107
Progress of Theoretical Physics, Vol. 14 No. 2, August 1955
Separation Energies and Nuclear Structures in Light Nuclei
Takao TATI
Institute for Theoretical Physics, Kanazawa Uniyersity, Kanazawa
(Received May 23, 1955)
The separation energies of light nuclei lO;SA;Sso are considered in relation to the nuclear struc·
ture. We can grasp the physical meaning of the complicated variation of the separation energies of
nucleons from nucleus to nucleus phenomenologically, by comparing them with the contributions to
the separation energies, from each term of the semi-empirical mass formula. The systematic deviation
of the separation energies from the semi-empirical mass formula disappears by the corrected mass formula by the uniform model which assumes the two-body interactions between symmetric pairs in
supermultiplet structure. This fact strongly supports that the symmetry effect of the nuclear binding
energy originates from the above mentioned two-body interactions. The deviations of the separation
energies from the corrected mass formula of the uniform model show the evidences to support the
independent particle model. In this mass region, the nucleus has, besides the dominant supermultiplet
structure, the j-j coupling shell structure. The comparison between the mass formula and the above
mentioned potential energy of two-body interactions which give the symmetry effects quantitatively, and
the requirement that the separation energy is nearly equal to the depth of the energy level of the last
nucleon measured from the top of the potential in the independent particle model, lead us to the conclusion that there are other kinds of nuclear potential energies which are not sensitive to the symmetry effect and to the " expansibility " of the nucleus, and bear about a half of the total potential
energy of the nucleus. And then, the depth of the average potential for the last nucleon is estimated
to be about 50 Mev and is a linearly decreasing function of X (X=Z-N for protons, X=N-Z
for neutrons) •
§ I. Introduction and summary
The separation energy of a proton or a neutron &om a nucleus is the energy required
to remove the most loosely bound proton or neutron adiabatically. The corresponding quantity in the atom is the ionization energy of an outermost electron. Ionization energies give
important knowledge about the interaction between the electron and the residual atom and
are intimately related to the atomic shell structure. The effects of the nuclear shell structure on the separation energies of nuclei have been studied by many authors, and the separation energies show the discontinuities at each of the shell edges/) although the magnitudes of the discontinuities have not been explained in relation to the nuclear structures.
The fact that both the electrons in atoms and the nucleons in nuclei are fermions gives a
similar character to the ionization and separation energies, but on the other hand, they have
quite different characters in many respects. In the atom there is only one kind of fermions,
the electrons, moving in the strong electric field of the nucleus, whereas in the nucleus
there are two kinds of fermions, the protons and the neutrons, interacting strongly with
each other. The force acting on an electron in the atom is, in fairly good approximation,
108
T. Tati
replaced by the average static force, and when an electron is removed from an atom, the
residual atom changes its structure only slightly.
As for the nuclei, the validity of the
approximation of the single particle model is not clear experimentally as well as theoretically
and when a nucleon is removed &om a nucleus, the residual nucleus changes its structure
appreciably (Section 4) . Therefore, the ionization energies vary monotonically with respect
to the numbers of electrons of the atoms except at the shell edges, whereas the separation
energies vary in a complicated way with the numbers of protons and neutrons. To understand these variations, we need a theory which gives the total binding energies of nuclei as
a function of the numbers of protons and neutrons, because we must take into account the
change of the nuclear structure when a nucleon is removed. Of course, at present there
is no theory except the semi-empirical ones.
The variation of separation energies is large in the region where the mass number is
For the region of mass
small, and becomes smaller as the mass number becomes large.
number 16$A$50, the analysis of this variation is tried at first, since the variation is
large and regular in the region. The results are summarized as follows.
1. We can grasp the physical meaning of the complicated variation of the separation
energies of nucleons from nucleus to nucleus phenomenologically, by comparing them with
the contributions to the separation energies, from each term of the semi-empirical mass
formula (with the coefficient of Fermi2>) (Section 2).
2. The systematic deviations of the separation energies from the values calculated &om
the semi-empirical mass formula explained excellently well if we use the corrected mass
formula by the uniform model3l which assumes the two-body interactions between symmetric
pairs in supermultiplet structure. This fact strongly supports the fact that the symmetry
effect of the nuclear binding energy originates from the above mentioned two-body interactions (Section 3).
3. The deviations of the separation energies &om the corrected mass formula of the
uniform model show the evidences to support the independent particle model (I. P.M.).
In this mass region, the nucleus has, besides the dominant supermultiplet structure, the j-j
coupling shell structure4> (Section 3).
4. The comparison between the mass formula and the potential energy of two-body
interactions which gives the symmetry effects quantitatively, and the requirement that theseparation energy is nearly equal to the depth of the energy level of the last nucleon measured
from the top of the potential in the independent particle model, suggest us that there are
other kinds of nuclear potential energies which are not sensitive to the symmetry effect
and to the " expansibility " of the nucleus which is defined by ( 28), and bear about a
half of the total potential energy of the nucleus. And then, the depth of the average
potential for the last nucleon is estimated to be about 50 Mev and is a linearly decreasing
function of X (X is defined in Section 3) (Section 4).
§ 2. Separation energies and semi-empirical mass formula
In Figs. 1-4, the separation energies of a proton and a neutron, SP and S,. &om the
Separation Energies and Nuclear Structures in Light Nuclei
109
MeV
14
12
10
8
6
4
z=5
t=
7
~
t=l3
;.·
2
0
40742
10
44
46
48
50
-2
a
11
f
-4
-6
Fig. 1. Plot of odd proton separation energies from nuclei (A, Z). Points of given Z are connected.
0 : Experimental value. X : Calculated value from the mass formula (1). Included for comparison
are curves of contributions of five terms of mass formula (1), v: volume, f: surface, s: symmetry,
c: coulomb and 6.
MeV
18
16
z=6
t=S
z=IO
z=l2
t= 14
14
12
10
8
6
4
2
0
-2
-4
-6
42
44
46
48
50
11
~~~~~,~=,=-----------------{
--'-
__,
Fig. 2. Plot of even proton separation energies from nuclei (A, Z) . Points of given Z are connected.
0: Experimental value.
X: Calculated value from the mass formula (1). Included for comparison
are curves of contributions of five terms of mass formula (1), v: volume, f: surface, s: symmetry,
c: coulomb and 6.
T. Tati
110
M.V
14
12
10
8
r· !) ; / /
:
I
I
f
2
I
/
6
4
N=.g)N=1l)N:l3)N=t5)N=)l7N~l9
1
I
I
I
•
,
/
'
I
I
I
I
,/
I
I
/
C
•
1
/
t
0 10
,/
f
/
/
I
/
I
:
/
/
//
.I
'
l
,.
I
./
•
N-=. 23 ;;N=25
,./
1'
,/
l
/ / . '/
/
/
N- 27
N= 21
,'/
,"
~,
/
- ..(.
·' .-..f/
,I
/
/
i"
_
,_
_,_
A
-2
-4
-6
Fig. 3. Plot of odd neutron separation energies from nuclei (A, N). Points of given N are connected.
0: Experimental value. X : Calculated value from the mass formula (1). Included for comparison
are curves of contributions of five terms of mass formula (1), v: volume, I: surface, s: symmetry,
c : coulomb and IJ.
MeV
20
N:10
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
Fig. 4. Plot of even neutron separation energies from nuclei (A, N). Points of given N are connected.
0: Experimental value.
X: Calculated value from the mass formula (1).
Included for comparison
are curves of contributions of 6.ve terms of mass formula (1), v : volume, I: surface, s: symmetry,
c: coulomb and IJ.
Separation Energies and Nuclear Structures in Light Nuclei
111
nuclei (A, Z) and (A, N) are analysed in terms of the contribution of each term of the
semi-empirical mass formula with the coefficients of Fermi, 2>
B(A, Z) =14.03A-13.03A 213 -77.27(A/2-Z) 2/ A
(1)
where a(A, Z) =0 for odd A and a(A, Z) = ±33.52A- 314 for odd Z/ even Z and even
A. The reference to the adopted experimental values are summarized in Appendix.
§ 3. Interpretation by the uniform model
We have seen from Figs. 1-4, in the preceding section that the magnitudes of the
separation energies of S,(Z, A) and S,.(N, A) depend mainly on the proton excess
(2Z-A) and the neutron excess (2N-A), apart from the charge effects and even-oddness
of Z and N. Hence it seems convenient to denote S,(Z, A) and S,.(N, A) as S,(XP, Z)
and S.. (X,., N) where X, and X,. are X,=2Z-A, X,.=2N-A. Further, we shall use
the word " excess X." X is defined as
X=X,=2Z-A
for protons,
X=X,.=2N-A
for neutrons,
(2)
where Z, N and A are the proton and neutron numbers and the mass number of the nucleus
from which the nucleon is removed. Then the separation energy S is specified by the
nucleon kind p or n which we denote with <, and the nucleon number Z or N which
we denote with L, and the excess X. Thus we can write the separation energy as S~ (X,
L). One can see from Figs. 1-4.
1. S~ (X, L)'s with given ' and X and given even-oddness of L do not depend
strongly on L.
2. S-c(X, L)'s with given X and given L are nearly equal if the coulomb energy
contributions are subtracted from both of them.
3. S-c(X, L)'s with given ' and given X depend strongly on even-oddness of L
4. S-c (X, L)'s with given ' and given L strongly depend on X.
In Fig 5. the deviations of the separation energies S~ (X, L) with the given X and odd
L from the calculated values with the mass formula (1) are plotted against L The deviations of two-nucleon separation energies from the nuclei of even (L+ 1)
S-c(X+1, L+1)+S"(X, L)
are also plotted at the same abscissa, to make easy the comparison of one and two-nucleon
separation energies which have the same residual nucleus.
The following features a and b in Fig. 5 show the systematic deviations of S, and Sn
from the mass formula (1), and suggest us to analyse the data by Wigner's uniform
model.*
*) T. L. Collins, A. 0. Nier and W. H. Johnson17) and R. E. Halsted!S) analysed the nuclear binding
energies by the Wigner's mass formula. They compared with the Wigner's formula the potential energy,
calculated from the nuclear mass and the kinetic energy given by the Fermi gas model, and obtained the
values fo two parameters. But our method of analysis is different from theirs and uses the formula (11).
T. Tati
112
MeV 2
~f~·------------------~~~~~~~
X=5
21
21
25
Full line : for one nucleon,
27 L
r
-~f~------=_;,;rl,==c:...Jr.·
__-__
Broken line : for two nucleons.
Mev 21
X=4
·.:a_
1~
- 2
21
23
25
27 L
~~
MeV
X=3 _
~1----===::s-__,
.,_. ,_,0::-=
•• ___""-c::
... y=/..-=~=----"-
-z
9
II
13
15
17
19
21
23
25
MeV
X=4
It
---S2:r-____,.6..,:s-
-~f--2
19
21
23
25
£
27
If
X=3:!1----:
--:-=~=l·~"'"-~--._-_
__./<=-----•-
MeV
9
II
13
15
17
19
21
23
25
27
£
27 L
~f?~
7
MeV
X=l
MeV 2
/
-4
-5
,.---. ---... -- .... --
.. ",
II
13
15
17
19
21 L
~
:}~1[~/
~-:=~~-~-~
- 4
7
9
II
13
15
17
19
21 L
"'\ ,/
7v-..LII-...113,.--='15:--:17!::--:1~9~21 (.
51
MeV 4
.. D""' ......Q,.....
~~
°
X=..-1
2!
1
9
7
9
II
13
15
17
19
L
..
0
MeV
31
-"
~~~"',/
---..____,...-
8
X=-2 0~-~---------~C----------~--------­
r:r'
23 L
-1
-2
7
MeV 3
X=-3
2~
1
o._...
_
9
II
13
15
17
19
'
21
'
23
L
~"--~-------~--;:.~~.......
.
13
,~
~
15
17
~.
_._~---:.--:~-;':;:--7::--~:-~
19
21
23 L
MeV 2r
MeV~f
~
X=-4 0~---------------,&~-------Fig. 5. Plot of d,t<l) (X, L) and d,t<2> (X, L),
deviations of S, (X, L) and S, (X+ 1, L+ 1) + S, (X,
L) from the values calculated with the mass formula
(1) against odd L. 0: for one proton, e: for
0 : for two protons, • : for two
one neutron,
neutrons.
~t-------""la::~:·:·;~·=·=-""~;=-=~r
X=- -4
- 1
~7
19
2~
23 1,
Fig. 6. Plot of J, <1>(X, L) and d;: <~> (X, L),
deviations of S, (X, L) and S, (X+ 1, L+ 1) +
(X, L) from the values calculated with the mass
formula (11) against odd L. 0 : for one proton,
for one neutron,
0 : for two protons,
• : for two neutrons.
s..
e:
Separation Energtes and Nuclear Structures in Light Nuclei
113
a.
The connected lines for one nucleon are too high by nearly the same amount except for X= 1, and are higher than those for the two-nucleons except for X= - 1.
b. The connected line for one nucleon of X= 1 is lower in comparison with others.
The connected line for two nucleons of X= 1 is lower, and that of X= - 1 is higher in
comparison with others.
Now we derive the correcteq mass formula given by (11) by the uniform model3>
which assumes that the Hamiltonian of the nucleus (A, Z) is given as
where x1 • • ·x., and x.+l" · ·xA are the co-ordinates of protons and neutrons, Ti is the kinetic
energy of the i-th particle, V. ( x,, xj) is the coulomb energy between i-th and j-th particle,
and V(x., xj) is the potential energy of the two-body nuclear force between the particles
i and j. The wave function of the ground state of the nucleus (A, Z) (/)OAZ satisfies
(4)
The uniform model assumes that the expectation values of the potential energies in the
state <PoAz are given as follows, when the anti-symmetric pairs make no contribution to the
potential energy,
(efr0 ) Z 2A- 1' 3,
xi) )OAz= -2n+ (A, Z) Lr A-1=V,
(1/2 ~V.(x., xj) )o..tz= (3/5)
(5)
(1/2~V(xi,
(6)
i,J
where - 21-r/ A is the average potential energy per symmetric pair in the nucleus A, and
n+ (A, Z) is the number of symmetric pairs in the nucleus (A, Z) and is given by3>5>
n+ (A, Z)
= (3/16)
A 2 + (3/4)A+ (5/4)- (1/2)~(A,Z),
= (1/2)
Tt2 +2JTtl + (13/4), for odd A nuclei,
(7)
where
~(A, Z)
= (1/2) T/+2JTtl +2 (5/2), for even-even nuclei,
(8)
= (1/2) Tt 2 +2JTtJ +4, for odd-odd N=f=Z nuclei,
=5,
for odd-odd N=Z nuclei.
The above n+ (A, Z) are derived assuming that the spin of each nucleon is a good
quantum number. In the light nuclei region, the spin-orbit coupling is not strong and,
according to Peaslee6>, in the nuclei of the mass number region considered here the supermultiplet structure'> is dominant. 8> But in heavier nuclei, the change in n+ (A, Z) resulting from the strong spin-orbit coupling must be scrutinized.
In the case J T 1:J <2, ~(A, Z) in (8) can be expressed almost exactly as
~(A, Z) =
where
1.75 Tt2 +3.95+.d+ .d0,
114
T. Tati
~=I
~.=I
0
0.81
-0.81
0.25
-0.63
0
for odd A nuclei,
for odd-odd nuclei,
for even-even nuclei,
(9)
for odd-odd N= Z nuclei,
for even-even N=Z nuclei,
for N=f=Z nuclei.
Then we have in the mass number region considered here,
M(A, Z) = (H(A, Z) )OAz
=ZM,C 2 + (A-Z) M,.C 2 + (2jTi)oAz
+ (3e2j5 ro)
Z2A-tfs+ V;Y+ V/,
where
V/=- ((3/8)A+ (3/2) +1.70 A-1)
~.
(10)
V, 8 is the part of the potential energy V, given by ( 6) which contributes to the symmetry
effect of the binding energy. V,A is the part of V, which depends only on A and does
not contribute to the symmetry effect. According to the discussions in the next section,
it seems likely that there exist other kinds of potential energies which depend on A
only.
Hence, if we assume that the part of potential energy which depend only on A is
determined empirically by the first and second term of the mass formula ( 1) , the empirical
mass formula corrected by the uniform model gives the following binding energy,
Bc(A, Z) =14.03A-13.03A 2' 3 - (3/5) (e2/ro)Z 2 A-113
- (1.75 T-r. 2 +L1+L10 )L1 A- 1•
(11)
In the formula ( 11) , the symmetry energy which comes from the kinetic energy term
(2j T;)oAz is omitted to be considered separately. The term 77.27 T// A in (1) has
explained very well the separation energy variations in Figs. 1- 4, so we determine Lr in
(11) by
and obtain
~=44.15
(12)
Mev.
L1 in (12) determines the value of L1 Llj A which corresponds to iJ(A, Z) in (1) as
Lr Llj A=± 35.7/ A
and the value of
Lr
Mev.
(13)
L10/ A which is the energy to be added to the N= Z nuclei as
Separation Energies and Nuclear Structures in Light Nuclei
.10 ~/A=
I
115
11.05/ A Mev for odd-odd nuclei,
(14)
-27.85/ A Mev for even-even nuclei.
The term (13) is appreciably smaller* than a(A, Z) in (1) and explains the systematic
deviation a, and the term (14) represents the extra energies of N=Z nuclei and explains
the systematic deviation b.
In Fig. 6, the deviations of S"'(X, L) of odd L from the values calculated with (11)
which are denoted by LJ'<Cll(X, L), and thedeviationsofS't(X,L)+S'<(X+I,L+1) from
the values calculated with (11), which are denoted by J'tC2l(X, L), are plotted against L,
in the same way as in Fig. 5.
Fig. 6 shows that the formula (11) represents the binding energies very well, and
the systematic deviations in Fig. 5 have disappeared. In this fit the contribution of the last
term in (11), which is determined by using only one parameter L10 is essential.
The variations of J'<Cll(X, L) and J'tC2l(X, L) with respect to L are well understood
as the variations of the kinetic part of S't (X, L) expected from the I. P. M., and are consi·
dered as evidences of the I. P. M. structure.
The magnitude of this
1. At L=9 and L=21, the value of J"'(l)(X, L) falls.
The fall of Ll, c2J (X, L)
fall becomes gradually small as X becomes small and negative.
is deeper than that of J"'Cll(X, L). The conspicuous exception is the point for Sn(3, 9),
and we want to discuss about Sn(3, 9) in a separate paper.
2. At L= 11, 13 (and 19), the values of Ll, CIJ (X, L) and Ll"' C2J (X, L) rise, when,
X is negative. These rises become obscure at X= 0, and disappear when X becomes positive. The rise of J"'C2l(X, L) is higher than that of LJ,C1l(X, L).
J, (lJ (X, L) is the difference between S't (X, L) and the value S't,c (X, L) calculated
with the B.(X, L) of (11) in which the kinetic energy is proportional to A as in the
Fermi gas model. If we denote the kinetic energy part of S,(X, L) and S"' .• (X, L) as
K, (X, L) and K't,c (X, L),
K,(X, L) =T(X, L) -T(X-1, L-1),
K,,. (X, L) =T.(X, L)- T.(X-1, L-1). ·constant
where T(X, L) are the kinetic energy of the nucleus of A=2L-X and [Z-Nl =X. In
the single particle model, the energy level distance D"'(X, L) between (L-1) th and Lth
r-nucleons in the nucleus (X, L) are large at
L=9, 21,
(A)
L= 10, 11, 12, 13, 14; 16; 18, 19, 20,
(B)
and small at
when the spin-orbit coupling exists.
Since in extreme j-j coupling scheme, L=8 and 20
*) A. E. S. Green and D. F. Edwards9) analysed in detail the pairing effects and the shell effects
of nuclei referring to their mass surface and emphasized that 36/A3 / 4 or 140/A exaggerater the average magnitude
of the pairing effect for light nuclides.
116
T. Tati
are magic numbers and three groups of nucleons, L=9, 10, 11, 12, 13, 14, L=15, 16
and L= 17, 18, 19, 20 are in the same configuration d512, s112 and p213 respectively. 4>
Hence we can expect that K., (X, L) - K., ,c (X, L) are large at L in (A) and small at
L in (B). Thus the falls at L=9, 21 and ri5es at L= 11, 13, (19) are considered to
be the consequence of the variations of D.,(X, L) such as (A) and (B). (shell effects)
The above effect acts doubly to Ll., c2>(X, L).
The decrease of the falls of Ll., ClJ and Ll., c2> at L in (A) with the decrease of X, and
the decrease of the rises of Ll., (lJ and Ll., C2> at L= 11, 13 with the increase of X would be
the consequence of that the D., (X, L) is the increasing function* of X. The shell effect which
is caused by the largeness of D., (X, L) at L in (A) decreases with the decrease of X.
The shell effect which is caused by the smallness of D., (X, L) at Lin (B) increases with
the decrease of X.
In Fig. 6, the connected lines of LJ,C1l(X, L) and LJ.,C2 l(X, L) coincide well on the
abscissa except for negative X, and this fact seems to show that the even-odd variations of
the binding energies are well explained by the term LIL1 / A in (10) with the value (13).
The large deviation of Ll., C2>(X, L) for negative X at L= 11, 13 (and 19, 23) are interpreted above by the kinetic energy part of the separation energy.
Green and Edwards 9 > emphasized from the analysis of the discontinuities in nuclear
mass surface that the expressions for the pairing effect involving shell quantum numbers
which were derived by Mayer using the shell model are not in consistent with the available
data. At L= 11, 13, 21, and 23, the £1, <2l (X, L) is certainly larger than .d., (1) (X, L) when
X is negative. It may be possible that the large difference between .d, c2>(X, L) and .d, Cll
(X, L) comes from the pairing effect involving j of the last two nucleons, because at L=
11, 13, 21, 23, the last two nucleons, for which J.,C2>(X, L) are considered, have high j values
in comparison with others. They are j=S/2, j=S/2, j=7 /2 and j=7 /2 respectively.
But then the fact that these large differences (£1.,'2>-J,Cll) disappear when X is positive
is not well understood. In view of the importance of the pairing effect in the shell model
theory, the even-odd variations of separation energies must be studied in more detail and
in more wide mass number region.
§ 4. Some theoretical considerations
The potential energy V, given by (6) with the value of Lr given by (12) has given
the excellent explanation about the symmetry effects on the separation energies.
If the
total potential energy is given by V., the kinetic energy per nucleon should have been (2J
T 1 )0A.z=(14.03-3/8 Lr)A+ .. ·=2.5A+ .. ·(Mev), and is too small. 5> There is another
reason which will be mentioned soon later, that the potential energy of the above mentioned
two-body interaction is smaller (about a half) than the total potential energy of the nucleus.
This fact suggests the existence of other kinds of forces which is insensitive to the symmetry
effects. We assume in this paper the existence of the potential energy which has the above
*) The change of D., (X, L) with X seems to be larger than what is expected from the fact that the
D-. (X, L) is a decreasing function of A in the case of square well potential.
Separation Energies and Nuclear Structures m Light Nuclei
117
nature and we denote it as VA.
The main parts of the fluctuations of Ll~ in Fig. 6 are explained by the variations of
the kinetic energies of nuclei which are expected from the I. P. M. We now consider the
separation energy by the I. P.M. which assumes that the wave function (/)O.A.z in (4) has
the form
(15)
a
where
represents the antisymmetrtzmg operator. The fruitful success of the nuclear shell
(In the
model has shown that this approximation is better than it has been expected.
shell model theory, (/)O.A.z in (15) is the zeroth order eigenfunction and the wave function
is an appropriate linear sum of (15), which is determined by the interactions between outer
equivalent particles). The I. P. M. is not inconsistent with the consideration by the uniform model in the previous section, when the I. P. M. wave function ( 15) gives the larger
probability of finding two nucleons in the nuclear force range for the symmetric pairs than
for the antisymmetric pairs. We write the wave function (15) as
(16)
where ¢~Az is the wave function of the last nucleon which is assumed here to be a neutron
and IJT~AZ is the wave function of the residual nucleus in the nucleus (A, Z), and P,
represents the exchange between x.A. and x,.
The Hamiltonian H (A, Z) of the nucleus of mass number A can be written as
H(A, Z) =H(A-1, Z) +T,.+M,.+V,.(A, Z),
(17)
where T,. and M,. are the kinetic energy and mass of the last neutron and V,.(A, Z) is
the interaction between the last neutron and the residual nucleus (A-1, Z). The variation principle gives the following equations for (/)o.A.z of the form (16) :
[T,.+ idrA-1 IJ1"~1z V,.(A, Z)(1- 2J P,) IJT~z
+iW~JzH(A-1, Z)IJT~...tza'TA-t+M,.-M(A,Z)]¢~.cz=O,
[H(A-1, Z)
(18)
+ idxA ¢~1z V,.(A, Z)(1- 2J P,)¢~Az
(19)
These equations express the interaction between the last nucleon and the residual nucleus
in the Hartree approximation. The difference between (/)OAz and Woc...t.-l)Z caused by the
removal of the last nucleon is considered here as caused by the difference of radii of nuclei
(A, Z) and (A- 1, Z) , since the effect by the change of radius is very large and the
constancy of the nuclear densities is known from the analysis of the nuclear coulomb energies. 5ltOl Then the difference
i IJT~lz H(A-1, Z)
IJTOAz drA-1- i (/)l~.t-l)z H(A-1, Z)
(/)o<...t-lJZ
drA-t=R{...tZl (20)
is estimated as follows. We consider the kinetic part T(A-1, Z) and potential part
V(A-1 1 Z) separately as
T. Tati
118
H(A-1, Z) =T(A-1, Z) +V(A-1, Z) +ZMv+ (A-1-Z)M,..
(21)
We assume that the kinetic energy of i-th nucleon in the nucleus A of radius RA is given
by atf (RA) 3"11, where a, and '1) are constants.
S 1Jf~1z T(A-1, Z) IJf~A.z drA-l- i (/)~c~-1)z T(A-1, Z)
A-l
(/)o(A.-l)Z
dr.A.-1
A-1
='2j(a,fR:"Il-a.jR'!;.'!..1) =2ja/(A-"Il- (A-1)-7!)
i-1
·-1
1
=-~2j a/(A-1)-'~- 1 =-'1)-A-1
A-1
•-1
A-1
_
2j a,jR~Yl=-lJT.
(22)*
>=1
T is the mean kinetic energy per nucleon. Next, we calculate the potential part of (20)
assuming that there is only V. energy in ( 6) which is inversely proportional to the volume
of the nucleus v A· c)
} 1Jf~1z V(A-1, Z) IJf~Az dr.A.-1- i (/)~A-l)Z V (A-1, Z)
= -2£,.' n+ (A-1, Z) +2L/
n+
VA
1
---
(A-1)
V.
is the mean
2n+
(A-1, Z) = -2L,.
(/)OA-lz
n+ (A-1,
dr A-1
Z) {A-1- (A-1)-l}
VA-l
(A-1, Z)
A-1
(23)
V. potential energy per nucleon.
R~z=
Then we have
-7jT-V,.
(24)
The Schrodinger equation for the last neutron (18) can be written as
(25)
where V,.,A.z (xA.) means the average potential energy for last neutron xA, averaged over all
co-ordinates of other nucleons, and when the exchanges P; are neglected it is :
(26)
The equation (25) means that the last neutron moves in the average potential
with the energy
En,OA.z=
10
E,.,.tz
v,...Az
Fig. 7.
-S,.(A, Z) -~z·
Vn,Az
(27)
Thus, as is shown in Fig. 7,** the level of the last nucleon
in I. P. M. is somewhat lower than -S,. (A, Z) by the
amount R~z· R~z=- (V.+7JT) has nearly constant value
for all nuclei.
R~z is the energy required when the nucleus expands
K. Woestelll noticed that these kinetic energy changes are appreciably large.
Fig. 1 in the Woeste's paperlll and Fig. 7 have the different meanings. In Fig. 7, V means
the potential which determines the wave function of the last nucleon and is not equal to the mean potential
energy per nucleon.
*)
**)
Separation Energies and Nuclear Structures in Light Nuclei
20
119
"'==41.6
15
5
10
15
20
30
35
Fig. 8. Mean energy per nucleon (measured from the bottom of the potential) of the nucleus which
consists of A nucleons moving independently in a square well potenital of the depth V and of the radius
1.45Xl0-13 Af/3 em. Here the effect on the wave function by the spin-orbit coupling is neglected. We
assumed that after Is and lp levels are filled, 12 nucleons fill (ld), and 4 nucleons fill (2s) and then 4
nucleons fill (ld) successively, where (ld), (2s) are levels without the spin-orbit coupling.
from the equilibrium volume to the volume which the nucleus with one more nucleon has
in the equilibrium. Since the nucleus has minimum energy at the equilibrium, R~z is
positive and we call in this paper E (A, Z) given by (28) as the coefficient of "ex·
pansibility " at the equilibrium.
E (A,
Z) =R'lz/(4/3)1r r 08
> 0.
(28)
V, is estimated from ( 10) and ( 12) as
V,.-- 3/8 L1 .--16.5 Mev.
(29)
T<24.7 Mev.
(30)
From (24), (28)
In (29), "1)=2/3 was assumed. The energy shift R'lz, of the last nucleon from S,. seems
to be not large but rather small, then the mean kinetic energy is nearly equal in absolute
magnitude to the mean V, potential energy.
T'$-2/3 V.,
T.-24 Mev.
(31)
T in (31) is considerably larger than the mean kinetic energy in the Fermi gas model*
*)
The kinetic energy increases if the position correlations are taken into account. 5)
120
T. Tati
( 13"" 14 Mev) . But the mean kinetic energy in the single particle model with in6nite
square well potential is 24'"'-29 Mev in the mass number region A=20--40.
For comparisons, the mean energies per nucleon Ev (A) in the single particle model with square
well potential of depth V= oo, 30 and 40 Mev were calculated graphically and Ev (A)V are plotted in Fig. 8. Since the total binding energy is 14.03A+ · · ·, the total potential energy per nucleon V is estimated from
T
V-T---14 Mev,
in ( 31) .
V---38 Mev,
Then we have
V-V,---21 Mev.
(32)
Here again we have the fact that the potential energy V, given by ( 6) in the nucleus
is only about a half of the total potential energy of the nucleus. We presumed besides
V., the existence of the potential energy VA which is insensitive to the symmetry effect
of the nucleus.
Here, we presume further that VA is also insensitive to the " expansibility " of the nucleus. We want to discuss the possible interactions of such a kind
in a separate paper. Of course, the estimation of (V- V,) is very crude and the surface
energy was neglected. And we assumed for the wave function the form ( 16) .
Next, we will estimate the magnitude of Vn,AZ(xA) in (25) assuming the square well
shape.
The contribution to Vn,AZ from v. is denoted as v~·~AZ and is from (6), (7) and (9)
v;:~Az=
0 lff'tA.z V(A-1,
= -2 (~/A)
(n+
Z) lffOAzd!'A-1)s
(A, Z) -n+ (A-1, Z))
= -2(~/ A) ((3/8)A+ (3/4)- (1/2)f(A, Z) +
=
where X=N-Z.
-~ ((3/4)
(1/2)f(A-1, Z))
+ (3/2) A- 1 -1.75 (X-1)A- 1 ± (LJA- 1 ) ± (L10 A-t, o) ),
(33)*
Since V,=-2~n+(A,Z)/A2 --3/8Lt> (33) is nearly equal to
2V,.
The V, potential energy difference between two nuclei (A, Z) and (A-1, Z)
is'"'- V,. When a neutron is removed from the nucleus, about a half of the potential energy
v;:~Az is consumed by the decrease of the potential energy of the residual nucleus according to (23). Using the value (12)
V~~>.u=-33+66A- 1 -77 (X-1)A- 1 =F~LJA- 1 =F
(L1 L10 A-t, 0).
(34)
(in Mev)
The contributions from VA to Vn,AZ is denoted by v~~1z·
VA---21 Mev,
VA is estimated from (32)
(35)
Since VA was assumed insensitive to the change of the volume, v~~1z is estimated to be
to
*) The signs and values of the last two terms in (33) and (36) must be chosen properly referring
the values in (9).
Separation Energies and Nuclear Structures in Light Nuclei
equal to the mean potential energy of VA·
square well potential Vrt.AZ is
121
Then estimated value for the depth of the
(36)*
V1,,Az is the same with (34) except for the coulomb energy.
I want to express my thanks to Professor S. Tom onaga
me when I showed him the figures in Section 2 last winter.
Sato for informing me the references on the nuclear binding
Hayakawa, Professor M. Sasaki and Dr. T. Tamura for the
of this work.
I am much indebted to Mrs. H. Tati for
graphs. I wish to express my thanks to Dr. S. Oneda and
institute, for the discussions in the course of this work.
for the encouragement he gave
I am indebted to Dr. M.
energy, and to Professor S.
discussions in the early stage
compiling data and making
Dr. S. Hori, members of this
References
14)
15)
16)
17)
J. A. Harvey, Phys. Rev. 81 (1951), 353; M. G. Redlich, Phys. Rev. 91 (1953), 328; N. S.
Wall, Phys. Rev. 96 (1954), 664.
E. Fermi, Nuclear Physics (1949).
E. P. Wigner, Phys. Rev. 51 (1937), 947.
M. G. Mayer. Phys. Rev. 75 (1949), 1969; 0. Haxel, J. H. D. Jensen and H. E. Suess, Phys.
Rev. 75 (1949), 1766.
J. N. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics, Chapter VII.
D. C. Peaslee, Phys. Rev. 95 (1954), 717.
E. P. Wigner, Phys. Rev. 51 (1937), 106.
I. Talmi, Phys. Rev. 91 (1953), 122.
A. E. S. Green and N. A Engler, Phys. Rev. 91 (1953), 40;
A. E. S. Green and D. F. Edwards, Phys. Rev. 91 (1953), 46.
B. C. Carlson and I. Talmi, Phys. Rev. 91 (1954), 436, B. G. Jancovici, Phys. Rev. 95 (1954),
389, T. Tati, Sci. Rep. Kanazawa Univ. 3 (1955), 45.
K. Woeste, Zeit. £ Phys. 137 (1954), 228.
H. Feshbach, C. E. Porter and V. F. Weisskopf, Phys. Rev. 96 (1954), 448, R. K. Adair, Phys.
Rev. 94 (1954), 737, N. Nereson and S. Darden, Phys. Rev. 94 (1954), 1678.
Li, Whaling, Fowler and Lauritsen, Phys. Rev. 83 (1951), 519.
C. W. Li, Phys. Rev. 88 {1952), 1041.
F. Ajzenberg and T. Lauritsen, Rev. Mod. Phys. 24 (1952), 321.
P. M. Endt and J. C. Kluyver, Rev. Mod. Phys. 26 (1954), 93.
T. L. Collins, A. 0. Nier and W. H. Johnson, Phys. Rev. 84 (1951), 717; 86 (1952), 408; 94
18)
19)
20)
R. E. Halsted, Phys. Rev. 88 (1952), 666.
N. Feather, Adv. Phys. 2 (1953), 141.
F. Ajzenberg, and T. Lauritsen, Rev. Mod. Phys. 27 (1955), 77.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
(1954) 399.
*) Feshbach, Porter and W eisskopf' s analysis12l of the data of the low energy neutron scattering from
nuclei by their rr.odel with the potential Vo+iWo shows that Vo=42 Mev gives better fits to the experiment
than Vo= 19 Mev. V,.,Az in (36) is not inconsistent with their V 0•
T. Tati
122
Appendix
The most of the separation energies are based on the Q,. values in the Ajzenberg and
Lauritsen's paper15> and the mass defects in the Endt and Kluyver's paper. 16l
Ajzenberg
and Lauritsen adopted the values of mass defects in the papers of Li, Whaling, Fowlder,
Lauritsen13l and Li, 14> which were derived from the reaction data. Endt and Kluyver adopted the values in Li's table14l and the weighted means of mass spectrographic data by Collins,
Nier and Johnson17l and several other authors. In the regions which are not included in the
papers of Ajzenberg and Lauritsen and Endt and Kluyver, the values of mass defects in
the tables of Li14l and Collins, Nier and Johnson6 l are adopted.* Dr. M. Sato kindly informed
me about the excellent summary of Feather16' about the separation energies, and from Feather's
table 9 Sp's and 16 Sn's are added to Tables I and II. Other Sp's and Sn's in Tables
I, II were compared with Feather's values, and most of them coincide within the limit of
errors.
16 Sp's and 21 Sn's in Tables I and II are not found in Feather's Table.
For
the discussions in this paper, the general distribution of separation energies is important
rather than the very precise value of each separation energy, so we use the values above
mentioned without further scrutiny.
*) Q,. values of (p, r), (n, r) etc., in the Ajzenberg and Lauritsen's paper coincide with Sp's and
Sn's calculated from the mass defects of the Li's table except for six values. Exceptions arise from the different values of the mass defects of Ql8) and Fl8) in two references. (-0.2 Mev) We adopted Ajzenberg
and Lauritsen's values. The mass defects derived from reaction date in the Endt and Kluyver's paper are
the same with the mass defects in Li's table except Mn27 for which we adopted the Endt and Kluyver's value.
Some of the mas~ defects in Collins, Nier and Johnson differ more than 0.1 Mev from the values in Endt and
Kluyver's paper which are weighted means of several mass spectrographic data. We adopted the latter values,
but the mass defects of Collins, Nier and Johnson are adopted without corrections which are not found in
Endt and Kluyver.
Table I
Sv from the mass formula (1)
A
10
10
11
11
12
z
5
6
5
6
6
-2.3
-2.7
-2.2
-2.7
-2.6
-6.0
6.0
-6.0
6.0
5.2
-4.7
5.0
-1.5
1.5
-4.0
-3.0
-2.6
-3.0
-2.9
-3.3
-5.2
5.2
-5.2
-4.7
4.7
-2.7
17.9
0.6
4.3
-3.6
-3.6
-3.5
-3.5
-3.4
4.2
-1.2
6.8
1.2
3.7
-2.9
-3.3
-2.8
-3.2
-3.2
-4.7
4.7
-4.2
4.2
4.2
-3.4
-3.4
-3.4
-1.1
5.9
1.1
-3.6
-3.1
-3.5
-4.2
3.8
-3.8
6.6
3.8
11.2
14.0
-4.1
-4.1
-4.0
-4.0
-3.8
-3.8
8.7
15.9
0.3
6
17.5
7
7
7.5
8
4.7
15
15
16
16
17
7
10.2
1.9
8
7.3
7
10.6
12.1
9
8
9
2.1
-5.8
6.0
-1.8
1.8
surf.
7
17
18
18
8
vol.
12
13
13
14
14
8
8
coul.
Sv
13.7
0.6
16.1
5.6
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
"
-3.7
-3.7
-3.7
-3.7
sym.
total
3.9
7.4
dev.
ref.
2.7
-3.6
3.4
-2.9
1.4
(15)
(19)
(15)
(15)
(15)
7.7
3.0
-0.4
1.3
3.2
-3.0
(19)
(15)
(15)
(15)
(19)
7.2
10.6
10.3
12.7
15.3
3.0
-3.3
0.3
-0.6
-1.6
{15)
(15)
(15)
(15)
(15)
1.7
17.3
4.5
-1.1
-1.2
1.1
(15)
(15)
(15)
7.8
11.6
14.5
Separation Energies and Nuclear Structures in Light Nuclei
19
19
9
10
"8.0
6.4
20
20
20
21
21
9
10
11
10
11
10.3
12.9
0.8
13.0
2.5
22
23
23
24
24
11
11
6.7
8.8
12
11
12
10.6
11.7
7.5
14.0
123
-3.2
-3.2
3.3
-1.0
-3.5
-3.8
-3.8
3.8
6.8
9.8
1.2
-3.4
(15)
(15)
-3.2
-3.2
-3.2
-3.2
-3.2
5.3
1.0
-2.9
2.9
-0.9
-3.4
-3.8
-4.1
-3.7
-4.1
-3.5
3.5
-3.5
3.5
-3.5
9.1
11.6
0.3
13.6
2.3
1.2
1.3
0.5
-0.6
0.2
(15)
(15)
(14) (16)
(15)
(16)
"
"
"
-3.1
-3.1
-3.1
-3.0
-3.0
0.9
2.7
-0.8
4.4
0.8
-4.1
-4.0
-4.3
-4.0
-4.3
-3.3
-3.3
3.3
-3.1
3.1
4.5
6.3
9.1
8.3
10.7
2.2
2.5
-1.6
2.3
1.0
(16)
(16)
(16)
(14)
(16)
"
"
-3.0
-3.0
-3.0
-2.9
-2.9
-2.4
2.4
-0.8
4.0
0.8
-4.6
-4.2
-4.6
-4.2
-4.6
-3.1
3.1
-3.1
2.9
-2.9
1.0
12.3
2.5
13.8
4.4
0.8
-0.2
-0.2
0.2
2.2
(16)
(16)
(16)
(16)
(16)
-2.9
-2.9
-2.9
-2.9
-2.9
2.3
-0.7
3.7
0.7
-2.0
-4.5
-4.8
-4.5
-4.8
-5.1
-2.9
2.9
-2.8
2.8
-2.8
5.9
8.4
7.6
9.8
1.2
2.4
-1.2
1.9
1.8
1.0
(16)
(16)
(16)
(16)
(16)
5.1
2.1
-0.7
3.4
0.7
-4.4
-4.7
-5.1
-4.7
-5.0
-2.8
2.8
-2.8
2.6
-2.6
9.1
11.3
2.7
12.7
4.3
1.3
1.0
0.0
0.9
1.2
(16)
(16)
(16)
(16)
(16)
"
"
"
"
24
25
25
26
26
13
27
27
13
14
28
28
28
14
15
29
29
29
30
30
13
14
15
14
15
10.4
12.3
2.7
13.6
5.5
-2.8
-2.8
-2.8
-2.8
-2.8
31
31
32
32
32
15
16
15
16
17
7.3
6.5
8.6
8.9
0.8
-2.8
-2.8
-2.8
-2.8
-2.8
2.0
-0.6
3.2
0.6
-1.8
-5.0
-5.4
-4.9
-5.3
-5.6
-2.6
2.6
-2.5
2.5
-2.5
5.6
7.9
7.0
9.0
1.4
1.7
-1.4
1.6
-0.1
-0.6
(16)
(16)
(16)
(16)
(16)
33
33
33
34
34
15
16
17
16
17
9..4
9.6
2.5
10.9
5.0
-2.7
-2.7
-2.7
-2.7
-2.7
4.4
1.8
-0.6
3.0
0.6
-4.8
-5.2
-2.5
2.5
-2.5
2.4
-2.4
8.3
10.4
2.8
11.5
4.1
1.1
-0.8
-0.3
-0.6
0.9
(16)
(16)
(16)
(16)
(16)
35
35
35
36
36
16
17
18
17
18
11.3
6.4
6.6
8.0
8.5
-2.6
-2.6
-2.6
-2.6
-2.6
4.1
1.7
-0.6
2.8
0.6
-5.1
-5.4
-5.8
-5.4
12.8
-5.7
2.4
-2.4
2.4
-2.3
2.3
7.4
6.5
8.5
-1.5
1.1
-0.8
1.5
0.0
(16)
(16)
(16)
(16)
(16)
37
37
37
38
38
17
18
19
17
18
9.2
8.8
2.5
9.6
10.2
,"
-2.6
-2.6
-2.6
-2.6
-2.6
3.9
1.6
-0.5
5.0
2.7
-5.4
-5.6
-6.0
-5.3
-5.7
-2.3
2.3
-2.3
-2.2
2.2
7.7
9.7
2.7
8.9
10.6
1.5
-0.9
-0.2
0.7
-0.4
(16)
(16)
(16)
(16)
(16)
38
39
39
39
40
19
18
19
20
18
5.1
10.8
6.5
"
"
,"
-2.6
-2.5
-2.5
-2.5
-2.5
0.5
3.7
1.5
-0.5
4.7
-5.9
-5.6
-5.9
-6.2
-5.5
-2.2
2.2
-2.2
2.2
2.1
3.9
11.7
5.0
7.0
12.8
1.2
-0.9
1.5
-1.3
-0.7
(16)
(16)
(16)
(16)
(16)
40
40
41
41
41
19
20
19
20
21
7.7
8.4
7.8
8.8
1.7
"
"
"
,
-2.5
-2.5
-25
-2.5
-2.5
2.5
0.5
3.5
1.5
-0.5
-5.8
-6.1
-5.8
-6.1
-6.3
-2.1
2.1
-2.1
2.1
-2.1
6.1
8.0
7.1
9.1
2.6
1.6
0.4
0.7
-0.3
-0.9
(16)
(16)
(16)
(16)
(17)
42
42
43
44
45
19
20
20
20
21
9.1
10.2
10.8
11.4
6.8
,
-2.5
-2.5
-2.5
-2.5
-2.5
4.4
2.4
3.3
4.2
3.2
-5.8
-6.1
-6.0
-6.0
-6.2
-2.0
2.0
2.0
2.0
-2.0
8.2
9.9
10.9
11.7
6.5
0.9
0.3
-0.1
-0.3
0.3
(16)
(16)
(16)
(16)
(17)
13
12
13
12
13
1.8
12.1
2.3
14.0
6-6
8.2
7.2
9.5
11.6
2.2
5.7
12.1
"
"
"
"
"
"
"
"
,"
-5.5
-5.2
-5.5
5.3
124
T. Tati
46
47
48
48
21
22
22
21
22
8.4
10.5
10.3
10.2
10.9
48
49
49
49
50
23
22
23
24
22
6.8
10.1
7.4
7.6
11.6
50
50
51
51
23
24
23
24
7.7
8.4
7.8
9.6
46
-2.4
-2.4
-2.4
-2.4
-2.4
4.0
2.2
3.0
5.7
3.8
-6.1
-6.5
-6.5
-6.1
-6.4
-1.9
1.9
1.9
-1.8
1.8
7.6
9.2
10.0
9.4
11.0
0.8
1.3
0.3
0.8
-0.1
(19)
(17)
(19)
(16) (17)
(19)
"
"
-2.4
-2.4
-2.4
-2.4
-2.4
2.1
4.6
2.9
1.2
5.4
-6.8
-6.4
-6.7
-6.8
-6.3
-1.8
1.8
-1.8
1.8
1.8
5.2
11.8
6.0
7.9
12.6
1.6
-1.7
1.4
-0.3
-1.0
(17)
(17)
(17)
(17)
(19)
,
-2.4
-2.4
-2.3
-2.3
3.7
2.0
4.4
2.8
-6.6
-6.8
-6.6
-6.8
-1.8
1.8
-1.8
1.8
6.9
8.6
7.8
9.5
0.8
-0.2
0.0
0.1
(17)
(19)
(17)
(17)
14.0
Table
n
S,. from the mass formula (1)
A
N
S,.
10
10
11
11
12
5
6
5
6
6
8.4
6.8
13.4
11.5
18.7
14.0
12
13
13
14
14
7
6
7
7
8
7
sur£
sym.
coul.
lJ
total
dev.
ref.
,
"
-4.1
-4.1
-4.0
-4.0
-3.8
2.0
-5.6
6.0
-1.8
1.8
0.3
0.2
0.3
0.2
0.3
-6.0
6.0
-6.0
6.0
5.2
6.2
10.5
10.4
14.5
17.4
2.2
-3.7
3.0
-3.0
1.3
(15)
(15)
(19)
(15)
(15)
3.4
20.5
5.0
10.6
8.2
"
"
"
,"
-3.8
-3.7
-3.7
-3.7
-3.7
-4.7
5.0
-1.5
1.5
-4.0
0.2
0.3
0.2
0.3
0.2
-5.2
5.2
-5.2
-4.7
4.7
0.5
20.8
3.9
11.2
2.9
-0.3
1.1
3.1
-0.3
(15)
(19)
(15)
(15)
(15)
"
",
,
,
,
,
"
,"
,
"
"
"
,"
,
,
,
,
-3.6
-3.6
-3.6
-3.5
-3.5
4.2
-1.3
-6.1
1.3
-3.5
0.4
0.3
02
0.3
0.3
-4.7
4.7
-4.7
4.2
-4.2
10.4
14.1
0.1
16.3
3.1
2.9
-3.3
2.1
-0.7
-0.5
(19)
(15)
(15)
(15)
(15)
-3.4
-3.4
-3.4
-3.4
-3.2
-1.1
-5.4
1.1
-3.2
-1.0
0.3
0.2
0.4
0.3
0.3
-4.2
4.2
-3.8
3.8
3.8
5.6
9.7
8.3
11.6
13.9
-1.5
-4.0
0.9
-3.5
-3.5
(15)
(19)
(15)
(15)
(15)
-3.2
-3.2
-3.2
-3.2
-3.2
-4.9
1.0
-2.8
2.9
-0.9
0.3
0.4
0.3
0.4
0.3
-3.8
3.5
-3.5
3.5
-3.5
2.4
15.7
4.7
17.7
6.7
1.8
1.2
1.9
0.9
0.1
(15)
(15)
(15)
(16)
(15)
-3.1
-3.1
-3.1
-3.1
-3.0
0.9
-2.6
-0.8
-4.0
0.8
0.4
0.3
0.4
0.3
0.4
-3.3
3.3
3.3
-3.3
3.1
8.9
11.9
13.8
3.9
15.4
2.1
-1.5
-1.4
1.3
1.2
(16)
(16)
(16)
(16)
(16)
-3.0
-3.0
-3.0
-3.0
-2.9
-2.4
2.4
-0.8
-3.7
0.8
04
0.5
0.4
0.4
0.4
-3.1
3.1
-3.1
3.1
-2.9
5.9
17.0
7.6
10.7
9.4
1.1
0.0
-0.3
-1.6
2.2
(16)
(16)
(16)
(16)
(16)
-2.9
-2.9
-2.9
-2.9
-2.2
-0.7
-3.5
0.7
0.4
0.4
03
0.5
2.9
2.9
-2.9
2.8
12.2
13.7
5.0
15.1
-1.1
-0.9
1.4
2.1
(16)
(16)
(16)
(16)
15
15
15
16
16
8
9
8
9
13.3
10.8
2.2
15.6
2.6
17
17
18
18
19
9
10
9
10
10
4.1
5.7
9.2
8.1
10.4
19
20
20
21
21
11
10
11
10
11
4.2
16.9
6.6
18.6
6.8
22
22
23
23
24
11
12
12
13
12
11.0
10.4
12.4
5.2
16.6
24
25
25
25
26
13
12
13
14
13
7.0
17.0
7.3
9.1
11.6
26
27
27
14
14
15
14
11.1
12.8
6.4
17.2
28
vol.
,
"
"
"
"
"
"
"
"
"
"
7.~
125
Separation Energies and Nuclear Structures in Light Nuclei
28
15
7.7
14.0
-2.9
-2.0
0.4
-2.8
6.7
1.0
(16)
28
29
29
29
30
16
14
15
16
15
8.6
17.7
8.5
9.4
11.3
"
"
"
-2.9
-2.8
-2.8
-2.8
-2.8
-4.6
2.1
-0.7
-3.2
0.7
0.3
0.5
0.4
0.4
0.5
2.8
2.8
-2.8
2.8
-2.6
9.6
16.6
8.2
11.1
9.8
-1.0
1.1
0.3
-1.7
1.5
(16)
(16)
(16)
(16)
(16)
30
31
31
32
32
16
16
17
16
17
10.6
12.4
6.6
14.8
7.9
-2.8
-2.8
-2.8
-2.8
-2.8
-1.9
-0.6
-3.0
0.6
-1.8
0.4
0.5
0.4
0.5
0.4
2.6
2.6
-2.6
2.5
-2.5
12.4
13.7
6.0
14.9
7.4
-1.8
"
"
0.6
-0.1
0.5
(16)
(16)
(16)
(16)
(16)
32
33
33
33
34
18
16
17
18
17
9.3
16.5
8.6
10.1
11.2
-2.8
-2.7
-2.7
-2.7
-2.7
-4.0
1.8
-0.6
-2.9
0.6
0.4
06
0.5
0.4
0.5
2.5
2.5
-2.5
2.5
-2.4
10.1
16.2
8.8
11.4
10.1
-0.8
0.3
-0.2
-1.3
1.1
(16)
(16)
(16)
(16)
(16)
34
34
35
35
36
18
19
18
19
18
11.4
6.6
12.8
7.0
14.7
-2.7
-2.7
-2.6
-2.6
-2.6
-1.7
-3.8
-0.6
-2.7
0.6
0.4
0.4
0.5
0.4
0.6
2.4
-2.4
2.4
-2.4
2.3
12.5
5.5
13.7
6.8
14.7
-1.1
1.1
-0.9
0.2
0.0
(16)
(16)
(16)
(16)
(16)
36
36
37
37
37
19
20
19
20
21
8.6
9.2
8.9
10.4
5.7
"
"
"
"
"
"
"
"
"
"
-2.6
-2.6
-2.6
-2.6
-2.6
-1.6
-3.6
-0.5
-2.6
-4.5
0.4
0.5
0.5
0.5
0.4
-2.3
2.3
-2.3
2.3
-2.3
7.9
10.5
9.1
11.7
5.0
0.7
-1.3
-0.2
-1.3
0.7
(16)
(16)
(16)
(16)
(16)
38
38
38
39
39
19
20
21
20
21
11.5
11.8
6.1
13.2
6.7
"
"
-2.6
-2.6
-2.6
-2.5
-2.5
0.5
-1.5
-3.4
-0.5
-2.4
0.6
0.5
0.5
0.5
0.5
-2.2
2.2
-2.2
2.2
-2.2
10.4
12.6
6.2
13.7
7.3
1.1
-0.8
-0.1
-0.5
-0.6
(16)
(16)
(16)
(16)
(16)
39
40
40
40
41
22
20
21
22
21
8.6
15.9
7.9
9.9
8.3
-2.5
-2.5
-2.5
-2.5
-2.5
-4.3
0.5
-1.4
-3.3
-0.5
0.4
0.5
0.5
0.5
0.6
2.2
2.1
-2.1
2.1
-2.1
9.9
14.6
8.5
10.9
9.6
-1.3
1.3
-0.6
-1.0
-1.3
(16)
(16)
(16)
(16)
(16)
41
41
42
42
43
22
23
22
23
23
10.0
6.1
11.4
7.4
8.0
-2.5
-2.5
-2.5
-2.5
-2.5
-2.3
-4.1
-1.4
-3.1
-2.2
0.5
0.4
0.5
0.5
0.5
2.1
-2.1
2.0
-2.0
-2.0
11.8
5.8
12.7
6.8
7.9
-1.8
0.3
-1.3
0.6
0.1
(16)
(16)
(16)
(16)
(16)
43
45
46
46
24
24
25
24
25
10.8
11.4
7.4
13.3
8.9
-2.5
-2.5
-2.5
-24
-2.4
-3.9
-3.0
-3.7
-1.3
-2.9
0.5
0.5
0.5
0.7
0.6
2.0
2.0
-2.0
1.9
-1.9
10.3
11.0
6.4
12.9
7.4
0.5
0.4
1.0
0.4
1.5
(16)
(16)
(16)
(19)
(19)
47
47
48
48
48
25
26
26
27
28
8.6
10.5
11.7
7.8
9.8
-2.4
-2.4
-2.4
-2.4
-2.4
-2.0
-3.6
-2.7
-4.2
-5.6
0.5
0.5
0.6
0.4
0.5
-1.9
1.9
1.8
-1.8
1.8
8.2
10.4
11.4
6.0
8.3
0.4
0.1
0.3
1.8
1.5
(17)
(19)
(17)
(19)
(17)
49
49
49
49
50
26
27
28
29
26
12.2
8.0
10.1
5.1
13.3
-2.4
-2.4
-2.4
-2.4
-2.4
-1.9
-3.4
-4.8
-6.2
-1.2
0.7
0.5
0.5
0.4
0.7
1.8
-1.8
1.8
-1.8
1.8
12.3
6.8
9.1
4.1
13.0
-0.1
l.l
1.0
1.0
0.4
(19)
(17)
(19)
(17)
(17)
50
50
51
51
27
28
27
28
8.0
11.0
9.1
11.1
-2.4
-2.4
-2.3
-2.3
-2.6
-4.1
-1.9
-3.3
0.5
0.5
0.6
0.6
-1.8
1.8
-1.8
1.8
7.8
9.9
8.7
10.8
0.2
1.1
0.4
0.3
(19)
(17)
(17)
(17)
44
"
"
"
"
"
"
"
"
"
"
"
-1.3