- Email: [email protected]
Optical model description of DIC?
Volume 91B, number 3,4
PHYSICS LETTERS
21 April 1980
OPTICAL MODEL DESCRIPTION OF DIC? B.V. CARLSON, M.C. NEMES 1 and M.S. HUSSEIN 2 Department of Physics, University of Wisconsin-Madison, Madison, W1 53706, USA Received 11 January 1980 Revised manuscript received 6 February 1980
We consider a simple one-dimensional model for a many-channel scattering problem that simulates some aspects of heavy ion deep inelastic collision processes. Applying the statistical considerations of Weidenmiiller, we obtain an averaged inelastic cross section given in terms of optical transmission coefficients.
1. Introduction. Statistical theories of deep inelastic collisions between heavy ions have been advanced [1,2] to treat those aspects o f DIC which seem to follow diffusion laws. In particular, the work o f Weidenmtiller and collaborators [1] attempts basically to relate the diffusion coefficients to some underlying microscopic interaction between the ions [2]. Similar comments can be made with regard to the work of N6renberg [2]. An important fact which one discovers in the above references is that the solution o f the collision problem is predetermined, namely the multidifferential cross section is assumed to be gaussian in its relevant final macroscopic variables. Classical equations of motion are then solved to obtain the mean values o f these macroscopic variables. The variances are obtained by solving F o k k e r - P l a n c k - t y p e equations. In this connection we would remind the reader of the work of Hofmann and collaborators [3] in which a similar procedure - albeit more phenomenological was adapted to calculate the cross section. Two important questions which are relevant to the discussion are the following: (1) To what extent is the gaussian form assumed
* Supported in part by the University of Wisconsin Alumni Foundation, the Tinker Foundation, the US NSF, the CNPq Brazil and FAPESP Brazil. 1 Present address: Max-Planck Institut fiir Kernphysik, Heidelberg, West Germany. 2 On leave from the University of S'ffoPaulo, S~o Paulo, Brazil. 332
for the distribution basically a consequence of the very large number of channels involved in DIC and not so much o f the actual nature of these channels? (2) Why do conventional optical quantities, e.g., optical potentials, transmission coefficients, etc. seem to have disappeared altogether in the final results o f the theories in refs. [1] and [2] ? Said differently, is it possible to calculate the spectra in DIC from a generalized optical potential or transmission coefficients? We should mention that in ref. [1], optical quantities do appear in the intermediate steps, although in the final solution they are de-emphasized. This, we believe, stems from the complicated nonlinear nature of the equations which have to be solved to obtain the optical potential. We shall attempt in this work to discuss the second question. We leave the answer to the first question to another future publication. Our analysis of the second question will be based on a simple one-dimensional model which mocks up to some extent some important aspects of the many-channel problem. We follow closely the statistical theory developed in ref. [1]. It is extremely gratifying that after some tedious and lengthy algebra we obtain a very simple and appealing form for the averaged cross section, namely 7i rf Oil ~ [Ti[26if + (1 + a6if ) ~ / T / ' where 7~. and r i are the optical diagonal scattering amplitude and transmission coefficient in channel i, re-
Volume 91B, number 3,4
PHYSICS LETTERS
spectively, and 1 + a is an elastic enhancement factor. The paper is organized as follows: in section 2 we introduce our model and calculate the relevant optical quantities following the averaging procedure of ref. [ 1 ]. In section 3 we deriw.~ the averaged cross section and finally, in section 4 we discuss our results and point out directions of continuation. 2. Statistical treatment o f a one-dimensional model A. The model we present in this section attempts to describe the interplay between two degrees of freedom through a separable interaction which permits an exact solution o f the Schr6dinger equation [4]. The hamiltonian has the following form: Hnn, = p2 / 2 m + qn6nn , + a6(r - a)Vnn, ,
(1)
where p is the momentum conjugate to the r-variable, which should be understood as the asymptotic separation between the centers of mass of the fragments involved in the collision. The term qnfnn, represents the spectra of possible q-values for the inelastic processes. They form a band o f N statistical levels with a distribution given by p(e)
p(O) e e/k T.
=
(2)
The distribution (2) could be used in general to generate any family of non-collective levels. For simplicity we use a family of N levels with regularly decreasing spacing, localized at energies e n such that
21 April 1980
where ki] = [(2mfli2)(E - e])] 1/26i], H +- = k - l / 2 e +-ika,
F = H - - H +,
G = F H +/2 i.
(5a, b) (6)
The flux factor has been included in the wave function so that S is unitary. We can rearrange this to obtain the Born expansion, oo
T-S-
1 = - ( H + ) -1 ~ ( G V ) i G ( 2 i k ) H - . i=1
(7)
Now, we will obtain optical quantities by averaging according to Weidenmfiller's rules [1 ]. If we let =~
i=0
(av)ea = a + a vava,
(s)
or Gi = Gi + GiGi ~ VijG ~r2 --j + GiG--2i Vii, /
(9)
then we have T= -(H+)-I(G - G)(2ik)H-,
(10)
and P = v&v.
(11)
ey/
The optical quantities are all diagonal. They form a self-consistent set in the sense that substitution of P in the original expression for S yields S = T + 1. One other optical quantity which will prove to be of interest is the transmission coefficient.
p(O) f l e e/kT de = n, 0 with p(O) given by the condition
o(o)f
emax e e/kT de = N.
r = 1 - ~t~.
(12)
0 The coupling matrix elements are assumed to be members o f an ensemble with a gaussian distribution with zero mean. They are assumed to be statistically independent, that is,
B. In order to obtain explicit analytical results, we made the further assumption that
v,+vk,
We performed several numerical studies and found that the results obtained using this expression deviated only slightly from those using the more general form
=
Vi~(SikSjl + 8ilSjk ).
(3)
For an arbitrary coupling matrix V in the ensemble, we have an S matrix as S
=
1 + (H+)-I(I
-
GV)-IGVG(2ik)H
-,
(4)
v~. = w.w.. q t ]
,
V i~ = W iWI. exp [ - ( e i - e/.)2/2A2].
(13)
(14)
With this separability assumption we find 333
Volume 91B, number 3,4
PHYSICS LETTERS
21 April 1980
C i = Yi/Wi,
(15)
Using our assumption on the averaged square of the coupling matrix element, eq. (13), we find
Ti = ( - 2 i k i ) e - 2iki(Yi - WiGi)/Wi,
(16)
Oi--~ = ITil 2~if
Vi = W i ( Y + Yi),
(17)
"ci = - 4 ( k i / W i ) Im(Y + Yi) I Yil 2,
(18)
where
Yi= 1 + {1 - [2WiGi/(1 - W i G i Y ) ] 2 ) 1/2'
2 k i IY.I 2 v.' ~'I-[Y/I 2
(26)
~i=7(1-1Yi[2)"
We can rewrite the cross section as a sum of "direct" and fluctuation contributions:
i
(20)
o = Tli)(ilTt
2WiGi/(1 - W i G i Y )
(27) = TIi)(ilT¢ + ( T -
i 1 + {1 - [2WiGi/(1 - WiGiY)] 2)1/2 Y is a complex number which must be found numerically by solving eq. (20). Once Y is found, however, all other quantities in the problem would be completely determined. The non-linearity of the problem appears in the possible roots of eq. (20). We have found that only one of these solutions has a negative imaginary part and thus leads to an absorptive optical potential.
3. The averaged cross section. The optical quantities which we have calculated so far, do not depend on a particular choice of an entrance channel. To calculate the cross section we must pick an entrance channel, li). The averaged cross section is then 6 = Tli)(ilTt.
(21)
Due to the form of T, we cannot write an equation for 6 itself. However, if we let oo
¢=~
(Yi = WiGi)'
2 IYil2 i l _ l Y / I 2'
(19)
Y = ~-Jri = ~-
(25)
otiSif)l)il;f/'},,
where
7=1_
2WiGi/(1 - W i G i Y )
+ (1 +
(G V)lG(2 ik)H-li)(ilH'~ ( - 2 ik) ~
]=0
l=0
( a t v)JG t , (22)
then,
¢ = G(2ik)H-li)(ilHt(-2ik)G
+ t~ l:~l:" (~?,
(23)
T)li)(i[(T-
T)?.
The "direct" contribution is the cross section due to scattering by the optical potential in channel i and corresponds to the term I:~i[28if in the cross section. The fluctuation term accounts for all flux that is lost by the average wave function and gives the factorized term. We have compared (25) numerically with the expression
Ti Tf 6if = ITi[26if + (1 + oti6if ) ~-,jT]"
(28)
We found the relative differences between the two to be about 10% for extremely weak coupling. The relative differences decrease to less than 1% for a coupling which is still dominated by the single channel potential scattering. As the coupling increases further, the differences decrease and the single channel potential effects are washed out (fig. 1). Numerically, we have found that the cross sections given by (25) preserve unitarity. This would suggest that the denominator in the fluctuation term in (26) be modified to ~-ffT"j+ OtiT"i. We have found that the changes in the cross sections induced by this modification do not appreciably change the comparisons given above. This is to be expected, as the above modification amounts to a reduction of the fluctuation cross section by a factor
~,T
O~Ti
and
-
6 = k¢ + Ii)(il(IT-iil 2 - 4k21G:12).
z.: + ~ r i z:.:/ It is interesting to observe that if we were to drop the
334
(24)
-
~
1 -
~- 1 -
aO(1/g).
Volume 91B, number 3,4 i
1.5
PHYSICS LETTERS
I
I
21 April •980
I
r• •
.22
WL. = 5 0 MeV
r
.20
x x Xx x
I
x x
l x x
.18
X X X
36 i
X
2-
X X
.14 .12
X
_~o5
,10 .08 x
.06 I
I.O
I
I
2.0
3.0
_
4.0
E W (MeV)
Fig. 1. The fluctuation cross section plotted versus channel excitation energy with twenty open channels and strong coupling. The exact result of (25) and the approximation of (28) are indistinguishable. The incident energy of 4.01 MeV is indicated by the arrow. The coupling strength in channel i is defined as Wi = WL{p(ri) J/2. Here, emax = 4.0 MeV, kT = 1.0 MeV and WL = 50 MeV. last term in eq. (9) for G, and the corresponding term in (23) for ¢~, then c~ becomes identically zero for all coupling strengths, and the fluctuation cross section uivi-/7 of eq. (25) is exactly 7"iT"f/Z]7"/. Although in the averaging we have thrown away terms o f order 1/iV, we have not lost unitarity. Thus, we suspect that the effect o f these terms would be corrections o f order 1/iV to the individual cross sections. The elastic enhancement factor 1 + a depends only weakly on the incident energy and entrance channel, and takes the limiting values o f 2 and 1 in the weak and strong absorption limits, respectively. 4. Discussion. It is clear from our final expression, eq. (22), that the spectrum o f final states is determined completely by the optical transmission coefficient rf(Ef). We show a typical transmission coefficient as a function o f energy in fig. 2. Our calculations were made possible with the use o f several approximations; however, 7"f(Ef) still displays the correct qualitative behavior. Although our expression for the cross section resembles in form the usual compound nucleus cross section, the physics involved is quite different. The states we are dealing with here are continuum (scattering) states, having as a good quantum number the linear momentum. In the H a u s e r - F e s h b a c h theory, the
.04 ,5
1
I
I
I
I
I
'[
Lo
~.5
2.0
2.5
3.0
3.5
4.0
E*(MeV)
Fig. 2. The transmission coefficient plotted versus channel excitation energy for twenty open channels and strong coupling. The incident energy is 4.01 MeV and the coupling parameters are emax = 4.0 MeV, kT= 1.0 MeV and WL = 10 MeV.
compound states, being basically resonant states, are labelled b y the total angular momentum. There, the transmission coefficient in channel i corresponds to the probability of compotind nucleus formation from channel i. In our theory there is no compound nucleus by construction and therefore r i is the probability that the system leaks out o f channel i into any other scattering state. Thus, the transmission coefficients entering in the two cases must be interpreted differently. It should be clear that our model does contain resonances (due to the delta potential). These, however, are potential resonances and they do appear in the ri's. They are not responsible for the factorizability of the cross section. Another important difference between our formula for the average cross section and the compound nuclear Hauser-Feshbach expression resides in the elastic enhancement factor, 1 + a. In the compound nucleus case the elastic enhancement factor attains the limiting values o f 3 and 2 in the weak and strong absorption limits, respectively. We have calculated our elastic enhancement factor 1 + a and the results are shown in fig. 3. It is clear that the maximum and minimum values o f 1 + a are 2 and 1, corresponding to weak and strong absorption, respectively. We should warn the reader, however, that the above conclusions about the factor a are based on retaining the last term in eq. (9), which is a factor I/iV smaller than the sec335
Volume 91B, number 3,4 2.5
I
PHYSICS LETTERS I
I
E = 4.OI(MeV) 20
~I 1.5 +
I.O
0oo5
I
I
I
Qo5
0.5
5.o
50.0
WE (MeV) Fig. 3. T h e elastic e n h a n c e m e n t f a c t o r , 1 + ~, p l o t t e d as a
function of the coupling strength, WL, in the case of twenty open channels. Numerically, the enhancement factor showed almost no dependence on the entrance channel. Here, the incident energy is 4.01 MeV, emax = 4.0 MeV and kT = 1.0 MeV.
21 April 1980
corrections to oif would result from including nonseparable terms of the type (e i - e j ) 2 / 2 A 2. Said differently, we have made the implicit assumption of many overlapping states and our results correspond to the strong absorption case. We are presently calculating some of the corrections mentioned above. Actual comparison of our theory with the results of Weidenmiiller's study of a one-dimensional model [ 1] would certainly be worthwhile. We believe that if collective effects (besides the trivial radial motion) were present, our results could easily be generalized to include them by simply considering the "ri's as being the transmission coefficients of the eigenchannels. The actual observed averaged cross section can then be constructed using, basically, an Engelbrecht-Weidenmfiller transformation [6]. The above generalization as well as the consideration of the more general three-dimensional case are presently being pursued.
References ond term in that equation. Without this last term, = 0, both for strong and weak coupling. Since the averaging procedure which led to our optical equations is based on retaining only the leading terms in what seems to be an expansion in powers of l / N , we therefore suspect that including at least the (1/N)terms which were dropped would change the value of OL
Corrections to our formula (25) would come from relaxing the separability assumption involving Vi2 (eq. (13)). Our approximation involves dropping the factor (e i -- e j ) 2 n / 2 A 2 in the exponential appearing in V~i/. This is equivalent to assuming an infinite average spreading width for the intrinsic states [5] , 1 . The , l The authors of ref. [5] show clearly that the gaussian factor exp[-(ei - ei)2/2Lx2] is actually an approximation to a lorentzian, (r2"/4)/[(ei - ej) 2 + F2/4], with r being the average spreading width of the intrinsic states.
336
[1] D. Agassi, C.M. Ko and H.A. WeidenmiiUer,Ann. Phys. (NY) 107 (1977) 140; C.M. Ko, D. Agassiand H.A. WeidenmfiUer,Ann. Phys. (NY) 107 (1979) 237,407. [2] S. Ayik and W. N6renberg, Z. Phys. A228 (1978) 401. [3] H. Hofmann and C. Ng6, Phys. Lett. 65B (1976) 97. [4] M.C. Nemes, Nucl. Phys. A315 (1979) 457. [5] B.R. Barrett, S. Shlomo and H.A. WeidenmiiUer,Phys. Rev. C17 (1978) 544. [6] C.A. Engelbrecht and H.A. Weidenmiiller, Phys. Rev. C8 (1973) 859.
PHYSICS LETTERS
21 April 1980
OPTICAL MODEL DESCRIPTION OF DIC? B.V. CARLSON, M.C. NEMES 1 and M.S. HUSSEIN 2 Department of Physics, University of Wisconsin-Madison, Madison, W1 53706, USA Received 11 January 1980 Revised manuscript received 6 February 1980
We consider a simple one-dimensional model for a many-channel scattering problem that simulates some aspects of heavy ion deep inelastic collision processes. Applying the statistical considerations of Weidenmiiller, we obtain an averaged inelastic cross section given in terms of optical transmission coefficients.
1. Introduction. Statistical theories of deep inelastic collisions between heavy ions have been advanced [1,2] to treat those aspects o f DIC which seem to follow diffusion laws. In particular, the work o f Weidenmtiller and collaborators [1] attempts basically to relate the diffusion coefficients to some underlying microscopic interaction between the ions [2]. Similar comments can be made with regard to the work of N6renberg [2]. An important fact which one discovers in the above references is that the solution o f the collision problem is predetermined, namely the multidifferential cross section is assumed to be gaussian in its relevant final macroscopic variables. Classical equations of motion are then solved to obtain the mean values o f these macroscopic variables. The variances are obtained by solving F o k k e r - P l a n c k - t y p e equations. In this connection we would remind the reader of the work of Hofmann and collaborators [3] in which a similar procedure - albeit more phenomenological was adapted to calculate the cross section. Two important questions which are relevant to the discussion are the following: (1) To what extent is the gaussian form assumed
* Supported in part by the University of Wisconsin Alumni Foundation, the Tinker Foundation, the US NSF, the CNPq Brazil and FAPESP Brazil. 1 Present address: Max-Planck Institut fiir Kernphysik, Heidelberg, West Germany. 2 On leave from the University of S'ffoPaulo, S~o Paulo, Brazil. 332
for the distribution basically a consequence of the very large number of channels involved in DIC and not so much o f the actual nature of these channels? (2) Why do conventional optical quantities, e.g., optical potentials, transmission coefficients, etc. seem to have disappeared altogether in the final results o f the theories in refs. [1] and [2] ? Said differently, is it possible to calculate the spectra in DIC from a generalized optical potential or transmission coefficients? We should mention that in ref. [1], optical quantities do appear in the intermediate steps, although in the final solution they are de-emphasized. This, we believe, stems from the complicated nonlinear nature of the equations which have to be solved to obtain the optical potential. We shall attempt in this work to discuss the second question. We leave the answer to the first question to another future publication. Our analysis of the second question will be based on a simple one-dimensional model which mocks up to some extent some important aspects of the many-channel problem. We follow closely the statistical theory developed in ref. [1]. It is extremely gratifying that after some tedious and lengthy algebra we obtain a very simple and appealing form for the averaged cross section, namely 7i rf Oil ~ [Ti[26if + (1 + a6if ) ~ / T / ' where 7~. and r i are the optical diagonal scattering amplitude and transmission coefficient in channel i, re-
Volume 91B, number 3,4
PHYSICS LETTERS
spectively, and 1 + a is an elastic enhancement factor. The paper is organized as follows: in section 2 we introduce our model and calculate the relevant optical quantities following the averaging procedure of ref. [ 1 ]. In section 3 we deriw.~ the averaged cross section and finally, in section 4 we discuss our results and point out directions of continuation. 2. Statistical treatment o f a one-dimensional model A. The model we present in this section attempts to describe the interplay between two degrees of freedom through a separable interaction which permits an exact solution o f the Schr6dinger equation [4]. The hamiltonian has the following form: Hnn, = p2 / 2 m + qn6nn , + a6(r - a)Vnn, ,
(1)
where p is the momentum conjugate to the r-variable, which should be understood as the asymptotic separation between the centers of mass of the fragments involved in the collision. The term qnfnn, represents the spectra of possible q-values for the inelastic processes. They form a band o f N statistical levels with a distribution given by p(e)
p(O) e e/k T.
=
(2)
The distribution (2) could be used in general to generate any family of non-collective levels. For simplicity we use a family of N levels with regularly decreasing spacing, localized at energies e n such that
21 April 1980
where ki] = [(2mfli2)(E - e])] 1/26i], H +- = k - l / 2 e +-ika,
F = H - - H +,
G = F H +/2 i.
(5a, b) (6)
The flux factor has been included in the wave function so that S is unitary. We can rearrange this to obtain the Born expansion, oo
T-S-
1 = - ( H + ) -1 ~ ( G V ) i G ( 2 i k ) H - . i=1
(7)
Now, we will obtain optical quantities by averaging according to Weidenmfiller's rules [1 ]. If we let =~
i=0
(av)ea = a + a vava,
(s)
or Gi = Gi + GiGi ~ VijG ~r2 --j + GiG--2i Vii, /
(9)
then we have T= -(H+)-I(G - G)(2ik)H-,
(10)
and P = v&v.
(11)
ey/
The optical quantities are all diagonal. They form a self-consistent set in the sense that substitution of P in the original expression for S yields S = T + 1. One other optical quantity which will prove to be of interest is the transmission coefficient.
p(O) f l e e/kT de = n, 0 with p(O) given by the condition
o(o)f
emax e e/kT de = N.
r = 1 - ~t~.
(12)
0 The coupling matrix elements are assumed to be members o f an ensemble with a gaussian distribution with zero mean. They are assumed to be statistically independent, that is,
B. In order to obtain explicit analytical results, we made the further assumption that
v,+vk,
We performed several numerical studies and found that the results obtained using this expression deviated only slightly from those using the more general form
=
Vi~(SikSjl + 8ilSjk ).
(3)
For an arbitrary coupling matrix V in the ensemble, we have an S matrix as S
=
1 + (H+)-I(I
-
GV)-IGVG(2ik)H
-,
(4)
v~. = w.w.. q t ]
,
V i~ = W iWI. exp [ - ( e i - e/.)2/2A2].
(13)
(14)
With this separability assumption we find 333
Volume 91B, number 3,4
PHYSICS LETTERS
21 April 1980
C i = Yi/Wi,
(15)
Using our assumption on the averaged square of the coupling matrix element, eq. (13), we find
Ti = ( - 2 i k i ) e - 2iki(Yi - WiGi)/Wi,
(16)
Oi--~ = ITil 2~if
Vi = W i ( Y + Yi),
(17)
"ci = - 4 ( k i / W i ) Im(Y + Yi) I Yil 2,
(18)
where
Yi= 1 + {1 - [2WiGi/(1 - W i G i Y ) ] 2 ) 1/2'
2 k i IY.I 2 v.' ~'I-[Y/I 2
(26)
~i=7(1-1Yi[2)"
We can rewrite the cross section as a sum of "direct" and fluctuation contributions:
i
(20)
o = Tli)(ilTt
2WiGi/(1 - W i G i Y )
(27) = TIi)(ilT¢ + ( T -
i 1 + {1 - [2WiGi/(1 - WiGiY)] 2)1/2 Y is a complex number which must be found numerically by solving eq. (20). Once Y is found, however, all other quantities in the problem would be completely determined. The non-linearity of the problem appears in the possible roots of eq. (20). We have found that only one of these solutions has a negative imaginary part and thus leads to an absorptive optical potential.
3. The averaged cross section. The optical quantities which we have calculated so far, do not depend on a particular choice of an entrance channel. To calculate the cross section we must pick an entrance channel, li). The averaged cross section is then 6 = Tli)(ilTt.
(21)
Due to the form of T, we cannot write an equation for 6 itself. However, if we let oo
¢=~
(Yi = WiGi)'
2 IYil2 i l _ l Y / I 2'
(19)
Y = ~-Jri = ~-
(25)
otiSif)l)il;f/'},,
where
7=1_
2WiGi/(1 - W i G i Y )
+ (1 +
(G V)lG(2 ik)H-li)(ilH'~ ( - 2 ik) ~
]=0
l=0
( a t v)JG t , (22)
then,
¢ = G(2ik)H-li)(ilHt(-2ik)G
+ t~ l:~l:" (~?,
(23)
T)li)(i[(T-
T)?.
The "direct" contribution is the cross section due to scattering by the optical potential in channel i and corresponds to the term I:~i[28if in the cross section. The fluctuation term accounts for all flux that is lost by the average wave function and gives the factorized term. We have compared (25) numerically with the expression
Ti Tf 6if = ITi[26if + (1 + oti6if ) ~-,jT]"
(28)
We found the relative differences between the two to be about 10% for extremely weak coupling. The relative differences decrease to less than 1% for a coupling which is still dominated by the single channel potential scattering. As the coupling increases further, the differences decrease and the single channel potential effects are washed out (fig. 1). Numerically, we have found that the cross sections given by (25) preserve unitarity. This would suggest that the denominator in the fluctuation term in (26) be modified to ~-ffT"j+ OtiT"i. We have found that the changes in the cross sections induced by this modification do not appreciably change the comparisons given above. This is to be expected, as the above modification amounts to a reduction of the fluctuation cross section by a factor
~,T
O~Ti
and
-
6 = k¢ + Ii)(il(IT-iil 2 - 4k21G:12).
z.: + ~ r i z:.:/ It is interesting to observe that if we were to drop the
334
(24)
-
~
1 -
~- 1 -
aO(1/g).
Volume 91B, number 3,4 i
1.5
PHYSICS LETTERS
I
I
21 April •980
I
r• •
.22
WL. = 5 0 MeV
r
.20
x x Xx x
I
x x
l x x
.18
X X X
36 i
X
2-
X X
.14 .12
X
_~o5
,10 .08 x
.06 I
I.O
I
I
2.0
3.0
_
4.0
E W (MeV)
Fig. 1. The fluctuation cross section plotted versus channel excitation energy with twenty open channels and strong coupling. The exact result of (25) and the approximation of (28) are indistinguishable. The incident energy of 4.01 MeV is indicated by the arrow. The coupling strength in channel i is defined as Wi = WL{p(ri) J/2. Here, emax = 4.0 MeV, kT = 1.0 MeV and WL = 50 MeV. last term in eq. (9) for G, and the corresponding term in (23) for ¢~, then c~ becomes identically zero for all coupling strengths, and the fluctuation cross section uivi-/7 of eq. (25) is exactly 7"iT"f/Z]7"/. Although in the averaging we have thrown away terms o f order 1/iV, we have not lost unitarity. Thus, we suspect that the effect o f these terms would be corrections o f order 1/iV to the individual cross sections. The elastic enhancement factor 1 + a depends only weakly on the incident energy and entrance channel, and takes the limiting values o f 2 and 1 in the weak and strong absorption limits, respectively. 4. Discussion. It is clear from our final expression, eq. (22), that the spectrum o f final states is determined completely by the optical transmission coefficient rf(Ef). We show a typical transmission coefficient as a function o f energy in fig. 2. Our calculations were made possible with the use o f several approximations; however, 7"f(Ef) still displays the correct qualitative behavior. Although our expression for the cross section resembles in form the usual compound nucleus cross section, the physics involved is quite different. The states we are dealing with here are continuum (scattering) states, having as a good quantum number the linear momentum. In the H a u s e r - F e s h b a c h theory, the
.04 ,5
1
I
I
I
I
I
'[
Lo
~.5
2.0
2.5
3.0
3.5
4.0
E*(MeV)
Fig. 2. The transmission coefficient plotted versus channel excitation energy for twenty open channels and strong coupling. The incident energy is 4.01 MeV and the coupling parameters are emax = 4.0 MeV, kT= 1.0 MeV and WL = 10 MeV.
compound states, being basically resonant states, are labelled b y the total angular momentum. There, the transmission coefficient in channel i corresponds to the probability of compotind nucleus formation from channel i. In our theory there is no compound nucleus by construction and therefore r i is the probability that the system leaks out o f channel i into any other scattering state. Thus, the transmission coefficients entering in the two cases must be interpreted differently. It should be clear that our model does contain resonances (due to the delta potential). These, however, are potential resonances and they do appear in the ri's. They are not responsible for the factorizability of the cross section. Another important difference between our formula for the average cross section and the compound nuclear Hauser-Feshbach expression resides in the elastic enhancement factor, 1 + a. In the compound nucleus case the elastic enhancement factor attains the limiting values o f 3 and 2 in the weak and strong absorption limits, respectively. We have calculated our elastic enhancement factor 1 + a and the results are shown in fig. 3. It is clear that the maximum and minimum values o f 1 + a are 2 and 1, corresponding to weak and strong absorption, respectively. We should warn the reader, however, that the above conclusions about the factor a are based on retaining the last term in eq. (9), which is a factor I/iV smaller than the sec335
Volume 91B, number 3,4 2.5
I
PHYSICS LETTERS I
I
E = 4.OI(MeV) 20
~I 1.5 +
I.O
0oo5
I
I
I
Qo5
0.5
5.o
50.0
WE (MeV) Fig. 3. T h e elastic e n h a n c e m e n t f a c t o r , 1 + ~, p l o t t e d as a
function of the coupling strength, WL, in the case of twenty open channels. Numerically, the enhancement factor showed almost no dependence on the entrance channel. Here, the incident energy is 4.01 MeV, emax = 4.0 MeV and kT = 1.0 MeV.
21 April 1980
corrections to oif would result from including nonseparable terms of the type (e i - e j ) 2 / 2 A 2. Said differently, we have made the implicit assumption of many overlapping states and our results correspond to the strong absorption case. We are presently calculating some of the corrections mentioned above. Actual comparison of our theory with the results of Weidenmiiller's study of a one-dimensional model [ 1] would certainly be worthwhile. We believe that if collective effects (besides the trivial radial motion) were present, our results could easily be generalized to include them by simply considering the "ri's as being the transmission coefficients of the eigenchannels. The actual observed averaged cross section can then be constructed using, basically, an Engelbrecht-Weidenmfiller transformation [6]. The above generalization as well as the consideration of the more general three-dimensional case are presently being pursued.
References ond term in that equation. Without this last term, = 0, both for strong and weak coupling. Since the averaging procedure which led to our optical equations is based on retaining only the leading terms in what seems to be an expansion in powers of l / N , we therefore suspect that including at least the (1/N)terms which were dropped would change the value of OL
Corrections to our formula (25) would come from relaxing the separability assumption involving Vi2 (eq. (13)). Our approximation involves dropping the factor (e i -- e j ) 2 n / 2 A 2 in the exponential appearing in V~i/. This is equivalent to assuming an infinite average spreading width for the intrinsic states [5] , 1 . The , l The authors of ref. [5] show clearly that the gaussian factor exp[-(ei - ei)2/2Lx2] is actually an approximation to a lorentzian, (r2"/4)/[(ei - ej) 2 + F2/4], with r being the average spreading width of the intrinsic states.
336
[1] D. Agassi, C.M. Ko and H.A. WeidenmiiUer,Ann. Phys. (NY) 107 (1977) 140; C.M. Ko, D. Agassiand H.A. WeidenmfiUer,Ann. Phys. (NY) 107 (1979) 237,407. [2] S. Ayik and W. N6renberg, Z. Phys. A228 (1978) 401. [3] H. Hofmann and C. Ng6, Phys. Lett. 65B (1976) 97. [4] M.C. Nemes, Nucl. Phys. A315 (1979) 457. [5] B.R. Barrett, S. Shlomo and H.A. WeidenmiiUer,Phys. Rev. C17 (1978) 544. [6] C.A. Engelbrecht and H.A. Weidenmiiller, Phys. Rev. C8 (1973) 859.