Coherence properties of two-neutron transfer reactions and their relation to inelastic scattering

Nuclear

Physics

Not to be

COHERENCE AND

A169 (1911) 225-238;

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by photoprint

PROPERTIES THEIR

@ North-Holland

or microfilm

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written

OF TWO-NEUTRON

RELATION

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REACTIONS

SCATTERING

R. A. BROGLIA The Niels Bohr Institute,

University

of Copenhagen,

Copenhagen,

Denmark

C. RIEDEL Zentralinstitut

fiir Kernforschung,

Rossendorf

near Dresden,

Germany

(DDR)

T. UDAGAWA Center for Nuclear Studies,

University

of Texas, Texas 78712

Received 21 January 1971

Abstract: The correlation

aspects of two-neutron transfer and inelastic scattering reactions are discussed from a unified point of view. The experimentally observed correlations between the two types of collective states and corresponding cross sections are qualitatively explained.

1. Introduction It has been noted that, when two-neutron transfer reactions (TNTR) are compared with inelastic scattering reactions (INR), rather conspicuous regularities are found [refs. ‘_“)I. For example, in many nuclei in which only one state (as a rule the lowest one) of spin and parity (2, (- 1)“) is strongly excited in ISR, this also holds true in TNTR. On the other hand, in nuclei for which several states are excited with comparable cross sections in ISR, many states are also excited in TNTR with comparable cross sections. In both TNTR and ISR a large number of 2p-2h and of particle-hole configurations (or in general, two-quasiparticle), respectively, contribute coherently to the reaction cross section. This is the basis for the observed regularities. It is the aim of the present paper to study the correlation aspects of both TNTR and ISR and to explain,

in a qualitative

way, the existence of the observed

2. Coherence properties of two-neutron transfer reactions

and of inelastic

similarities. scattering

reactions Particle-hole correlation aspects that play a decisive role in the nuclear spectrum are the multipole correlations, which scatter particle-hole pairs coupled to angular momentum L (in spherical nuclei) and with parity (- 1)“. As an example we can consider the quadrupole correlations, which are responsible, among other features, for the existence of the well-known quadrupole vibrations around closed-shell nuclei and for the existence of stable quadrupole deformed nuclei far away from closed shells. 225 July

1971

226

R. A. BROGLIA et nl.

Particle-particle correlations which aIso play an important role in the nuclear spectrum are pairing correlations. These correlations scatter pairs of particles coupled

““5”

Fig. 1. Inelastic scattering cross sections of 2 +,3 -, and 5 - states are compared with the corresponding (t, p) and (p, t) cross sections. The ISR cross sections are given in terms of single-particle units and are taken from refs. “) (42Ca(cx, cc’)), 18) (1’6*1*8Sn(p, p’)) and lp) (206*ZasPb@, p’)). The TNTR data were taken from refs. 20) f40Ca(t, pf*“Ca), 21) (lzo* “*Sn(p, t)) and 22) (zo4-206Pb (t, P)).

COHERENCE

to angular

momentum

of correlation

227

PROPERTIES

;1 and with parity (- 1)“. The best-known

example

of this type

is the one with 1 = 0, which is responsible, among other things, for of pair vibrations 5- ‘) around closed-shell nuclei and for permanent

the existence superfluid distortions for nuclei far away from closed shells. A natural consequence of the existence of these correlations is that some nuclear states are excited with enhanced cross sections in ISR and TNTR, respectively. What is surprising, however, is that in most of the known cases, there exists a very conspicuous correspondence between both sets of states. This correspondence can be stated in the following way: for nuclei in which only one state, usually the lowest, of spin J = A and parity rr = (- 1)” is strongly excited in ISR, this also holds true in TNTR. On the other hand, in nuclei for which several states are excited with comparable cross section in ISR, several states are also excited in TNTR and their corresponding cross sections are of comparable magnitude. We display in fig. 1 some of the pertinent

data. The TNTR test the particle-particle correlation features of the residual force, whereas inelastic scattering processes probe the particle-hole correlation aspects of the residual interaction. In principle then, one should expect only accidental correlations to take place between enhanced cross sections in ISR and TNTR. A phenomenological interpretation of the experimental systematics (examples of which are shown in fig. 1) was obtained by studying the correlations in space of the particle-hole and particle-particle motion, respectively “). Yoshida ‘) analysed the same problem using the pairing-plus-quadrupole model in the random phase approximation (RPA) for describing the collective states and making simplifying assumptions about the reaction mechanism (PWBA). In the present section, we recast Yoshida’s discussion in the DWBA and allow the particle to interact both through multipole particle-hole and multipole pairing forces [refs. 8*‘)I. The total Hamiltonian is H = H,,,.+

c H’“‘(p-h)+ 1

c H’“‘(P-P), I

(1)

where H c.p.

=

2 Ej bi’,bjm> j, m

H’“‘(p-h)= - +xn 1 Q,, Q,,

=

-

121+1 j~2{(jlll~ahlljzj[b~ J H’“‘(p-p)

= - G,(21+

QA-',,

bj,l~+
(2) (3)

(3a) (4)

(44

R. A. BROGLIA

228

et al.

where b& is the creation operator of a particle in a quantum state (jm). In eq. (3a) sj, > .sr and Ej, < sr, whether in eq. (4a) Ej, and Ej, > EF or .sj, and sj2 < E,+ where sr is the Fermi level. The Hamiltonian [eq. (I)] is diagonalized in the RPA for both the normal

and the superfluid

cases. For more details see refs. *, ‘).

The reaction process is described in the DWBA, taking proper account of the correlations in the motion of the transferred pair of neutrons. For this purpose we use Glendenning’s formulation of TNTR mechanism ’ “). The corresponding form factor is U,(R)

=iTZB(l, /

2; J)g(l,

2Kjr

jZJILSJ)

1
U,,(R),

(5)

where 1 E (n,, Z,,j,). The quantum numbers L, S and J label the orbital angular momentum, spin and total angular momentum of the transferred pair. The relative and c.m. motion of the di-neutrons is characterized by the principal quantum numbers n and N, respectively. The variable R is the c.m. coordinate. The transformation coefficient from jj to LS-coupling is denoted by (j, j, JI LS J>; (nONL ; Lln, I, n2 I, ; L) is a Moshinsky transformation bracket. The factor g(1, 2) = 42 for (n,Z, j,) # (It2 I, jz) and is equal to unity otherwise. The factor 52, is the overlap of the wave functions describing the relative motion of the two neutrons in the triton and in the final nucleus. The function UNL(R) describes the motion of a particle in a harmonic oscillator. It is matched to a Hankel function whose asymptotic behaviour is fixed by the Q-value of the transition under consideration (separation-energy method prescription). The spectroscopic information concerning the structure of the states connected by the reaction is carried by the spectroscopic amplitudes R(l? 2; 4 where

= <4J,I[4Ji

4Ji and 4Jf are wave functions

of the initial

r$,( 1, 2) is the wave function of the transferred pair. The DWBA transition amplitude for a two-neutron TAB = $Do

c L

and final nuclear

states,

stripping

is given by

reaction

and

x

The corresponding

(6)

4,(1,2)1,,)~

Id&&(-’ differential

(k,, AR) cross section

i-“Y~*(QR)UL($R)&+)(kt,

R).

(7)

is then equal to

& k rn~rn~D~ _da = _a da k, (27~nh~)~ (2J,+

TAB12 1)(2S,+

1) ’

(8)

The functions 41’) and 4k-j are the distorted-wave functions describing the relative motion in the initial and final reaction channels. The vector R defines the position of the c.m. of the captured di-neutron. The relative momentum of particles a and b in the c.m. system is denoted by k, and k,, respectively, and m* stands for the corre-

COHERENCE

PROPERTIES

229

sponding reduced mass. The quantity Do is the absolute normalization factor. We discuss below different approximate expressions of the form factor given in eq. (5). 2.1. THE

OS APPROXIMATION

FOR

THE

TNTR

FORM

FACTOR

The transformation coefficients (j,j, JI LSJ) can be connected matrix element + of the irreducible tensor operator

c
TJM = (j,

where N,,,

= max(n, n(l,2)

j,LSILSJ)z

$1, 2)(-1)‘2+Nmax(ljlrLTJ112),

+ rz2+ &(1, + Zz-L))(n

2)

Z200)LO)(n,

J(21, +1)(21,+1)(1,

U,(R)

only S = 0 (singlet)

= c B(1,2;

(9)

= 0), and

$&l,

= abs

In what follows we consider

‘) with the reduced

transitions.

J)(-1)NmaX+‘2n(l,

Z,lrLln, Z2)

(10) ’

In this case,

2)(111rLYL112)

122

x {Q0<00~,ax -G Lb1 I1n2 12;L)UN,,_,:,dR> +n;oQ,
(11)

From fig. 2 it can be seen that Q,, >> Q, even for large values of A. Although important for quantitative calculations, we can neglect the sum over IZin eq. (11) in the present qualitative discussion. This approach is called OS approximation. The function

\h\

1.0 i

---

“‘Pb

-

?a

I \\

0.5

‘\

\ ‘0 \

\ \

\

\ ‘0.

_ N-0

I\ 0

----O--1

2

3

4

n

Fig. 2. The overlap integral n, between the harmonic oscillator wave function, &, describing the relative motion of the two neutrons in the triton and the harmonic oscillator wave function, &,,, describing the motion of the two neutrons in the final nucleus. The value of the integral is proportional to the difference of the frequencies between the triton and the nuclear harmonic oscillator potential to the power an, where 11is the principal quantum number of the relative motion of two neutrons. t The Condon-Shortley

phase

convention

is used throughout

the paper.

230

R. A. BROGLIA

et al.

BIEX)

(O,O) A,

Fig. 3. Schematic representation of the inelastic scattering and two-neutron transfer processes for systems around closed-shell nuclei A”. The pairing phonons (#) and particle-hole phonons (7) carry quantum transfer quantum number a = 2 and 0, respectively, and are of multipolarity il.

iCP) *---

_

~ Fig. 4. Schematic representation of the inelastic scattering and two-neutron transfer processes for superfluid systems. The collective phonon (5) receives contributions both from the multipole pairing and particle-hole residual interactions, as the distinction between particles and holes is lost here. The phonons are completely characterized by the m~tipol~i~ 1.

COHERENCE

PROPERTIES

231

uN,.,,L(R) has its last maximum around the nuclear surface, where the TNTR takes place. In this approximation and assuming that all two-particle configurations are built from states of a single major shell t, we can write UL(R) = st UK_*, L(R),

(12)

where S); z C n(l, 2)(00N,,,L;

Lln, I, n2 I,; L>B(l, 2; J)(1JlrLYLl12)(-

I)‘*.

(13)

122

We can now write da”’ = k, m:m:D: & dQ k, (2i’Ch’)’ (25‘4+ 1)(2s, + i)

c MAMB

lTio8’12,

(14

where Tiz is equal to the magnitude defined in expression (7) but where the form factor U, is replaced by UN_,,=. The magnitude S, can be interpreted as the spectroscopic factor associated with the reaction. It is important to emphasize that in the general formulation of TNTR (i.e., eqs. (7) (8) and (ll)), 1‘t IS not possible to define a kinematic-independent magnitude (reduced transition rate). This is possible only if simplifying assumptions for the form factor like the one introduced above (OSapproximation) are used. Expression (13) is similar to the corresponding one in the PWBA, the difference being the factor rr(E,S) and the Moshinsky bracket (OON,,,L;Lln, I, n2f2;L). TABLE 1

Cross sections for L = 0 (t, p) transitions for some pure two-particle configurations jja(J = 0)) pertinent to the Ca region, calculated using both the complete form factor (second column) given in eq. (5) and the OSapproximation form factor (third column) Single-particle configuration ______.

o(j&

L = 0, 05, (mbisr)

$?2

0.102 0.200 0.041

P*2 4” p$z gq2

0.949 0.150 0.474 0.416

S.*Z

Qo)

dos) (jl j,;L =

0,

0;. QQ)

a’“S’

@b/sr)

0

0.091 0.024 0.094 0.915 0.070 0.465 0.164

0.88 0.58 0.47 0.96 0.47 0.96 0.39

The cross sections are given in mb/sr and were calculated for 0; = 1 (in units of IO4 - MeVZ - fm3), Q. = 10 MeV and a triton incident energy of Et = 12 MeV. f If two-particle configurations from several major shells contribute to the form factor, eq. (12) is no longer correct, and we must include a sum over N,,,.., as N,,,.r takes more than one value. This situation is, of course, present in most cases. The first term in eq. (11) shows that Uc(R) receives constructive contributions from the sum over NmaXas sign { UN~,,~,L(&)} = (-l)Nmar, where R. is the nuclear radius. This constructive coherence in N,,,.. is particularly important for deformed nuclei, in which a Nilsson state has components belonging to different major shells in the spherical representation rl).

R. A. BROGLIA

232

et al.

An idea of the range of validity of the OSapproximation can be obtained from table 1, where the cross sections for several two-particle configurations are compared using this approximation and the correct expression (5) for calculating the form factor. With these approximations the magnitude St plays the same role in TNTR cross sections as the reduced matrix element of the multipole operator Q, defined in eq. (3a), plays in the definition

From the expression

of the total electromagnetic

for (OON,,,L;L

transition

probability:

InI I, nz I, ; L), it can be shown “) that

(OON,,, L; Lln, 1, n2 1,; L) 2 0.

(16)

We can then replace the Moshinsky bracket by an average value over all the twoparticle configurations without altering the phase relation between the different members of the sum in eq. (13). As the factor n(1, 2) is always positive, we can also replace it by its average value without altering the phase relation of the sum in eq. (13). With these assumptions S,3 c~r~~B(l, ,

2; J = L)(l]]r”Y,]]2)(-l)“,

(17)

which is equivalent to Yoshida’s expression ‘). Whereas the OS approximation [eq. (13)] can be successfully used in calculating cross sections t (see table I), the use of eq. (17) for the same purpose is certainly incorrect, as (OON,,,L; L In, I, n2 I, ; L) can change its value by an order of magnitude for different two-particle configurations. On the other hand, the coherence aspects of the form factor defined by eqs. (17) and (12) are still the same as that holding for the complete form factor given by eq. (5) (apart from the coherence implied by the neglected sum over n). It is interesting to point out that, with the help of the OS approximation, one can understand the experimentally observed 13) independence of the angular distribution shapes with respect to the nuclear structure. This is because the radial dependence of the form factor (see eq. (12)) is the same for all two-particle configurations belonging to the same shell. With the help of eqs. (12), (14) and (17), we can obtain insight into the empirically observed correlation between two-neutron transfer reactions and inelastic scattering processes. We have to distinguish between two limiting situations, namely normal and superfluid systems. t The agreement final excited states

between displaying

the OS approximation strong configuration

and the total mixing.

cross

section

is

quite good for

COHERENCE 2.2. NORMAL

PROPERTIES

233

SYSTEMS

For closed-shell nuclei (A 0), come the single-particle energy These nuclei are called normal In this case, two types of states

the pairing correlations are not strong enough to overgap and to produce a permanent pairing distortion. (the equilibrium value of the gap parameter is A = 0). with J # 0 and rr = (- I)” are excited in both TNTR

and ISR: (a) The first type of states belongs to the A, 52 systems (two-particle (hole) like states in the TDA). Its characteristic feature is to be strongly excited in a TNTR from the A 0 system “). One can then conclude that that part of the wave function, which corresponds to the situation in which the two particles move in relative OS states and in which its cm. carries angular momentum i, is definitely enhanced as compared to the same component of the uncorrelated two-particle wave function. This type of correlation can be described in terms of multipole pairing fields which in turn can be generated by multipole pairing forces. In refs. 4-9* 14, i5), the adequacy of this interpretation is shown, and consequently the states in question can be viewed as pair addition and pair subtraction of the AA system ground state with angular momentum, parity and transfer quantum numbers /z = J, rc = (- 1)” and CI = + 2, respectively. In this model, the pair addition and pair subtraction modes are the eigenvalues of the Hamiltonian ‘) H,,,.+H’“‘(p-p) [see eqs. (2) and (4)], namely (Hs.p.+H(L)(p-p))]CI

= *2,1;

n) = Wn(CX= &2, n>jc( = -L-2,1; n),

(18)

where I = +2 labels states of the A, +2 and c( = -2 of the A, -2 systems respectively, and is called the transfer quantum number 14). In the RPA, the spectroscopic factor defined in eq. (17) associated with states in the (A 0 + 2) system is equal to Si((E = 0, jl = 0; iI = 1) --f (a = 2, ;1; n)) E 1

l~j,Ilr”Y~llj,)12(-1)P”‘2’“’

1>2

I

where atlj, = ajl+aj,(]jl -j,l 2 d 5 j, +j,) and (- 1)f1+12 = (- 1)“. By the index i we denote states below the Fermi level and by the index k states above the Fermi level. For n = 1 (i.e. the state with lowest energy of the given angular momentum A), the phase (-1) ‘(I, ” ‘=l’ = 1 for any combination of single-particle states (1, 2). The corresponding reduced matrix element of the multipole operator [eq. (3a)] connecting the ground state of the system A 0 + 2 with the state with spin 1 of the same system is equal to (c( = 2,1; nl]Qillcl = 2, ;t = 0; n = 1> cc c

[a( j:(O))l( - ljP(1~2’n)l( j,]]r”Y,l] j,)]’

132

X

Cl ____~_ E:,j,

k)Q2 -

W,(cL

k) =

2, n)

-

Ql

El, jz

iP(j2

i) (20)

+

Wn(cI

~)

.

234

R. A. BROGLIA

et al.

where a(j:;O) is the amplitude of the j:(O) configuration in the ground-state wave function of the A, +2 system. Again, for IE = 1, (- l)“l? *in) = I. The symbol S(ji k) indicates that the summation x1 should be carried over states of the type k. namely

states with &k> sF.

If we neglect ground-state correlations (TDA), which corresponds to disregarding the second term in both eqs. (19) and (20) (contributions of the backwards going amplitudes to the corresponding matrix elements), both equations display the same type of coherence, and consequently large values of St imply constructive coherence for the matrix element of QA,. However, it is clear that the enhancement factor for TNTR cross sections must be larger than the corresponding factor for the B(E1,) value associated with the reduced matrix element [eq. (20)], as this last process is of higher order than the TNTR process. A schematic graphical representation of this statement is given in fig. 3. The role played by ground-state correlations is discussed in sect. 3. (b) The second type of states belongs to the A, system (particle-hole excitations). The characteristic feature of these states is to display very enhanced B(Eil) values. These states are well understood in terms of particle-hole fields i “). Consequently we use eigenfunctions of the Hamiltonian H_,, + H’“‘(p-h) defined in eqs. (1) and (3) for describing these states. The spectroscopic amplitude and reduced matrix element of the multipole operators are, in the RPA, equal to s;((fx = 2, A = 0; n=l)+(C(=O,&n))

(cc = 0, A;

nllQnllcc = 0,1

ccl~2

= 0; n = 1)

(

fJLW(j24

I(jlll~“Ynllj2)lZ(_1)P(‘~2~“)

, i

E:,

jz

-

+

Wn(OIL)

6(jl+Gk))]. &i,

j,

+

)Vn(On)

(22)

Large values of the reduced matrix element defined in eq. (22) imply that the contributions to the sum in eq. (21) are also constructive. In this case, however, the TNTR process is of higher order than the electromagnetic decay (see fig. 3) and consequently one expects the enhancement factor for this type of reaction to be smaller than the corresponding electromagnetic-decay enhancement factor. 2.3.

SUPERFLUID

SYSTEMS

In systems with a large number of particles outside closed shells, the pairing interaction (1 = 0) can produce a permanent static deformation of the Fermi surface (superfluid nuclei). This deformation is measured by the energy-gap parameter A. In this case, the distinction between particles and holes is lost (i.e., the transfer number tl is not any more a good quantum number). For this reason, the particlehole and particle-particle residual interactions are responsible for the correlations of the same two-quasiparticle states. States of type (a) and (b) are mixed and the vibra-

COHERENCE

tional excitations

235

PROPERTIES

(i.e., 2+, 3-, etc.) represent

collective

modes in both channels,

and

are strongly excited in both ISR and TNTR. For pedagogical reasons, let us still keep artificially the distinction. (a) The TNTR spectroscopic amplitude and reduced matrix element of the multipole operator Q,,, associated with wave functions (RPA) which are eigenstates of the Hamiltonian

st@.= 0;

Hs.+. + #“‘(p-p)

are equal to

n = 1) + (A; n)) cc C 1>2

(U, U,-V,

V,)(-1)P’1~2~“~l<~ll~~~~l12>12

i Ul

Vl v2

((2; dllQi@

= 0; n = 1)) cc C

u2

'

Ef2 - W,(A) + E:, + W&I)

(U, U,-VI

V2)(-1)P”~2;“)j(lllrAY~~)2))2

3

(23)

.

c24

132 x

(Ul

v2+u,vii,) _-

E:2-K(4

(Ul

V2+U2V1)

E:2+Wd4

The magnitude Et2 which takes the place of et2 is equal to the sum of the quasiparticle energies E, and Es. Disregarding the contribution of the ground-state correlations (corresponding to the term with the energy denominator Et2 + lVn(;l)), we are left only with the forwardgoing amplitude contribution. We see that in this case both eqs. (23) and (24) have the same coherence properties, and again (- 1) p(192; n, = 1 for n = 1. It can also be seen from eqs. (23) and (24) (and fig. 4) that both processes are of the same order, and consequently the corresponding enhancement factors are expected to be of similar magnitude. The role of ground-state correlations is discussed in sect. 3. (b) The TNTR spectroscopic amplitude and reduced matrix element of the multipole operator Q, associated with wave functions Hamiltonian Hs+ + H’“‘(p-h), are equal to s;((n

= 0; n = 1) -+ (A; a)) “,Lz, ,

(( U, V,+V,

(RPA)

which are eigenstates

U2)(-1)P”~2~n)~(l~~r~YA~/2)~2 Vl v2

'

(6% ~>liQ,ll(L = 0; n = l)> cc C

of the

Ul

u2 3

Et2 - Wn(A) - J% + KG)

(25)

(U, V,+ Uz V~)2(-1)P”~2;“)~(1~(rAYA~~2)12

122

1 ’

Ei2 - W,(Aj +

1 Et,+

KM

1)

.

(26)

The same considerations as in case (a) apply to the present situation. We want to point out that the main results obtained in sects. 2.2 and 2.3 can also be obtained using wave functions of a realistic force. This is because the above results are based on the phase structure of the eigenstates describing the initial and final

236

R. A. BROGLIA

et al.

states connected by the corresponding reaction, and this phase structure is the same for the main components of the two types of wave functions. 3. The role of ground-state correlations In the preceding sections we have explained the rather conspicuous correspondence found between ISR and TNTR. The great difference of the physical picture underlying both types of processes is best exemplified by the role played by the ground-state correlations in both cases. One has to point out, however, that in all known cases, the backward-going amplitudes (which are a measure of the ground-state correlation) are much smaller than the forward-going amplitudes. 3.1. NORMAL

SYSTEMS

In this case the particle-particle and particle-hole correlations act separately in first order. (a) As can be seen from eqs. (19) and (20) the contributions of the backward-going amplitudes to the TNTR cross section are constructive, whereas they are destructive in the case of ISR. This result is clearly illustrated in the adiabatic approximation (only one collective state exists and its energy W,(2,l) < E;,~, for any combination (j,, j,)). In this limit, eq. (19) becomes equal to S~((~ = 0, Iz = 0) --) (a = 2, ~))adiab ~Kj111r”411jJ12 cc ’

(S(jl

k>G2

k)+%j,

+%2

k)W2

k) - %A ihW2

i)>,

(194

iI>.

(204

4h

and eq. (20) equal to
IT214jI ’

= 0)adia.b ~~2~11<~,11~“~~11~2~12 _ (Kh Ejh

This result shows that the contributions of the ground-state correlations to the ISR process can lead to an almost complete cancellation of the two-particle contribution. (b) In this case one finds a situation different from the previous one.The groundstate correlations induced by the particle-hole correlations enhance ISR cross sections and contribute destructively to the TNTR cross section, the overall cancellation effect depending on the single-particle energy distribution. This can be clearly seen from the adiabatic limit of eqs. (21) and (22) which is equal to s+((cc = 2,A = 0) -+ (cz = 0, A))adiab cc

11Z214.iI ~~2~ll<~,ll~“~~ll.i2~12 ’ -&id,

(S(jI

k>%2

i)-@jl

M.i2

k)),

(21a)

COHERENCE

3.2. SUPERFLUID

PROPERTIES

237

SYSTEMS

(a) The ground-state corrdations in this case give a constructive contribution the TNTR cross section and a destructive one for the case of ISR cross sections. the adiabatic approximation, eq. (23) obtains its largest value, which is equal to C (ul

$((A = 0) 4 (A)),,,,, cc c%

UZ- vi

to In

Wl(111r”U12>12 (234 Et2

The same limit for eq. (24) is equal to

<(4ilQill(~

= O))adiab 0~ C (VI u2-K

v2)(uI v2+ CJ2~I)l(~llr”~l12)12

= 0. @a)

152

This last result is a consequence of the fact that the factor (U, V2 + U2 V,) is strongly peaked around the Fermi surface, in which case U, U, x V, V,. (b) The opposite situation is found in this case, as the ground-state correlations tend to decrease the TNTR cross section while increasing the ISR cross section. The adiabatic limits of eqs. (25) and (26) are equal to St((l

= O) +

Cn))adiab

K

C (VI 182

u2-K

v2)(Ul

C

((‘)llQ~ll(~

(Ul

V,+

v2 +

= O))adiab CCEP

~2)l
= 0, (254

~)2i(~llrA~l12>12 p

(264

Vl

U2

12

As pointed out in sect. 3.3, both types of states are mixed and one has to describe the system by means of eigenvalues of the total Hamiltonian Hs+ +H’“‘(p-h)+ H’“‘(p-p). In this case, both types of ground-state correlations are present at the same time and contribute to the same state. As seen before, they produce opposite effects on the TNTR and ISR cross sections. They are also exclusive *), and their relative importance depends on the value of the ratio xI/GL. One may hope to obtain information on this ratio by comparing the result of calculations using the residual interactions (3) and (4), with the results obtained using effective forces derived from two-nucleon scattering data.

4. Conclusions It is possible to understand in a qualitative way the empirically observed correlation between TNTR cross sections and ISR cross sections by studying the structure of the corresponding form factors. Remarkable similarities between them (or be-

238

R. A. BROGLIA

et al.

tween the matrix elements of the multipole particle-hole and two-particle operators) are found. There is, however, no simple direct relation between the two processes because of the different role played by the ground-state correlations induced by p-p(h-h) and p-h types of residual correlations. The interplay between types of correlations depends on the mass region under consideration.

these

two

The authors wish to acknowledge many illuminating discussions with Professors A. Bohr, B. Mottelson and D. B&s. They also want to thank Professor A. Bohr for having carefully read the manuscript.

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