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Bremsstrahlung in heavy-ion molecular resonances
Nuclear Physics A440 (1985) 173-188 @North-Holland Publishing Company
BREMSSTRAHLUNG
IN HEAVY-ION MOLECULAR RESONANCES’ 0. TANIMURA and U. MOSEL
i~s~i~~t~~r Th~~relisch~ P&w&,
U~iuer~it~f Giesstn, 6300 Giessen, E R. Germuy
Received 12 October 1984 Abstract: The bremsstr~lun8 cross section for the quadrupole photon emission in a heavy-ion collision is fo~u~at~d on the basis of the coupi~-charnel method. For the branching ratio of the radiative decay width to the total width for the transition between the structures at 25.6 MeV and 19.3 MeV in the 12C + 12C collision, we predict (3 - 7) X lo-‘, which is roughly comparable with existing measurements. The radiation from the mutual 2’ excitation channel is found to yield 70% of the total branching ratio.
1. In~~uction
Prominent structures have been found in the excitation functions of the elastic and inelastic 12C +‘*C cross sections’-4). These structures have often been interpreted in terms of the “molecular” resonances5-8), since they appear near the grazing angular momenta in the weak abso~tion window. There have been attempts to obtain a deeper understanding of these states from a microscopic viewpoint 9-11). Chandra and Mose19), on the basis of the two-center shell model, have predicted the existence of quasi-molecular di-carbon configurations in the excited states of 24Mg [see also Ragnarsson et al. “)I. They have suggested that these states correspond to the “sticking” configuration which is closely associated with the enhancement of the mutual 2+ channel. The recent spin-alignment measurement by Konnerth et al. 12) indicates that this configuration seems indeed to be favoured for the 25.6 MeV structure of the mutual excitation channel. If these states form a rotational band with a well-defined intrinsic state, one expects a large enhancement of the quad~pole photon emission as in the rotational nuclei. measurements of the E2 transition during the collision of two carbon nuclei have been carried out by Metag et al. 13-14)and McCrath et al. 15) independently. Metag et al. 14) have deduced a branching ratio, I’,/I’,,,, of (6 & 3) x lop7for the photon emission between the initial energy of 25.7 MeV and the final energies around 19.3 MeV. Since this y-decay width is much smaller than the rigid-rotor limit, these authors have concluded that an interpretation of the gross structure in terms of the molecular band is unlikely. McGrath et al. 15) have obtained an upper limit for ’ Work supported by BMFT and GSI. 173
174
0. Tanimura,
(1. Mod
/
Bremsstrahlung
the ratio of (2 - 8) X 10-6, which also shows no collective y-emission between the intermediate resonances but is consistent with a potential resonance interpretation. Nyman16) Feshbach
has and
calculated
Yennie’s
the
bremsstrablung
formula”)
cross
for this transition
section
on
and obtained
the
basis
of
the value
of
7 X lo-’ dE,/E, as the number of bremsstrahlung events per elastic scattering, which is comparable to the measured branching ratio14). This estimate, however, neglects the effects of the collective excitation channels. Since the molecular-like states decay to the single 2+ and mutual 2+ channels with considerable probabilities through the strong-coupling interaction, it is essential to take the couplings to these channels into account. Langanke and van Roosmalen”) have estimated the B(E2) values of the combined 12C-12C system for 1~ 10 in a microscopic model, and obtain values similar to the predictions by the rigid-rotor model for two touching nuclei. The purpose of this paper is to get information of the molecular-like structures of the 12C-12C system through the analysis of y-decay measurements. On the basis of the coupled-channel method we calculate directly the bremsstrahlung cross sections for E2 transitions during the collision in the same manner as Blair and Sherif 19) did in the DWBA framework. The initial and the final scattering wave functions are determined so as to fit the measured cross sections in the elastic, the single 2’ and the mutual 2+ excitation channels.
2. Evaluation of bremsstrahlung cross sections The
bremsstrahlung
cross
section
for the 2’-pole
photon
emission
has
been
formulated by Alder ef al. 20). For a heavy-ion collision, the double-differential cross section of the Eh radiation per solid angle s2 and photon momentum q is given as 20) 2 ZlAE;
+
(4
’
CI(K,lrXY~,(P)IIUi>12,
(-)*Z,A:
+A2Y
(1)
P
where we have used the same notations as in ref. 20). The indices i and f specify the initial and final channels, respectively. In using eq. (1) the assumption is made that the intrinsic quadrupole moment of the interacting nuclei can be neglected in calculating the y-emissions. This is justified because the quadrupole moment due to the relative distance between the nuclei is much larger than the intrinsic value. Since we are interested in the interaction of nuclei with strongly excited collective states, we couple these states to the ground channel states to obtain the relevant wave functions. In treating the coupled-channel wave functions it is convenient to use the r-representations of IK,) and IK,) and write the boundary conditions
explicitly, i.e. TP$I$K~, r> and ~~~~~~~~,~) where Q denotes z-component ( Inif) of the aucleus in the a~~~tot~~ region. The wave function may be written as 2s)
the
spin
and its
with
The quantity E is taken to be 1 for ~d~~ti~al nudei and 0 otherwise+ The wave function is normalized as
where c = ff112) and c’= ( ItI;)
specify the initial and final ebannek. By using the expression of the couplgd~~ba~~el wave function (Z), we find the total b~ern~st~~u~~ cross section for the EA radiation as
176
0. Tanimura, U.
Mosel/ Bremsstrahlung
K( K ‘).
For quadrupole radiation emitted during the scattering with the ground-state spin Ii = 0 this equation reduces to
of identical
nuclei
00) with ---
X
__
c
c_ - i(
(jj,)’
~02O$'O)W(2z'J~;
iJ’)j-u$;!+‘r=z$+)dr
=,
(I,J,)I, L. P
01) where we have used the relation E, = Acq. The final channel spins 1; and 1; enter the index I’ of the final relative wave function. Note that the emission of a photon does not cause a change of the channel states because of orthogonality among the channel intrinsic states due to the assumption of spherical 12C. It is quite obvious that a simple numerical integration over r in eq. (9) leads to a divergence, since the wave function does not drop off at infinity. One can avoid this difficulty by using the conventional technique that is used in the evaluation of dipole radiation “) and basically consists of introducing radial exponential shielding factors. Applying this technique to the quadrupole radiation, we find
+ 4
u~~H,~u~--~H,&u$‘u~ k k’
1
r2dr,
(12)
where H,$ represents the coupled-channel hamiltonian for J (see eq. (A.3)). For the detailed derivation of eq. (12) see the appendix. In the first term of this equation the exponent of r is reduced by one unit and in the second term by asymptotic region, where the coupling potential T/I, = 0 and the potential U, = 0; see eqs. (A.5) and (20)). Further repeating this times, we find that all terms are reduced by more than 5 powers of we denote the initial and final wave functions with indices i and f, results
are as follows:
two units (in the diagonal nuclear procedure three r. For simplicity respectively. The
117
0. Tunimurcr, U. Mosel / Bremssrmhlung with
F,=E,I3rZ+2alh’(4rI+r21’)+2a2{(rh”+3h’)h’+2(rh”‘+4h”)h},
(14)
F, = 4a2(21h + h-h + I’rh)’ + 2a*I(rh”+
(15)
F3 = 2E,a*(lr)“+
2a3(rh”+
3h’),
3h’)“,
(16)
F4= [E~r2+E~lrZ+Ey{2a(rh’+2h)+f2r2}+4a(2~h+Irh’+I’rh)]A~~ +4a[E;+Ey(Ir)‘+a(rh”’
+4h”)]AV2
+2a[E,fr+Eylr+a(rh”+3h’)](AV;-2AV3),
(17)
where a = A’/2p, I=a[Li(Li+l)-L,(L,+l)]/r*,
(18)
h=a[Li(Li+1)+_LL,(L,+l)]/r2+2Z2e2/r-(Ei+Ef),
(19)
and AVI = (Ui - U,) UfUi + ufC
vi,uj - C vf,“[ui
AV, = ( Ui + U,) UfUi + ufC
vij”J + C vf,“,ui
(20)
9
(21)
I
J
At’, = U,u,u; + U&q
1
I
J
+ u;c i
vjuJ + c Vf,u,u~. I
(22)
Here lJi and vij denote the on- and off-diagonal nuclear interactions, respectively (see eq. (A.3)). The right-hand side of eq. (13) is now evaluated numerically. Details of the integration (13) are given in the appendix.
3. Bremsstrahhng
cross section in 12C + 12C collision
We calculate the bremsstrahlung cross section for the quadrupole photon emission in ‘*C + **C scattering for an incident cm. energy of 25.6 MeV and final energies around 19.3 MeV. One realizes from the measured inelastic excitation functions3) that at 25.6 MeV the mutual 2+ channel is strongly enhanced, whereas the single 2+ one is suppressed around this energy, while at 19.3 MeV the mutual 2+ channel is quite weak and intermediate structure is observed in the single 2+ channel. Supposing that the gross structures form a quasi-molecular band, we expect an enhancement for the E2 transition between neighbouring states. We first determine the initial- and the exit-channel wave functions taking into account the couplings to the single and mutual 2+ channels. We have searched sets
118
0. Tanimuru, U. Nmel f 3re~~srru~~lu~g
of parameters of the folded interactions in the same way as described in ref. 22) so as to fit the angular dist~bution and excitation function measurements by Cormier et al. 3*4), Then we obtain the wave functions at each energy using the resulting potential parameters. We note that the initial wave functions are the solutions with the elastic boundary condition, while the exit ones are the solutions with the respective boundary conditions. This point is realized in eq. (11).
Fig. 1. Angular d~st~bu~ons of the elastic and single 2 + excitation cross sections at ewrgies around 19.3 NeY and at 25.625 McV. The data points are taken from ref. 4 ).
0. Tunimuru, U. Mosel / Bremsstruhlung
Integrated
cross
sfngle
179
sectlon
2’
Ec,m(MeV) Fig. 2. Integrated cross sections for the single 2+ and the mutual 2+ excitations at energies around 19.3 MeV. The data of the mutual excitation are taken from ref.3). The single 2+ data are obtained by integrating numerically over the data points of the angular distributions in ref. 4).
Fig. 1 shows the resulting fits to the elastic and the single 2+ channels at 25.6 We have obtained the same qualities of MeV. Fig. 2 shows the integrated cross parameters
for the single and mutual
measured angular distributions4) for the MeV and at 13 energies around 19.3 MeV. fits at energies of 18.3-18.5 and 20.1-20.3 sections calculated with the same sets of
2+ channels.
In this figure we see that our
model reproduces both measured cross sections including the intermediate structures. The integrated cross sections at 25.6 MeV are found to be 94 mb and 70 mb for the single and mutual 2+ channels, respectively, which are comparable to the measurements 3). We have also checked that our folding-model parameters give general agreement with the measured individual m-state populations, PI,,,,,,,, at 25.6 MeV [refs. 12,23)]. The predicted values averaged over 6’,,,,= 40”-90” are 0.03, 0.11, 0.24 and 0.62 for p 0.0’ pi,,, P2.0 and P2.2, respectively, for which the measured ones are roughly equal to 0.05, 0.15, 0.25 and 0.55, respectively. In view of these results, we believe that the resulting wave functions are realistic.
___ I?
0. Tunimuru, U. Mosel / Bremsstrcrhlung
180
INTERNAL 11
'2
CONTRIBUT '1
‘2
IONS
11
2+
‘2
---
2+
T
2+
0' Fig. 3. Diagrams
0’
0’
0’
0’
_-___ -_-
2+
0+
of the internal bremsstmhlung for three different processes. The horizontal lines denote the nuclear interaction and the photon radiation, respectively.
and curled
We have two kinds of y-radiation, i.e. the “internal” and the “external” contributions “). In the internal (external) contribution a photon is emitted inside (outside) the nuclear interaction. In fig. 3 the three possible kinds of internal emission are sketched. Note that the external emission is associated with the outer line in this figure. As mentioned in the previous section, a photon is emitted only between the same channel states because of the orthogonality of the intrinsic states. We note that the bremsstrahlung cross section formulated in eq. (11) contains both internal and external contributions, since we have used the complete wave functions ranging into the asymptotic distance where the integral of eq. (13) converges (- 30 fm). The grazing angular momentum at 25.6 MeV is equal to 14 and/or 16. From the inspection of the S-matrix for the initial wave the J = 14 wave is found to yield the dominant contribution of the structure of the mutual 2* channel, where u14(2+, 2+) = 28 mb and 016(2+, 2+) = 3 mb. Thus in the estimation
of the branching
ratio we
assign J = 14 to the 25.6 MeV structure. We note that the doublet-like structure around 25 MeV in the mutual 2+ excitation function, one at 24.3 MeV and the other at 25.6 MeV, may have different in the single 2+ channel with origins: the lower member has a large population J, = 16, which has been determined from the energy behavior of the S-matrix24), while the higher member is found to be dominated by J = 14 from the analysis described above. Fig. 4 shows the bremsstrahlung cross sections for J = 14 together with the contributions for the individual exit channels. One can see that the structure at 18.7 MeV is caused by the matching of the single 2+ and the elastic exit channels, while that at 19.6 MeV is caused by the matching of the mutual 2+ and the elastic exit channels. Although, for Ji = 14, the exit angular momenta of 12,14 and 16 can contribute to the cross section, most of the magnitude (more than 90%) is found to be contributed by the Jf= 12 component.
181 I
’
I
’
’
I
’
I
Bremsstrahlung cross section Ji=lL
Ei’25.6
MeV
TOTAL ---10: 0’) --A i 2: 0’) ___-.__[ 2: 2’)
Fig. 4. The total bremsstrahlung cross sections summed over exit channels for J; = 14 and their contributions in the respective exit channels for the quadrupole photon emission from an incident energy of 25.6 MeV to the final energies around 19.3 MeV.
I
I
I
Bfem~strahlu~ Ji=16
,
I
I
L
I
19 Ef(MeVt
I
cross section E,=25.6MeV -
18
I
TOTAL
I
I
20
21
L
Fig. 5. The total bremsstrahhmg cross sections for Ji = 16 for the same transition as in fig. 4.
182
0. Tunimuru,
Fig. 5 shows the cross sections
U.
Mosel/ Bremsstruhlung
for Ji = 16 as a function
of final c.m. energies.
We
see that the cross sections are an order of magnitude smaller than those of Ji = 14, which is consistent with our later assumption that the 25.6 MeV structure in the mutual
2 + channel
is due to the J = 14 resonance.
One can see from eq. (11) that the three kinds interfere with one another in the same exit channel.
of processes shown in fig. 3 For the purpose of getting a
rough estimate for the individual yields of the intermediate states, however, we separate these processes in the following. Fig. 6 shows the estimates of the individual processes summed over all exit channels, where the difference of dot-dashed and dashed curves and that of solid and dot-dashed ones correspond to, respectively, the “single + y ” and the “mutual + y ” contributions. The “mutual + y ” process obviously dominates the total yield (about 70% of the total yield). Fig. 7 shows such estimates for individual exit channels, where the intermediate channel state emitting a photon is taken to be the same as the exit state. Again the
I
I
I
Bremsstrahlung
I
cross
Ji=lL
sectlon
Ei = 25.6 MeV elastic+ Y sing.2++ Y Imutu.Z++ Y
-
___
elastic+
Y
I sing. 2++ Y elastic+
_I’ 1
18
I
I
19
I
I
20
21
Y
Ef(MeVl Fig. 6. Estimates of the contributions to the bremsstrahhmg cross sections from the individual processes sketched in fig. 3; the channels given in the figure are the intermediate states. The cross sections are summed over all exit channels.
0. Tanimuru,
U. Mosel/
Contribution Ji=lh,
183
Bremsstruhlung
of each Eiz25.6,
process exit
__--
_-_
ch. + Y
elastic
+ Y
sing.
2++ Y
mutu.
2++ Y
I/\’ I!
19
20
Ef (MeVI Fig. 7. Estimates of the contributions to the bremsstrahlung cross sections from the individual processes sketched in fig. 3 for individual exit channels. The exit channels are taken to be the same as the states emitting the photon.
the dominance of the large enhancement of the “mutual + y ” process indicates mutual 2 + excitation in the 25.6 MeV structure. Figs. 6 and 7 reveal that the mutual excitation is of essential importance around 19.6 MeV. In view of the fact that the mutual 2+ excitation channel has an intermediate structure at 19.6 MeV [ref. 3)], we may interpret the calculated rather narrow peak in terms of a well-matching of initial and exit angular momentum windows within the respective excitation channels. On the other hand, the single 2+ channel plays a minor role in the total cross sections, although it is enhanced at the exit energies. This may be caused by a reduction of the overlap integral (1) due to strong suppression of the single 2* channel at the initial energy. We have tried to obtain an estimate for the relative importance of external versus internal bremsstrahlung by identifying the former with the contributions to the transition integral in eq. (11) from the region with r > 10 fm. Again neglecting
184
0. Tanimura, Il. Mosel / Bremsstrahlung
interference
between
sion contributes internal
the two types of y-radiation
we find that the external
less than 5% of the total cross section. This strong dominance
y-emission
reflects the fact that the transitions
take place primarily
y-emisof the between
states in which both 12C nuclei are excited to their first 2+ state. In these the wave functions
for the relative
plus centrifugal
potentials
motion
that enter eq. (11) are quasi-bound
and, therefore,
localized
near the contact
in the nuclear region.
4. Branching ratio of quadrupole radiation We estimate the branching ratio of the E2 radiation width to the total width by assuming that the initial 25.6 MeV structure and the final 19.3 MeV one are due to the resonances with J = 14 and 12, respectively. The branching ratio can be written as 14)
(23) where Pee and P,,,, represent the probability of the decay of the initial resonance and the decay of the final resonance accompanied by photon emission. I’,, is the sum of the partial widths of the elastic and inelastic channels. Since the probabilities can be replaced by the resonant part of the cross sections, we find
(24) where A represents the width of the y-radiation to the final state. u;‘,I denotes the resonant part of the cross section summed over the elastic and inelastic channels. In the present
problem
we have u;‘,I = &r(el)
+ uJr(2+)
+ uJr(2+,2+)
(25)
with
(26) rtot=re,+rc.n.+r2+~r2+,2+.
(27)
where the symbol m in eq. (26) denotes the channels el, c.n., 2+ and (2+, 2+). We estimate cr? according to the method of Cormier et al. 3). We assume u14(2+,2’) = 35-55 mb and u14(2+.0+) = 5-15 mb from the measurements at 25.6 MeV [ref. 3)]. We take for the compound resonance cross section 50 mb as in ref. 3). Solving the simultaneous equations spanned by eqs. (26) and (27) we obtain u-‘r(el) = lo-20 mb.
0. Tunimuru, II Mosef / l3remsstruhlung
185
From fig. 4 we have a rough estimate of the resonant bremsstrahlung cross section averaged over a 2 MeV wide y-energy window: /(do’r/dElfd,!$=
3.5 X 10m5 mb.
A
(28)
Taking for the second factor of eq. (23) the value of 0.7-1.0 that Metag et al. 14) used, we obtain the branching ratio of + t
1
,=(3-7)x10-7. tot 1
(29)
This value is comparable to the measurementsi4) and within the upper limit deduced by McGrath et al. I’). If we assume J, = 16 for the 25.6 MeV structure, we obtain a branching ratio of less than 4 x 10-8, which is too small compared with the data. That this spin assignment is unrealistic is also indicated by the analysis of the S-matrix (see sect. 3). Taking only the “elastic + y ” process with the elastic exit channel as in the optical-model analysis, we obtain a branching ratio that is smaller by a factor of $ than the measurements (see fig. 7). This means that the coupled-channel effect is of key importance in this radiation.
5. Discussions We have estimated the branching ratio for the E2 transition between the structures at 25.6 MeV and at 19.3 MeV using optimally-adjusted coupled-channel wave functions. We have obtained a ratio r,/rtot of (3 - 7) X 10F7 which is comparable with the measurements 14,15).We have furthermore found that the mutual 2+ channel plays a dominant role in the E2 radiation, which may reflect the strong enhancement of the “sticking” configuration for the 25.6 MeV structure. The new value for I$J‘~,,, is significantly smaller than the older estimates that were based on the Bohr-Mottelson rigid-rotor model with a quadrupole moment corresponding to the configuration of two touching nuclei. The reason for the lowering of the branching ratio, or equivalently the B(E2) value, is that the radial wave functions are significantly different for the initial and the final states in the same channels, so that the assumption of a molecular band with an inert intrinsic structure is not justified. This can also be seen from the experimentat fact that the channels that are enhanced are different at the initial and final energies. The mechanism of the lowering of the y-transition obtained in the present calculation is thus the nuclear analog of the Franck-Condon principle25) that is well known to determine the transition strength between the different states in a molecule. The transition rate in the case studied here is determined by two counteracting effects; first there is an enhancement in the intermediate (2+, 2’) channel because of the better overlap of quasi-bound radial wave functions. This enhancement is partly
186
counteracted
by the fact that the partial
excitation
channel
compared
with
“molecular”
U. Mosel / Bremsstmhlung
0. Tunimura,
is very
0.95
and
small 0.55
first has to make
a transition
(< 0.15 MeV
system, after emitting
width rzt2+ for decay into the mutual for
MeV) r,+
around
and
EC,,,= 19 MeV,
Tel, respectively3).
the y-ray from the intermediate
into another
one before
2+
to be
Thus
the
(2+, 2’) channel,
it can decay into the final
channel (compare figs. 4 and 6). Since rzt2+ is quite large both at EC,,-
31 and 25 MeV, one expects that in an experiment at an initial energy of 31 MeV the decay into the final channel is not hindered but that the y-yield is increased. However, this expected enhancement will be counteracted by the fact that at the higher energy the channel wave functions are less well “quasi-bound” and that, therefore, the overlap in the radial transition integral becomes small, so that the overall transition strength is expected to be the same order of magnitude as at Ei = 25.6 MeV. This expectation is indeed borne out by exploratory calculations for Ei = 31 MeV using a DW method. The authors would like to express their sincere thanks R. Vandenbosch for valuable discussions.
to Prof. V. Metag and Prof.
Appendix We show here how to reduce the power of r in the integration of eq. (9). We first set up the basic formula. For a certain well-behaved (except at r = 0) function g(r), we consider the following integral: I=
E,
J
where E, = Ei - E,. The wave functions
g(r)u,uidr,
(A.11
u: and uf’ satisfy
~(E,6,,-H~k)u,=0
(n=iandf),
(A.2)
with H,fk=(--ad2/dr2+aL.,(L,+1)/r2+Z2e2/r+U,)6,,+V,,,,
(A.31
where U,, and V,, represent the diagonal and the off-diagonal Using eq. (A.2), we can rewrite the integral I as Z =
ajg( u;Iui
where the prime on integration, we have
-
u,u[‘) dr + /g( ufFHiruk
u means
the derivative
about
nuclear
- 5: H,,u,ui) r on
interactions.
dr,
u. By making
(A.41
partial
(A-5)
0. Tu~rimuru, U. Mosel / Bremsstrddung
187
where I(r) and AI’,(r) are given by eqs. (18) and (20). Here we have dropped out the integrated values at infinity, since the electric field is neutral there. One sees that the exponent of r is reduced by one unit in the integrand of the first term and by two units in the second, while the last term contains only a finite-range interaction and thus does not cause difficulties. The first term of eq. (AS) is further reduced by the same procedure as
+a/g”( u& +
/
+
z&) dr + 2Jgti&dr
g(AV; - 2AV,)dr,
G4.6)
where we have used the symbols defined by eqs. (18)-(22). One can reduce the power of r to any amount by repeated use of eqs. (AS) and (A.6). In the bremsstrahlung calculation for the system of 12C + “C, the following numerical methods have been used. The integrations were made by means of a 5-point Newton-Cotes method (closed type) over the range from 0.3 fm to 28.0 fm with an interval of 0.05 fm. The final results are numerically stable with respect to the upper boundary within 1% in the total bremsstrahlung cross sections. The differentiations for all quantities have been computed by using a 7-point Taylorexpansion method with equal interval.
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U. Mosel / Bremsstruhlung
E.M. Nyman, Phys. Lett. 136B (1984) 143 H. Feshbach and D.R. Yennie, Nucl. Phys. 37 (1962) 150 K. Langanke and O.S. van Roosmalen, Phys. Rev. C29 (1984) 1358 J.S. Blair and H. Sherif, annual report 1981, Nucl. Phys. Lab., University of Washington, Seattle K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 H.J. Fink, W. Scheid and W. Greiner, Nucl. Phys. A188 (1972) 259 0. Tanimura and R. Wolf, Phys. Lett. 12SB (1983) 275 W. Trombik, private communication 0. Tanimura, R. Wolf and U. Mosel, Phys. Lett. 132B (1983) 264 See e.g. Quantum chemistry, ed. W. Kauzmann (Academic, New York, 1957) p. 664
BREMSSTRAHLUNG
IN HEAVY-ION MOLECULAR RESONANCES’ 0. TANIMURA and U. MOSEL
i~s~i~~t~~r Th~~relisch~ P&w&,
U~iuer~it~f Giesstn, 6300 Giessen, E R. Germuy
Received 12 October 1984 Abstract: The bremsstr~lun8 cross section for the quadrupole photon emission in a heavy-ion collision is fo~u~at~d on the basis of the coupi~-charnel method. For the branching ratio of the radiative decay width to the total width for the transition between the structures at 25.6 MeV and 19.3 MeV in the 12C + 12C collision, we predict (3 - 7) X lo-‘, which is roughly comparable with existing measurements. The radiation from the mutual 2’ excitation channel is found to yield 70% of the total branching ratio.
1. In~~uction
Prominent structures have been found in the excitation functions of the elastic and inelastic 12C +‘*C cross sections’-4). These structures have often been interpreted in terms of the “molecular” resonances5-8), since they appear near the grazing angular momenta in the weak abso~tion window. There have been attempts to obtain a deeper understanding of these states from a microscopic viewpoint 9-11). Chandra and Mose19), on the basis of the two-center shell model, have predicted the existence of quasi-molecular di-carbon configurations in the excited states of 24Mg [see also Ragnarsson et al. “)I. They have suggested that these states correspond to the “sticking” configuration which is closely associated with the enhancement of the mutual 2+ channel. The recent spin-alignment measurement by Konnerth et al. 12) indicates that this configuration seems indeed to be favoured for the 25.6 MeV structure of the mutual excitation channel. If these states form a rotational band with a well-defined intrinsic state, one expects a large enhancement of the quad~pole photon emission as in the rotational nuclei. measurements of the E2 transition during the collision of two carbon nuclei have been carried out by Metag et al. 13-14)and McCrath et al. 15) independently. Metag et al. 14) have deduced a branching ratio, I’,/I’,,,, of (6 & 3) x lop7for the photon emission between the initial energy of 25.7 MeV and the final energies around 19.3 MeV. Since this y-decay width is much smaller than the rigid-rotor limit, these authors have concluded that an interpretation of the gross structure in terms of the molecular band is unlikely. McGrath et al. 15) have obtained an upper limit for ’ Work supported by BMFT and GSI. 173
174
0. Tanimura,
(1. Mod
/
Bremsstrahlung
the ratio of (2 - 8) X 10-6, which also shows no collective y-emission between the intermediate resonances but is consistent with a potential resonance interpretation. Nyman16) Feshbach
has and
calculated
Yennie’s
the
bremsstrablung
formula”)
cross
for this transition
section
on
and obtained
the
basis
of
the value
of
7 X lo-’ dE,/E, as the number of bremsstrahlung events per elastic scattering, which is comparable to the measured branching ratio14). This estimate, however, neglects the effects of the collective excitation channels. Since the molecular-like states decay to the single 2+ and mutual 2+ channels with considerable probabilities through the strong-coupling interaction, it is essential to take the couplings to these channels into account. Langanke and van Roosmalen”) have estimated the B(E2) values of the combined 12C-12C system for 1~ 10 in a microscopic model, and obtain values similar to the predictions by the rigid-rotor model for two touching nuclei. The purpose of this paper is to get information of the molecular-like structures of the 12C-12C system through the analysis of y-decay measurements. On the basis of the coupled-channel method we calculate directly the bremsstrahlung cross sections for E2 transitions during the collision in the same manner as Blair and Sherif 19) did in the DWBA framework. The initial and the final scattering wave functions are determined so as to fit the measured cross sections in the elastic, the single 2’ and the mutual 2+ excitation channels.
2. Evaluation of bremsstrahlung cross sections The
bremsstrahlung
cross
section
for the 2’-pole
photon
emission
has
been
formulated by Alder ef al. 20). For a heavy-ion collision, the double-differential cross section of the Eh radiation per solid angle s2 and photon momentum q is given as 20) 2 ZlAE;
+
(4
’
CI(K,lrXY~,(P)IIUi>12,
(-)*Z,A:
+A2Y
(1)
P
where we have used the same notations as in ref. 20). The indices i and f specify the initial and final channels, respectively. In using eq. (1) the assumption is made that the intrinsic quadrupole moment of the interacting nuclei can be neglected in calculating the y-emissions. This is justified because the quadrupole moment due to the relative distance between the nuclei is much larger than the intrinsic value. Since we are interested in the interaction of nuclei with strongly excited collective states, we couple these states to the ground channel states to obtain the relevant wave functions. In treating the coupled-channel wave functions it is convenient to use the r-representations of IK,) and IK,) and write the boundary conditions
explicitly, i.e. TP$I$K~, r> and ~~~~~~~~,~) where Q denotes z-component ( Inif) of the aucleus in the a~~~tot~~ region. The wave function may be written as 2s)
the
spin
and its
with
The quantity E is taken to be 1 for ~d~~ti~al nudei and 0 otherwise+ The wave function is normalized as
where c = ff112) and c’= ( ItI;)
specify the initial and final ebannek. By using the expression of the couplgd~~ba~~el wave function (Z), we find the total b~ern~st~~u~~ cross section for the EA radiation as
176
0. Tanimura, U.
Mosel/ Bremsstrahlung
K( K ‘).
For quadrupole radiation emitted during the scattering with the ground-state spin Ii = 0 this equation reduces to
of identical
nuclei
00) with ---
X
__
c
c_ - i(
(jj,)’
~02O$'O)W(2z'J~;
iJ’)j-u$;!+‘r=z$+)dr
=,
(I,J,)I, L. P
01) where we have used the relation E, = Acq. The final channel spins 1; and 1; enter the index I’ of the final relative wave function. Note that the emission of a photon does not cause a change of the channel states because of orthogonality among the channel intrinsic states due to the assumption of spherical 12C. It is quite obvious that a simple numerical integration over r in eq. (9) leads to a divergence, since the wave function does not drop off at infinity. One can avoid this difficulty by using the conventional technique that is used in the evaluation of dipole radiation “) and basically consists of introducing radial exponential shielding factors. Applying this technique to the quadrupole radiation, we find
+ 4
u~~H,~u~--~H,&u$‘u~ k k’
1
r2dr,
(12)
where H,$ represents the coupled-channel hamiltonian for J (see eq. (A.3)). For the detailed derivation of eq. (12) see the appendix. In the first term of this equation the exponent of r is reduced by one unit and in the second term by asymptotic region, where the coupling potential T/I, = 0 and the potential U, = 0; see eqs. (A.5) and (20)). Further repeating this times, we find that all terms are reduced by more than 5 powers of we denote the initial and final wave functions with indices i and f, results
are as follows:
two units (in the diagonal nuclear procedure three r. For simplicity respectively. The
117
0. Tunimurcr, U. Mosel / Bremssrmhlung with
F,=E,I3rZ+2alh’(4rI+r21’)+2a2{(rh”+3h’)h’+2(rh”‘+4h”)h},
(14)
F, = 4a2(21h + h-h + I’rh)’ + 2a*I(rh”+
(15)
F3 = 2E,a*(lr)“+
2a3(rh”+
3h’),
3h’)“,
(16)
F4= [E~r2+E~lrZ+Ey{2a(rh’+2h)+f2r2}+4a(2~h+Irh’+I’rh)]A~~ +4a[E;+Ey(Ir)‘+a(rh”’
+4h”)]AV2
+2a[E,fr+Eylr+a(rh”+3h’)](AV;-2AV3),
(17)
where a = A’/2p, I=a[Li(Li+l)-L,(L,+l)]/r*,
(18)
h=a[Li(Li+1)+_LL,(L,+l)]/r2+2Z2e2/r-(Ei+Ef),
(19)
and AVI = (Ui - U,) UfUi + ufC
vi,uj - C vf,“[ui
AV, = ( Ui + U,) UfUi + ufC
vij”J + C vf,“,ui
(20)
9
(21)
I
J
At’, = U,u,u; + U&q
1
I
J
+ u;c i
vjuJ + c Vf,u,u~. I
(22)
Here lJi and vij denote the on- and off-diagonal nuclear interactions, respectively (see eq. (A.3)). The right-hand side of eq. (13) is now evaluated numerically. Details of the integration (13) are given in the appendix.
3. Bremsstrahhng
cross section in 12C + 12C collision
We calculate the bremsstrahlung cross section for the quadrupole photon emission in ‘*C + **C scattering for an incident cm. energy of 25.6 MeV and final energies around 19.3 MeV. One realizes from the measured inelastic excitation functions3) that at 25.6 MeV the mutual 2+ channel is strongly enhanced, whereas the single 2+ one is suppressed around this energy, while at 19.3 MeV the mutual 2+ channel is quite weak and intermediate structure is observed in the single 2+ channel. Supposing that the gross structures form a quasi-molecular band, we expect an enhancement for the E2 transition between neighbouring states. We first determine the initial- and the exit-channel wave functions taking into account the couplings to the single and mutual 2+ channels. We have searched sets
118
0. Tanimuru, U. Nmel f 3re~~srru~~lu~g
of parameters of the folded interactions in the same way as described in ref. 22) so as to fit the angular dist~bution and excitation function measurements by Cormier et al. 3*4), Then we obtain the wave functions at each energy using the resulting potential parameters. We note that the initial wave functions are the solutions with the elastic boundary condition, while the exit ones are the solutions with the respective boundary conditions. This point is realized in eq. (11).
Fig. 1. Angular d~st~bu~ons of the elastic and single 2 + excitation cross sections at ewrgies around 19.3 NeY and at 25.625 McV. The data points are taken from ref. 4 ).
0. Tunimuru, U. Mosel / Bremsstruhlung
Integrated
cross
sfngle
179
sectlon
2’
Ec,m(MeV) Fig. 2. Integrated cross sections for the single 2+ and the mutual 2+ excitations at energies around 19.3 MeV. The data of the mutual excitation are taken from ref.3). The single 2+ data are obtained by integrating numerically over the data points of the angular distributions in ref. 4).
Fig. 1 shows the resulting fits to the elastic and the single 2+ channels at 25.6 We have obtained the same qualities of MeV. Fig. 2 shows the integrated cross parameters
for the single and mutual
measured angular distributions4) for the MeV and at 13 energies around 19.3 MeV. fits at energies of 18.3-18.5 and 20.1-20.3 sections calculated with the same sets of
2+ channels.
In this figure we see that our
model reproduces both measured cross sections including the intermediate structures. The integrated cross sections at 25.6 MeV are found to be 94 mb and 70 mb for the single and mutual 2+ channels, respectively, which are comparable to the measurements 3). We have also checked that our folding-model parameters give general agreement with the measured individual m-state populations, PI,,,,,,,, at 25.6 MeV [refs. 12,23)]. The predicted values averaged over 6’,,,,= 40”-90” are 0.03, 0.11, 0.24 and 0.62 for p 0.0’ pi,,, P2.0 and P2.2, respectively, for which the measured ones are roughly equal to 0.05, 0.15, 0.25 and 0.55, respectively. In view of these results, we believe that the resulting wave functions are realistic.
___ I?
0. Tunimuru, U. Mosel / Bremsstrcrhlung
180
INTERNAL 11
'2
CONTRIBUT '1
‘2
IONS
11
2+
‘2
---
2+
T
2+
0' Fig. 3. Diagrams
0’
0’
0’
0’
_-___ -_-
2+
0+
of the internal bremsstmhlung for three different processes. The horizontal lines denote the nuclear interaction and the photon radiation, respectively.
and curled
We have two kinds of y-radiation, i.e. the “internal” and the “external” contributions “). In the internal (external) contribution a photon is emitted inside (outside) the nuclear interaction. In fig. 3 the three possible kinds of internal emission are sketched. Note that the external emission is associated with the outer line in this figure. As mentioned in the previous section, a photon is emitted only between the same channel states because of the orthogonality of the intrinsic states. We note that the bremsstrahlung cross section formulated in eq. (11) contains both internal and external contributions, since we have used the complete wave functions ranging into the asymptotic distance where the integral of eq. (13) converges (- 30 fm). The grazing angular momentum at 25.6 MeV is equal to 14 and/or 16. From the inspection of the S-matrix for the initial wave the J = 14 wave is found to yield the dominant contribution of the structure of the mutual 2* channel, where u14(2+, 2+) = 28 mb and 016(2+, 2+) = 3 mb. Thus in the estimation
of the branching
ratio we
assign J = 14 to the 25.6 MeV structure. We note that the doublet-like structure around 25 MeV in the mutual 2+ excitation function, one at 24.3 MeV and the other at 25.6 MeV, may have different in the single 2+ channel with origins: the lower member has a large population J, = 16, which has been determined from the energy behavior of the S-matrix24), while the higher member is found to be dominated by J = 14 from the analysis described above. Fig. 4 shows the bremsstrahlung cross sections for J = 14 together with the contributions for the individual exit channels. One can see that the structure at 18.7 MeV is caused by the matching of the single 2+ and the elastic exit channels, while that at 19.6 MeV is caused by the matching of the mutual 2+ and the elastic exit channels. Although, for Ji = 14, the exit angular momenta of 12,14 and 16 can contribute to the cross section, most of the magnitude (more than 90%) is found to be contributed by the Jf= 12 component.
181 I
’
I
’
’
I
’
I
Bremsstrahlung cross section Ji=lL
Ei’25.6
MeV
TOTAL ---10: 0’) --A i 2: 0’) ___-.__[ 2: 2’)
Fig. 4. The total bremsstrahlung cross sections summed over exit channels for J; = 14 and their contributions in the respective exit channels for the quadrupole photon emission from an incident energy of 25.6 MeV to the final energies around 19.3 MeV.
I
I
I
Bfem~strahlu~ Ji=16
,
I
I
L
I
19 Ef(MeVt
I
cross section E,=25.6MeV -
18
I
TOTAL
I
I
20
21
L
Fig. 5. The total bremsstrahhmg cross sections for Ji = 16 for the same transition as in fig. 4.
182
0. Tunimuru,
Fig. 5 shows the cross sections
U.
Mosel/ Bremsstruhlung
for Ji = 16 as a function
of final c.m. energies.
We
see that the cross sections are an order of magnitude smaller than those of Ji = 14, which is consistent with our later assumption that the 25.6 MeV structure in the mutual
2 + channel
is due to the J = 14 resonance.
One can see from eq. (11) that the three kinds interfere with one another in the same exit channel.
of processes shown in fig. 3 For the purpose of getting a
rough estimate for the individual yields of the intermediate states, however, we separate these processes in the following. Fig. 6 shows the estimates of the individual processes summed over all exit channels, where the difference of dot-dashed and dashed curves and that of solid and dot-dashed ones correspond to, respectively, the “single + y ” and the “mutual + y ” contributions. The “mutual + y ” process obviously dominates the total yield (about 70% of the total yield). Fig. 7 shows such estimates for individual exit channels, where the intermediate channel state emitting a photon is taken to be the same as the exit state. Again the
I
I
I
Bremsstrahlung
I
cross
Ji=lL
sectlon
Ei = 25.6 MeV elastic+ Y sing.2++ Y Imutu.Z++ Y
-
___
elastic+
Y
I sing. 2++ Y elastic+
_I’ 1
18
I
I
19
I
I
20
21
Y
Ef(MeVl Fig. 6. Estimates of the contributions to the bremsstrahhmg cross sections from the individual processes sketched in fig. 3; the channels given in the figure are the intermediate states. The cross sections are summed over all exit channels.
0. Tanimuru,
U. Mosel/
Contribution Ji=lh,
183
Bremsstruhlung
of each Eiz25.6,
process exit
__--
_-_
ch. + Y
elastic
+ Y
sing.
2++ Y
mutu.
2++ Y
I/\’ I!
19
20
Ef (MeVI Fig. 7. Estimates of the contributions to the bremsstrahlung cross sections from the individual processes sketched in fig. 3 for individual exit channels. The exit channels are taken to be the same as the states emitting the photon.
the dominance of the large enhancement of the “mutual + y ” process indicates mutual 2 + excitation in the 25.6 MeV structure. Figs. 6 and 7 reveal that the mutual excitation is of essential importance around 19.6 MeV. In view of the fact that the mutual 2+ excitation channel has an intermediate structure at 19.6 MeV [ref. 3)], we may interpret the calculated rather narrow peak in terms of a well-matching of initial and exit angular momentum windows within the respective excitation channels. On the other hand, the single 2+ channel plays a minor role in the total cross sections, although it is enhanced at the exit energies. This may be caused by a reduction of the overlap integral (1) due to strong suppression of the single 2* channel at the initial energy. We have tried to obtain an estimate for the relative importance of external versus internal bremsstrahlung by identifying the former with the contributions to the transition integral in eq. (11) from the region with r > 10 fm. Again neglecting
184
0. Tanimura, Il. Mosel / Bremsstrahlung
interference
between
sion contributes internal
the two types of y-radiation
we find that the external
less than 5% of the total cross section. This strong dominance
y-emission
reflects the fact that the transitions
take place primarily
y-emisof the between
states in which both 12C nuclei are excited to their first 2+ state. In these the wave functions
for the relative
plus centrifugal
potentials
motion
that enter eq. (11) are quasi-bound
and, therefore,
localized
near the contact
in the nuclear region.
4. Branching ratio of quadrupole radiation We estimate the branching ratio of the E2 radiation width to the total width by assuming that the initial 25.6 MeV structure and the final 19.3 MeV one are due to the resonances with J = 14 and 12, respectively. The branching ratio can be written as 14)
(23) where Pee and P,,,, represent the probability of the decay of the initial resonance and the decay of the final resonance accompanied by photon emission. I’,, is the sum of the partial widths of the elastic and inelastic channels. Since the probabilities can be replaced by the resonant part of the cross sections, we find
(24) where A represents the width of the y-radiation to the final state. u;‘,I denotes the resonant part of the cross section summed over the elastic and inelastic channels. In the present
problem
we have u;‘,I = &r(el)
+ uJr(2+)
+ uJr(2+,2+)
(25)
with
(26) rtot=re,+rc.n.+r2+~r2+,2+.
(27)
where the symbol m in eq. (26) denotes the channels el, c.n., 2+ and (2+, 2+). We estimate cr? according to the method of Cormier et al. 3). We assume u14(2+,2’) = 35-55 mb and u14(2+.0+) = 5-15 mb from the measurements at 25.6 MeV [ref. 3)]. We take for the compound resonance cross section 50 mb as in ref. 3). Solving the simultaneous equations spanned by eqs. (26) and (27) we obtain u-‘r(el) = lo-20 mb.
0. Tunimuru, II Mosef / l3remsstruhlung
185
From fig. 4 we have a rough estimate of the resonant bremsstrahlung cross section averaged over a 2 MeV wide y-energy window: /(do’r/dElfd,!$=
3.5 X 10m5 mb.
A
(28)
Taking for the second factor of eq. (23) the value of 0.7-1.0 that Metag et al. 14) used, we obtain the branching ratio of + t
1
,=(3-7)x10-7. tot 1
(29)
This value is comparable to the measurementsi4) and within the upper limit deduced by McGrath et al. I’). If we assume J, = 16 for the 25.6 MeV structure, we obtain a branching ratio of less than 4 x 10-8, which is too small compared with the data. That this spin assignment is unrealistic is also indicated by the analysis of the S-matrix (see sect. 3). Taking only the “elastic + y ” process with the elastic exit channel as in the optical-model analysis, we obtain a branching ratio that is smaller by a factor of $ than the measurements (see fig. 7). This means that the coupled-channel effect is of key importance in this radiation.
5. Discussions We have estimated the branching ratio for the E2 transition between the structures at 25.6 MeV and at 19.3 MeV using optimally-adjusted coupled-channel wave functions. We have obtained a ratio r,/rtot of (3 - 7) X 10F7 which is comparable with the measurements 14,15).We have furthermore found that the mutual 2+ channel plays a dominant role in the E2 radiation, which may reflect the strong enhancement of the “sticking” configuration for the 25.6 MeV structure. The new value for I$J‘~,,, is significantly smaller than the older estimates that were based on the Bohr-Mottelson rigid-rotor model with a quadrupole moment corresponding to the configuration of two touching nuclei. The reason for the lowering of the branching ratio, or equivalently the B(E2) value, is that the radial wave functions are significantly different for the initial and the final states in the same channels, so that the assumption of a molecular band with an inert intrinsic structure is not justified. This can also be seen from the experimentat fact that the channels that are enhanced are different at the initial and final energies. The mechanism of the lowering of the y-transition obtained in the present calculation is thus the nuclear analog of the Franck-Condon principle25) that is well known to determine the transition strength between the different states in a molecule. The transition rate in the case studied here is determined by two counteracting effects; first there is an enhancement in the intermediate (2+, 2’) channel because of the better overlap of quasi-bound radial wave functions. This enhancement is partly
186
counteracted
by the fact that the partial
excitation
channel
compared
with
“molecular”
U. Mosel / Bremsstmhlung
0. Tunimura,
is very
0.95
and
small 0.55
first has to make
a transition
(< 0.15 MeV
system, after emitting
width rzt2+ for decay into the mutual for
MeV) r,+
around
and
EC,,,= 19 MeV,
Tel, respectively3).
the y-ray from the intermediate
into another
one before
2+
to be
Thus
the
(2+, 2’) channel,
it can decay into the final
channel (compare figs. 4 and 6). Since rzt2+ is quite large both at EC,,-
31 and 25 MeV, one expects that in an experiment at an initial energy of 31 MeV the decay into the final channel is not hindered but that the y-yield is increased. However, this expected enhancement will be counteracted by the fact that at the higher energy the channel wave functions are less well “quasi-bound” and that, therefore, the overlap in the radial transition integral becomes small, so that the overall transition strength is expected to be the same order of magnitude as at Ei = 25.6 MeV. This expectation is indeed borne out by exploratory calculations for Ei = 31 MeV using a DW method. The authors would like to express their sincere thanks R. Vandenbosch for valuable discussions.
to Prof. V. Metag and Prof.
Appendix We show here how to reduce the power of r in the integration of eq. (9). We first set up the basic formula. For a certain well-behaved (except at r = 0) function g(r), we consider the following integral: I=
E,
J
where E, = Ei - E,. The wave functions
g(r)u,uidr,
(A.11
u: and uf’ satisfy
~(E,6,,-H~k)u,=0
(n=iandf),
(A.2)
with H,fk=(--ad2/dr2+aL.,(L,+1)/r2+Z2e2/r+U,)6,,+V,,,,
(A.31
where U,, and V,, represent the diagonal and the off-diagonal Using eq. (A.2), we can rewrite the integral I as Z =
ajg( u;Iui
where the prime on integration, we have
-
u,u[‘) dr + /g( ufFHiruk
u means
the derivative
about
nuclear
- 5: H,,u,ui) r on
interactions.
dr,
u. By making
(A.41
partial
(A-5)
0. Tu~rimuru, U. Mosel / Bremsstrddung
187
where I(r) and AI’,(r) are given by eqs. (18) and (20). Here we have dropped out the integrated values at infinity, since the electric field is neutral there. One sees that the exponent of r is reduced by one unit in the integrand of the first term and by two units in the second, while the last term contains only a finite-range interaction and thus does not cause difficulties. The first term of eq. (AS) is further reduced by the same procedure as
+a/g”( u& +
/
+
z&) dr + 2Jgti&dr
g(AV; - 2AV,)dr,
G4.6)
where we have used the symbols defined by eqs. (18)-(22). One can reduce the power of r to any amount by repeated use of eqs. (AS) and (A.6). In the bremsstrahlung calculation for the system of 12C + “C, the following numerical methods have been used. The integrations were made by means of a 5-point Newton-Cotes method (closed type) over the range from 0.3 fm to 28.0 fm with an interval of 0.05 fm. The final results are numerically stable with respect to the upper boundary within 1% in the total bremsstrahlung cross sections. The differentiations for all quantities have been computed by using a 7-point Taylorexpansion method with equal interval.
References 1) I2.A.Bromley, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)
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(1961)
Phys.
1983,
Stony
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U. Mosel / Bremsstruhlung
E.M. Nyman, Phys. Lett. 136B (1984) 143 H. Feshbach and D.R. Yennie, Nucl. Phys. 37 (1962) 150 K. Langanke and O.S. van Roosmalen, Phys. Rev. C29 (1984) 1358 J.S. Blair and H. Sherif, annual report 1981, Nucl. Phys. Lab., University of Washington, Seattle K. Alder, A. Bohr, T. Huus, B. Mottelson and A. Winther, Rev. Mod. Phys. 28 (1956) 432 H.J. Fink, W. Scheid and W. Greiner, Nucl. Phys. A188 (1972) 259 0. Tanimura and R. Wolf, Phys. Lett. 12SB (1983) 275 W. Trombik, private communication 0. Tanimura, R. Wolf and U. Mosel, Phys. Lett. 132B (1983) 264 See e.g. Quantum chemistry, ed. W. Kauzmann (Academic, New York, 1957) p. 664