- Email: [email protected]
Pairing reduction due to the blocking effect
Nuclear Physics A421 (1984) 12%~~140~ North-Holland, Amsterdam
PAIRING REDUCTION
125c
DUE TO THE BLOCKINGEFFECT
J. Y. Zeng The Niels Bohr Institute, DK-2100 Copenhagen 0, Denmark and Physics Department, Peking University, Beijing, China T. S. Cheng. L. Cheng and C. S. Wu Physics Department,
Peking University,
Beijing,
China
Pairing reduction in deformed nuclei due to the blocking effect is treated exactly with the particle-number-conserving method. Merely due to the blocking effect the gap parameter A becomes configuration-dependent. The pairing reduction 6(vD) depends sensitively on the location of the blocked level v relative to the Fermi surface, X, and decreases rapidly with The pairing reduction 6(vD) depends also sensitively on the lC"O - 21. In some special single-particle level distribution near the Fermi surface. cases pairing reduction may be negative. With an increasing number of the blocked levels (above the Fermi surface) the qap parameter A decreases If the blocked levels are below the Fermi surface the situdramatically. ation is suite different and as the number of the blocked levels becomes sufficiently large the pairing reduction may vanish.
1. I NTRODU~TION There are two kinds of anti-pairing (i) the
Coriolis anti-pairing
effects
in nuclei:
effect' and (ii) the blocking effect*.
The
fact that the blocking effect can lead to pairing reduction is well known3. Let A denotes the gap parameter neighbouring
for an even-even
nucleus and A(vo) that for the
odd-A nucleus, where vD is the single-particle
the odd nucleon.
level occupied
A(wD) < A. The pairing reduction due to blocking
(1.11 is defined as
&(uD) = A - A(v&
(1.2)
It is a special kind of even-odd difference. differences
are essentially
Example 1.
by
It is found that in general
The even-odd
In fact, the various even-odd
due to the blocking effects. shift in the moment of inertia$J4.
Experiments
show that in general the moment of inertia of odd-A nucleus is larger than that of the neighbouring
even-even
7 (odd-A nucleus) Example 2.
The even-odd
nucleus, >p(g.s.b.
of even-even
shift in bandcrossing
U3~5-9474/g4/$~3.~0 0 Elsevier Science Publishers B.V. ~No~h-H~~and Physics ~blishing Division)
nucleus).
frequency, hwC.
(1.3) Recently,
it
L Y. Zeng et al. j Pairing Reduction due to the Biocking Effect
126c
was established
that5
hwC(odd-A
nucleus)
< hoC(even-even
nucleus).
(1.4)
Let 6hwC = %wC(e-e)
- hwC(Odd-A).
il.51
It is found that in most cases for the rare-earth nuclei 6RwCjexp These two examples
are closely
the moment of intertia
hw
C’
the pairing
The analysis
connected
and the bandcrossing
value of the gap parameter calculations,
% 40 kev.
A.
By making
reduction
(1.6) with the pairing frequency
reduction,
use of the cranked shell model
can be extracted
because
depend sensitively
from the even-odd
shows that for most cases in rare-earth
on the (GM)
shift in
nuclei the pairing
reductions 6 % d/2, where d is the average
(1.7)
spacing of the Nilsson single-particle
300 kev for the rare-earth
nuclei.
However,
levels and d 2,
the simple BCS estimate
gives
6 Q, d/4 {blocking effect being neglected). The main reason is that the blocking AS pointed
effects
have not been considered
out by Rowe7 that while the blocking
is very difficult duce different
to treat them exactly
quasiparticle
to make comparison
between
(1.8)
in the BCS formalism
bases for different the various pairing
properly.
effects are straightfo~ard,
blocked
because
levels.
reductions
it
they intro-
So it is hard
due to different
blocking. bn the contrary,
the blocking
effects
in the particle-number-conserving
ment only the single-particle free parameters
the observed effect, surface,
for treating
the nuclear pairing
The details of this method can be found in ref8.
Correlation.
tional
are taken into account automatically
(PNC) method
even-odd
especially
A is assumed
The pairing
are involved. mass difference.
the blocking
is very important'.
meter, A, becomes
level (e.g. Nilsson)
configuration-dependent.
However,
G is determined
levels near the Fermi effect,
the gap para-
in the usual BCS treat~nt
and the quasiparticle
energy
is
as E=m
(1.9)
V
Another
by
show that the blocking
due to the blocking
to be configuration-independent
simply expressed
strength
Calculations
of the sinle-particle
Merely
In PNC treat-
scheme is needed and no addi-
interesting
problem in the forefront
of high-spin
State research
is
f 27c
J. Y. Zeng et al. / Pairit?g Reduction due to the Blocking Effect
the residual quasiparticle the residual
quasiparticle
interaction
the residual quasiparticle
interaction
(i)
The usually neglected H&
which do not appear
to calculate
In BCS formalism
in the BCS formalism.
may come from two sources 3 higher order terms
= Hi0 + Hi, + H;2,
in the PNC treatment,
even in the perturbational (ii)
It is also very difficult
interaction.
approach
(1.10)
It is very difficult
to treat them
in the BCS formalism.
The blocking effects.
This problem has been studied
in detail in another
paper
10
.
2. BRIEF REVIEW OF THE PNC METHOD As usual, the pairing
hamiltonian
of a deformed nucleus
is expressed
as
follows
H = l,ev(ata, + aia:) - Gla+aza-a
(2.1)
p" l-rlJ \f v'
where Ev stands for the single-particle of v and G the average
state and the pair-excitational wave functions
energy and < is the time-reversal
strength of the pairing
states of the even-even
take the following Azn(Oz =
1 o,*.*on
interaction.
state
For the ground
nucleus
(K"=O+) the PNC
form8
vr S+ Pl"'Pn pl"'on
(2.2)
IO>
s+ :
i=1,2,**.,n.
oi
Pl*{_Pn (v;l...on)2 =1. a=O(g.s.),1,2,*** The corresponding valence pairs.
eigenenergies The eigenequation
(pair-excitational
are denoted
states)
by Eon, where n is the number of
reads
on
+*
l
l +zv” x q!9--4
]=O (2.3)
where
5-q“‘0, It is interesting ing hamiltonian in a certain
to note
are either
= %,
t
8.0
f
(2.41
*
pn
that the off-diagonal -G or 0.
E
matrix elements
The diagonalization
of the pair-
of such a hamiltonian
truncated
configuration
space (E
of G can be determined
unambiguously
by the observed
The value
even-odd mass difference
128c
J, Y. Zeng et al. j P&ring Reduction
due to the Blocking
Effect
eq. (2.61).
(see It
is
ration
worthwhile
cation convention, the practical because
a cut-off of the COnfigu-
i.e. a cut-off of the single-particle
point of view we prefer the truncation
the problem concerned
adopt the truncation tant configurations tively
note that in the PNC treatment
to
energy is adopted, which is different from the usually adopted trun-
important
is essentially
of single-particle
energy levels.
of configuration
of a many-body
character.
energy, while many relatively
of rather high energy are involved,
configurations
From
energy If we unimpor-
a lot of other rela-
of lower energy are omitted.
A simp?e illus-
trative example will be given in the appendix. As we know, the realistic the average
value of the pairing strength,
spacing of the single-particle
levels, d3.
to study the ground state and the low-lying that the number of the important number of configurations
Renormalization
the average
mass difference.
configuration truncated
energy EC.
scheme
Of course,
configuration
pairing
However,
strength G is determined
energy EC is sufficiently
large (EC>>G),
spectra will be obtained
G is renormalized
is given in fig. 1.
An uniformly
appropriately. distributed
K=R vO
single-particle
the BCS formalism
mass difference,
the
An illustrative
single-particle
level
is expressed
as
is denoted by Eon(vO), where VC is the
by the odd nucleon.
single-particle
If the odd nucleon is ex-
level v, the wave function
which may be considered
state oG/O>in
it can be
provided
vO
eigenenergy
level occupied
cited to another
even-odd
,n=n
If the
in the calculation.
For an odd-A nucleus the ground state wave function
and the corresponding
by the observed
the value of G depends on the truncated
(average spacing d=l, see fig. 2) is adopted
A&,(u)/O>
the larger EC,
The larger EC, the smaller the G value.
pairing strength
example
solution.
So it is not diffi-
solution.
shown that nearly the same low-lying average
is not very large8; e.g. the
> 1%) is about 10.
accurate
the obtained
If we confine ourselves states, it can be shown
of G
In principle, even-odd
configurations
(probability
cult to obtain a sufficiently the more accurate
intrinsic
G, is smaller than
as the counterpart
is then denoted by
of the one-quasiparticle
if the blocking effect is neglected.
The
P, is then given by
P = @Eon
+ E
Gn+l)
- ‘On~‘O) 1
(2.6)
J. Y. Zeng et al./ Pairing Reduction
7.63 7.32 7.09
759 730 7 01
7.56 7.29 7.03
7.62 Z28 7.09
5.17 4.90
5.25 4.94
5.23 4.98
5.22 4.99
3.10
3. IO
3.10
3. IO
0
Ec=8
129~
due to the B~o~k~~g Effect
0
Ec=lO
G = 0.655
Ec=12
0
0
Ec=lS 6=0.5006
G=0.558
G= 0.600
Fig. 1 Renormalization of the pairing strength G. EC is the truncated configuFor a71 four cases G is chosen so that the first 07 levels are ration energy. located at the same position (EO+=3.10). 1 By means of the obtained intrinsic
properties
even-even
nucleus
wave functions
can be calculated*.
the occupation
by a pair of particles
probabilities
for each single-particle
level
(vO 12,u=l,Z,..-. Pl “‘Pn_lV
1
pl"'Pn_l
for an odd-A nucleus
1 o1 "'P"_l
A is defined A = G&Vv,
(2.7)
in the state A&.,(~O)~O>,
v~(~o) = The gap parameter
all the nuclear
for the ground state of
are given by
v;= Similarly,
and eigenenergies For example,
(vO(~O) f2, .“Pn_lV Pl
v=l,Z,***.
(2.8)
by (U; = 1-V;).
(2.9)
V
Thus, by means of the Vzls obtained the pairing
reduction
in the PNC treatment
due to blocking exactly 6(vo) = A - Aho),
where
we can calculated
(2.10)
13oc
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
Aho) = G~Uw,f@‘bD).
(2.11)
V
If v,, is close to the Fermi surface, Similarly,
for an even-even A~n_l(~OvO)D~>=ai K=Q
reduction
in the pair-broken
+Q
,II=Il
~(~oVo)
S+
ol"'"Qn_l
ol "'on_1
_ IO>, (2.121
II 'Jo vo'
v.
can be calculated
from A.
state
vO(~0~0)
a+ 1 "0 '0 ol***on_l
no the pairing
A(v0) may be quite different
nucleus
as follows
= A - A(~oVo),
(2.13)
where A(uovo)
= GIU (u v )V (P v ). vv 00 L, 00
A&i-l (II 9 v 0,f/D> maY be considered state a
a
IO>in
no vo
3. CALCULATED
the KS
as the counterpart
formalism
RESULTS AND
(2.14) of the two quasiparticle
if the blocking effects
are neglected.
DISCUSSIONS
We use a schematic (uniformly distributed)
single-particle
level scheme
(average spacing reduction.
d=l, see fig. 2) to illustrate the features of the pairing The chosen truncated configuration energy is EC=16. In this case,
8 pairs of particles average
and over 16 single-particle
pairing strength
mined by the observed
G=O.5.
even-odd
given in the following
levels are involved.
(For a realistic mass difference3.)
tables and depicted
The
nucleus G should be deterThe calculated
in figs. 3 and 4.
results are
It can be seen
that: (if blocked
The pairing reduction level v. relative
iifv,) depends
sensitively
on the location
to the Fermi surface and decreases
of the
rapidly with
- XI. For example, while for the nearest single-particle level "9", 1% 6(97 tt d/Z, for the distant single-particle level "16", 6(16) 2 0.03d. It can be seen from fig. 3 that the pairing a certain
region around the Fermi surface
for the rare-earth (ii)
nuclei.
With an increasing
face) the gap parameter particle exceeds
reduction
is significant
(>d/lO) only in
IE - hjg3d, which is about 1 VO
number of the blocked levels
A decreases dramatically.
Rev
(about the Fermi sur-
For the schematic
Single-
level scheme it can be seen that if the number of the blocked levels 3, the value of
A becomes rather small.
being taken into account, tional frequency seems possible
Thus, the blocking effects
the pairing collapse would occur at much lower rota-
than would be the case without
that several
the blocking effects.
(say, three or more) blocked
So it
levels above the Fermi
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
surface, which
is equivalent
lead to pairing
collapse.
to a large gap in the single-particle In this case the intrinsic
formed nucleus may change significantly, proach
the rigid-body
(iii)
properties
scheme, may of the de-
e.g. the moment of inertia may ap-
value.
It should be emphasized
Fermi surface,
131c
the situation
that if the blocked levels are below the
is quite different
(see fig. 4).
though the blocked levels may reduce the pairing
correlation
In this case, on the one-hand,
some active pairs are formed above the blocked levels on the other hand (because the number of particles relation may be enhanced particle
level scheme,
large the pairing Fermi surface
in various
reduction
For the schematic
may vanish, because
to a higher location
hence the pairing cor-
the only effect
and all the blocked
reduction
6 depend sensitively
6 tends to zero.
level distribution As an important
on the other hand.
out that the usually accepted A(odd-A
nucleus)
A and the pairing
near the Fermi surface special
case it should be
concept
< A(neighbouring
e-e nucleus)
may not be true for some special hole-excited
configuration
in certain nucleus.
In other words, the pairing reduction may be negative, 6~0. bandcrossing
frequency hwC(odd-A)
nucleus
level scheme and there exist several densely a hole excitation
active pair above the Fermi surface example
Some realistic
examples
For example,
is located in a gap in the Nilsson distributed
single-particle
levels
in the odd-A nucleus will create an
and leads to a larger A.
is given in table 5, in which a schematic
is adopted.
In this case, the
may be larger than hmC(e-e).
if the Fermi surface of an even-even
above the Fermi surface,
is to shift the
levels lie far below the
on the location of the blocked levels on the
hand, and on the single-particle (shell effect)
single-
levels becomes sufficiently
It also should be noted that the gap parameter
reduction
pointed
degrees.
if the number of the blocked
Fermi surface and so the pairing (iv)
should be conserved),
An illustrative
single-particle 10 .
can be found in ref.
level scheme
9
A 6
: 1
8 7 6 5 4
1.396
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
L
0.910 0.486
0.9618 0.9789 0.9877 0.9927 0.9957 0.9980 1.0000
0.0020 0.0043 0.0073 0.0123 0.0211 0.0382 0.0785 0.9x215
9
10
1.148 0.248
0.9347 0.9673 0.9817 0.9893 0.9938 0.9964 0.9972
0.8343 0.2:44
0.0022 0.0045 0.0080 0.0136
12
1.238 0.168
0.9241 0.9623 0.9790 0.9876 0.9924 0.9947 0.9973 1.286 0.110
0.9190 0.9602 0.9771 0.9865 0.9910 0.9919 0.9974
1.318 0.078
0.9158 0.9580 0.9761 0.9851 0.9911 0.9950 0.9975
0.0x240 0 0436 0'0883 0:7883 0'2224
1.339 0.057
0:9144 0.9568 0.9743 0.9852 0.9912 0.9950 0.9975
0 0:38 0'0241 0'0437 0'0894 0'7841 0'2235
O.OG23
0.0023 0.0045
1.356 0.040
0:9125 0.9543 0.9746 0.9854 0.9913 0.9951 0.9975
0.0182 0 0139 0'0243 0'0440 0'0893 0'7820 0'2255
15
1.367 0.029
0:9083 0.9548 0.9748 0.9855 0.9914 0.9951 0.9975
0.0x047 0 0083 0'0141 0'0245 0'0440 0'0892 0'7839 0'2239
16
blocking
14
due to different
0.0023 0.0045 0.0081
13
reduction
0.0x428 0.0x857 0 0873 0:8073 0 2181 0'2201 0:7956
0.0022 0.0044 0.0080 0.0135 0.0233
11
of the pairing
0.0021 0.0044 0.0079 0.0131 0.0227 0.0407
The PNC analysis
&Jo>
Table 1
J. Y. Zeng et al. /Pair&g
Table 2
The PNC analysis
A0+8/0’1 17 16 1:
0
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
Reductiotz
of the pairing
due to the Blocking
reduction
133c
Effect
due to different
blocking
2
3
4
5
6
7
8
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9857 0.9915 0.9952 0.9975 x
0.0025 0.0049 0.0086 0.0145 0.0252 0.0452 0.0917 0.2160 0.7761 0.9108 0.9560 0.9755 0.9859 0.9917 0.9953
0.0025 0.0049 0.0087 0.0146 0.0254 0.0457 0.0875 0.2180 0.7745 0.9107 0.9562 0.9757 0.9861 0.9918
0.0025 0.0050 0.0088 0.0148 0.0257 0.0432 0.0865 0.2159 0.7765 0.9106 0.9563 0.9759 0.9862
0.0025 0.0050 0.0089 0.0149 0.0239 0.0420 0.0842 0.2117 0.7776 0.9117 0.9564 0.9760
0.0026 0.0051 0.0090 0.0135 0.0229 0.0400 0.0810 0.2044 0.7799 0.8127 0.9572
0.0027 0.0053 0.0076 0.0124 0 0210 0.0377 0.0759 0.1927 0.7818 0.9143
0.0028 0.0036 0.0062 0.0107 0.0183 0.0327 0.0653 0.1657 0.7856
l.O"oO
0.9x977 1.000
1.000
0.9;67 0.9x864 0.9865 0.9xs19 0.9920 0.9920 0.9956 0.9955 0.9955 0.9978 0.9978 0.9977 1.000 1.000 1,000
0.9x593 0.9773 0.9869 0.9921 0.9956 0.9979 1.000
1.396 0
1.367 0.029
1.356 0.040
1.340 0.056
1.318 0.078
1.286 0.110
1.148 0.248
0.9955 0.9977
.A 6
1.396
16 I4 13 I2 II IO 9 8
-----------
7 6 5
4 "2I+ V
Fig. 2 A schematic single-particle level scheme.
x
1.238 0.158
134c
J, Y. Zeng et al. / Pairing Reduction due to the Blocking Effect Table 3
A&IO> 0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975 1.396
9
9,lO
9,10,11
9,10,11,12
9,10,11,12,13
0.0020 0.0043 0.0073 0.0123 0.0211 0.0382
0.0016 0.0036 0.0065 0.0108 0.0185 0.0334
0,0012 0.0029 0.0053 0.0091 x x x
0.0011 0.0027 0.0049 x x x x
0.0785
x
0.0014 0.0032 0.0058 0.0098 0.0168 x x
0.9x215 0.9618 0.9789 0.9877 (1.9927 0.9957 0.9980 1.0000
0.9;66 0.9815 0.9892 0.9935 0.9964 0.9984 1.0000 1.0000
0.9x823 0.9902 0.9942 0.9968 0.9986 1.0000 1.0000 1.0000
0.9x909 0.9947 0.9971 0.9988 1.0000
1.oooo
0.9x951 0.9973 0.9989 1.0000 1.0000 1.0000
1.0000 1.0000
1.0000
0.910
0,598
0.397
0.256
0.155
1 .oooo
Table 4
48Io> 8 ::
8 0
8,7 i
ii
0 0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
: 0028 0:0036 0.0062 0.0107 0.0183 0.0327 0.0653 0.1657 0.?856
1.396
ii 0028 0'0055 0:0080 0.0114 0.0189 0.0330 0.0676 0.1739 0.7877 0.9174 x
8,7,6
8,7,6,5
8,7,6,5,4
:
0
0.0026
0 0027
0.0027 0 0053
0.0090 0.0051
0'0096 0'0054 0:0143 0.0224 0.0360 0.0714 0.1846 8.7846 0.9156 0.9592 x x
0.0152 0.0264 0.0447 0.0843 0.2024 0.7767 0.9123 0.9577 0.9768 0.9868 x x x x
0.9x593 0.9773 0.9869
0.9;73 0.9926
0.9X96T
0'0093 0'0157 0:0252 0.0419 0.0786 0.1911 0.7796 0.9143 0.9586 0.9777 x x x l.o"OO
0.9921 0.9956 0.9979 1.000
0.9959 0.9980 1.000 1.000
0.9981 1.000 1.000 1.000
1.000 1.000 1.000
do0 1.000 1.000
1.148
1.129
1.157
1.203
1.275
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
A-A(%) t 0.5 -
d
0.4 0.3 0.2 -
Fig. 3
number of the blocked levels number of the blocked levels below the Fermi surface of above the Fermi surface of the even-even system the even-even system Fig. 4
13%
136c
Table 5
12
0
0
~.0~92
0.0955
71
0
0
0.0131
0.1737
70
0.0055
0,074o
0.0207
0.7403
9
0.0129
0.1128
0.0381
0.1914
8
0.026?
0.1594
0.0994
0.2992
7
0.0370
0.1888
0.2019
0.4014
6
0.0423
0.2013
0.2652
0.4414
5 4 3 2 1
137c
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
ACKNOWLEDGEMENT The authors
would like to express
Bohr, B. R. Mottelson
APPENDIX.
their sincere
and J. D. Garrett
ON THE TRUNCATION
levels is often
used.
Let us consider
A cut-off of single-particle
three pairs of particles
a simple illustrative
configurations
is E=18.
However,
the number of these omitted
configurations
amounts
configurations
convention.
is essentially
to be
to 33 (see tab. 6). (E&18) is much larger
of a many-body
are relatively
character,
more important
(e.g. (356), (456), etc.) involved
no such trouble occurs.
than
in the usual
Instead of a cut-off of single-
levels, 21~~ - Xlg6, we may choose the truncated
EC=lO, i.e. all the excited
configurations
In this case, five pairs of particles Fermi surface
rations
in such a truncation
convention.
In the PNC treatment
figurations
concerned
The lowest configu-
region, 2(~~ - A)<-6, are assumed
lower configurations
those higher configurations
particle
levels near the Fermi sur-
in tab. 6).
omitted
the problem concerned
many of these omitted
truncation
i.e.
below E=18, apart from these 20 configurations,
In fact, the total number of these omitted Because
(see fig.
by (123) and the highest one is (456), of which the configu-
Even when the pairs below the truncated
than 33.
example
Thus the number of configurations
there are a great many configurations
frozen,
energy
levels, 21~~ - X166, is assumed,
and six single-particle
is (6,)=20, (see the bracketed
ration energy
A.
and cotrnnents.
a cut-off of single-particle
energy
face, A, are taken into account.
ration is denoted
discussions
CONVENTION
In the usual shell model calculation,
5).
thanks to professors
for valuable
(i.e. ba12345678)
and 10 single-particle
are involved.
considered
in the cut-off of single-particle
these configurations tions involved
low-lying
are relatively
spectra,
levels near the
with the number of configulevels,
calculation
more important
in the usual truncation
energy
The total number of these con-
is 19 (see tab. 7), which is comparable
dealing with the nuclear
configuration
below E=lO are taken into account.
convention.
21Ev
-
X1<6.
In
shows that many of
than those higher configura-
138~
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
. . . 2’ 1' 0 9 8
7
--
6 5 4 -_-_--_____ 3 -----p2 I
l
-
x
Iw II c;< 6 c\r
--
F
Fig. 5
A schematic single-particle level scheme.
b
Table 6 E 18
configuration 122' 131' 230
140
159
168 167
16
121' 130
239
149
158
14
120
139
238
148
157
12
129
138
237
147 (156)
10
128
137 (236 )(146)
a
(E<18) 249
258
248
257
247 (256) (246)
267
348
357 (456)
347 (356) (346) (345)
(245)
127 (136)(235 ')(145)
6
(126)(135)(234
4
(125)(134)
2
(124)
0
(123)
1
Table 7 E
configuration
10
ba128
ba137
ba236
ba146
a
ba127
ba136
ba235
ba145
6
ba126
ba135
ba234
4
ba125
ba134
2
ba124
0
ba123
ba245
b1235 b1234
al234
J. Y. Zeng et al. / Pairing Reduction due to the Blocking Effect
139c
REFERENCES 1) B. R. Mottelson
and J. G. Valatin,
2) A. Bohr and B. R. Mottelson, 1975) ch.5.
Phys. Rev. Lett. 5 (1960) 511.
Nuclear Structure,
vol. 2 (Benjamin
New York,
3) S. G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. $?_ (1961) no. 16. 4) J. Y. Zeng, Physica
Energiae
Fortis et Physica
Nuclearis
3_ (1979) 102.
5) J. D. Garrett et al., Phys. Rev. Lett. 47 - (1981) 75. 6) B. Bengtsson
and S. Frauendorf,
7) D. J. Rowe, Nuclear collective pp. 194-195.
Nucl. Phys. A327 (1979) 139. motion
(Methuen,
London,
1970) ch. 11,
8) J. Y. Zeng and T. S. Cheng, Nucl. Phys. A405 (1983) 1. L. M. Yang & J. Y. Zeng, Acta Physics Six20 (1964) 846. 9) J. Y. Zeng, T. S. Cheng, L. Cheng and C. S. Wu, Nucl. Phys. A411 (1983) 49. 10) J. Y. Zeng, T. S. Cheng, L. Cheng and C. S. Wu, Nucl. Phys. A, in press.
PAIRING REDUCTION
125c
DUE TO THE BLOCKINGEFFECT
J. Y. Zeng The Niels Bohr Institute, DK-2100 Copenhagen 0, Denmark and Physics Department, Peking University, Beijing, China T. S. Cheng. L. Cheng and C. S. Wu Physics Department,
Peking University,
Beijing,
China
Pairing reduction in deformed nuclei due to the blocking effect is treated exactly with the particle-number-conserving method. Merely due to the blocking effect the gap parameter A becomes configuration-dependent. The pairing reduction 6(vD) depends sensitively on the location of the blocked level v relative to the Fermi surface, X, and decreases rapidly with The pairing reduction 6(vD) depends also sensitively on the lC"O - 21. In some special single-particle level distribution near the Fermi surface. cases pairing reduction may be negative. With an increasing number of the blocked levels (above the Fermi surface) the qap parameter A decreases If the blocked levels are below the Fermi surface the situdramatically. ation is suite different and as the number of the blocked levels becomes sufficiently large the pairing reduction may vanish.
1. I NTRODU~TION There are two kinds of anti-pairing (i) the
Coriolis anti-pairing
effects
in nuclei:
effect' and (ii) the blocking effect*.
The
fact that the blocking effect can lead to pairing reduction is well known3. Let A denotes the gap parameter neighbouring
for an even-even
nucleus and A(vo) that for the
odd-A nucleus, where vD is the single-particle
the odd nucleon.
level occupied
A(wD) < A. The pairing reduction due to blocking
(1.11 is defined as
&(uD) = A - A(v&
(1.2)
It is a special kind of even-odd difference. differences
are essentially
Example 1.
by
It is found that in general
The even-odd
In fact, the various even-odd
due to the blocking effects. shift in the moment of inertia$J4.
Experiments
show that in general the moment of inertia of odd-A nucleus is larger than that of the neighbouring
even-even
7 (odd-A nucleus) Example 2.
The even-odd
nucleus, >p(g.s.b.
of even-even
shift in bandcrossing
U3~5-9474/g4/$~3.~0 0 Elsevier Science Publishers B.V. ~No~h-H~~and Physics ~blishing Division)
nucleus).
frequency, hwC.
(1.3) Recently,
it
L Y. Zeng et al. j Pairing Reduction due to the Biocking Effect
126c
was established
that5
hwC(odd-A
nucleus)
< hoC(even-even
nucleus).
(1.4)
Let 6hwC = %wC(e-e)
- hwC(Odd-A).
il.51
It is found that in most cases for the rare-earth nuclei 6RwCjexp These two examples
are closely
the moment of intertia
hw
C’
the pairing
The analysis
connected
and the bandcrossing
value of the gap parameter calculations,
% 40 kev.
A.
By making
reduction
(1.6) with the pairing frequency
reduction,
use of the cranked shell model
can be extracted
because
depend sensitively
from the even-odd
shows that for most cases in rare-earth
on the (GM)
shift in
nuclei the pairing
reductions 6 % d/2, where d is the average
(1.7)
spacing of the Nilsson single-particle
300 kev for the rare-earth
nuclei.
However,
levels and d 2,
the simple BCS estimate
gives
6 Q, d/4 {blocking effect being neglected). The main reason is that the blocking AS pointed
effects
have not been considered
out by Rowe7 that while the blocking
is very difficult duce different
to treat them exactly
quasiparticle
to make comparison
between
(1.8)
in the BCS formalism
bases for different the various pairing
properly.
effects are straightfo~ard,
blocked
because
levels.
reductions
it
they intro-
So it is hard
due to different
blocking. bn the contrary,
the blocking
effects
in the particle-number-conserving
ment only the single-particle free parameters
the observed effect, surface,
for treating
the nuclear pairing
The details of this method can be found in ref8.
Correlation.
tional
are taken into account automatically
(PNC) method
even-odd
especially
A is assumed
The pairing
are involved. mass difference.
the blocking
is very important'.
meter, A, becomes
level (e.g. Nilsson)
configuration-dependent.
However,
G is determined
levels near the Fermi effect,
the gap para-
in the usual BCS treat~nt
and the quasiparticle
energy
is
as E=m
(1.9)
V
Another
by
show that the blocking
due to the blocking
to be configuration-independent
simply expressed
strength
Calculations
of the sinle-particle
Merely
In PNC treat-
scheme is needed and no addi-
interesting
problem in the forefront
of high-spin
State research
is
f 27c
J. Y. Zeng et al. / Pairit?g Reduction due to the Blocking Effect
the residual quasiparticle the residual
quasiparticle
interaction
the residual quasiparticle
interaction
(i)
The usually neglected H&
which do not appear
to calculate
In BCS formalism
in the BCS formalism.
may come from two sources 3 higher order terms
= Hi0 + Hi, + H;2,
in the PNC treatment,
even in the perturbational (ii)
It is also very difficult
interaction.
approach
(1.10)
It is very difficult
to treat them
in the BCS formalism.
The blocking effects.
This problem has been studied
in detail in another
paper
10
.
2. BRIEF REVIEW OF THE PNC METHOD As usual, the pairing
hamiltonian
of a deformed nucleus
is expressed
as
follows
H = l,ev(ata, + aia:) - Gla+aza-a
(2.1)
p" l-rlJ \f v'
where Ev stands for the single-particle of v and G the average
state and the pair-excitational wave functions
energy and < is the time-reversal
strength of the pairing
states of the even-even
take the following Azn(Oz =
1 o,*.*on
interaction.
state
For the ground
nucleus
(K"=O+) the PNC
form8
vr S+ Pl"'Pn pl"'on
(2.2)
IO>
s+ :
i=1,2,**.,n.
oi
Pl*{_Pn (v;l...on)2 =1. a=O(g.s.),1,2,*** The corresponding valence pairs.
eigenenergies The eigenequation
(pair-excitational
are denoted
states)
by Eon, where n is the number of
reads
on
+*
l
l +zv” x q!9--4
]=O (2.3)
where
5-q“‘0, It is interesting ing hamiltonian in a certain
to note
are either
= %,
t
8.0
f
(2.41
*
pn
that the off-diagonal -G or 0.
E
matrix elements
The diagonalization
of the pair-
of such a hamiltonian
truncated
configuration
space (E
of G can be determined
unambiguously
by the observed
The value
even-odd mass difference
128c
J, Y. Zeng et al. j P&ring Reduction
due to the Blocking
Effect
eq. (2.61).
(see It
is
ration
worthwhile
cation convention, the practical because
a cut-off of the COnfigu-
i.e. a cut-off of the single-particle
point of view we prefer the truncation
the problem concerned
adopt the truncation tant configurations tively
note that in the PNC treatment
to
energy is adopted, which is different from the usually adopted trun-
important
is essentially
of single-particle
energy levels.
of configuration
of a many-body
character.
energy, while many relatively
of rather high energy are involved,
configurations
From
energy If we unimpor-
a lot of other rela-
of lower energy are omitted.
A simp?e illus-
trative example will be given in the appendix. As we know, the realistic the average
value of the pairing strength,
spacing of the single-particle
levels, d3.
to study the ground state and the low-lying that the number of the important number of configurations
Renormalization
the average
mass difference.
configuration truncated
energy EC.
scheme
Of course,
configuration
pairing
However,
strength G is determined
energy EC is sufficiently
large (EC>>G),
spectra will be obtained
G is renormalized
is given in fig. 1.
An uniformly
appropriately. distributed
K=R vO
single-particle
the BCS formalism
mass difference,
the
An illustrative
single-particle
level
is expressed
as
is denoted by Eon(vO), where VC is the
by the odd nucleon.
single-particle
If the odd nucleon is ex-
level v, the wave function
which may be considered
state oG/O>in
it can be
provided
vO
eigenenergy
level occupied
cited to another
even-odd
,n=n
If the
in the calculation.
For an odd-A nucleus the ground state wave function
and the corresponding
by the observed
the value of G depends on the truncated
(average spacing d=l, see fig. 2) is adopted
A&,(u)/O>
the larger EC,
The larger EC, the smaller the G value.
pairing strength
example
solution.
So it is not diffi-
solution.
shown that nearly the same low-lying average
is not very large8; e.g. the
> 1%) is about 10.
accurate
the obtained
If we confine ourselves states, it can be shown
of G
In principle, even-odd
configurations
(probability
cult to obtain a sufficiently the more accurate
intrinsic
G, is smaller than
as the counterpart
is then denoted by
of the one-quasiparticle
if the blocking effect is neglected.
The
P, is then given by
P = @Eon
+ E
Gn+l)
- ‘On~‘O) 1
(2.6)
J. Y. Zeng et al./ Pairing Reduction
7.63 7.32 7.09
759 730 7 01
7.56 7.29 7.03
7.62 Z28 7.09
5.17 4.90
5.25 4.94
5.23 4.98
5.22 4.99
3.10
3. IO
3.10
3. IO
0
Ec=8
129~
due to the B~o~k~~g Effect
0
Ec=lO
G = 0.655
Ec=12
0
0
Ec=lS 6=0.5006
G=0.558
G= 0.600
Fig. 1 Renormalization of the pairing strength G. EC is the truncated configuFor a71 four cases G is chosen so that the first 07 levels are ration energy. located at the same position (EO+=3.10). 1 By means of the obtained intrinsic
properties
even-even
nucleus
wave functions
can be calculated*.
the occupation
by a pair of particles
probabilities
for each single-particle
level
(vO 12,u=l,Z,..-. Pl “‘Pn_lV
1
pl"'Pn_l
for an odd-A nucleus
1 o1 "'P"_l
A is defined A = G&Vv,
(2.7)
in the state A&.,(~O)~O>,
v~(~o) = The gap parameter
all the nuclear
for the ground state of
are given by
v;= Similarly,
and eigenenergies For example,
(vO(~O) f2, .“Pn_lV Pl
v=l,Z,***.
(2.8)
by (U; = 1-V;).
(2.9)
V
Thus, by means of the Vzls obtained the pairing
reduction
in the PNC treatment
due to blocking exactly 6(vo) = A - Aho),
where
we can calculated
(2.10)
13oc
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
Aho) = G~Uw,f@‘bD).
(2.11)
V
If v,, is close to the Fermi surface, Similarly,
for an even-even A~n_l(~OvO)D~>=ai K=Q
reduction
in the pair-broken
+Q
,II=Il
~(~oVo)
S+
ol"'"Qn_l
ol "'on_1
_ IO>, (2.121
II 'Jo vo'
v.
can be calculated
from A.
state
vO(~0~0)
a+ 1 "0 '0 ol***on_l
no the pairing
A(v0) may be quite different
nucleus
as follows
= A - A(~oVo),
(2.13)
where A(uovo)
= GIU (u v )V (P v ). vv 00 L, 00
A&i-l (II 9 v 0,f/D> maY be considered state a
a
IO>in
no vo
3. CALCULATED
the KS
as the counterpart
formalism
RESULTS AND
(2.14) of the two quasiparticle
if the blocking effects
are neglected.
DISCUSSIONS
We use a schematic (uniformly distributed)
single-particle
level scheme
(average spacing reduction.
d=l, see fig. 2) to illustrate the features of the pairing The chosen truncated configuration energy is EC=16. In this case,
8 pairs of particles average
and over 16 single-particle
pairing strength
mined by the observed
G=O.5.
even-odd
given in the following
levels are involved.
(For a realistic mass difference3.)
tables and depicted
The
nucleus G should be deterThe calculated
in figs. 3 and 4.
results are
It can be seen
that: (if blocked
The pairing reduction level v. relative
iifv,) depends
sensitively
on the location
to the Fermi surface and decreases
of the
rapidly with
- XI. For example, while for the nearest single-particle level "9", 1% 6(97 tt d/Z, for the distant single-particle level "16", 6(16) 2 0.03d. It can be seen from fig. 3 that the pairing a certain
region around the Fermi surface
for the rare-earth (ii)
nuclei.
With an increasing
face) the gap parameter particle exceeds
reduction
is significant
(>d/lO) only in
IE - hjg3d, which is about 1 VO
number of the blocked levels
A decreases dramatically.
Rev
(about the Fermi sur-
For the schematic
Single-
level scheme it can be seen that if the number of the blocked levels 3, the value of
A becomes rather small.
being taken into account, tional frequency seems possible
Thus, the blocking effects
the pairing collapse would occur at much lower rota-
than would be the case without
that several
the blocking effects.
(say, three or more) blocked
So it
levels above the Fermi
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
surface, which
is equivalent
lead to pairing
collapse.
to a large gap in the single-particle In this case the intrinsic
formed nucleus may change significantly, proach
the rigid-body
(iii)
properties
scheme, may of the de-
e.g. the moment of inertia may ap-
value.
It should be emphasized
Fermi surface,
131c
the situation
that if the blocked levels are below the
is quite different
(see fig. 4).
though the blocked levels may reduce the pairing
correlation
In this case, on the one-hand,
some active pairs are formed above the blocked levels on the other hand (because the number of particles relation may be enhanced particle
level scheme,
large the pairing Fermi surface
in various
reduction
For the schematic
may vanish, because
to a higher location
hence the pairing cor-
the only effect
and all the blocked
reduction
6 depend sensitively
6 tends to zero.
level distribution As an important
on the other hand.
out that the usually accepted A(odd-A
nucleus)
A and the pairing
near the Fermi surface special
case it should be
concept
< A(neighbouring
e-e nucleus)
may not be true for some special hole-excited
configuration
in certain nucleus.
In other words, the pairing reduction may be negative, 6~0. bandcrossing
frequency hwC(odd-A)
nucleus
level scheme and there exist several densely a hole excitation
active pair above the Fermi surface example
Some realistic
examples
For example,
is located in a gap in the Nilsson distributed
single-particle
levels
in the odd-A nucleus will create an
and leads to a larger A.
is given in table 5, in which a schematic
is adopted.
In this case, the
may be larger than hmC(e-e).
if the Fermi surface of an even-even
above the Fermi surface,
is to shift the
levels lie far below the
on the location of the blocked levels on the
hand, and on the single-particle (shell effect)
single-
levels becomes sufficiently
It also should be noted that the gap parameter
reduction
pointed
degrees.
if the number of the blocked
Fermi surface and so the pairing (iv)
should be conserved),
An illustrative
single-particle 10 .
can be found in ref.
level scheme
9
A 6
: 1
8 7 6 5 4
1.396
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
L
0.910 0.486
0.9618 0.9789 0.9877 0.9927 0.9957 0.9980 1.0000
0.0020 0.0043 0.0073 0.0123 0.0211 0.0382 0.0785 0.9x215
9
10
1.148 0.248
0.9347 0.9673 0.9817 0.9893 0.9938 0.9964 0.9972
0.8343 0.2:44
0.0022 0.0045 0.0080 0.0136
12
1.238 0.168
0.9241 0.9623 0.9790 0.9876 0.9924 0.9947 0.9973 1.286 0.110
0.9190 0.9602 0.9771 0.9865 0.9910 0.9919 0.9974
1.318 0.078
0.9158 0.9580 0.9761 0.9851 0.9911 0.9950 0.9975
0.0x240 0 0436 0'0883 0:7883 0'2224
1.339 0.057
0:9144 0.9568 0.9743 0.9852 0.9912 0.9950 0.9975
0 0:38 0'0241 0'0437 0'0894 0'7841 0'2235
O.OG23
0.0023 0.0045
1.356 0.040
0:9125 0.9543 0.9746 0.9854 0.9913 0.9951 0.9975
0.0182 0 0139 0'0243 0'0440 0'0893 0'7820 0'2255
15
1.367 0.029
0:9083 0.9548 0.9748 0.9855 0.9914 0.9951 0.9975
0.0x047 0 0083 0'0141 0'0245 0'0440 0'0892 0'7839 0'2239
16
blocking
14
due to different
0.0023 0.0045 0.0081
13
reduction
0.0x428 0.0x857 0 0873 0:8073 0 2181 0'2201 0:7956
0.0022 0.0044 0.0080 0.0135 0.0233
11
of the pairing
0.0021 0.0044 0.0079 0.0131 0.0227 0.0407
The PNC analysis
&Jo>
Table 1
J. Y. Zeng et al. /Pair&g
Table 2
The PNC analysis
A0+8/0’1 17 16 1:
0
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
Reductiotz
of the pairing
due to the Blocking
reduction
133c
Effect
due to different
blocking
2
3
4
5
6
7
8
0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9857 0.9915 0.9952 0.9975 x
0.0025 0.0049 0.0086 0.0145 0.0252 0.0452 0.0917 0.2160 0.7761 0.9108 0.9560 0.9755 0.9859 0.9917 0.9953
0.0025 0.0049 0.0087 0.0146 0.0254 0.0457 0.0875 0.2180 0.7745 0.9107 0.9562 0.9757 0.9861 0.9918
0.0025 0.0050 0.0088 0.0148 0.0257 0.0432 0.0865 0.2159 0.7765 0.9106 0.9563 0.9759 0.9862
0.0025 0.0050 0.0089 0.0149 0.0239 0.0420 0.0842 0.2117 0.7776 0.9117 0.9564 0.9760
0.0026 0.0051 0.0090 0.0135 0.0229 0.0400 0.0810 0.2044 0.7799 0.8127 0.9572
0.0027 0.0053 0.0076 0.0124 0 0210 0.0377 0.0759 0.1927 0.7818 0.9143
0.0028 0.0036 0.0062 0.0107 0.0183 0.0327 0.0653 0.1657 0.7856
l.O"oO
0.9x977 1.000
1.000
0.9;67 0.9x864 0.9865 0.9xs19 0.9920 0.9920 0.9956 0.9955 0.9955 0.9978 0.9978 0.9977 1.000 1.000 1,000
0.9x593 0.9773 0.9869 0.9921 0.9956 0.9979 1.000
1.396 0
1.367 0.029
1.356 0.040
1.340 0.056
1.318 0.078
1.286 0.110
1.148 0.248
0.9955 0.9977
.A 6
1.396
16 I4 13 I2 II IO 9 8
-----------
7 6 5
4 "2I+ V
Fig. 2 A schematic single-particle level scheme.
x
1.238 0.158
134c
J, Y. Zeng et al. / Pairing Reduction due to the Blocking Effect Table 3
A&IO> 0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975 1.396
9
9,lO
9,10,11
9,10,11,12
9,10,11,12,13
0.0020 0.0043 0.0073 0.0123 0.0211 0.0382
0.0016 0.0036 0.0065 0.0108 0.0185 0.0334
0,0012 0.0029 0.0053 0.0091 x x x
0.0011 0.0027 0.0049 x x x x
0.0785
x
0.0014 0.0032 0.0058 0.0098 0.0168 x x
0.9x215 0.9618 0.9789 0.9877 (1.9927 0.9957 0.9980 1.0000
0.9;66 0.9815 0.9892 0.9935 0.9964 0.9984 1.0000 1.0000
0.9x823 0.9902 0.9942 0.9968 0.9986 1.0000 1.0000 1.0000
0.9x909 0.9947 0.9971 0.9988 1.0000
1.oooo
0.9x951 0.9973 0.9989 1.0000 1.0000 1.0000
1.0000 1.0000
1.0000
0.910
0,598
0.397
0.256
0.155
1 .oooo
Table 4
48Io> 8 ::
8 0
8,7 i
ii
0 0.0025 0.0048 0.0085 0.0144 0.0249 0.0446 0.0901 0.2246 0.7754 0.9099 0.9554 0.9751 0.9856 0.9915 0.9952 0.9975
: 0028 0:0036 0.0062 0.0107 0.0183 0.0327 0.0653 0.1657 0.?856
1.396
ii 0028 0'0055 0:0080 0.0114 0.0189 0.0330 0.0676 0.1739 0.7877 0.9174 x
8,7,6
8,7,6,5
8,7,6,5,4
:
0
0.0026
0 0027
0.0027 0 0053
0.0090 0.0051
0'0096 0'0054 0:0143 0.0224 0.0360 0.0714 0.1846 8.7846 0.9156 0.9592 x x
0.0152 0.0264 0.0447 0.0843 0.2024 0.7767 0.9123 0.9577 0.9768 0.9868 x x x x
0.9x593 0.9773 0.9869
0.9;73 0.9926
0.9X96T
0'0093 0'0157 0:0252 0.0419 0.0786 0.1911 0.7796 0.9143 0.9586 0.9777 x x x l.o"OO
0.9921 0.9956 0.9979 1.000
0.9959 0.9980 1.000 1.000
0.9981 1.000 1.000 1.000
1.000 1.000 1.000
do0 1.000 1.000
1.148
1.129
1.157
1.203
1.275
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
A-A(%) t 0.5 -
d
0.4 0.3 0.2 -
Fig. 3
number of the blocked levels number of the blocked levels below the Fermi surface of above the Fermi surface of the even-even system the even-even system Fig. 4
13%
136c
Table 5
12
0
0
~.0~92
0.0955
71
0
0
0.0131
0.1737
70
0.0055
0,074o
0.0207
0.7403
9
0.0129
0.1128
0.0381
0.1914
8
0.026?
0.1594
0.0994
0.2992
7
0.0370
0.1888
0.2019
0.4014
6
0.0423
0.2013
0.2652
0.4414
5 4 3 2 1
137c
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
ACKNOWLEDGEMENT The authors
would like to express
Bohr, B. R. Mottelson
APPENDIX.
their sincere
and J. D. Garrett
ON THE TRUNCATION
levels is often
used.
Let us consider
A cut-off of single-particle
three pairs of particles
a simple illustrative
configurations
is E=18.
However,
the number of these omitted
configurations
amounts
configurations
convention.
is essentially
to be
to 33 (see tab. 6). (E&18) is much larger
of a many-body
are relatively
character,
more important
(e.g. (356), (456), etc.) involved
no such trouble occurs.
than
in the usual
Instead of a cut-off of single-
levels, 21~~ - Xlg6, we may choose the truncated
EC=lO, i.e. all the excited
configurations
In this case, five pairs of particles Fermi surface
rations
in such a truncation
convention.
In the PNC treatment
figurations
concerned
The lowest configu-
region, 2(~~ - A)<-6, are assumed
lower configurations
those higher configurations
particle
levels near the Fermi sur-
in tab. 6).
omitted
the problem concerned
many of these omitted
truncation
i.e.
below E=18, apart from these 20 configurations,
In fact, the total number of these omitted Because
(see fig.
by (123) and the highest one is (456), of which the configu-
Even when the pairs below the truncated
than 33.
example
Thus the number of configurations
there are a great many configurations
frozen,
energy
levels, 21~~ - X166, is assumed,
and six single-particle
is (6,)=20, (see the bracketed
ration energy
A.
and cotrnnents.
a cut-off of single-particle
energy
face, A, are taken into account.
ration is denoted
discussions
CONVENTION
In the usual shell model calculation,
5).
thanks to professors
for valuable
(i.e. ba12345678)
and 10 single-particle
are involved.
considered
in the cut-off of single-particle
these configurations tions involved
low-lying
are relatively
spectra,
levels near the
with the number of configulevels,
calculation
more important
in the usual truncation
energy
The total number of these con-
is 19 (see tab. 7), which is comparable
dealing with the nuclear
configuration
below E=lO are taken into account.
convention.
21Ev
-
X1<6.
In
shows that many of
than those higher configura-
138~
J. Y. Zeng et al. /Pairing Reduction due to the Blocking Effect
. . . 2’ 1' 0 9 8
7
--
6 5 4 -_-_--_____ 3 -----p2 I
l
-
x
Iw II c;< 6 c\r
--
F
Fig. 5
A schematic single-particle level scheme.
b
Table 6 E 18
configuration 122' 131' 230
140
159
168 167
16
121' 130
239
149
158
14
120
139
238
148
157
12
129
138
237
147 (156)
10
128
137 (236 )(146)
a
(E<18) 249
258
248
257
247 (256) (246)
267
348
357 (456)
347 (356) (346) (345)
(245)
127 (136)(235 ')(145)
6
(126)(135)(234
4
(125)(134)
2
(124)
0
(123)
1
Table 7 E
configuration
10
ba128
ba137
ba236
ba146
a
ba127
ba136
ba235
ba145
6
ba126
ba135
ba234
4
ba125
ba134
2
ba124
0
ba123
ba245
b1235 b1234
al234
J. Y. Zeng et al. / Pairing Reduction due to the Blocking Effect
139c
REFERENCES 1) B. R. Mottelson
and J. G. Valatin,
2) A. Bohr and B. R. Mottelson, 1975) ch.5.
Phys. Rev. Lett. 5 (1960) 511.
Nuclear Structure,
vol. 2 (Benjamin
New York,
3) S. G. Nilsson and 0. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. $?_ (1961) no. 16. 4) J. Y. Zeng, Physica
Energiae
Fortis et Physica
Nuclearis
3_ (1979) 102.
5) J. D. Garrett et al., Phys. Rev. Lett. 47 - (1981) 75. 6) B. Bengtsson
and S. Frauendorf,
7) D. J. Rowe, Nuclear collective pp. 194-195.
Nucl. Phys. A327 (1979) 139. motion
(Methuen,
London,
1970) ch. 11,
8) J. Y. Zeng and T. S. Cheng, Nucl. Phys. A405 (1983) 1. L. M. Yang & J. Y. Zeng, Acta Physics Six20 (1964) 846. 9) J. Y. Zeng, T. S. Cheng, L. Cheng and C. S. Wu, Nucl. Phys. A411 (1983) 49. 10) J. Y. Zeng, T. S. Cheng, L. Cheng and C. S. Wu, Nucl. Phys. A, in press.