1. D. 2.
Nuclear
Physics
8 (1958)
237-249;
Not to be reproduced by photoprint
ROTATIONAL
STATES
©
North-Holland
Publishing
Co
Amsterdam
or microfilm without written permission from the pubiishen
IN EVEN ATOMIC
NUCLEI
A. S. DAVYDOV
and G. F. FILIPPOV
Aloscow State University
Received
29 March
1958
A theory of the energy states and the electromagnetic
Abstract:
transitions between them
is developed for nuclei which do not possess axial symmetry. It is shown that violation
of axial symmetry does not significantly
change the rotational states of axial nuclei and
leads to the appearance of new energy states. The reduced probabilities for E2 and All
transitions between various rotational
states are computed.
1. Introduction
In some previous papers of the authors 1-4) energy levels of non-spherical
involving
to
violation of
not
nuclei corresponding
collective excitations
basis
the
investigated
the
of
generalized
on
nuclei were
axial symmetry of
5.6).
It
Mottelson
B.
by
A.
Bohr
was shown that
nuclear model proposed
and
is
the rotational-vibrational
states
a
excited
nuclear
energy of collective
function of only two parameters, viz., of the frequency of nuclear surface
deformation
to
the
the
zero
the
equilibrium
vibrations and of
ratio of
these
inquire
to
to
It
extent
what
vibration
seems natural
amplitude.
results are applicable if one takes into account possible violation of axial
symmetry of the nucleus.
The problem of violation of axial symmetry of nuclei has been qualitatively
treated in a number of papers 5,7,8). Recently it has even become usual to
9.10).
This
type
the
to
of
so-called y-vibrations
ascribe nuclear excited states
levels
these
the
is
based
the
and
of
on
values
spin
on
usually
assignment
large probability of electromagnetic transitions which confirms the collective
has
been
theory
levels.
No
the
of
y-vibrations
quantitative
nature of
)roposed.
In the present paper we investigate energy levels corresponding to rotation
its
internal
It
does
the
of
changes
entail
state.
not
will
nucleus which
of
be shown that violation of axial symmetry of even nuclei only slightly
affects the rotational spectrum of the axial nucleus although some new
do appear.
rotational states with total angular momenta of 2,3,4
...
If the deviations from axial symmetry are small these levels lie very high
and are not excited. As the deviations from axial symmetry are increased
some of the additional levels become much lower. Thus, for example, the
237
October (1) 1958
238
A.
S. DAVYDOV
AND
G. F.
FILIPPOV
ratio of the second'excited state with spin 2 to the first state (which also
exists in axial nuclei), varies from infinity to two. The probabilities for
electric transitions between rotational levels in non-axial nuclei are evaluated
in sec. 3. The probabilities for magnetic transitions are computed in sec. 4.
data
From comparison of the proposed theory with the experimental
(sec. 5) it can be concluded that the properties of the experimentally
explained by
observed energy states of even nuclei can be satisfactorily
assuming that these nuclei do not possess axial symmetry.
2. Energy States in Non-Axial
Proceeding
Nuclei
from
the generalized nuclear model, consider the energy
states of an even nucleus corresponding to rotation of the latter as a whole
with no change of its internal state. The operator corresponding to the
rotation energy of the nucleus has the form
A-i 2
sinn
(y-
3
a1
where A= h2/4Bß2 is a quantity which has the dimension of energy;
y varies between 0 and 7v/3and determines the deviation of the shape of the
JA
from
the
nucleus
axial symmetry;
are operators of the projections of the
nuclear angular momenta on the axes of a coordinate system connected
with the nucleus. The commutation rules for these projections differ from
the rules for the projections in a space-fixed coordinate system by the signs
in the right hand side.
According to (1.1), for y00
or 7r/3 the nucleus should be regarded as an
asymmetric top. In stationary states of the asymmetric top not one of the
projections of the total angular momentum on axes 1,2,3 of the body-fixed
coordinate system has a definite value and hence the energy levels cannot be
specified by the values of K= J8. Each value of the total angular momentum
in the asymmetric top corresponds to 2J+ 1 different energy levels. These
levels can be classified with respect to the irreducible representations of
group D2 with symmetry elements C21, C22, C2 corresponding to rotation
through r about the coordinate axes 1,2,3, because operator (1.1) and the
commutation between the JA are invariant with respect to this transformation group. Thus the energy levels of an asymmetric top split up into four
types of levels which correspond to the four irreducible representations of
group D2 (see ref. il), § 101 and ref. 12)).
In virtue of the symmetry conditions imposed on the wave function b),
in even nuclei of the 2J+1 different levels only those energy levels
with a given J can exist which correspond to a completely symmetric
ROTATIONAL
IN
STATES
EVEN
ATOMIC
239
NUCLEI
the
Rotation
D2.
required symmetry will
of
states
representation of group
for
J=3,
for
j=2,
Two
if
j=1.
one
exist
will
such states
not exist
four for j=6,
two for J=5,
etc.
three for J=4,
If the energy is expressed in units of A= ? 2/4Bß2 the energy of two levels
by
for
j=2,
defined
the expression
the
are,
symmetry
required
of
9(1-V1
X1(2) -
sinn (3y))
sin29 (3y)
The energy of a level with
sin2 (3y))
sin (3y)
9(1+-%/1-1
(2) _
2
angular
momentum
j=3
The three spin 4 energy levels are the roots of the third
48
E3
sin2 (3y)
82+
sin4 (3y)
640
[27+26sinz(3y)]E-
is given by
(1.3)
e(3) =18
sin (3y)
90
(1.2)
(3y)
sin4
degree equation
[27+7sin2(3y)]=0.
The two spin 5 energy levels are given by the formula
E,(s) =
45+91/9-8
sine (3y)
(1.4)
sin2(3y)
if a plus is
In (1.4) r=1,
if a minus sign is before the root and i=2,
before it. The energy levels of states possessing an angular momentum
equal to 6 are defined by a fourth degree equation which we shall not present
here.
The following simple relation between the spin 2 and spin 3 energy levels
follows from (1.2) and (1.3)
El
+
The energy levels of even nuclei computed on basis of the preceding forthe energy spectrum
mulas are plotted in fig. 1 as a function of V. For y=0
is identical to that of an axially-symmetric
nucleus. For a fixed value of
ß violation of axial symmetry of the nucleus leads to an increase of the energy
of the levels belonging to the axial nucleus. This increase of the level energy
corresponds to a decrease of the effective moment of inertia of the nucleus or
of the effective deformation parameter Bett. For the first excited state of
spin 2 the effective deformation parameter can be determined from the
formula
C4
sine (3y)
Bett =ß9_,
'81-72
sin2(3y)
Besides the comparatively small change of the level energies of an axially
symmetric nucleus, violation of axial symmetry of the nucleus leads to the
appearance of some new energy levels e2(2), e(3), e2(4) etc. By using the
240
A.
S. DAVYDOV
AND
G. F.
FILIPPOV
dependence of 62(2)/el(2) on y, one can determine the corresponding value of
y from the experimental value of the ratio. From fig. 1 the energy sequence
of the spins and the energy values of the other nuclear levels can then be
determined. Comparison of the theory with the experimental results will be
performed in sec. 5.
70
E
ßi2
60
48)3
24
4
5
50
3
40
6
30
20
42
10
22
n
5
10
15
20
25
ý -"-
30
Fig. 1
3. Electric Transitions
between Rotational
Nuclear States
As is well known, measurements of the transition probabilities between
electric states in a nucleus yield important information on the nature of the
excited states. In particular, for elucidation of the nature of the second
excited state of spin 2 in an even nucleus one may study the relative probability for transition from this level directly to the ground state or to the
first excited state with a spin of 2. It has been assumed in
a number of
investigations
that the first two spin 2 levels observed experimentally
refer to one-phonon and two-phonon vibrations of the nuclear surface. In
this case transition from the second state to the ground state can take place
only as a result of violation of the oscillator approximation.
Assuming that both spin 2 levels can be assigned to rotational levels, we
compute the ratio of the reduced transition probabilities for 82(2) -* 81(2)
and 82(2) -* e(0) as a function of the parameter y and hence as a function of
the ratio 82(2)/81(2)since the latter depends on V.
STATES
ROTATIONAL
IN
EVEN
ATOMIC
241
NUCLEI
In order to calculate the probability for E2 type transitions between the
rotational levels the operator of the nuclear quadrupole moment
z
ý`
4n
02µ
2e
=
5G
yi2Y2µ(tigi)
i-1
Eulerian
in
defining
be
the
terms
the
of
angles
orientation
expressed
should
in
terms of the collective coordinates with respect to axes
the
nucleus
and
of
connected to the nucleus. Employing the transformation
of the spherical
functions
Y219(OT) _
D2
V
corresponding to transition to a coordinate system connected with the
nucleus and expressing the proton coordinates r{ 0; 99jin this system (assuming
the protons to be uniformly distributed within the nucleus) through the
collective coordinates a,, where ao =ß cos y, ai = a_1 = 0, a2 = a_2 =
ß sin y/V2 by aid of the formula
Z (r)2
{, Y2, (O'
i 9ýc)
4n
5
a!
we obtain the following expression for the µth component of the operator of
the electric quadrupole moment
D2+D,
(Docos+
-2
= eQ2
(2.1)
sin y/
where
Qo =
3ZR2 ß
(2.2)
_/
is the intrinsic electric quadrupole moment of an axial nucleus with a deformation
(which
parameter ß; Dµ,, are generalized spherical functions
depend on the Eulerian angles) defining the unitary transformation
from a
coordinate system fixed in space to a coordinate system fixed to the nucleus.
The wave functions of the rotational states of a non-axial nucleus can be
written in the form
1
Vo -
1ý8n299A
Y),
[aiDo+bi2+_21
(lam.
ýV21M
-
9(ßY)
1
VV22M =
\8nal
9' (ßy)
(2.3)
/2
22
[a2Dco+b
DM2ý2
z
m,-al
,
where q'(ß, y) is a function corresponding to the internal state of a nucleus
which is assumed to be the same in all three rotational states
242
a1N1 =b1N1 =3
S. DAVYDOV
A.
AND
G. F.
FILIPPOV
[sin y sin(3y)+3
cos y cos(3y)+V'9-8sin2(3y)],
sin y cos(3y)-cos y sin (3y),
N12 = 2''9-8sin2(3y)[V9-8
_
9-8 sinn (3y)-sin
a2N2 =
b2N2 =3 sin y cos(3y)-cosy
N22 = 21/9-8
sin (3y) + sin y sin(3y)+3
(3y),
cos
cos
y sin(3y)-3
y
sin (3y),
cos y cos (3y)],
(3y)-3
sin
y
cosy cos (3y)].
sin2(3y)[1'9-8
sin2(3y) -sin
(2.4)
As was previously noted, y varies between 0 and n/3 and determines the
deviation of the shape of the nucleus from axial symmetry. The axes of the
ellipsoid employed to approximate the shape of the nucleus can be expressed
through y and ß by the formula
Rx =R
11
j
ý-ß
(ycos
ý,
3
1],
A=1,2,3.
If y=0
the nucleus is an elongated ellipsoid of revolution with a symmetry
is
the
3 axis. If y=
an oblate ellipsoid of revolution with a
nucleus
t/3
defined
(1.1)
by
The
2
states
of
a
nucleus
operator
rotational
axis.
symmetry
for
between
transitions
them are the
the
electromagnetic
and
probabilities
We
therefore
the
for
present
values of the various
and
a/3-y1.
same
yl
between
0 and r/6.
interval
in
the
quantities
In connection with the preceding it should be mentioned that one cannot
decide on basis of measurements of the rotational energy of the nuclei or of
the electromagnetic transitions between them whether the nucleus is an
The
is
this
to
to
ellipsoid.
only
way
answer
question
oblate
elongated or
measure the mean values of the quadrupole electric moments in stationary
In
J).
(J,
M=
even nuclei the mean values of the quadrupole
states
moments in the ground state are equal to zero. In the first excited state with
spin 2 the mean value of the quadrupole moment is
Qi =
-Qo
6 cos (3y)
(3y)
72
where Q0 is defined in (2.2). In the second excited state of spin 2 the mean
value of the quadrupole moment has a different sign
Q2 =
-Q1"
The reduced probability for the electric quadrupole transition Ji
averaged over the initial polarization states of the nucleus is
B(E2;Jr -* J'i') =5I
I(J'r'miQ2,.
IJTM)12.
16a(21+ 1) M.,&
that in
(2.5)
Since we assume
this transition the internal state of the nucleus does
not change, the reduced transition probability can be expressedthrough the
ROTATIONAL
STATES
IN
EVEN
ATOMIC
243
NUCLEI
help
(2.5).
in
Inserting
(2.3)
ß
the
state
with
of
and
qß(ß,
y)
y
mean value of
into (2.5) and using (2.4) we find the following values for the reduced electric
in
transition
expressed
probabilities
e2Q02/16n units. This unit
quadrupole
corresponds to the reduced electric quadrupole transition probability in an
levels of spin 2 and 0.
axially symmetric nucleus between rotational
b(E2; 21-ý0)=
B(E2; 21 -. 0)
=
l
eaQo2ý1+
167v1
b(E2; 22 -* 0) -
3-2
sin2(3y)
ý9-8
3-2
1-
sin2(3y)
9-8
ý,
(2.6)
sin 2 (3y)
l
(2.7)
sing (3y)J
sin' (3y)
9-8sina (3y)
b(E2; 22
21)
_
--*
(2.8)
It is interesting to note that
b(E2; 21
1.
0)
22-+
0)+b(E2;
=
->
The ratio e2(2)/El(1), the relative reduced transition probabilities
(2.6)
and (2.8) and the ratio of the reduced electric quadrupole transition probabilities b(E2; 22 -> 21)/b(E2; 22 -. 0) are listed in table 1 for several
TABLE
ell (2)
E1(2)
0
5
10
15
20
22.5
24
25
26
x
64.2
15.9
6.85
3.73
2.93
2.59
2.41
2.26
b(E2; 21 .
1.000
0.093
0.972
0.947
0.933
0.937
0.948
0.955
0.968
I
0) b(E2;
22
0
0.0074
0.028
0.053
0.067
0.0625
0.052
0.0425
0.0324
0) b(E2; 22
0
0.011
0.051
0.143
0.357
0.563
0.782
0.865
1.01
28
2.07
0.99
0.010
29
2.01
0.996
0.004
1.28
1.41
30
2.00
1.000
0
1.43
b(E2; 22 -> 21)
21)
b(E2; 22 -* 0)
1.43
1.49
1.70
2.70
5.35
19.02
15.1
20.6
31.2
126
363
00
values of y. From the data in table 1 it follows that the reduced probability
(2.6) for transition from the first excited state to the ground state only
slightly changes when axial symmetry of the nucleus is violated. The
(2.7) for transition from the second excited state of
reduced probability
spin 2 directly to the ground state vanishes for y= 0° or 30° and comprises
about 5-7 % of the corresponding reduced transition probability between
the first excited and ground energy states when y= 15-24°. The reduced
E2 transition probability from the second excited spin 2 level to the first
244
A.
S. DAVYDOV
G. F.
AND
FILIPPOV
increases
for
level
is
0
excited
and subsequently rapidly
very small
y ti
40 % of the reduced ground state
20° equals approximately
and for y
transition probability in the axial nucleus and about 140 % for y s;
ýe30°. Of
b(E2;
21)/
22
interest
is
the
the
reduced probabilities,
ratio of
special
-a
b(E2; 22 -. 0) as this quantity does not depend on the degree of population
of level 82(2) and can be directly measured.
With help of the explicit form of the wave function for energy level E(3)
733 (fly)
[Dm2- Dm,
T
V31m= i'
-21,
16;12
the reduced electric quadrupole
(in our units)
transition
25
b(E2 3
22)
-- 28
-X
b(E2;
(
probabilities
3-2
1ý-
1/ 9-8
3-2
3 -ý 21) -- RAI' 128
9-8
can be computed
sin2 (3y)
J
(3y)
sine
sin2 (3y)
sine (3y)i/
.
The values of these probabilities and their ratio are presented in table 2.
These reduced probabilities satisfy the relation
b(E2;
3 -* 22)+b(E2;
3
TABLE
b(E2; 3>
]0
15
20
25
29
30
21)
21)
-*
2.
2
b(E2; 3 -> 22)
0.0132
0.095
0.12
0.079
0.0696
0
b(E2;
3
21)
b(E2; 3
22)
1.77
1.69
1.67
1.70
1.72
1.78
4. Magnetic Dipole Transitions
between Rotational
0.0075
0.056
0.072
0.044
0.039
0
States
According to Bohr and Mottelson 5) the operator of the magnetic dipole
moment corresponding to collective motions in an even nucleus is defined
by the formula
f
(3.1)
R(r)V(rYlµ)dr
l(lfp)
_ 4u0gn
where po = e1/2Mc is the nuclear magneton; gR is the gyromagnetic ratio
corresponding to collective motion of the nucleons in the nucleus;
R(r) = BRö-5
is the angular momentum density.
A
ax[r0(r2Y2A)]
ROTATIONAL
Introducing
STATES
IN
i[rV]
L=-
the operator
EVEN
246
NUCLEI
ATOMIC
we can write
V--i
[rV(Y2Y2A)] = Ir2LY2A and V(rY1, ) _ (-1)! '
Inserting
these values
into
(3.1) and employing
6 (21,1+µ,
(eAL)Y2A = (-1)PN/
form
(3.1)
the
to
we reduce
the relation
-4I2))Y2,
(111)+Jii(1p)+
TZ (1/t) _
4n
e
A+µ,
(3.2}
...
where
gx
moo
,
Jo(lu)
J1(lu)
(21du-v,
7/
(3.3)
JJ,
=2-
5V6
= Pogx
Vi
*-,
J,.
vll, u)a,
(3.4)
Here the J, are projections of the total angular momentum which in the
classical theory are expressed through the coordinates a,. (in a space-fixed
deviation
the
these
of the shape
coordinate system
coordinates characterize
of the nucleus from a spherical one) and through the corresponding velocities
by the formula
Jv = i'ßß(-1)vB
where
vector
action
(2.3)
a, aµ+v(21, i+v,
&
,
-vI2y),
B is the mass parameter of the theory; (21, µ+v, -v12, u) is the
,
The
J,
In
the
theory
addition coefficient.
are operators.
quantum
of these operators on wave functions DMK, in terms of which functions,
of an asymmetric top are expressed, is defined by the equality
JPDMK= (-1)" -VJ(J+1)(J1M+v, -viJ, M)DM+V,
K.
The reduced magnetic dipole radiation probability from the state 22 to
state 21 defined by functions (2.3) is
I(21m'j9i(1, u)j22m)I2.
(3.5)
It is easy to verify that operator (3.3) does not contribute
ability. Inserting (3.4) and (2.3) into (3.5) we get
to this prob-
B(M1;
22
21) =b
mim
sine (3y)
2
aß2
B(M1; 22
21)
gR
uo
= 49; ,
-9-8sin2 (3y)
c2
90
(3.6)
From (3.6) it follows that in an axial nucleus (y = 0) the reduced magnetis zero and increases with y reaching a
ic dipole radiation probability
maximal value at y= 30°.
If the value of y is determined from the ratio E22/E21of the energies of
two spin levels, one can evaluate the value of the gyromagnetic ratio gR for
246
A.
S. DAVYDOV
AND
G. F.
FILIPPOV
collective nucleon motion by using (3.6) and the results of measurement of
the reduced magnetic dipole transition probabilities.
If one, furthermore, takes into account that according to (2.8) the
reduced electric quadrupole radiation probability for transitions between
the same levels is
B(E2;
10e2 Q02
22 --ý--21)
sin2 (3y)
[9-8
7n
sine (3y)]
one may determine the ratio of the reduced probabilities
B(M1;
22--21)
B(E2;
22 -*- 21)
2
__
e 'Ro2
where Ro is the nuclear radius. It is interesting
dependent on y and P.
to note that
(3.7)
(3.7) is not
From (3.7) the ratio of the intensities of y quanta emitted in both types of
radiation is
T(MI)
80
IQ
0.21k a eZRo J'
T(E2)
(3
(3.8)
where
E22-E21
k=
lic
5. Comparison
with
Experiment
The results obtained in the preceding sections were based on the assumption that during rotation of a nucleus its internal state does not change.
This assumption can be only approximately true and is the more accurate
the farther the rotational energy levels are located from levels corresponding
to excitation of the internal states of the nucleus and possessingthe same J
values, parities, etc.
Experimental values of the ratio of the energy of the second spin 2 level
to that of the first level are listed in table 3. The value of parameter y can be
computed from this ratio with help of formulas (1.2). Employing (2.7) and
(2.8) and making use of this value of y one can evaluate {b (E2; 22
/
21)
-.
b(E2; 22
0)}theor. Comparison of these ratios with the experimental
ones (column 7) indicates that the theory satisfactorily describes the experimentally observed rapid variation of the ratio of the reduced probabilities for transitions from a nucleus to one with a different y value.
As can be seen by comparing columns 4 and 5 of table 3 for many nuclei
the sum of the energies of the two spin 2+ levels is equal to the energy of the
spin 3 level, as required by equation (1.5), with an accuracy to 1 %. The
deviation of 5% observed in the Cd114nucleus can be explained by the effect
of three other levels located near level 2 and possessing spins of 4,0 and 2.
ROTATIONAL
STATES
IN
EVEN
ATOMIC
247
NUCLEI
From fig. 1 it can be seen that for y< 21.5° the second excited 2+ level
lie
below
level.
for
it
level
this
he
21.5°
4+,
should
whereas
above
y>
should
The level 3+ should always be higher than level 4+. These rules are valid for
TABLE
Experimental
3
keV
(2)
Nucleus
Pt19'
Pt192
Xe18'
Tel'
Cdl1'
Se7'
Te122
Hg1°8
Gd'5'
W'8'
Sm162
Dv160
W182
Pu238
Pu240
81(2)
1.97
1.94
2.0
2.14
2.17
2.2
2.25
2.66
3.01
8.08
8.9
11
12.1
23.4
23.7
Y°
30
30
30
27.5
26.75
26.6
26.5
23.2
21.4
13.9
13.48
12.2
11.6
8.13
8.0
E (2)
+82(2)
1040
929
1445
2040
1771
1760
1760
1500
1133
1000
1206
1051
1322
1074
1063
(3)
920.9
1860
-
1140
1000
1226
1047
1331
1076
1060
b(E2; 22 -+ 21)
b(E2; 22
0)
theor.
oc
a
a
104
78
68
66
13
7.2
2.0
1.9
1.9
1.6
1.2
1.2
ref.
exper.
> 2500
163
197
200
77.9
23.5
78.2
25
1.7
2.38
1.59
1.3-1.5
-
9,24
13
14
23
15
16
17
23
18
19
18,22
10
20
25
19
all the nuclei in table 3 for which the positions of levels with spins of 2,4,3
are known.
It is interesting to note that for y= 30° the theory predicts an equidistant
spacing of energy levels el(2), e2(2), e(3). A level spacing of this type can
for the energy of surface
also be derived in the oscillator approximation
degenebe
levels
However
in
this
the
vibrations.
should
case
s2(2) and e(3)
rate with spin values of respectively 0,2,4 and 0,2,3,4,6.
Unfortunately,
the experimental data which can be used to determine the
ratio b(E2; 3 -* 21)/b(E2; 3 -* 22) are quite sparse. We know this ratio
only for Kral 21) for which it equals 0.016. This corresponds to a value of y
which slightly exceeds 29. A value y> 29 agrees with the observed value
in
if
hat
it
into
be
1.9
this nucleus adiabatic
taken
accountt
s2(2)/81(2) =
fulfilled.
conditions apparently are not rigorously
From formulas (2.6) and (2.7) one can compute the ratio of the reduced
it
b(E2;
b(E2;
21-*
the
0);
22
0)/b(E2;
0 -. 22)/
equals
ratio
probabilities,
for
inverse
b(E2; 0
the
transitions
21)
the
of
probabilities
reduced
which
-This
Coulomb
during
the
nuclei.
of
ratio is maximal for
excitation
appear
(the
between
lying
25°
15°
variation of y in this interval correand
y values
is
2.4)
6.28
to
>
>
and
very small for y< 5° and
sponds
E2(2)/sl(2)
b(E2;
The
0 -* 22)/b(E2; 0
29°.
the
in
21)
values
of
ratio
y>
presented
--).
ref. 23) and derived from Coulomb excitation data and also the theoretical
248
A.
S. DAVYDOV
AND
F.
G.
FILIPPOV
from
(2.0)
(2.7)
for
the corresponding energy
computed
and
values of y
that
from
in
It
is
table
the
table
4.
apparent
ratio, s2(2)/s1(2), are presented
the theoretical values satisfactorily
ones.
agree with the experimental
TABLE
p2(2)
Nucleus
h(E2; 0 --> 22)
b(E2; 0
21)
y
eß(2)
20
23.7
24
25.05
25.07
26.5
27
27
2.9
2.55
2.56
2.35
2.24
2.2
2.18
2.17
Os190
Ru10°
Ru'°4
Cd116
Rn102
Pd106
Pd108
Cd"4
4
exper.
theor.
; ze0.04
0.024
0.013
0.021
0.015
0.017
0.01
0.02
0.072
0.057
0.055
0.044
0.043
0.027
0.02
0.02
In order to compare the experimental ratio of the intensities of magnetic
dipole and electric quadrupole radiations in the 2+ - 2+ transition with the
values which formula (3.8) yields, we put yo = 5.05 x 10-24 erg " gauss-1
The
T(Ml)/T(E2)
in
10-13
Ro
this
1.2AI
0.4;
X
cm.
ratios
computed
gR =
=
(third
The
intensity of the E2 transition
in
5
table
column).
way are given
TABLE
5
I
Nuclens
Se96
TeLaa
Os186
Os188
E22-E2,
643
693
627
480
T(M1)
keV
T(E2)
9.8 x
1.9 x
6.5 x
1.04 x
10-'10-2
10-8
10-2
exper. % of E2
transition
98±1
92+4
99±1
99.6
,
from
intensity
derived
the
total
the
to
22
the
21
transition
of
relative
-data of Lindvist and Marklund 25)is given in the last column of the table.
Comparison of the predictions of the theory with the experimental data
presently known to us thus confirms the assumption that some even nuclei
do not possess axial symmetry.
References
1)
2)
3)
4)
5)
6)
7)
8)
G.
A.
A.
A.
A.
A.
L.
C.
F. Filippov, Doklady AN SSSR 115 (1957) 4
S. Davydov and G. F. Filippov, JETP 33 (1957) 723
S. Davydov and A. A. Chaban, JETP 33 (1957) 547
S. Davydov and B. M. Murashkin, JETP 34 (1958) 1619
Bohr, Mat. Fys. Medd. Dan. Vid. Selsk., 28 (1952) no. 14
Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 1F
Wilets and M. Jean, Phys. Rev. 102 (1956) 788
Marty, Nuclear Physics 1 (1956) 85; 3 (1957) 193
ROTATIONAL
9)
10)
11)
12)
13)
14)
15)
18)
17)
18)
19)
20)
21)
22)
STATES
IN
EVEN
ATOMIC
249
NUCLEI
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0. Nathan, Nuclear Physics 4 (1957) 125
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D. Dennison, Revs. Mod. Phys. 3 (1931) 280
L. Baggerly, P. Marmer, F. Boehm and J. Du Mond, Phys. Rev. 100 (1955) 1364
N. Benczer, B. Farrely, L. Koerts and C. S. Wu, Phys. Rev. 101 (1955) 1027
H. Motz, Phys. Rev. 104 (1956) 1353
B. Murray and M. Sakai, Phys. Rev. 98 (1955) 674
J. Kurbatov,
M. Glaubman, Phys. Rev. 98 (1955) 645
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Physics
Nuclear
548
and M. Waggoner,
Spectroscopy,
(Leningrad,
1958)
Conference
L. Peker,
on Nuclear
Phys. Rev. 95 (1954) 864
F. Boehm,
P. Marmier
and J. Dumond,
R. Waddell,
and E. Jensen, Phys. Rev. 102 (1956) 816
T. D. Nainan,
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S. K. Bhattcherjee,
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Nathan
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501
23) M. Jean and J. Touchard, Jour. Phys. Radium 19 (1958) 56
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and I. Marklund, Nuclear Physics 4 (1957) 189
Cimento
7