397
Progress of Theoretical PhYbICS, Vol 29, No 3, Match 1963
Phase Transition and Level Density of Atomic Nuclei
Mitsuo SANG and Shuichiro YAMASAKI
Department of Physlcs, Osaka Umversity, Osaka
(ReceIved November 24, 1962)
The effect of the paIrIng mieractlOn on nuclear spectlum IS btudied WIth the aid of the
modern theory of superconductivIty developed by B C Sand BogollUbov As the states
near the ground sLate have been !:>tuchecl by many authorb, we employ d btailbtlcal dynamIcal apploach to examllle the grobs behavIOur of the excIted btates For the nucleI of
medlUm weIght, a phase trdnbltlOn Ib prechcted at the twnsltlOn energy of some 5 to 10 Mev
The level denSIty below thib energy IS greatly dIfferent from that expected on extrapolatlllg It from hIgher enel gy blde It Ib dIbcubsed that the Imaglllary part of the optical
potentIal can be much smaller than that obtallled by the Fel mi gas model
§ 1.
Introduction
The importance of the paIrmg Interaction m atomIC nucleI was known for a
long time This was suggested most pronouncedly by the smglet nature of all
of the observed ground states of even-even nucleI The even to odd alternatIOn
of the binding energy, or the paIring energy, also supports the paInng couplmg
scheme. I )
The shell model studies of vanous propertIes of nucleI were successful and
for the treatment of the excIted configuratIOns the semonty couplmg scheme
was introduced 2) StatIstIcal dynamics or thermodynamIcs of nucleI, however,
was stIll studIed wIth the md of the mdependent partIcle model or the FermI
gas model. Let us consider the most typical example, the level denSIty
In
thIs case the painng mteraction was recalled only in the phenomenologIcal IntroductIOn of the even-odd correction .~)
Recently the successful method of the theory of superconductivIty4) is
employed In the study of nuclear structure."') The method proposes essentIally
a generalIzatIOn of Racah's ~emonty couplmg scheme StudIes m thIS field are
lIkeWIse focussed mainly upon the ground states or the low lymg excIted states,
and the investIgatIOn of the over all character of nuclear spectra IS stIll lackmg
The authors, thus, have comlldered It worth whIle to apply the method to
hIgher states, and to dISCUSS Its phYSIcal consequences
In § 2 we give the resume of the statistIcal dynamIC!:> of the system with
the paInng mteractIOn Thls is applIed m § 3 where the thermodynamIcal propertIeS of nucleI are studIed. In thIS sectIOn the phase transitIOn IS predIcted
at ~omL' 0 to ] 0 Mev for the nucleI of medIUm weIght In § 4 the level denSIty
398
M. Sana and S. Yamasaki
is discussed with the aid of the thermodynamic functions in § 3. It is pointed
out that the mere correction of the VVeisskopf formula by a constant amount of
energy is insufficient below the transition energy. In fact the level density is
much smaller than that obtained by the extrapolation of this formula.
In § 5
we discuss that the imaginary part of the optical potential is affected greatly
by the change of the level density due to the pairing interaction.
§ 2.
Pair in.!?; interaction and therlllodynamical functions
We discuss a system of nucleons interacting with the pairing force.
Hamiltonian of this system is written as follows,
The
where Ek and A are the single particle energy and the chemical potential respectively. The spin index s distinguishes between two single particle states
connected by the operation of time reversal. The nucleon creation and annihilation operators aL and aJ..;fgf are subjected to the usual anticommutation relations,
(2)
The last term in (1) represents the pairing potential and a negative sign IS
introduced in order to emphasize the attractive nature of this potential. The
thermodynamical functions of this system are derived from the grand potential
,Q,
,Q = - () log Sp e- H / IJ ,
(3)
where () is the temperature of the system in Mev.
It is convenient to rewrite the Hamiltonian in the form
H=HO+I-P,
HO = ~J..;s (EJ..; - },) als elks + ~kkf G 1ck f (Xk
HI = - ~lclcf Gkkf (a r -Ic- aL.
+ Xk)
akf+ a_kf_
+ Xlc f a~k_ aL + Xk
Xlc f) ,
(akf:_ a_J..;!- + Xkf).
(4)
Bogoliubov et al. have shown that the asymptotically exact evaluation of ,Q is
greatly facilitated for a large system, if Xlc is so chosen as to satisfy the following equation
Sp {e- 1iO / 8 (ak+ a-lc- + Xlc) }
o.
=
(5)
This follows from the perturbational expansion of the grand potential
1/8
log Sp e-
Il 8
/
=
log Sp e-
HO IJ
/
tn -
J
+ log[ 1 + ~~. dt 1 dt2"'~' dt n ~Xn
o
where
I
II
0
0
J'
(6)
Phase Transition and Level Density of Atomic Nuclei
399
and
In fact, if (5) is substituted in (6) it is found that, for a very large system,
the last term of the latter equation remains finite while the first term is proportional to the large volume of the system. Thus, if Eq. (5) is satisfied, the
asymptotically exact evaluation of the grand potential is possible simply with
the aid of RO. Neglecting in this way the last term in the right-hand side of
(6) we have
Q = - () log Sp e- H010 •
Equation (5)
formation
IS
(7)
more easily handled with the aid of the Bogoliubov trans-
alc+
=
a_k_
where the coefficients Ule and
Vk)
Uk a!c+
=
+ Vk a-~k_ ,
Uk a_ k - -
(8)
Vlc ak+,
subjected to the restriction
(9)
are so determined as to make J-]O diagonal with respect to the newly introduced
operator ales' This condition implies the relation
Jle
(10)
V (Ele _).)2+ Jki~'
or
(11)
where
(12)
In this way the new diagonal Hamiltonian RO can be written as follows
HO = L-k (Ele -). - E le ) + 'Ele, lei Glele Xle Xk + L-k, s Ele aL aleS)
l
l
(13)
where
The equation satisfied by Xle can now be readily obtained.
and (13) into (5) we have
Putting (8)
400
M. Sano and S. Yamasaki
(15)
This equation has similarities to the momentum space representation of the two
particle equations in references 6) and 7). The same equation can be transformed into an equation for Li k , namely,
Li k = ~ ~kl Gkkl tanh (Ekl/2(})Li kf'
2
Ekf
(16)
Equations (15) and (16) have, besides the trivial solution Xk = Llk = 0, also a
non trivial solution in the region of temperature bellow the critical temperature.
The critical temperature (}c can be determined from the solution of the eigenvalue problem,
(17)
This equation is obtained from (15) and (14) by dropping Llk in (14).
By using (7) and (13) we can calculate the grand potential
It is
side
fixed
state
interesting to note that, if (12) and (14) are substituted, the right-hand
of (18) is stationary with respect to arbitrary variations of Xk for the
values of () and A. The mean occupation number nk.~ of the single particle
k,s also follows from (13),
t·_
nks =
~ -( 1- Ei~ it_ tanh if-)
.
(19)
The total number of particles in the system is given by
N = ~ks llks = -
If we put Xk = Llk =
f}
fJi- Q.
(20)
°
into (18) and (17), we obtain formulae familiar to the
Fermi gas model of atomic nuclei.
Thus the non-trivial solution implies our
departure from the latter model.
Thermodynamical functions of our interest, such as the energy E, the
entropy S, and the specific heat C, can be derived from (18) as follows:
(21)
(22)
(23)
Phase Transition and Level Density of Atomic Nuclei
§ 3.
401
Thermodynamical properties of atomic nuclei
In this section we discuss some characteristic features of the thermodynamical functions with the aid of the following simplifying assumptions:
a) Glelef is a constant G within an energy interval (j) around the Fermi
surface
(24)
and is equal to zero otherwise.
b) The sums over 1~ can be approximated by integrals with the single
particle level density set equal to a constant y. In every case of our interest
the assumption gG<l is justified. Even under these rather crude assumptions
a significant departure from the predictions of the Fermi gas model is implied
and qualitative features of the pairing system can be examined.
i)
Ground state energy gap and critical tetnj)erature
The energy gap LI (0) in the ground state is determined by the zero-temperature limit of Eq. (16)
LI le
=~}~ ~lef CleM -V(E~f !r)2+.;t~~2 .
(25)
With the aid of assumptions (a) and (b) we see that the energy gap is equal
to zero outside of the region (24), while it IS gIven by the solution of the
following equation inside of the regIOn (24)
~=E-)..
(26)
This equation gIves
LI (0) =~;-~ _(1}~ __~ ;
smh 1jgG
and as gC<l
IS
(27)
valid m our case we have the approximate equality
LI (0) = 2(j) exp ( --ljgC).
(28)
Equation (17) determining the critical temperature becomes similarly
(29)
from which we obtain
Oc = 1.14(j) exp ( -ljgG).
(30)
From Eqs. (28) and (30) follows the relation obtained by B.C.S.,
2L1 (0) jOe = 3.50.
(31)
402
M. Sana and S. Yamasaki
)
5
o Neutron
.. Proton
4-
o
a
o
., 0
o
"
o
00
_________ Q
'J
0
®
0
"
0 0~eooo
0
00 0
0
c 0
(;)
00
0
e
2
0°
a "
____________
"
a
oe o
a
@
°1;)°0
8
e
G
€l
0: oe~
e
.. ..
a
'il g
'"
___________
~_~
0
@
1
'----"------L--"-----'---='!'---'----'------'---~__'_____" __ ~_~~_'____L_'______"____"_____.J
o
.30
](X)
150
2CXl
:\Iass numLer Jl
Fig. 1.
Experimental data on the value of the pairing energy.
It is interesting that in the approximation employed, Be is determined by L1 (0) .
The ground state energy gap was calculated by Emery and Sessler8) without recourse to the crude assumptions (a) and (b), and similar calculation is available
also for the critical temperature. g) It was found, however, that the result is
much sensitive to the parameters employed and an agreement with experiment
can hardly be claimed. Therefore in the estimation of Be we proceed by using
empirically known results for L1 (0) rather than the parameters G and g themselves.
In this respect Eq. (31) is a great help to us.
The energy gap' L1 (0) is related to the pairing energy by the relation
P=2.:1(0).
(32)
The experimental values of pairing energy is available for a number of nuclides,
and it is seen that P has a general trend to decrease with the increase of the
mass number. The mean value of P for a fixed value of the mass number A
can be roughly represented bylO)
(33)
Though the empirical values are scattered above and below this value, we use
this relation for the purpose of the first orientation.
ii)
E.T.citation energy and temperature
The energy of the system with temperature 0 IS gIven by
Phase Transition and Level IJensity of Atomic Nuclei
403
(34)
This follows from Eq. (21) by the substitution from Eq. (16). The ground
state energy Es (0) is obtained by putting 0 = 0 into the above equation. Under
the assumptions (a) and (b) we have
ES(O)=L:k
Ek(l-
=EN(O)
+
V(Ek~~)~+L1k2)- -~--L:kV(Ek~i;2_~.dk2
L:
Ek -
Ek>A
L::
E/c--:Ek
Ek<A
- Ek--.J..
V(ElG-J..)2+L1lG2
E/c--
~L::k
2
f1k~~.cc~__ _
V(Ek-J..)2+A/c 2
_c .•
(35)
where EN(O) is the normal ground state energy expected on grounds of the
Fermi gas model. Thus, the excitation energy of the nucleus corresponding to
the temperature 0 is given by
G=E(O) -E(O)
=I::Ek(l- Ek--J.. tanh1l.k.)_l I::kL1k2_tanh-_Ek_E~)+1_g,:j2(O).
Ie
Ek
20
2
Ek
20
:2
(36)
Above the critical temperature Oc, the energy expression is much simplified
because Eq. (15) or (16) possesses but a trivial solution. In fact jn the normal
phase we have, putting L1 = 0 into (21),
E=2L;k
. _~lc
.-..
1 + exp {( Elc - A) If)}
This is the well-known expression for the energy of the Fermi gas.
the number distribution given by (19) becomes
(37)
Similarly
I.e. also the well-known Fermi distribution. Equation (37) allows the usual
low temperature expansion in terms of 0, and the excitation energy is given by
(39)
where
a=
77:
2
3
g.
(40)
404
M. Sana and S . Yamasaki
This equation not only shows the quadratic dependence of the energy on tenlperature but also reflects the lowering of the ground state energy due to the
pairing interaction.
The transition energy ee, corresponding to the critical temperature Be, IS
given by
(41)
G)
:>.
bJJ
H
Q)
c>
0
c
P
IV
P
.S
is
'13
:>j
~
1
2
-g.d2 (O)
,
/
1
/
Super
Fig. 2. This figure gives a schematic illustration of the level scheme. The complete spectrum consists of two parts.
One is the levels in the superconducting phase shown in the left half of
the figure, the other is the levels in
the normal phase shown in the right
half. The levels depicted by broken
lines, in fact, are absent in the presence
of the pairing interaction. The system
is not found in these states, rather, it
is found in the superconclucting state
with energy corresponding to the same
temperature. The clotted line connects
two levels, normal and superconducting, whose energies correspond to the
same temperature.
Normal
10
"?
C)
6
>,
en
8~
OJ
:~
;)
~
'"
'-<
[-<
Mass
!lumber A
Fig. 3. Transition energy plotted against the atomic mass number A.
p= 3.36(1-A/400), a=AjlO.
Parameters:
405
Phase Transit-lon and Level Density of Ato.rnic Nuclei
Below this energy the nucleus is in the superconducting phase. The transition
from super to normal phase is illustrated in Fig. 2. The left half of this figure
represents the levels of the system in the superconducting phase, while the
remaining part represents the same expected for the normal phase. The broken
lines join the levels with the energies corresponding to the same temperature.
The difference in the energy of corresponding levels decreases with the increase
of temperature as the correlated nucleon pairs are broken, or in other words,
as the number of elementary excitations increases. The difference vanishes at the
critical temperature where· the energy gap also vanishes, and above this temperature the normal phase is reached. The excitation energy c corresponding
to this temperature is given in Fig. 3 as the function of the mass number;
using the relation (33) and the empirically known value of a.
e
30 .
30:
~
6"'"
20
C,,)
:0-,
to/)
H
Q)
::::
Q)
::::
..8
E
'ux
f.il
·10
10,
:~
:
Ii
)
r ,
!
!
1
o
o (Mev)
Fig. 4. Excitation energy plotted as the function
of the temperature. Parameters: A=60,
P=3.36(1-A/400), a=AjlO. ----- Fermi
gas model; ------ Fermi gas model, where
only the ground state is lowered by the
pairing correlation (Eq. (39»; - - prediction from our theory. The high and
low energy regions of this curve were
calculated by using Eqs. (39) and (43) respectively, while the intermediate portion
was interpolated in a reasonable manner.
Similar steps were taken in drawing the
following figures.
?
2 /
o (Mev)
Fig. 5. Entropy plotted as the function of ·the
temperature. Parameters: A=60,
P=3.36(1-Aj400), a=AjlO.
406
M. Sano and S. Ya'masaki
Below the critical temperature, the functional relationship of the excitation
energy and the temperature is not so simple as the relation (41) holding above
the critical temperature. For low energies, however, we have
J «()) - J (0) ~ - 2.1 (0) Ko (.1 (0) I()),
(()<.1 (0»
where Kn
IS
(42)
the modified Bessel function of the second kind.
Thus we have
In Figs. 4 and 5 the excitation energy as well as the entropy are plotted as
the function of the temperature.
iii)
Entropy and specific heat
The entropy S, of the system is given by
(44)
In the normal phase this equation is much simplified because Ek = ItEl,; - AI and
the usual expansion in terms of () gives
S=Sn=2a().
(45)
Combining Eqs. (39) and (45) we have
(46)
In the superconducting phase the entropy is a much more complicated function
of the energy, and no relation as simple as (46) is available. For temperatures
sufficiently low as to satisfy .d (0) I(» 1 we have similarly to (42) and (43)
S = Ss = 2-rt-.d 2 (0) [Ko (.d (0) I() + K2 (J (0) I()
()
The
Eqs.
Fig.
The
J.
(47)
entropy is determined as a function of the excitation energy by combining
(43) and (47). We have plotted the entropy versus the temperature in
5. The same quantity is plotted versus the excitation energy in Fig. 6.
specific heat (23) can be rewritten as
(48)
Above the critical temperature this reduces to
c= Cn~2a(),
(49)
as is evident from (39).
Below the critical temperature, the ~ame approximatio_1 as used for (43) and
(47) gives the result derived by B.C.S.,
407
Phase Transition and Level Density of Atomic Nuclei
15
10
CI:)
:>..
p.,
I
j
I
~
5
I
')
I
2
2.42
ve
Fig. 6. Entropy plotted against the excitation energy. Parameters: A=
60, a=A/10, P=3.36(1-A/400). -- ... Fermi gas model; ----.- Fermi
gas model, where only the ground state is lowered by the pairing correlation (Eq. (39»; - - Prediction from our theory .
.Ji(O)
.
c= cs ~ g........
[3Kl (iJ (0) If) + Ka (iJ (0) If) J.
f)2
The discontinuity of the specific heat at the transition temperature
(50)
IS
given by
(51)
where
~(-.iI.j2
f)c
df)
)
= -.9.4.
O=0r,
Thus the ratio of the specific heats just below and just above the critical temperature is given by
(53)
This ratio IS common to the nucleon system and the electron system in metals.
F or the sake of the latter convenience we note the jump in the derivative
of the specific heat at the critical temperature.
\OJ The slight and unfortunate disagreement with the
result in reference 4) stems perhaps
rum the numerical evaluation of the relevant integral.
408
M. Sano and S. Yamasaki
(54)
§ 4.
Level density
We discussed in the preceeding section the thermodynamical properties of
atomic nuclei. In our case many of the thermodynamical quantities are not
measured directly in contrast with the case of the superconducting metals.
These properties, however, are expected to manifest themselves in various aspects
of nuclear reactions.
The nuclear level density is the most characteristic
quantity in this respect as it can be inferred from nuclear reaction experiments.
The level density w (E) of the system of N nucleons is related to the
thermodynamical functions by the following relation,
e~F/3 =
f
w (E) e- E/3 dE,
{3=1/f},
(55)
where
F=Q+J.N
is the Helmholtz free energy, and the integral extends from the ground state
energy to infinity. The inverse Laplace transformation of (55) glve~
'Y+ ioo
weE)
=
-~J
27Cz 'Y-too
.
e(3(E~F) d/1,
(56)
This integral can be evaluated by the method of steepest descent at the saddle
point of /1 given by
o
E=-
0/1
=
(oJ.) N+J.N
0
({3F)N=~-·-((3Q)N+{3N
-
0(3
of]
a ((3 Q) A + AN.
fiij
(57)
Thus we have
eS
(3(E-F)
w (E) =
v=2n~~2/8(32)
((3l!)
N
V'2ntr C'
J
(58)
where, using (20), (21), and (57), the exponent is identified with the entropy
(22) , and (8 2/8(32) ((3F) N is identified with ~ 02 times the specific heat (23). In
accordance with the usual convention we consider the level density as the function of the excitation energy and write w (e) as no confusion is likely. As we
discuss in the following the level dE:nsity w (e) predicted by Eqs. (58), (44),
and (48) differs in many respects from that expected, on grounds of the Fermi
gas model.
Above the transition energy, the situation is very much alike to the case
Phase Transition and Level Density of Atomic Nuclei
409
where the pairing interaction is absent. In fact thermodynamical functions are
in accordance with the prediction of the Fermi gas model with the only exception that the relation between the excitation energy and temperature is di:6:erent . from that predicted by this model. As we discussed already in § 3 this
relation is given by
(39)
The last term in the right-hand side of this equation, representing the effect of
the pairing interaction, is the principal cause of the failure of the conventional
formula known as the Weisskopf formula. In view of the importance of the
the pairing energy in nuclear spectroscopy, phenomenological attempts have been
made to introduce the even-odd difference in the nuclear level density.3) The
even-odd difference in this case was attributed to the pairing interaction which
was supposed to affect only the ground state of even nucleus. Accordingly,
the energy temperature relation for even nucleus obtained was written as
0=a(P+P,
(P=211 (0) ) .
Contrary to this approach, the second term in (39) is quadratic
(59)
III
the paIrIllg
20
Cl
Cl
10
;\Iass Number .!l
Fig. 7. Empirically determined values of a in the level density formula Wee) =c exp(2vae)
derived from the Fermi gas model.
--a=A/10.
X Igo and Wegner, Phys. Rev. 102 (1956), 1364.
6
Igo and Wegner, Phys. Rev. 102 (1956), 1364.
o Swenson and Cindro, Bull. Am. Phys. Soc. 5 (1960), 76.
o Thompson, thesis (Los Alamos Scientific Laboratory, 1960), unpublished.
•
Mead and Cohen, Phys. Rev. 125 (1962» 947.
410
M. Sana and S. Yamasaki
energy. Moreover, the discussions of § 3 show that the effect of the paInng
interaction is not limited to the ground state but extends up to the transition
energy. In our case the even-odd difference, if necessary, should be attributed
to the fine structure of the second term in the right-hand side of Eq. (39).
Below the transition energy, as we have discussed in the preceeding section,
the entropy, the specific heat, and the excitation energy are ratb.er complicated
functions of the temperature. At low temperature, however, Eqs. (43), (47),
and (50) are available. In Fig. 8 we give the calculated level density for a
specific example. It is seen that, below the transition energy, the level density
is smaller than that expected by the Fermi gas model.
~---~"""'-!~-.~--~~~~-~'~-
.f
loge'wCe)
§
;
f
II
I
4'
fI
II
fI
10
fI
f
I
$
I
1/
I
I
!
:
:
I
8.04- -
,
:
:
7.58-
5
o
o
1
1.36
Fig. 8. Logarithmic level density plotted against the square
root of the excitation energy.
Parameters: A=60, a=AjIO, P=3.36(I-Aj400)
- - - - - Fermi gas model, -.---- Fermi gas model, where only
the ground state is lowered by the pairing correlation
(Eq. (39)); - - Prediction from our theory.
Another fact of our interest is the discontinuity of the level density at the
transition energy. This is due to the jump, Eq. (51), of the specific heat
coming in the denominator of (58). The discontinuity of the slope of th(
logarithmic level density is given by
- d
deS'(:-
log'w (eS'c -)
I·deS'/!+
d log
'LV
(eS'/)
Phase Transition and Le'vel Density of Ato'mic Nuclei
(dC,. _ dCn )} (_9p_)2
dO
dO
Cs
411
+2 ~IC ]}/[l- 43-a1o--c-] ,
s
(60)
where
(61)
This ratio differs from unity by only a few percent. Thus, the level density
curve experiences no violent change of slope at the transition energy. This is
so because we choose the energy as the abscissa. The curve should be bent
at the transition point if we choose the temperature as the abscissa.
§ 5.
Imaginary part of the optical potential
VVe discuss in this section the imaginary part of the optical potential. It
is expected that this quantity, representing one of the important characteristics
of atomic nuclei, is affected by the pairing interaction. We would like to point
out here that the imaginary part of the optical potential is affected greatly by
the change of the level density due to the pairing interaction which we discussed in the previous section. Until now, theoretical evaluations of the optical
potential have been based on the Fermi gas model and proved to agree with
experiments. The imaginary part, however, should deviate largely from these
predictions if the superconducting states of our interest extend to fairly high
energIes.
According Feshbach et al.,l1) the optical potential is so defined that the
amplitude of the potential scattering does not interfere with the amplitude coming
from the compound process. The transition matrix element of the elastic scattering IS
Taa=
iti{ <([)al UI¢~+)
+ <¢~-)IV - UI¢~+) + ~I'<¢~-)IV -
UI/l)E=\v -<fJ.IV - UI¢'al-) ] .
I'
(62)
Here V is the interaction potential of the incoming particle in the field of the
target nucleus, U is the trial distortion potential which is suitably chosen in
the following to give the optical potential.
([)a is the wave function of the
system in the absence of the interaction, ¢~+) is the wave function of the initial
state distorted by the potential U.
WI' is the complex energy eigenvalue of
the compound state I jJ. ). In the right-hand side of (62), the first term represents the amplitude of the potential scattering, while the remaining part gives
the amplitude of the compound elastic scattering. The optical potential U is
defined by the condition that the interference of these two parts should vanish
on the average taken over u small energy interval including still a number of
. <f
M. Sano and S. Yamasaki
412
close lying resonances.
equation,12)
f·' -"".., ...
This condition can be expressed by the
U = <0 IVI 0> + :E,,<O IV - VI 11> ___
1 -(Ill V - UIO)
E-Wjt
?
following
(63)
where the bar represents the energy average, and 10> represents the channel
in which the target nucleus is found in the ground state.
If we assume the widths of compound levels to be small as compared with
the level spacings, the imaginary part of the optical potential can be written
as follows:
(64)
In this way the imaginary part of the optical potential is related to the level
density. As the imaginary part of the optical potential is expected to be small
as compared with the real part, we can proceed with the iteration process in
the evaluation of (64), in which case the first approximation for 1m U consists
in replacing U by <OIVIO> in the right-hand side of (64). Although the transition matrix element between superconducting state 10 > and 111> would be
different from that expected by the Fermi gas model, it is natural to consider
this will make little difference in view of the great difference introduced in the
level density as was discussed in the previous section. Thus, if the transition
energy is sufficiently large, and the energy of the incident particle is so small
that the compound nucleus is formed in the superconducting phase, the imaginary part of the optical potential describing the elastic scattering would be
much smaller than it has been considered up to date. Thus it is possible that
resonances of small widths are present in this energy regIOn. This must also
be reflected in the energy sensitiveness of the angular distribution as well as
of the total cross section of nuclear reactions.
§ 6.
Discussions
We have discussed the effect of the pairing interaction on the gross structure of nuclear spectrum. The main conclusion is the existence of the transition energy of about 5 to 10 Mev below which the Fermi gas model is invalidated.
In fact in this superconducting phase, the energy-temperature relation is much
different from that expected for the latter model, and the level density is much
smaller than is expected by the extrapolation from the normal phase. This
fact must be reflected in the smaller values of the imaginary part of the optical
potential.
It must be remembered here that our estimation is of statistical nature.
Moreover,· the nucleus is assumed to be sufficiently large despite the fact that
the real nucleus is finite.
Indiviclualities of existing nuclei are incooperated only when the shell
Phase Tt"ansition and Level Density of Atomic Nuclei
413
structure of the single particle levels is taken account. Though we do not go
into this stage of the calculation here, we would like to mention that the shell
structure of nuclei can partly be allowed for by using the value of parameter
P given in Fig. 1 deviating significantly from the mean values.
Another important problem left is the effect of the angular momentum. As
the angular momentum is a good quantum number for atomic nuclei, the statistical dynamical considerations can, in principle, be applied also to statistical
ensembles with definite values of angular momentum. Or, we can introduce
another parameter corresponding to the angular momentum or the angular
velocity in addition to the temperature. Thus the critical temperature or the
transition energy is a function of the angular momentum. Our results for 0 c
correspond to the case of vanishing angular velocity. It must be mentioned,
moreover, that for sufficiently large values of the angular momentum, the Coriolis
force prevents the formation of the coupled pair.I3) This corresponds to the
case of the external magnetic field applied to the superconducting electron
systems whose pairing correlation is decoupled by the field. Thus it is expected
that there is a critical value of the angular momentum, i.e. the upper bound of
the angular momentum such that the nucleus can be superconducting. The study
of the relationship between the transition energy and the angular momentum
is now in progress and we wish we are able to report on this interesting topic
in the near future.
Acknowledgements
The authors would like to express their sincere thanks to Prof. K. Husimi
for his kind interest and encouragement throughout this work and to Drs. T.
Kammuri and T. Yamazaki for many helpful discussions.
References
1)
2)
3)
4)
5)
6)
M. G. Mayer, Phys. Rev. 75 (1949), 1969; 78 (1950), 16.
Haxel, Jensen and Suess, Phys. Rev. 75 (1949), 1766; Z. Physik 128 (1950), 295.
G. Racah, Phys. Rev. 63 (1943), 367.
J. P. Elliot and A. M. Lane, "The Nuclear Shell-Model" in the Encyclopedia of Physics,
edited by S. FIUgge (Springer-Verlag, Berlin), Vol. 39 (1957).
H. Hurwitz and H. A. Bethe, Phys. Rev. 81 (1951), 952.
T. D. Newton, Can. J. Phys. 34 (1956), 804.
D. W. Lang and K. J. Le Couteur, Nuclear Phys. 14 (1959), 21.
N. N. Bogoliubov, D. N. Zubarev and Iu. A. Tserkovnikov, Doklady Akad. Nauk SSSR
117 (1957), 788.
J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957), 1175.
A. Bohr, Mottelson and Pines, Phys. Rev. 110 (1958), 936.
S. T. ,Belyaev, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medd. 31 (1959), No. 11.
L. S. Kisslinger and R. A. Sorensen, Kgl. Danske Videnskab. Selskab, Mat.-fys. Medel.
32 (1960) , No.9.
V. M. Galickij, JETP 34 (1958), 151.
414
M. Sana and S. Yamasaki
7)
8)
9)
10)
11)
12)
13)
F. Iwamoto, Prog. Theor. Phys. 23 (1960), 871.
V. J. Emery and A. M. Sessler, Phys. Rev. 119 (1960), 248.
Y. Wada, Prog. Theor. Phys. 26 (1961), 321.
V. J. Emery and A. M. Sessler, Phys. Rev. 119 (1960), 43.
A. Stolovy and J. A. Harvey, Phys. Rev. 108 (1957), 353.
H. Feshbach, C. E. Porter and V. F. Weisskopf, Phys. Rev. 96 (1954), 448.
M. Sano, S. Yoshida and T. Terasawa, Nuclear Phys. 6 (1958), 20.
G. E. Brown, Rev. Mod. Phys. 31 (1959), 893.
B. R. Mottelson and J. G. Valatin, Phys. Rev. Letters 5 (1960), 511.