IC/66/106

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

PERTURBATION OF RESONANCE STATESAND ENERGY SHIFT IN MIRROR NUCLEI

J. HUMBLETAND

G. LEBON

1966PIAZZA OBERDAN

TRIESTE

IC/66/1O6

IHTERMTIONAL ATOMIC ENERGT AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

PERTURBATION OF RESONANCE STATES ATTD 3HERCT

SHIFT IK MIRROR JJUCL3T*'

J . HUMBL3T**

Internat ional Centre for Theoretical Physics

and

Theoretical Nuclear Physics, University of Liege (Lie'ge)

G. LEBO1I

Theoretical Nuclear Physics, University of Lie'ge

TRIESTE

November 1966

:'; To be submitted to Nuclear Physios

** Permanent addressj Institute de Mathematique, 15 Avenue des Tilleuls,Lie'ge, Belgium. , •

ABSTRACT

Defining the reaonanoe states of a nucleus according

to the S-matrix theory of nuclear reactions, the perturbation

of-their positions and widths are calculated to the first order

for a given perturbation of the Hamiltonian* The results are

applied to the computation of the energy shift of some levels

of light mirror nuclei.

I ETURBATIOIT 0? R3S0]T.\IIC: STATES AiID SiranCff SHIFT I1J JIIRROS JIUCLSI

1. INTRODUCTION

THien the Hamiltonian of a system of A nucleons i s perturbed,

the bound s ta tes of the corresponding nuclei are shifted in energy

according to the Sohrodinger perturbation theory. The energy shift

of the resonance s ta tes cannot bo evaluated according to the same

theory because the unperturbed wave functions belonging to such s ta tes

arc not normalizable. To obtain a perturbation formula applicable to

the resonance s t a t e s , one must return to the i r def in i t ion .

If one defines them according to the S-matrix theory of nuclear

react ions , i . e . , as the decaying s ta tes of the compound nucleus, one

has to solve a problem of perturbation of energy-dependent boundary

conditions, i t s general solution for the many-channel case i s given in

Sections 3 and 4 to the f i r s t order in the perturbation. Introducing

single—particle approximations, these r e su l t s have besn applied to the

computation of the energy shi f t s of low-energy levels of the following

pairs of light nuclei: S(k-rLi, ^ . - ' f c , V - ^B, '\~ 'SN, 'V*F\In Section 5» however, numerical data are given for the G— If levels

only.

The same problem has been raised earlier in the It-matrix

theory of nuclear reactions by 'SlCIftlATnT and TII01IAS w . The l a t t e r

derived an implicit equation from which the energy-shift known as the

"Thomas-Shrmann shift" could be derived from the experimental reduced

widths for a few levels in C~ N and O ~ F. These reduced widths

are strongly dependent on the channel radii and it seems that a fair

aiTreement between the computed and the measured shifts could only be

reached for rather large channel radii. Recently, >EID3BIULLElrl ; has

extended the validity of the Thomas-Bhrraann shift to the S-matrix

theory, i.e., assuming that the resonances and the reduced widths

are defined according to the latter theory.

Here, we rather derive an explicit expression for the energy

shift in terms of physical quantities, such as the partial widths,

which are a-independent, i.e., independent of the channel radii.

- 1 -

2 . NOTATION AMU KHFIJTITIOZTS

Except when otherwise stated, we follow the notation of ref.

4); let us briefly recall some of the definitions.

Disregarding the three-body "break-up, let us consider the

usual division of the confimiration space into an interior region CO

bounded by a surface-O , and an orterior region where the A nucleons

are divided into two fragments n ^ , r\^ .Let o^ be the channel radii;

then when the distance Z^ of two fragments A^ , f\d is larger than

a, , the only interaction between those fragments is the Coulomb

repulsion. The surface-O is composed of partial surfaced) and we

define

(1)

where <x i and. 0L0 . (i = Ij2) respectively refer to the relative angular^ *" ' A

position of the two fragments, and to the internal co-ordinates of r\^t .

The so-called surface factor ep of channel C describes the relative

angular position of the two fragments and their individual states in

that channel; e.g., in the c:hannel-spin coupling, we have

l c ~ 1 o<,«^-4 fc •»'*«' ' -. (2a)

or, in the total spin coupling (J,M),

where the indices Q(, , ()(„ characterize 'the internal'energies i, £

of the fragments.

. Neglecting the overlapping of surface factors belonging to

two different fragmentations, the set of functions iS> is orthonormal

and complete on the "boundary surface; we have

- 2 -

t-i •LlC *

where f ornally d./£> - L.O.V,. Hence, on /Owe can expand the to ta l wave

function X and i t s normal derivative as

(5a)

(5b)

where t designates a different distance 1^ on each partial surface •o^

of A , while the "radial factors" & , S' are defined by

is

(6b)

In a state of total energy Q , the channel wave numbers K

defined according toC , a r e

011 (T)Ti7f °

If in particular one considers a wave function x °f

total system with only one incoming channel, namely c, then its radial

factors satisfy the following boundary conditions;

(8)where

L. ^ I O :(9)

-3 -

Oc is the outpoint solution of the Coulomb radial equation, such

as defined in ref. 6), and

• (10)

is the energy—independent Coulomb parameter. If I£ designate the

corresponding incoming solution, then the collision matrix element4)for the reaction c —» c is

Let us also recall that a resonance state of energy G = EL," 5

and channel wave numbers kc » k , ... is a pole of the collision

matrix, i.e., that it satisfies the boundary condition

f = L w

As in ref. 4)> ne index n will be used to indicate that the considered

quantity ir, taken at the resonance energy c = t , with *-JL = , i. . s ^d„„."^

and ^Jy,^ 0 for all d, and the index -n when h = «„ with, fc, » &~ fe,^^ <L Jit

— i d - i-TTj * Accord ing t o e.qs. ( 8 ) and ( 1 2 ) , a t r esonance e n e r g y ,•I'M *'*

we liave

(13)

3. PERTURBATION OP RESONANCE STAT38

Let us consider the wave equations of the A nucleons at energy

Q for the unperturbed Hamiltonian H = T + V, and at an energy £ for

the perturbed Hamiltonian H = T + V, namely,

(14a)

-4-

*"•" : .V|j. '**' "*''' ""•

where

V = V + 3

(15)

we assume that V and V, are hermitian and of the same order of mag-

nitude, while X is a real parameter satisfying the relation

(16)

From now on, the notation ~¥ will always be used to designate a

quantity which reduces to P for A = 0.

As for the channel radii, however, we may assume that a = a,

for any fragmentation ct ; should it not be the case, one could always

choose as a common channel radius the largest of the two radii <x. a ,

because th<c__T>hysical quantities are a-independent; hence co E €j

and ^5 3 -a •

Under such conditions, the hermiticity of the potentials

V, V and the expansions (5)» easily give the following Green's relation

(17)

In the last term of this equation, let us consider the inte/jral

A = V-P P CL-^J when the channels c and c* are not identical. If' theJ I tV

spin and angular quantum numbers of channels c and c1 are not equal,then A = 0 because of the orthogonality properties of the factors

which in cP CP correspond to the spin and angular co-ordinates.It |c'

~T

Hence, when ok c', A is different from zero only when the spin and

angular quantum number are identical in c and c', but when the

excitation of the fragments is different in c and c', i.e., -when

simultaneously o^ f- oL^ and ^ lfl C*. . Under such conditions,

the non vanishing A are proportional to

This product 'is of the order of A , since each of its factors is of the

order of "X . Hence, neglecting only second order terms in A , eq.. (16)

simplifies into

(19)

Up to now, this relation is valid at any enert y ^ and at any energy

From now on, let us however assume that * \ T : are the wave

functions "belonging respectively to a resonance located in the left-

hand half of a complex k-plane and to its perturbed symmetri o

the right-hand side:

• (20b)

Accordingly, we must also substiimta for %> and & respectively *

We neglect the difference J.I -4! ; , see end of this section,

(21b)

For any channel d, these energies satisfy the following equations

(22b)

the former being equivalent to /

As a consequence of the relation (16), let us now assume

that

F = F +- XF1 + OCX1-) } (23)

when P is any1 of the following quantities;

Under such conditions, neglecting only second order terms in X , eqa,

(19) and (22) give

L -L ) . (24)

—7—

where

Assuming that kCn ^ 0, we have

4

where

U7)are respectively the unperturbed and perturbed internal energies of

the fragments in channel c.

Finally, introducing as in the general theory ' the a-

independent quantity V defined according to

* (28)

the eq. (24) is easily given the following form:

In general, the right-hand side of eq. (29) is a complex

quantity and i t s real and imaginary parts, respectively, give the

perturbation of the position and total width of the resonance accord

ing to

- 8 -

(30a)

For a given separation of the configuration space into in-

terior and exterior regions, eq, (29) must clearly be interpreted

as follows: The first term gives the contribution of the perturbation

of the potential in the interior region;, the second has in each

channel, contribution proportional respectively to the perturbation

of the internal energies of the fragments and to the perturbation of

their Coulomb repulsion. In principle, however, such an individual

interpretation of each term should be considered as qualitative only,

since these terms are dependent_on the channel radii. Only their sum,

i.e., the first order in X of G K — G ,is exactly a-independent, as

will be proved in the Appendix._ the lar^nt part o.r

This is the term in & ltd c ~ l«t c which contains* the effect

of the proton-neutron mass difference when V, is a charge-exchange

operator. In the latter case)however, we have completely neglected

the much smaller effect of the proton-neutron mass difference on the

reduced mass of the fragments in a channel. It is easily evaluated

when eq. (14b) is written with a kinetic energy operator T, rather

than T, Then, in the last tern of eq. (17), Mc# should read I'IC> and

henco in eq. (24) the factor ( L» h~ -c*i)/Me i-s ^° e replaced ~bj

L /W — L / M . Finally, an extra term appears in the right-

hand side of eq. (29), namely )

l(31).(31)

where, for light mirror nuclei, (Mt-Mcl/Mt i s °? ^ e order of 10

or less (10 for the application given in Section 5).

-9-

4. ALTERNATIVE FORK OF TIIE PERTURBATION FORMULA

At an energy5 > a channel c iB said to be open (o+) or

closed (c~) according to '•?)

crv(c open)

(32a)

I = - closed)

(32b)The radial factor of x in channel c, namely u (r , k ), where X > at

being proportional to exp (ik£<ftrc)» is exponentially increasing in

an open channel and exponentially decreasing in a closed one.

Accordingly, because of the a-independence of /theintegration in eq. (29) can be extended to the region It composed of

the interior region a) and of the exterior region of the ohanne3s which

are closed at the energy B^: • ."

<:*)

(33a)

where

tx (33b JThese expressions are readily rewritten in terms of the ob-

servable partial widths P° and real phases Q defined according"<• * tn en

to wen

(34)

o-10-

4

and

2© =c-n CYL

(35)Introducing also the notations

£

(36a)

we have

TV

JSI en

, 5 , . . , ^ , (37a)

. ( 3 T b )

'*'• When, in p a r t i c u l a r , a l l the channels are olosed a t the

energy 6 • "we have 5 < 0 for a l l c and P a 0 (see r e f . 6 ) ) . Thenctv

to i s neoesaarily a "bound state and K" a 0 for a l l o. HenceU c-n.

-TV

and the eq,s. (37) reduce to the Schrodinger perturbation

A'K flft (39)

11

- 1 1 -

T •

5. APPLICATION

The application of the results of Sections 3 and 4 to specific

nuclear problems requires in principle the determination of \ f in the

interior region of the configuration space and that of the radial

factors <p . The l a t t e r could also be deduced from the observed part ialre. *widths I and phases^ *) given by an S-matrix analysis of the

experimental data, but, at present the available analyses are not

sufficiently complete.

Under such conditions, the applications made so far of the

above general results have been completely derived from a much simpli-

fied theoretical model. They are concerned with the computation of the

energy shift of 13 levels of the following pairs of mirror nuclei:5 He- s Ll /LC-" ' e>e / B e - 9 b / i C - i a t f , 1 7 0 - 1 ' F , under the assumption

that their wave functions can be approximated by a single-particle or

one-channel model. These applications will be found in the thesis of

one of the authors (G.L.) • ' ' and, hereafter, we only briefly describe

the model and give some numerical results for the C - N pair .

Let G be the unperturbed nuclei and let us consider i t s

ground state ("i , X = l ) and i t s f i r s t three excited states at

E^=.3.09(i* ,*»<>)> a . t S t i ' ^ i ^ J . S S ^ . i * * ) McV. Disregarding a l l

the channels but one ^^ / .: £+1t,let the wave function be that of a

square well of range a and depth V = *n ^ / ( ^ . . A which is •

determined for each level in order that i t s energy E = t\ %

sat isf ies the boundary condition (22c), namely

\k d Q^

(40)

where \k and Q^. are the usual spherical Bessel and Hankel functions;

the origin of the energy scale is taken at the neutron threshold, i.e.,

at 3 - 4«95 MeV. The range a is given three different values,

The phases Q are related to the phases of the expansion of

the collision matrix according to the relation

* » * >

-12-

namely, 2.75, 3.00, 3.25 fm, corresponding respectively to ra » 1.20,

1.31, 1.42 fm when a = r A* with A =* 12.o

Let us now consider the perturbed nuclei N to beassume that in the interior region (-t<a.), the Coulomb

potential is constant and equal to its value on the boundary r = a.

With our notations of Section 3, we have

>w = 0.312. iU"' (41)

(42)

(43)Under such conditions, numerical computations the details of which

are explained in ref. 9), give for £-6 "the results given in Table I .

Although, except the ground state, the levels of H" are

above the l C + |u threshold (E = 1.94 MeV), i t was a priori obvious

that the one-channel approximation would give a real perturbation

•^-•g for eaoh level. Indeed, in this approximation, we are dealing"*• ^ - u *

with real * when the unperturbed states are bound rather than being

proper resonances; hence we necessarily have H^sO when ("^ = o .

It is also true,however,that in the present case the measured total

widths, namely ^rn= 31»55>61 keV, are muoh smaller than the shift of the

positions of the levels given in the last column of Table I .

Let us also notice that obtaining different results for

different channel radii does not violate the general property of a-

independenoe of \-&ni this is only related to the fact that the per-

turbation potential V - V is different for each value of the range a,

as indicated by eq. (43).

6. COUCLUSIOHS

The results we have obtained still call for a few remarks.

As far as the general formulae of Sections 3 and 4 are conoerned, they

have been deduced from a systematic approach of the perturbation

-13-

problem, rather than from an often imprecise distinction between large

and small effects; they are a-independent and valid for resonance and

bound states as well. In contrast with other results •"'on the same

problem, our formulae are explicit in & -c and they are valid for broad,

non-isolated resonances as well as for narrow, isolated ones. Finally,

one should also notice that it is the partial widths, rather than the

reduced widths Ji \ which have been introduced in eqs. (37); since

this is unrelated to any expansion in terms of poles, it brings a new

argument to the fact that the partial widths have in general a more

fundamental character than the reduced widths

Turning to the applications1 reported here in Section 5 and

in the detailed paper to come 9)? it is clear that we cannot claim

that the quantitative agreement is very good. But there is, however,

no point in choosing for ja the range which gives the closest results

to the experimental data, because of the crudeness of the modeli we

completely neglected the spin and electromagnetic effects, besides the

second-order terms in the perturbation. Moreover, a consistent com-

parison between,theory and experiment should also imply an S-raatrix

analysis of the experimental data. Finally, let us also point out

that,concerning the range a, our results suggest that it should bet

a » 1 A* as in the shell-model calculations, rather than the con-

ventional channel radius a = r (A3 + A3 )» at least when A = 1 .

The authors thank Dr. C. Mahaux for many valuable discussions

and Professor E. Ricoi for useful comments.

One of the authors (JH) is also grateful to Professor Abdus

Salam and the IAEA for the hospitality extended to him at the International

Centre for Theoretical Physics, Trieste,

-14-

APPENDIX

Let us prove here that the approximations which have been

introduced in the derivation of eq. (29) led us to an expression

which is rigorously independent of the channel radii, just like the

exact eigenvalues G^ g^ of eqa. (14). The a-independence of "^ having

been proved elsewhere p\ only the a-independence of the factor

following 1/0 in eq. (29) remains to be proved here. Since this is

an expression -which is completely symmetrical with regard to the

channel index, this can be done by increasing the radius of only one

channel in the right-hand side of eq. (29) and verifying that JL - ~&

remains unchanged. Let this channel be o_ and let CO be that part of

the configuration space in channel c which is bounded by the surfaces

t "CL and t^al' with tn > <u .Under suoh conditions. £ - £ will

remain unchanged if the following relation is satisfied:

U {tit) c c * c * » 7 c / J(44)

On the one hand, because of the very definition of a

channel, in CO we have

where (b (A. )/ Q (fl ) is in fact the a-independent amplitude of a

purely outgoing wave; hence

(47)On the other hand, recalling the notations (36) and multiplying

both sides of eq. (44) by the quantity 0 / W the latter

equation reads , f

(48)

Combining the following equation

(49)with those resulting from its differentiation according to k

£1*.

and *Y\ we get

(50b)

Introducing these two relations into the left-hand side of eq* (48),

one sees immediately that the latter equation is indeed an identity.

-16-

LEVELS

•

Ground

state

First

excited

state

(i.= o)Secondexcitedstate

third

excited

state

Eex('3C)

(MeV)

0

3.086

3.686

3.85

EXPERIMENTAL DATA

E ( < 3 l Oexv '

(MeV)

0

2.367

3.508

3.555

0.471 i

0.289 i

0.236 i

0.222 i

TABLE

(MeV)

1.7H

0.990

1.541 .

1.421

I

THEORETICAL R33ULT3

a

2.753.00

3.25

2.753.00

3.25

2.753.00

3.25

2.753.00

3.25

(tteV)

1.6271.400

1.201

I.O84. 0.931

0.816

1.5511.3291.102

1.3341.2311.077

Experimental data and theoretical results on the energy shift of the first four levels ofthe mirror nuclei ^C -^Jf . '

REFERENCES

1) J . B . EHHKAHN. Phys. Rev. 81,412 (1951)

2) R.G. THOMAS. Phys. Rev. 88, 1109 (1952)

3) H.A. ¥ELDEMULLER. Hucl. Phys. jS9, 113 (1965)

4) J . HUMBLET and L. ROSEHFELD. Nucl. Phys. _26, 529 (I96l)

5) J . HUMBLET. Nucl. Phys. j j l , 544 (1962)

6) J . HUMBLBT. JTuol. Phys. j>0, 1 (1964)

7) J . HDMBLBT. Nuol. Phys. _5J, 386 (1964)

8) J . HUMBLET and G. LBBON. Journal de Physique, 2£, 885 (1963)

9) G. LEBOU. Thesis, University of Liige, '1966j to be published

in Mem. Soc. Roy. So. de Liege

10) J . HUMBLET. Lectures on the S-matrix theory of nuclear resonance

react ions , Trieste,1966; to be published by the IAEA, •

Vienna,1967, in Proceedings of the Internat ional Course

on Nuclear Physios

11) I . TALMI and I . tTNBA. Annual Review of Nuclear Sciences

^ 0 , 353 (I960)

P . GOLDHAMMEH. Rev. Mod. Phys. ,35., 40 (1963)

-18-

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