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Statistical intermediate structures in compound nucleus reactions
Volume 103B, number 4,5
PHYSICS LETTERS
30 July 1981
STATISTICAL INTERMEDIATE STRUCTURES IN COMPOUND NUCLEUS REACTIONS W.R. GIBBS Los A lamos National Laboratory, Theoretical Division, Los Alamo& NM 87545, USA
Received 6 April 1981
A study of the 12C + 12C and 160 + 160 elastic scattering reactions is made in terms of a general stochastic theory of nuclear excitation functions. It is shown that a general understanding of the cross section and a quantitive description of the position and widths of the structures observed is achieved in these terms. Calculations of average elastic widths for the relevant angular momenta are presented and compared with data.
The H a u s e r - F e s h b a c h theory of statistical nuclear reactions has been the mainstay o f compound nucleus calculations for a number of years. This theory has the virtues of stationarity and ergocity. A general theory of the nuclear excitation function as a stochastic process need not have either o f these useful and pleasant features. In this case such a (more general) theory is based on ensemble averages alone. A number of such theories have been considered. For the inelastic reactions unitarity appears not to be a serious constraint because o f the small cross sections. In fact, ergotic theories have been developed in this case (such as the H a u s e r - F e s h b a c h theory for average properties and Ericson's technique for the treatment o f fluctuations about the mean). For the elastic cross section, however, unitarity provides a powerffd constraint and thereby strong modifications are imposed on the ensemble averages. As a result, non-stationary effects may appear. When one looks at gross average quantities these effects may not appear to be important. However, when certain energy regions are investigated or specific quantities demanded (e.g. behavior of phase shifts), these once seemingly trivial modifications may dominate the important physics. Aside from the tenant o f regarding the excitation function as a general stochastic process the principal modification considered here is due to unitarity and alters the basic H a u s e r - F e s h b a c h formula relating the
width to the transmission coefficient: 2,rP/D = T .
(1)
This formula was derived originally under the assumption o f small T. It is generally admitted that this relation has no validity for T ~ 1. The corrections to this expression for large T appear to be related to unitarity. An alternate expression relating the width to T may be obtained using the R-matrix formalism [1 ] (either in a "picket fence" approximation or an ensemble of level spacings), or a product representation of the Smatrix [2], or, in fact, using the original Blatt and Weiskopf [3] derivation but not making the assumption of small T [2]. The common feature of all o f these derivations, although they use additional approximations of various types, is that unitarity is strictly preserved. A remarkable feature is that they all lead to the same expression, viz. 2rrF/D = - l n ( 1 - T).
(2)
Devaney [41 has shown that this formula is indeed very accurate for the single-channel case where it can be tested in detail. Due to the slow growth o f the logarithm function the difference between these two expressions is not very great in most cases. In fact, in the calculation of total widths they give very similar results [5]. A unique opportunity for the testing of the validity of eq. (2) occurs in the collisions of heavy ions. In this 281
Volume 103B, number 4,5
PHYSICS LETTERS
+1 As quoted in ref. [10].
~Oo' '|I I' ' 'll, o.... / \
50
"~
I ....
I i
L ....
IZC + 12C Elastic 90° Cr°ss Secti°n
^l l
q,
?,
n
'
-3,o
kA
/ u'\l t
,,,8 l" ~ ~, I0
~
1]~. I
k-. . . . . . [5
70
-- 6.0 --5.0
j,7 ~
1981
For the carbon case strong evidence has recently been obtained showing that the sharp features observed in the elastic scattering are correlated with reaction channels exiting from the 12C + 12C entrance channel [7], and that the gross structures show a consistent single angular momentum [ 11 ]. Straight-forward calculations using eq. (2) and the philosophy of a general stochastic process provide an understanding of both of these features. Fig. 1 shows the measured 90 ° cross section of the carbon elastic scattering and the calculated average elastic width as a function of CM bombarding energy. The elastic widths are calculated using eq. (2) and the optical potential of Rickersen +~. (The potential was modified slightly by taking a radius of 5.90 instead of 6.18. This shifts the peak of the gross structure by ~0.5 MeV.) The energy dependent potential of ref. [12] gives similar results. Level densities are taken from Gilbert and Cameron (13) except for the spin dependence which is calculated using a rigid moment of inertia [5] (R = 3.30). The absolute values of the widths would appear to be rather small but, while the positions and general shape of gross structures are insensitive to the
case the effective optical potential is rather weak so that T approaches the value 1 very closely. For the collision of identical heavy ions only even partial waves contribute so that the features predicted by eq. (2) become well separated and are observable. Standard optical model calculations show that T rises from zero for small energy (being suppressed by Coulomb and angular-momentum barriers) to a value close to 1 and then drops again slowly (see e.g. ref. [5]. The effect ofeq. (2) is to augment very strongly the region around T ~ 1. Thus the rate of entry into the compound nucleus (or fusion) state is not limited by eq. (1). Transparent "windows" may appear. Such features have been considered for inelastic reactions with some success [6]. To show the effect of eq. (2) two popular low-energy heavy-ion elastic-scattering cross sections have been examined. These are 12C + 12C [7] and 160 + 160 [8]. A number of sharp structures have been seen in these cross sections and groupings into larger scale features have been identified. The excitation function of 12C has been treated in the theory of Ericson fluctuations with reasonable success [9]. More often some evidence has been seen for "quasi-molecular" states in the gross structure [10].
60
30 July
,° ...---..
jill', / |l
I
/J~l
\
//
k ', /"
~-~J--'~'20
-20 "\
'~,--"
16
i-'--'--e~-~----'l---i---'r~ N---t--25
30
Ecru (MeV)
Fig. 1. Elastic scattering o f 12C on 12C. T h e solid curve is a representation of the data [7 ] and is referred to at the left scale. The dashed curves axe calculations of t h e expectation value of the elastic widths (labeled by angular m o m e n t u m ) as described in the text. T h e y axe referred to at the right scale.
282
Volume 103B, number 4,5
PHYSICS LETTERS
level parameters, decreasing the value of " a " (the single-particle level density) will increase the width substantially. The fine structures are taken to be due to large values of the elastic widths which are much more probable when a large value of the average for the (presumed P o r t e r - T h o m a s ) distribution is present. These large (elastic) width states will tend to cause a large cross section in all reaction channels fed from the 12C + 12C entrance channel as observed [7]. Note that a correlation with reactions not involving this entrance channel would not be expected. While a large value of the ensemble average of the width leads to an expectation of a large elastic cross section the relation is fat from linear: (a) The actual value of the width must be drawn from a distribution based on the average value. The probability of finding a large width is highly nonlinear in the average width. (b) The cross section is proportional to the square of the resulting width and (c) depends on the ratio to the total width for the given angular m o m e n t u m and energy. This total width would seem to have a gentle J dependence but its exact value is a matter of some uncertainty. It is for these reasons that no complete cross
f
r
r
p
I
j
p
,
I
30 July 1981
sections are estimated but only expectation values o f elastic widths are given. It may be noted that the grouping o f states around 20 MeV has recently been verified to have J = 12 and that around 25 MeV to have J = 14 [11], Indeed the identification of structures, as such, is not completely clear without this correspondence. Fig. 2 shows the results for the oxygen scattering. Here the structures are very clear and correspond extremely well with the maxima of the average width. The optical potential used here is essentially the same as for the first case but with a radius of 6.7. The moment of inertia was calculated with R = 3.50. In this case the use o f a = 2.72 (instead of the 3,36 prescribed by ref. [13] and used in the calculation of fig. 2) leads to an increase of the average width by a factor o f 5 but leaves the shape essentially unchanged. It should be mentioned that the use of eq. (2) may well have profound influence on the interpretation of anomalies such as that seen in the 12C(13C, n)24Mg [14] or 13C(160, 170)12C [15] reactions. Perhaps it is worthwhile to emphasize the strength and weakness of such a statistical approach. The strength is that it allows the prediction of ge-
l
[
~
~
p
I
I
'
J
i
I
160 + 160 Elostic 25
9 0 ° Cross Section
20
,~;-',
gm II
,6,-,
;
I\l-,l /i l vl , \
5p
b,' k 15
I/
/i ,;,\ 1;
20
0.8
I
I
<~
i
l a,
b
-
/'~ ,
k
,;/ I,, / I',,
'1
I\
I', ,'F t"..'.l
o6
,8/',
/
/
t',/ ¢t/t.x t .i.. j L - " r
25
50
\ /
,,'" -.. 35
Ecru (MeV)
Fig. 2. Elastic scattering of 160 on 160, The solid curve is a representation of the data [8 ] and is referred to at the left scale. The dashed curves are calculations of the expection value of the elastic widths (labeled by angular momentum) as described in the text, They are referred to at the right scale. 283
Volume 103B, number 4,5
PHYSICS LETTERS
neral features and make more or less firm predictions about physical phenomena. It may be very useful in the forecasting of interesting energy domains for experiments. The principal weakness is that it does not provide a microscopic description of the process and should not discourage a more detailed excursion into a precise theory of the underlying structure. This work was supported by the US DOE
References [1] P.A. Moldauer, Phys. Rev. Lett. 18 (1967) 249; 19 (1967) 1047; Phys. Rev. 157 (1967) 907; 171 (1968) 164. [2] W.R. Gibbs, Phys. Rev. 181 (1969) 1414. [3] J.M. Blatt and V.T. Weisskopf, Theoretical nuclear physics (Wiley, New York). [4] J.J. Devaney, Phys. Rev. C6 (1972) 1162.
284
30 July 1981
[5 ] R.B. Leachman, P. Fessenden and W.R. Gibbs, Phys. Rev. C6 (1972) 1240. [6] R.L. Phillips, K.A. Erb and D.A. Bromley, Phys. Rev. Lett. 42 (1979) 566. [7] E.R. Cosman, R. Ledoux and A.J. Lazzaxini, Phys. Rev. C21 (1980) 2111. [8 ] J.V. Maher, M.V. Sachs, R.H. Simssen, A. Weidingerand D.A. Bromley, Phys. Rev. 188 (1969) 1665. [9] J. Bondorf, Nucl. Phys. A202 (1972) 30. [10 ] N. Cindro, ed., Nuclear molecular phenomena, Pro c. Intern. Conf. on Resonances in heavy ion reactions (Hvar, Adriatic Coast, Yugoslavia, 1977). [11] E. Cosman, R. Ledoux and M. Bechara, Bull. Am. Phys. Soc. 26 (1981) 26; and M.I.T. preprint. [12] D. Shapira, R.G. Stokstad and D.A. Bromley, Phys. Rev. C10 (1974) 1063. [13] A. Gilbert and A.G.W. Cameron, Can. J. Phys. 43 (1965) 1446. [14] K.N. Geller and R.V. Kollartis, Phys. Rev. Lett. 37 (1976) 279. [15] P.T. Debevec, H.J. Korner and J.P. Schiffer, Phys. Rev. Lett. 31 (1973) 171.
PHYSICS LETTERS
30 July 1981
STATISTICAL INTERMEDIATE STRUCTURES IN COMPOUND NUCLEUS REACTIONS W.R. GIBBS Los A lamos National Laboratory, Theoretical Division, Los Alamo& NM 87545, USA
Received 6 April 1981
A study of the 12C + 12C and 160 + 160 elastic scattering reactions is made in terms of a general stochastic theory of nuclear excitation functions. It is shown that a general understanding of the cross section and a quantitive description of the position and widths of the structures observed is achieved in these terms. Calculations of average elastic widths for the relevant angular momenta are presented and compared with data.
The H a u s e r - F e s h b a c h theory of statistical nuclear reactions has been the mainstay o f compound nucleus calculations for a number of years. This theory has the virtues of stationarity and ergocity. A general theory of the nuclear excitation function as a stochastic process need not have either o f these useful and pleasant features. In this case such a (more general) theory is based on ensemble averages alone. A number of such theories have been considered. For the inelastic reactions unitarity appears not to be a serious constraint because o f the small cross sections. In fact, ergotic theories have been developed in this case (such as the H a u s e r - F e s h b a c h theory for average properties and Ericson's technique for the treatment o f fluctuations about the mean). For the elastic cross section, however, unitarity provides a powerffd constraint and thereby strong modifications are imposed on the ensemble averages. As a result, non-stationary effects may appear. When one looks at gross average quantities these effects may not appear to be important. However, when certain energy regions are investigated or specific quantities demanded (e.g. behavior of phase shifts), these once seemingly trivial modifications may dominate the important physics. Aside from the tenant o f regarding the excitation function as a general stochastic process the principal modification considered here is due to unitarity and alters the basic H a u s e r - F e s h b a c h formula relating the
width to the transmission coefficient: 2,rP/D = T .
(1)
This formula was derived originally under the assumption o f small T. It is generally admitted that this relation has no validity for T ~ 1. The corrections to this expression for large T appear to be related to unitarity. An alternate expression relating the width to T may be obtained using the R-matrix formalism [1 ] (either in a "picket fence" approximation or an ensemble of level spacings), or a product representation of the Smatrix [2], or, in fact, using the original Blatt and Weiskopf [3] derivation but not making the assumption of small T [2]. The common feature of all o f these derivations, although they use additional approximations of various types, is that unitarity is strictly preserved. A remarkable feature is that they all lead to the same expression, viz. 2rrF/D = - l n ( 1 - T).
(2)
Devaney [41 has shown that this formula is indeed very accurate for the single-channel case where it can be tested in detail. Due to the slow growth o f the logarithm function the difference between these two expressions is not very great in most cases. In fact, in the calculation of total widths they give very similar results [5]. A unique opportunity for the testing of the validity of eq. (2) occurs in the collisions of heavy ions. In this 281
Volume 103B, number 4,5
PHYSICS LETTERS
+1 As quoted in ref. [10].
~Oo' '|I I' ' 'll, o.... / \
50
"~
I ....
I i
L ....
IZC + 12C Elastic 90° Cr°ss Secti°n
^l l
q,
?,
n
'
-3,o
kA
/ u'\l t
,,,8 l" ~ ~, I0
~
1]~. I
k-. . . . . . [5
70
-- 6.0 --5.0
j,7 ~
1981
For the carbon case strong evidence has recently been obtained showing that the sharp features observed in the elastic scattering are correlated with reaction channels exiting from the 12C + 12C entrance channel [7], and that the gross structures show a consistent single angular momentum [ 11 ]. Straight-forward calculations using eq. (2) and the philosophy of a general stochastic process provide an understanding of both of these features. Fig. 1 shows the measured 90 ° cross section of the carbon elastic scattering and the calculated average elastic width as a function of CM bombarding energy. The elastic widths are calculated using eq. (2) and the optical potential of Rickersen +~. (The potential was modified slightly by taking a radius of 5.90 instead of 6.18. This shifts the peak of the gross structure by ~0.5 MeV.) The energy dependent potential of ref. [12] gives similar results. Level densities are taken from Gilbert and Cameron (13) except for the spin dependence which is calculated using a rigid moment of inertia [5] (R = 3.30). The absolute values of the widths would appear to be rather small but, while the positions and general shape of gross structures are insensitive to the
case the effective optical potential is rather weak so that T approaches the value 1 very closely. For the collision of identical heavy ions only even partial waves contribute so that the features predicted by eq. (2) become well separated and are observable. Standard optical model calculations show that T rises from zero for small energy (being suppressed by Coulomb and angular-momentum barriers) to a value close to 1 and then drops again slowly (see e.g. ref. [5]. The effect ofeq. (2) is to augment very strongly the region around T ~ 1. Thus the rate of entry into the compound nucleus (or fusion) state is not limited by eq. (1). Transparent "windows" may appear. Such features have been considered for inelastic reactions with some success [6]. To show the effect of eq. (2) two popular low-energy heavy-ion elastic-scattering cross sections have been examined. These are 12C + 12C [7] and 160 + 160 [8]. A number of sharp structures have been seen in these cross sections and groupings into larger scale features have been identified. The excitation function of 12C has been treated in the theory of Ericson fluctuations with reasonable success [9]. More often some evidence has been seen for "quasi-molecular" states in the gross structure [10].
60
30 July
,° ...---..
jill', / |l
I
/J~l
\
//
k ', /"
~-~J--'~'20
-20 "\
'~,--"
16
i-'--'--e~-~----'l---i---'r~ N---t--25
30
Ecru (MeV)
Fig. 1. Elastic scattering o f 12C on 12C. T h e solid curve is a representation of the data [7 ] and is referred to at the left scale. The dashed curves axe calculations of t h e expectation value of the elastic widths (labeled by angular m o m e n t u m ) as described in the text. T h e y axe referred to at the right scale.
282
Volume 103B, number 4,5
PHYSICS LETTERS
level parameters, decreasing the value of " a " (the single-particle level density) will increase the width substantially. The fine structures are taken to be due to large values of the elastic widths which are much more probable when a large value of the average for the (presumed P o r t e r - T h o m a s ) distribution is present. These large (elastic) width states will tend to cause a large cross section in all reaction channels fed from the 12C + 12C entrance channel as observed [7]. Note that a correlation with reactions not involving this entrance channel would not be expected. While a large value of the ensemble average of the width leads to an expectation of a large elastic cross section the relation is fat from linear: (a) The actual value of the width must be drawn from a distribution based on the average value. The probability of finding a large width is highly nonlinear in the average width. (b) The cross section is proportional to the square of the resulting width and (c) depends on the ratio to the total width for the given angular m o m e n t u m and energy. This total width would seem to have a gentle J dependence but its exact value is a matter of some uncertainty. It is for these reasons that no complete cross
f
r
r
p
I
j
p
,
I
30 July 1981
sections are estimated but only expectation values o f elastic widths are given. It may be noted that the grouping o f states around 20 MeV has recently been verified to have J = 12 and that around 25 MeV to have J = 14 [11], Indeed the identification of structures, as such, is not completely clear without this correspondence. Fig. 2 shows the results for the oxygen scattering. Here the structures are very clear and correspond extremely well with the maxima of the average width. The optical potential used here is essentially the same as for the first case but with a radius of 6.7. The moment of inertia was calculated with R = 3.50. In this case the use o f a = 2.72 (instead of the 3,36 prescribed by ref. [13] and used in the calculation of fig. 2) leads to an increase of the average width by a factor o f 5 but leaves the shape essentially unchanged. It should be mentioned that the use of eq. (2) may well have profound influence on the interpretation of anomalies such as that seen in the 12C(13C, n)24Mg [14] or 13C(160, 170)12C [15] reactions. Perhaps it is worthwhile to emphasize the strength and weakness of such a statistical approach. The strength is that it allows the prediction of ge-
l
[
~
~
p
I
I
'
J
i
I
160 + 160 Elostic 25
9 0 ° Cross Section
20
,~;-',
gm II
,6,-,
;
I\l-,l /i l vl , \
5p
b,' k 15
I/
/i ,;,\ 1;
20
0.8
I
I
<~
i
l a,
b
-
/'~ ,
k
,;/ I,, / I',,
'1
I\
I', ,'F t"..'.l
o6
,8/',
/
/
t',/ ¢t/t.x t .i.. j L - " r
25
50
\ /
,,'" -.. 35
Ecru (MeV)
Fig. 2. Elastic scattering of 160 on 160, The solid curve is a representation of the data [8 ] and is referred to at the left scale. The dashed curves are calculations of the expection value of the elastic widths (labeled by angular momentum) as described in the text, They are referred to at the right scale. 283
Volume 103B, number 4,5
PHYSICS LETTERS
neral features and make more or less firm predictions about physical phenomena. It may be very useful in the forecasting of interesting energy domains for experiments. The principal weakness is that it does not provide a microscopic description of the process and should not discourage a more detailed excursion into a precise theory of the underlying structure. This work was supported by the US DOE
References [1] P.A. Moldauer, Phys. Rev. Lett. 18 (1967) 249; 19 (1967) 1047; Phys. Rev. 157 (1967) 907; 171 (1968) 164. [2] W.R. Gibbs, Phys. Rev. 181 (1969) 1414. [3] J.M. Blatt and V.T. Weisskopf, Theoretical nuclear physics (Wiley, New York). [4] J.J. Devaney, Phys. Rev. C6 (1972) 1162.
284
30 July 1981
[5 ] R.B. Leachman, P. Fessenden and W.R. Gibbs, Phys. Rev. C6 (1972) 1240. [6] R.L. Phillips, K.A. Erb and D.A. Bromley, Phys. Rev. Lett. 42 (1979) 566. [7] E.R. Cosman, R. Ledoux and A.J. Lazzaxini, Phys. Rev. C21 (1980) 2111. [8 ] J.V. Maher, M.V. Sachs, R.H. Simssen, A. Weidingerand D.A. Bromley, Phys. Rev. 188 (1969) 1665. [9] J. Bondorf, Nucl. Phys. A202 (1972) 30. [10 ] N. Cindro, ed., Nuclear molecular phenomena, Pro c. Intern. Conf. on Resonances in heavy ion reactions (Hvar, Adriatic Coast, Yugoslavia, 1977). [11] E. Cosman, R. Ledoux and M. Bechara, Bull. Am. Phys. Soc. 26 (1981) 26; and M.I.T. preprint. [12] D. Shapira, R.G. Stokstad and D.A. Bromley, Phys. Rev. C10 (1974) 1063. [13] A. Gilbert and A.G.W. Cameron, Can. J. Phys. 43 (1965) 1446. [14] K.N. Geller and R.V. Kollartis, Phys. Rev. Lett. 37 (1976) 279. [15] P.T. Debevec, H.J. Korner and J.P. Schiffer, Phys. Rev. Lett. 31 (1973) 171.