Document text
Specific Density Of Binding Energy Of Core
In p - Stable Nuclei Is 2.57 MeV/fm 3
Abstract
G. K. NIE
Institute of Nuclear Physics, Ulugbek, Tashkent 702132, Uzbekistan
o
sx-
Recently an a-cluster model based on the pn-pair interactions with using the
isospin invariance of nuclear force has been proposed. According to the model the
excess neutron pairs fill out the free space in the core determined by the difference
in the charge and matter radii of the a-clusters. Then the number of excess neu-
trons in /3-stable nuclei depends on the number of the core a-clusters. In such a
representation the specific density of binding energy of core p is the only parameter
to fit the experimental binding energies of /5-stable nuclei and it turned out to be a
constant value equal to 2.57 MeV/fm 3 . Knowing the value p allows one to estimate
^r) ■ the size of a nucleus from its experimental binding energy.
>
0^ , Key words: nuclear structure; alpha-cluster model; Coulomb energy; surface
| tension energy, binding energy; charge radius.
PACS: 21.60.-n; 21.60.Gx; 21.10.Dr; 21.60.Cs.
o
o
>
•1— I ,
^ ■ 1 Binding Energy Of Core
The idea that the main properties of nuclei can be described from a simple
representation (like liquid drop [1] or some regular forms of a-cluster structure
[2]) has been popular since the very beginning of nuclear physics. The main
features of nuclear force, its strength and the short range, allow one to find
some simple formulas to describe size and binding energies of nuclei.
Recently an a-cluster model based on pn-pair interactions with using isospin
invariance of nuclear force has been proposed [3,4,5]. In the framework of
this model new formulas to calculate radii and binding energies of /3-stable
nuclei have been found. Also the model provides some reasonable explanation
Email address: [email protected] (G. K. NIE).
Preprint submitted to Elsevier
1 February 2008
of existing excess neutrons in stable nuclei. The spinless nn-pairs of excess
neutrons fill out the free space in the core which is determined by the difference
in the volumes occupied by the charge and by the matter of the a-clusters.
The proton charge radius is bigger than the neutron radius due to isospin
invariance of nuclear force [6].
In such a representation the value of the charge radius R of an even Z nucleus
can be obtained from an estimation of the volumes occupied by the alpha-
clusters of the core N corea and the peripheral a-clusters N pa = N a — N corea
(N pa = 1 - 5) , N a = Z/2,
R 3 = r 3 a N corea + r 3 He N pa , (1)
where the radius of a peripheral a-cluster equals the experimental radius of the
nucleus 4 He r4 He = 1.71 fm [7]. In case of N corea = 0, N a = N pa , R = r-4 Re N^ 3
[ 3-5 ]. The value of the radius of an a - cluster of core r a = 1.60 fm is obtained
from fitting the experimental radii of the nuclei with N corea » N pa with the
formula R = r a N^ 3 .
For odd Z\ — Z + 1 the number of a-clusters N la in the nucleus is 7V lQ =
N a + 0.5 and the number of peripheral a-clusters is N lpa = N pa + 0.5. Then
the formula is
= r 3 a N corea + r 3 He N lpa . (2)
Whereas the nuclear radii are calculated by means of estimation of the vol-
umes occupied by the a-clusters of core and by the a-clusters of periphery,
the binding energy is calculated on the total amount of a-clusters N a [5] dis-
regarding to the fact that some of them belong to the core and the others are
of the nucleus periphery. In this paper the model [5] is developed to have a
consistency between the ways of how the radii and the binding energies are
calculated.
In the representation with dividing nucleus for core and periphery the binding
energy of a nucleus is to be the sum of the binding energy of the core E core
and the internal binding energy of the compound peripheral cluster E^ pa con-
sisting of Npa a-clusters minus the Coulomb energy of the compound cluster
interaction with the core a-clusters E^ aNcorea = 2N pa 2N corea e 2 / R p , where R p
is the radius of the last alpha-cluster position in the nucleus (in the center of
core mass system) [5] (Rpi is the radius of the single pn-pair's position in the
odd Zi — Z + 1 nucleus),
Rp = 2.1Q8(N a - 4) 1/3 ; R pl = 2.168(iV a + 0.5 - 4) 1/3 (3)
2
So the new formula to calculate binding energy is to be as follows
E = E core + E Npa - E NpaNcorea . (4)
The energy of the peripheral compound cluster E^ pa consisting of N pa a-
clusters is taken equal to the experimental binding energy of the nucleus 8 Be
(N pa = 2) Es Bc = 56.5 MeV, 12 C (N pa = 3) Ei2 C = 92.2 MeV, 16 (N pa = 4)
Em = 127.6 MeV, 20 Ne (N pa = 5) E 20Ne = 160.6 MeV.
In this representation a nucleus A(Z, N + AN) with even Z, N = Z and
AN is an even number of excess neutrons, has the same core as the nucleus
Ai(Zi, iVi + AN + 1) with Z 1 = Z + 1, N 1 = Z x . Then A x = A + 3. In case of
the odd Z\ one excess neutron is glued to the single pn-pair, which is bound
with the three nearest peripheral clusters [4, 5]. The long range Coulomb
energy of the single pn-pair interaction with the core is compensated with its
contribution to the surface tension energy ( see section 2). So for the odd-odd
nuclei the binding energy is calculated as follows
Ex = E core + E Nlpa - E NpaNcorea , (5)
where E Nlpa is the experimental binding energy of the nucleus 7 Li (N lpa = 1.5)
Er u = 39.2 MeV, n B (N lpa = 2.5) Eu B = 76.2 MeV, 15 N (N lpa = 3.5) £is N =
115.5 MeV, 19 F (N lpa = 4.5) Ei9 F = 147.8 MeV and 23 Na (N lpa = 5.5) £ 23Na =
186.6 MeV.
The binding energy of a core occupying the volume V core = 4/37ir^N corea can
be expressed by a formula with using the specific density of binding energy p
[MeV/fm 3 ]
E C ore VcoreP' (6)
The binding energy of N corea a-clusters Ejsr corea is easily calculated in the
framework of the a-cluster model (see section 2), where the energy of short
range nuclear force E nuc , the energy of surface tension E ST and the Coulomb
energy E c are calculated on the number of a-clusters. Then
E AN = E core - E Ncorea . (7)
The binding energy of excess neutrons E AN depends only on the number of
the excess nn - pairs N nn [5] (see also section 2), which means that only some
particular number of AiV = 2N nn can have place in the core. This allows one
to find the correspondence between AiV and N pa . It brings a result that the
specific density of core biding energy is a constant value for all nuclei of the
/5-stability valley and its vicinity.
3
2 Binding Energy Of Core a - Clusters
The a-cluster model [3,4,5] has been developed on the basis of the fact that
the radii of the most abundant isotopes are determined by the number N a [8]
and that the binding energies of symmetrical even Z nuclei are calculated by
the formula [9]
E = N a e a + 3(N a - 2)e aa , (8)
where e a = 28.296 MeV is the experimental energy of the nucleus 4 He. The
value 3(N a — 2) is considered as the number of short range bonds between
nearby alpha-clusters and e aa = 2.425 MeV. In case of an odd Z 1 = Z + 1
symmetrical nucleus the single pn-pair is glued to the three nearby peripheral
clusters with 6 bonds with their six pn-pairs. Thus, for the symmetrical odd
nuclei [9] the binding energy E\ is calculated as follows
E\ = E + e pn + Qe pnpn , (9)
where e pn = 1.659 MeV and e pnpn =2.037 MeV. What was remarkable in [9]
that in (8) and (9) the energy portions e Q , e aa , e pn and e pnpn were obtained from
analysis of the lightest nuclei with Z < 6. The binding energy of nuclear force
of an a-cluster e™ uc = 29.060 MeV was found from the relation e a = e™ uc — e£
where e% = AE np = 0.764MeV , AE np is the difference between the binding
energies of the last neutron and the last proton in the nucleus 4 He. The energy
of nuclear force interaction e™ c = 4.350 MeV and the Coulomb energy e^ a =
1.925 MeV in a short range bond between two nearby a-clusters were obtained
from the analysis of the experimental binding energies of the lightest nuclei
using the relation e aa = e™ c - e% a [5] .
The Eqs. (8) and (9) mean that the long range Coulomb interactions between
alpha-clusters in the nuclei with N a > 5 (for the nuclei with N a < 4 there are
only short range interactions) must be compensated by the surface tension
energy. From this assumption some formulas to calculate the radius of the
last a-cluster position in the nucleus (3), the Coulomb radius of the nucleus
R c = 1.869iV 1 / 3 and the formula to calculate the Coulomb energy of the
charge sphere of the radius R c were obtained [5].
E c = 1.849(7\y 5/3 ; E? = 1.849(7V Q + 0.5) 5/3 . (10)
The surface tension energy E ST is the sum of the square radii of the N a — 4
alpha-clusters' positions ( Eq. (9) in [5]).
4
A phenomenological formula to calculate the binding energy of the excess
neutron pairs E A ^ was found from fitting the experimental separation energies
of 27 nn-pairs, see Eq. (13) in Ref. [5].
Thus, the formula to calculate binding energy is [5]
E th = E nuc + E ST -E c + E AN . (11)
The Eq. (11), obtained from analysis of a reduced amount of nuclei (the sym-
metrical nuclei with Z < 22 for which the experimental values of AE np are
known), turned out to be good for all /3-stable nuclei. The accuracy is a few
MeV, which is the same as that of Weizsacker formula, although the ways of
calculations are different.
One of the important conclusions of the model is that the energy of excess
neutrons E A ^ depends only on the number of excess neutron pairs N nn =
AN/2. To make the calculations easy we propose a simple approximation to
the phenomenological formula (13) in Ref. [5]
E AN = (21.93 - 0.7Q2N*l 3 )N nn . (12)
The values of E AN are given in Fig. 1 in comparison with the values calculated
by the Eq. (13) of Ref. [5]. In the figure some empirical values obtained from
the Eq. E exp - (E nuc - E c + E ST ) where the values (E nuc - E c + E ST ) are
calculated in the framework of the model are given too. One can see from the
figure that the empirical values of E AN depend only on the number of N nn
disregarding to Z.
The binding energy E^ corea of core alpha -clusters N corea must include the
energy of nuclear force of the core a-clusters , the Coulomb energy
EN corea an d surface tension energy E ST . The energy of nuclear force of the core
a-clusters includes N corea e nuc +3(N corea — 2)e"^ c . Also the energy of the number
of bonds A between the core alpha-clusters and the peripheral compound
cluster consisting of N pa alpha-clusters must be taken into account
A = 3(N a - 2) - 3(iV corea - 2) - 3(N pa - 2). (13)
One can see that for N pa > 2 A bonds = 6. So the nuclear force energy Ejl^
is calculated as follows
E N c L a = N CO r ea e nuc + 3(N corea - 2)eZ° + Ae™. (14)
The Coulomb energy of core is equal to the Coulomb energy of N corea a-
clusters, which is calculated as 1.849A^/ r 3 ect due to (10) plus the Coulomb
5
Fig. 1. The values E exp - (E nuc - E c + E ST ) for all /3-stable even-even nuclei with Z
= 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 (41 square points), where E nuc - E c + E ST
is calculated on the a-cluster model [5]. The solid line indicates the values E^n
calculated on the Eq. (13) in [5]. The dashed line denotes the values Ean (12) of
this paper.
energy of the A bonds (13)
E c Ncorea = lMWl£ ea + Ae c aa . (15)
Using the relation e aa = e"^ c — e^ a the formula to calculate the binding energy
of the N corea alpha - clusters is as follows
E Ncorea = N corea eT + 3(iV coreQ - 2)eZ° ~ l^9N 5 J ea + Ae aa + E ST '.(16)
We use here the simple function proposed in [5], which provides a good approx-
imation to the phenomenological formula of calculation of the surface tension
energy E ST
E ST = (N a + l.7)(N a - 4) 2/3 ; Ef T = E ST + 1.1 (iV Q - 4) 2/3 . (17)
The value of l.l(A^ a — 4) 2 / 3 is the contribution of the single pn-pair into the
surface tension energy. But the energy of the Coulomb interaction of the pn-
pair with the a-clusters of the core 2N corea e 2 / R p i almost compensates it. For
N a < 59 the difference is within few MeV. Therefore to calculate the binding
energy of the odd-odd nuclei in Eq. (5) the Coulomb energy E^ paNcorea is used.
To fit experimental data for heavy nuclei one has to use a representation
that the peripheral compound cluster consists of two compound clusters of
smaller size. In the formulas the values of A is changed. For example, for two
peripheral compound clusters N pa \ = 2 and N pa2 = 3 with the total amount
of alpha-clusters in them N pa = 5 the total number of bonds is 1 + 3 = 4. In
6
(13) the number 3 (5 - 2) = 9 is changed for 4, which leads toA = 6 + 5=ll.
In (4) E Npa = E Napl + E Nap2 where E Napl = Es Be and E Nap2 = Ei* c -
In calculation of the energy E% N one has to take into account the long
range interactions between N ap i and N ap2 clusters. The algorithm looks simple
if one uses a two dimensional matrix E c (N a , N pa ) for N a = 9 -=-59 and N ap =
1 8, which is
E c (N a , N pa ) = 2N pa (2N a - 2N pa )e 2 /(2.168((7V Q - 4)^). (i 8 )
Then the Coulomb energy for the case is easily expressed as
E N vaNcorea = E c (N a , 2) + E c (N a - 2, 3). (19)
So, the binding energy of the nucleus A(Z, N + AN) is calculated by the
following way. For AiV the value N pa is calculated from Eq. (7) and Eq. (12)
taking into account that the number of nn-pairs in core N nn is integer. Then
the binding energy is calculated by (4) and (5). The radii are calculated by
(1) and (2). In this approach the expected accuracy is a few MeV, because
the single particle effects (shell effects) are not taken into account. The value
p = 2.57MeV/fm 3 fits the binding energies of all /3-stable isotopes and not
stable isotopes of the vicinity. In Table 1 the results of calculation by (4) and
(5) are given for some nuclei. The values of E th (11) are also presented.
If the value of the specific density of binding energy is known, it gives an op-
portunity to estimate the size of the isotopes from their experimental binding
energies. Then the value N pa is calculated by (4) where instead of E the value
E exp is used. For example, the nucleus 142 Nd 60 has E exp = 1185 MeV. The
value = 2-^5 are tried and only Npa = 4 corresponds to the p = 2.57
MeV/fm 3 . So the radius (1) R = 5.02 fm and for 145 Pm 61 (2) R = 5.05 fm.
Table 1. Binding energies and radii calculated for /3-stable isotopes. For
stable nuclei the most abundant isotopes have been selected. Also the results
are presented for the corresponding nucleus having the same core (if it is a (5
- unstable nucleus, it is marked by *).
7
z
AN
A
iV
pet
E exv 10
£(4,5)
E th (U)
^[11,12,13]
i?(l,2)
Zx
AN+1
Ai
MeV
MEV
MeV
fm
fm
10
20
5
161
161
158
2.992(8)
2.92
11
1
23
5.5
187
187
185
2.94(6)
3.02
12
24
4
198
199
196
3.075(15)
3.04
13
1
27
4.5
225
219
223
3.06(9)
3.13
14
28
4
237
238
234
3.14(4)
3.18
15
1
31
4.5
263
258
260
3.19(3)
3.26
16
32
4
272
277
271
3.240(11)
3.31
17
1
35
4.5
298
297
297
3.388(17)
3.39
18
36
5
307
306
307
3.327(15)
3.43
19
1
39
5.5
334
332
333
3.408(27)
3.53
18
4
40
2
344
343
349
3.393(15)
3.38
19
5
43*
2.5
369
363
369
3.45
20
40
5
342
345
343
3.482(25)
3.57
21
1
43*
5.5
367
371
369
3.63
20
2
42
3
362
370
364
3.505
3.52
21
3
45
3.5
388
394
390
3.550(5)
3.59
24
4
52
3
456
453
452
3.645(5)
3.73
25
5
55
3.5
482
477
477
3.680(11)
3.79
28
2
58
5
506
502
502
3.760(10)
3.96
29
3
61*
5.5
532
528
529
4.01
28
4
60
4
527
520
524
3.812(30)
3.94
29
5
63
4.5
551
540
548
3.888(5)
3.99
30
4
64
4
559
560
557
o m 0/1 1 \
3.918(11)
A no
4.02
31
5
67*
4.5
583
580
583
4.07
30
6
66
3
578
578
577
3.977(20)
4.00
31
7
69
3.5
602
602
602
4.05
8
Table 1. Continued.
z
AN
A
£(4,5)
E th (U)
i2exp[ll,12,13]
R(l,2)
Z\
A7V+1
A
M
MeV
MEV
MeV
fm
fm
40
10
90
3
784
788
778
4.28(2)
4.39
41
11
93
3.5
806
810
802
4.317(8)
4.43
50
18
118
3
1005
999
1002
4.72
51
19
121
3.5
1026
1022
1024
4.63(9)
4.76
50
20
120
2
1021
1026
1018
4.630(7)
4.71
51
21
123
2.5
1042
1046
1040
4.74
60
22
142
4
1185
1181
1177
4.993(35)
5.02
61
23
145
4.5
1204
1201
1198
5.05
70
32
172
4
1393
1390
1391
5.28
71
33
175
4.5
1412
1410
1410
5.378(30)
5.31
80
42
202
2+2
1595
1580
1587
5.499(17)
5.51
81
43
205
2+2.5
1616
1615
1605
5.484(6)
5.52
90
50
230
2+3
1755
1752
1756
5.74
91
51
233*
2+3.5
1772
1776
1773
5.76
100
52
252
3+4
1879
1881
1881
5.95
101
53
255*
3+4.5
1896
1901
1900
5.98
110
60
281
2+3+3
2031
2030
2037
6.15
111
61
283*
2+3+3.5
2047
2045
6.17
116
72
304
2+3+3
2145
2147
6.26
117
73
307(*?)
2+3+3.5
2168
2157
6.28
3 Conclusion
The alpha - cluster model [3,4,5] has been developed to find some formulas for
calculation of the binding energies of f3 - stable nuclei with using the notion of
core. This brings to a discovery that the specific density of the binding energy
of core for the nuclei of /3-stable valley and those which are in its vicinity
can be a constant value equal to 2.57 MeV/fm 3 . The idea that the number of
9
excess neutrons is determined by the amount of a-clusters of the core, which
has been approved before in terms of charge radii of nuclei [3], is approved
now in terms of the binding energies.
In the formulas (1) and (2) to calculate radii the radius of an a-cluster of the
core r a = 1.60 fm is the only fitting parameter to describe the experimental
radii. So is the value p = 2.57 MeV/fm 3 to describe the experimental binding
energies of all stable isotopes having core by the formulas (4) and (5). Thus,
it is clearly seen here that the radius of one a-cluster, as well as the specific
density of binding energy of core, are the constant values. Then one has an
opportunity to estimate the size of a nucleus from its experimental binding
energy.
In the heavy nuclei the growing Coulomb energy pushes out small compound
clusters from the core to the periphery of the nucleus. If for the stable nuclei
the total number of peripheral a-clusters is within 2 4- 5, for the nuclei with
Z > 80 the number is within 5 4 8.
The work is supported by international grant STCU 3081.
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10