Download Influence of the Pairing Interaction on the Nuclear Energy Level

CHINESE JOURNAL OF PHYSICS
VOL. 3, NO. 1
APRIL, 1965
Influence of the Pairing Interaction on the
Nuclear Energy Level Density
J E N N- LIN H W A N G (3.W$)
Department of Physics, National Taiwan University, Taipei, Taiwan
(Received May 30, 1965)
The modern theory of superconductivity developed by BCS and BZT is
emp!oyed to examine the pairing interaction on the nuclear energy level density.
A statistical model of nucleus is assumed to obtain the over-al1 character of
the excited states. The empirical formula for the pairing energy proposed by
Nemirovsky and Adamchuck and the semi-empirical formula for the single
particle level density proposed by Abdelmaleck and Stavinsky are used to
ca!cuJate the transition temperature, excitation energy, entropy, specific heat
and the total level density. Except for the extremely light and heavy nuclei, a
phase transition is predicted all at an energy of 2.73 Mev or so for a wide
range of the mass number. This is remarkabIy different from the one ca!culated
by Sano and Yamasaki. The level density below the transition energy is far
smaller than that expected on extropolating it from the higher energy side.
§ 1. INTRODUCTION
pairing correlation effect in nuclei was first considered by A. Bohr, Mot7r HEtelson
and Pines(‘) and Belyaev@) with the aid of the method used in the
theory of superconductivity. An important consequence was a gap in the spectrum
of even nuclei. Based on the existance of this energy gap Ericsonc3’ obtained a
qualitative understanding of the so-called even-odd effect, namely that the level
densities for odd nuclei are larger than the density for odd-mass or even nuclei.
The ground state energy gap was calculated by Emery and SessIer(4), and recently
by Kennedy, Wilets and Henley (5) for different types of internucleonic potentials.
Through lengthy and tedious calculations it was found that the result is much
sensitive to the parameters employed and an agreement with observed values
for finite nuclei can hardly be claimed.
Kluge (‘1 studied the influence of the magnitude of energy gap on the level
density by employing the Fermi gas model. Canuto and Garcia-Colin”) treated a
nucleus as a binery mixture of non-interacting gases of nucleons and counted
( 1) A. Bohr, B. R. Mottelson and Pines, Phys. Rev. 110, 936 (1958).
( 2 ) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 35 No. 11 (1959).
( 3 ) T. Ericson, Nuclear Phys. 8, 265 (1958) ; Advan. Phys. 9, 425 (1960).
( 4 ) V. J. Emery and A. M. Sessler, Phys. Rev. 119, 248 (1960) ; 119, 43 (1960).
( 5 ) R. Kennedy, L. Wilets and E. M. Henley, Phys. Rev. 113, B1131 (1964).
( 6 ) G. Kluge, Nuclear Phys. 55 41 (1964).
(7) V. Canuto and L.S. Garcia-Colia, Sl, 177 (1965).
35
.
~__~__.
JENN-LIN
36
HWANG
observed variation of level densities for odd, even and odd-mass nuclei.
In their calculation the energy gap was chosen as an independent free parameteter.
Sano and Yamasaki,@’ used empirically known results for the energy gap to study
the gross behavior of the excited states. They predicted that the phase transition
would occur at the energy of some 5 to 10 Mev. The experimental data adopted
by them were, however, limited only to light and medium nuclei.
Recently Nemirovsky and Adamchuk (‘I have determined the pairing energy
from the difference of the masses of odd-mass and adjacent even nuclei, tanking
into account the corrections for the difference in the surface and coulomb energies
as well as the symmetry energies in these nuclei. On the other hand Abdelmalek
and Stavinsky”‘) have obtained an accurate semi-empirical formula for the single
particle level density from the Erba’s results. It is the purpose of this paper to
recalculate the transition energy, excitation energy, entropy, specific heat and
the level density using these new data of the pairing energy and the single
particle level density. It is found that the transition energy of the phase change
is far smaller than the one predicted by Sano and Yamasaki@) and that the
variation with the mass number is dissimilar with theirs.
for the
92. PAIRING FORCE AND THERMODYNAMIC FUNCTIONS
In the Bardeen-Cooper-Schrieffer”” (BCS) theory of superconductivity the
thermodynamic properties were treated by the variational method, and in a different
approach developed by Bogoliubov, Zubarov and Tserkovnikov”*) (BZT) they were
treated by the perturbation method in the statistical mechanical perturbation
theory. In the present paper only some important relations which will be referred
to in the subsequent calculations are quoted.““)
The Hamiltonian of a syste.m of nucleons interacting with the pairing force
can be written in the same form as that of an electron system in a superconductor
as follows:
H=% (Q--D) (C&t +Cc,fC,+) -2,~+ C-K~G~CV,C-~L
(1)
where Ek and fl are the single particle energy and the chemical potential respectively. Jkv is the interaction matrix element and is a small positive quantity
_
(8) M. Sano and S. Yamasaki, Prog. Theoret. Phys. (Kyoto), 29, 397 (1963).
(9) P. E. Nemirovsky and Yu. V. Adamchuk, Nuclear Phys. 39, 551 (1962).
(10)
(11)
(12)
(13)
(14)
N. N. Abdelmalek and V. S. Stavinsky, Nuclear Phys. 58, 601 (1964).
J. Bardeen, I,. N. Cooper and J.S. Schrieffer, Phys. Rev. 108, 1175 (1957).
N. N. Bogoliubov, D. N. Zubarev and Iu, A. Tsarkovnikov, Doklady Akad. Nauk SSSR 117, 788 (1957).
C. T. Chen-Tsai, Chin. J. Phys. 3, (1965) (this issue) A nice general survey of BZT approach has
been furnished in the introductory part of this paper.
52 and $3 of this paper are written partly for an heuristic purpose, and are mainly based on the
references 8,ll and 13.
c
PAIRING INTERACTION ON THE NUCLEAR ENERGY LEVEL DENSITY
37
symmetric in k and K’ and nonvanishing only for those k’ and k whose corresponding energy levels Ek and Ek’ locate in the neighborhood of the Fermi surface. The
creation and annihilation operators CAD and Cka satisfy the usual anticommutation
relations. By the BZT approach or by the Feynman diagram evaluation method
of the Bloch perturbation series (Is) the grand partition. function
Z=e;pP=Trace [e-PHI
(2)
can be evaluated, and yields an asymtotically exact expression for the Gibbs
function
where
fi=ck
(Ek-/-d)-+g
(l+e-p=k)
+$k
tanh
+,8Ek
17
(3)
is the energy spectrum of the quasiparticle and
the energy gap that should be determined from the equation
Ek=d (&k-p)‘+
Ai
dk
is
(4)
B is the reciprocal of the temperature 0 in units of Mev. The transition temperature Bc can be determined from the ‘solution of the equation
Jf
( 5 )
2
k’ v
which is obtained from (4) by dropping Ak in Ek.
The mean occupation number nk of the single particle state follows from (3),
1
2(Ek-~)dk=tanh--B(Ek-~)C * Ltk’,
nk=$(l-K&!! tanh+B&),
(6)
and therefore the total number of particles and the Helmholtz function are given
N=F nk= -$S2 a n d F=SJ+,uN
by
(7)
respectively. Other thermodynamic functions are found from (3) and (7) as
follows :
Entropy
L
=2F
[10g(l+t?-8Ek)
+2,8Ek/(l+dEk)].’
Ek
[j2Ek-Bsk-].
3(1/B) cash’+ /%
( 8 )
(9)
y tanh -+BEk) - 2$:-tanh-;.-BEa] (10)
$3. THERMODYNAMIC PROPERTIES WITH SIMPLIFYING
ASSUMPTIONS AND LEVEL DENSITIES
As mentioned in §l, the solution of (4) is difficult to obtain even for the
JENN-LIN HWANG
38
ground state. Therefore we use, as done by BCS, the following simplifying assumptions :
[ a ] Jw/V can be replaced by a constant I within a region of interval w
around the Fermi surface --L<EL-P<W, and by zero outside this region.
[ b ] The sum over k can be replaced by integrals with single particle density
set equal to a constant g, and in every case gI<l is assum,ed.
With these assumptions Eqs (4)-(10) may be simplified and the quantities in
which we are intersted can be evaluated.
( I ) Ground state energy gap and transition temperature
The ground state energy gap A(0) is determined by the zero-temperature
limit of (4). With the simplifying conditions it is found from
o
2
_=
E=E-P,
(4)’
SI s-.P+k,
A (0) = w/sinh ( l/g11 ,
to be
(4)”
A(O)=Zw exp (-l/gI)
or
in the limit gI<l.
The transition temperature 6, is determined similarly from the simplified
f0r.m of (5)
2 -.f “tanh-&-
gr
-0
c
l
-$k,
(5)’
(5)”
6,=1.14w exp (-l/gI).
as
Combining (4) V with (5)” yields the relation obtained by BCS,
(11)
24(0)/0,=3.50.
( II ) Excitation energy
The ground state energy U,(O) is obtained by putting 6=0 in (10). After the
simplification,
zz EF(O) -+gA'(O),
(10)’
where&(O) is the normal ground state energy expected fro.m the usual Fermi
gas model. Above the transition temperature Bc we have, putting A=0 in (lo),
the usual expression for the Fermi gas,
(10)”
U,( 6) =2F Ek/[l +eQ(ck -p)] = I&(O) +ao2,
where
a=Ag_
3
The excitation energy in the normal state is thus given by
E”=U.(0)-US(O)=a02+-~-gA2(O).
(12)
(13)
.,
+
PAIRING INTERACTION ON THE NUCLEAR ENERGY LEVEL DENSITY
39
Below the transition temperature, U,(0) is not expressible in a simple form
as (lo)“, and only for temperatures sufficiently low as to satisfy d(O)/B>l we have
d(8)---d(O)=--24(0)&(/3d(0)),
and
~,=K(~)-K(0)=2g~2(~)Eo(B~(O)) +&(B~(o))l,
(14)
where K,, is the modified Bessel function of the second kind.
The transition energy E c, corresponding to the transition temperature Bc is
given from (43) by
E,=&q + +gd2(0).
(15)
(III 1 Entropy
The entropy S of the system (8) can be rewritten as
In’ the normal state where d=O, the usual expansion in terms of 0 gives
S,z2ae.
Using (13) this can be expressed in terms of the excitation energy as
(16)
(16)’
. L=2da[&--- (1/2)gd2(0)1.
In the superconducting state, (8)’ is a complicated function of 0, and only for
low temperature such that d(O)/e>l we have
(17)
&=2~s~“(0)lX(B~(0)) +&(B~(O))l.
(IV) Specific heat
The specific heat (9) becomes
c= -;-J-1
sech’% 8r/[2+ P[p2(E2+ A2) --&Bdd2/d(1/8)]&.
(9)’
L
In the normal state this reduces to
C,z2aB.
(18)
In the superconducting state we have similarly to (14) a n d ( 1 7 )
(19)
C,zB2gd3(0)[3~(Bd(O)) +K3@~(0))1.
The discontinuity of the specific heat at the transition temperature is found from
(9) ’ to be
(cs-c~)~=@c=-$ F sech2w (g)B_-B
l
c
=
c.
-C
(20)
The derivative dA2/dB can be obtained from the relation suggested by Buckinghamo5)
A=3.070c
[l.-+ 1 ’
c
(15) M.J. Buckingham, Phys. Rev.
_
t
lOl, 1431 (1956).
40
JENN-LIN HWANG
1 dAz
~=-9.4 and (C,-CC,)e=e,=9.4gflc.
(21)
0, ( d0 >o=e,
The ratio of the specific heats just below and above the critical temperature is
C,(e,)/C,,(6,) =2.42.
given by
(22)
( V ) Level density
The level density w(E) of the system is related to the thermodynamic
functions by
e-BF= Omw (E)e-8 EdE,
(23)
J
where F is the Helmholtz function (7). The inverse Laplace transformation of
(22) gives
and we find
This integral can be evaluated by the method of steepest descent at the saddle
point of B determined from
Thus we have
w(E) =ea(E-F)/[-27i.(d2BF/dB2) ~1: = es//2#C_
(24)
Following the usual convention we consider the level density as the function of
the excitation energy and write it w(E). In (16)-(19), the entropy and specific
heat are expressed in terms of the temperature 6. On the other hand 0 is related
with the excitation energy % through (13) and (14), so that the right handed
member of (24) is formally expressible in 6.
$4. NUMERICAL CACULATIONS
The energy gap A(0) is related to the pairing energy by the relation
Pz2A(O).
(25)
Fig. 1 shows the pairing energy of two nucleons deduced from even-odd
The mean variation with
mass differences by Nemirovsky and Adamchuk.“)
mass numbe A can be fitted to an A-o*551 law, that is
Pz2A(O) =16.2/A”.551 Mev.
(26)
The empirical values are scattered above and below this curve. Neverthless, for
the purpose of testing the gross behavior of nuclei we use this curve in the
present calculation.
The parameter a in (10)” connects the excitation energy with the temperature
E=ae2,
in the form
and appears in. the level density formula
w(Z) =C exp (ada% )
for the Fermi gas model. If B is much smaller than the Fermi energy, then
a=$g.
(12)
PAIKING INTERACTION ON THE NUCLEAR ENERGY LEVEL DENSITY
41
neutrons
. - protons
0-
3.01
2.5 2.0 1.5:
i.0:
0.5 -
Weiyhf -.A
A f omit
Fig. 1. Pairing energy of two nucleons deduced from even-odd mass difference,
reproduced from Nemirovsky and Adamchuk (Ref. 9)
30
.
t
p0
I
1
N-82
N=-50
W Q’
I
,
5
,
,
0
,
1
100
I
,
,
- Average fit
I
I
I
N-l%
I
I
I
150
200 A
Fig. 2. Dependence of Ievel density parameter a on mass number A, reproduced
from Abdelmalek and Stavinsky (Ref. 10)
Fig. 2 shows the dependence of a on mass number A. Abdelmalek and Stavi”“
risky ” have obtained a semi-empirical formula
a= (0.095 f 0.007)AZ’* (jz jly + 1) Mev-’
(27)
on the basis of the experimentally observed order of filling of the single particle
level. In this formula jz and jN are, respectively, the average values of the total
angular momentum j for protons and for neutrons over the shells near the Fermi
+
42
JENN-LIN HWANG
level. For the present purpose we also neglect the local variation of d and take
the smoothed fitting values.
The transition energy &, is plotted against the mass number A in Fig. 3
according to (15), where the- relations (ll), (12), (26) and (27) are utilized. The
result of Sano and Yamasaki is also shown in the same figure. They used
P=3.36 (l-A/400) Mev and d=A/lO Mev-’
instead of (26) and (27). Therefore both curves have different character not
only in magnitude but also in shape. It is impressive that the present calculation
yields a plateu of about 2.74 Mev over a wide range of the mass number
5O<A<160.
1O -
- Present Calculation
---- Sane-Yamasaki’s
Calculation
S-
4-
2-
L
I
I
1
1
I
I
100
Mass number A
I
I
I
-_
200
Fig. 3. Transition energy plotted against the mass number A. (Eq. (15)).
Parameters used by Sano and Yamasaki:
P-3.36 (l-A/400), a=A/lO.
Parameters used in the present paper:
P= 16.2A0.551, a = 0.095 A*/* (jz +j~ + 1)
In Fig. 4 and Fig. 6 the excitation energy and specific heat are given as the
functions of temperature respectively. Since (14) and (19) hold only for low temperature, the intermediate portion below the transition temperature is interpolated
in a reasonable manner. In Fig. 5 and Fig. 7 the entropy and logarithmic level
PAIRING INTERACTION ON THE NUCLEAR ENERGY LEVEL DENSITY
43
density are respectively calculated in terms of the square root of the excitation
energy. In the superconducting phase the portion near the transition energy is
also interpolated suitably. In these four figures the nucleus of mass number
A=100 is taken as a specific example. Dotted line in each case represents the
Fermi gas model, in which only the ground state is lowered by the pairing
interaction. It is evident that the Fermi gas model is invalidated in the superconducting region and the level density there is much smaller than is expected
by an extrapolation from the normal region. No significant different character
can, however, be found from these figures as compared with those illustrated by
Sano and Yamasaki.
‘I” -
(hf.+]
-
s
30 -.
%
b
El
20
-
10 -
Temperature 0
Fig. 4.
Excitation energy as the function of
temperature (Eqs. (13), (14)).
@i;Teters: A=lOO, P=1.28 Mev, a=14.1
---------‘Fermi gas model, -a-- Fermi gas
model, where only the ground state is
lowered by the pairing interaction.
Fig. 5.
Entropy plotted against the sqnare root
of the excitation energy.
Parameters:
A = 100, P= 1.28 Mev-i,
a~14.1 Mev-*.
--------- Fermi gas model, --- Fermi gas
model, where only the gronnd state is
lowered by the pairing interaction.
- Present theory.
JENN-LIN
44
HWANG
30-
G
zo-
s
B
”
!5
“x
CA
_’
10 -
Fig. 6. Specific heat plotted against the temperature (Eqs. (18), (19)). Discontinuity at
the critical temperature is given by
(Cs-Cn)8=,qc=9;4 g8,, where g=-$-a.
Parameters: A=lOO, P=1.28 Mev, a=14.1
Mev-‘.
Fig. 7. Logarithmic level density plotted against
the square root of the excitation energy
(Eq. (24)) _ _ _ __ _ _ _ _ Fermi gas model, -_- Fermi gas
model, where only the ground state is
lowered by the pairing correlation.
Parameters: A =lOO, P= 1.28 Mev. a= 14.1
Mev-‘.
ACKNOWLEDGMENT
The author wishes to thank the National Council on Science Development for
a financial support.