Investigation of quadrupole deformation of nucleus and its surface
dynamic vibrations
Igor Boboshin1, Boris Ishkhanov2, Sergei Komarov1, Vadim Orlin1, Nikolai Peskov1,
Vladimir Varlamov1
1
Skobeltsyn Institute of Nuclear Physics of Moscow State University, 119992 Moscow, Russia,
2
Physics Faculty of Lomonosov Moscow State University, 119992 Moscow, Russia
The new unique and having no analogues complete database (Chart) of nuclear deformation parameters
was developed using various sources of related information. Database is realized as system analogue to wellknown Chart of nuclides (various nuclide properties sorted in N and Z coordinates). The color and intensity
solution of this Chart individual nuclide element gives to one possibility for comfortable overview of nuclei of
different (prelate and oblate) shapes.
Using this new Chart of nuclear quadrupole deformation parameters the systematical comparative
analysis of those from various sources was carried out for many nuclei. The clear systematical disagreements of
those parameters obtained from nuclear quadrupole moments (“Q-type”) data and from reduced transition
probability B(E2) for 0+  2+1 transitions (“B-type”) data were revealed. It was found out that all (with only
few exceptions) nuclides have been investigated can be clearly separated into 2 large and 1 small groups: 1) for
isotopes of Ti, Cr, Zr, Nd, Sm, Gd, Dy, Er, W, Os, Ra good agreement is observed for data of both types; 2) for
isotopes of C, Si, Ar, Ca, Fe, Ni, Zn, Ge, Se, Kr, Sr, Mo, Ru, Pd, Cd, Sn, Te, Ba, Yb, Hf, Pt, Pb “B-type” 2
values are systematically (and in many cases significantly) larger than “Q-type” ones (in cases of only few
exceptions mentioned above (Mg, Xe, Hg, U) both types of disagreement are combined – group 3).
It was shown that two types data difference could be explained in the frame of assumption about the not
negligible role of dynamical vibrations of nucleus surface, by other words assumption about deformation
parameter beta2 value dependence on surface vibration of nucleus in ground state. From this point of view “Qtype” 2 values obtained did not take into account surface vibration, but “B-type” ones did. Therefore “B-type”
data reflect not only static nuclear deformation (nucleus shape), but dynamic deformation also. It was pointed
out that comparison of static and total nuclear deformations could be a source of information on such nucleus
properties as mass and hardness parameters balance.
INTRODUCTION
Atomic nucleus matter space distribution is one of most important fundamental physical
features to which nuclear physics has a first priority interest. At last time amount of experimental
researches devoted to studying nuclei shapes is increasing permanently, new experimental methods are
developed, large volume nuclear structure parameters compilations are created. All those give to one
possibility to obtain of new data on various nuclear deformations. This work is devoted to partial one
case of very important atomic nucleus quadrupole deformations.
Among the most systematical collections of data concerned to quadrupole deformations as
more complete and carefully obtained two ones can be pointed out which give to one possibility to
obtain nuclei quadrupole deformation parameters. The first one is the experimental data on electrical
quadrupole moment Q collection [1], included more than 1300 data for about 440 nuclides (obtained in
more than 52 experiment types), continuing and adding analogous previous collections [2, 3]. The
second one [4] is collection of reduced transition probability В(Е2) values for 0+→21+ transitions
(1765 В(Е2) values from 133 publications, 281 nuclides) also continuing and adding previous
collections.
1
Both type collections of data on quadrupole deformation parameters have been used at the
Russia Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics Centre for
Photonuclear Experiments Data (CDFE) for development of new relational database - Chart of Nuclear
Quadrupole Deformations (http://cdfe.sinp.msu.ru/services/nsr/defchart/defmain.html) [5]. Such
complete database is powerful tool for system analysis of known data and for overview of unknown
agreements or, vice versa, disagreements of data obtained by various ways or for founding out new
nuclear objects and investigation their new features. Analogous complete database of nuclear reaction
cross sections (http://cdfe.sinp.msu.ru/exfor/index.php) was used at the CDFE for investigations of
consistency between total and partial reactions investigated using different methods in various
laboratories [6]. Many new (not classic traditional) magic nuclei has been found out and many their
features
investigated
using
the
complete
nuclear
structure
database
(http://cdfe.sinp.msu.ru/services/ensdfr.html) [7]. New Chart of Nuclear Quadrupole Deformations at
first time revealed the existence of clear and in many cases significant systematical disagreements
between quadrupole deformation parameters obtained using two methods described above.
TWO MAIN METHODS FOR NUCLEUS QUADRUPOLE DEFORMATION
PARAMETER OBTAINING
Nucleus quadrupole deformation parameters from its quadrupole moments.
As is well-known (for example, from [6]) intrinsic quadrupole moment of evenly charged
ellipsoid can be described by equation
Q0 = 2/5 Z (b2 – a2),
(1)
where b is large, and a is small ellipsoid axises.
Since quadrupole deformation parameter  (degree of nucleus shape difference from sphere) can
be written as
 = 0.3 (b2 – a2)/2 <r>2,
(2)
where average mean-squared charge distribution radius is equal to
<r2> = (b2 + 2a2)/5,
(3)
equation (1) would be the following
Q0 = 4/3 Z <r2> .
(4)
Hense, using experimental quadrupole moment values [1] the nucleus quadrupole deformation
parameter values  could be calculated by the formula
 = 0.75 Q0 (Z r2).
(5)
2
Moment Q0 values were recalculated [7] from static electric quadrupole moment Q values
measured in laboratory system. It was supposed than near spherical nucleus shape vibrations amplitude
is small in comparison to nucleus equilibrium deformation and therefore the equation
Q = Q0 (3K2 – I(I + 1)) ((I + 1)(2I + 3)),
(6)
where I is spin of state studied – the member of rotational band based on nucleus ground state, K is I’s
projection on symmetry axis (the joint characteristics of all band members).
Value of r2 was evaluated using the following expressions
r2 = 0.6R02(1 + 10/3(a0/R0)2)(1 + (a0/R0)2) (А  100),
r2 = 0.6(1.2 A1/3)2 (А > 100),
(7)
(8)
which takes into account effects of light nuclei surface diffusion properties.
Parameters of radial Woods-Saxon potential form-factor (R0 = 1.07A1/3 fm and a0 = 0.55 fm)
were obtained from the data on fast electrons scattering.
Nucleus quadrupole deformation parameters
from the values of reduced transition probability B(E2) for 0+  2+1 transitions .
In [4] nucleus quadrupole deformation parameter was obtained using the following equation
2 = (4/3 Z R02)[B(E2; 0+→21+)/e2]1/2,
(9)
where B(E2; 0+→21+) is reduced probability for Е2- transition from ground 0+ to first excited 2+1 state
and R02 = (1.2 A1/3 sm)2.
The values under discussion,  (5) and 2 (9), are connected as  ≈ 0.95β2 and therefore further
for simplicity the only symbol β2 will be used.
COMPARISON OF EXPERIMENTAL DATA ON NUCLEUS
QUADRUPOLE DEFORMATION PARAMETERS OBTAINED FROM
DIFFERENT SOURCES
3
Nucleus quadrupole deformation parameters, obtained using two different methods described
above (“Q-type” data from experimental quadrupole moments Q and ”B-type” data from reduced
transition probabilities В(Е2)) were compared systematically. It was found out that all nuclei
investigated are clearly separated into three groups (“1”, “2”, and “3”):
1)
for nuclei of “group 1” (ядер Ti, Cr, Zr, Nd, Sm, Gd, Dy, Er, W, Os, Ra) “Q-type” data and ”Btype” data are very close to each other (small differences are of order of statistical uncertainties);
two examples of comparison of both type data for W and Er isotopes are presented on Figs. 1
and 2 correspondingly (here and further thin line - “Q-type” data, thick line - ”B-type” data);
2)
for nuclei of “group 2” (C, Si, Ar, Ca, Fe, Ni, Zn, Ge, Se, Kr, Sr, Mo, Ru, Pd, Cd, Sn, Te, Ba,
Yb, Hf, Pt, Pb) ”B-type” data have values clearly larger than those for “Q-type” data; two
examples of comparison of both type data for Ge and Sn are presented on Figs. 3 and 4
correspondingly;
3)
for nuclei of “group 3” (Mg, Xe, Hg, U) both systematics found out are mixed (for several
isotopes ”B-type” data have values larger than those for “Q-type” data, but for others – vise
versa, “Q-type” data have values larger than those for ”B-type” data; an example is presented on
Fig. 5 for Xe isotopes.
QUADRUPOLE DEFORMATION OF NUCLEUS
AND ITS SURFACE VIBRATION
The difference between 2 data derived from В(Е2) and Q values founded out can be
explained in the fame of assumption about the connection of quadrupole deformation parameter with
vibration of surface of nucleus in ground state. Such assumption was formulated, for example, in
private communication of Prof. Yuri Gangrsky from Joint Institute of Nucleus Research. It is based on
assumption that “Q-type” data do not take into account such nucleus surface vibrations, but “B”-type
data do that. In other words from that point of view “B-type” data are affected not only by static
4
nucleus deformation dependent on its shape, but also by dynamical nucleus deformation that came
from its surface vibration.
Rotating nucleus model.
In the frame of rotating nucleus model it is supposed that rigid nucleus has an static
deformation and its excitation is the result of nucleus rotating. It was shown [8] that value of reduced
probability В(Е2) of transition between any states of rotating band can be expressed as
B(E2; KJ1→KJ2) = (5/16π)e2Q02‹J1K20‫׀‬J2K›2,
(10)
where J is state spin, K is spin projection to the axis of symmetry, Q0 is intrinsic nucleus quadrupole
moment, ‹J1K20J2K› is vector summation coefficient
Partially, for Е2 transition 0+  2+1
B(E2; 0+ → 2+) = ((3/4π)eZR02)2(β2)2,
(11)
which completely corresponds to the expression (9).
Vibrating nucleus model.
Because nucleus is not absolutely tough (rigid) object, its surface vibrates and amplitude of such
vibrations depends on 1) excitation state energy and 2) nucleus toughness (rigidity). It is obvious that
if such vibrations occure the nucleus shape is changing. It is important that such vibrations can occur
not only in excited states but in ground state also.
Changing of nucleus shape (shape vibrations) occur relative some “equilibrium” shape that can
be partially spherical as in case of magic nuclei. It is important to underline that in such enough simple
cases though in average nucleus shape is sphere, at different times initial photons are interacted with
nuclei of different shapes. It is important that independently of the sign of deformation – prolate or
oblate nucleus – at the same absolute deformation parameter value interaction probability is the same.
Both processes contributions are summed and therefore nucleus spherical in average can have not zero
probability for transition 0+  2+1 and hence not zero deformation parameter.
For simplicity: harmonic oscillator with spherical equilibrium shape the average amplitude of
nucleus shape difference from spherical one in ground state is
<0  0> = 0,
(12)
where  are collective variables describing nucleus surface motion, and  = 2 for harmonic oscillator
(that is true [9] for N-th excited vibration state).
5
Squared deformation parameter , characterized vibrating nucleus is defined [8] as average of
the sum   2
 =

5  2 N  ,
2 B22
(13)
where В2 is mass coefficient, 2 is vibration frequency and N is phonon number.
It must be underlined that for nucleus ground state (N = 0)
 =
5
,
2B2 2
(14)
and   0 (!). This phenomenon has very simple explanation: dynamical deformation of nucleus in
ground state occurs ((9), (11)) because not deviation of nucleus shape from “equilibrium” is averaged,
but deviation squared. It must be underlined that both formulae ((9) and (11)) though being very
similar (near identical) were obtained in quite different models and describe different physical
processes taking place in nucleus at excitation of in it of 2+ state. The theory that could simultaneously
describe both kind motions (vibration and rotation) of nucleus is very complicated and really in
essence has not been developed.
But at the same time it must be pointed out that in vibration model probability В(Е2) of
transition 0+  2+1 is connected [8] with introduced squared deviation  of nucleus shape from
equilibrium as
B(E2; n2 = 0 → n2 = 1) = ((3/4π)eZR02)2(β0)2.
(15)
This expression textually (letter-to-letter) coincides to the expression (11)
B(E2; 0+ → 2+) = ((3/4π)eZR02)2(β2)2,
(11)
which gives to one possibility to interpret introduced  as “deformation” namely but “dynamical
deformation” - not traditional “static” one.
It is obvious that the only simplest case was analyzed. In reality vibrations occur relative notspherical equilibrium shape and therefore much moreo complicated interaction of two (or more?) types
of moving takes place. As a result relatively simple and strict solution can not be obtained. But it is
6
obvious also that effective interaction of two degrees of freedom will lead to effective increasing of
deformation.
Dynamical deformation of nucleus ant its toughness
So one can see that:
There are many nuclei for which both-type parameter of deformation data are of the same value;
There are many nuclei for all isotopes of which B-type data have values larger (in some cases significantly (factor
1.5 – 2.0)) larger than Q-type data;
iii)
There are only few nuclei for which the intermediate situation is observed;
iv)
There are absolutely no nuclei for which Q-type data are larger than B-type ones.
The physical explanation of phenomenon observed can be the following. For group 1 (B-type data are near to Qtype data) the 2+-level excitation is pure rotational. In such case vibrations of nucleus in ground state (“zero vibrations)) are
weak and the shape of such nucleus can be treated as static.
Vice versa, for group 2 (B-type data are larger than B-type ones) the “zero vibrations” play important role – there
are the superposition of static and dynamical deformations which leads to effective increasing of nucleus deformation. As a
matter of fact in this case the rotational model can not be employed. But because the formulae connected B(E2) are
formally the same in rotational and vibrational models it can be waited the B-type parameter of deformation is reflected the
effective increasing of nucleus deformation. Therefore such nuclei can be treated as trembled.
Several isotopes were founded out for which Q-type parameter of deformation has value approximately equal to
zero, but B-type one is enough large – for example,16O, 40Ca, 94Mo, 96Ru, 124Sn. For those isotopes the vibration model in
pure version can be applied and therefore some new parameters of nuclear matter can be calculated, for example nucleus
surface rigidity coefficient (table).
i)
ii)
Table. Several parameters characterized nucleus surface features.
Nucleus
(2)B
(2)Q
C2, MeV
C2, MeV
C2, MeV
a
a
16
O
0.362
 0.0
132
–––
–––
–––
–––
18
O
0.355
0.085
39.3
–––
–––
–––
–––
Ne
0.562
0.585
–––
?
49.4
?
0.21
Mg
0.606
0.443
–––
82.2
35.7
0.20
0.24
28
0.407
-0.352
–––
84.1
185
0.17
0.14
30
Si
0.316
0.094
56.1
–––
–––
–––
–––
40
Ca
0.122
 0.0
655
–––
–––
–––
–––
42
0.247
0.212
–––
36.5
63.2
0.16
0.14
50
0.293
0.307
–––
?
127
?
0.11
52
0.224
0.067
71.2
–––
–––
–––
–––
0.217
0.128
–––
–––
–––
–––
–––
0.095
 0.0
293
22
24
Si
Ca
Cr
Cr
102
Ru
124
Sn
7
144
Nd
0.131
 0.0
102
–––
–––
–––
–––
150
Nd
0.285
0.351
–––
43.1
68.6
0.088
0.079
Sm
0.341
0.319
–––
12.7
14.6
0.208
0.201
0.353
0.335
–––
?
10.4
?
0.217
Er
0.338
0.338
–––
16.3
7.4
0.193
0.236
Hf
0.280
0.269
–––
19.7
18.8
0.174
0.177
182
W
0.249
0.268
–––
210
240
0.052
0.050
186
Os
0.200
0.200
–––
205
107
0.051
0.060
U
0.286
?
–––
23.3
25.4
0.141
0.138
Pu
0.289
?
–––
20.6
22.6
0.144
0.141
154
160
Gd
168
178
238
240
It is obvious that nucleus’s inclination to be dynamically deformed must be naturally depended
on its property such as «toughness (rigidity)» or vise versa, «laxity (friability)». Therefore in the frame
of supposition about influence of nucleus dynamical deformation upon its shape nucleus «toughness
(rigidity)» can be qualitatively estimated by separation of all nuclei into two groups – tough (rigid) and
not tough (rigid), lax (friable). From that point of view the comparison of static and dynamical nucleus
deformations could be the source of information on some new nuclear matter features, for example for
estimation of mass and rigidity parameters.
That can be illustrated very clear by the case with tin isotopes the data for which are presented
on Fig 4. One can easily see that for all isotopes the static deformation depended on quadrupole
moment is near zero, but at the same time the values of dynamical deformation depended on reduced
transition probability B(E2) are enough large - ~ 0.095 – 0.123. For most interesting case of 124Sn the
parameter of static quadrupole deformation is equal to zero - 2 = 0, and therefore this isotope
deformation is completely determined by its dynamical part with parameter 0 = 0.095. From the
presented article point of view that means that
124
Sn isotope deformation is of pure vibration (mean8
root-square) type. If that really takes place, the equation (15) could be used for such squared
deformation parameter  determination. For that case and for phonon number N = 0 (14) using wellknown equation for vibration frequency
ω2 =
C2 / B2 ,
(16)
where В2 is so-called mass coefficient and C2 is nucleus rigidity coefficient, one can estimate for 124Sn
the product of those coefficients
В2С2 = (75891 ± 3819)  2 .
ω2 =
C2 / B2 .
(17)
(16)
Authors acknowledge Prof. Yu.P.Gangrsky from Joint Institute of Nuclear Research for very important ideas and
fruitful comments and discussions.
Research is supported by the President of Russia Grant N SS-1619.2003.2.
REFERENCES
1.
N.J.Stone, Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments, Atomic Data
and Nuclear Data Tables, 90, 75 (2005)
2.
N.J.Stone. Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments. Preprint
Oxford Physics, Clarendon Laboratory OXFORD OX1 3PU U.K., 2001.
3.
P.Raghavan, Table of Nuclear Moments. Atomic and Nuclear Data Tables. 1989. V. 42. P.189.
4.
S.Raman, C.W.Nestor, P.Tikkanen. Transition Probability from the Ground to the First-Excited
2+ State of Even-Even Nuclides. Atomic Data and Nuclear Data Tables. 2001. V. 78. P. 1.
5.
I.N.Boboshin, V.V.Varlamov, S.Yu.Komarov, N.N.Peskov, V.N.Orlin. Nuclear Quadrupole
Deformation and Surface Vibration: New Global View. The International Conference “Nuclear
Structure and Related Topics” (Dubna, June 13 – 17, 2006). Contributions. ISBN 5-9530-01134, JINR, 2006, p. 23.
6.
I.N.Boboshin, B.S.Ishkhanov, V.V.Varlamov. New Data on Nuclear Subshells Obtained from
the Analysis of the Information from the International Database on Nuclear Structure ENSDF.
Physics of Atomic Nuclei, 67, N10 (2004) 1846 – 1850.
7.
V.V.Varlamov, N.N.Peskov, D.S.Rudenko, M.E.Stepanov. Consistent evaluation of
photoneutron reaction cross sections using data obtained in experiments with
quasimonoenergetic annihilation photon beams at Livermore (USA) and Saclay (France) in
9
Articles Translated from Journal Yadernye Konstanty (Nuclear Constants). INDC(CCP)-440,
IAEA NDS, Vienna, Austria, 2004, pp. 37 – 85.
8.
I.M.Kapitonov. Introduction into Nuclei and Particles Physics. 2002. «EDITORIAL URSS,
Moscow.
9.
B.S.Ishkhanov, V.N.Orlin. Yadernaya Fizika, 2005. Т. 68. С. 1197.
10.
O.Bohr, B.Mottelson. Structure of Atomic Nuclei. V. 2. 1977. «MIR»», Moscow.
11.
I.Hizenberg, V.Greiner. Nuclear models. Collective and single-particle phenomena. 1975.
«ATOMIZDAT», Moscow.
10
2
0,400
0,350
0,300
0,270
0,250
0,252
0,200
0,268
0,241
0,249
0,224
0,235
0,202
0,164
0,150
0,151
0,100
0,088
0,050
0,000
180
0,013
A
182
184
186
Fig. 1. Comparison of quadrupole deformation parameters 2 derived from В(Е2) and Q for W
(“A”- group) isotopes.
11
2
0,600
0,500
0,429
0,399
0,400
0,388
0,358
0,342
0,322
0,300
0,323
0,311
0,325
0,281
0,296
0,336
0,338
0,277
0,271
0,200
0,100
0,000
162
A
164
166
168
170
Fig. 2. Comparison of quadrupole deformation parameters 2 derived from В(Е2) and Q for Er
(“A”- group) isotopes.
12
2
0,350
0,283
0,262
0,300
0,242
0,250
0,204
0,172
0,200
0,150
0,069
0,100
0,048
0,050
0,130
0,098
0,016
A
0,000
66
68
70
72
74
76
Fig. 3. Comparison of quadrupole deformation parameters 2 derived from В(Е2) and Q for Ge
(“B”- group) isotopes.
13
2
0,140
0,120
0,123 0,119
0,112
0,111
0,108 0,104
0,095
0,100
0,080
0,060
0,043
0,040
0,020
0,008
0,000
112
0,020
0,013
0,005
114
116
118
0,000
0,012
120
A
122
124
Fig. 4. Comparison of quadrupole deformation parameters derived 2 from В(Е2) and Q for Sn
(“B”- group) isotopes. Special case of 124Sn (2 = 0) is indicated.
14
2
0,300
0,249
0,264
0,265
0,244 0,245
0,250
0,231
0,200
0,221
0,218
0,214
0,188
0,184
0,169
0,150
0,153
0,141 0,120
0,120
0,086
0,100
0,136
0,065
0,062
0,045
0,050
0,036
0,096
0,037
0,031
A
0,000
114 116 118 120 122 124 126 128 130 132 134 136 138 140 142
Fig. 5. Comparison of quadrupole deformation parameters derived from В(Е2) and Q for Xe
(“C”- group) isotopes.
15
16
QUADRUPOLE DEFORMATION OF NUCLEUS
AND ITS SURFACE DINAMIC VIBRATIONS
I.N.Boboshin1, B.S.Ishkhanov2, S.Yu.Komarov1, V.N.Orlin1, N.N.Peskov1,
V.V.Varlamov1
1
Skobeltsyn Institute of Nuclear Physics, 2Physics Faculty,
Lomonosov Moscow State University, Moscow, Russia
Using the new Chart (relational database) of nuclear quadrupole deformation parameters 2
the systematical comparative analysis of those from various sources was carried out for
many even-even nuclei. The clear systematical disagreements of 2 values obtained from
nuclear quadrupole moments Q data and from reduced transition probability B(E2) data
were revealed: it was found out that all nuclides (with only few exceptions) can be clearly
separated into 2 groups. In one good agreement is observed for data of both types, but in
another deformation parameters 2 obtained from B(E2) are systematically (and in many
cases – significantly) larger than those obtained from Q. It was shown that two types data
difference could be explained in the frame of assumption about the not-negligible role of
dynamical vibrations of nucleus surface, by other words - assumption about deformation
parameter 2 value dependence on surface vibration of nucleus in ground state. From this
point of view 2 values obtained from B(E2) reflect not only static nuclear deformation
(nucleus shape), but dynamic deformation also.
17