R-matrix description of threshold phenomena

HAL Id: jpa-00208400

https://hal.archives-ouvertes.fr/jpa-00208400

Submitted on 1 Jan 1976

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

R-matrix description of threshold phenomena

C. Hategan

To cite this version:

C. Hategan. R-matrix description of threshold phenomena. Journal de Physique, 1976, 37 (2), pp.45-

47. �10.1051/jphys:0197600370204500�. �jpa-00208400�

45

R-MATRIX DESCRIPTION OF THRESHOLD PHENOMENA

C. HATEGAN

Institute for Atomic

Physics, Bucharest,

^Romania

(Rep

^{le 27 mai}

1975, accepte

le 13 octobre

1975)

Résumé. ²⁰¹⁴ On présente ^une description ^des

phénomènes

de seuil à l’aide de la matrice R, ^en considérant, ^à part ^le cusp de Wigner, les anomalies

produites

^{par une} résonance dont

l’énergie

correspond ^au seuil. On discute aussi les phénomènes ^{de seuil} concernant les ondes s et p.

Abstract. ^- A R-Matrix

description

of threshold phenomena ^is

presented.

^It includes, in addition to the

Wigner

cusp, anomalies caused

by

^{a resonance at the} threshold. Threshold phenomena ⁱⁿ

s- and p-waves ^are discussed.

LE JOURNAL DE PHYSIQUE TOME 37, ^FtvRiER 1976,

Classification

Physics ^Abstracts

4.305

The

theory

of threshold

phenomena

^{in nuclear}

reactions was first

developed by Wigner

in framework of R-matrix

theory,

e.g.

[1].

^He

obtained,

^{in the zero}

energy limit

k. -+ 0,

^for the collision U-matrix the

following

^formula

Here a and b denote _open channels of the reaction system ; n ^{is a} neutral electrical channel which _opens at

kn

⁼

0 ; U°

is the collision matrix at the threshold

(E.

⁼

kn

⁼

0).

Because of the behaviour for the s-wave,

Uan ~ k n 1/2 , d6ab/dEn

^is infinite in the limit

En --> 0,

and hence the name cusp is used for such threshold

phenomenon.

In the above

treatment

no resonance near the threshold of the n-channel is assumed. In this

letter,

we expose ^a R-matrix

description

^{of threshold}

pheno-

mena which is a

generalization ^{of Wigner}

^cusp

^theory.

It

includes,

in addition to the cusp

phenomena

^ano-

malies caused

by

^{a resonance at} the threshold.

The collision matrix can be

expressed in

^{terms of}

R-matrix functions

by

^{the formula}

where bi

is real and

positive,

^{P the}

penetrability

^matrix

and I Wi I

⁼

1, (for l = ^0,B~ k’ /2

^and

where a is the channel

radius).

^{In the}

following

^we

will work with the U-matrix defined

by

which is

unitary

^{because 0 and uIL are}

unitary.

^We

split U,

B and R matrices in the form

where the first N x N matrices refer to the _open channels of the

system

^and

U.,

rn,

bn

^{refers to the new}

n-channel. The

coupling

^between

^the

^N open channel

system

^{and the new} n-channel is

represented by II Uan II

^{= u matrix}

elements;

^{it is due to r}

and,

^for

direct

reactions,

^{to modified}

boundary

conditions

[2].

The row matrix

uT(rl)

^{is the} transpose ^{of the}

u(r)

column matrix. The

components

^{of the U-block} matrix are

In absence of the

coupling

^between open channels and the new

n-channel,

^{ie. u = 0 or r =}

0,

^{we obtain} the collision matrices for N

independent system

^{and n}

independent

^{channel :}

Obviously

the threshold contribution is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197600370204500

46

where a new

Q

^{matrix is} introduced

by

^formula

The evaluation of

d UN

is reduced to calculation of

Q ;

in order to do this we represent q ^{and A as}

and after some

calculations,

^{we obtain}

Now

by using

^the

identity (P, qp) p = (P

^x

p.q) p,

we obtain another form for the column matrix ^{u :}

The

following

final result is obtained for the threshold contribution

Because _eq.

(10)

^was derived with no restrictions

on k

dependence

^of the U-matrix

elements,

it is valid not

only

^{in the}

vicinity

of the threshold

(like

^eq.

( 1 )),

and even if there ^are resonances in the _energy interval considered. In the

following,

^{as a}

specific application

of the result

(10),

^{we establish} necessary conditions for

experimental

observation of threshold

phenomena

in s- and _p-waves.

Let us consider that the resonances occur far _away from the

threshold;

^{then at E =}

Ethr,

^{the R-matrix}

terms r, rn, ^{r are}

monotonically

energy

dependent, while b’ - k;’+ 1,

^and

consequently Un -+

^{1. It results}

for the threshold contribution

This is the

Wigner

cusp

phenomenon (1) ;

it is difficult to observe it

experimentally

^{even for s-waves because}

of its small value.

A _necessary condition for

experimental

^obser-

vation of a threshold

phenomenon (AUN :0 0)

is

Un -+ -

1 in order to

compensate (in

^formula

(10))

the

lowering

^{value of u x u}

(u

^-+

0,

^for

k,, -+ 0).

This condition means a resonance at the

threshold;

in the

following

^we discuss this condition for

compound

nucleus and

single particle

resonances.

A

compound

^{nucleus resonance located at the}

threshold of the

n-channel,

means r, rn,

r - kj ^2.

^This

implies

^{in limit of narrow}

resonances,

Un

- >- 1 +

2 i(i

^{+ const.}

k n 21-1)-l ;

^for

1 ⁼

0, Un -+ - 1,

while for I >

1, Un

^{-+ 1. It results}

that a

compound

^{resonance located at} the threshold energy, induces a non-zero threshold contribution

only

^{for s-waves.}

Now we consider

single particle

resonances, which

can be

incorporated

in the above formalism

by using

the Bloch

procedure [3],

^as described in

[4].

^The

single particle

resonance case is described

by

the R matrix

where K is the hamiltonian of the system,

split

^into

(Ko : single particle hamiltonian ;

^{V :} direct interaction

potential),

^and

(Qa :

^{channel wave}

functions ;

_{Mp :} ^{radial wave}

func-

tion of a set of

single particle

^{states of}

^nucleon,

defined for the

potential Vo

^of the hamiltonian

Jeo

^{= T +}

Vo, subject

^to

given boundary conditions).

Now

expanding (JC 2013 E) -1

ⁱⁿ powers of V

1.

and

assuming

^{in the n-channel}

only

a p

single particle

resonance, it results for RI and R matrices

R °aa :

^monoton energy terms;

An identical result could be obtained

by using Wigner procedure

^to include direct reactions in R-matrix formalism

[2].

^At every level of truncation of R series

(except

^{the first one, which means no direct}

coupling

^of

channels),

^{we obtain}

Un

^{-+ - 1 for}

I ⁼ 1.

(This

truncation ^conserves the U-matrix uni-

tarity

and this is a vital condition for treatment of threshold

phenomena.)

^{So the}

explanation

^{of a}

threshold

anomaly

in p-wave

requires,

in addition to a zero energy

single particle

^{resonance in the}

n-opening channel,

^a direct interaction

coupling

^{of the n- and a,}

b-channels of interest. In this framework ^we could

explain

^the

(d, p) anomaly

^{observed with}

targets

A ~

90,

^{at the threshold of the}

(d, n) analogue

channel;

for A N 90

nuclei,

^a p-wave

single particle

47

resonance occurs at zero energy

[5].

^{Also the most}

experience,

^{is that an} observable threshold

pheno- significant experimental

^{data from}

low-energy

^{nuclear menon}

requires

a resonance at the threshold of the

physics

^are connected with a resonance at the threshold

opening

^{channel : a narrow}

compound

^{nucleus one}

of the new n-channel

[6].

^{for s-waves and a}

single particle

^one and direct The

conclusion,

^{which seems to be}

supported by

interaction effects for p-waves.

References

[1] WIGNER, E. P., in Dispersion Relations and their Connection with Causality, ^Course XXIX, Proc. of the Enrico Fermi Int. School of Physics, Varenna, Italy; ed. E. P. Wigner (Academic Press, ^N. Y.) 1964, p. 40.

[2] WIGNER, ^E. P., ⁱⁿ Group Theory ^{and its} Applications, ^{ed. E.}

Loebl (Academic Press, ^N. Y.) ^1968, p. 119.

[3] BLOCH, C., ^Nucl. Phys. ⁴ (1957) ^503.

[4] LANE, A. M. and THOMAS, R. G., Rev. Mod. Phys. ³⁰ (1958) ^257.

[5] MOORE, ^{C. F. et al.} Phys. ^{Rev. Lett. 17} (1966) ^926.

[6] HATEGAN, C., ^{Ph. D.} Thesis, unpublished, ^1973.