289

5.4956 0.00233 0.00709

4.3827 0.00494 0.01368

3.2196 0.01021 0.02286

~tjetj ~ [~+, 1 ]a

BE (MeV) J = ~ total

0.00100

0.00392

0.00942

0.01862

0.03307

BE (MeV) J=~

8.6050 0.00004 0,00012 0.00133

8.7973 0.00014 0.00017 0.00499

8.9764 0.00024 0.00034 0.01050

9.1546 0.00033 0.00077 0.02044

9.3023 0.00041 0.00142 0.03099

total

0.00149

0.00530

0.01108

0.02154

0.03282

BE ( M e V ) J=½

9.2890 0.0002• 0.00003 0.00018 0.00016

10.3307 0.00071 0.00021 0.00085 0.00553

11.3957 0.00124 0.00060 0.00240 0.01820

12.4268 0.00176 0.00122 0.00494 0.01687

13.4043 0.00221 0.00206 0.00829 0.02223

total

0.00058

0.00730

0.02244

0.02479

0.03479

P a r a m e t e r s for the R o s t code are: Vo = 55 MeV, A = 25.0, a = 0.65, A = 22, N BE is the binding energy. Values o f { C t S2} are calculated f r o m eq. (4.3.4).

12, Z = 10;

[~+,l]a

[½+, 2]~

TABLE 2 C o n t i n u u m in the single-particle d e f o r m e d levels, Cij s~, using the potential Vb [Mr, vk ]

fl

0.10

0.20

[]+, l]b

BE (MeV) J=~

2.9809 0.00537 0.00046 0.00105

3.1479 0.01640 0.00043 0.00419

total

0.00688

BE (MeV) J = ½

[~+, 2]b

9 total

0.30

0.40

0.50

3.3126 0.02831 0.00054 0.00931

3.4651 0.03907 0.00083 0.01616

3.6023 0.04805 0.00135 0.02473

0.02102

0.03816

0.05606

0.7413

3.6019 0.00098 0.00388 0.00019 0.00120

4.5162 0.00363 0.01526 0.00061 0.00411

5.4332 0.00672 0.02861 0.00166 0.00818

6.3150 0.00957 0.04060 0.00332 0.01298

7.1482 0.01200 0.05009 0.00554 0.01824

0.00625

0.02361

0.04517

0.06647

0.08587

P a r a m e t e r s for the R o s t code are: Vo = 45 MeV, A -- 25.0, a = 0.65, A = 22, N = 12, Z ~ 10; BE is the binding energy. Values o f C~j~ are calculated from eq. (4.3.4),

MICROSCOPIC THEORY

433

where ho is spherical, A V is due to deformation, and k,

q~k.vk = ~ (°'irk(r) ~ j . lj

(4.3,2)

r

The functions (ok)~ are the solutions of the coupled equations ( h o - E k , ~k)(cokJ~k/r) = -- E (~l~JlZlVl~/~v')(~°f'3r"/r)

•

(4.3.3)

l'j"

It was shown in eq. (3.3.3) that the functions r,,k, ",,ij vk involve the continuum of the scalar operator ho. We define (cf. eq. (3.3.3)) n k , vk lj ~

TM

-

~ n

lak, v~ 2 a'tnlj

f o ~ d e t I %k, v k ( e )t 1 ,2

~lk, Vk ~,tj -

dr[co

Vk

]{~k, Vk 3_/"~k, 'Ok (r)] 2 = ~'lj ~'~tj

(4.3.4)

The quantities B and C are the bound and continuum contents of /,~k* ~,zjVk, N is the probability (spectroscopic factor) that the single particle in the deformed level has angular momenta l and j. The total continuum content of (Pk,~, is calculated numerically from the expression /,~k, Vk

•~,j

= 1-

~

..

b |j fo 0(o,j(k,, r)eo~jV~(r)dr 2,

(4.3.5)

t,~k, vk is calculated and the continuum content cRick of the unnormalized function ~,,j from the last equality in eq. (4.3.4). The coefficients C~j *k are calculated in tables 1 and 2 for some positive-parity levels, starting from two unperturbed (fl = 0) configurations, i.e. spherical potentials V~ and Vb. The single-particle level [lg~) is not bound in V~, and the levels Ild~) and [lg~) are not bound in Vb. From the numerical results* we conclude that: (i) The levels in V, get mostly g~ continuum admixtures while the levels in Vb get mostly d~ continuum admixtures followed by g~ continuum admixtures. (ii) When the potential is deformed, the magnetic quantum number degeneracy is removed. Independently of whether the split level comes closer to threshold or gets more deeply bound, its continuum admixture increases as the strength fl of the deformation is increased. Examples are [k, Vk]: [{, 1]~, [½, 2]~, [½, 2]b. (iii) Even for a level with considerable binding energy, the continuum admixture is still important for spectroscopic factors. For example, the total continuum in the level [½, 2]b with binding energy 7.15 MeV (fl = 0.5) is about 10 ~ . + D e f o r m e d levels o) --ljk,vk were calculated with the Rost code. Spherical levels (obntj were calculated with the code "bound" written by Dr. H. Yoshida.

A. S E V G E N

434

TABLE 3 N o n - o r t b o g o n a l i t y a m o n g c h a n n e l states ZI j j

I

J

n-}-2°Ne Llla

nq-24Mg n + 22Ne ,dlb

~dXa

zllb

½

0

½

0.0018

0.0096

0.0018

0.0096

½

2

]

0.0018

0.0096

0.0018

0.0096

½

2

~

0.0018

0.0096

0.0018

0.0096

½

4

5

0.0018

0.0096

0.0018

0.0096

½

4 0

~ ~

0.0018 0.0061

0.0096 0.0203

0.0018 0.0061

0.0096 0.0398

3

9

2

½

0.0012

0.0460

0.0012

0.0398

2 2

~ ~

0.0061 0.0002

0.0203 0.0058

0.0062 0.0005

0.0398 0.0392

2

~

0.0078

0.0261

0.0080

0.0400

4

~

0.0010

0.0348

0.0011

0.0404

4

~

0.0044

0.0145

0.0046

0.0396

0

~

0.0034

0.0026

0.0075

0.0058

2

~

0.0080

0.0062

0.0153

0.0119

2

~

0.0003

0.0003

0.0039

0.0031

4

~

0.0003

0.0002

0.0167

0.0135

Values o f zll are calculated from eq. (3.3.10).

TABLE 4 N o n - o r t h o g o n a l i t y o f BSEC a n d channel states

[N zr, vN ]

fl

[k+, 3]a

BE (MeV) ,£12

[~+, 1 ]a

0.2

0.3

5.7070 0.00149

5.6326 0.00412

5.5925 0.00840

5.6272 0.01425

5.6816 0.02196

BE (MeV) Az

7.5203 0.00100

6.5443 0.00392

5.4956 0.00942

4.3827 0.1862

3.2196 0.03307

[]+, 1]a

BE (MeV) LI2

8.6050 0.00149

8.7973 0.00530

8.9764 0.01108

9.1546 0.02154

9.3023 0.03282

[½+, 2]a

BE (MeV) ~J2

9.2890 0.00058

10.3307 0.00730

11.3957 0.02244

12.4268 0.02479

13.4043 0.03479

[,~-+, 1]b

BE (MeV) zl2

2.9809 0.00688

3.1479 0.02102

3.3126 0.03816

3.4651 0.05606

3.6023 0.07413

BE (MeV)

3.6019 0.00625

4.5162 0.02361

5.4332 0.04517

6.3150 0.06647

7.1482 0.08587

[½+, 2]b

Az

0.1

z~J2

Va: Vo = 55 MeV, A = 25.0, a = 0.65, A = 22, N -- 12, Z = 10. Vb: Vo = 45 MeV, A = 25.0, a = 0.65, A = 22, N = 12, Z = 10. BE is the binding energy. Values o f Az are calculated from eq. (3.3.20).

0.4

0.5

MICROSCOPIC THEORY

435

Let us again stress the fact that the continuum admixtures are very sensitive to the unperturbed configuration, that is to the mass number A. 4.4. NUMERICAL VALUES OF AI AND A2 FOR DOUBLY EVEN ROTATIONAL TARGETS We can now calculate A t using tables 1 and 2, and eq. (3.3.10). In table 3, we give A t for the potentials Va and Vb (denoted by A ta and A lb ) for neutron scattering from 2°Ne, 22Ne and 24Mg. The deformation of the target was taken to be fl = 0.4. The potential Vb is too shallow for the mass region considered because it has a d~ resonance. Therefore A x~ gives an overestimate of the non-orthogonality of channel states. From table 3 we see that max dla < 0.02 and m a x Alb < 0.05 SO that compared to unity we can neglect d 1. Since A x can be neglected, we can use eq. (3.3.20) to calculate A2, which is nothing but the total amount of continuum admixture in the single-particel level ~0N.~. Values for A 2 are given in table 4 for the levels in the deformed potentials obtained from Va and Vb- We see that some levels (e.g. [½+, 2]b), even when far from threshold, contain a considerable amount of continuum admixture. For some levels A 2 will rapidly increase when the level approaches threshold*. Therefore the conclusion for doubly even nuclei is that A t can be neglected, but J when BSEC of type ~:~,~N (eq. (3.2.4)) exist, in general A a cannot be neglected. In sect. 5, which is concerned with the S-matrix, we will discuss two different ways of handling A 2. 4.5. dl AND d2 FOR THE ODD-EVEN DEFORMED TARGETS For the doubly even system At can be neglected. This is because the ground and low-lying excited states of' the target are described by configurations where the nucleons occupy bound levels with binding energies larger than about 8 MeV. If the deformed target nucleus is odd-even, its low-lying states can be written as

=

E

" v~(e)P,rar,~,~[Xo), , t Cr,

(4.5.1)

K,vK

where e = 1, 2 . . . denotes the states with quantum numbers (I, re) in the order of increasing excitation energy. Here Xo is the intrinsic H F determinant of the doubly even system. If for the odd-even system A 1 is to be neglected too, only those few levels (k, Vk) just above the Fermi surface of the doubly even core should contribute to the sum (4.5.1). If, on the other hand. the probabilities [CrtU,~J2 in eq. (4.5.1) are considerable for levels (k, Vk) close to threshold, or if the excitation of the core particles is important, then At cannot be neglected. It is difficult to establish a one-to-one correspondence between the phenomenological and microscopic descriptions of the target states. However, one can roughly say * For example levels evolving from lid) (see fig. 3) will pick up continuum admixtures very rapidly; the one that evolves from 12s) (fig. 4) will be less influenced by the continuum. See also subsect. 6.2.

436

A. SEVGEN

that eq. (4.5.1) corresponds to the effect of the Coriolis coupling in the macroscopic approach. Due to the Coriolis coupling, the probability of finding the odd particle is spread out to orbitals a few MeV above the lowest available single-particle level. If there is a low-lying narrow resonance over the unperturbed potential (fl = 0), it can be strongly pulled down into the deformed well and the levels (k, vk) occupied by the odd particle share its strength, so that A1 may be appreciable. Thus, we conclude that, for the odd-even nuclei, we can neglect A ~ if the Coriolis coupling is weak and especially if there are no low-lying resonances over the unperturbed well. As usual A2 cannot be neglected.

5. The scattering matrix We calculate the S-matrix with the assumption that there are BSEC of type 4~J N, v N (eq. (3.2.4)) so that A2 cannot be neglected. The S-matrix is calculated in subsect. 5.1 after orthogonalizing the basis functions and in subsect. 5.2 directly in a non-orthogonal system. Each method has its own advantage depending on the case under consideration. One may ask whether the use of projection operators P and Q which project onto the subspaces {X~} and {q~} respectively, is still a powerful tool to obtain the S-matrix when the two subspaces are non-orthogonal. The fact that P and Q are non-hermitian in the space of functions {X[, ~} does not allow the usual handling of the problem 5, 6). Properties of such projection operators are briefly discussed in appendix B. 5.1. C A L C U L A T I O N OF T H E S - M A T R I X A F T E R C O N T I N U U M S U B T R A C T I O N F R O M OJ N , V/~,

Let us suppose we have a set of channel states for which A 1 can be neglected, a set of BSEC ~ss,,~ (eq. (3.2.11)) andonly one BSEC t of type ~.v~,. s The requirement for ~sN, v ~ to be linearly independent of channel states is that the single-particle level ~PN,~N (eq. (3.2.2)) must have spherical bound components also. We assume that this ~J is the case. Let us define the state I~N, vN) such that

(5.1.1)

pJ

where p is the unit operator in the channel subspace {X/~} and is given by p=

dE'

,

.

(5.1.2)

c"

Since the states obtained by using the full projection integrals mainly add to the numerical complexity rather than to the dynamical features, we will for simplicity use the adiabatic approximation to the projection integrals or, equivalently, the adiabatic wave functions (appendix A).

MICROSCOPIC THEORY

437

We make use of eqs. (3.1.3), (3.2.3), (3.3.3) and (3.3.15) and write q~' as

IJN I JNI

Bstj ~¢a{OOtg(hs,ra)~Oc}.(5.1.3)

The state Iq~') has the following properties: (i) It is normalized to unity. (ii) By construction p~' = 0. (iii) In the adiabatic approximation, ( ~ ' l ~ s ,~) = 0. s If there are several BSEC of type ~N,~,, then after continuum subtraction those states will no longer be orthogonal, i.e. ,.J ,.J (q~N,~,,l~u,(,.~),) 4: 6~,, (vN)'.

(5.1.4)

Therefore one must use an orthogonalization procedure and obtain the orthonormal set {~s}. The states {~s.} are orthogonal to cb~,~ in the adiabatic approximation. Now that the states are all orthogonal the S-matrix can be obtained easily. We first diagonalize pHp where p is given by eq. (5.1.2); pHp can be written as

pHp = Ho+ V,

(5.1.5)

where Ho is defined by eq. (3.1.5) and the residual interaction V is given by a-I

V =- ~, v(i, A)- U(ra).

(5.1.6)

i

Let us denote the scattering states ofpHp by ~}. If the residual interaction is strong enoughpHp can also have bound states, which are denoted by ~,,. At very low energies those bound states of pHp which have small binding energies will be important for the description of scattering. We define the new projection operators P and Q such that P = ~£~°l~cE')dE'({ I~' E' c

~c

S , VS

m

(5.1.7)

=- Z I~,><~',1. i

We can now write down the S-matrix, sc,c(E) : Sc.c Bo - 2 , ~ ~ <~(-qVl~k>[D-1]k,(a',lVl¢~(+)>.

(5.1.8)

k,l

The functions ~ can be written as c

~e

~

E ~(C') = c"

= Z da{f~(ra, c'

t

c

t

dE dE(c, E')X~: c"

c')~Pc,}

(5.1.9t

438

A. S E V G E N

In the expansion eq. (5.1.9) one must include in principle all the channels, in practice those (whether open or closed) that are strongly coupled to the elastic channel. The background scattering matrix in eq. (5.1.8) is given by,

=

-e),

(5.1.10)

and is non-diagonal in the channel indices. The inverse of the propagator appearing in eq. (5.1,8) is

Dkt =

E " ((~k VI~E")(~E" V I a l )

E3kt--(~klHl(15t)--

c,

E( +)--E ''

where

~k[ V l ~ ) -~ ( ~ [ ~ v(i, A ) - U(A)I(A) ¢ ~ f ~ ( r A , C')¢Pc,). i

(5.1.1l)

c"

We note that once ScS,~ has been found numerically, the rest of the problem can be treated algebraically. It is worthwhile to note the following four points: (i) The functionsf~(r, e') (eq. (5.1.9)) are solutions [see chapter 5 of ref. 1)] of the coupled integro-differential equations

(Dt,,i,, + sc,, - E)f• c (, . A' C't) +(FC"(ra, r , , , . . . . rA,)rZ v(i', A ' ) - U ( A ' ) I ( A ) ~ Z f£(rA,, c')qgc,) = 0. i'

(5.1.12)

¢' C'~£"

The integration is over the primed variables and

FC" - ~ A ' { ~ ( E A-- I'A') - Z ('ob'j"( FA' ~s)O)b'J"( I'A''

/~s)]~0c"(

s

rl''

" " "'

r( A -

1)',

PA')},

(5.1.13)

and Dt,,j, is the differential operator of which u~.,; is the continuum solution (c.f. eq. (3.1.3)). 'c . The functions JE(~, e') are orthogonal to the bound-state wave functions of the potential U, i.e. Jo~dr ' JE(~ r', c ,)c%,v~,( ' ) = 0.

(5.1.14)

If there is a single-particle resonance over the deformed H F well which has as its components mainly spherical bound levels, then such a resonance cannot be produced in the channel subspace. Again we delay the discussion of such resonances to sect. 6. d (ii) In the case where q~N,~N is linearly dependent on {X~} we do not include it among the basis vectors. Because of the way ~b~,~, is constructed, it is very likely that it will approximate well a weakly bound level ~0 o f p H p or a low-lying resonance in ~c In most cases however, the level cpu,~N = ir,la t 10/\ (eq. (3.2.2)) will have {;E}. \ ~ ~.~,~ appreciable spherical bound components also and therefore #t¢,~, s will be linearly independent of the channel states. It will then be important to include it in order to have an accurate description of the resonances (see subsect. 6.3).

MICROSCOPIC THEORY

439

(iii) The experimental data indicate the importance of the inelastic background scattering, i.e. S~B,~ in our model. The populations of the 19F(~+) and 19F(½-) levels 22) in the reaction Z°Ne(d, 3He)19F, of the 23Na(~-) level 23) in the reaction Z4Mg(d, aHe)Z3Na and of the 25Mg(7+) level 14, 15) in the reaction 24Mg(d, p)25Mg are strong indications of this fact. Also the neutron strength function S ° in the deformed region indicates that the direct coupling of the channel states via the residual interaction, which is responsible for S~,~, BQ is rather strong. The graph of S ° as a function of mass number A has two peaks around A1 and Az (A1 < A2). We shall see in sect. 6 that by using channel states {X~} we can explain only the peak around A2. The fact that this resonance occurs far from the positions of the unperturbed resonances 16) contained in {X~} tells us that some inelastic channels, although closed, are directly and strongly virtually excited. (iv) We can write the full scattering wave function ~- cE( + ) (eq. (3.4.2)) as

=Z

E

i

C'

(5.1.15)

where the functions p~ are not to be confused withf~ (eq. (5.1.9)): the former are influenced by the BSEC whereas the latter are not. The asymptotic behaviour of p~ gives the full S-matrix. In the shell-model approach, use of the lp-lh model simplifies the complicated integro-differential equations satisfied by p~[see chapter 5 of ref. 2)]. For the rotational nuclei, however, the terminology of particle-hole states applies only to the intrinsic states and not to the projected states which are complicated. It can be shown that such a lp-lh model in the intrinsic system does not simplify the general integro-differential equations for p~. 5.2. CALCULATION OF THE S-MATRIX IN A NON-ORTHOGONAL BASIS The method of subsect. 5.1 for calculating the S-matrix becomes impractical when there are many BSEC of type ~JN, v,, so that constructing and orthonormalizing the set (~'} (eq. (5.1.1)) proves to be tedious. Therefore we calculate the scattering matrix in the non-orthogonal basis {X~, 4}. The states {X~} and (~) are orthonormal in their respective subspaces but they are not orthogonal to each other. We assume that all states are linearly independent t. For simplicity we neglect the channel-channel coupling. The coefficients a~(c', E') and bCr(i) (eq. (3.4.2) completely determine the scattering wave function 7t~~+~. We find these coefficients from the following coupled equations:

( (x :i

\ %/1

'+') =

O.

(5.2.1)

The asymptotic part of --E ~ucc+) yields the full scattering matrix

Sc,c(E) = exp (i6~ + itSc,)[tS~c,-2i7r ~, V~,'(E)[D-~ t-Iu V[(E)],

(5.2,2)

kl

t The case of linear dependence of basis vectors was discussedin subsects. 3.2, 3.3.3 and 5.1.2.

440

A. S E V G E N

where Vt~(E) -- <~zlV[X~)

- <@,[ ~ v(i, A ) - U(A)I(A)~X~>, i

(D,)~, = E a ~ , - < ~ I H I ~ , > - V (~dE' ~.1~,

D~(Z)D'~'(E'),"-'" E(+)-E '

D~'(E') -- (E'--E)< X};I~k> + V["(E').

(5.2.3)

For a single BSEC or for well-isolated narrow resonances only the shift and not the width is affected by the terms arising from non-orthogonality of X~ and ~. For the overlapping resonances non-orthogonality also affects the widths. 6. Resonant coupling of the single-particle motion to rotational states and the s-wave neutron strength function 6.1. I N T R O D U C T I O N

In this section we discuss how to treat low-energy scatteringprocesses where singleparticle resonances occur using our basis states {X~:, ~0}. We study this for a familiar example, the neutron strength function. It is interesting to consider this example, because: (i) The available calculations of the strength function in the deformed region ' 6 - , 8) use complex deformed phenomenological optical potentials t. We reproduce the wellknown deformation effects as an " o u t p u t " from the microscopic theory, which involves the rotationally invariant many-body Hamiltonian and the fully antisymmetric many-body states {X~, ~}. (ii) Since we use a mixed representation it is of some technical interest to produce the correct number of resonances. We first define the neutron strength function. For low-energy neutron scattering, the resonances in the shape elastic (a s'e') and absorption (a ab) cross sections are due to the broad resonances in t, where brackets indicate energy averaging over an energy interval L For very low-energy scattering, if the (n, 7) process can be neglected, compound elastic scattering is the only mechanism contributing to a ab. Then, o.ab ~--. ~ 2re2 F, (1 -I
S ° =- f~/O =- -(Fn(2)/[E,(2)]½)~t[Eo] -~, D

* T h e j u s t i c a t i o n o f this p h e n o m e n o l o g i c a l a p p r o a c h will be a t t e m p t e d from analysis in the following paper.

(6.1.2) a microscopic

MICROSCOPIC THEORY

441

where E,().) is the neutron energy at resonance 2, F,().) is the width of the resonance 2, {2} are all the resonances that are coupled to the incoming channel c and E o is the normalization energy usually taken to be 1 ev. Since the absorption cross section a,b is approximated very well by S ° only at low energies, a,b is observed as a function of mass number A and not energy. In the spherical shell model, the channel states {X~} describe naturally the lowenergy shape resonances. Therefore we consider first whether we can reproduce the shape resonances by using our channel states. 6.2. C A N S ° BE E X P L A I N E D

BY {XEC}?

In the region o f deformed nuclei the experimentally obtained curve for S ° has two peaks. Our purpose in this section is to show that we cannot account for both peaks by using the functions {X~:} alone. In the schematic calculation below, the channel-channel coupling is taken into account in the Born approximation, only to illustrate how a resonance in one channel appears in another channel c' via coupling and obtains a width due to the coupling to the complicated BSEC. The spin of the projectile is neglected since the inclusion of a spin-orbit coupling with reasonable strength does not change in an essential way the gross structure given by calculations which omit it ~s, 19). Let us consider the solutions ~ ofpHp (eq. (5.1.5)). We denote the potential resonances in channels X~ by R c. If the channels X~ are coupled via the interaction a resonance R c shows up in other channels c' also. The resonant solutions le) of the full channel space are essentially linear combinations of R ¢. We take the rotational energies to be degenerate since thresholds are close in energy [e2 + - e 0 + < 1 MeV] whereas the inelastic scattering potential* has a strength of about several MeV, so that the inelastic channel, even though closed, will be virtually strongly excited. Let us consider two J = 0 channels c o dan c2 such that Co is the elastic channel (swave neutron coupled to target ground state) and c 2 is the inelastic channel (d-wave neutron coupled to 2 ÷ rotational state). The solution ~ ° ~ ~which satisfies the equation

(E-pHp)~ °'+' =

0

(6.2.1)

has components ~°(co) and ~°(c2) (eq. (5.1.9)). The background S-matrix is due to ~°(Co) since ~°(Cz) either decays exponentially (ez ÷ ~ eo +) or for degenerate channel thresholds has negligible amplitude in the asymptotic region. The background Smatrix is given in the Born approximation by

,G (E) = exp(2i~o) S ....

( 1-2irt(X~e°lVIf f IX~r~')dE'(XE~'IVIX~°)'~ ~E(+)_E ./.

(6.2.2)

¢2

The shape resonances are contained in S ~ ° (eq. (6.2.2)). If we consider S ~° as a function of mass number A, at the appropriate mass number either 6~o resonates or t In o u r model, the d o m i n a n t potential that operates only on [Y,,j(2 + I V ( i , A ) I 0 + ) + exchange]. in the c o m p l e x energy plane are

t e r m in the expression for the single-particle inelastic-scattering the c o n t i n u u m orbitals is given for the 0 + --~ 2 + transition as This f o r m is correct only when the c o m p o u n d resonance poles shifted d o w n by an a m o u n t 1, I ~> D.

442

A. S E V G E N

there is a d-wave resonance in X~~ which shows up in (~°(Co) due to coupling. When there is a d-wave resonance close to threshold we make a single-pole approximation to the propagator in eq. (6.2.2) and obtain

,o ~ exp (2i6¢o) ( 1 Scoco

E -iF'~ ~ / _] '

(6.2.3)

where FI" is a measure of the channel coupling strength and is given by V~" = 2n(X~ °] 1/1W~¢2)2,

(6.2.4)

where W~¢. is the many-body Gamow state [chapter 2 of ref. 1)] and ~ is the complex resonant energy in channel c2. Of course the resonances R ¢° and R ¢~ are broadened by the coupling of complicated BSEC to the channels. Due to the resonances in (¢°(co), the coupling strength to the BSEC is increased and in the energy average of the full Smatrix a broad resonance-doorway phenomenon is seen whose spreading width is roughly equal to 2n r + m D ((~°(c°)[Vk°z)2' (6.2.5) where q);~is an eigenstate of the matrix with elements (~0k[Hlq~l) with eigenvalue e;.; D is the average distance between e;. and q + 1. The bar over the square of the matrix element indicates the ensemble average. When channel coupling is strong the Born approximation does not suffice. The resonances R c°, R ~2 of the two channels get mixed and the resulting combinations el, e2 have a greater energy difference than the unperturbed resonances R *°, R c2. In fact the shift is so large for rotational nuclei that, if the low-lying unperturbed potential resonances R c°, R ~2 occur on mass Ao, then in the low-energy neutron scattering they cannot be observed: e L is bound and e2 is pushed up in energy. That is pHp (eq. (5.1.5)) has a bound level r(~) and (~° does not resonate at low energies. To find e 1 (e2), an experiment is done on mass A~ (A2) such that A 1 < A o (Ao < A 2 ) and e~ (e2) is nearly bound t. When e~ and e 2 occur as low-lying resonances over masses A 1 and A2 respectively, configurations (depths of the spherical and deformed potentials, number of particles etc.)are different. The question is then whether the following relation holds: [e(A,)) - Z da{w~(rA, c', A/)(pc,} C"

? ~ ( ~ 7 dE'd[(c', E ,)Xw(Ai), c,

i = 1, 2,

(6.2.6)

where we denote e~ by e(A~) to emphasize that they are observed now on different target nuclei. If the resonant single-particle state w~ defined above can be expanded in terms of continuum solutions only, then eq. (6.2.6) holds. Let us now remember * I f e2+ :;~ eo+, A2 m u s t be such that the resonance e 2 is pulled below the threshold of channel c2.

MICROSCOPIC THEORY

443

our discussion in subsect. 3.4 about the completeness of the set {X~, 4}. The strength with which a bound spherical level tOb~j was pushed out of the deformed well (hence the lost strength) was given by 6,1J (eq. (3.4.5)). If w~ is made up out of such bound levels that are pushed up, the relation (6.2.6) does not hold. One can convince oneself that the relation (6.2.2) holds for e(A ~) (where R ~°, R ~ are in the continuum) but not for e(Az) (where R ~° and R ~ are bound)*. From our schematic model we can conclude that {X~} can explain resonances of type e(A1) but not those of type e(A2). However if w[ in resonance e(A~) has spherical bound components also, then the set {X~} will not be able to account for this resonance completely. Therefore to explain J ~, (subsect. 3.2). the resonance e(A2) and in general e(A 1) we must use doorways 4u, This is the subject of the next section. 6.3. S ° E X P L A I N E D BY {~¢, vN}

If w~(Ai) (eq. (6.2.6)) can be approximated fairly well by ~o~¢. . . . e(A~) by 4N, s v,, (eq. (3.2.3)) provided 4N, J vN are linearly independent

we account for of {X~}. Doorways 4N, s vN can represent the case when the low-energy resonance is just bound in the deformed potential well. For low-energy narrow resonances which are close to threshold there is little difference if these resonances are treated in the continuum or as very weakly bound states 2o). We consider e(A2) and e(Al) separately. (i) As for e(A2), we have discussed in subsect. 6.2 that (X~(Az)} is unable to account for it; it can be described by the doorway 4 u= J o, v~,=z using the methods of subsect. 3.2, where J [4o, z> oc pS=Oa*k=O,~=2[Xo, 2>, (6.3.1) and Xo, 2 is the K = 0 intrinsic state of the doubly even target A 2 . (ii) Since in our schematic model e(A1) can be explained by (X~(A1) }, we do not construct the doorway 4 s, i if it is linearly dependent upon the set (X~:(A~)}. However in a realistic description of the problem spin-orbit interaction splits the single-particle levels and also deformation mixes states of different major shells so that the wave function tpo, 1 = in general will have spherical bound levels with large amplitudes also. Therefore in this case we must construct 4 s=° o, for an accurate description of the resonance. Whenwe construct state 4tJ=O o,~ (eq. (5.1.1.)) from 4~J=° O, 1 to construct the S-matrix, channel-state contributions to 4 J0, ~ are subtracted off and 4' is that component of the resonance which cannot be explained in terms of (X~(A 1)}. Doorway states 4 J=° o,~ and 4 J=° o,2 have large escape widths due to the similarity of their structure with the entrance channel. They gain additional width since they are coupled to a great number of complicated compound states. * One can see this from figs. 3 and 4 for Rlao and R2so, and from our discussion in subsect. 4.2 about R,tm curves. We have seen that the deformed level evolving from the spherical 1d (2s) level has large (small) continuum admixtures. When the 1d (2s) level is nearly (weakly) bound, and the deformation is large, the deformed level evolving from l d (2s) is b o u n d (unbound) and has mainly spherical continuum (bound level) components.

444

A. SEVGEN

It is interesting to note that the peaks seen in the strength function in the deformed region are not proportional to the s-wave strength distributed over the compound le,mls but they are proportional to the strength with which the doorway states ~0s_l° and ~b0, J =2o are shared by compound states. Hence the name "s-wave strength function" is somewhat misleading in the deformed-mass region. 6.4. DISCUSSION In sect.6, we have shown in a schematic model that we can account for the neutron strength function with our microscopic basis states {X~, ~b}. Therefore the low-energy single-particle resonances in our problem, by which we mean the resonant coupling of the continuum particle to the rotational states of the target, can be conveniently handled. We have shown that the shape resonances can be represented by means of doorway states cbsN, ~/q where care must be taken to assure that these functions are linearly independent from {X~}. In general they will be. Resonances of type e(At) can also be explained in terms of channel states {X~(A~)}; however this description may be incomplete in the sense that not all components which make up the resonance are taken into account. In this schematic calculation we used states which are antisymmetric in the coordinates of all the nucleons, and the microscopic Hamiltonian H which is rotationally invariant, as it must be, contrary to the operator used in the phenomenological calculations.

7. Summary and conclusions This paper is devoted to the construction and analysis of the formalism by which we can treat the scattering of nucleons by deformed nuclei in the frame of a microscopic approach. We have expanded the fully antisymmetric many-body scattering wave function in terms of basis states {X~:, ~} and we have shown that the choice of X~ required some care. On the one hand, a complex deformed potential as a channelcoupling interaction is a phenomenological operator. It has no place in a microscopic theory. On the other hand, continuum solutions of the microscopic deformed H F Hamiltonian do not conserve angular m o m e n t u m so that meaningful channel states cannot be constructed. Therefore in our choice of basis states (subsect. 3.1) we have carefully avoided the use of a " d e f o r m e d " potential of any kind for the single-particle continuum states. The compound states in our model (which are linear combinations of BSEC, i.e. {~}) are random and complicated. Rotational motion is a property of the low-lying states and most compound states which are highly excited states of the A-particle system do not have this property. The occurrence of BSEC of the type • J which represent binding of the continuum nucleon by exciting only the collective states of the target, is characteristic of the deformed region subsect. 3.2. Dealing with fully antisymmetric many-body wave functions and a mixed representation forced us to consider the non-orthogonality of {X~}, which among themselves are measured by A~, and among (X~} and {~b} are measured by ,4 2 subsect. 3.3. We

MICROSCOPIC THEORY

445

presented numerical calculations for A 1 and Az in sect. 4. The fact that A 1 is negligible indicates that our formulation is also practical for a numerical calculation. When doorways ~JN,~ are excited A2 is not negligible. In subsects. 5.1 and 5.2 we presented two different ways of calculating the scattering matrix in the case of non-zero A2. Even though a mixed representation is used, in sect. 6 we showed that the low-energy resonances can be correctly treated. As an example the splitting of the s-wave neutron strength function S ° was considered. In conclusion we believe that especially for the doubly even deformed targets or for those odd-even target nuclei where Coriolis coupling effects are small, we have given a consistent description of inelastic scattering processes which treats all nucleons on an equal footing. The applications of this approach will be considered in the next paper. There we shall investigate: (i) The microscopic analysis and justification of the phenomenological channelchannel coupling models which involve the real and imaginary parts of the phenomenological optical potential and their derivatives. (ii) Correlations of the decay widths of random compound states to those channels which involve the target in one of its rotational states. Corrections to the HauserFeshbach expression for the energy-averaged fluctuation cross-section and the experimental consequences of such correlations. (iii) The extension of the formalism to study scattering from those spherical nuclei which have excited deformed levels. I am very grateful to Prof. H. A. Weidenmtiller for suggesting the topic and for many interesting discussions, advice and criticism during the course of the work and also for reading the manuscript. I would like to thank the directors of the MaxPlanck-Institute for Nuclear Physics, Heidelberg, Professors W. Gentner, U. SchmidtRohr and H. Weidenm~iller for the financial support. I have enjoyed and benefitted from visits to the Niels Bohr Institute in Copenhagen in the summers of 1969 and 1970. I would like to thank the director Prof. A. Bohr for his kind hospitality and him and Prof. B. Mottelson for a discussion. Prof. R. Lemmer brought the strength function phenomena to my attention. It is a pleasure to thank Dr. H. Yoshida for letting me use his subroutine and for his help in programming. The financial support of Yale University and the Turkish Scientific Research Council during various phases of the work is gratefully acknowledged.

Appendix A THE ADIABATIC APPROXIMATION In subsects. 3.2 and 3.3 we used for simplicity the adiabatic approximation to projection integrals which we briefly explain. The matrix element M of a tensor

446

A. SEVGEN

operator T~a between projected determinants can be written z a) as:

M ==-[NKNK.]
'

,t

o-

"7 C

K-z

Z z

dfl sin fld~_~,r,(fl)(K, vrl T~I exp (-iflJ,)lK', v~,)

+ (- )"- ""f

_

"

x (K, VKITfl exp (--iflJ,) exp (--ircJ,)lK', VK,)},

(A.I)

where [N~] -~ is the normalization coefficient. If the coordinate integrals are maximum around the angle fl = flo, then analytical expressions can be obtained by inserting the unity operator

1 = • IS, Vs>
(A.2)

S, VS

in the indicated places in the eq. (A.1), and approximately, evaluating the matrix element (S, vslexp (-iflJy)[K',vr,) for fl around flo usually making the Gaussian approximation. The adiabatic approximation is defined by the relation: (K, VKIexp (-iflYy)lK', vK') ~ fiKK,6~v,,,fi(COS fl--1).

(A.3)

In this limit, the projected states with K =P 0 can be written as lf2~) = [2/(2• + 1)]~P~KIK, vK)("ei"b"t'~),

(A.4)

indicating that the coordinate integrals will be approximated as shown in eq. (A. 3). Making the adiabatic approximation to projection integral is equivalent to using adiabatic wave functions from the beginning. Adiabatic wave functions are Slater determinants in the "intrinsic" system multiplied by the rotation matrix elements, and the product is suitably symmetrized, e.g. for K ¢ 0,

[O~) = [½(2I+I)]~[D~* +(--1)'+KD'~,*_K exp (-ircJ,)]lK, vK).

(A.5)

We note that the wave function Out given by eq. (A.4) cannot be written in the form given by eq. (A.5); however they give the same result in the evaluation of matrix elements so that they are equivalent. We emphasize that (A.3) and (A.5) are only approximations for the evaluation of matrix elements in the "microscopic" theory; they have nothing to do with the phenomenological approach and phenomenological wave functions. Appendix B PROJECTION OPERATORS P A N D Q F O R THE N O N - O R T H O G O N A L SET {Xec, (it'}

Since in the spherical shell-model approach these operators are widely used, it is interesting to consider their properties for our choice of basis states {X~) and {~).

MICROSCOPIC T H E O R Y

447

As usual one can define projection operators P and Q onto channel and BSEC subspaces such that: p 2 = p, Q2 = (2, P + Q = 1, PQ = QP = O,

(B.1)

and formally !

P =

c"

c'i

, Zf d f d t

E IX~,>gE,<¢, + c', ~

ec,

c'c"

£c,

c"

c'c"

c 'l

~c ' ,

dE Iq~i>ge, (XE,],

Q = ~ I,/,,>g'i(~jl + i,j

~t

E IX~,>g~,E,,
i,c" J,ec,

where g~P = [g-1]~p, g~a being the metric tensor. The operators P and Q are hermitian in their respective subspaces: c

¢

t

= = fi~¢,f(E-E ), < ~ i l 0 1 ~ > = = '~j.

(B.4)

However in the full space they are not hermitian, = O, but

= # 0.

(B.5)

Even if channel-channel coupling via residual interaction is neglected, PHP stiU couples channels:

= E 6 ~ , ~ ( E - E ' ) + ~ g~,~.

(B.6)

J

However P*HP is diagonal. Since P and Q are non-hermitian in the total space {X~, q~}, the S-matrix should be calculated as in subsect. 5.2.

References 1) C. Mahaux and H. A. Weidenmiiller, Shell model approach to nuclear reactions (North-Holland, Amsterdam, 1969) 2) A. Arima, H. Horiuchi and T. Sebe, Phys. Lett. 24B (1967) 129 3) Y. Akiyama, A. Arima and T. Sebe, Nucl. Phys. A138 (1969) 273 4) I. R. Afnan, Phys. Rev. 163 (1967) 1016 5) H. Feshbach, Ann. of Phys. 5 (1958) 357 6) H. Feshbach, Ann. of Phys. 19 (1962) 287 7) R. C. Braley and W. F. Ford, Phys. Rev. 182 (1969) 1174 8) W. L. Wang and C. M. Shakin, Phys. Lett. 32B (1970) 421 9) K. Gottfried, Quantum mechanics (Benjamin, New York, 1966) sect. 45 10) E. Rost, Phys. Rev. 154 (1967) 994

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