NUCLEAR PHYSICS A

~:t.~EVtER Nuclear Physics A583 (1995)291-294

P O T E N T I A L R E S O N A N C E A F F E C T E D BY N O N - A D I A B A T I C T R A N S I T I O N

IN N U C L E A R H E A V Y - I O N C O L L I S I O N S

T.Tazawa, N. Takigawa ~ and Y. Abe ~ Department of Physics, Faculty of Liberal Arts, Yamaguchi University

Yamaguchi 753, JAPAN °Department of Physics, faculty of Science, Tohoku University

Sendai 980, JAPAN 2~ Yukawa Institute for Theoretical Physocs, Kyoto University

Kyoto 606-01, JAPAN

1. Introduction and summary Landau-Zener non-adiabatic transition has been expected to play an important role in

nuclear heavy-ion collisions in low energies from the energy diagrams of the theory of two-center shell model 1. There have been several attempts to find definite evidence of the Landau-Zener transition in nuclear heavy-ion reactions such as fusion reactions and inelastic scattering ~. The discovery of peculiar resonancelike behaviors by Strasbourg group 4 and Abe and Park's suggestion that the resonancelike behaviors occur due to the Landau-Zener transition 5 stimulated theoretical investigations on the occurrence of the resonancelike enhancements. We have also investigated whether resonancelike enhancements predicted by the classical or semi-classical approaches really survive or not, using a very useful analytic formula for the T-matrix derived in quantum-mechanical way within a two-state model under the assumptions that diabatic potentials are linear and that the coupling is finite 6. However, We can not draw a definite conclusion because of defects in the model such as use of linear diabatic potentials, the neglect of inner potential effects and of absorption effects.

We here propose a new approach to remove the defects mentioned above within a two-state model where two adiabatic energy surfaces have a single avoided crossing point. The semi-classical S-matrix is analytically derived for two important cases where the avoided crossing distance is larger and less than the barrier top distance,using transition amplitude formula at the avoided crossing derived in generalized adiabatic representation 7 and alalalyin~ semt-classtcal approach for complex potential scattering developed by Brink and Takigawa to two-state problem. The dynamical relation between non-adiabatic transition and potential resonance is derived. Qualitative discussions for both of the weak and strong coupling limits are given.

2. Semi-Classical Formulae We consider a two-state model where two adiabatic energy surfaces have a single avoided

crossing point. In nuclear heavy-ion collisions, there are two important cases which depend on whether the avoided crossing distance r x is larger than the barrier top distance r B or not.

Let us consider the case shown in Fig. 1 as an illustrative example. We divide the whole space into four regions. Region I lies outside of the avoided crossing distance r~. Region II covers from the avoided crossing distance rx to barrier top. Region III covers from the barrier top to some internal domain. Region IV is the most internal region. The semi-classical S-matrix can be derived as follows. The amplitudes which correspond to the ingoing and outgoing waves for the four regions are connected by using the wave propagation method 9 except for the crossing distance. Employing the transition matrix derived in non-adiabatic semi-classical theory 7 and the matching condition at the avoided crossing distance leads to

0375-9474/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved. SSDI 0375-9474(94)00675-X

292c T. Tazawa et al. / Nuclear Physics A583 (1995) 291-294

the connection condition between amplitudes inside and outside the avoided crossing distance. The details of semi-classical derivation for the S-matrix will be published elsewhere. In this way, the semi-classical S-matrix elements for the case where r~>r B and E>E~ are given by

IV III I/ I

i r(2) i ( 2 ) ] ~ ~ r ! 2 ) E

' V i r//~r~")i(.1 i

r ~

v

Fig. 1 rx

• . (1) (2)

St1 = (1 -p) 171 e2/0, + P //2 e2'05+ -~' +oo)

,(+:,_+++,++,o++,) -,(<+'?-8? +oo+,+)I S12 ---- ~ 01 e -172 e f ,

(1)

-26 where p = e , <:r 0 +i8 = l~x[k2(r)-kl(r)lar,

= - '+ '+ - ~r - -~ + 4 Arg F 1+ ,

(2)

r , is an complex solution of the equation k2(x)=k;(x ), F(x) the Gamma function, and #s the so-called Stokes phase. The notations O, and 6~ +) are the extension of the notations to the two-channel problem of Ref. 8. The factor 17i have the same form as the semi-classical expression for the scattering amplitude derived by Takigawa and Brink s. The factor 17i (i=l or 2) are, therefore, interpreted as the scattering amplitude from the adiabatic potential without non-adiabatic transition.

Next we consider the case where rx<r B. In this case we can get the rather complicated expressions for the S-matrix in the same way as previous case

(1-p~l+N(ie,)e2(3~ *'))(N(te2)+e 2(~2 *,)

l - - 2 " ) t D . ( I) (2)

St, e 2""~?+ ~.+p(l+ N( iet )e ,(s3. +s.2 -,~o ) )( N(ie: ) + e, (+. + +2 +<~+ ) ) J = (3)

(a-p)(N(iel)+e~(~l'+',)~Niie2)+e2'(sl~'-',)) l

,+,(¢ +s ~,, ,,o) ,,(s+~, <,+<,0) +p(N(ie,)+e " +- ~N(iez>+e )J

T. Tazawa et al. / Nuclear Physics A583 (1995) 291-294 293c

$12 -- e '~%-~ )+'(8'c~' +~')+'(s~'~'-s~'~' +*,-~° ) ~I x

(I- N(ie z)N(iez))(e='( s~'-" ) - en( s~',' ` s:~' +~o))

] (1-p)(N(iea)+ei(S~>÷¢')~N(iez)+e2i(s~)-¢')) l

+ p(N(iel) +e~( ~" +s.,-aO) ~N(iez)+eZ(S,~'+s.] +~ro))J

(4)

where the notations e~ ~0 and S (° , : are the extension of the notations to the two-channel

problem of Ref. 8. The functions N(ie) and N(ie) are also defined in Ref. 8.

3 .The D y n a m i c a l R e l a t i o n B e t w e e n N o n - A d i a b a t i c T r a n s i t i o n a n d P o t e n t i a l R e s o n a n c e l_~t us investigate how non-adiabatic transition, for example the Landau-Zener transition,

influence the potential resonance. We consider two limitting cases i.e. weak coupling and strong coupling limits for both r,>r B and rx< r B cases. Here the weak and strong couplings mean 5--->0 and ~-->0o, respectively.

3.1. r~>r 8 e a s e Equation (1) indicates that there occurs a resonance when the following condition is

satisfied in the weak coupling ~--->0 i.e. p--->l, t~0--->0

N(iez) + e 2~ ' = 0 (5)

This is equal to the resonance condition for the diabatic channel 1 without coupling. On the other hand, in the strong coupling 5--->00 i.e. p--->0, equation (1) gives the equivalent resonance condition for the adiabatic channel 1. These results are easily understood physically, because non-adiabatic transition in the outer region can not affect the resonance condition determined by the behaviour of the potential in the inner region.

3.2. rx< r B ease For simplicity of discussion, we assume that EBs<<E<<Em, where EBI and EB2 are

barrier-top energies of the potentials for the adiabatic channel 1 and 2, respectively. By taking the conditions into account, resonance condition becomes

- P)~ Lv~'e2)~'e-~[~'::~ ~_ :~(s~'-,, )~j + p[N( ie2 )+ e2'(st~+s~+"° " J = 0 . (l (6)

We first consider the weak coupling 5--->0 i.e. p--->l, t~0--->0, ~8--->~/4. Expanding the

left-hand side of the equation (6) around an resonance energy ~0 d~ - i ~ and using the

approximation ( p - J - l ) - 2 6 and also neglecting the energy dependence of the phase of

N(ie), we can easily get the resonance energy and the width

294c T. Tazawa et al. / Nuclear Physics A583 (1995) 291-294

h6, ~a, cos2 E R = E~do ) + ~-~2) e 1, f l

~R-1 F - ~1~o 1-('0 + T'~d) ~lh6"( _e%a, sin 2ill

w h e r e tJ = J7 - r = q " ) - ~.~)_ l r (~) i 2 n0 ,

(7)

A,r = . r ~ ) -,r~a2 ) , "1 7 " "32' i = a d or d.

Here the channel number 2 for both diabatic and adiabatic channels is abbreviated and x32 is the one way transit time between the inner turning points r 3 and r 2. For Simplicity of notation, suffix of quantum number n is also abbreviated with respect to the variables fl, y ,

A'r, ~Jo and I7'~ 0 . As can be seen from equation (7), the second terms in equations explicitly show the effects of non-adiabatic transition on the potential resonance. It is to be noted that both E R and FR/2 show the oscillatory behavior as a function of quantum number n.

We secondly consider the strong coupling limit ~--+,,* i.e. p--+0, ~--+0. In the same way as the weak coupling case, we can also get the relations as follows

ER = F(~) _ h -2a.+2-La, • _z. + ',, ~.o 2,c~) e h sm2~p tr0)

--2~ 2 1F 1,~Oa). he "[. _zra,_ R = - t . o "~" ~--~-~' /x-e, ~ 2(fl+Go) }

2 2"t'32 t

(8)

The definitions of 1~ and y are the same as those in the weak coupling case except for the replacement of notations EtaJ---)E t~a~ and ~a~-~F°a). The same kind of behavior as in the weak coupling case can be seen in the resonace energy and the width.

The details of the derivation of the above expressions will be published elsewhere and the further investigation based on the numerical calculations in the appropriate model will be carried out in future.

References 1. K.Pruess and W. Greiner, Phys. Lett. 33B, 197(1970); J.Y.Park, W. Greiner and W.Scheid,

Phys. Rev. C21,958(1980). 2. D.Glas and U. Mosel, Phys. Lett. 49B, 301(1974); Nucl. Phys. A264, 268(1985); W.Cassing,

Nucl. Phys. A433,479(1985). 3. B.Imanishi and W. von Oertzen, Phys. Rep. 155, 29(1987) 4. R.M. Freeman, C. beck, F. Haas, B. Heusch and J.J. Kolata, Phys. Rev. C28,437(1983);

C. Beck, R.M. Freeman, F. Haas, B. Heusch and J.J. Kolata, Nucl. Phys. A443, 157(1985). 5. Y. Abe and J.Y. Park, Phys. Rev. C28, 2316(1983). 6. T. Tazawa and Y. Abe, Phys. Rev. C41, R17(1990); Prog. Theor. Phys. 8 5 , 567(1991). 7. H. Nakamura, Phys. Rev. A26, 3125(1982), T. Tazawa and M. Nogami, Prog. Theor. Phys.

6tl, 1739(1978); O. Tanimura and T. Tazawa, Phys. Rep. 61,253(1980) and Ref. 3. 8. D.M. Brink and N. Takigawa, Nucl. Phys. A279, 159(19"T7). 9. N. Takigawa, MSUCL-449, Talk presented at the 7th Oatepec Symposium on Nuclear

Physics, Mexico, Jan. 4-6, 1984.