On the factorization of the wave function and the green function in the region of isolated poles of the S-function

Nuclear Physics A104 (1967) 564--576; (~) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmw~thout x~rtttenpermissionfrom the publisher

ON THE AND THE GREEN

FACTORIZAT1ON FUNCTION

OF THE

WAVE FUNCTION

IN THE REGION

OF ISOLATED

POLES

OF TIlE S-FUNCTION H.-J. UNGER t Nuclear Research lnstttute, Moscow State University, Moscow

Received 26 June 1967 Abslract: The Brelt-Wigncr formula for tsolated levels for scattering on diffuse edge potentials is deduced. The most effective choice of parameters necessary for the determination of energy eigenvalues is discussed. We also obtain the factorlzatlon of the wave functton and the Green function of the continuous spectrum in the region of isolated poles of the S-function. In the range of low energy, an approximation of the penetrability and the shift factor for potentials with a diffuse edge are deduced.

1. Introduction

F o r the solution o f some p r o b l e m s in nuclear physics (e.g. configuration mixing with c o n s i d e r a t i o n o f the c o n t i n u o u s spectrum m e n t i o n e d by Balashov et al. 1) a useful factorization o f the energy and the radial d e p e n d e n c e o f the wave function and the G r e e n function is necessary. O f course, this kind o f factorization is only possible in limited regions o f space and energy. O f p r i m a r y interest is the factorization in the region o f isolated poles o f the Sfunction in the c o m p l e x k-plane. As in the R - m a t r i x theory, a set o f energy eigenvalues is defined by b o u n d a r y conditions at a m a t c h i n g radius which are satisfied by the wave function regular in the p o i n t r = 0 on the real axis of the c o m p l e x k-plane. T h e n we o b t a i n an a p p r o x i m a t i o n o f the Jost function in the vicinity o f these levels. It is possible to get all the well-known relations from the R-matrix theory, even w i t h o u t the a s s u m p t i o n t h a t the potential has a cut-off. The level widths thus o b t a i n e d are energy-dependent. W i t h the a p p r o x i m a t i o n o f the Jost function, we get all the factorization formulae. Assuming, that the m a t c h i n g radius lies on the edge of the potential, we find a simple relation between the diffuse-edge potential penetrability and the penetrability o f the centrifugal barrier in the low-energy region. In this p a p e r the results o f ref. z) on the theory of potential scattering are needed. W e use the same n o t a t i o n here. t On leave from the ZfK Rossendorf, near Dresden. 564

S-FUNCTION

565

2. Some general relations from the theory of potential scattering The Schrbdinger equation describing the motion of a particle with spin in a potential, which includes spin-orbit coupling, has the form (E-H)M'

=

0,

(1)

where the Hamiltonian is

H

=

1"$

T + U , ( x ) - -~, U2(x). h"

(2)

After decomposition of the wave function into partial waves, we obtain for each channel J, s

k,

[k s

2 _ v(x)]

k, x )

0,

(3)

2 (X)].

(4)

--

where 2 = l+X2 and

g(x) = 2m_m[ U1 (x) - ½{J(J ÷ 1) - h2

( 2 2 - - &) 4 --

S(S 31-1)} 0

In the following we drop the explicit denomination of the channels J, s. If the potential V(x) fulfills certain requirements 2) (which are not fulfilled by the Coulomb potential) then solutions (p()l, k, x) and f(2, k, x) of eqs. (3) exist with the following boundary conditions: ~0(2, k, x) = x z+½

if x -+ 0,

(5)

f(2, k, x) = e z", k x

if x --+ oo.

(6)

For potentials real for x >__0, the following equations are applicable: [~o(2, k, x)]* -- tp(2*, k*, x),

if(2, k, x)]* = f(2*, - k*, x).

(7)

Furthermore, the following equation is satisfied:

~0(~, k, x) = ~o(~, - k , x). If we suppose, that the potential V(x) satisfies the requirement 2) that an energy range exists, in which the solutions with different boundary conditions exist simultaneously, we can define the Jost function f(;t, k) = W(f, q~) = f ( 2 , k, x)q¢0., k, x ) - f ' ( 2 , k, x)¢p(~, k, x).

(8)

Applying the Jost function we find linear relations between the solutions with different boundary conditions 2). It is possible to represent the Jost function as f(2, + k) = z(2, k)e +'~(~' k)~:,½,(~--~).

(9)

566

H.-J.

UNGER

Then we get for the S-function

.I-(44

S(& k) = ____

e izcn-a, =

erza(n,k)

(10)

f-0, -k)

The solution of eq. (3) is normalized on the b-function of the energy and has the following asymptotical behaviour: te’ia(A*k) sin (kxi-&c(A-+)+@A,

k)),

(111

it can be represented as ke+%

k)

{_&I, k)f(x%, -k))”

cp(4k 4,

(12)

In the following we assume that the angular momentum has only the physical values I = Z++, and that potential is real for x 2 0. 3. holated 3.1. APPROXIMATION

resonance level

FOR THE JOST FUNCTION

We define a set of energy eigenvalues E,, by the condition

Initially there is an arbitrary radius x = a and B an arbitrary constant. Later their exact values will be defined. We decompose the logarithmic derivation of the function q(l, k, a) into powers of (E-E,,), in which Ep is an isolated level of the set En. We take into consideration only the linear term of the decomposition

It is now advisable to define a reduced width y as well as its dimensionless equivalent e2 = ma2y2/h2 ~“(1, k, a) = -

A= = ii&

Y~(A k, a) a q=V, k, x)dx

(15)

s0

Then the Jost function can be represented in the vicinity of Is, in the following manner:

(16)

S-FUNCTION

567

where the level width is defined by

F(Z, k, a)

=

2P()., k, a)72(2, kp, a)

(17)

and the level shift by d(2, k, a) = (B-S(2, k, a))?2()., kp, a).

(18)

We call the expression

ak P(Z, k, a) = [~0.,-k, a)[~,

(19)

the penetrability and the expression S(2, k, a) =

a

d

If(2, k, a)I 2 -da If()., k, a)] 2,

(20)

the shift factor. Taking into consideration (9), we obtain from (16) an expression for the phase shift in the vicinity of Ep 8(2, k) = a r c t g

F

2[e,,+A-E]

+q5(2, k, a).

(21)

The function ~b(2, k, a) is the phase shift for scattering at the potential, which for x > a coincides with the potential V(x) and which is infinite for x ~ a. From the definition (10) and from (16), we get an expression for the S-function in the neighbourhood of Ep S(Z, k) = e i2÷ta' k. , ) E - E o- A - i½r.

(22)

E - E o - A + i)F Expressions (21) and (22) are well known from the R-matrix theory for scattering at a potential with cut-off. 3.2. FACTOR1ZATION OF THE WAVE FUNCTION IN THE CONTINUUM Introducing expression (16) into eq. (18), we obtain

O~f)(2, k,x)=e±,~(;.,k)V

F

"2~[(E-Eo-A)2+¼F2]

l/~ "

a

(p(2, k,x) ~p(2,k,a)"

(23)

The boundary condition of the function q~(A, k, x) is independent of the energy. Therefore the function ~p(2, k, x) is nearly constant in the energy range AE around the point Ep, which meets the requirement hE

IAE[ <<, E p - 2m V(x) .

(24)

~(2, k, x) ~ ~(2, k,, x).

(25)

Therefore we obtain

568

H.-J. LINGER

Condition (24) is fulfilled for broadly spread intervals A E only in the range of strong interaction of the potential. With this approximation, we obtain for the wave function

Og-~(;., k, x) = e +-'`"'k~ 1/

r

:o(Z kp, x)

~[(~-Eo--A)2+¼r =]

. (26)

~2(;¢, k,, x')dx' *

3.3. C H O I C E O F T H E P A R A M E T E R S a A N D B

The structure of the formal theory is independent of the choice of the parameters a and B. The most effective values of the parameters must be chosen in order to get a maximum profit from the formulae. We suppose that the potential V ( x ) has a diffuse edge, i.e. there exists a mean interaction radius R of the potential and a parameter b defining the thickness of the diffuse edge, so that hE I V ( x ) l x < g - b >> IV(x)I~>R+b,

bz < . . . .

2mlVm.xl

(27)

It can be seen that the edge is to be narrow, and the potential converges rapidly towards zero outside the edge. The Woods-Saxon potential is an example for such a potential 2m

V(x)- h2

1%

x-R"

(28)

1 + exp . . . . .

b In refs. 3-s), the choice of the parameters a and B is discussed. In order to separate phenomena which are caused by properties of the potential in the range of weaker interaction, from phenomena which are caused by properties in the range of stronger interaction, it is advisable to choose the parameter a within the limits of the potential edge. The precise position of the radius a in the limits of the edge is less important, because all effects caused by the diffuse edge are taken into consideration. We choose the parameter a corresponding to the condition d2V(a) - 0 da 2

(29)

in the edge range. With this choice of a, factorization (26) of the wave function is possible in the range

X~-~a. It is now clear that the penetrability (19), the shift factor (20) and the hard-sphere phase shift q~(2, k, a) depend on the properties of the potential in the range of weaker interaction (x > a), and the reduced width (15) depends on the behaviour of the potential only in the range of stronger interaction (x < a).

S-FUNCTION

569

Only the level shift depends essentially on the choice of the parameter B. We choose B in such a manner, that the value Ep lies in the limits of the width of the observed resonance Er, i.e. A < F. With B = S(2, k,, a),

(30)

this last mentioned condition is fulfilled. The Er corresponds to the centre of the resonance

E,-Ev-A(2,

kr, a) = 0.

(31)

3.4. REMARKS ON THE CASE OF SEVERAL LEVELS Thus far we have only considered one level in the resonance region. It is possible, that the cross section in the resonance region depends on several levels of the set E,. For the derivation of corresponding formulae, it is useful to define the R-function

(acp'(2, k, a) R(2, k, a) = \ -~(~, lc.a)

B

)-t

.

(32)

It is easy to show 3) that it is possible to represent the R-function in such a manner that ~2

n(;~, k, a) = ~

)'" E ~-E. ,

(33)

where 72 = 7z(2, k,, a). Taking into consideration in (33) only two levels, we obtain, employing (32), the following expression for the Jost function:

f ( ) . -T-k) = f(2, T- k, a)(p()., k, a) (E, - E)(E 2 - E) + (E z - E)(A a -T-½iF,) + (E, - E)(A 2 -T-kit2) X

.

(34)

It is possible to evaluate all interesting quantities in the two-level approximation by using expression (34). Now we can estimate the application range of the decomposition (14). From the requirement that the terms of second order in decomposition (14) are negligible compared with the linear term, it follows by using (32) and (33) that

I - oi >>I z

E. - E,

(35)

n.'# O

if the level Ep is isolated, then decomposition (14) is applicable in a sufficiently broad energy interval.

570

H.-J. UNGER

4. Bound level in the neighhourhood of the threshold

We have derived an expression for the factorization of the energy and the radial dependence of the wave function in the resonance region of the continuous spectrum. Sometimes it is interesting to know how a bound state near the threshold energy affects the wave function in the continuous spectrum. Also in this case it is possible to factorize the wave function. The derivation of corresponding expressions can be obtained as in the case of resonance. We consider only energies satisfying the following conditions:

IEI << IV(x)l,

(36)

x < a.

The bound energy Eo also satisfies these conditions. We choose the radius a as in the case of resonance. Therefore we can write ~2K2

~0(;., k, x) -- ~0(;:, i~, x),

x =< a,

eo . . . .

(37)

2m

The Jost function f(2, - k ) has a vanishing point k = ix (~ > 0). Decomposing the &fference of the logarithmic derivations of the functions ~0(2, k, a) and f(2, - k , a) into powers of ( k - ix), we confine ourselves to the linear term. Then we get with (37)

f(;t, -k) = -f(2, - k,

a)(k-ix)2ix 7(2, i1¢,a)

t~,2I., r|°°402(2, i~c, x')dx'}:".'

t2ma 3o

!

(38)

The reduced width of bound levels is defined by

aq?(L ix, a)

ma 2

02(2, i~c, a) = -~--72(2, i~c, a) 2

(39)

~p2().,iK, x')dx'

Considering only real values for k and introducing expression (38) in eq. (12), we get

07+ ~(~, k, ~) = e * ' ~ " ~

1/- --

_r _

_

~o(2, ix, X)

"2n41E°l(E+'E°J) [f~o~2~2

x < = a " (40)

The width F(2, k, a) in the numerator of the energy-dependent factor in formulae (26) and (40) is responsible for the correct behaviour of these approximations in the limit E ~ 0. The denominator of this factor has a resonance behaviour like the S-function. The normalized wave function contains only the radial dependence. 5. The factorizafion of the Green function

We know the expression of the Jost function in the neighbourhood of the poles of the S-function. Now we easily obtain an approximation for the Green function.

S-FtmCTION

571

The Green function can be represented in the following manner: 2m 1 t f()., - k , x)rp(2, k, y), h 2 f ( t , --k) If(Z, - k , y)q~(2, k, x),

G(2, k, x, y) =

x - r > > y

y -- r >

>

X,

(41)

where 0 < arg k _< }rc. After some transformations (k = k*), we get

G(2, k, x, y) = - 2--m ik rp0l' k, x)rp()., k, y) h2 f()l, k)f(2, - k)

×

(1+"

1

p'0., k,

_Re J'(), k, ,>)])

i- )~(2, k, r>) Lq~(2, k, r>)

f(2, k, r>)J

"

(42)

If the following relation is fulfilled

Im f'(2, k, x)

I

f ( ) . k, x)

]~b(2, k, a)l << arctg - -

] i

q¢(~t, k,x). q~(,~, k, x)

_

(43)

Re f'()'' k, x) f(), k, x).

then the second term in round brackets of the Green function (42) is independent of r>, because it represents the ratio between the real and imaginary parts of the Jost function. Therefore we introduce r> = a in eq. (42), and employing expressions (14) and (24), we get G(2, k, x, y) -

1

E-Ep-A+i-'~-£

q~(2, kp, x)(p()., k o , Y ) ,

x, y =< a.

(44)

fo

q~2(2, kp, x')dx'

Introducing x = a in (43), we obtain the application condition of expression (44) kb()., k, a)l << i arctg 1" I 2(E-Ep-A) - -

--,,,

I I

(45)

i.e. the hard-sphere phase shift has to be small compared with that part of the phase shift caused by the resonance. Approximation (44) is applicable in the limits of the resonance. Similarly we can find an expression for the Green function on the real axis of the k-plane if a bound state exists near the threshold energy. With the condition kb(2, k, a)l << arctg--F

t

I

41Eol . '

(46)

572

H.-J. UNGER

it follows that G(2, k, x, y) = -

i½-F-2lEo[ q~(2,ix, x)cp(;t, i~c,Y), 41Eol(E + leol) fo oq~2(2, itc, x')dx'

x, y _< a.

(47)

Formulae (44) and (47) are useful for the application, since the dependence on the energy and on the x- and y-coordinates is separated. 6. Approximation of the penetrability and of the shift factor for low energies All expressions, which we have obtained, depend on the penetrability and on the shift factor. The quantities are determined by the external (x > a) behaviour of the potential. Tables of the penetrability P(2, k, a) and of the shift factor S(), k, a) do not exist for diffuse-edge potentials. Because there exist simple expressions of the penetrability Po(,;~, k, a) and the shift factor S0(2, k, a) of the centrifugal barrie.r 3) (which are the conventional penetrability and shift factor), an approximate relation between the penetrability Po(2, k, a) of a centrifugal barrier and the penetrability P()., k, a) of a diffuse-edge potential is of interest. Similar questions are discussed by Vogt 4). He has shown by calculations with the Woods-Saxon potential that the penetrability of the centrifugal barrier must be multiplied by a reflection factor to obtain the penetrability of a diffuse-edge well. Such a factor essentially depends on the thickness of the diffuse edge but does not depend on the orbital momentum nor on the energy in the low-energy region. It is easy to get an analytical expression of the reflection factor. The wave function f(2, k, x) satisfies the integral equation f ( ) , k, x) = fo(2, k, x)(1 +AfO,

k, x)),

(48)

where

Af(2, k, x) = f ; f20.dx'k, x') f ° • fOfo()-,k,k, x") x") V(x")f2(2' k, x")dx",

(49)

fo(2, k, x) is a solution of the homogeneous SchrSdinger equation with the same boundary conditions as f(2, k, x). We define the reflection factor 1 F(2, k, x) = I i + A f l 2 .

(50)

PO, k, a) = Po(;., k, a)F()., k, a),

(51)

Then the penetrability is

and the shift factor is S(2, k, a) = So(2,

k, a)+ aF(2, k, a ) ~da (r(2, k, a)) -1

(52)

If the potential tends to zero in the region x >__a, then the reflection factor tends to

S-FUNCTION

573

one, and the penetrability P and the shift factor S tend to the expressions of the centrifugal barrier Po and S o. In the region of low energies, we can obtain a useful approximation of the reflection factor. The potential V(x) fast approaches zero outside the potential edge (x > a). Therefore the expression

Af(2, k, x) ,~ f;.. fg().,dx'k:

•

k, x")V(x")dx",

x :> a

is a good approximation of the function A f If the energy is so low that the wavelength of the functionfo(2, k, x) is large compared with the thickness of the potential edge, then Af(2, k, x) ,,~

f; f/ dx

V(x")dx",

x >= a.

(54)

In this approximation the reflection factor F does not depend on the orbital momentum (in the case of central potentials) or on the energy. For attractive potentials F > 1, i.e. the centrifugal barrier only reflects much more of the incident wave than a barrier built up from the centrifugal and an attractive diffuse-edge potential. In the Woods-Saxon potential (28), it is possible to calculate the integral (54) co

iv01,1,=~l(_). = +

Af(2, k, a) = - b 2 2_m2_mh

exp

(_haiR) /,/2

, a>R,

Vo < 0. (55)

For a = R,

Af()., k, R) = - b 2 nz 2m

(56)

Here too we have found that the reflection factor essentially depends on the thickness of the diffuse edge. Because we have established relation (27) for the parameters b and Vo, it is certain that the reflection factor is limited in the approximation (56).

7. Numerical examples We have calculated some wave functions for the scattering of neutrons from 160 to show that we can get useful results with the proposed approximations. The interaction was described by the Woods-Saxon potential

V(x)=2m

I

2(/. s)t¢ d I Vo . . . . h2 x ~ l+expX--R___. ' b n = 0.295 fm, b = 0.5 fm,

h2-

R = 1.4 A ¢ fm, Vo = - 4 2 . 5 M e V

| "[-

for

l = 2;

Vo = - 4 6 . 6 M e V

for

l=0.

574

H.-J. UNGER

In the s~ channel a bound level lies at - 3 . 2 7 MeV, in the d~ channel at - 4 . 1 4 MeV and a resonance level lies in the d~. channel at 0.94 MeV. The exact wave functions and their approximations from (26) and (40) are represented in figs. 1 and 2. We have used the linear approximation of the expression [

2 dS\

The penetrability and the value dS/dEr are calculated in approximation (54) of the reflection factor. Assuming that the reflection factor and the value dS/dEr are parameters of the theory, which must be determined by comparison with the exact

0,¢ i:

5 ~ , x o 3 , z s f~,

ii

,2 0,06

, +/

.... :

I

I'

1

J t ~ -

z

3

/

,

~

~ - ~4 , " Z W"

J~

-

s --

6

t

[,~v

Fig. 1. The exact continuous spectrum wave functions (sohd lines) and their approximations (broken hnes) by expressions (26) for d~ waves and by expression (40) for s~ and d.I. waves dependent on the energy.

wave functions, the accuracy of the approximation can be improved in certain energy regions. We note that the Coulomb potential does not satisfy the requirements to the potential V(x). But the Coulomb potential can be taken into consideration by using wave functions with the corresponding asymptotic behaviour. The author thanks Professor V. V. Balashov, N. M. Kabatshnik and G. J. Korenman for useful discussions and N. M. Kabatshnik for permission to use the exact solutions of his calculations.

a)

V,

4~ ; / '

o..!.. \ .

I

/ e: 6 ,,'.,,v

-i ~v

b) , -v, ifaV

°,2 l

/

4f~ 1i i

,;'

4i2 i . . . . 4~

o z ~:v

•- . . . . , ~ _ .

"-.

o.o8)I J

20

//... /I

40+ I

. 2 ..',\ --.

....

."

!

E:St&Y

"

-\

2

3

R

÷ ~::~3~ - -

"C.~

~,&M<,/~ l

c) ~ -

-v M
o,o~!

/,' \ ",

~o~. I ............

~3o : o,02i

"

,;"--"

i',' i,,

/'

-.

\

", \ ~ - ' \

".t

"

",,e.6M
""

Fig. 2. The exact c o n t i n u o u s spectrum wave functions (solid lines) and their approximations (broken lines) by expressions (26) and (40) dependent on the radius for a) s~ waves, b) d.i waves and c) d I_ waves. The potential is represented by the dot-and-dash curve.

516

H.-J.

UNGER

References 1) 2) 3) 4) 5)

V. V. Balashov et al., Yad. Flz. 2 (1965) 643 V. de Alfaro and Regge, Potential scattering (North-Holland Publ. Co., Amsterdam, A. M. Lane and R. G. Thomas, Revs Mod. Phys. 30 (1958) 257 E. Vogt, Revs. Mod. Phys. 34 (1962) 723 T. Teichmann and E. P. Wagner, Phys. Rev. 87 (1952) 123

1965)