Parametrization of strongly overlapping nuclear resonances

Volume 148B, number 1,2,3

PHYSICS LETTERS

22 November 1984

PARAMETRIZATION OF STRONGLY OVERLAPPING NUCLEAR RESONANCES Sadhan K. ADHIKARI

Departamento de Ffsica, UniversidadeFederal de Pernambuco, 50.000 Recife, Pc, Braz~ Received 15 July 1984

We compare the parameters involved in two approaches to the study of overlapping nuclear resonances, the first one by Kawai, Kerman, and McVoyand the second one by the present author. We also compare the parameters of these two approaches to those of the approach of Agassi,WeidenmgUer,and Mantzouranis.

Recently, Hussein, Kerman and McCoy [1 ] have compared two different parametrizations of overlapping compound nuclear resonances [ 2 - 4 ] . Since then an alternative formulation of such resonances has appeared in the literature [5]. This last formulation has certain advantages in the study of multistep compound reactions [5,6]. Using this alternative approach we calculate the fluctuation cross section and the energy autocorrelation function which as in the approach of Kawai, Kerman and McVoy (KKM) are shown to be parametrized in terms of a width and a spacing of such resonances [2]. It is shown that for strongly overlapping resonances this width and spacing are the same as the corresponding width and spacing of KKM in the limit of many open channels. Though these characteristic widths are supposed to be different from the average width of physical resonances, contrary to a recent conjecture in ref. [1], the characteristic spacings of overlapping resonances of these two approaches are shown to be equal to the average spacing of physical resonances. Finally, connections are established between these parameters and those involved in the approach of Agassi, Weidenmiiller and Mantzouranis [4] following the discussion of rel. [1]. The striking similarity between the fluctuation amplitudes of refs. [2,5 ] allows us to essentially repeat the analyses of refs. [2,3,7] to the case of the fluctuation amplitude of ref. [5] in order to calculate the fluctuation cross section and energy autocorrelation function [3]. We start with this calculation, as 0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)

this is essential for our analysis. The fluctuation amplitude of KKM is given by [2,5 ] KKM T~l,ab KKMV~Q~ KM(1 +GopU)10b) , =(~al(1 + UGop)V~p~KMGQQ

(1) where

v = v +/4< aoo>ztt,

(2a)

Gop = P(E - Hpp - HpQ(GQQ)IHQp)-IP ,

(2b)

V~e~"a~= HpQ B(I/ 2 )
(2C)

VI~o~,M= [i(I/2 )
(2d)

GKKM QQ = Q(E - HQQ -- V~QpIGMGoppV~QM)-IQ,

(2e)

G QQ = Q(E - HQQ) -1 Q,

(20

where 0b and ~ba are initial and final channel states, V is the channel interaction, and H is the full hamiltonian. In eq. (2) ( )l denotes energy averaging over an energy interval I with respect to a lorentzian function and HpQ = PHQ,HQQ = QHQ,Hpp = PHP, HQp = QHP with P + Q = 1, where P and Q are projectors onto the open and the dosed channel spaces, respectively. In the alternate (A) formulation of ref. [5 ] the same fluctuation amplitude is represented by A = (•al(1 + VGpp)V~QGQQV~p(1 A A A T/lob + GppV)lOb),

(3)

Volume 148B, number 1,2,3

PHYSICS LETTERS

where

gqc = (0c IUalq a>,

Gpp = P(E - Hpp)-lP,

(4a)

VAQ = HpQ [i(I/2 )(G~Q)I] 1/2,

(4b)

V~p = [i(1/2)(a~Q)t] 1/2HQp,

(4c)

G~Q = Q(E - HQQ - HotgTppHpQ )-1Q.

(4d)

In eqs. (I) and (3) the interval I is much larger than the characteristic widths F KKM and F A of resonances re~aresented by the resolvent operators G ~ M and G~Q, respectively. Also, both these types of resonances are assumed to be strongly overlapping. These two conditions are expressed by I>> I"KKM>>D KKM'

uKKM = (1 + U G o p ) V £ KM , e = a,b,

and the property (T~I)1 ~- O. The advantage of using (7) is that one can get rid of the constraints of analytic unitarity and employ average unitarity on the real energy axis and as a consequence can neglect the level-level correlation term in the fluctuation cross section [2,7]. Then for a sufficiently large I and for many open channels N the fluctuation cross section o~ can (in the notation of ref. [7]) be written as (O?I) --~X~aaX~bb + X~abX~ba -- 2(1 -D~/Trra)l Y~b 12,

I >> F A >)>D A ,

(5b)

where D KKM and D A are the mean spacing of these two types of resonances. These widths and spacings provide a convenient parametrization of physically "measurable" quantities such as the fluctuation cross section and the energy autocorrelation function among others. The equivalence between the two forms TKKM and TA given by eqs. (I) and (3) was established in ref. [5]. When averaged over the averaging interval I each of these fluctuation amplitudes average to zero. In order to proceed further we employ the following eigenfunction expansions of the resolvent operators G ~ M and G~Q [21:

G~QQ= ~ ~[qaXq°' I , q E - eq

(6)

where in eq. (6) and in the following t~ stands for KKM and A, respectively. In eq. (6) Iq a) and (qal are the biorthogonal set of eigenstates of the effective hamiltonian appearing in G~Q belonging to complex eigenvalue %. Using expansion (6) in eqs. (1) and (3) we have [2,3] Ot

ot

~fl,ab = ~ --,gqagqb qE-e;

u A = ( 1 + VGpp)VAQ,

(5a)

and

with

22 November 1984

(7)

(8)

where the hermitean matrices X a and ya are defined by [7]

X~ab = ( 2,r/PaDa) 1/2(gaaga~ q q )q ,

(9a)

and = (ff/Da)(g~lagqb) q ,

(9b)

Eq. (8) is identical to eq. (3.23) of ref. [7] for large I. In eq. (9) ( )q represents the average on the states q contained in the averaging interval 1. It is interesting to recall that for overlapping resonances and for I>> 1~ , Y~abof eq. (8) is a small quantity: (Tfi,ab)I ~ Y~ab~- 0.

(10)

In the same limit, N ~ 1, the energy autocorrelation function C(e) is given by [1,3]

C(e) ~- ( o~I)I'~/(F a - ie).

(11)

As the expression for Tfl given by eqs. (1) and (3) are equivalent and as eqs. (8) and (11) are supposed to produce identical results for the two approaches eq. (11) immediately yields pKKM -.. 1-,A.

(12)

Next in order to find a relation between the D's, X's, and Y's of the two formulations we note that in eq. (8) the terms involving the X's are an order of magnitude larger than the last term involving Y, and among the two pieces involving I YI 2, I YI2 >> [Da/ (rrl-~)] 1YI 2, as 1-~ >>D a. Hence, as eq. (8) is supposed to yield identical results for the two approaches for

Volume 148B, number 1,2,3

PHYSICS LETTERS

both elastic and inelastic processes one must have xKKM ~ XAb,

yKKM ~ YAb,

(13a,b)

and DKKM](rd-'KKM) ~--DA](TrFA).

(14)

Eqs. (13) and (14) follow by considering eq. (8) as an expansion containing terms of different orders of magnitude with the value of each of these terms being controlled by a different factor. For example, general properties of energy averaging are responsible for the smallness of I YI 2, and the strongly overlapping nature of resonances is responsible for an even smaller value of [Da/(,rlXX)] IYI 2 . Eq. (14) together with (12) yields D KKM ~- D A .

(15)

So under the conditions given by eq. (5) and for N >> 1 the fluctuation cross section of eq. (8) becomes [2] (a~l) ~ X~aaX~bb+ ~ a b ~ ,

(16)

with X given by e q; (9a), where F KKM ~-- l~A, D KKM ~- DA,XKabKM ~--XaAb. These conditions imply

KKM KKM- ~(g~qag~qb)q It is interesting to note that though the parameters

gqa,gqb, and e~ characterizing the individual resonances of the two approaches are not equal the , parameters D a, 1~ , and X ~ denoting some average properties of the two approaches are equal. It has been pointed out in ref. [1 ] that F KKM is larger than the "true" width of the compound nuclear resonances. The same property holds for F A as F A "" F KKM. It has also been conjectured [1] t h a t D KKM is expected to be different from the "true" average spacing of the compound nuclear states. We, however, provide arguments why D KKM is expected to be equal to the true spacing of resonance poles, which is essentially D A. This is because the physical resonances are usually defined by the resonance poles of the resolvent operator given by eq. (4d) [8]. Then it is intuitively easy to understand why the average spacing D KKM of the KKM poles contained in eq. (2e) is the same as the average spacing of the true poles D A contained in eq. (4d). In order to establish this in an

22 November 1984

alternative semi-qualitative way we note that the last term containing Gpp or Gop of the effective harniltonians of the resolvent operators of eqs. (2e) or (4d) are slowly varying on the averaging interval I and hence these two effective hamiltonians differ by a slowly varying function on this scale. Hence, if we rewrite eq. (4d) as

G~.Q = Q(E - H~Q)-Io,

(17a)

with H~Q = (HQQ + HQpGppHpQ), then eq. (2e) can be rewritten as GKKM OO = Q(E

- H~.Q - HQQ)-IQ,

(17b)

where /4QQ is slowly varying on the averaging interval I. Now we recall that the correct energy averaging requires [9] I to be much smaller than the extent of the resonance poles. Hence, as HQQ is slowly varying the diagonalization of the effective KKM hamiltonian a(n~o +ffoo)Q will not essentially change the number of strongly overlapping resonances contained in the averaging interval I. On the average the number of resonances that move out of the interval I as a result of including HQQ will be essentially equal to the number of resonances that enter this interval. As a result the pole spacings D a of the two approaches are expected to be equal, and equal to the true average pole spacing. The average widths of resonances of the two approaches are, however,not expected to be equal, because the presence Of HQo will, in general, change the width of resonances. At this point one should note that this fact does not contradict eq. (12) as 1-~ is not really the average width of the resonances of these approaches, but rather F a = 1/( 1/F~)q [2,7], where I~q represent the width of the resonances of eq. (7): I~q = - 2 lm(e~/), where Im represents the imaginary part. Finally, in the light of ref. [1 ] one has for the correlation width f,AWM of Agassi, Weidenmiiller, and Mantzouranis [4] : FAWM ~ pA ~ FKKM.

(18)

The characteristic spacing D AwM of the formulation of ref. [4] is not, in general, related to D A or D KKM except in the strong coupling limit when the resonances are non-overlapping and one has [1 ] D AwM ~ D A ~ D KKM.

(19)

Volume 148B, number 1,2,3

PHYSICS LETTERS

In conclusion, we have established that for strongly overlapping nuclear resonances in the limit o f many open channels the characteristic widths and spacings o f the resonances o f the formulations o f refs. [2,5 ] are equal. We establish that the characteristic spacing of these formulations is equal to the true average pole spacing. The characteristic width o f these formulations is, however, expected to be different from true average pole width and to be equal to the correlation width of Agassi, Weidenmiiller, and Mantzouranis. This work is supported in part by the CNPq and the FINEP o f Brazil.

22 November 1984

References [1 ] M.S. Hussein, A.K. Kerman and K.W. McVoy, Phys. Lett. 131B (1983) 8. [2] M. Kawai, A.K. Kerman and K.W. McVoy, Ann. Phys. (NY) 75 (1973) 156. [3] W.A. Friedman, M.S. Hussein, K.W. McVoy and P.A. MeLlo,Phys. Rep. 77 (1981) 47. [4] D. Agassi, H.A. W¢idenmilller and G. Mantzouranis, Phys. Rep. 22C (1975) 145. [5 ] S.K. Adhikari, Phys. Rev. C28 (1983) 2013. [6] S.K. Adhikari, Phys. Rev. Lett. 51 (1983) 1834. [7] A.K. Kerman and A. Sevgen, Ann. Phys. (NY) 102 (1976) 570. [8] H. Feshbach, Ann. Phys. (NY) 5 (1958) 357. [9] P.A. Moldauer, Phys. Rev. Lett. 23 (1969) 708.