Excitation of unnatural parity states and proximity of atomic nuclei to the point of the π-condensate instability

1.02.L

NuclearPhysicsA326 (1979) 463-496 @ North-Holland Publishing Co ., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

EXCITATION OF UNNATURAL PARITY STATES AND PROXIMITY OF ATOMIC NUCLEI TO THE POINT OF THE ir-CONDENSATE INSTABILITY S . A . FAYANS, E. E . SAPERSTEIN and S . V. TOLOKONNIKOV

I. V. Kurchatov InstituteforAtomic Energy, Moscow, USSR Received 6 November 1978 (Revised 27 February 1979) Abstract: It is indicated that inelastic nucleon-nucleus scattering leading to the excitation of unnatural parity states may be a sensitive test of the proximity of nuclei to the point of a-condensate instability. Transition densities for low-lying unnatural parity states in 2c8Pb are calculated using the theory of finite Fermi systems, emplicitly taking into account the one-pion exchange amplitude in the annihilation channel . In the Born approximation, these transition densities yield the differential cross sections with resonance-like enhancement at momentum transfers in the region of the Fermi momentum. This enhancement becomes the more prominent the closer the situation approaches the critical one . The calculated transition densities are tabulated so they may be used for obtaining the differential cross sections in the framework of DWBA and other methods.

1. Introduction

The possibility of vr-condensation in atomic nuclei, discussed in recent years 1-a), has caused greater interest in the study of the spin-dependent terms of the effective irreducible NN interaction amplitude F. When analyzing this possibility within the framework of the theory of finite Fermi systems (FFS), these components are taken as the sum Fo +F of the local (Landau) amplitude Fo and the one-pion exchange amplitude F in the annihilation channel s) : Fo = Co(g+g'T1

- T2)01

' 02,

Fn = Cot(k 2)(01 - k)(02 - k)T, -r2,

where t(k

(r ) 2

(1 - 2es)2 2+

(2a)

Co = ire /PFM *, g and g' are the constants of the local amplitude Fo, 6, is the renormalization parameter for the spin-isospin vertex, gN is the TrN interaction constant (g 2 N/4vr -14.6), P, is the polarization operator associated with the virtual excitation of the A33 resonance and nucleon hole, k is the momentum transfer, m(m *) is the bare (effective) nucleon mass, PF is the Fermi momentum and m is the 463

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S. A. FAYANS et al.

pion mass. Units are f< = c = 1. Using the experimental data of Carter et al .6), 3 .s) be roughly approximated by the expression Pa(k2)= -

0.9(1-a)k2 1 +0 .23(k/m)

Po can

(2b)

Here the constant a is introduced to allow for a possible renormalization of the 1rNA vertex in nuclei. A more refined formula for Pa that takes into account the off-shell effects may be found in ref. 4), but it does not practically differ from eq . (2b) in the region of k - pF which especially interests us . At the given F, the problem of the stability of nuclear matter against 7rcondensation is solved with the help of the equation that determines the dispersion (a2= 0) 2 relation (k) for the spin-isospin sound (excitation with the wa rp symmetry). The appearance of the complex frequencies w in this equation at k = ko -., pp indicates the appearance of the a-comdensate instability . The fact that the momentum transfer ko at which this phenomenon occurs is of the order of the Fermi momentum pF justifies to some extent the parametrization of the amplitude F in the form of eqs. (1) and (2) when only term F is taken out of all k -dependent terms, while the other terms are approximated by the constants g and g' . Indeed, the dependence F on k is characterized by the parameter m =12AF, while the dependence of all the other terms on k is determined by the parameter 1 /rc - 2PF (rc being the core radius) and allowance for it in FO brings about arelatively small correction of order kó/(2pF) 2 that can be effectively taken into account by renormalization of the constants g and g' . The first estimations 1) with the interaction parameters obtained from the analysis of the experimental data without introducing F [ref.')] showed that the situation in actual nuclei is close to the critical one but gave no definite answer as to whether the ir-condensation had already taken place or the nuclei should be somewhat compressed for it. The latter version is of greatest interest as it suggests the existence of 4 .8) in which the a-comdensate phase transition have occurred and abnormal nuclei density is higher than that of the normal nuclei . This transition gives some whose additional gain in energy which can compensate for the loss due to nuclear compression . 9 .10) Analysis of the stability conditions in finite nuclei shows that here the situation is very much like the one in an infinite system . Though the critical values of the interaction constants can differ somewhat from one another (mainly due to the shell effects) depending on the particular nucleus and on the angular momentum and parity of the excitation analyzed, they are quite close to the critical parameters for infinite matter . 5 .11 .12) o More detailed investigations f the magnetic properties of nuclei and spectra of the low-lying unnatural parity states with F introduced show that the most probable situation is that with no a-comdensate in nuclei but with the nuclear

a-CONDENSATE INSTABILTTY

465

parameters close to the critical ones. This conclusion is confirmed by the analysis of the data on the interaction of slow pions with nuclei 13) . Unfortunately, analyzing the spectroscopic data we cannot determine the parameters of the amplitude F accurately enough to ascertain the degree of the nuclear proximity to the critical point. This is so because the parametrization for F used is somewhat crude since F contains, as was mentioned above, other terms dependent upon k (and also upon the momentum PI -P2 transferred through the scattering channel). Though their k -dependence is more smooth than that of F, the influence of these additions is that g and g' in eq. (1) are not strictly speaking the universal constants: in the theoretical description of spectroscopic data they may somewhat change depending on the state and the nature of the phenomenon analyzed . These variations become most conspicuous when calculatingthe spectra. It is rather difficult to select the "pure" cases where the effective constants which correspond to the given state are the same as in the equation that defines the stability conditions. And though analysis of the spectra 12) yields g'= 0.8-0.9, which exceeds the critical value 9") by 10-20% (the difference between the critical density and the density of the normal nuclei being of the same order), it is desirable to have a more reliable confirmation of these conclusions and a more precise determination of the degree of proximity of nuclei to the point of the ir-condensate instability. For these purposes the study of nuccear reactions with excitation of unnatural parity states in which the momentum transfer may be changed "by hand" offers, as we believe, greater advantages. The simplest reactions of this kind are inelastic scattering of nucleons by nuclei and charge-exchange reactions (p, n) and (n, p). The nature of the effect expected 14) is readilyillustrated in the Born approximation when the scattered nucleon acts on the nucleus as the external field Vo whose component that can excite unnatural parity states is proportional to Q,TB e't ''. The appearance of the ir-condensate instability can be interpreted as the vanishing of the stiffness C(k) for such a field at k = k o -_ p F . This stiffness is proportional to the function ár2(k) in the denominator of the pion propagator in nuclear matter, which may be written as D-1(k, w) =w 2 -(Z 2 (k, w), and at w =0 one gets 4) á2(k, w=0)sm 2 (k)=mvv +k 2 +Pa (k 2) -2

PFm * \gwNl2 (1 - 2e.)2,p(k/2pF) k2~ 2m 1+2g'¢(k/2pF)

(3)

where O(x) is the Lindhard function, i.e., the particle-hole propagator taken at w = 0 and normalized by the condition 0(0) =1 : 1 (1+ 1_ X2

In 1 1 +x l).

For reasonable parameters g', a and C, which do not noticeably depart from the realistic ones the function ár2(k) has its minimum at some value of k(= ko) which is

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determined by the condition [Gi 2 (t)]' = 0. Near this minimum, w2 can be expanded as Z CÛ (k) _ ~min + [ai 2 (k)]k-1cp2(k 2- ko)2, (3a)

where the prime indicates the derivative with respect to k2. The turning of (5 Zm;n into zero signifies the appearance of or-condensate instability. In normal nulcear matter this happens, for instance, at 0.05, ac = 0, m* / m = 0.9 and g c' = 0.61, ko in this case being equal to 1 .9m (=pF) . If the situation is close to the critical one, w2mm [ = m2(ko)] slightly differs from zero and is proportional to the deviations of the interaction parameters from the critical values . By varying eq . (3) and by evaluating the factors involved numerically we have c5 m;n = M2 [1.8(a - aj + 2.6(g'- gc) + 13(f.(4)

So, near the instability point, the stiffness [i.e. (5 2 (k - ko)] is small, and it means that the response to the external field Vo (k) at k -., k o is expected to have a resonance-like behaviour: the effective field V(k) is proportional to Vo(ko)kó/[m2m, + ~k_kO2(k2- kó)2] and correspondingly the Born differential cross section (do,/d12)H, which is proportional to I V(k)12, must have its maximum at k = ko"~ PF . We see that this resonance enhancement does not depend on what particular parameters are close to the critical ones but is only determined by the value of mom. For simplicity, we fix here the parameters f. and a and also assume m*/m = 0.9 and PF =1 .9m (Co = 360 MeV - fm3), letting only the strength constant g' change . In numerical calculations for 2o8Pb we also fix the parameter g assuming g = 0.65 . The analysis of the magnetic and ß-decay properties of nuclei yields 0 < 6. < 0.1 while reasonable estimates of ref. `) for a give - 0.2 < a < 0.3. We take ~. = 0.05 and a = 0; if we change these parameters at fixed cw nun, then ko may be shifted, the position of minimum of the function (52 being mainly influenced by a, but the resonance-like picture will be the same . With the chosen parameters we have (4a) w2(k= ko)=m .2a[2.6(g' - gc)+1 .3(k 2- kó)2/2kó], where, as the above discourse shows, g', = 0.61 and k o = pF =1.9m, The wave distortions and other effects (indirect processes, etc.) certainly smother this picture but if the nuclei are close enough to the v-condensate threshold the anomalies in the cross sections must remain . Besides, even if these anomalies are not seen "by eye", one can hope to extract the parameters of the effective interaction [eqs. (1)-(2b)] from the experimental examination ofthe region of k - pip if we have a reliable theory of nuclear reactions. Now there is some experimental information available on (p,p') reactions in which unnatural parity states are excited. The most satisfying data were obtained for the nucleus 2o8Pb by Wagner et al. is) who carried out a precise experiment with 35 MeV protons and measured, in particular, the differential cross sections for excitation of nine such states (three for 2-, three for4- and three for 6-) whose energies are within the region of 3-6 MeV. They also made DWBA calculations using the transition

w-CONDENSATE INSTA 3ELITY

467

density matrices ptr(r, r') = (sI'P(r) P+(r')10) for excitations of states Is) of two kinds: (i) those calculated in the framework of RPA by Gillet etaL 16) and (ii) the so-called experimental p,,, obtained from the data on electromagnetic and ß-transactions 17) . However all these p,, have, as we shall see, a serious drawback because they are constructed on a truncated quasi-particle basis which is inadequate for a description of theFourier transform p,,(k) at greater transfer momenta, particularly in theregion of k - ., pF . The states considered are not collective : among the matrix elements drdr'O A (r)ptr(r, r')<.1 (r') J (OA being the quasiparticle wave functions) only one (or several) is great and the rest are much smaller. But there always are many small matrix elements (pt,),, , between single-quasiparticle states IA) and IA') that are far from the Fermi surface. We intend to show that with realistic parameters of effective interaction F these matrix elements make a coherent contribution into the Fourier transform of p,, at k - ko which is completely lost in the calculations mentioned above . Their coherent sum at k - ko is always comparable with the contribution of the main term and even dominates the latter near the point of the a-condensate instability. Here we only deal with the calculations of transition density matries p and do it for two reasons . First, treating the nucleon-nucleus inelastic scattering from the point of view of the FFS theory in sect . 2 one can obtain the expression for the matrix element in which two factors are separated, one being determined only by the reaction mechanism, and the other by the structure of the state excited. The latter is actually p,r which can be calculated in the framework of the FFS theory, but the former involves its own uncertainties so it is reasonable to calculate these factors in matrix elements separately, in two stages. Second, pt, itself can be used for the analysis of the other reactions and also for the description of nucleon-nucleus inelastic scattering at high energies, e.g. for the (p, p') reaction at energy Ep - 1 GeV, using Glauber theory. Due to the parametrization of the amplitude F by expressions such as (1) and (2) which do not contain velocity-dependent terms the crosssection is determined by the transition density ptr(r), that is by the matrix pr,(r, r') taken at coinciding points . As is seen from the calculations 15) the velocity-dependent forces are most important at small scattering angles when the momentum transfer is small. For the effect considered here they are not so essential, so we shall mainly deal with the transition densities pa(r) . We calculate them in the framework of FFS theory using the coordinate representation and precise particle-hole propagators (the response functions). This method is free from drawbacks caused by a truncation of the single-particle basis and seems most satisfactory for studying the phenomena that take place at momentum transfers k - PF. In sect . 2 we describe the nucleon-nucleus inelastic scattering in a microscopic version of DWBA proceeding from FFS theory . In sect. 3 we give an approximate Wtr)AA -=

468

S. A. FAYANS et al.

analytical solution of the equation for the transition density p r which shows that pv has a resonant behaviour near k = ko-pF . In sect. 4 we give the results of detailed calculations of pr, which verify the qualitative analyses . The calculated p,,(r) for the first states with Jv = 2-, 4-, 6- in 2°sPb are tabulated as functions of the radial coordinate r for several sets of interaction parameters . They may be used for DWBA-type calculation of differential cross sections. Since the effect under consideration is due to the components of p,,(r) quickly oscillating in space, these tables seem an adequate form to represent the results of structure calculations . Sect . 4 also demonstrates that a truncated basis is unsuitable for a proper description of these components of pr,, i.e. of the Fourier transform of p,,(r)) at k -PF. In sect. 5 we discuss the choice of the effective interactions and set forth an approximate method of allowing for the contribution ofvelocity-dependent terms which can be used while calculating the differential cross sections . The appropriate DWBA calculations will be dealt with in our next paper. 2. Microscopic FFS version of DWBA The matrix element MO, for the excitation of the state Is) in an even-even nucleus in the process of inelastic nucleon scattering can be represented by the diagram MO. =

f

= (V,*fIvtI V"ó,

where P; (Pf) is the wave function of incoming (outgoing) nucleon in the nuclear optical potential and v, is the vertex where state Is) is created. If the state Is) is of the particle-hole nature, then v, is determined by the homogeneous equation 18) v.(r r2; E)=

1 U(r1, r2, r3, r4; E, E', tor)G(r3, rs, E I -2Ws)

_de x v.(rs, r6 ; s ')G(r6, r4, E' +?fi>,)dr3dr4drsdr62M.

(6)

Here U is a block of NN interaction diagrams irreducible in the particle-hole channel and G is the Green one-particle function . For brevity we omit the spin and isospin indices. Eq. (6) is not very suitable for practical calculations of v,. Bearing in mind that the structure of the state Is) mainly depends on the quasiparticle component GQ of the Green function G, we conventionally represent G as a sum of G4 and the regular part GR(G = aG4 + GR) and then carry out the ordinary renormalization procuedure 18) splitting GG into two parts: GG = a2A+B,

(7)

where A(r r2, r3, r4; E, w)=21riS(e - jt)

G4(r1, r2; E~ - ~)GQ(r4, r3;

d6 27r1

(8)

or-CONDENSATE INSTABILTTY

469

Here a is the renormalization factor and W is the chemical potential. Now introducing in the conventional way the renormalized amplitude, F(e, e ' , w) = a 2 U(E, E ' , (0)+~ a 2 U(E r a1, w)B(e1, to)F(e1, E',

we get from eq . (6)

w)

v, (e) =F(e, li, wa)A(wr)v,(W)

de,

1r1'

(10)

(writing down only the energy variables) . The amplitude F in eqs. (9) and (10) differs from the effective interactionbetween quasiparticles introduced in the FFS theory by the energy dependence and turns into the latter at E = E' =,u . The vertex v,(W ) coming in the r.h.s. of eq . (10) is determined by the conventional equation of FFS theory : (11) v,(W)=F(W,W)A«o,)v.,(W), which is already completely formulated in terms of quasiparticles and can be solved by standard methods. It must be supplemented by the normalization condition which is given by V,W( 3

V, (A»

(12)

where the common brackets mean integration over all coordinates and summing up over the spin and isospin indices. Introducing the transition density matrix pt,(r, r') = (s'I'Pá> P(r')10) for the excitation of state Is) which is connected with the amplitude v,(W) by the relation we finally get

Ptr(rl, r2) = J dr3dr4A(rl, r2, r3, r4i w,)v,(r3, r4i NW), vr(rl, ri ; E) =

f F(rl, ri , r2, r2 ;

E,

u, W,)Ptr(r2, r2)dr2drz "

(14)

This equation can be used in combination with eq . (5) as a basis for describing the reaction, all the information concerning the nature of the process being contained in the interaction amplitude F(e, W, m,). For example, if the coupling with other channels is important, then the amplitude F essentially depends on the frequency m, and in this case it is more correct to consider several channels simultaneously and to solve a system of equations of the type of eq . (10). In our case when we consider the negative parity states in 2°8Pb with energies -4-5 MeV, the coupling of the channels seems non-essential because the lowermost two-phonon negative parity state which could in principle be mixed with them lies at the energy - (o3-+ m2' - 6.7 MeV. As to the transition density matrix that comes in eq . (14) it contains the information on the structure of the state Is) and is an object of the structure calculations. These are just what we carry out here using FFS theory .

470

S. A. FAYANS et al.

3. Qualitative analysis For qualitative analysis itis useful to introduce the quasiparticle wave functions Oa and to express the particle-hole propagator A in eqs. (11)-(13) as A(r1, ri, r2, rz ; W) = E

n -nx,

1A' EA - EA w

&(r1)
(15)

where n are the occupation numbers and E  are the energies of the single-particle levels . In the case of non-collective states it is convenient to write eq . (51) as A =AO +A',

(16)

including in the term Ao only those particle-hole components that play the main role in the wave function of the state Is). All the other components that enter A', taken separately, are small and only their coherent sum is important. Eq. (11) in view of (16) may be rewritten in the form v. = P(w.)Ao(w.)v.

(17)

r(w',)=F+FA'(w,)P(w,).

(18)

where the primed amplitude P is defined by the equation

In the qualitative analysis, F(u, p,) will be considered to have the form of eqs . (1) and (2) and F(E,,u) not to contain velocity-dependent terms. Then the problem involves the one-point form factor v,(r, E), the transition density p,,(r) and the two-point propagator A(rl, r2; w) which comes from eq. (15) at ri = r1 and r2' = r2. Furthermore, for simplicity we shall ignore the smooth dependence of F(6, ,U) on E assuming F(E, W) =F(Fa, u) . Then the form factor v, (r, E) for excitation of the state Is) has no dependence on energy E and coincides with the vertex v,(r,,u.) which is defined by eq. (11) or eqs. (17) and (19). In the momentum representation eq . (17) becomes v.(k)=

1 P(k, k' ; w.)X12(k')X z(k~v.(k~ w - w12 J

N

dk,

(19)

where W12 = E 1 - E 2 is the energy difference of the particle-hole component included in AO and dr X12(k) - J 0* (r)¢2(r)e'k' .

Now the approximation of ref S) can be used. We are interested in the region of k-pF where P(k,k' ;w,) can be approximated as (2V)3T(k)S(k-k') with r(k) being the corresponding NN interaction amplitude in the infinite matter . Disregarding the term A' in the normalization condition (12) (which is justified in the case of the non-collective state Is) ; for details see subsect. 4.2) and separating the angular

ir-CONDENSATE INSTABILITY

47 1

variables as described in the appendix, we get from eq . (19) vi(k) =

~

~ 1L-

47r

L'

r G./ T 1

where

' -Lri`' (k)X 2 (k)(illiilTjL, 1 iij212),

(21)

X12 (k) = jL(kr)R1(r)R2(r)r2dr, J 1 ;L ' (k)=Co{[G(k)+G'(k)Ti "

(20)

T2]SLL , + C~~T(k)k 2 T1

T2},

(22)

G(k)=g/[1+20(k)g],

(23)

G'(k) = g'1[1 +20(k)g' ],

(24)

T(k) ={1

t(k 2) +2
(25)

Using eqs. (2a) and (3), the amplitude T(k) can be written as T(k)

=

C(1 - 2ej2

-l;2 (k)[1+2g

2,


(25a)

where C = CO ' (9 N1m)2 = 1 .15 . According to the discussion set forth in the intro duction, the denominator of eq . (25a) near the ir-condensate threshold, when g'-g~« 1, may be expressed at k-rko - pF as wo+ß(k-k0)2, where moo is small [mó~PB(g'-g~)] and ß is a number of the order of unity [e.g., see eq . (4a)]. As a result, T(k) and along with it v; (k) from eq . (20) have a resonance-like behaviour near ko if X12 (k) does not accidentally cross zero in this region . The form factor for the excitation of the unnatural parity sate Is) with a total spin J can be expanded as v,(k)=i'-'v;(-)(k)[Yr-1(n)®Q ]' +fJ+lvác+'(k)[Yj+1(n)©Q ],

(26)

where n is the unit vector along k and the label r stands for the type of excitation : for the natural excitations T = 0 [reactions (p, p') and (n, n')] and for the charged ones r =1 [reaction (n, p)] and r = -1 [reaction (p, n)]. To simplify the notations we shall mostly omit this label but every time specify the excitation we deal with . Figs. 1 and 2 shows the results of the qualitative calculations based on eqs . (20)-(25) for the r = 0 (p, p') excitation of the first 0 - state in 208pb [ref . 14)] . Full curves in fig. 1 represent T(k)k 2 for three values of the constant g': curve 1 for g'= 0.73, curve 2 for g' = 0.77 and curve 3 for g'= 0.81 . The other parameters of the interaction amplitude F [see eqs. (1) and (2)] have realistic values : g = 0.65, C, = 0.05 and a = 0. At such an interaction, the critical constant g,' calculated for infinite matter is equal to 0.61 . It should be noted that in finite nuclei g' somewhat depends upon the particular nucleus and on the excitation mode, i.e. on r and J, but is quite close to this value 9'1°). The broken curve in fig. 1 represents kX12(k) = k fji(kr)R1(r)R2(r)r 2 dr where j1(kr) is the Bessel spherical function and R 1(r) and

47 2

S. A. FAYANS et aï_ T(k)k2 a (k) 3 k Y(k) 2

0

0.5

1 .0

1 .5

2 .0

2 .5

Fig. 1. Functions used in qualitative analysis of the first 0- state excitation in 208Pb . Solid curves show the amplitudes T(k)k 2 for three values of the interaction constant g' : curve 1 for g'=0 .73, curve 2 for g'-0.77 and curve 3 for g'=0.81 . The dotted curve shows G'(k) for g'=0 .77. The broken curve represents kx(k)=k f ji(kr)R3n,n(r)R%,n(r)r2dr .

300 v o (k) M 200

0 0

0 .5

1

1 .5 k (ral-I )

2

2 .5

Fig. 2. Theform factors for the excitation of thelowest 0- state in 20 Pb calculated from eqs . (19)-(25) for various g' values. Curve numbers have the same meaning as in fig. 1.

ir-CONDENSATE INSTABILITY

47 3

R2(r) are the radial wave functions for the single-particle neutron states 3pt/2 and 4s1í2 obtained in the Saxon-Woods shell-model potential with the parameters given by Chepurnov 19)t. The dotted line in this figure corresponds to G'(k) calculated for g'= 0.77 (G' depends on g' but slightly). In fig. 2 we have drawn the form factor for the excitation of the 0 - state in 2°8 Pb by protons which is proportional to [G(k) -G'(k)-T(k)k2]X12(k). In this case, only one component v:+) is retained in eq . (26). The calculations are carried out for the same three values of the constant g', and the curve numbers have the same meaning as in fig. 1 . As we see, the form factor v ( +) in the region of k -ph is really very sensitive to the interaction parameters, especially if they are close to the critical ones . A similar conclusion is true for excitation of other states of unnatural parity. At .7" 0 0- , both components are contained in the form factor v,. In fig. 3, these

Fig . 3 . The form factors for the transition from the ground 0+ state in Z°spb to the ground 5 + state in 2 "Bí calculated according to eqs . (19)-(25) . The solid curves 1 and 2 correspond to the orbital angular momentum transfers L = 4 and L = 6 for g'- 0 .78 . The broken curves 3 and 4 represent the same form factors for g'- 0.93 . t To properly reproduce the experimental single-particle levels near the Fermi surface we slightly vary the well depths for the various lj.

47 4

S. A. FAYANS et al.

components are shown for the transition between the 0 + ground state in z°8Pb and the 5 + ground state in z°g Bi [an example of the mode with T = -1 that can be excited in (p, n) reaction]. Thesolid curves 1 and 2 represent the form factors v;-~ (the orbital angular momentum transfer L = 4) and v ( +) (L = 6), respectively, calculated for g'= 0.78. The broken lines represent the same form factors for g'= 0.93. In this case the picture looks more complicated, since the resonance dependence in T(k) is noticeably modulated by the functions X12(k) (L=4, 6) which involve the singleparticle wave functions 403.1',2(r) and 04,2(r) [and the spherical Bessel function j4(kr) or j6(kr)] and which change their signs near pF (contrary to the previous case where X12(k) had no zero in this region). Still at k lyingin the vicinity of pF there is an obvious tendency for the form factors to sharply increase when g' approaches gc'. It should be stressed that for the qualitative conclusions drawn here the assumption that the interaction amplitude F(e, u) coincides with F(EL, u) defined by eqs. (1) and (2) is not important . The calculations performed actually determine the quasiparticle vertex v, (1A) . Neglecting the dependence of Fon e the form factor v, (e) for the excitation of the state is) becomes equal to this vertex v, (u). In a general case, the form factor v,(s) is determined by eqs. (13) and (14) where the vertex v,(w) is assumed to be known from the structure calculations. Since the Fourier transform of the kernel FA is a smooth function of k in the region of pP, the resonance-like dependence of the vertex v, (FL) on the momentum transfer results in a similar resonant behaviour of the form factor v,(e). Now we proceed to detailed calculations which will show whether the effects ascertained here by qualitative analysis will remain or not. 4. Detailed calculations 4.1 . DIRECT METHOD OF SOLUTIONS To find a precise solution of eqs. (11)-(13) and to properly reproduce the form factors v, and transition densities p,, at momentum transfers k -., pF one must allow for the contributions of the single-particle states A lying far from the Fermi surface, which makes the A -representation, where A is defined by eq . (15), practically unsuitable . Instead of it we use a coordinate representation in whichthe particle-hole 9.2° .21) . propagator A is calculated precisely in accordance with the method of refs. At fixed angular momentum and parity J'(= 0 - ,1 +, 2 - . . .), after separating the angular variables from eq . (11) with the interaction amplitude F defined by eqs. (1) and (2), we get

o~ (r, ~) (rl, E FiL' (r, rl)A1 1 rz ; w)vi2 (rz; m)ridrlrZdr2, L1L2 J J

(27)

the isospin indices being omitted for brevity. Here L, L1 and L2 take the values of J t 1. The expressions for F; 1L2 and for Á;1L2 are given in the appendix .

-r-CONDENSATE INSTABILrrY

47 5

A detailed description of the method for the solution of eq . (27) in coordinate space iscontained in ref. 2 ) . Briefly, it boils down to the following. At first the kernel of this integral equation is computed by integrating the product FA over the radial coordinate rl by the trapezoid rule with the step of 0.2 fm. The integral over r2 is replaced by the sum which contains the values of this kernel at points spaced apartby 0.4 fm, its contributions at intermediate points being included by means of a simple interpolation. As a result, eq . (27) turns into a system of homogeneous algebraic equations solved by conventional methods. Thus the energy eigenvalues w, are in fact the zeros of the determinant D(w) of a square matrix whose typical dimensions are 100 x 100 (mind that eq. (27) contains four unknown functions for neutral excitations, i.e. two components of form factors v; for each kind of nucleon). Then the eigenvectors v; (r ; w,) normalized by the condition (12) are computed and finally the transition densities which in an explicit form are Wtr(ri ws) ]! = y L'

AJ~I (r~ rI

!'(r% ws)r12dr'

(28)

To show that a precise solution proves the resonant behaviour of the form factors versus the transfer momentum k revealed bya qualitative analysis, we solved eq . (27) in accordance with the above procedure for the same cases that were dealt with in the previous section. Only now we somewhat changed the values of the constant g' responsible for the proximity to the point of ir-condensate instability, bearing in mind that in finite nuclei the critical values g~ of this constant differ from that for the infinite system [(g') . = 0.61] and slightly depend upon JA (see refs . 9'1°)). At the same time, the form factors mainly depend on the difference g'- gL, i .e. on the degree of proximity to the point of instability. For the neutral 0- excitation and for the charged (n, p)5+ excitation in 2°8Pb considered the critical values of g' are practically the same and equal to 0.70, i.e. they differ from (g') by 0.09. That is why the values of g' in figs . 4a and 4b exceed similar values in figs. 2 and 3 by 0.09. We see that both the precise and the qualitative calculations reveal the resonance-like behaviour of form factors v, in the region of k -- pF when g' approaches the corresponding critical value, v, obtained by various methods differing in this region by 1.5-2 times. Shown in figs. 5a and 5b are the corresponding Born differential cross sections which are calculated according to the expression ( ád) d

2J+l

B = -~-~ (4,7r)

(2m )2(I V(l)

)12+

-)(k

)12)

E-w.

for E = 35 MeVwith w, = 5.28 MeV for 0- neutral excitation and with w, = 3.7 MeV for 5+ charged excitation . The resonance-like enhancement at k - PF(B - 70°) is very clearly seen in these cross sections . Then we calculate the characteristics of unnatural parity states in 2°sPb with excitation energies s 5 MeV. All the calculations are made for three values of the constant g'(0.8, 0.85 and 1.0) on which the proximity to the ir-condensate instability

0

0 .5

1 .0

1 .5 2.0

2 .5

Fig.4a. The form factorsfor theexcitation of the lowest 0- state in 2o8Pb obtained by precise solution of eq. (27) forvarious g' values: curve 1for g' = 0.82. curve 2 for g'= 0.86 and curve 3 for g'= 0.90.

-100

0

100

200

300

400

vs (k)

0

0 .5

1 .0

1 .5 l k(fm )

2 .0

2.5

Fig. 4b. The form factors for the transition from the ground 0+ 4tate in 2°sPb to the ground 5+ state in 208Bi obtained by precise solution of eq . (27) . The solid curves 1 and 2 correspond to the angular momentum transfers L = 4 and L = 6 for g'= 0.87. The broken curves 3 and 4 represnet the same form factors for g'= 1.02.

-25

0

25

50

100

V@ (k)

w-CONDENSATE INSTABILITY

477

Fig. 5a . The Born differential cross sections for the excitation of lowest 0- state in Z0spb by protons calculated with the form factors from fig. 4a .

EP = 35 MeV

10 -2 300

60 0

900

1200

e

Fig. 5b. The Born differential crosssections forthe (n, p) reaction accompanied by the excitation between ground states 0+ ( Z°spb) and 5+(so8Bi) for E = 35 MeV calculated with the form factors from fig. 4b .

478

S. A. FAYANS et al.

point depends; in cases of negativeparity the critical constant at which this instability appears in 208Pb is equal to gc' = 0.70. For the other parameters we take realistic values : g = 0.65, 0, = 0.05, a = 0. Table 1 contains the energy eigenvalues w, (which are the roots of the equation D(w) = 0), the corresponding single-particle energy differences w (ro) and experimental energies wl . Irrespective ofour calculations, attention should bepaid tothe fact that the shift dw, of the eigenfrequency wr relative to the particle-hole difference owJ°) is as a rule very small and in some cases dw r is negative . We believe that this phenomenon is accounted for by a considerable compensation of the spin-spin repulsion in the channel with isospin T =1 and the one-pion attraction which is enhanced by a proximity to the point of the ir-eondensate instability and which even sometimes dominates over the repulsion. TAHLF- 1

Experimental energies, corresponding configurations, particle-hole energy differencies and calculated positions of some low-lying unnatural parity states in 2°sPb for various values of the spin-isospin interaction constant g'

J

m

4i 42 6i 43 62 2i 63 22 0,

3.48 3.92 3.99 4.13 4.21 4.23 4.39 5.04 5.28

eza

Main configuration n(3pij22ga/2) n(2f5j2299n) n(2f3j22g9/2) p(3sij21h9/2) n(3pij2lilln) n(2f3/22 99n) n(3p3j22ggi2) n(3pij23ds/2) n(3pij23s1/2)

W(o) 3.43 4.00 4.00 4.21 4.21 4.00 4.33 5.00 5.46

m

u om

g'=0.80

g'=0.85

g'=1 .00

3.51 4.05 4.04 4.12 4.22 4.13 4.34 4.97 4.80

3.52 4.09 4.04 4.20 4.24 4.27 4.42 5.13 5.23

3.54 4.12 4.05 4.28 4.27 4.39 4.49 5.19 5.51

The energies are given in MeV. The calculations are performed with g - 0 .65, a - 0 and & -0.05.

As is seen from the analysis carried out in ref. 12), the "pure" states of all those considered are 2 i and 0 i (in the sense the term "pure" is used in the introduction). They are best described with the constant g'= 0.85 [with the same value the 1+ level with the energy of - 7 MeV is reproduced fairly well, just as are the magnetic momenta and probabilities B(Ml)] . We consider this value to be the most realistic. However, it is impossible to simultaneously reproduce precisely all levels whatever the value of g'. The reasons for this were discussed in the introduction . Tables 2-4 contain the transition densities pr, for three states, namely 2-1 , 4-1 and 6-1 . They can be used for calculating differential cross sections of various reactions bringing about the excitation of these states. Their Fouriertransforms as well as form factors v, (k) have a resonant behaviour near k---pF, the intensity of the resonance

a-CONDENSATE INSTABILITY

479

TABLE 2 Transition densities (fm -3) between ground 0 + state and first 2 - state in Z°sPb for three values of g' g'- 0.80 neutron

proton

R(fm)

L=1

L=3

L=1

L=3

0 .2 0 .6 1 .0 1 .4 1 .8 2 .2 2 .6 3 .0 3 .4 3 .8 4 .2 4.6 5 .0 5 .4 5 .8 6.2 6.6 7.0 7.4 7.8 8 .2 8 .6 9 .0 9.4 9.8

2.416-03 8.608-03 1 .397-02 1 .737-02 1 .725-02 1 .326-02 6.676-03 -2.405-04 -5 .205-03 -6 .953-03 -5 .745-03 -3 .021-03 -4 .581-04 9 .178-04 1 .019-03 3 .983-04 -2 .788-04 -6 .328-04 -6 .484-04 -4 .922-04 -3 .139-04 -1 .790-04 -9 .472-05 -4 .740-05 -2 .257-05

1 .456-05 4.150-04 1 .358-03 3 .140-03 5 .661-03 8 .447-03 1 .078-02 1 .186-02 1 .106-02 8 .360-03 4 .545-03 8 .665-04 -1 .649-03 -2 .598-03 -2 .225-03 -1 .216-03 -2 .881-04 2 .070-04 3 .197-04 2 .543-04 1 .591-04 8 .831-05 4.622-05 2 .360-05 1 .200-05

-3 .671-03 -1 .079-02 -1 .503-02 -1 .614-02 -1 .440-02 -1 .051-02 -5 .436-03 -4.637-04 3 .073-03 4 .464-03 3 .939-03 2 .328-03 5 .605-04 -6 .496-04 -1 .021-03 -7 .751-04 -3 .625-04 -8 .447-05 2 .096-05 3 .466-05 2 .309-05 1 .192-05 5 .467-06 2 .360-06 9 .872-07

-1 .100-05 -3 .174-04 -9.854-04 -2.316-03 -4.408-03 -7 .032-03 -9.476-03 -1 .073-02 -1 .008-02 -7.625-03 -4 .186-03 -8 .160-04 1 .600-03 2 .599-03 2.322-03 1 .426-03 6 .002-04 1 .439-04 -1 .236-05 -3 .451-05 -2 .327-05 -1 .197-05 -5 .547-06 -2 .468-06 -1 .088-06

g' = 0.85 neutron R (fm)

L=1

0 .2 0 .6 1 .0 1 .4 1 .8 2.2 2.6 3.0 3.4 3.8 4.2

8 .263-04 3 .391-03 6 .150-03 8 .410-03 8 .879-03 6 .842-03 2 .759-03 -1 .851-03 -5 .176-03 -6 .089-03 -4 .699-03

proton L=3 4.081-06 1 .261-04 4.113-04 1 .087-03 2.289-03 3 .909-03 5 .530-03 6.519-03 6 .292-03 4 .717-03 2 .308-03

L-1

L=3

-1 .647-03 -4.850-03 -6.924-03 -7.646-03 -6.990-03 -5 .144-03 -2 .527-03 1 .389-04 2 .025-03 2 .657-03 2 .151-03

-3 .518-06 -1 .064-04 -3 .324-04 -8 .762-04 -1 .880-03 -3 .291-03 -4 .719-03 -5 .530-03 -5 .223-03 -3 .802-03 -1 .765-03

S. A. FAYANS et al.

480

TABLE 2 (cont.) g' - 0.85 neutron

proton

R(fm)

L=1

L=3

L=1

L=3

4.6 5.0 5.4 5 .8 6.2 6.6 7.0' 7.4 7.8 8.2 8.6 9 .0 9.4 9.8

-2 .169-03 4 .578-05 1 .024-03 7 .573-04 -1 .106-04 -8 .676-04 -1 .165-03 -1 .057-03 -7 .730-04 -4 .929-04 -2 .882-04 -1 .597-04 -8 .572-05 -4 .511-05

-3 .088-05 -1 .517-03 -1 .848-03 -1 .267-03 -3 .652-04 3 .075-04 5 .536-04 4 .958-04 3 .367-04 1 .964-04 1 .054-04 5 .434-05 2 .765-05 1 .412-05

1 .038-03 -6 .324-05 -6 .924-04 -7 .294-04 -4 .125-04 -8 .754-05 7 .156-05 9 .277-05 6 .315-05 3 .341-05 1 .553-05 6 .754-06 2 .848-06 1 .189-06

1 .881-04 1 .461-03 1 .776-03 1 .317-03 6 .034-04 9 .036-05 -1 .057-04 -1 .117-04 -6 .754-05 -3 .257-05 -1 .411-05 -5 .858-06 -2 .416-06 -1 .012-06

g' -1 .00 neutron

proton

R (fm)

L-1

L=3

L=1

L-3

0.2 0 .6 1 .0 1 .4 1 .8 2 .2 2.6 3 .0 3 .4 3 .8 4.2 4.6 5 .0 5 .4 5 .8 6.2 6.6 7.0 7.4 7.8 8 .2 8 .6 9 .0 9 .4 9 .8

-1 .267-04 2 .776-04 1 .423-03 2.923-03 3 .751-03 3 .059-03 8.303-04 -1 .987-03 -4.059-03 -4.502-03 -3 .348-03 -1 .422-03 1 .887-04 7 .949-04 4 .364-04 -3 .405-04 -9 .699-04 -1 .192-03 -1 .064-03 -7 .831-04 -5 .073-04 -3 .028-04 -1 .717-04 -9 .452-05 -5 .117-05

-1 .661-06 -3 .325-05 -1 .1ló-04 -8 .462-05 2.702-04 1 .049-03 2 .068-03 2 .892-03 3 .058-03 2 .387-03 1 .136-03 -1 .494-04 -9 .476-04 -1 .035-03 -5 .661-04 6 .111-05 4 .891-04 6 .019-04 4 .993-04 3 .338-04 1 .958-04 1 .066-04 5 .602-05 2 .908-05 1 .514-05

-4.496-04 -1 .328-03 -2.021-03 -2 .407-03 -2.375-03 -1 .882-03 -1 .012-03 -2.515-05 7 .176-04 9.717-04 7 .468-04 2 .542-04 -2 .272-04 -4 .752-04 -4 .348-04 -2 .339-04 -5 .003-05 3 .554-05 4 .604-05 3 .059-05 1 .576-05 7 .142-06 3 .042-06 1 .265-06 5 .264-07

3 .931-07 5 .847-06 1 .408-05 -7.362-05 -3 .818-04 -9.456-04 -1 .613-03 -2.066-03 -2.016-03 -1 .414-03 -4.878-04 4.023-04 9.314-04 9 .559-04 6.014-04 1 .733-04 -8 .605-05 -1 .451-04 -1 .052-04 -5 .515-05 -2 .425-05 -9 .760-06 -3 .810-06 -1.499-06 -6 .098-07

or-CONDENSATE INSTABILITY

48 1

TABLE 3

Transition densities (fm -3) between ground 0 + state and first 4- state in 2(*Pb for three values of g' g' = 0.80 neutron

proton

R(fm)

L=3

L=5

L=3

L=5

0 .2 0 .6 1 .0 1 .4 1 .8 2 .2 2 .6 3 .0 3 .4 3 .8 4 .2 4 .6 5 .0 5 .4 5 .8 6 .2 6 .6 7 .0 7 .4 7 .8 8 .2 8 .6 9 .0 9 .4 9 .8

5 .760-06 1 .584-04 6 .885-04 1 .687-03 2 .967-03 4 .038-03 4 .298-03 3 .405-03 1 .574-03 -4 .532-04 -1 .826-03 -2 .092-03 -1 .450-03 -5 .319-04 6 .998-05 1 .479-04 -1 .259-04 -4 .394-04 -5 .887-04 -5 .577-04 -4.339-04 -2 .995-04 -1 .925-04 -1 .188-04 -7 .170-05

2 .559-08 6.520-06 2 .988-05 9.798-05 2 .673-04 6 .159-04 1 .189-03 1 .910-03 2 .522-03 2 .694-03 2 .246-03 1 .306-03 2 .377-04 -5 .644-04 -8 .693-04 -7 .277-04 -3 .985-04 -1 .252-04 9.964-06 4.587-05 4.116-05 2 .834-05 1 .782-05 1 .090-05 6.671-06

-5 .259-06 -1 .389-04 -5 .577-04 -1 .263-03 -2 .092-03 -2 .753-03 -2 .906-03 -2.356-03 -1 .250-03 -2 .926-05 8 .516-04 1 .160-03 9.383-04 4.473-04 1 .525-05 -1 .739-04 -1 .620-04 -8 .598-05 -3 .035-05 -6.449-06 3 .112-07 1 .242-06 8.969-07 5 .063-07 2.618-07

-2 .672-08 -6.421-06 -3 .442-05 -1 .264-04 -3 .531-04 -7 .853-04 -1 .425-03 -2 .130-03 -2 .630-03 -2 .665-03 -2 .145-03 -1 .205-03 -1 .524-04 6 .448-04 9 .564-04 8 .380-04 5 .423-04 2 .849-04 1 .318-04 5 .741-05 2.442-05 1 .024-05 4.234-06 1 .729-06 6.956-07

g'= 0.85 neutron R(fm)

Lai

0.2 0.6 1 .0 1 .4 1 .8 2.2 2.6 3.0 3 .4 3 .8 4.2 4.6

2 .911-06 8 .408-05 3 .976-04 1 .064-03 2 .015-03 2 .882-03 3 .114-03 2 .347-03 7 .551-04 -9 .670-04 -2 .023-03 -2 .028-03

proton L=5 1 .075-08 2 .850-06 9 .463-06 2 .679-05 8 .652-05 2 .524-04 5 .813-04 1 .043-03 1 .465-03 1 .605-03 1 .319-03 6 .912-04

Lai

L=5

-2 .843-06 -7 .491-05 -3 .042-04 -6 .963-04 -1 .161-03 -1 .528-03 -1 .591-03 -1 .238-03 -5 .617-04 1 .578-04 6 .396-04 7 .508-04

-1 .399-08 -3 .363-06 -1 .810-05 -6 .777-05 -1 .946-04 -4 .447-04 -8 .236-04 -1 .246-03 -1 .537-03 -1 .529-03 -1 .170-03 -5 .561-04

482

S . A. FAYANS et al. TABLE 3 (cont.) g' = 0 .85 neutron

proton

R(fm)

L=3

L=5

L=3

5 .0 5 .4 5 .8 6 .2 6.6 7 .0 7.4 7.8 8 .2 8 .6 9 .0 9 .4 9 .8

-1 .239-03 -3.235-04 1 .536-04 7.096-05 -3.107-04 -6 .476-04 -7 .608-04 -6 .759-04 -5 .064-04 -3 .413-04 -2 .159-04 -1 .318-04 -7 .897-05

-1 .440-05 -5.056-04 -6 .266-04 -4 .439-04 -1 .692-04 2 .243-05 9 .196-05 8 .784-05 6 .197-05 3 .865-05 2 .299-05 1 .355-05 8 .050-06

5 .405-04 1 .965-04 -6 .583-05 -1 .485-04 -1 .070-04 -4 .325-05 -7 .011-06 3 .931-06 4 .367-06 2 .701-06 1 .393-06 6 .676-07 3 .115-07

L=5 9.833 135 5 .500-04 6 .682-04 5 .247-04 3 .061-04 1 .449-04 6 .200-05 2 .649-05 1 .166-05 5 .161-06 2 .242-06 9 .470-07 3 .865-07

g' =1 .00 neutron

proton

R(fm)

L=3

L=5

L=3

L=5

0.2 0.6 1 .0 1 .4 1 .8 2 .2 2 .6 3 .0 3 .4 3 .8 4 .2 4 .6 5 .0 5 .4 5 .8 6.2 6.6 7.0 7.4 7 .8 8 .2 8 .6 9 .0 9 .4 9 .8

8 .314-07 2 .784-05 1 .627-04 5 .177-04 1 .111-03 1 .729-03 1 .963-03 1 .500-03 4 .344-04 -7 .425-04 -1 .452-03 -1 .407-03 -7 .991-04 -1 .315-04 1 .785-04 5 .973-05 -2.771-04 -5 .666-04 -6 .683-04 -6 .001-04 -4 .549-04 -3 .100-04 -1 .980-04 -1 .219-04 -7 .355-05

4 .327-10 2 .868-07 -3 .812-06 -1 .886-05 -3 .278-05 1 .353-06 1 .423-04 3 .964-04 6.678-04 7.955-04 6.713-04 3 .304-04 -6.493-05 -3.249-04 -3.517-04 -1 .991-04 -1 .150-OS 1 .007-04 1 .229-04 9 .760-05 6 .414-05 3 .864-05 2 .257-05 1 .318-05 7 .806-06

-9 .099-07 -2 .409-05 -1 .025-04 -2 .473-04 -4 .344-04 -5 .981-04 -6 .462-04 -5 .147-04 -2 .329-04 7.788-05 2.842-04 3.175-04 1 .995-04 2 .548-05 -9.537-05 -1 .187-04 -8 .035-05 -3 .606-05 -1 .076-05 -1 .522-06 5 .650-07 6 .211-07 3 .758-07 1 .958-07 9 .816-08

-4 .499-09 -1 .083-06 -6 .050-06 -2 .439-05 -7 .692-05 -1 .910-04 -3 .768-04 -5 .943-04 -7 .482-04 -7 .406-04 -5 .393-04 -2.028-04 1 .410-04 3.519-04 3.682-04 2 .518-04 1 .208-04 4 .292-05 1 .374-05 5 .863-06 3 .371-06 1 .934-06 9 .979-07 4 .656-07 1 .981-07

ir-CONDENSATE INSTABILITY

48 3

TASus 4 Transition densities (fm3 ) between ground 0+ state and first 6 - state in 2°s Pb for three values of g' g'= 0.80 neutron

proton

R(fm)

L=5

L=7

L=5

L=7

0 .2 0 .6 1 .0 1 .4 1 .8 2 .2 2 .6 3 .0 3 .i 3 .8 4 .2 4 .6 5 .0 5 .4 5 .8 6 .2 6.6 7 .0 7 .4 7.8 8.2 8.6 9.0 9.4 9.8

-1 .653-09 -3 .789-07 -6 .224-06 -4 .039-05 -1 .575-04 -4 .275-04 -8 .703-04 -1 .391-03 -1 .790-03 -1 .855-03 -1 .501-03 -8 .446-04 -1 .604-04 2 .870-04 3 .924-04 2.391-04 7 .324-06 -1 .572-04 -2 .109-04 -1 .873-04 -1 .361-04 -8 .821-05 -5 .349-05 -3 .126-05 -1 .793-05

1 .577-12 -5 .362-09 5 .142-07 5 .737-06 2 .656-05 7 .097-05 1 .159-04 8 .751-05 -1 .049-04 -4.865-04 -9.612-04 -1 .337-03 -1 .420-03 -1 .135-03 -5 .959-04 -4 .434-05 3 .134-04 4 .268-04 3 .760-04 2 .684-04 1 .698-04 1 .003-04 5 .726-05 3 .221-05 1 .807-05

1 .486-09 3 .476-07 5 .434-06 3 .263-05 1 .181-04 3 .060-04 6.133-04 9.826-04 1 .276-03 1 .346-03 1 .135-03 7.042-04 2 .067-04 -1 .688-04 -3 .030-04 -2 .313-04 -9 .368-05 -4 .867-08 3 .032-05 2 .692-05 1 .621-05 8 .198-06 3 .798-06 1 .686-06 7 .348-07

9 .563-12 2 .230-08 1 .730-07 7 .562-07 3 .564-06 1 .661-05 6.082-05 1 .686-04 3 .629-04 6.283-04 9.025-04 1 .092-03 1 .101-03 8.902-04 5 .365-04 2 .038-04 1 .134-05 -4 .473-05 -3 .648-05 -1 .843-05 -7 .425-06 -2 .673-06 -9 .250-07 -3 .252-07 -1 .221-07

g' = 0.85 neutron R(fm) 0.2 0.6 1 .0 1 .4 1 .8 2.2 2.6 3.0 3.4 3.8 4.2 4.6

L=5 1 .304-09 2 .672-07 4 .825-06 3 .263-05 1 .300-04 3 .550-04 7 .181-04 1 .128-03 1 .415-03 1 .413-03 1 .073-03 5 .150-04

proton L=7

L=5

L=7

-1 .438-11 -2 .490-08 -9 .515-07 -8 .758-06 -3 .960-05 -1 .120-04 -2 .177-04 -2 .952-04 -2 .506-04 -2 .956-05 3 .213-04 6 .559-04

5 .421-10 1 .302-07 2 .743-07 -4.919-06 -3 .345-05 -1 .145-04 -2 .687-04 -4.743-04 -6.528-04 -7 .083-04 -5 .922-04 -3 .335-04

-1 .345-12 -2 .998-09 -8 .447-09 -4.858-08 -3 .976-07 -5 .702-06 -2 .897-05 -9.181-05 -2 .101-04 -3 .730-04 -5 .374-04 -6 .402-04

484

S . A . FAYANS et al. TABLE 4 (cont.) g'= 0 .85 neutron

proton

R(fm)

L=5

L=7

5 .0 5.4 5.8 6.2 6 .6 7 .0 7.4 7 .8 8 .2 8 .6 9 .0 9 .4 9 .8

-1 .971-05 -3 .184-04 -3 .231-04 -1 .360-04 7 .859-05 2 .072-04 2 .308-04 1 .907-04 1 .336-04 8 .471-05 5 .070-05 2 .939-05 1 .677-05

7 .987-04 6 .566-04 2 .965-04 -9 .538-05 -3 .504-04 -4 .193-04 -3 .594-04 -2 .554-04 -1 .619-04 -9 .593-05 -5 .489-05 -3 .093-05 -1 .738-05

L=5

L=7

-2 .887-05 1 .953-04 2 .587-04 1 .880-04 8 .001-05 8 .306-06 -1 .638-05 -1 .611-05 -9.895-06 -5 .016-06 -2.315-06 -1 .021-06 -4.423-07

-6.220-04 -4 .654-04 -2 .312-04 -3 .072-05 6.494-05 7 .118-05 4 .415-05 2 .057-05 8 .107-06 2 .931-06 1 .031-06 3 .690-07 1 .394-07

g' =1 .00 neutron

proton

R(fm)

L=5

L=7

L=5

L=7

0 .2 0 .6 1 .0 1 .4 1 .8 2 .2 2 .6 3 .0 3 .4 3 .8 4 .2 4 .6 5 .0 5 .4 5 .8 6 .2 6 .6 7 .0 7 .4 7 .8 8 .2 8 .6 9 .0 9.4 9 .8

-1 .875-09 -3 .396-07 -5 .182-06 -3 .063-05 -1 .113-04 -2 .839-04 -5 .389-04 -7 .913-04 -9 .209-04 -8 .438-04 -5 .617-04 -1 .769-04 1 .477-04 2 .909-04 2.458-04 9.711-05 -5 .364-05 -1 .435-04 -1 .632-04 -1 .385-04 -9 .962-05 -6 .471-05 -3 .950-05 -2 .326-05 -1 .343-05

1 .713-11 3 .173-08 9 .159-07 8 .119-06 3 .689-05 1 .070-04 2 .179-04 3 .228-04 3 .426-04 2 .222-04 -2 .015-05 -2.854-04 -4.363-04 -3 .844-04 -1 .609-04 1 .064-04 2 .926-04 3 .499-04 3 .077-04 2 .249-04 1 .461-04 8 .826-05 5 .127-05 2 .923-05 1 .657-05

-2.228-09 -5 .275-07 -5 .023-06 -1 .802-05 -3 .598-05 -4.051-05 -4.898-06 8.094-05 1 .870-04 2.545-04 2.358-04 1 .259-04 -3 .075-05 -1 .580-04 -1 .972-04 -1 .533-04 -8 .040-05 -2 .628-05 -1 .970-06 3 .906-06 3 .349-06 1 .898-06 9 .211-07 4 .169-07 1 .832-07

-2.404-12 -5 .784-09 -4.791-08 -2 .556-07 -4.686-07 1 .737-06 1 .427-05 5 .035-05 1 .194-04 2 .142-04 3 .070-04 3 .592-04 3 .366-04 2.331-04 9.066-05 -2 .257-05 -6.718-05 -5 .814-05 -3 .314-05 -1 .459-05 -5 .448-06 -1 .857-06 -6 .151-07 -2 .092-07 -7 .755-08

a-CONDENSATE INSTABILITY

48 5

greatly depending upon g'-g .. Therefore one can hope that the analysis of the reactions where these transition densities are used will add to our knowledge of the degree of the proximity of nuclei to the point of the or-condensate instability. 4 .2 . RATIONAL METHOD OF SOLUTION

To find w, with sufficient accuracy the determinant D(w) must be computed as a function of the frequency w at several points . This may take a lot of computer time. To obtain the energy eignevalues w the form factors v, and the transition densities p,, we also used another method saving the time of computation and yielding in the case of non-collective states results without noticeable losses of accuracy. This method is based on eqs. (16)-{18) which are equivalent to eq . (11). As we shall see below this method is also convenient for performing the calculations when the velocity-dependent terms are included in the amplitude F. We thus write here the generalformulae assuming F tobe an arbitrary four-point function . The equation for w, is obtained from eq. (17): Ws ~ (1)

where

12+

129

(29)

1 [P(w.) 12 =

J
(30)

1012 = el -e2 is the energy difference nearest to w, associated with the particle-hole component included in Ao, and 01 and 102 are the corresponding single-particle wave functions. Amplitude P satisfies eq. (18). If the energy shift w, - wlz is small and the absolute values of the energy denominators w, - s,, - ea- that enter A' are much greater than 110, -(0121 as is usually true for the non-collective states, then P very slightly changes upon variations of w near 1012. In the first approximation, we can assume m, =1012 in the right-hand side of eq . (29) to find w;1) =1012 + [r(W12)112~

(31)

Obviously, in the next approximation and so on. The convergence of this iterative procedure is very high (see table 5), and as a rule good accuracy is reached at the first step . Eq. (17) can be rewritten in the form (y.)12 w, - 1012

where (Vs)12 =

J

v. (r rz»1(r1)0z* (r2)drldr2,

(33) (34)

486

S. A. FAYANS et nl. 5 Examples of convergence of the iterative procedure for obtaining energy eigenvalues TABLE

JR

w(0)

w(1)

w(2)

w(3)

w(4)

w(5)

21 4i

4.002 3.432

4.210 3.514'

4.287 3.523

4.275 3.522

4.277 3.522

4.277 3.522

4.230±0 .002 3.475±0 .001

The energies are given in Mev. The calculations are done with g'- 0.85, g = 0.65, a - 0 and & = 0.05.

rl2 (rl, r2; W.) = ƒ r(rl, r2, r3, r4; WJ 4P1 (r3)Y'2(r4)dr3dr4 ,

(35)

Eq. (33) gives a solution minus normalization which can be carried out according to eq. (12). When we consider the condition (12) it is convenient to use the A -representation because the relevant sums have a good convergence near the Fermi surface. If we write the transition density matrix as

then from eq. (12) we get

ptr(rl, r2) = Y, PAA'Oá (rl) Ç6a'(r2), )u'

(36)

ÄAEl (n - na-)IPAA')IPAA'1 2 =1.

(37)

The matrix elements paa , and (v,)AA' are connected by the relation ni - ni, el - EA' - W.

(38)

It should be noted that in the case of the non-collective state Is) one of the coefficients pa,,' in eq . (36), namely P12 is almost equal to unity, and all the others are small as are their contributions into the normalization sum (37). However, they cannot be neglected in p,,,(rl, r2) [eq. (36)] for, as is seen from the Fourier transform of p,r, their contribution at k -pip is comparable with that of P12, and is often even greater. This fact is responsible for the effect discussed. For the interaction amplitude F not containing any velocity-dependent forces, as for example in eqs. (1) and (2) used in our calculations, the reduced amplitude l' can be written as .)8(rl - r2)8(r3 - r4), r(rl, r2, r3, r4; W.) = r(rl, r3; w (39) and the form factor v, as

v.(rl, r2)=v.(rl)S(rl - r2).

(40)

Then the quantity r12 that enters eq . (33) is equal to ri2 (rl, r2; W.) = 8(rl -r2) J r(rl, r3 ; WOO * (r3)102(r3)dr3 -r'12(rl ; W.)s(rl-r2) .

(41)

a-CONDENSATE INSTABILITY

487

The quantity rig (r; w,) obeys the equation following from eq. (18) and having the same form as the conventional one for the effective field in FFS theory : r'12 (r ; ws )=F12(r)+ J F(r, r1)A'(r1, r2; w.)r'12(r2; w,)drl dr2 ,

(42)

where F12(r) is defined similar to F12 (r; w,) [eq. (41)]. As is seen from eq . (33), the amplitude r12 (r ; w,) is in fact the unnormalized form factor v, (r). Thus using the scheme suggested here one can obtain the transition density pv by solving eq . (42) only once where w, can be assumed equal to w12 with good accuracy (or to the experimental value), the subsequent normalization being done according to eqs. (37) and (38). Angular variables in eq . (42) are separated as in eq. (27), and the ensuing equation is solved in coordinate space by a computer as described in subsect. 4.1 . Table 6 contains the matrix elements pAA " between one-particle states A and A' whose energy differences are close to w, (only these matrix elements are significant, the remaining ones, as was repeatedly mentioned, being small and making a noticeable contribution as a coherent sum, while their role in the normalization factor is negligible). The results are compared against "phenomenological" wave functions 17) and against those calculated within the framework of RPA [ref. 16)]. As we see, all the cases yield one conspicuous matrix element P12 ~ 1. This proves that the states considered are non=collective . The other small terms considerably depend upon the details of the interaction amplitude F. To demonstrate this, we carry out two calculations for the constants g' equal to 0.65 and 0.3 which are rather far from the instability point. While the main terms change but slightly, some of the small ones do by 2-3 times. As a whole, the results of our calculations are related to the "phenomenological" wave functions just as the RPA results. It is not surprising that certain small matrix elements may not be reproduced precisely since the parametrization used for the amplitude F is somewhat crude, and some pAA' may be sensitive, for example, to the contribution of the velocity-dependent forces . 4.3 . COMPARISON WITH CALCULATIONS IN TRUNCATED BASIS

To demonstrate that the A -representation with a truncated single-particle basis cannot be used while solving the RPA-type equations to properly reproduce the form factors v, at momentum transfers k -., pF, we have carried out the calculations where for the particle-hole propagator Awe have taken eq . (15) with states k and A' belonging only to two main shells adjacent to the Fermi surface. Such a procedure practically repeats the calculations of ref. 16) . Shown in fig. 6 are the results obtained in the case of 0 state in 208 Pb at g'= 0.52 (for such a constant we get the experimental energy wo, = 5.28 MeV), at g'= 0.40 (5 .05 MeV) and at g'= 0.65 (5.48 MeV). Then gc. becomes equal to - 0.27, which is less than gc' for the complete basis by ^-1 .0. A comparison with the precise solution (fig. 4) shows that the maximum of v,(k) is shifted towards lower k ; now its position is mainly determined

TABLE 6

2g9/z 2fs/"2

-0.980 0.954 0.927 -0 .158 -0 .071 -0 .111 -0 .871 0.995 0.992

Wave function

phenom . RPA FFS phenom. RPA FFS phenom. RPA FFS -0 .921 0.988 0.985

299/2 3piz

-0 .245 0.121 0.088 0.101 0.074 0.089

2g9/z 3pa%z

0.046 0.030 0.024

-0 .080

0.117 0.153 -0.060

289/2 2f7/"2

0.329

liii/z 3piiz -0 .210 -0 .179 -0 .244

3d5""/z 3piiz

-0 .015

0.025 0.073 -0 .039

5/2

ad s5/z -

0.025

0.113

0.045

ads/z 3ps%z

0.012

-0.036 0.346

0.119

2d3fz

lh9/z -

-0.054

-0.023

3s i/2

1h9/z -

0.002

0.130 -0 .076

27/2 2d3/2

where

*22 -

=

Rnn

E Ci1

Cj~3/Iml Tlllnlnl

W/.-

MI-2

1,

1- In3n2 s W

-m1

yi,,x ,.

=

TheFFS wave functions are calculated for g'= 0.85, l', = 0.05, a = 0 and g =0.65. The RPA and phenomenological wave functions are taken from ref.'s). The FFS wave functions are obtained by expansion of pv in terms of basic vectors

6i

4i

2i

J.

Comparison of the main components of the calculated FFS wave function againstthoseof the RPA andphenomenological wave functions for the lowest 2-, 4' and 6- states of z°sPb

r-CONDENSATE INSTABILITY

48 9

2°sPb calculated in truncated basis as Fig. 6. The form factors for the excitation of lowest 0- state in explained in the text . Curve 1 is obtained for g'= 0.40, curve 2 for g'= 0.52 and curve 3 for g'= 0.65. by the behaviour of functions X12(k) at small k and its value depends upon variations of g' noticeably less than in fig. 4, these variations provoking similar changes of the form factor v, (k) throughout the region of k sPF . All these defects of calculations with the truncated basis are easily understood if we come back to the qualitative analysis of sect . 3. The only approximation in sect . 3 boils down to using the expression pertaining to an infinite system for the reduced particle-hole propagator A' . This approximation is reasonable at greater momentum transfers k and actually comes as the following replacement: A'(k,w)-~

dp~ n,+k-n, - psm*
(43)

At k << pF, the main contribution into the integral in eq . (43) is made by the quasiparticles near the Fermi surface with the momenta p within the region [p -pFJ ^- k, and therefore in this case the summing up in eq . (15) only over the shells adjacent to the Fermi surface does not cause serious errors . At the same time at k -pF the integration in eq. (43) embraces regions far from the Fermi surface, but if

49 0

S. A. FAYANS et al.

the basis contains only two shells then in eq . (15) we take into account only a small portion of all states (literally -,-A -1/3) that make an essential contribution into the particle-hole propagator. This means that in the denominators of eqs. (23)-(25) there appears a value oforder A-1/3 instead of ¢ (k, 0) - 1 . As aresult, the pole of the total amplitude r(k) is shifted to the position that corresponds to the interaction parameters lying far from the realistic values, and the resonant dependence of v,(k) on the constant g' is smoothed out. 5. On the effective interactions for calculating differential cross sections ; velocitydependent forces To calculate the matrix element (5) for the inelastic scattering of a nucleon with energy E by a nucleus using the expression (14) for the form factor v one mustknow the effective interaction F(e, g) at e -.. E. What in fact do we know about it? At e - 'M amplitude F coincides with the conventional interaction amplitude F(EL, g) of the quasiparticles introduced in FFS theory and when e -> oo it approaches the interaction amplitude F' of free nucleons. Therefore at E -~ 0 the effective interaction F may be as in the FFS theory and at E >> eF (eF being the Fermi energy) F should be taken equal to that in vacuum as is done in Glauber theory . At medium energies when E - eF the situation is rather uncertain and probably the simplest way to describe this region is to use some interpolation formula which must yield F(ft,, A) at small e, and F"°° at high e. Things become to a certain degree easier because the particular spin-dependent components of the amplitude F(EL, ,u) are close to the corresponding components of F"`° at e - eF: the constants g and g' in eq . (1) are close to those calculated from F' for these energies 18), and F can be quite well substituted for by F' . obtained from eqs. (2) and (3) at C, = 0 and P, = 0 since the effect of the factor (1 -2C,)Z compensates to a great extent the influence of Pe. Therefore it is reasonable to use as the first approximation in this case F in form of eqs. (1) ad (2). Then the form factor v,(e) coincides with the quasiparticle vertex v,(u) for creation of the state Is). The worst disadvantage of taking F as it comes in eqs. (1) and (2) is that it makes no allowance for the velocity-dependent forces, i.e. for the dependence of the interaction amplitude F on the momentum p of scattered nucleon and on the momentum p' of a nucleon in the nucleus. In the literal sense formula (1) means that it contains the zero harmonies go and go from the function g(k, p, p') and g'(k, p, p') with respect to the angle betweenp andp'. But this formula canbeviewed in asomewhat different light. Ifwe deal with problems involving integrals f g(k, p, p')f(p')dp' (and that with g replacingg') where the weight functions f(p') are ofthe same kind, then go and go can be regarded as some effective constants corresponding to the integrals (sums of harmonics) of a certain type. Such effective constants derived from various data (magnetic moments, probabilities B(ML), spectra of collective and non-collective states etc.) may be different. The fact that the constants go and go of FFS theory

w-CONDENSATE INSTABILITY

49 1

obtained from various nuclear data are actually close to each other shows that the contribution of the non-zero harmonics is small. Therefore their influence can be observed under special conditions when the contribution of the zero harmonics is suppressed . This is the case when we calculate the form factors v; at small k. As is easily seen, if we neglect the velocity-dependent forces v; turns to zero at k --> 0 as k L. At the same time, if we take these forces into consideration, vi at k -> 0 can actually become finite . Hence, when E ~ EF and the situation is close to the Born one, the non-zero harmonics of F are only essential at small scattering angles . The effect that we consider, i.e., the resonant enhancement of the cross section, takes place at k -PF, so in the Born case the approximation for F defined by eqs. (1) and (2) (or a similar one in the case of F"t within the angles that interest us is justified. Then the problem will only involve the reduced form of pa(r, r') the transition density ptr(r) that we have calculated. What is now essential is that the dependence of the matrix element Me on the momentum transfer k = pi -pr is completely determined by the Fourier transform of the transition density p,r(k) which appears as a sharp function at k-pF . The amplitude F(e, k) in eq . (14) is a more smooth function of k and therefore affects only the absolute value of the cross section. If E :S I VI - SF (V is the depth of the optical potential), the distortions of the scattering wave functions are great, and the behaviour of v,(k) at small k can influence the differential cross sections even when the scattering angles are great [though as is seen from the calculations carried out in ref.' s) the velocity dependentforces are as a rule important here only at small scattering angles]. Apart from that, the effect we analyze cannot be now observed so to say visually, and the resonance peak is spread over a wide region of k. Therefore the information that interests us is contained in the quantitative description of the differential cross sections over all scattering angles . This problem is quite complicated, and for its solution the velocity forces must be taken into account. This can be done using the method of calculating pt., described in subsect. 4.2. Let us represent the transition density matrix in eq . (14) as it comes in eq . (36) . Then v, (r, r'; a) = 2: F,> -(r, r' ;

E )pÄk -,

(44)

where Fu ,(r, r' ; E) is defined as 1'11 2 in eq . (35). Now let us express F as F=F+SF,

(45)

where F does not depend upon p and p', and SF is the velocity-dependent component of F (minus the term averaged in a appropriate way which must be included in F). Mind that in the approximation considered pt2 m 1 and that each of t All the arguments here are the same, only the averaged constants go and go have different values . If F'(k, p. p') is known they can be calculated explicitly .

49 2

S. A. FAYANS et al.

the remaining coefficients p ,' is small, so what is essential is their great sum which is  ^. pF . coherent at k Then we have from eq . (44) U,(r, r' ; E)=F(r ; e)P12S(r - r')+SF12(r, r' ; E)P12

+

E

(a.a')0(1 .2)

[F,,-(r; E)p,u'S (r - r') + SFaa -(r, r' ; e )pAÄ'].

(46)

The last term is a sum over a great number of states A ; this sum is a universal one (and at k -p F it can be approximately replaced by an integral). In this sum the term containing SF can be considered equal to zero . This is in fact the definition of F and amplitude F of exadtly this kind. appears in the problem dealing with the ircondensation and is usually approximated by the expressions (1) and (2). Then we obtain vs(r, r' ; E) = LF12(r ; E)P12+

E

AA'pá (1 .2)

F>.A 'p"'l 8(r -r' )+SF12(r, r'; E)P12 " (47)

Since the expression in the square brackets is the complete sum over AA', we finally get v, (r, r' ; E)=8(r-r')

J

F(r, r1 ; E)Ptr(rl)dr1+SF12(r, r' ;e)P12 .

(48)

ü the function F(s, )u.) and the averaging method (singling out of F) are known, this expression solves the problem. The quantity p,,(r) in the integrand of the first term is the transition density that we have calculated . The main contribution to SF is made by the term F corresponding to one-pion exchange in the scattering channel (fig. 7) . For great s it may be taken as the ordinary one-pion exchange amplitude:

F,v. = 1 .15Co(Q l " N&2 " l)(12 +mÁ) -1 (Tl " T2),

(49)

where 1=pl-p2+k. The Pauli matrices Q and T are defined here for the cross channel, their transformation into the direct (annihilation) channel yielding 'rl * 72-*2 - 271 * 72,

(50 )

(Ql " 1)(Q2 . 1) -i(ol' 1)(Q2 . 1)-zl2(Q1 . Q2)+i1 2.

(51)

We are interested only in the first two terms of this expression.

P.+ k

Ps

Fig. 7. Diagram of one-pion exchange in the scattering channel .

ir-CONDENSATE INSTABILTTY

493

We suggest a simple ansatz for the averaging procedure: after substituting expression (50) into eq . (49) we replace the expression 01101" by 3(Ol ' O2As and take the zero harmonic from the expression obtained, which is

This harmonic is equal to fo=1+

m~ In l(P1-P2)2-k2+m  l 4PlP2 (Pl+P2) -k 2 +m 

where P2 = PF, P1= m 2J E+F F. a result, we have As SF=-1 .15Co{

2 (Ol ' 1)(O2 . I) - z1 (Ql . a2) 12+M2

3f0(Ol ' O2) ((2 - 271 ' 72) .

(52)

For small E, the averaging procedure is the same, only Fv` must be replaced by a more complicated expression including the polarization effects, V which may be done according to the scheme developed in sect . 3.

6. Conclusion In FFS theory, the inelastic scattering of nucleons by even-even nuclei accompanied by the excitation of particle-hole states can be described by an expression with explicitly separated factors, one depending upon the reaction mechanism and the other upon the structure of the excited state. The latter is actually the transition density matrix . Now, if the contribution of the indirect processes is small we get, just as we often do, the formulae of the microscopic DWBA approach. It is shown that the conventional calculations of ptr which use the A -representation and a truncated single-particle basis are unsuitable for describing unnatural parity state excitations because they inadequately reproduce components of ptr at momentum transfers k -pF. This inaccuracy is due to the neglect of the contributions of quasiparticle levels lying far from the Fermi surface which are coherent at k -PF for the realistic values of the interaction parameters. We calculate ptr within the framework of FFS theory using the coordinate representation where the particle-hole propagators (the response functions) are obtained precisely. We believe this method is the most suitable for the effect considered. This is not the only case when the A -representation brings about inadequate results. A similar situation is observed for low-lying (collective) states of normal parity for which the use of a truncated basis within the framework of both FFS theory and the RPA approach yields the transition densities ptr with a not very distinct and

494

S. A. FAYANS et ai.

broad surface peak . This does not agree with the quantum-hydrodynamical nature of a collective movement in such states 2''22) according to which pa, must be close to the classical transition densities p~ which are proportional to the derivative ôp/ar where p is the nuclear density. Any enlargement of the basis causes increasing and narrowing of the surface peak, while a coordinate representation, with a precisely calculated particle-hole propagator A, naturally gives pt, which differ from p~ only by relatively small quantum corrections. We believe it important that the simple approximation used for the effective interaction fairly well reproducesthe energy levels and the main components ofwave functions for unnatural parity states at parameters close to the critical ones . The main thing that is demonstrated here is a high sensitivity of form factors for excitation of unnatural parity states to the degree of proximity tothe -tr-condensation threshold: at the transfer momentum ^. pF a sharply defined maximum appears in them near this threshold (and hence in the Born cross sections). Of course, in reality the resonance-like picture may be blurred by undesirable effects, in particular due to wave distortions which can be allowed for by the DWBA method that uses transition densities calculated here. Such calculations will appear in our next paper. Preliminary results show that at ^- 35 MeV proton energy for which Wagner et al. '15) got excellent experimental data, the distortions are so great that the specific Born picture (see e.g., figs . 5 and 5a) leaves no trace; still the proximity to the instability noticeably increase the cross sections, but this takes place over a wide region of k. To reveal this effect explicitly, one should use a 100 MeV protons. It would be interesting to carry out such experiments in order to clarify the role of pion degrees of freedom in (p, p') reactions and the possibility to approximate in a simple way the corresponding effective NN interaction, and in order to reliably select its parameters and thus to assess the proximity of nuclei to the ir-condensate instability. The authors are greatly indebted to A. B. Migdal, V. M. Galitsky, V. A. Khodel, V. I. Man'ko, I. N. Mishustin, A. A. Ogloblin, V. A. Timofeev, M. A. Troitsky and D. F. Zaretsky for valuable discussions. 7. Appendix Interaction amplitude and particle-hole propagator ín coordinate space with separated angular variables Here are the expansions of the interaction amplitude F and propagator A in terms of the spin-angular tensors TJLBM (n, a)= ~ CzM M_  ,.Yzr_ . (n)LQmr, J

where C JLmM,, .m2 is the Clebsch-Gordan coefficient, Yt, (n) is the spherical function and [zr j is equal to the Kronecker symbol Smo if s = 0 or to the spherical Pauli

a-CONDENSATE INSTABILTTY

495

matrix a m if s =1. For the case of unnatural parity when s =1 we have F(r1, rz) _

JL1L2M

F,`1 L2 (rl, r2)TI11M(A1, Q1)TJL21M(h2, Q2)-

(A .2)

The components Ai'L2 (rh rz ; w) that enter eq . (27) are defined in a similar way. If the amplitude F is taken as in eqs . (1) and (2), we get Fl1L2 (rl, r2) = ,*L2 (ri, r2)+FJjL2 (ri, r2),

where

FouxL2 FA}L2

(r r2)

= Co

S (ri-r2) r l r2

(A.3)

(é+g'T ' T2)SL1L2,

(rl, r2) = -1 .15Co(1-2Ca)zCi1

f~1~T1 .

(A .4)

Tz,

Here the angular factors C; 1L2 are

(A.5)

J+1.J+1 = J + 1 CJ 2J + l'

J-l.J-1 __ J CJ 2J + l'

' CÎ-1 .J+1 = L,;+l.J-1 = - J( 2J+ l

(A .6)

and the quantity fLI L2 is defined by (for -1.4 < a < 0.7) fr1L2(rl,rz) =s( 1r2r2)-

r

2

Re {ab2[kL1+i(br1)

x IL 2+1(br2)0(rl - r2) + kLl+} (br2)IL1+} (brl) 9 (r2 - rl)JI,

(A .7)

where I (z) isthe modified Besseí function and k(z ) isthe MacDonald function . The constants a and b depend upon theparameters of the amplitude F [eqs. (3) and (4)] : a

where

_

1-bo

2~

bo= !B - VID~- 0.23,

b

z __ rn 

0.23

bo, o,

B=1+0.23-0.9(1-a).

(A.9)

In the same way we have LL'(")

1JIT

U111 TJL1IV 'l') QIII TJL'll U 'l') 2J+1

XSY l n

knl

n!/r(r1)Rn1J*(rz)Gi~' (r i, r2 ; se -~)

+Y_ kn'rj'r'Gll(rl, r2 ; sn'l'i'r - co)R .'ry'T (rl)Rn'1'l'r'(r2)

n'

(A.10)

496

S. A. FAYANS et al.

where k,ah is the occupation factor of a single-particle level with the quantum numbers nliT, Rnljr the radial wave function of this level, Gii the Green function obeying the radial Schr6dinger equation in a self-consistent potential well and UIII Tn .jjV'l') a conventional reduced spin-angular matrix element. The method calculating G'11 in coordinate space is described in ref. z°) . References 1) A. B. Migdal, ZhETF (USSR) 61 (1971) 2209 ; 63 (1972) 1993 2) R. F. Sawyer, Phys. Rev. Left. 29 (1972) 382; D. J. Scalapino, Phys. Rev. Lett. 29 (1972) 386 3) A. B. Migdsl, O. A. Markin and I. N. Mishustin, ZhETF (USSR) 66 (1974) 443; 70 (1976) 1592 4) A. B. Migdal, Rev. Mod. Phys . 50 (1978) 107; Fermions andbosons in strong fields (Nauka,Moscow, 1978) 5) E. E. Saperstein and M. A. Troitsky, Yad. Fiz. 22 (1975) 257 6) A. A. Carter, J. R. Williams, D. V. Bugg, P. J. Bussey andD. R. Dance, Nucl . Phys. B26 (1971) 445 7) V. M. Osadchiev and M. À. Troitsky, Phys. Lett . 26B (1968) 421 8) A. B. Mgdal, G. A. Sorokin, O. A. Markin and I. N. Mishustin, Phys. Lett . 6519 (1976) 423; ZhETF (USSR) 72 (1977) 1246 9) E. E. Ssperstein, S. V. Tolokonnikov and S. A. Fsyans, ZhETF Pis'ma (USSR) 22 (1975) 529; Yad. Fm 25 (1977) 959 10) S. A. Fsyans, E. E. Saperstein and S. V. Tolokonnikov, J. of Phys . G3 (1977) L51 11) E. E. Saperstein and M. A. Troitsky, ZhETF Pis'ma (USSR) 21 (1975) 138 12) E. E. Saperstein, S. V. Tolokonnikov and S. A. Fsyans, Izv. Acad. Nauk SSSR (Ser. fiz.) 41 (1977) 1573 ;41(1977)2063 13) M. A. Troitsky, M. V. Koldaev and N. I. Chekunsev, ZhETF (USSR) 73 (1977) 1258 14) E. E. Ssperstein, S. V. Tolokonnikov and S. A. Fayans, ZhETF Pis'ma (USSR) 25 (1977) 548 15) W. T. Wagner, G. M. Grawley, G. P. Hammerstein and EL McManus, Phys. Rev. C12 (1975) 757 16) V. Gillet, A. Green and E. Sanderson, Nucl . Phys . 88 (1966) 321 17) A. Heusler and P. von Brentano, Ann. of Phys. 75 (1973) 381 18) A. B. Migdal, Theory of finite Fermi systems and properties of atomic nuclei (Wiley, NY, 1967) 19) V. A. Chepurncv, Yad. Fiz. 6 (1967) 955 20) E. E. Saperstein, S. V. Tolokonnikov and S. A. Fayans, Preprint IAE-2571, Moscow (1975) 21) E. E. Ssperstein, S. A. Fayans and V. A. Khodel, Preprint IAE-2580, Moscow (1976) 22) V. A. Khodel, Yad. Fiz.19 (1974) 792