Coupled channels calculations with complex eigenvalues (II)

Nuclear Physics Al64 Not

(1971) 246-256;

@

North-Hoilond Publishing Co., Amsterdam

to be reproduced by photoprint or microfilm without written permission from the publisher

COUPLED

CHANNELS

WITH COMPLEX A. LEJEUNE

t AND

CALCULATIONS

EIG~~ALUES M. A. NAGARAJAN

(II) r

Theoretical Nuclear Physics, University of Lidge, Sart Tilman, 4000 Likge I, Belgi’iron

Received 21 December

1970

Abstract: A soluble coupled square-well model is analysed by the use of Kapur-Peierls dispersion theory. The dependence of the complex eigenvalue on the channel radii is studied and a unitarized one-level approximation to the S-matrix theory is suggested. It is shown that the lack of unitarity is confined only to the elastic scattering channel. A comparison between the predicted and exact inelastic scattering cross section is made with a model with three coupled square wells.

1. Introduction In an earlier paper ‘) (to be referred to as I), we had studied the validity of a onelevel approximation to the S-matrix in the framework of the Kapur-Peierls dispersion theory. We had considered a model of coupled square wells and adjusted the parameters till we obtained a narrow resonance. The narrow resonance was associated with a bound state in the closed channel whose width was proportional to the weak coupling to the open channel. It was chosen so that there was a strong correspondence between the exact pole of the S-matrix and the Kapur-Peierls eigenstate over a wide range of the strength of coupling. It was shown that the variation of the eigenvalue with the channef radii was very small and that this made for the desirabil~~ of the Kapur-Peierls theory in the analysis of narrow resonances. It was found, however, that the one-level approximation to the S-matrix was violently non-unitary, We had argued that this lack of unitarity arises from an improper description of the elastic scattering state and hence would be confined to the elastic channel. In this paper, we make a systematic study of the dependence of the eigenvalue on the channel parameters. In sect. 2, we modify the theory to allow for an independent variation of the two channel radii, and derive a unitary S-matrix. We consider the contribution of various levels to the background giving rise to the unitary S-matrix. In sect. 3, we study the model of three coupled square wells at energies below the second inelastic threshold. We thus have the elastic and one inelastic channel open. We study the behaviour of the cross sections for different strength of coupling between the square wells. t Chercheur I.I.S.N. 246

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CALCULATIONS

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247

2. The model 2.1. THE S-MATRIX

The model is identical

to the one studied in I. The Schrodinger

g$ where 2 is a unit matrix,

+E)

h-y.]

e is a diagonal

equation

is

(2.1)

YE(r) = 0,

matrix with

eij = eiaij, where e, is the threshold model assumes that

(2.2)

energy for the ith channel, Vii

and W” is a square matrix.

= Vij

(J” I R)

= 0

(r > R),

where Vij is a constant. The matrix YE(r) is a column matrix given by

The

(2.3)

V is real and symmetric.

The exact solution

(2.4) where the functions the origin and that

!@(r)

$2)(r)

=

satisfy the boundary

(Vi)-3[eXp

for r 2 R, where Vi is the velocity

conditions

(-ikir)dil-Si,

exp

that they are regular

(ikir)],

(2.5)

in the ith channel vi = hk,/M,

and ki is the wave number

at

in the ith channel

(24

given by

ki = [2M(E_ei)/h’]‘.

(2.7)

The numbers Si, are the elements of the scattering matrix. The Kapur-Peierls eigenstates are solutions of the equation

[(& -$cb.)

I-e-V]

where Gn, E(r)

is a column

d&(r)

=

0,

(2.8)

matrix,

(2.9)

A. LEJEUNE AND M. A. NAGARAJAN

248

where the functions qSlf)r(r)are regular at the origin and obey the boundary conditions dp!?Ar) -&I,=,$ = ik,cp$jaJ.

(2.10)

We have allowed for the possibility of different channel radii in eq. (2.10). One can show that the functions a>&r) belonging to different eigenvalues cY~are orthogonal. We choose to normalize them such that the orthonormality relation reads i~~~~~~~(~)~~~~($)

= &I, f

(2.11)

If the eigenvalue cY,,is written as G, = E,-*if*,

(2.12)

it can be shown that (2.13)

where there are M open channels. It can be noticed that the direct contribution to the width arises mainly from the open channels, while the closed channels contribute to the normalization integral in the denominator. Expanding the exact wave function Y, in terms of the Kapur-Peierls eigenstates @n,E, one can obtain the S-matrix, given by (2.14) where the width amplitude yc)Eis defined by r!f’E = (~Zki/M)f4D~‘E(ai). 2.2. NUM~RrCAL

EXAMPLE

(2.15)

OF TWO COUPLED SQUARE WELLS

We shall consider the same values as in I. The square wells have a range of 6 fm and their depths are V,, = -31 MeV,

V,,

e, =o,

= -41

Vi, = -0.1 MeV, MeV, e2 = 6 MeV.

The S-matrix has a pole at the energy (3.05 -&i 0.00388) MeV. The Kapur-Peierls

“resonance energy” was obtained as (3.05 -+i 0.00542) MeV,

COUPLED CHANNELS CALCULATIONS

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249

when both the channel radii were chosen to be equal to 6 fm. The width obtained is about 40 y0 larger than the exact width. However, the energy dependence of the eigenvalue compensates for the discrepancy. For instance, if we make a one-level approximation to the S-matrix, S,,(E)

and c?,, is energy-dependent,

z e-2ik’a*

(l+i E)

(2.16)

i.e. (2.17a)

and (2.17b) we can rewrite S1 1(E) as S,,(E) = ewzikiai

(2.18)

l+i

The observed resonance energy ER and the width fR have to be compared with the real and imaginary parts of the quantity (E, -$zT,)/(l f CI-J&). In fig. 3 of I, the energy dependence of the eigenvalue E,, is shown. The imaginary part r, is almost energy independent while the real part E,, was linearly dependent on energy with a = 0.397. Hence the predicted width is ra = r,J(l+c1)

= 54211.397 = 3.88 keV,

which agrees with the exact width. In the R-matrix theory, one finds the same effect from an energy dependence of the level shift. In I, we had also studied the dependence of the eigenvalue on the channel radii, where the channel radii were kept equal. It was found that the real part E,, was independent of the radii while the imaginary part decreased with increasing radii. In fig. 1, we show the same except that we keep the open channel radius constant at 6 fm, and vary the closed channel radius. It is seen that for a value larger than about 12 fm for the closed channel radius, the width r, attains the minimum value of 3.88 keV, and remains constant thereafter. From the discussion of the renormah~tion of the width due to the energy dependence, this implies that if the closed channel radius is varied, the energy dependence of E,, should also vary until eventually c1=0 for radii larger than 12 fm. In fig. 2, we show the variation of the energy dependence for different values of the closed channel radius. It shows the constancy of the predicted width. The eigenvalue &,, was found to be independent of the open channel radius a,. In I, it was noticed that the one-level approximation (eq. (2.16)) was non-unita~. It was seen that the reduced width $2 was related to the r, through the relation

250

A. LEJEUNE

1

6

AND

M. A. NAGARAJAN

I

I

I

I

I

I

I

7

6

9

10

11

12

13

Fig.

1 I Dependence

of En and r.

on the closed

channel

I oq(fm)

radius

14

a,.

:1-w c

(3)

(2)

(1)

30. 3.0

3.05

E(MeV)

3.1

Fig. 2. Variation of the energy dependence of En with the closed channel radius az. The values are a~ = 6 fm for (I), 8 fm for (2) and 10 fm for (3). For values of az larger than 10 fm, E, is energy independent.

for narrow resonances. It was the fact that the phase 4, was non-zero which caused the non-unitarity of the S-matrix. If the lack of unitarity is due to the elastic channel, the phase 4, must be dependent upon the open channel radius al. In fig. 3, we show the dependence of the phase 4, on the radius a,. The phase (p, was found to be independent of energy in the region 3 to 3.1 MeV. It was also found to be independent of the

COUPLED

closed channel

CHANNELS

CALCULATIONS

251

(II)

radius a,. Fig. 3 shows that 4. has a linear dependence

of a, and be-

comes zero for a value of a, = 11.44 fm. Hence, if we choose a, = 11.44 fm and a, larger than 12 fm, we would obtain a unitary S-matrix with energy-independent resonance energy and reduced width. These parameters could be identified with the Smatrix parameters of Humblet and Rosenfeld 3), except for the weak energy dependence hidden in the background phase shift (k, al). The fact that the phase 4, could be reduced to zero for an appropriate choice of the channel radius a, indicates that one has to seek for the correct background contri-

Fig. 3. Variation of A, the phase of the reduced width, with the open channel radius al.

bution

to the S-matrix.

In the case of Kapur-Peierls

theory as well as the S-matrix

theory of Humblet and Rosenfeld, the background term is necessary to obtain a unitary S-matrix. From the discussions above, it can be seen that one can very simply obtain a unitary one-level approximation by writing eq. (2.16) as “) (2.16’) where the implication is that the distant levels contribute a term of the type i( 1 - ie”‘?‘“) to the S-matrix. In fig. 4, we show the comparison between the unitarized cross section and the exact cross section. The number of such levels necessary to obtain this contribution defines the unitarity correlation length. In order to study how many levels we need to include to obtain the unitarity, we looked for the neighbouring Kapur-Peierls eigenstate. This was observed to be at an energy of 4.648 MeV with a width of 6.83 MeV. This level corresponds to a pole of the S-matrix for scattering of a particle by

252

A. LEJEUNE

AND M. A. NAGARAJAN

the first square well I’,,. It is very weakly affected by the coupling to the second square well and has a large partial width in the elastic channel. Hence, its width is very sensitive to the open channel radius. To calculate its contribution to the Smatrix, around

we have to calculate its eigenvalue in the vicinity of the sharp resonance, i.e., an energy of 3.05 MeV. The eigenvalue obtained was (4.55 -4i 6.14) MeV 1

Ia’

L= 1.25-

loo-

\

0.50-

\

\

‘\ \

‘.

I ---w

N.

.,__I2

I I I

'\ '\

I

025‘\

' \ \ \ \ \

I 3025

I 31

3075

E(MeV)

Fig. 4. Comparison of the one-level Kapur-Peierls approximation with the exact elastic scattering cross section. The full curve is the exact cross section, the curve marked (2) is the non-unitarized onelevel cross section and the curve marked (1) is the unitarized one-level cross section.

and the width yi corresponding C$ M 0.02 radians). If we write

to this state was found to be almost real (with a phase

e2i4i3

- l+iYR, _

&,-E where &R is the energy of the resonance state, we obtain a value of 1.01 for &. This compares very well with the value 1.14 for 4,, the phase of yi. Thus the main contribution to the background matrix comes from the single broad state in the neighbour-

COUPLED CHANNELS

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253

hood of the narrow resonance. This is reasonable because the next closest neighbour occurs at about 37 MeV. Lejeune and Mahaux “) had analysed the same model using R-matrix theory. All the above results agree strikingly well with their conclusions. They had found also that by a suitable choice of the boundary condition in the open channel, they could cause two of the levels to come close to each other and thus obtain a fit to the cross section with a two-level approximation. They chose the two channel radii to be equal to 6 fm, the boundary value parameter in the closed channel to be -2.262 and that in the open channel to be 1.290. The two R-matrix levels were located at 2.959 MeV and 3,153 MeV respectively and their partial widths were all of the same order of magnitude. We used the expression of the S-matrix in terms of the level matrix “) S, = Bi

nj[6ij

+

i C Anp J?$i J..P

rij],

(2.19)

where 52, = exp(-&a{)

for open channels,

(2.20) (2.21)

r$ = (2k,Uj)fu,

(2.22)

A = B-1, where 3 A&

=

(E~-E~~~~-i~l~Y~iY,i,

L!j zz -BP + ik, ai, = -Ikilai-Bi,

for open channels for closed channels.

(2.23)

In the above equations yzi is the reduced width amplitude of the Ievef ;kin channel i, defined by (2.24) YV = ~~2~2~ui)~~~i(~j)* One can diagonalize the matrix A and obtain a form for the S-matrix similar to the Kapur-Peierls expression (2.16). We considered the two-level, two-channel case and obtained the eigenvalues c”%= (3.0493 -_liO.OOS) MeV, 82 = (4.538-4i6.14)

MeV,

corresponding to the narrow and broad Kapur-Peierls eigenstates, The agreement is good. However, one should be careful when considering such a transformation between the R-matrix and Kapur-Peierls theory. An orthogonal transformation to eq, (2.19) will obtain a unitary &‘-matrix. Bence, it should be compared to a unitary version of the Kapur-Peierls theory, It appears as if one obtains extremely good resonance energies in the Kapur-Peierls theory at the expense of unitarity. One would not thus normally expect a simple one-to-one correspondence between its eigenstates and the R-matrix eigenstates.

254

A. LEJEUNE

AND M. A. NAGARAJAN

3. Model with three coupled square wells To study the prediction of inelastic scattering cross section, we considered a model with three coupled square wells. The model is defined by eq. (2.1). The square wells were assumed to have the same range. The parameters used were the following: VI1 = -31 MeV,

V,, = -40 MeV,

e2 = 1 MeV,

V,, = -41 MeV,

e3 = 6 MeV.

The radii of the square weUs were 6 fm. The coupling interactions V13, YI, and V,, were varied over a wide range of values. In the calculations for the cross sections, a

Fig. 5. Same as fig. 4 in the three-channel model. Only the unitarized one-level approximation is shown. The parameters are VI 1 = -31 MeV, Vzz = -40 MeV, Vs3 = -41 MeV, VI, = 0, VI3 = Vz3 = -0.3 MeV.

unitarized form for the elastic scattering cross section was used. The normalization factor

calculated was never different from unity by more than 4 y0 even in cases where the resonance width was of the order of 100 keV. This was an essential requirement for the predictions of the inelastic cross sections.

COUPLED

CHANNELS

CALCULATIONS

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Since the agreement between the exact cross section and the one-level KapurPeierls calculation was extremely good, we show only a few of the results. In fig. 5, we show the elastic cross section in the case where there is no direct coupling between the two open channels and the coupling of the two channels to the closed channel is given by V,, = V,, = -0.3 MeV. In fig. 6, we show the comparison between the predicted and exact inelastic scattering cross sections. The agreement in both cases is extremely good.

21 L”

,

,

,

3.0 25

3,050

3.075

3.1 IO EtMeV)

Fig. 6. The comparison for inelastic scattering cross sections. The parameters are the same as in fig. 5.

We shall merely quote the effects of the different coupling terms on the resonance parameters. The effect of direct coupling is to make the resonance width narrower. If the coupling between the inelastic scattering channel and the closed channel is increased, the resonance is shifted to a larger energy and becomes wider. On the other hand, if the coupling between the elastic scattering channel and the closed channel is increased, the level shift is negative and the resonance becomes wider. 4. Summary and conclusions In sect. 2, we analysed the dependence of the Kapur-Peierls resonance parameters on channel radii. It is seen that the resonance energy and width are independent of

256

A. LEJEUNE AND

M. A. NAGARAJAN

the open channel radius. They are sensitive to the closed channel radius. As the closed channel radius is increased, the difference between the predicted width and the exact width decreases and eventually vanishes when it takes on a large value. At the same time the energy dependence of the Kapur-Peierls eigenvalue also becomes smaller and vanishes as the predicted width becomes identical to the exact width. This effect is related to the renormalization correction in the R-matrix theory “). The phase of the reduced width, however, depends only on the open channel radius. By a suitable choice of the open channel radius, the phase could be reduced to zero. It is also seen that this phase is primarily due to one of the Kapur-Peierls states being confined to the open channel, which could be considered as a wide single-particle resonance in the open channel. For narrow resonance, it is seen that one can unitarize the one-level approximation to the S-matrix by an introduction of a background term which is determined by the phase of the reduced width. The unitarized version of the S-matrix predicts an elastic scattering cross section which agrees very well with exact calculations. In sect. 3, we showed the comparison between the one-level approximation to a three-channel model with one inelastic scattering channel open. It is shown that it is sufficient to introduce a background term in the elastic scattering channel and none in the inelastic channel. The agreement between the exact cross section and the onelevel approximation is very good even for resonances of width of the order of 100 keV. The above results indicate that in the analysis of isolated resonances, it would be useful to apply Kapur-Peierls dispersion theory in the unitarized form. The energy dependence of the resonance eigenvalues is more than compensated by the fact that in the case of narrow resonances, the contribution from distant levels can be very well approximated by a knowledge of the phase of the reduced width. In addition, it appears as if the “dependence” of the parameters on the radius is compensated by the appropriate energy dependence of the resonance parameters, so that one can make an arbitrary choice of the radii. We thank Professor J. Humblet for his kind interest. References 1) 2) 3) 4) 5) 6)

A. Lejeune and M. A. Nagarajan, Nucl. Phys. A154 (1970) 602 P. L. Kapur and R. E. Peierls, Proc. Roy. Sot. Al66 (1938) 277 J. Humblet and L. Rosenfeld, Nucl. Phys. 26 (1961) 529 C. Mahaux, Bull. Roy. Sot. SC. Liege 32 (1963) 62 A. Lejeune and C. Mahaux, Nucl. Phys. A145 (1970) 613 A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 259