Virtual states and resonances

Nuclear Physics A l l 5 (1968) 481-----494; ( ~ North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

VIRTUAL STATES AND RESONANCES K. W. McVOY t

Department of Physics, University o/' Wisconsin, Madison, Wisconsin 53706 Received 25 March 1968 Abstract: The distinction between an s-wave virtual state and a resonance is explained by exhibiting the characteristic phase shifts and scattering cross-section shapes attributable to each. They are described in both R-matrix and S-matrix terms, and the relation between these two parametrizations is explained. As an application, in the two succeeding papers we investigate the energy dependence of the s-wave neutron strength function and show how it is affected by the presence of a virtual state in the complex optical potential. Such a state is expected to occur for A ~ 50 and A ~ 150, where the 3s and 4s optical states are slightly unbound.

1. Introduction Consider a system with a scattering resonance near the threshold of a channel to which the resonance is strongly coupled. Provided the resonance does not actually "overlap" threshold, it will produce a normal peak (or dip) in any open-channel cross sections to which it is coupled, no matter what its angular momentum. Furthermore, if we imagine the resonance to be " m o v e d " in energy toward the threshold, it may narrow somewhat and become slightly asymmetric but otherwise maintain a normal resonance shape, even as it passes through the threshold energy, provided the orbital angular momentum of its decay products in the channel under consideration is not zero. However, if it is s-wave in this channel and is moved toward threshold from below, its shape will be converted, just as its center reaches threshold, from a rounded Lorentzian to a pointed cusp centered exactly at threshold. This is shown clearly in the example of fig. 1 taken from ref. 1). The rounded Lorentzian will re-appear above threshold, if at all, only when the state is " u n b o u n d " by more than about half of its width, ½F (which can be 1 MeV or so for a nuclear resonance, if the state has a singleparticle width). A slightly unbound s-state corresponding to the cusp shape (e.g. the Hez molecule or the singlet state of the deuteron) is called a virtual or threshold state to emphasize its unusual properties. Neither the R-matrix descriptio n of a virtual state nor a detailed discussion of its effect on the phase shift seems to have appeared in the literature. Since they play an important role in the investigation of neutron giant resonances (see the succeeding two papers), we offer a brief description of them here. ?

Work supported in part by the National Science Foundation. 481

July 1968

482

K.W. McVOY

The simplest form of the virtual state is found at the lowest threshold, therefore for brevity we confine most of our discussion to the elastic case (i.e. the case of only one open channel). 10

OA

oc 50

__L___ l 5.2 .5.4 (w- M)/m~

56

Fig. 1. Illustration o f the effect o f an s-wave resonance near a threshold taken f r o m ref. 1). The solid curves are proportional to the elastic cross section, the dashed ones to the inelastic cross section into the channel which opens at 5.4. In curve 4, the level occurs below this threshold and produces a peak in the elastic cross section at 5.25. In curve 3 it has been moved closer to threshold, and in curves 2 and 1 it is a virtual state at an energy slightly above threshold. Its principal effect on the elastic cross section in these cases is an enhancement o f the size o f the threshold cusp.

2. Elastic virtual states The reason that virtual states occur only for l = 0 can readily be understood from the threshold energy-dependence of resonance widths t. If we imagine a level in an arbitrary/-wave at an energy E above threshold to be " m o v e d " downward in energy toward threshold (with its reduced width held constant), the energy of the state decreases in proportion to k 2, while its width decreases (because of the centrifugal barrier) like k 2t+ 1. Hence, if I > 0, the width decreases faster than the central energy; the level never "overlaps threshold", therefore there can be no ambiguity as to whether it occurs above threshold (a resonance) or below it (a bound state). If l = 0, however, there is no centrifugal barrier, and the width of the level decreases only like the square root of the energy, so that as the level is moved down in energy, it will necessarily "overlap threshold" before it becomes a bound state. Depending on how great this overlap is, it may not be clear whether the center of the level is above or below threshold, and it is to describe this ambiguous situation that the term virtual state is used. t Virtual levels only occur if at least one of the two decay products is neutral, therefore the energy dependence o f the width is not influenced by a C o u l o m b barrier.

VIRTUAL STATES

483

If the reduced width of an s-wave state is of single-particle magnitude, its effect on the scattering cross section is practically undetectable (as we show explicitly below) unless its energy is low enough to make it virtual; single-particle resonances (as distinct from virtual states) of neutral particles are nearly non-existent from an experimental point of view. If the reduced width of the state is much smaller than single particle, on the other hand, it can easily produce a sharp rise in the phase shift when its energy ER is large compared to its width F, and in this case it is of course called an s-wave resonance.

(o)

ER:

(c)

(b)

57 =

E R : y=

ER=OIOy

=

<{

I'-

-3

IZ5

2~0

~

~O0

f~5

i

50

.r25

25O

,3?~

-5

50O

~5

.~0

~5

~00

,

,~

625

(INVERSE FERMI'S)

2O

3OO

O"

I00

250

500

7~

tOO

,2~

•

--~o

~oo

~

,oo

~25

MeV

MeV

. (k) In, (k)

Ira(k)

f . 5 0,5

,o. ,s I~"]R,(.)

:

}(

O5

I0

Fig. 2.

: Re (k)

,

(k)

484

McVOY

K. W.

(d)

ER:OO7

(e)

y'~

(f)

E R = 005 71

Ea=

.3

.125

250

37~

~00

IR5

-005

yl

-3

-$

~25

250

3?5

~JOO

"5o

i~

250

37'5

500

-4

ooo

ooo

o"

6"

2 •

$

Im

2

2OO

(k}

. MeV

'&

.I)2

&

ER• -.371

~

e MeV

NkrV

Ira

(k)

Ira (k)

t

t ..... ' I:

i

~ Re (k)

I-

1 2 Fig. 2. P h a s e shifts a n d cross sections c o r r e s p o n d i n g to a m o d e r a t e l y n a r r o w (72 = ~Ts.p. ) s-wave state near the elastic threshold. T h e R - m a t r i x energy E R (or m o m e n t u m k R) is indicated by a dot a n d is decreased successively f r o m a ) - f ) ) ; R e ( E s) is indicated by a cross as is the c o m p l e x m o m e n t u m o f t h e S-pole in the lower figures, a) s h o w s a n o r m a l resonance, 10 widths above threshold. In 2d) a n d 2e) it h a s b e c o m e a virtual state, a n d in 2f) it is b o u n d . T h e h a r d - s p h e r e r a d i u s a o f eq. (1) is t a k e n as 5 f m a n d the particle m a s s is t h a t o f a nucleon, so thaWs.p. 2 = (maS) - t = 1.6 MeV. N o t e that the scales e m p l o y e d are n o t u n i f o r m in all c o l u m n s . T h e cross section o f 2d) is repeated in 2e) for c o m p a r i s o n . All m o m e n t a in f m -1.

3. Narrow ("compound nucleus") resonances A distinction between resonances and virtual states, in other words, can really be drawn only for a n a r r o w level, and we consider this case first. The distinction is

VIRTUAL STATES

485

clearly not a sharp one, but the empirical motivation for introducing it can perhaps be made clear by the example of fig. 2, which shows the cross sections and phase shifts produced by an elastic resonance whose partial width is ~o the single-particle value (~p. = h2/ma2). These phase shifts are computed from the single-pole Rmatrix approximation fi(E) = - k a + a r c t g

( kay2 ~, \E R --

(1)

E/

(which thus includes a hardsphere background), with curves (a) through (f) differing only in the choice of ER, i.e. all levels shown have the same reduced width. When the state is sufficiently far above threshold (fig. la), its cross section has the familiar s-wave shape including the "interference" zero just below ER, where the phase shift rises through zero (or any multiple of ~). If ER is decreased until the state is nearly bound, however, it does indeed begin to overlap threshold. In fig. 2b the zero of the cross section has just reached threshold, and in fig. 2c almost half of the resonance peak has "vanished" into the negative-energy region. Perhaps the strangest behavior of all occurs just after the maximum of the cross section passes below threshold (which occurs when ER is still positive, ER ~ 0.75 ?,/?2.p.). One might expect the "other half" of the resonance shape to follow as E R is decreased further, but that would require the value of the cross section at E = 0 to decrease, and this would contradict its familiar preference to increase without limit (in proportion to the square of the scattering length) as the state becomes bound. In fact, it is readily verified from eq. (1) that o'(E=0)=4=a 2

-1

,

(2)

which does increase monotonically as E R decreases below ),2. What actually happens (figs. 2d-f) is that after half the resonance curve has vanished below threshold, further decrease in ER causes the remaining half to grow in height and appear to broaden, as though the resonance were attempting to "back u p " to a higher energy t; tr(0) becomes infinitely high as the bound state appears (ER = 0), and thereafter the positive energy tail of the state is finally pulled into the negative-energy region by the receding bound state. It is during this last "backing u p " stage before the appearance of the bound state, when the cross section is a monotonic decreasingfunction of the energy ("half a cusp"), that the level is called virtual. Its essential characteristic and the reason for the name is that the cross section displays no positive energy feature which can be used to * Said s o m e w h a t m o r e carefully, t h e value o f t h e p h a s e shift increases at each energy as E R decreases, c a u s i n g #(E) to increase if 6 < ½~ at this energy, a n d to decrease if ~ > ½n. F o r a n a r r o w resonance, 7 2 << 7s.p), 6 = ½~ at E ~ ½ER so a ( E ) increases only o u t to this energy w h e n ER is decreased. F o r a single-particle resonance, o n the o t h e r h a n d , c5 < ½~z at all energies, so decreasing E R increases ~ at all energies.

486

K.w. MeVOY

mark the energy of the state; although the level is unbound, it affects the cross section most strongly at zero energy (rather than at a finite positive energy). In fact, it is impossible to tell from the shape of the cross section alone whether the level is virtual or bound, the two cases being distinguished at positive energies by the sign of the scattering length. An additional distinction between resonances and virtual states can be drawn in terms of the shape of the phase shift curve 6(k). If in the low-energy region, we write 6( k ) ,~ ctk + flk a,

(3)

the sign of fl(ER) determines whether t~(k) curves upward from its zero-energy slope (figs. 2a-c) or downward (figs. 2e and 2f). In the first case, the maximum derivative d6/dk (which is the maximum spatial delay of the wave packet and hence marks the resonance energy) occurs at a finite positive energy, whereas in the second case this maximum derivative (i.e. the maximum influence of the level) occurs at k = 0. Thus by this criterion also, the energy of the level appears to go negative at about fig. 2d (fl = 0 in this figure), marking the transition from a positive-energy resonance to a virtual state. Further details regarding fig. 2 are given in the appendix. Finally it should be recalled that a compound-nucleus level is far more likely to be a resonance than a bound state, since the chance of its occurring with in F (say, a few mV!) of neutron threshold is vanishingly small. Only single-particle virtual states are of practical interest in the nuclear context. 4. The S - m a t r i x poles

If E R is well above threshold, it accurately marks the center of the resonance, but we note from fig. 2 that as E~ is decreased the apparent energy of the level decreases faster, and in the virtual state range it even seems to go negative while E R is still positive. Thus E R is no longer an accurate measure of the level position when the state is virtual, and as an alternative description of the state we briefly consider the corresponding S-matrix pole. In the elastic case, the S-matrix element is given in terms of R by S(k) = e 2~

=

e -

2ika

1 + ikaR(E) 1 - ikaR(E)

= e-2i~a E - E R - ika~ 2 E _ Ert + ika)p2 '

(4a) (4b)

the second form following from the one-pole approximation to R

R(E)-

(5)

which gave eq. (1). The corresponding poles of S ( k ) are thus the complex momenta

VIRTUAL STATES

487

at which

E-Ex+ikay 2

= 0

(6)

is satisfied. I f we define t kR -- (2mER) ½, this can be written as the quadratic equation kx2 = 0,

(7)

kRE+_~/1-22-i2],

(8)

2 = maT2/krt = Frt/4ER,

(9a)

FR = 2krtaT 2.

(9b)

k 2 + 2imay2k-

whose solutions are

ks

=

with

These pole positions are shown for several representative values o f ER in the third line o f fig. 2. The corresponding positions in the energy plane are Es = ER~I -- 222 ~ 2i2x/1 --2z].

(10)

I f E R >> ¼FR, so the resonance is well above threshold, the solutions are k s = _ k R iJ'na72, the right-hand one lying just below kR in the k-plane tt. The corresponding energy is E s = ER--½iFR, indicating that it describes a resonance at E = E R o f width F R. As E R is decreased, E s also moves to the left, along the parabolic path shown in fig. 3, but it moves faster than ER, and its real part even goes negative when ER = FR/ 2X/2 = ~4/~.p., where Ts2.p. = (ma 2 ) - 1 (fig. 2c). It then curves u p w a r d until it reaches the negative real axis at E s = - E R = ¼FR = --~4/2T2.p. (fig. 2e), thereafter returning along the real axis (virtual state, at a negative energy) to the right, reaching E = 0 just as E R does, signalling the appearance o f the b o u n d state. This part o f the trajectory has been on what is technically k n o w n as the " s e c o n d sheet" of the energy surface (merely meaning that Ira(ks) < 0), which is why this negative-energy pole for 0 < Eli < ~4/2~s2.p. does not correspond to a b o u n d state. As E R becomes negative, however, E s moves t h r o u g h the branch point at E = 0 (i.e. Im(ks) becomes positive), and E s also moves left along the negative real axis, exactly at the energy of the b o u n d state ttt A l t h o u g h it is actually more appropriate to the single-particle levels discussed below, fig. 4 m a y be enlightening in this context, for it shows the rate at which b o t h t We simplify the algebra by choosing units in which h = 1. tt In other words, as eq. (8) shows explicitly, holding the reduced with constant means that ks moves along the horizontal line Ira(ks) = --maT,2, as kit is varied (provided 3. < 1, i.e. kR > maT~). ttt It is amusing to compare the momentum-plane poles of fig. 2 with the complex eigenfrequencies of a damped oscillator. Such a system, described by an equation like m~ = --mco~x--TY¢, has exponential solutions x(t) ~ x - i ~ t only for the complex frequencies to = ± lto~--~S/4rn2]~--iT/2m. If 7 < 2mtao, the system is underdamped, and the eigenfrequencies lie in the lower half of the coplane, symmetric about the imaginary axis, like the k-plane poles of fig. 2a which describe a sharp resonance. If 7 > 2mtoo, it is overdamped, with frequencies on the lower imaginary axis above and below --/coo; the upper one bears an interesting formal analogy to the virtual state pole in the kplane shown in fig. 5c.

488

K . w . MevoY

the R and S-poles move as a function o f the potential strength B = [ 2 m V o a 2 ] ~ for the 3s state in a square potential well 2). N o t e that E s = ER = 0 when B = ~Tr, corresponding to the 3s state at zero energy. Im (El

4

},

-2

~

,b M , vRe(E)

~

-6

I

-lO

MeV

Fig. 3. Trajectory traversed in the complex E-plane by the 3s pole of the S matrix for a nucleon in a (real) square potential well 50 MeV deep. The numbers along the path are the values of the potential strength parameter B = (2mVoa2/h2)½,which equals ~n = 7.85 when the state becomes bound. The real, negative values of E s for a slightly u n b o u n d level describe the virtual state.

4O ER

3O

20

>

10

~

~

Bound State

~

State

-I0

I

I

i

I

I

i

6.0

6.5

7.0

7.5

8.0

8.5

[2mVo Fig. 4. ER and Re(Es) as a function of the potential strength parameter B for the 3s state of the square well described in fig. 3. The two energies are quite different unless the state is well above threshold or exactly at threshold.

VIRTUAL STATES

489

5. Threshold branch points

This rather bewildering difference between the trajectories of ER (straight along the real axis) and E s (wandering through the complex energy plane) is clearly due to the presence of the threshold t. It reflects an essential distinction between the properties of R(E) and S(k), namely, that R(E) is a meromorphic function of the energy, which contains no threshold information whatever, while S[k(E)], considered as a function of E, has a square root branch point at E = 0 (k = (2mE)½), which provides S(k) and hence ~5(k) with their appropriate threshold properties (S ~ 1 +2i~k and ~ ,,~ 0tk for small k). The entire virtual state phenomenon is a threshold effect, and it is because the S-matrix "knows about threshold" while the R-matrix does not, that the Smatrix pole provides a more accurate representation of the position of the virtual state than does the R-pole. In particular, if the level is only very slightly unbound (like the singlet state of the deuteron), the S-pole is located at the negative energy _ (2m~t2)-1 (in terms of the scattering length 0c), while the corresponding R-pole is at the positive energy E R = +?2~t/(a+ ct). Unfortunately, the "energy of the virtual state" is often confusingly described in the literature 3) as + (2m0t2) - 1. 6. Single-particle virtual states

For the neutron giant resonance application considered in the following papers, it is a single-particle virtual state (i.e. a virtual state of the optical potential) which plays a central role. In the nuclear context, a level of this type typically becomes virtual when ER is as much as 1 or 2 MeV above neutron threshold. The phase shifts and cross sections corresponding to a single-particle state in a real potential well [i.e. ~2 = lima 2 in eq. (1)] are shown in fig. 5 for several values of E R. The essential difference between this and the previous case of a narrow resonance is that before the level becomes virtual, its S-pole in the k-plane is a distance a-1 (rather than (y2/~2v.)a-1) from the real axis, so that the maximum slope it can contribute to the phase shift is (d6/dk)res = + a. But this is only enough to just balance the slope of - a from the background phase; whereas in the previous case (fig. 2a), the resonance produced a large phase-rise in spite of the descending background phase, now the resonance is so broad it merely provides a momentary halt in the descending phase (fig. 5a). Decreasing E R to E R = ~2.v. moves this "flat spot" in the phase down to E = 0, therefore the scattering length vanishes (fig. 5b). The zero-energy phase curvature da6(0)/dk is still positive for this ER, therefore 6(k) does rise slightly in this case, producing a very low maximum in tr(E), followed by a zero where the phase decreases through zero. Further decrease of E R causes this low maximum to vanish into the negative-energy region and for all smaller values of ER, tr(E) decreases monotonically from its zero-energy value (fig. 5c), corresponding to the level being virtual; tr(0) is given in terms of ER by eq. (2) with y2 = lima 2. t The specific path followed by Es depends in addition, of course, on the presence or absence of a centrifugal barrier.

(a)

ER:

(b)

I0),

e

ER

5

~3

= ),t

3 I

-I W <"1-[ - 3 0-

-3

-5

i

0

.250

i

.~5o ,leo

.500

,'~5

-5

2.0

0.4

1.0

o" 0.2

.k(x)

.~r-~o i'.~

,'.~

;,

6

~o

0.3 (Bo~ns)

6 MeV)

2

4

6

8

0,1

2

I0

MeV

8

MeV

Im(k)

Im(k) .2 I .2

.4

I .6

: :

: Re(k)

- - .S -x----

, -

•

I

i

~ Re (k)

l x- \ ~x'

'-.4

Fig. 5. P h a s e shifts, cross sections a n d pole positions for a n s-wave single-particle state 7 3 = 7s.p. ~, t h e r e m a i n i n g p a r a m e t e r s being t h e s a m e as in fig. 2. a) the state is well above threshold, in c) it is virtual a n d in d) b o u n d . T h e principal difference between these p h a s e shifts a n d t h o s e o f fig. 2 is t h a t t h e descent o f the b a c k g r o u n d p h a s e ( - - k a ) across t h e width o f the r e s o n a n c e is a b o u t t e n times as great in this case as in fig. 2.

(c)

3

(d)

F"R : 0.12 ) ' =

5'

E R = -0.5

y'

3

I -I

-I

-3,

-3

"5

I 0

~

! ,750

I

.500

I

-5

I

1.00

1.25

0

ZO

! 2.50

I .500

~f

I .750

I LO0

k

k

I 12.5

20'

15,

15 o"

o" I0,

I0 5

5' __~R = 0 . S T I

2

4

6 MeV

MeV

Im (k)

Im(k)

|

I

I

i

,

i Re (k)

F i g . 5.

i

i

i Re (k)

492

K.W.

McVOY

In summary, an s-wave single-particle level well above threshold is necessarily so broad it produces hardly any visible effect on the cross section. It becomes somewhat more visible as a virtual state but primarily through its influence on the scattering length, which makes the zero-energy cross section large as the state becomes bound. In particular it can be seen from fig. 5 that the maximum phase-rise it produces is ½~z, so an s-wave single-particle state can never cause a maximum in the cross section due to the phase rising through 90 ° . This is closely related to the fact (established below) that an s-wave giant resonance, described by a single-particle optical potential, fails to produce a m a x i m u m in the energy dependence of the s-wave strength function. 7. Inelastic virtual states Inelastic virtual states or threshold states have received exhaustive treatment in discussions of the enhancement of threshold cusps by s-states occurring near thresholds 1,4). These discussions have been restricted to virtual or bound states very near threshold (i.e. one S-pole approximation to figs. 2e, 2f or 5d), but the properties of levels farther above threshold are nearly the same whether or not the threshold is the lowest one and thus can be implied from our above considerations. I f there is another channel open below the threshold concerned, the essential change from, e.g. fig. 5c, is that k is to be interpreted as the relevant channel momentum, and the S-pole

x

xI

(a)

(b)

Fig. 6. The S-matrix poles in a channel-momentum plane for an s-state which is slightly below (fig. 6b) the corresponding threshold. Because other channels are open in this energy range, the pole does not lie on the imaginary axis, as it does, e.g. in figs. 5c and 5d. representing a state which is virtual relative to this channel occurs not on the lower imaginary k-axis but slightly to the left of it as in fig. 6a. Since k = [2m(E-Erh)] ~ is positive real for E > Erh and positive imaginary for E < Erh, this pole is closest to the "physical region" o f k at k = 0, i.e. exactly at threshold, which is why it enhances the cross-section cusp there. I f the state occurs below threshold, on the other hand ( " b o u n d " relative to this channel), the pole is to the left of the k-axis in the upper k-plane (fig. 6b). This is near a real energy below Erh and thus produces a normal resonance effect there, rather than at ETh; cf. fig. 1. Appendix The following trivial consequences of eq. (1) are of considerable help in understanding figs. 2 and 5.

VIRTUAL

493

STATES

a) The S-pole positions. In the k-plane, (A.1)

ks = kR[___x/l--~-i2], with 2 --- may2/kR = FR/4ER, or

=

).

In the E-plane,

i[0-2x2)-2a41-

Es =

].

(a.2)

b) Scattering length and zero-energy curvature of 6(k). If at low energy ,5(k) ~ ~zk + 3 k 3,

the scattering length 0~ is given in terms of the R-parameters by

= a

(';)

(y2y~v.] ( 3 = ½a 3 ~--~-R2] I

-1

, 2

(A.3)

,f4

).

3 El~-~v.

(A.4)

Hence the phase starts out negative from zero energy if E! > ~2, positive if Ea < ~2. c) If tr(E) = 47r£2 sin26,

and so vanishes for the two values of ER at which e = 0 (El = ~2) and fl = e3/6 (E R ~ 0.75 ~/41~1,.). --

~

a

m

1;

(A.6)

Era, ~ = E l { [ 4 - 2y4/Ely2p.] * - 1},

(A.7)

dk

[ i E l _ - - ~ +--~,,/V~p"

(d6/dk) reaches its maximum value as a function of E at

4 ,/3 2Y~.p., which is when fl = 0, provided E~,, > 0; E=,, reaches zero when El = 2 Y as it should be. The values of these quantities are given for the various stages of both figs. 2 and 5 and in table 1. We have used the convenient abbreviations 2 p = y 2 /~.p.,

(A.8)

ko = p/a = may 2 = 2kl,

(A.9)

k o being the value of - I m ( k s ) when Re(k~) ~ 0.

494

K. w. MeVOY TABLE 1 (Fig. 2, )'2 = ~)'s.p. 1 2) Description

Eit

kR

Re(Es)

(a) Resonance far above ~>)'2 threshold (b) ~ = 0

)"

E of Re(ks) 6'(k)max

~,

fl

da(0) dE

kit

--a

+

+

eit = )'2./~Wp k0

0

Eit x/--~ko

ER=)'2

Eit

(c) R e ( k e ) = - - I m ( k s ) , or R e ( E s) = 0

p)'2

~/2ko

0

0.4ER= 0.4 try,2

ko

(d) fl = Emsx = 0

]p)'2

4~ko

_½p)'~

0

ko/V/6

(e) k s turns corner (f) state is b o u n d

½p)'2

ko

_½p)'2

< 0

0

+½a 3

0

×i(1-tp) p +alp

+½aap -2

+

+]a/p

0

--

+ 2a/p --~aap -~

--

TABLE 2 (Fig. 5, Description

(a) resonance far above threshold (b) ~ = Re(Es) = 0 (c) (d) state is b o u n d

Eit

~ ) )'2 )'2 ~)'2

kit

~/2a-1 (2a) -1

),2 = )'s.p.a)

Re(Es)

E of Re(ks) t~'(k)max

E0 0 --0.07Eit

Eo 0.4)' 2 < 0

kit kit 0

~

--a 0 +7a

fl

+ ~a a < 0

dtr(0) dE

+ 0 --

References I) 2) 3) 4)

W. R. H. M. J. M. H. A.

Frazer and A. W. Hendry, Phys. Rev. 134 (1964) B1307 Nussenzveig, Nucl. Phys. 11 (1959) 499 Blatt a n d V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) p. 68. Weidenmiiller, Nucl. Phys. 69 (1965) 113