Universal Journal of Physics and Application 1(1): 18-25, 2013
DOI: 10.13189/ujpa.2013.010104
http://www.hrpub.org
Stability and Mass Parabola in Integrated Nuclear Model
N.Ghahramany*, H.Sarafraza, E. Yazdankish
Physics Department, Shiraz University, Shiraz, Iran
*Corresponding Author: [email protected]
Copyright © 2013 Horizon Research Publishing All rights reserved.
Abstract
Following our previous introduction of
nuclear quark-like model, in this paper, a more precise
formula is presented for nuclear binding energy in the
context of modified Integrated Nuclear Model (INM). INM
is based upon quark-like model with three basic assumptions
from which the nuclear binding energy and magic numbers
are easily obtained. INM is modified here to give the nuclear
mass parabola and most stable nuclei in each nuclei group.
Our findings are compared with Liquid Drop Model (LDM),
unmodified Integrated Nuclear Model (INM) and
experimental data for most stable nuclei.
Keywords
Stability, Mass Parabola, Integrated Nuclear
Model, Binding Energy
1. Introduction
Nuclear physicists have been struggling hard for so many
years to present a simple and complete nuclear model from
which the characteristics of nuclei can be explained and
comprehended .The existing successful nuclear models can
only explain certain characteristics of the nuclei and make no
comments about other nuclear properties. For example
Liquid Drop Model (LDM) was presented by Von
Weizsacker [1] and then was extended by Bohr and Wheeler
[2].This model give the nuclear binding energy formula in
terms of A and Z [3] as follow:
𝐵𝐵(𝐴𝐴, 𝑍𝑍) = 𝑎𝑎𝑣𝑣 𝐴𝐴 − 𝑎𝑎𝑠𝑠 𝐴𝐴2⁄3 − 𝑎𝑎𝑐𝑐 𝑍𝑍(𝑍𝑍 − 1)𝐴𝐴−1⁄3 − 𝑎𝑎𝑎𝑎 (𝑁𝑁 −
𝑍𝑍2𝐴𝐴−1±𝛿𝛿+𝜂𝜂
(1)
In which five experimentally determined constants are
introduced.
Liquid Drop Model (LDM) has been successful in the
calculation of binding energy, mass parabola and most stable
isobars. However this model fails to predict other properties
of nuclei such as, the magic numbers and nuclear magnetic
moments. On the other hand Nuclear Shell Model [4] which
is based upon Schrodinger equation solution with selected
potential such as rounded edge potential well , predict the
magic numbers and nuclear magnetic moments by using
spin-orbit couplings in a relatively complicated
manners[5,6].
We have presented the Integrated Nuclear Model (INM)
based upon nuclear quark-like model from which all magic
numbers are easily obtained and a new magic number is
predicted [7,8]. This model is also used to find the magnetic
dipole moment of deuteron with greater precision [9]. Using
INM, the nuclear binding energy formula was also obtained
from quark-like model of nuclei [10] and is given as follow:
𝐵𝐵(𝐴𝐴, 𝑍𝑍) = �𝐴𝐴 − �
�𝑁𝑁 2 −𝑍𝑍 2 �+𝛿𝛿 (𝑁𝑁−𝑍𝑍)
3𝑍𝑍
+ 3�� ×
0, 𝑁𝑁 ≠ 𝑍𝑍
In formula (2) 𝛿𝛿(𝑁𝑁 − 𝑍𝑍) = �
1, 𝑁𝑁 = 𝑍𝑍
𝑀𝑀𝑁𝑁 𝑐𝑐 2
𝛼𝛼
𝐴𝐴 > 5
(2)
and 𝑀𝑀𝑁𝑁 𝑐𝑐 2 is the mass of nucleon instead of up-quark mass
used in reference [10]. The coefficient 𝛼𝛼 is a
dimensionless constant defined to have a range from 90 to
100 [10].
Formula (2) provides the nuclear binding energy for most of
the stable nuclei in term of only one coefficient namely 𝛼𝛼
which is simpler than LDM with several coefficients.
However this formula needs to be modified to give us the
mass parabola in the same way as LDM.
In this paper the modified nuclear binding energy
formula is presented in order to find the nuclear mass
parabola and the stability of isobaric groups of nuclides.
The determination of the mass parabola itself is an
indication of the validity of the INM which is based upon
the quark structure of the nuclei instead of nucleon
structure.
2. Modified Nuclear Binding Energy
Formula in INM
In the binding energy formula (2) presented in INM a
coefficient 𝛼𝛼 is introduced which varies between 90 to 100
for all stable nuclides with 𝐴𝐴 > 5. The coefficient 𝛼𝛼 may
be called “nuclear stability coefficient”.
A careful investigation of the stability coefficient for
many stable nuclides indicates the fact that 𝛼𝛼 depends
upon atomic number (Z) and mass number (A). In fact for
isobar nuclides, the stability coefficient 𝛼𝛼 is proportional
to atomic number (Z) whereas for isotopic nuclides the
stability coefficient 𝛼𝛼 is inversely proportional to the mass
number (A). Therefore
Universal Journal of Physics and Application 1(1): 18-25, 2013
𝑍𝑍
𝐴𝐴
Further analysis of most of the stable nuclides allows proper
modification of formula (2) as follow:
𝛼𝛼 ∝
𝐵𝐵(𝐴𝐴, 𝑍𝑍) =
�𝐴𝐴 − �
�𝑁𝑁 2 −𝑍𝑍 2 �+𝛿𝛿(𝑁𝑁−𝑍𝑍)
3(𝑍𝑍−𝑘𝑘)
+ 3�� ×
𝐴𝐴𝑛𝑛 +𝑠𝑠 𝑀𝑀𝑁𝑁 𝑐𝑐 2
𝐴𝐴 > 5
126 (𝑍𝑍−𝑘𝑘)
(3)
In which the coefficient k, s and n are defined as
0.0003,
𝑠𝑠 = �
−0.0003,
𝑁𝑁, 𝑍𝑍 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒
𝑘𝑘
𝑁𝑁, 𝑍𝑍 𝑜𝑜𝑜𝑜𝑜𝑜
2,
𝑍𝑍 ≤ 118
=�
0,
𝑍𝑍 > 118
𝑛𝑛 = 0.87 𝑡𝑡𝑡𝑡 0.88
As can be seen, only one free coefficient namely, n exist
that has a limited range of fine tuning and all other
coefficients are known constants.
Now with this modified formula (3), one can find the mass
parabolas and stability line.
First we write the nuclear mass as follow:
𝑀𝑀(𝐴𝐴, 𝑍𝑍) = 𝐴𝐴𝑀𝑀𝑁𝑁 + (𝑀𝑀𝐻𝐻 − 𝑀𝑀𝑁𝑁 )𝑍𝑍
− �𝐴𝐴 − �
(𝑁𝑁 2 − 𝑍𝑍 2 ) + 𝛿𝛿(𝑁𝑁 − 𝑍𝑍)
𝐴𝐴𝑛𝑛+𝑠𝑠 𝑀𝑀𝑁𝑁
+ 3�� ×
126(𝑍𝑍 − 𝑘𝑘)
3(𝑍𝑍 − 𝑘𝑘)
5
= 𝐴𝐴𝑀𝑀𝑁𝑁 + (𝑀𝑀𝐻𝐻 − 𝑀𝑀𝑁𝑁 )𝑍𝑍 − � 𝐴𝐴 −
3
𝐴𝐴2 +𝛿𝛿(𝑁𝑁−𝑍𝑍)
3(𝑍𝑍−𝑘𝑘)
− 3� ×
𝐴𝐴𝑛𝑛 +𝑠𝑠 𝑀𝑀𝑁𝑁
126 (𝑍𝑍−𝑘𝑘)
(4)
Differentiating 𝑀𝑀(𝐴𝐴, 𝑍𝑍) with respect to 𝑍𝑍 and equating it to
zero give us 𝑍𝑍𝐴𝐴 at which 𝑀𝑀(𝐴𝐴, 𝑍𝑍) is minimum. After
carrying out the standard calculation, we end up with an
equation of power of three in Z such as:
𝑎𝑎𝑍𝑍 3 + 𝑏𝑏𝑏𝑏 − 𝑐𝑐 = 0
(5)
19
Where
5
𝐴𝐴𝑛𝑛+𝑠𝑠 𝑀𝑀𝑁𝑁
�;
𝑎𝑎 = (𝑀𝑀𝐻𝐻 − 𝑀𝑀𝑁𝑁 ); 𝑏𝑏 = �� 𝐴𝐴 − 3�
126
3
𝑐𝑐 = ��
𝐴𝐴2 + 𝛿𝛿(𝑛𝑛 − 𝑧𝑧) 𝑛𝑛+𝑠𝑠
� 𝐴𝐴 𝑀𝑀𝑁𝑁 �
189
Equation (5) is solved with Maple program and we get one
real solution:
𝑍𝑍𝐴𝐴 = 𝑘𝑘 +
−
1
6
4𝑏𝑏 3 + 27𝑐𝑐 2 𝑎𝑎 2
��108𝑐𝑐 + 12√3�
� 𝑎𝑎 �
𝑎𝑎
2𝑏𝑏
4𝑏𝑏 3 +27𝑐𝑐 2 𝑎𝑎 2
�𝑎𝑎 �
��108 𝑐𝑐+12√3 �
𝑎𝑎
1 ⁄3
𝑎𝑎
1 ⁄3
(6)
Fig.(1) shows the mass parabolas for odd A (𝐴𝐴 = 135) and
for even A (𝐴𝐴 = 102) in which 𝑍𝑍𝐴𝐴 is shown by down ward
arrows. For odd A with 𝑠𝑠 = 0 only one parabola is
obtained; for even A isobars we get two parabolas.
As shown in fig.(1) for 𝐴𝐴 = 135 only one stable isobar
exist at 𝑧𝑧 = 56 which is very close to our finding at
𝑍𝑍𝐴𝐴 = 55.8 but for even A nuclides depending upon the
distance between two obtained parabolas, we get several
stable even-even isobars. Here for 𝐴𝐴 = 102 there are two
stable isobars at 𝑍𝑍 = 44 and 𝑍𝑍 = 46 and for 𝑍𝑍 = 44 we
get the most stable one, since this is closer to the calculated
value namely, 𝑍𝑍𝐴𝐴 = 44.3 . For LDM for 𝐴𝐴 = 135 ,
𝑍𝑍𝐴𝐴 = 55.7 and for 𝐴𝐴 = 102, 𝑍𝑍𝐴𝐴 = 44.7 [11]. It is seen
that our calculated values of 𝑍𝑍𝐴𝐴 is closer to the
experimental data as compared to the LDM.
Figure 1. The isobars mass parabolas a) for even A nuclides b) for odd A nuclides
Using equation (6) , we plot 𝑍𝑍𝐴𝐴 verses N to obtain the stability line shown in figure (2).
20
Stability and Mass Parabola in Integrated Nuclear Model
Figure 2. Stability line in modified INM
Finally in table (1) a comparison is made between our findings of nuclear binding energy and the results of other models
and experimental data.
3. Conclusion
Determination of the stability mass parabola and the nuclear stability line in modified INM and comparison of the most
stable nuclei obtained from this model, with the experimental data and LDM is an indication of the validity of the modified
INM which is based upon quark-like model of nuclei. Also comparison of nuclear binding energy obtained from modified
INM and experimental data, results in a closer match than LDM for most of the light, medium and heavy nuclei. Other
characteristics of nuclides are being tested by INM in our research group and the results are promising.
Universal Journal of Physics and Application 1(1): 18-25, 2013
21
Table 1. Comparison of nuclear binding energy with different model
Z
Nucleus
A
B(EXP)
MeV
B(LDM)
MeV
1
2
3
3
4
5
5
6
6
7
7
8
8
8
9
10
10
10
11
12
12
12
13
14
14
14
15
16
16
16
17
17
18
18
19
19
20
20
21
22
22
22
22
22
23
H
He
Li
Li
Be
B
B
C
C
N
N
O
O
O
F
Ne
Ne
Ne
Na
Mg
Mg
Mg
Al
Si
Si
Si
P
S
S
S
Cl
Cl
Ar
Ar
K
K
Ca
Ca
Sc
Ti
Ti
Ti
Ti
Ti
V
1
4
6
7
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
37
36
40
39
41
40
44
45
46
47
48
49
50
50
0
28.296
31.994
39.244
58.165
64.751
76.205
92.162
97.108
104.659
115.492
127.619
131.763
139.807
147.801
160.645
167.406
177.77
186.564
198.257
205.588
216.681
224.952
236.537
245.011
255.62
262.917
271.781
280.442
291.839
298.21
317.101
306.717
343.811
333.724
351.619
342.052
380.96
387.848
398.193
407.073
418.7
426.842
437.781
434.794
-26.461
21.9452
27.64
38.3835
56.6316
63.0939
75.0627
87.749
93.629
99.6605
112.2803
123.7138
130.9744
141.24997
149.6775
160.15493
168.363
179.44476
188.0092
196.68558
205.5993
217.2668
224.1192
233.089
242.5576
254.6751
260.9052
269.23215
279.1541
291.6321
297.29795
317.675
305.02832
346.7388
333.24958
354.4661
340.41858
383.66084
390.6604
399.52474
408.53494
419.9271
427.26614
437.02429
434.6673
Table (contd.)
B(INM)
MeV
B(Modified INM)
MeV
0
28.27305
30.10864
33.58272
54.71667
65.03467
75.73481
92.19658
96.69506
104.82993 115.06522
127.94649 131.23772
139.15385
147.91676
160.75488
167.29278
178.76377
186.66229
198.70651
206.02692
217.4098
225.38779
236.6431
244.7457
255.17021
261.40635
271.82458
280.5625
292.09649
299.71709
317.5582
309.3663
343.29829
334.60819
350.20724 346.90367
379.8567
389.4375
398.22944
409.04261 420.01354
426.51645
437.60839 435.44378
0
28.27305
31.7322
38.9815
59.3114
65.2428
77.4225
91.8896
96.9228
104.2934
113.8012
127.5033
130.1135
138.5735
145.9720
158.8530
167.4783
177.7011
186.7211
197.1147
206.2926
214.5963
222.5858
237.1105
246.7711
255.7162
264.1575
272.0444
280.8788
288.9729
297.3346
313.3686
304.5650
343.6705
335.6762
352.6519
342.5981
382.8547
391.2262
398.8393
407.5979
419.2554
427.4059
436.8867
436.8867
22
Stability and Mass Parabola in Integrated Nuclear Model
Z
Nucleus
A
23
24
24
25
26
26
27
28
28
29
29
30
30
30
31
31
32
32
32
33
34
34
35
35
36
36
36
36
37
37
38
38
38
39
40
40
40
40
41
V
Cr
Cr
Mn
Fe
Fe
Co
Ni
Ni
Cu
Cu
Zn
Zn
Zn
Ga
Ga
Ge
Ge
Ge
As
Se
Se
Br
Br
Kr
Kr
Kr
Kr
Rb
Rb
Sr
Sr
Sr
Y
Zr
Zr
Zr
Zr
Nb
51
52
53
55
54
56
59
58
60
63
65
64
66
68
69
71
70
72
74
75
78
80
79
81
82
83
84
86
85
87
86
87
88
89
90
91
92
94
93
Table (contd.)
B(EXP)
MeV
B(LDM)
MeV
B(INM)
MeV
B(Modified INM)
MeV
445.845
456.349
464.289
482.075
471.763
492.258
517.313
506.459
526.846
551.385
569.212 559.098
578.136
595.387
601.996
618.951
610.521 628.686
645.665
652.564
679.99 696.866
686.321
704.37
714.274
721.737
732.258
749.235
739.283
757.856
748.928
757.356
768.469
775.538
783.893
791.087
799.722
814.677
805.765
445.416
455.5536
463.39155
481.1957
468.80385
490.5518
516.2997
502.61932
524.93075
550.7451
569.4077
558.6975
576.4769
596.72888
603.8851
620.99784
612.65537
631.17806
647.61204
655.4886
682.06496
697.35903
689.2439
705.81675
715.7677
722.76424
731.78274
746.12903
739.5233
755.05569
748.7453
756.09715
765.42723
772.4887
781.019
788.70082
798.32021
814.04797
804.7359
446.8546
455.28655
462.20892
482.30496
470.09324 494.95007
516.9666
508.02694
529.02083
550.87284
572.5846
562.37022 578.59789
594.627
600.3837
616.54873
611.89844
628.65957
645.29982
650.43133
679.32321
696.67873
683.54105
701.12258
712.55477
721.61748
730.68841
748.85597
742.09493
760.81913
745.0301
754.44401
772.1865
774.67303
784.95789
794.88848
796.30213
815.99203
806.51142
445.7475
455.6228
464.0847
482.4365
471.9696
492.0997
517.1329
506.0674
526.6298
551.6894
568.8508
558.7607
577.7257
595.8342
602.4001
618.9171
611.0470
628.2117
644.4527
652.5537
679.6684
695.4010
687.4491
703.7878
714.8280
723.0077
730.9761
747.9164
739.0302
756.5981
748.3195
756.6489
768.2167
776.0138
783.4294
791.9112
800.2144
814.4118
805.9039
Universal Journal of Physics and Application 1(1): 18-25, 2013
23
Z
Nucleus
A
B(EXP)
MeV
B(LDM)
MeV
B(INM)
MeV
B(Modified INM)
MeV
42
42
42
42
44
44
44
44
44
45
46
46
46
46
46
47
47
48
48
48
48
48
49
49
50
50
50
51
51
52
52
52
53
54
54
54
55
56
56
Mo
Mo
Mo
Mo
Ru
Ru
Ru
Ru
Ru
Rh
Pd
Pd
Pd
Pd
Pd
Ag
Ag
Cd
Cd
Cd
Cd
Cd
In
In
Sn
Sn
Sn
Sb
Sb
Te
Te
Te
I
Xe
Xe
Xe
Cs
Ba
Ba
92
95
96
98
99
100
101
102
104
103
104
105
106
108
110
107
109
110
111
112
113
114
113
115
116
118
120
121
123
126
128
130
127
129
131
132
133
137
138
796.508
821.625
830.779
846.243
852.255
861.928
868.73
874.844
893.083
884.163
892.82
899.914
909.474
925.239
940.207
915.263
931.727
940.646
947.622
957.016
963.556
972.599
963.094
979.404
988.684
1004.955
1020.546
1026.325
1042.097
1066.369
1081.439
1095.941
1072.577
1087.651
1103.512
1112.448
1118.528
1149.681
1158.293
793.09932
820.5965
830.48545
846.8345
851.80259
861.94355
869.4767
878.87415
894.40381
885.0934
892.71256
900.53543
910.189
926.29915
941.11778
915.80492
932.86781
940.7986
948.22807
957.45559
964.25463
972.84985
963.41231
979.7077
987.89353
1003.8319
1018.5973
1025.6927
1040.1801
1063.5988
1076.7653
1088.937
1070.8771
1085.9613
1100.1915
1107.9009
1115.3434
1144.0675
1151.5421
795.94785
817.88079
827.97895
848.34648
857.39063
858.94508
869.40782
870.67943
891.54317
880.17114
889.22905
899.97871
910.81159
922.81499
944.64934
919.77497
932.11781
940.98553
952.19342
953.35906
964.54063
975.82258
962.12798
984.90679
983.20639
1006.25447
1018.68817
1027.54763
1040.04681
1061.3598
1085.47929
1097.80741
1069.98274
1090.45272
1103.42321
1115.85749
1124.19407
1145.90412
1158.66304
795.4930
821.6087
829.9908
846.2422
852.8166
861.2846
869.5955
875.7178
893.5445
885.1863
892.3174
900.7118
908.9562
924.9800
940.3585
916.1749
932.5995
939.9041
948.0861
956.1212
964.0060
971.7369
963.5100
979.5262
986.9973
1005.2137
1020.0715
1027.0924
1043.4808
1066.7824
1082.4686
1097.6144
1073.9530
1089.1120
1105.2131
1113.0778
1117.6565
1148.4568
1158.9851
Table (contd.)
24
Stability and Mass Parabola in Integrated Nuclear Model
Z
Nucleus
A
B(EXP)
MeV
B(LDM)
MeV
B(INM)
MeV
B(Modified INM)
MeV
57
57
58
58
59
60
60
60
60
62
62
63
63
64
64
64
65
66
66
66
67
68
68
68
69
70
70
70
71
71
72
72
72
73
74
74
74
75
75
La
La
Ce
Ce
Pr
Nd
Nd
Nd
Nd
Sm
Sm
Eu
Eu
Gd
Gd
Gd
Tb
Dy
Dy
Dy
Ho
Er
Er
Er
Tm
Yb
Yb
Yb
Lu
Lu
Hf
Hf
Hf
Ta
W
W
W
Re
Re
138
139
140
142
141
142
143
144
146
152
154
151
153
156
158
160
159
162
163
164
165
166
167
168
169
172
173
174
175
176
177
178
180
181
182
183
184
185
187
1155.774
1164.551
1172.692
1185.29
1177.919
1185.142
1191.266
1199.083
1212.403
1253.104
1266.94
1244.141
1258.998
1281.598
1295.896
1309.29
1302.027
1324.106
1330.377
1338.035
1344.256
1351.572
1358.008
1365.779
1371.352
1392.764
1399.131
1406.595
1412.106
1418.394
1425.185
1432.811
1446.297
1452.24
1459.335
1465.525
1472.937
1478.341
1491.877
1151.2653
1159.105
1167.8856
1181.675
1174.1085
1181.5574
1188.3505
1196.7746
1211.0565
1253.5745
1266.5934
1245.4718
1260.0164
1281.9731
1295.468
1308.1556
1301.9809
1323.6067
1329.4064
1336.7617
1343.3234
1351.0262
1357.0604
1364.6314
1370.6194
1391.7813
1397.6674
1405.069
1411.0893
1416.9087
1424.3349
1431.9414
1444.9382
1450.9735
1458.1089
1464.0784
1471.526
1477.0087
1490.2904
1153.85554
1166.73445
1174.39252
1187.94784
1181.66244
1188.56759
1189.38183
1202.61474
1216.51726
1258.49943
1272.46549
1250.92515
1252.0625
1286.04737
1300.35239
1314.5448
1306.82008
1327.56172
1327.81106
1342.0879
1348.27368
1354.12255
1354.45782
1368.95728
1374.66657
1395.18194
1395.41458
1410.24421
1415.66989
1415.85338
1421.10223
1436.13156
1451.43158
1456.56802
1461.40522
1461.62301
1476.98029
1481.67469
1497.36935
1155.7274
1136.5646
1170.7473
1186.3789
1177.7014
1184.4452
1192.5988
1200.6467
1213.3904
1252.2646
1267.1604
1243.8002
1259.4052
1281.7931
1296.8405
1308.1365
1300.7987
1322.9576
1330.3749
1337.6889
1345.0398
1352.1740
1359.0660
1367.0472
1370.7159
1392.6360
1399.9972
1407.2110
1410.8362
1418.0879
1425.2612
1431.4239
1446.5547
1453.2859
1459.8504
1464.6875
1471.2362
1477.8280
1492.8904
Table (contd.)
Universal Journal of Physics and Application 1(1): 18-25, 2013
25
Z
Nucleus
A
B(EXP)
MeV
B(LDM)
MeV
B(INM)
MeV
B(Modified INM)
MeV
76
76
76
77
77
78
78
78
79
80
80
80
80
80
81
81
82
82
82
83
90
92
92
Os
Os
Os
Ir
Ir
Pt
Pt
Pt
Au
Hg
Hg
Hg
Hg
Hg
Tl
Tl
Pb
Pb
Pb
Bi
Th
U
U
189
190
192
191
193
194
195
196
197
198
199
200
201
202
203
205
206
207
208
209
232
235
238
1505.007
1512.799
1526.116
1518.088
1532.058
1539.577
1545.682
1553.604
1559.386
1566.489
1573.153
1581.181
1587.411
1595.165
1600.87
1615.072
1622.325
1629.063
1636.431
1640.23
1766.687
1783.864
1801.69
1503.2513
1510.5523
1523.036
1516.0377
1529.0561
1536.145
1541.8699
1549.0352
1554.5095
1561.0472
1566.9765
1574.3316
1579.9487
1586.9882
1592.4387
1604.989
1611.9829
1617.5005
1624.4233
1629.8385
1769.1834
1785.8988
1804.1704
1502.04693
1517.73611
1533.53906
1522.14121
1538.08144
1542.34012
1542.38144
1558.40616
1562.51729
1566.36429
1566.48393
1582.67354
1582.63325
1598.99688
1602.80964
1619.26432
1622.92633
1622.8084
1639.51406
1643.0243
1764.19153
1786.7108
1803.59006
1503.9947
1510.5094
1525.0828
1517.1743
1531.9626
1538.6738
1546.0526
1553.3518
1560.1086
1566.7061
1574.1315
1581.4803
1588.7515
1595.9441
1602.7903
1617.1500
1624.0399
1631.2074
1638.2975
1640.8417
1767.0470
1782.1712
1802.6499
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