THE THEORY OF THE T-ODD CORRELATIONS IN TERNARY FISSION
1
Bunakov V.E.1, Kadmensky S.G.2
PNPI, Gatchina 188300, 2Voronezh State University, Voronezh 394006
Abstract: The quantum version of the unified description is presented for the two T-odd
effects observed recently in ternary fission induced by polarized neutrons. The effects are
explained by the influence of the rotating fissioning system Coriolis interaction on the
angular distributions of the light charged particles in the interier and exterier nuclear regions.
The quantum approach is compared with the classical trajectory one.
1.Introduction.
A few words on the hystory of the problem. In our previous papers [1-3] we developed the
theory which explained the earlier observed (see e.g. [4]) T-odd correlation of the type:
  
 n [k LF  k ]


in ternary fission induced by polarized neutrons. Here  n is the neutron spin, while k LF and

k are the momenta of the light fragment and the ternary particle (usually alpha) emitted in
ternary fission. This correlation, which was called TRI-correlation, was described by the
differential cross-section of the type:

d 2
 
 B0  D1   n [k LF  k ]
d LF d
(1)
The experimental geometry was chosen in such a way that the directions of the unit vectors



 n and k LF were parallel to the y and z axes, while the vector k varied in the (x,y) plane. The
effect measured was defined as:
  
,
(2)
D 
 
where σ+ and σ- stand for the differential cross-sections with the neutron beam positive and
negative helicities. The magnitude of the effect for the 233U target was about -3·10-3. It is
important to point that this magnitude and the sign of the effect were practically independent of


the angle θ between the vectors k LF and k in a wide range of angles around θ≈900.
However, the recent measurements [5] for the 235U target demonstrated the existence of the
new effect, which was called ROT. Contrary to the TRI, this effect does not change with the


inversion of either k LF or k but changes its sign in the vicinity of θ≈900. This effect is rather
well reproduced [6] in the classical trajectory calculations of the alpha-particle emission in the
rotating Coulomb field of the fission fragments with the angular momentum of about few ħ,
equal to the total spin J of the polarized fissioning nucleus 236U which appears after the
absorption of the polarized neutron by the target nucleus. The inversion of the neutron helicity
causes the inversion of the J direction and, therefore to the inversion of the system’s rotation
direction. This leads to the slight shifts of the alpha-particle angular distributions with respect to

the direction k LF of the light fragment emission. The effective angular velocity of the fissioning
system’s rotation was estimated in [6] as:

R
(3)
 cl  P( J )   n ,

where  is the moment of inertia of the fissioning system and P(J) is the compound-nucleus
polarization, resulting from the absorption of the neutron with polarization pn:
J 1
 2I  3
p

pn
for
J  I  1/ 2
n
 3(2 I  1)
3
J
P( J )  
(4)
1
  pn
for
J  I  1/ 2
 3

The magnitude of the vector R (called the angular momentum of the nuclear collective rotation)
was related to the values of the total nuclear spin J and its projection K on the nuclear symmetry
axis (which coincides in our coordinate choice with the direction of the fragments emission, i.e.
with z axis):
R   J ( J  1)  K 2
(5)
Already here one can see some inconsistency. Indeed, one might expect that after the
absorption of the neutron with spin directed along the y-axis, the total spin J (and not R!) is
polarized along this axis:
P( J ) 
 Jy 
(6)
J


Since for K≠0 the directions of J and R can not coincide, equation (3) seems to be rather
questionable. Even in the case of K=0 one might expect the rotation velocity of the system
around the y-axis to be:
y 
 Jy 


 Jy  J
J
  P( J ) ,
J


(6)
rather than (3), (5).
The classical trajectory calculations of the alpha-particle angular distributions in ternary
fission without the fissioning system’s rotation usually reproduce the experimental data fairly
well. However the precision of those calculations in the description of the quantum system’s
rotation with the angular momentum of the order of ħ and the rotation angle δθ<10 might be
questionable.
A more detailed analysis of the experimental data indicates that both TRI and ROT effects
contribute simultaneously. However the reason why the TRI-effect dominates for 233U target,
while the ROT one prevails in the case of 235U, remains unknown. The experimentally observed
lack of correlation between the signs of both effects [5] also can not be explained in the
framework of the classical approach.
In the present paper we shall demonstrate that our earlier developed approach [1-3] (see
also [7]) allows to give the quantum description of both effects and to show the possible reason
of the difference between the effects observed in the two uranium isotopes. We shall also show
how the difficulties of the sign correlation and of eq. (3) are resolved in this approach.
2. The products’ angular distribution for the ternary fission induced by
polarized neutrons with account of the Coriolis interaction
The basic idea of our approach [1-3] was exactly to consider the influence of the fissioning

system collective rotation (fixed by the  n direction) on the ternary particles’ angular
distribution. In the intrinsic coordinate system of the fissioning nucleus this means considering
the Coriolis interaction in addition to the nuclear and Coulomb ones. In this case the total
Hamiltonian of the ternary particle interaction with the rotating system in classical mechanics
would be:
 
 
J l
H  H0    l  H0 

(7)


Here l is the ternary particle angular momentum,  is the angular velocity of the system’s

rotation, J is the system’s angular momentum and  is its moment of inertia. One can see
already from (7) that by the inversion of the neutron beam polarization (i.e. by the inversion of

the vector J sign) one makes the emission of the ternary particle either easier or more difficult.
This might serve the quasi-classical explanation of the TRI-effect.
In quantum mechanics the Coriolis interaction operator is usually [8] written as:

2    
(8)
H Cor  
( J  l  J l )
2


The operators J  and l  change the projections K or Kℓ of the angular momenta J and l on the
J
deformed nucleus symmetry axis while acting on the function DMK
( ) and the spherical
function Yl , K 
( ) :
 J
J  DMK
( )  ( J  K )( J  K  1) DMJ ( K 1) ( )
(8a)

lYlK ( )  (l  K  )(l  K   1)Yl ( K 1) ( )
Using the technique of [1] in order to take into account the Coriolis interection in the firstorder perturbation theory, we obtain for the ternary fission differential cross-section:
d 3
 B0 ( LF ,  )  B Cor ( LF ,  )
(9)
d LF d d
Here the quantity  is related to the ternary particle energy E3 by the equation (see [1]):
  arccos x , where x  E3 E3max , while E3max is the maximal energy of this particle
E3max  Qc  A1  A2  A , A1,2 are the fragments’ masses, A is the mass of the fissioning
nuclei and Qc is the total kinetic energy of the fission products’ relative motion in the c channel.
The first term in (9) describes the angular distributions for the reaction with unpolarized neutrons
and is proportional to the modulus squared of the amplitude А0 of this reaction. The second term
is defined by the interference of the А0 amplitude with the amplitude ACor of this reaction with
account of the Coriolis interaction. It is convenient to choose the coordinate system which


corresponds to the experimental geometry – the vectors  n and k LF directed along the y and z


axes, while the vector k lies in the (x,y) plane. Then θ is the angle between the vectors k LF and

k , while φ=0. With this choice we obtain for (9) the expression:
d 2
d 2 0
d 2 Cor


d  d d  d d  d
The part of the differential cross-section d
polarization is given by the expression:
(10)
 0 d  d , which is independent of the neutron
2
d 2 0

Js
| hsJ s || hsJ's ' | bsK
b J s '  (2 J s  1) 


2
s s ' K s ' cK s
d  d  2(2 I  1)kn ss ' J s J s 'cK s
 | d cl ( ) || d cl ' ( ) |Yl 0 ( )Yl '0 ( )cos( sJ s s ' J s '   cl ( )   cl ' ( )).
ll '
(11)
Here kn is the neutron wave-vector, I is the target spin, the factor

hsJ s
snJ s
E  EsJ s
 isJ s
/2
| hsJ s | exp{i sJ s }
(12)
is the capture amplitude of the neutron with energy E into the resonance with spin Js, energy
EsJ s , partial neutron width  snJ s and total width  sJ s . The fission width of the neutron
resonanance into the channel c is:
Js 2
sJ s , cK s | bsK
| cK s ,
(13)
s
where cKs is the transition probability from the pre-scission state with the quantum number Кs
into the channel с, while the quantity
Js
Js Js
bsK
 asK
csK
s
s
(14)
s
J
contains the probability amplitude asKs s to find the component with the quantum number Ks in
J
the wave-function of a given neutron resonance and the amplitude c sKs s of the transition from this
resonance via the transition state with quantum numbers Js , Ks to the scission point. Due to the
J
complete K-mixing [9] in neutron resonances the amplitude asKs s has random sign and the
1
average value  (asKs )  ( 2 J s  1) . The coefficients d l ( ) and the phsase shifts  l ( )
s
define the ternary particles’ angular distribution for the reaction with unpolarized neutrons.
The phase shifts  sJ s s ' J s ' of the interfering neutron resonances are defined as:
J
2
 sJ s s ' J s '   sJ s   s ' J s '
(15)
Introducing the amplitude Ac ( ,  ) of the ternary particles’ angular and energy distribution in
the absence of the Coriolis interaction:
Ac0 ( ,  )  | d cl ( ) | exp i cl ( )Yl 0 ( ) | Ac0 ( ,  ) | exp{i c0} , (16)
0

l
one can write eq.(11) in the form:
d 2 0

Js

| hsJ s || hsJ's ' | bsK
b J s '  (2 J s  1) 

2
s s ' K s ' cK s
d  d  2(2 I  1)kn ss ' J s J s 'cK s
(17)
 cos( sJ s s ' J s ' ) | A ( ,  ) | .
0
c
2
The amplitude Ac ( ,  ) of the alpha particles’ angular and energy distributions which takes
into account the Coriolis interaction can be represented as a sum of the two parts corresponding
to the even and odd values of the alphas’ angular momenta ℓ:
Cor
AcCor ( ,  )  | dclCor ( ) | e
 clCor ( )
Y
l ,1
( )  Yl ,1 ( )  
l
 A
Cor
c even 
(18)
( ,  ) exp{i
Cor
c even 
} A
Cor
c odd 
( ,  ) exp{i
Cor
c odd 
},
where the coefficients d l  and the phase shifts  l '
take into account the Coriolis interaction.
It is convenient to choose the coordinate system which corresponds to the experimental geometry



– the vectors  n and k LF directed along the y and z axes, while the vector k lies in the (x,y)
Cor
Cor
plane. With this coordinate choice one can represent the differences Yl ,1 ( )  Yl ,1 ( )  of
the spherical functions in (18) as:





 n [k LF  k ]{a0  a1 (k LF  k ) 2  ...}
for the odd ℓ values, and






(19)

 n [k LF  k ](k LF  k ){b1  b3 (k LF  k ) 2  ...}
(20)
for the even ℓ values. Then the amplitude (18) can be written in the form:

AcCor ( ,  )   n [k LF
 

i cCor
Cor
 k ](k LF  k ) | Fc (even) ( ,  ) | e ( ev) 

i cCor
 
( odd )
  n [k LF  k ] | FcCor
|
e
,
( odd )
(21)
where the functions Fc even  ( ,  ) and Fc odd  ( ,  ) coincide with the amplitudes
Cor
AcCor
 even  ( ,  )
and
Cor
AcCor
 odd  ( ,  )
provided
one
substitutes
the
differences
Yl ,1 ( )  Yl ,1 ( )  of eq.(18) by the expressions in the curly brackets of eqs.(19) and (20).
Now by using the technique of [1] we can obtain:
pn
d 2 Cor
J s J s'

| hsJ s || hsJ' s ' | bsK
b

(2 J s  1)(2 J s '  1) 

2
s s ' K s ' cK s
d d 2(2 I  1)k n ss ' J s J s ' cK s

 
0
Cor
(22)
 g K s J s J s ' | Ac0 ( ,  ) | {| FcCor
|
sin(





)

[
k

k
( odd )
sJ s s ' J s '
c
c ( odd )
n LF
]

 

0
Cor 
 | FcCor
|
sin(





)

[
k

k
](
k

k
( even)
sJ s s ' J s '
c
c ( ev )
n LF

LF
 )}.
Cor
/ d l ) ≈10-3 here defines the magnitude of the observed effect caused by the
Coriolis interaction (8) . The factor g K s J s J s ' coming from the non-diagonal part of the density
The ratio (d l
matrix
Js Js '
MM
' (see [1]) has the form:
g Ks J s J s '  A  J s , J s '  


 J s  K s  J s  K s  1CJJ 1K K 11 
 J s  K s  J s  K s  1C
s' s
s
J s 'Ks
J s 1 K s 11
s
(23)
,

where

Js
A J s , J s '    Js ,Js ' 


 2  J s  1 J s , J 

2Js  1
2Js  1

 J s , J s ' 1 
 J , J 1 ,
2Js
2  J s  1 s s '

Js 1
 Js ,J  

2Js

(24)
with J   I  1/ 2 , J   I  1/ 2 .
As pointed earlier (see, e.g. [10]) the conservation of the K quantum number implies that
the fissioning system can not be thermalised on its way from the saddle-point to scission.
Therefore the superfluid correlations in the neck enhance the formation of alpha-clasters with the
orbital momentum ℓ=0. The admixture of the states with ℓ≠0 is caused by the long-range
Coulomb potential of the fragments (pre-fragments) acting on the alpha-particle [11]:
VCoul 
2( Z  2)e 2 8( Z  2)e 2  Z
2A  r 



  P1 (cos ) 
R
R
Z

2
A

4

 R 
16( Z  2)e 2  r 
0
d
Q
  P2 (cos )  VCoul  VCoul  VCoul
R
R
2
(25)
Here R is the distance between the fragments’ centers, r is the distance between the alphaparticle center and the center of mass of the whole system; Z and A are the charge and mass of
the fissioning nucleus. The quantities ΔZ=Z1-Z2 and ΔA=A1-A2 characterise the charge and mass
asymmetry of the fission fragments.
d
Now the dipole component VCoul of this potential dominating in the interior region r<<R adds
Q
the claster states with the orbital momentum ℓ=1. The quadrupole component VCoul which
dominates in the exterior region r≈R adds the states with the orbital momentum ℓ=2.
The first sum in the curly brackets of (22) corresponds to the odd ℓ values and is proportional to
sinθ. Therefore it has a maximum at θ=900 and changes only slightly (from 1 to 0.87) in the
range 600≤θ≤1200. This sum is responsible for the TRI effect.
The second sum corresponds to the even ℓ values and is proportional to cosθ. Therefore it
changes sign at θ=900. It is responsible for the ROT effect.
Thus, if the Coriolis forces act in the interior region (r<<R), where the dominant dipole term of
(25) admixes ℓ=1 to ℓ=0, then we have the TRI effect which was described in our earlier
publications. If, however, the Coriolis forces act in the exterior region r≈R then the dominant
quadrupole term of the Coulomb potential admixes the even ℓ values to ℓ=0. This leads to the
ROT effect.
As seen from (22) both effects are T-odd. However the TRI effect changes sign with the

 
inversion of each of the three vectors (  n , k LF and k ), in agreement with the experimental
observations. This, as well as the form of dipole term in (25) shows that the TRI effect is caused
by the charge and mass asymmetry of the fission fragments. The ROT effect changes sign only


with the inversion of the neutron beam polarization (  n   n ) since it is connected with the
Q
quadrupole term VCoul , which is independent of the charge and mass asymmetry. Exactly this is
observed in the experiments [5].
In general both correlations should contribute to the observed effect (and, as pointed above,
the experimental data do indicate this). However the TRI correlation dominates for the 233U
target, while the ROT one dominates in the case of 235U. Both fissioning compound nuclei (234U
and 236U) are the neighbouring even-even isotopes whose fission modes q and fission channels c
practically does not differ. The charge and mass fragments’ asymmetries are also practically the
same. Even if the K quantum numbers of their transition states are different, our exp. (22) shows
that this would only affect the common K-dependent factors in front of both correlations, thus
enhancing or hindering both effects simultaneousely.
Therefore the question arises – why are the ratios of TRI to ROT contribuions to the BCor values
so different for those two targets?
The only possible source of this difference might be caused by the different energies and other
parameters of the inerfering neutron resonances contributing to the effects. This would lead to
the appearance in eq.(22) of the different phases  sJ s s ' J s ' coming from the interfering resonances.
If the phase difference in the arguments of the sines is about zero for the leading even l -values
the ROT effect is suppressed and we have the dominant TRI one, and vice versa.
3. The special case of isolated resonance and comparison with the classical
approach
Consider now the special case of isolated neutron resonace which is a rather rare
phenomenon for the fissioning nuclei where the typical ratio of the total width to the resonance
spacing is (Γ/D)~0.3. Then only one term s, J s  s ', J s ' contributes to the sums in (17), (22) and
those differential cross-sections are:
d 2 0

Js 2

| h J s |2 | bsK
| cKs (2 J s  1) | Ac0 |2
2  s
s
d  d  2(2 I  1)kn cKs
(26)
d 2 Cor
pn
Js 2

| h J s |2 | bsK
| cKs (2 J s  1)g Ks J s J s | Ac0 | 
2  s
s
d  d  2(2 I  1)kn cKs

 
0
Cor
 { n [k LF  k ]FcCor
( odd ) sin( c   c ( odd ) ) 
(27)
 

 
Cor
0
Cor
  n [k LF  k ](k LF  k ) Fc (even) sin( c   c (even) )}
Here the phases  sJ s s ' J s ' =0 and
g Ks J s J s  A( J s , J s )[ ( J s  K s )( J s  K s  1)CJJss1(KsK s 1)1 
 ( J s  K s )( J s  K s  1)C
J s Ks
J s 1( K s 1) 1
(28)
.
It is interesting to compare these results with the classical approach [6] where the angular
velocity Ωcl which defines the magnitude of the ROT effect was estimated by the eq.(3) with the
polarization of the compound-nucleus spin given by eq.(4):
J 1
 2I  3
p

pn
n
 3(2 I  1)
3
J
P( J )  
  1 pn
 3
for
J  I  1/ 2
(29)
for
J  I  1/ 2
We shall show now that this expression does not take into account the specific features of
J
the fission channel for the deformed nucleus with the axial symmetry. The wave function  MK
of this nucleus takes into account the fact that its eigenstates are always doubly-degenerate in the
sign of K. Therefore it contains with equal weights the contributions of the functions
J
DMK
  and DMJ  K   depending not only on the total spin J and its projection M on the
laboratory z-axis, but also on its projection K on the nuclear symmetry axis. The average values
of the system’s spin projections on the axes i=(х, у, z) are defined as:
J
J
J
ˆ
 J i ( K )    
'  MK | J i | MK 
(30)
 
Here
J
 
' is the density matrix defined in ref. [1]. With our coordinates choice the laboratory z-
axis coincides with the direction of the light fragment, while the y-one coincides with the
direction of the neutron soin polarization. Now we can choose the Euler’s angles  to be zero,
so that the axes of the laboratory and internal systems coincide. Then we can replace the
operators Jˆi of the spin projections on the laboratory axes by the projections on the internal
axes in (30) and use the expressions similar to (8a). As one should expect, the only non-zero
projection with this coordinate choice would be the y-one:
 J ( K )  J y ( K ) 
g KJJ
pn 
2
(31)
where g KJJ is defined by eq.(28) and contains the explicit dependence on the K-values. This
means that the fissioning system rotates around the y-axis with the angular velocity:
1
J
(32)
 J ( K ) 
pn  g KJJ  PK ( J )

2

The quantity PK ( J ) here means the polarization of the system with the fixed values of J and K:
 J ( K )  pn  g KJJ
(33)
PK ( J ) 

J
2J
q ( J , K ) 
Using now (31) and (28), we obtain:
 J s ( J s  1)  K 2 
 pn

J
2

s

gK J J 
q (J s , K s )  s s s pn  
2
 J ( J  1)  K 2 
 s s
 pn

( J s  1)
2
for
J s  I  1/ 2
(34)
J s  I  1/ 2
for
Cor
As mentioned above the coefficients dcl take into account the Coriolis interaction. Therefore
2
our quantum exp. (27) contains the coefficient
/ 2 and is proportional to the angular
velocity q ( J , K ) in complete analogy with the system’s rotation angle in the classical
approach [6].
If several transition states with different K-values contribute to fission then it follows from (27)
that the effective rotation velocity is:
J 2
J 2
eff
q  | bsK | cK q ( J , K )  | bsK | cK
K
K
J
PK ( J )

(35)
If all the K-values contribute to fission (i.e. all the coefficients | bsKs | | asKs csKs | equal
s
s
s
J
2
J
J
2
1
their average value (2 J s  1) ), then the polarization PK ( J ) of eq.(33) is averaged over all
the K-values and
1
 PK ( J )  P( J ) ,
(2 J  1) K
(36)
where P(J) is defined by (29) used in [6]. It is well known that neutron resonances in deformed
nuclei posess no definite value of the K quantum number, as a result of the complete K-mixing
caused by the dynamical enhancement of the Coriolis interaction [9]. Therefore for isolated
resonances, induced by polarized neutrons their decay into various neutron and gamma channels
is governed by the factor (29). However for sufficiently low neutron energies the saddle- point
in the fission channel selects only one-two values of K corresponding to the lowest collective
excitations. Exactly this specific feature of the fission channel prohibits the use of eqs.(3)-(5).
Notice that the K-dependence of the angular velocity (34) is rather strong. Indeed the value of
 q  J , K  changes by the factor of (2J+1) for K changing from K=0 to K=J. This leads to the
additional enhancement of the small K contributions to the TRI and ROT effects.
The ratio of the corresponding angular velocities (34) to (3)
 J

2
q 3

J ( J  1)  K 2   ( J  1)
 cl 2
 1
 J
for
J  I  1/ 2
(28)
for
J  J  1/ 2
varies by the numerical factor of 0.5—2 for J=3, 4, while their signs coincide. However the sign
of the ROT effect is defined not only by the sign of Ω but also by the function
sin( c0   cCor
( even) ) in (27)whose sign is defined by the leading even ℓ-values contributing to
the effect.
But most important is that the isolated resonance approximation can not explain the
difference of the effects in the two neighbouring uanium resonances. As already pointed this
difference can be explained only by the neutron resonances’ interference in eq.(22).
4. Summary
1. The account of Coriolis interaction in the framework of the quantum ternary fission theory
allows to explain both observed T-odd effects.
2. Both TRI and ROT contribute always, but their relative contribution varies (confirmed by the
preliminary experimental data).
3. The changing ratio TRI/ROT arizes ONLY FROM THE RESONANCE INTERFERENCE
EFFECTS and can’t be explained in classical approach.
J
4. This interference affect the value and the SIGN of both effects (via the signs of bsKs ,
s
g Ks J s J s ' and of the phases' differences in the sin arguments of (22)). Therefore in general the
sign of TRI is not correletaed with the sign of ROT.
5. Contribution of different transition states affects the value of both effects simultaneously but
does not change the TRI/ROT ratio
6. The relative TRI/ROT ratio (as well as their relative sign) should vary for the same target
with the variation of the incident neutron energy En which considerably changes the phase shifts
 sJ s s ' J s '   sJ s   s ' J s ' of the interfering resonances. (Therefore this energy variation should
be larger or of the order of the energy spacing between the neighbouring neutron resonances, i.e.
of the order of eV).
This work was supported by INTAS (grant № 03-51-6417) and by RFBR (grant № 0602-16668) .
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