Angular Distribution of Products of Ternary Nuclear Fission Induced by ...

c Pleiades Publishing, Ltd., 2008. ISSN 1063-7788, Physics of Atomic Nuclei, 2008, Vol. 71, No. 11, pp. 1887–1906. c V.E. Bunakov, S.G. Kadmensky, S.S. Kadmensky, 2008, published in Yadernaya Fizika, 2008, Vol. 71, No. 11, pp. 1917–1936. Original Russian Text

NUCLEI Theory

Angular Distribution of Products of Ternary Nuclear Fission Induced by Cold Polarized Neutrons V. E. Bunakov1)* , S. G. Kadmensky** , and S. S. Kadmensky Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006 Russia Received January 10, 2008; in final form, March 27, 2008

Abstract—Within quantum fission theory, angular distributions of products originating from the ternary fission of nuclei that is induced by polarized cold and thermal neutrons are investigated on the basis of a nonevaporative mechanism of third-particle emission and a consistent description of fission-channel coupling. It is shown that the inclusion of Coriolis interaction both in the region of the discrete and in the region of the continuous spectrum of states of the system undergoing fission leads to T -odd correlations in the aforementioned angular distributions. The properties of the TRI and ROT effects discovered recently, which are due to the interference between the fission amplitudes of neutron resonances, are explored. The results obtained here are compared with their counterparts from classic calculations based on the trajectory method. PACS numbers: 25.85.-w, 25.85.Ec DOI: 10.1134/S1063778808110069

1. INTRODUCTION In [1], it was found that angular distributions of products of the ternary fission of 233 U nuclei that is induced by polarized cold neutrons feature a T -odd correlation, which was later called a TRI correlation. This correlation has the form (σn , [k3 , kLF ]) , where σn is the neutron spin, while kLF and k3 are unit wave vectors of, respectively, a light fragment and a third particle that are emitted in the ternary-fission process. In [2, 3], a T -odd correlation, which was later called an ROT correlation, was discovered in angular distributions of products of the ternary fission of 235 U nuclei that is induced by polarized cold neutrons. The ROT correlation has the form (σn , [k3 , kLF ]) (k3 , kLF ) . In contrast to the TRI correlation, the ROT correlation does not change sign upon the inversion of one of the vectors kLF and k3 and vanishes at θ = 90◦ , where θ is the angle between the vectors kLF and k3 . On the basis of methods of quantum-mechanics ternary-fission theory [4, 5], the nature of T -odd correlations belonging to TRI and ROT types was explained in [6–9] via taking into account the effect of 1)

Petersburg Nuclear Physics Institute, Russian Academy of Sciences, Gatchina, 188300 Russia. * E-mail: [email protected] ** E-mail: [email protected]

rotation of the entire fissile system on the angular distributions of ternary-fission products. Owing to this rotation, the Hamiltonian describing it in the bodyframe system develops Coriolis interaction, which is responsible for the emergence of T -odd correlations being discussed. The objective of the present study was to consider in more detail angular distributions of products of ternary nuclear fission by means of a procedure that employs the results of the studies reported in [10– 13] and devoted to the quantum-mechanical theory of ternary nuclear fission. Here, particular attention is given to analyzing the Coriolis interaction effect both in the region of the discrete and in the region of the continuous spectrum of states of a fissile system with allowance for the coupling of ternary-fission channels and to solving the problem of the Coriolis interaction effect on the orbital motion of not only a third particle but also fragments originating from ternary nuclear fission. 2. ANGULAR DISTRIBUTIONS OF PRODUCTS OF TERNARY NUCLEAR FISSION IN (n, f ) REACTIONS WITHOUT ALLOWANCE FOR CORIOLIS INTERACTION Let us consider an (n, f ) reaction of ternary fission induced in an axisymmetric deformed target nucleus of spin I and its projection MI onto the z axis in the laboratory frame by polarized cold neutrons and

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accompanied by the emission of an alpha particle as a third particle. As was shown in [6], the reaction being studied proceeds predominantly through the formation of s-wave neutron resonances of a compound nucleus in the first well of the deformation potential, which are described by the wave functions ΨsJs M characterized by the spin Js ; its projection M onto the z axis in the laboratory frame; and other quantum numbers s, including the atomic weight A and charge Z of the compound nucleus. We now consider the compound-nucleus effective wave function ϕIms MI that is determined by a superposition of the wave functions for the neutron resonances excited in the (n, f ) reaction being studied and whose normalization takes into account the properties of the entrance neutron channel; that is, hJs s CIJs1M ΨJs s M , (1) ϕIms MI = M m 2

sJs

where

hJs s =

I

s

associated with the change in its deformation parameters, its axial symmetry being conserved, describes the dynamics of the fission process. The collective vibrational component having the dynamical amplis tude eJsK and taking into account the structure of s the fission barriers and transition fission states goes Js M for a specific fission over to the wave function ΨK s mode corresponding to the prescission configuration of the nucleus undergoing fission. The above scheme of the evolution of the neutron-resonance wave function ΨsJs M can be represented as s bJsK ΨJKssM , (3) ΨJs s M → s Ks s bJsK s

where the amplitude is given by the product of the above amplitudes; that is, s s = aJsK cJs eJs . (4) bJsK s s sKs sKs of the fissionThe shell-model component ΨJKssM

sh

ΓJsns

E − EsJs + iΓJs s /2

.

(2)

Here, ΓJsns is the amplitude of the neutronic width with respect to the decay of an s-wave neutron resonance; E is the total energy of the fissile system in the c.m. frame; and EsJs and ΓJs s are, respectively, the energy of the s-wave neutron resonance and its total decay width. The smallness of phase shifts for coldneutron potential scattering was taken into account in (2). It was shown in [14, 15] that, because of the effect of the dynamical enhancement of Coriolis interaction in a neutron resonance state, the wave function for this state, ΨJs s M , is a superposition of Js M for the the multiquasiparticle wave functions ΨsK s compound nucleus at fixed values of the projections Ks of the spin Js onto its symmetry axis, which is coincident with the z axis of the body-frame system of this nucleus. The coefficients in this superposition, s , has a stochastic character; the average value of aJsK s s 2 −1 their absolute value squared aJsK is (2Js + 1) ; s and their sign is random. In order to describe the fission of a compound nucleus from the state desM , one can isolate scribed by the wave function ΨJsK s in this function, a collective vibrational component s , whose modulus having a stochastic amplitude cJsK s 2 s squared cJsK has the average value N −1 , where s N is the number of multiquasiparticle components Js M . This component, which, in the wave function ΨsK s in the evolution of the nucleus undergoing fission, is

Js M can be represented in the mode wave function ΨK s form [16] 2Js + 1 Js M = (5) ΨK s 16π 2 sh Js × (1 − δKs ,0 ) DM Ks (ω) χKs (ξ)

Js (ω) χ (ξ) + (−1)Js +Ks DM ¯ Ks −Ks

√ J , + δKs ,0 2DMs 0 (ω) χ0 (ξ) Js where DM Ks (ω) is a generalized spherical harmonic depending on the Euler angles ω = (α, β, γ) that characterize the orientation of the body-frame axes of the nucleus undergoing fission with respect to the axes of the laboratory frame. The function χK¯ s (ξ) coincides with the time-inverse intrinsic wave function χKs (ξ), which depends on the set of intrinsic nuclear coordinates ξ. In addition to the shell-model Js M component, the fission-mode wave function ΨK s involves the cluster component ΨJKssM , which is cl

given by [12] Js M Js M Js M = G |H| Ψ . ΨK Ks s cl

sh

(6)

Here, H is the total Hamiltonian of the fissile system and GJs M (x, x ) is the multiparticle Green’s function, which depends on the total sets of coordinates x = ω, ξ and x = ω, ξ of the fissile system and which describes the evolution of this system in the

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ANGULAR DISTRIBUTION OF PRODUCTS OF TERNARY NUCLEAR FISSION

cluster region, where fission products have already been fully formed. In the case of ternary fission, the Green’s function GJs M with allowance for strong fission-channel coupling can be represented in the form [12] (7) GJs M x, x GαJs M + (x> ) + iFαJs M + (x> ) FαJs M − (x< ), =π α

where the index x> (x< ) corresponds to the set of coordinates x or x that includes the larger (smaller) of the hyperradii ρ and ρ appearing in these sets of coordinates. In order to derive an explicit expression for the Green’s function GJs M (7), the regular and irregular wave functions (FαJs M ± and GαJs M ± , respectively) normalized to a delta function in energy that describe the potential scattering of fission products on each other can be represented in the form of an expansion in the channel functions UαJs M as FαJs M ± =

J (±)

UαJs M

α

GJαs M ±

=

α

J (±)

fαsα (ρ) , ρ5/2

(8)

Js (±) Js M gα α (ρ) Uα , ρ5/2

J (±)

where fαsα (ρ) and gαs α (ρ) are, respectively, the regular and irregular radial form factors that are solutions to a set of coupled differential equations with the boundary conditions determined in [12]. The form J (−) J (−) factors fαsα (ρ) and gαs α (ρ) coincide with the J (+)

J (+)

time-inverse form factors fαsα (ρ) and gαs α (ρ), respectively. The channel function UαJs M appearing in (8) has the form

Js M = ΨJK11M1 (ω, ξ1 ) ΨJK22M2 (ω, ξ2 ) (9) Uα F MF

L l × i YLML (ΩR ) i Ylml (Ωr ) L0 ML0

× Ψ3 (ξ3 )

Js M

PLlλ (ε) . sin ε cos ε

Here, the radius vectors R and r are defined as R = R1 − R2 and r = R3 − (A1 R1 + A2 R2 )/(A1 + A2 ), respectively, where R1 and R2 are the coordinates of the centers of mass of fission fragments, R3 is the coordinate of the center of mass of the alpha particle (third particle), and A1 A2 ; α = cβLlλ, c = σ1 K1 σ2 K2 , and β = F L0 J1 J2 ; and the solid angles ΩR and Ωr determine the directions of the radius vectors R and r in the laboratory frame. The coordinates R and r are related to the hyperspherical coordinates ρ and ε [17], and the asymptotic energy of the third PHYSICS OF ATOMIC NUCLEI Vol. 71

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particle, E3 , is determined by the angle ε (0 ≤ ε ≤ π/2) via the relation √ ε = arccos x, (10) where x = E3 /E3max ; here, E3max is the maximum energy of the third particle, E3max = Qc (A1 + A2 )/A, with Qc being the total energy of the relative motion of ternary-fission products in channel c. The functions PLlλ (ε) are expressed in terms of Jacobi polynomials [17]. Together with the spherical harmonics YLML (ΩR ) and Ylml (Ωr ), these functions form a complete orthonormalized basis in the space of the solid angles ΩR and Ωr and the angles ε. In (9), Ψ3 is the alpha-particle wave function, while ΨJKiiMi (ω, ξi ) is the wave function for the ith axisymmetric fission fragment (i = 1, 2). The latter is given by (5) upon the substitution of the indices Ji Ki for the indices Js Ks and the corresponding fission-fragment intrinsic wave functions χKi (ξi ) for the intrinsic wave functions χKs (ξ). In constructing the channel function UαJs M (9), use was made of the result reported in [4, 18], where it was shown that the deviations of the Euler angles ωi characterizing the orientation of the body-frame coordinate axes of fission fragments with respect to the laboratory-frame axes from the Euler angles of ω characterizing the analogous orientation of the body-frame coordinate axes of the nucleus undergoing fission are small; therefore, the approximation ωi = ω can be employed. With the aid of expressions (7) and (8), the asymptotic behavior of the fission-mode wave function (6) at large values of ρ can be represented in the form [12] U Js M Js M α (11) ΨK s 5/2 as ρ α J ˜ ΓKss α Lπ , × exp i kc ρ − 2 υc ˜ = L + l + 2λ + 3/2; kc = √2Mc Qc /2 = where L Mc υc /, with Mc being the effective mass of the

ternary-fission products in channel c; and ΓJKss α is the complex-valued amplitude of the partial fission width, √ ΓJKss α = 2π (12) f Js (−) (ρ) Js M Js M αα U Ψ . × H Ks α sh ρ5/2 α Transforming the functions YLML (ΩR ) and Ylml (Ωr ) into the body-frame system of the fissile

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nucleus by means of the Wigner transformations [16] L YLML (ΩR ) = DM (ω) YLKL ΩR , (13) L KL KL

Upon employing, for Clebsch–Gordan coefficients, the orthonormality condition LK ¯ L0 K0 CLlKL¯L Kl CLlK = δL,L (17) ¯ 0 δKL ¯ ,K0 , L Kl Kl KL

we can recast the channel function (9) into the form (2J1 + 1) (2J2 + 1) Js M = (14) Uα 16π 2 Js iL il DM × K (ω) YLKL ΩR KK0 KF KL Kl

PLlλ (ε) Js K L0 K0 CF L0 KF K0 CLlK × YlKl Ωr L Kl sin ε cos ε F × χK1 χK2 CJF1K J2 K1 K2 J1 +J2 +K1 +K2 F KF + (−1) χK¯ 1 χK¯ 2 CJ1 J2 −K1 −K2 Ψ3 ,

KK0 Kl KF

where, in the braced expression on the right-hand side, we have written only two terms arising upon the multiplication of the fission-fragment intrinsic wave functions for Ki = 0 that have the form (5) and which correspond to the projections of the channel spin F that are equal to (K1 + K2 ) and −(K1 + K2 ). In expressions (9) and (14), the fission-fragment asymptotic relative orbital angular momentum L takes into account, through the law of conservation of the total spin of the nucleus undergoing fission, the effect of the relative orbital angular momentum l of the emitted alpha particle. Therefore, it is natural to go over to the representation where, in the fission-fragment relative orbital angular momentum L, one singles out the ¯ that is independent of the alpha-particle component L orbital angular momentum l and which corresponds to it in the case of binary nuclear fission. For this, we go over, in the spherical harmonic YlKl (Ωr ), to the reference frame associated with radius vector R, by means of a Wigner transformation of the form (13), l DKl K ΩR YlKl (ΩrR ), (15) YlKl Ωr = Kl

l

where ΩrR is the solid angle characterizing the direction of the radius vector r in the above reference frame, and employ the relation [16] l (16) DK ΩR YLKL ΩR K l l L 2L + 1 l DKl K ΩR DK ΩR = L0 l 4π LK ¯ ¯ 2L + 1 L¯ LK l DK ¯ K ΩR CLlKL¯L Kl CLl0K = . L l l 4π ¯ LKL ¯

we can recast the channel function UαJs M (4) into the form (2J1 + 1) (2J2 + 1) Js M Uα = (18) 16π 2 2L + 1 L l Js L0 i i DM K (ω) DK × 0 Kl 4π L0 K × ΩR YlKl (ΩrR ) CFJsLK0 KF K0 CLl0Kl l F KF J1 +J2 +K1 +K2 × χK1 χK2 CJ1 J2 K1 K2 + (−1)

Ψ P (ε) 3 Llλ F . × χK¯ 1 χK¯ 2 CJF1K J2 −K1 −K2 sin ε cos ε In constructing the amplitude of the ternaryΓJKss α (12), it is necessary fission partial width, to take into account the following. First, the radius vector R specifying the direction of escape of fission fragments is parallel or antiparallel to a high precision to the symmetry axis of the fissile nucleus both in the case of binary and in the case of ternary fission; therefore, the projections K0 and KL of the fissionfragment relative orbital angular momenta L0 and L onto the symmetry axis of the nucleus undergoing fission are zero. Second, the projections Kl and Kl of the alpha-particle orbital angular momentum l onto, respectively, the symmetry axis of the fissile nucleus and the radius vector R are also equal to zero because of the effect of superfluid nucleon–nucleon correlations on the formation of an alpha particle in ternary fission. By using methods developed in [4, 12] for the most probable channels c of binary and ternary fission and expression (16) for the channel function UαJs M , we can then represent the amplitude ΓJKss α (12) in the form (2J1 + 1) (2J2 + 1) Js (19) ΓK s α = a 2Js + 1 √ s C F KF C L0 0 d × 2L + 1 CFJsLK0 K F 0 J1 J2 K1 K2 Ll00 clλ Js × Θ (Lmax − L0 ) exp iδK ΓK s c , sα

where, because of the parity-conservation law, (−1)L0 = ππ1 π2 , with π being the parity of the parent nucleus and π1 and π2 being the parities of fission fragments; dclλ are real-valued coefficients determining the alpha-particle angular and energy distribution


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normalized to unity; Θ (x) is the Heaviside step function, which depends on the maximum value Lmax of the relative orbital angular momentum L0 Js of binary-fission fragments; and δK is the potential sα fission phase shift for channel α. In employing expression (19) in the subsequent analysis, it is necessary to take into account the fact that, because of the smallness of the centrifugal potential in relation to the kinetic energy of the relative motion of fission fragments at the point where the fissile nucleus undergoes scission, the dependence of the potential Js on the orbital angular mofission phase shift δK sα menta L0 and L and on the spins Js , J1 , J2 , and F is negligible [4]; therefore, we can treat this phase shift as a function of only the channel indices clλ Js = δclλ . At the same and represent it in the form δK sα time, fulfillment of the condition Lmax l, in which case the characteristic values of the fission-fragment relative orbital angular momenta L and L0 are much larger than the characteristic values of the alphaparticle orbital angular momentum l, implies that we can replace, in the Jacobi polynomial PLlλ (ε), which appears in the definition of the channel function ˜ l. in (18), the quantity L by some averaged L In this case, the normalization constant a in (19) is determined from the condition that the positive real quantity ΓKs c coincides with total ternaryfission width of the prescission state with respect to channel c: 2 Js (20) ΓK s c = ΓK s α .

J1

s CFJsLK0 KF K0 CFJsLK0 K F0

F

where the factor fLmax (ΩR ) has the form YL0 0 ΩR cL0 πL0 , fLmax ΩR = dLmax

L

with

as

=

c

= δKl ,0 ,

2Js + 1 16π 2

(22)

Js × DM Ks (ω) χK1 χK2

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πL0 = 1 + (−1)L0 ππ1 π2 /2, cL0 = ⎡ =⎣

2L0 + 1Θ (Lmax − L0 ) , dLmax

(24)

(25)

⎤−1/2

(2L0 + 1) πL0 Θ (Lmax − L0 )⎦

.

L0

In (22), the factor gc (ΩrR , ε) determines the alpha-particle angular and energy distribution normalized to unity dcl (ε) Yl0 (ΩrR ), (26) gc (ΩrR , ε) = l

where dcl (ε) =

dclλ eiδclλ PLlλ ˜ (ε)

(27)

λ

= |dcl (ε)| eiδcl (ε) , |gc (ΩrR , ε)|2 dΩr dε =

d2clλ = 1.

(28)

lλ

The spherical harmonic Yl0 (ΩrR ), which appears in the definition of the factor gc (ΩrR , ε) in (26), can be recast into the form ∗ 4π YlKl Ωr YlK ΩR , Yl0 (ΩrR ) = l 2l + 1 (29)

we can find for K1 = 0, K2 = 0 fission channels, in which case Ks = K1 + K2 , that Js M ΨK s

(23)

L0

Kl

(2F + 1) = δK,Ks δK0 ,0 , (2Js + 1)

L0 Kl L0 0 (2L + 1) CLl0K CLl00 l (2L0 + 1)

1891

Js + (−1)Js +Ks DM ¯ 1 χK ¯ 2 Ψ3 −Ks (ω) χK gc (ΩrR , ε) eikc ρ ΓKs c × fLmax ΩR , sin ε cos ε ρ5/2 υc

βLlλ

Substituting expressions (16) and (17) in the definiJs M in the asymptotic tion (11) of the wave function ΨK s region of large values of ρ and using the relations 2 (2J + 1) FK 1 = 1, (21) CJ1 J2FK1 K2 (2F + 1)

where the spherical harmonics YlKl (Ωr ) and ∗ (Ω ) are associated with the body-frame system YlK R l of the fissile nucleus. However, formula (29) will not change if, in the summand on its right-hand side, one employs spherical harmonics dependent on the solid angles Ωr and ΩR of the radius vectors r and R in an arbitrary system of Cartesian coordinates. This means that Yl0 (ΩrR ) is a scalar function and that expression (22) takes correctly into account conservation laws for the total spin and parity of a fissile nucleus in its ternary fission. The above consideration confirms the validity of formula (22),

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which was used previously in [10−12] in describing angular distributions and P -odd, P -even, and T -odd asymmetries in angular distributions of products of ternary nuclear fission. At high values of Lmax , the form factor fLmax (ΩR ) (23) is different from zero in a narrow region of angles around θ = 0 and around θ = π; to a high θR R R precision, we can therefore represent this form factor as 1 − 1 δ ϕR (30) fLmax ΩR = √ δ˜ cos θR 2

+ 1 δ ϕR + π + (−1)L0 δ˜ cos θR

1 ˜ δ (cos θR − cos β) δ (ϕR − α) . =√ 2 + (−1)L0 δ˜ (cos (π − θR ) − cos β)

× δ (ϕR + π − α) , where δ˜ (x) is a smeared delta function. We align the z axis of the laboratory frame with the direction R1 along which a light ternary-fission fragment escapes. By employing the results obtained in [13], we can show that, because of the smallness of the alpha-particle mass and kinetic energy in relation to the masses and kinetic energies of fragments originating from ternary fission, the angle ∆θRR1 between the direction R1 in the c.m. frame of the nucleus undergoing fission and the radius vector R has a small value not exceeding 3◦ , which corresponds to the maximum alpha-particle energy of Eα = 30 MeV. In this case, the vector R is parallel to the z axis of the laboratory frame to within small corrections associated with the effect of recoil of the fissionfragment center of mass because of alpha-particle emission. The laboratory frame can then be chosen to be coincident with the reference frame associated with the radius vector R. Because of the axial symmetry of the system undergoing fission, the third Euler angle γ appearing in the set of Euler angles ω characterizing the orientation of body-frame axes with respect to the laboratory-frame axes is excessive and can therefore be set to γ = 0 [16]. The square of the form factor (30), which depends only on the Euler angles α and β, can then be replaced by the form factor f 2 (ω), which includes the delta function δ (γ) and whose integral with respect to all Euler angles ω is equal to unity. In view of the fact that, in constructing the function fL2 max (ΩR ), δ˜2 (x) can be replaced to a high precision by δ (x) (this corresponds to the Bohr limit, in which Lmax → ∞), the form factor f 2 (ω) can be

represented as 1 [δ (cos β − 1) δ (α) 2 + δ (cos β + 1) δ (α − π)] δ (γ) . f 2 (ω) =

(31)

(1)

The first term f 2 (ω) in (31) corresponds to the situation where the body-frame system of the fissile nucleus coincides with the laboratory system. In this J J case, the Wigner function DM K (ω) = DM K (α, β, γ) corresponds to the Euler angles of α = β = γ = 0 and proves to be [16] J DM K (0, 0, 0) = δM,K .

(32)

(2)

The second term f 2 (ω) in (31) corresponds to the case where the body-frame system of the fissile nucleus is obtained upon the inversion of all axes of the laboratory frame. It follows that, since the total fissilenucleus wave function ΨJM K has a specific parity π and, upon the inversion of all coordinates, goes over to the function πΨJM K , any combinations formed by and ΨJM the fissile-system wave functions ΨJM K K , contained in the definition of the (n, f ) cross section together with the second term in (31), and character ized by the structure

dωf 2

(2)

(ω)ΨJM ΨJM K K

∗

are

equal to the analogous combinations including the first term in (31) and having the form (1) JM ∗ dωf 2 (ω)π 2 ΨJM K (ΨK ) (1) ∗ = dωf 2 (ω)ΨJM , ΨJM K K since π 2 = 1. In follows that, in calculating the (n, f ) cross section, we can represent the form factor f 2 (ω) (31) as f 2 (ω) = δ (cos β − 1) δ (α) δ (γ) .

(33)

Substituting expression (29) for the function Yl0 (ΩrR ) into the form factor gc (ΩrR , ε) (26) and using the properties of the form factor fLmax (ΩR ) (30), we can recast (22) for the asymptotic wave expression function ΨJKssM

into the form 2Js + 1 Js M = ΨK s 16π 2 as c Js × DM Ks (ω)χK1 χK2 as

(34)

Js + (−1)Js +Ks DM (ω) χ χ ¯ ¯ K 1 K 2 Ψ3 −Ks × fLmax ΩR Pl θR Yl0 Ωr


l

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×

ρ

dcl (ε) eikc sin ε cos ε ρ5/2

ΓK s c , υc

In constructing formula (39), we replaced the form factor fL2 max (ΩR ) by the form factor f 2 (ω) (33). Ex-

) is a Legendre polynomial, which is where Pl (θR related to the function Yl0 (ΩR ) by the equation 2l + 1 Yl0 ΩR = Pl (θR ). 4π Substituting the wave function (34) into expression (1), introducing the multidimensional operator jρ for the particle-density flux in the direction of the hyperradius ρ, and performing integration with respect to the intrinsic coordinates ξ1 , ξ2 , and ξ3 of fission products and with respect to the Euler angles ω, we obtain the (n, f ) double-differential cross section in the form 0 d2 σnf 4π = 2 B 0 (ΩR , ΩrR , ε), (35) dΩr dε kn

J J

pression (39) involves the spin density matrix ρMs Ms taking into account the interference between fission amplitudes for different s-wave neutron resonances sJs and s Js that was constructed in [6] for the case where the neutron polarization vector pn is parallel or antiparallel to the y axis of the laboratory frame. Specifically, this spin density matrix has the form J J

=

B 0 (ΩR , ΩrR , ε) sJ

sJs s Js Ks c

In (36),

sJ s J DKss s sJ

DKss

s J

s

DKss

and

s J

+

×

MM

sJ

ΓKs c BcKss

s

(43) A (Js , Js ) = δJs ,Js Js Js + 1 δJs ,J< − δJs ,J> × 2 (Js + 1) 2Js 2Js + 1 2Js + 1 δJ ,J −1 δJs ,Js +1 + − 2Js 2 (Js + 1) s s

.

with J> = I + 1/2 and J< = I − 1/2.

sJ s J BcKss s

are given by Js Js Js s = bJsK b h h s s , s s Ks sJ s Js

s

BcKss

=

1 16π 2 J J

(37) (38)

sJ s J

s (2Js + 1) (2Js + 1)ρMs Ms AcKss M M .

sJ s J

s The quantity AcKss M M , which appears in (38), has the form Js∗ sJs s Js Js (ω) D (39) AcKs M M = dω DM Ks M Ks (ω)

Js∗ Js + (−1)Js +Js +2Ks DM −Ks (ω) DM −Ks (ω) |dcl (ε)| |dcl (ε)| Pl θR × f 2 (ω)

ll

Yl0 Ωr

ipn 2 (2I + 1)

where the factor A (Js , Js ) has the form

(36) s J

1 δJ ,J δM,M (42) 2 (2I + 1) s s J M Js M A (Js , Js ) CJss1M + C 1 Js 1M −1 ,

ρMs Ms =

where kn is the incident-neutron wave function and the quantity B 0 (ΩR , ΩrR , ε) has the form

1893

× " ! × exp i δsJs s Js + δcl (ε) − δcl (ε) ,

which stems from the second term in (42) and which has only off-diagonal components in the indices M and M , does not contribute to the quantity in (38) or, hence, to the (n, f ) differential cross section (35). Upon applying the Wigner transformation inverse to the transformation in (13), spherical harmonics belonging to the same type as Yl0 (Ωr ) in (39) can be transformed into the laboratory frame: l∗ Ylml (Ωr ) Dm (ω). (44) Yl0 Ωr = l0 ml

sJ s J

where the phase δsJs s Js is (40)

while the phase δsJs , which is associated with the neutron resonance sJs , is given by (41) hJs s = hJs s eiδsJs . PHYSICS OF ATOMIC NUCLEI Vol. 71

J J

dices M and M . But in the spin density matrix ρMs Ms , σ J J the neutron-polarization-dependent part ρMs Ms ,

Performing integration with respect to Euler angles dω in (39) with the aid of formulas (44), (32), (33),

Pl θR Yl 0 Ωr

δsJs s Js = δsJs − δs Js ,

Upon taking into account relation (32), it can be sJ s Js proven that the factor AcKss M M is diagonal in the in-

No. 11

and (42), we can recast the quantity BcKss s in (38) into the form (2Js + 1) δJs ,Js sJ s J (45) BcKss s = (8π 2 ) · 2 (2I + 1) |dcl (ε)| |dcl (ε)| Yl0 (Ωr ) Yl 0 (Ωr ) ×

2008

ll

× cos (δsJs s Js + δcl (ε) − δcl (ε)) .

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BUNAKOV et al.

In this case, the (n, f ) double-differential cross section (35) reduces to the form 0 d2 σnf

1 (46) dΩr dε 2πkn2 sJs s Js (2Js + 1) DK × s 2 (2I + 1) ss Js cKs |dcl (ε)| |dcl (ε)| Yl0 (Ωr ) Yl 0 (Ωr ) × =

ll

× cos (δsJs s Js + δcl (ε) − δcl (ε)) ΓcKs . If the amplitude A0c (θr , ε) of the angular and energy distribution of alpha particles is defined as |dcl (ε)| eiδcl (ε) Yl0 (Ωr ) (47) A0c (θr , ε) ≡ l

× A0c (θr , ε) eiδc (θr ,ε) , then expression (46) can be recast into the form 2 1 0 (θ , ε) (48) A r c ¯ dΩr dε 2πkn2 (2Js + 1) sJs s Js ΓcKs , DK cos (δsJs s Js ) × s 2 (2I + 1) 0 d2 σnf

=

ss Js cKs

where use was made of the fact that, for ternaryfission fragments possessing close charge and mass 2 asymmetries, the quantity A0c (θr , ε) is only slightly dependent on the channel index c and can therefore be factored outside the sign of the sum in (48) over c at c equal to its most probable value c¯. As follows from experimental data on ternary nuclear fission induced by unpolarized cold neutrons [1, 2], 2 the quantity A0c¯ (θr , ε) has a Gaussian form whose maximum is shifted with respect to π/2 by a relatively small angle determined by the effect of the charge and mass asymmetries of the most probable ternaryfission fragments. 3. FISSILE-NUCLEUS WAVE FUNCTIONS WITH ALLOWANCE FOR CORIOLIS INTERACTION In order to describe T -odd correlations in angular distributions of products of ternary fission induced in unpolarized target nuclei by polarized cold neutrons, it was proposed in [6] to take into account the Hamiltonian of Coriolis interaction, HCor , in the body-frame system of the nucleus undergoing fission. The general form of the Hamiltonian HCor is [16] HCor = −

2 (J+ j− + J− j+ ) , 2J0

(49)

where J and j are the operators of, respectively, the total and the intrinsic spin of the fissile nucleus, while J± = Jx ± iJy , j± = jx ± ijy , (50) with x and y being the indices of the axes of the body-frame system. Expression (49) involves the fissile-system moment of inertia J0 , which changes as fission products move apart, from the moment of inertia of the fissile system in the prescission configuration to about Mc R2 for R → ∞, where Mc is reduced mass of fission fragments in the fission channel c being considered. Upon taking into account the Coriolis interaction (49), the Hamiltonian of the fissile system can ˜ = H + HCor , where be represented in the form H H is the Hamiltonian used above in (6) for the fissile system without allowing for Coriolis interaction. Since, on the energy scale, the Hamiltonian HCor is much smaller than the operators appearing in the unperturbed Hamiltonian H, the Coriolis interaction effect on the ternary-fission process can be taken into account in the first order of perturbation theory. In this case, all terms in (6) that describe the fission process without allowance for Coriolis interaction—namely, the Green’s function GJs M , the fissile-system HamilJs M tonian H, and the shell-model component ΨK s

sh

of the fission-mode wave function—must be replaced by their counterparts distorted by the Coriolis interaction in the first order of perturbation theory; that is, ˜ = H + HCor , ˜ Js M = GJs M + ∆GJs M , H G ˜ Js M = ΨJKssM + ∆ΨJKssM . Ψ sh

sh

sh

The unperturbed cluster component of the fission

Js M mode wave function, ΨK s

(6), is then replaced cl ˜ Js M that is perby the cluster component Ψ Ks cl

turbed interaction and which is given by Coriolis Js M Js M Js M ˜ = ΨK s + ∆ΨK , the quantity by ΨKs s cl cl cl Js M being defined as ∆ΨK s cl Js M (51) ∆ΨK s cl = GJs M HCor GJs M H ΨJKssM sh Js M Js M + G HCor ΨKs sh Js M Js M + G H ∆ΨKs sh Js M Js M = G HCor ΨKs cl


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Js M + GJs M HCor ΨK s sh Js M Js M . + G H ∆ΨKs sh

It was shown in [10–12] that, in the ternary-fission process, an alpha particle is formed prior to the scission of the fissile nucleus in a bound superficial cluster state that isdescribed in terms of the corresponding α

Js M of the shell-model wave funccomponent ΨK s Js M . It is precisely this component that detion ΨK s sh

termines the properties of the ternary-fission process accompanied by alpha-particle emission. In this case, the operator representing of the total intrinsic spin of the fissile nucleus, j, and appearing in the Coriolis interaction Hamiltonian (49) can be represented in the form (52) j = J0 + J3 + l, where J0 is the operator of the total intrinsic spin of the fissile-nucleus core formed upon the separation of an alpha particle having an orbital angular momentum l and an intrinsic spin J3 = 0 from the nucleus undergoing fission. The action of the operators J± , which appear in the Hamiltonian HCor specified by Eqs. (49) and (50), on the generalized spherical harJ monics DM K (ω), which describe the rotation of the fissile nucleus as a discrete unit, and the action of the operators J0± and l± on the intrinsic spin functions ΨJ0 K0 for the fissile-nucleus core and on the spherical harmonics YlKl (Ωr ), which describe the motion of the alpha particle in the space of angles in the bodyframe system, are given by [16] J J± DM K

leads to the appearance of core intrinsic wave functions that are orthogonal to the unperturbed intrinsic wave function for the core. that component Therefore, Js M in (51) which is of the wave function ∆ΨKs cl

associated with the last two terms in (51) is orthogonal to the unperturbed wave function ΨJKssM

= [(J0 ∓ K0 ) (J0 ± K0 + 1)]1/2 ΨJ0 K0 ±1 , l± YlKl Ωr = [(l ∓ Kl ) (l ± Kl + 1)]1/2 Yl(Kl ±1) Ωr . Since the alpha-particle intrinsic wave function is associated with the spin of J3 = 0, the application of the operator J3± to this function yields zero, as follows from (53). Because of the effect of superfluid nucleon–nucleon correlations [10], the alpha-particle orbital angular momentum l in the superficial cluster state of the nucleus undergoing fission is zero. It follows from (53) that the application of the term l± in the operator j± (52) to the alpha-particle component Js M of the shell-model wave function ΨKs sh therefore

No. 11

, with

the result that, in the (n, f ) cross section, the term that stems from the interference between the aforementioned components of the functions ∆ΨJKssM cl and of the function ΨJKssM vanishes. It follows cl

that, in (51), one can retain only the first term, in which the Coriolis Hamiltonian acts directly on the cluster component ΨJKssM

cl

of the wave function for

the nucleus undergoing fission where ternary-fission products have already been fully formed. Upon employing expression (7) for the Green’s function GJs M , the function in (51) in the asymptotic region can be recast into the form (54) ∆ΨJs M (x) as

=π GJαs M (x) + iFαJs M (x) α

. × FαJs M x HCor x ΨJKssM x In the cluster region, the operator J0 in (52) can be represented in the form (55) J0 = J1 + J2 + L,

J0± ΨJ0 K0

yields zero.

cl

cl

J = [(J ± K) (J ∓ K + 1)]1/2 DM (K∓1) (ω) ,


By using the methods developed in [7, 8], we can show that the application of that term in the operator HCor specified by Eqs. (49) and (51) which is associated with operator J0 of the intrinsic spin of the fissile-nucleus core to the core intrinsic wave function αinvolved in the definition of the wave function Js M and not perturbed by Coriolis interaction ΨK s

(53)

(ω)

1895

where J1 and J2 are the intrinsic-spin operators for ternary-fission fragments and L is the operator of their relative orbital angular momentum. By using the methods developed in [10, 11], one can show that the application of the operators J1± and J2± to fission-fragment intrinsic wave functions not perturbed by the Coriolis interaction leads to the appearance of fission-fragment intrinsic wave functions that are orthogonal to unperturbed intrinsic wave functions for the fission fragments being considered. It follows that, in the (n, f ) cross section, the term that the function from the interference between stems Js M and that part of the function ∆ΨJKssM ΨK s cl

cl

which is caused by the effect of the operators J1± and J2± vanishes. Therefore, we can retain only the

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BUNAKOV et al.

operators L and l in expression (52) for the intrinsic spin j of ternary-fission products, which appears in the definition of the Coriolis Hamiltonian HCor (49), (l) (L) and represent HCor in the form HCor = HCor + HCor , (l)

(L)

where the operators HCor and HCor are determined by those terms in (49) that are associated with the operators l± and L± , respectively. With allowance for the fact that total asymptotic the fissile-system J M ˜ Js M ˜ s has the form Ψ = wave function Ψ as as Js M + ∆ΨJs M , the flux jρ of fission prodΨK s as

as

ucts in the direction of the hyperradius ρ in the first order of the Coriolis interaction can be represented in the form jρ = j0ρ + ∆jρ , where the quantity j0ρ is Js M determined by the unperturbed function ΨK s

2Js + 1 Js DM (Ks +m) (ω) χK1 χK2 bm Ks Js 16π 2 cL0 lm Js m + (−1)Js +Ks DM (ω) χ χ b ¯1 K ¯ 2 −Ks Js K (−Ks +m) l (l + 1) Ylm Ωr × Ψ3 Pl θR ΓKs c ϕcl (ρ, ε) , × c˜L0 YL0 0 ΩR υc ρ5/2 sin ε cos ε m where m = ±1 and the quantities bm Ks Js and b−Ks Js have the form −1 (57) b+1 Ks Js = b−Ks Js = (Js − Ks ) (Js + Ks + 1); ×

as

and the quantity ∆jρ , which characterizes the Coriolis interaction effect on the angular and energy distributions of fission products, is generated by the Js M ΨK s

interference between the functions ∆ΨJs M .

(x)

and

as

as

In order to construct the function ∆ΨJs M (54), as Js M the structure of the cluster component ΨKs (x) cl

of the fissile-nucleus wave function not perturbed by the Coriolis interaction is studied here at the point x for a finite value of the hyperradius ρ on the basis of the transformationsused above to derive the asymptotic

Js M (x) in (34). In this case, wave function ΨK s as Js M (x) can be represented by (34) the function ΨK s cl

upon replacing the function dcl (ε) eikc ρ by the function ϕcl (ρ, ε) and the amplitude cL0 in the expression for fLmax (ΩR ) by the amplitude c˜L0 (R), which differs from cL0 by the introduction of Lmax (R), 1 Lmax (R) Lmax , in the Heaviside function instead of Lmax . It is noteworthy that, in the asymptotic limit ρ → ∞ (R → ∞), the function ϕcl (ρ, ε) and the quantity Lmax (R) reduce to their limiting values ϕcl (ρ, ε) → dcl (ε) eikc ρ and Lmax (R) → Lmax . Let us first consider the action of theoperator Js M on the wave function ΨK (x) intros

(l) HCor (x)

duced above. We have (l) HCor (x) ΨJKssM (x) = − cl

cl

(Js + Ks ) (Js − Ks + 1).

can now (l) calculate the behavior of the function We J sM (x) as with allowance for the evolution of ∆Ψ (l) the function HCor (x ) ΨJKssM (x ) (56), which was cl

rearranged under the effect of the Coriolis interaction, in the region x > x determined by the overlap integral of these functions and the Green’s function GJs M (x, x ), which takes into account only the potentials of fission-product interaction. Because of the axial symmetry of these potentials, the projections m = ±1 of the alpha-particle orbital angular momenta onto the fissile-nucleus symmetry axis, which are involved in expression (56), are also conserved in the Green’s function GJs M (x, x ) as integrals of the motion over the entire region x > x . By employing the methods above in constructing the func developed Js M (34), which was not perturbed by tion ΨKs (x) as (l) Js M (x) the Coriolis interaction, the function ∆ΨK s can be represented in the form (l) 2 2Js + 1 Js M =− ¯ ∆ΨKs 16π 2 as 2J0 × fLmax ×

ΩR

2J0 (R)

(56)

2 2J0 (R)

as

(58)

c

ΓK s c eikc ρ υc ρ5/2 sin ε cos ε

Js m DM (Ks +m) (ω) χK1 χK2 bKs Js

m

2

× (J+ l− + J− l+ ) ΨJKssM (x) = − cl

=

+1 b−1 Ks Js = b−Ks Js

Js m + (−1)Js +Ks DM ¯ 1 χK ¯ 2 b−Ks Js (−Ks +m) (ω) χK

× Ψ3

Cor l (l + 1) dcl (ε)

l


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2008


× Ylm Ωr

iδCor (ε) Pl θR e cl ,

× c˜L0 YLm ΩR

where the complex-valued amplitude dCor cl (ε) = Cor iδCor (ε) d (ε) e cl takes into account the character of cl the motion of the alpha particle in the region x > x in the Coulomb and nuclear fields of ternary-fission fragments for the case where the projection of the alpha-particle orbital angular momentum l onto the symmetry axis of the nucleus undergoing fission is fixed at m = ±1. This amplitudes differs markedly from the analogous amplitude dcl (ε) involved in the Js M (x) function ΨK s

as

(34), which is not perturbed

by the Coriolis interaction, and associated with the projection m = 0 of the alpha-particle orbital angular momentum. Expression (58) features the fissile-system average moment of inertia J¯0 , which takes effectively into account its dependence on the distance R between fission fragments. In order to take into account the action of the (L) operator HCor (x), which involves the operators L± depending on the solid angle ΩR of the radius vector R body-frame system, on the wave function in the Js M ΨK s

cl

L

For the function

(L) HCor (x)

Js M ΨK s

then obtain the the expression (L) Js M (x) =− HCor (x) ΨK s

(x) , we can cl

(60)

2J0 (x) × (J+ L− + J− L+ ) ΨJKssM (x) cl 2 2Js + 1 =− 2J0 (x) 16π 2

cl

Js m + (−1)Js +Ks DM ¯ 1 χK ¯ 2 b−Ks Js (−Ks +m) (ω) χK cL0 YLm ΩR × Ψ3

×

l

LL0

4π L0 2 ¯ CL0 l00 dcl (ε) Yl0 Ωr , 2l + 1

where the amplitude d¯cl (ε) differs from the amplitude dcl (ε) (27) by allowance in the former for alphaparticle interaction with fission fragments occurring in the states where the projections of their relative orbital angular momentum L are KL = m = ±1 rather than in the state where the respective projection is KL = 0. now consider that the wave function If we Js M (x) (34), which is not perturbed by the ΨK s as

Coriolis interaction and which (L)in the is involved Js M (61), interference with the function ∆ΨKs (x) contains the factor fLmax (ΩR ) (30) possessing the , properties of a delta function smeared in the angle θR then, in the interference terms # (L)∗ Js M ∆ΨK ΨJKssM s

as

Js m + (−1)Js +Ks DM ¯ 1 χK ¯ 2 b−Ks Js (−Ks +m) (ω) χK

4π Yl0 Ωr L (L + 1) 2l + 1


as

(L) ∗ $ Js M Js M + ∆ΨKs ΨK s ,

cL0 lm

(61)

m

as

Js m × DM (Ks +m) (ω) χK1 χK2 bKs Js

2 × Ψ3 CLL00 l00

this

Js M ∆ΨK s

as

2

cl

ΓKs c ϕcl (ρ, ε) . υc ρ5/2 sin ε cos ε

represented in the form (L) 2 2Js + 1 Js M =− ¯ ∆ΨK s 16π 2 as 2J0 c ΓK s c eikc ρ × υc ρ5/2 sin ε cos ε Js m DM × (Ks +m) (ω) χK1 χK2 bKs Js

, we isolate, in this function, the quantity

) depending on the angles Ω and YL0 0 (ΩR ) Pl (θR R recast it into the form (59) YL0 0 ΩR Pl θR 4π YL0 0 ΩR Yl0 ΩR = 2l + 1 L0 2 4π . YL0 ΩR CL0 l00 = 2l + 1

1897

case, the asymptotic function (L) (x) , which is related to the function as (L) Js M (x) (60) by Eq. (54), can then be HCor (x) ΨK s

In

No. 11

as

(L) Js M at it is necessary to take the function ∆ΨK s as

= 0 or π. But at these angles, the angles close to θR spherical harmonic YLm (ΩR ) at m = ±1 vanishes, which leads to the disappearance of the interference terms being considered. Therefore, we can disregard the effect of the Coriolis interaction on the orbital

2008

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BUNAKOV et al.

motion of ternary-fission fragments and take into account the effect of this interaction only on the orbital motion of the alpha particle. 4. ANGULAR DISTRIBUTIONS OF PRODUCTS OF POLARIZED-NEUTRON-INDUCED TERNARY FISSION OF UNPOLARIZED NUCLEI WITH ALLOWANCE FOR CORIOLIS INTERACTION (l) Adding the function ∆ΨJs M as (58) to the un Js M (34), substituting the perturbed function ΨK s as ˜ Js M function Ψ obtained in this way into (3),

Cor × exp i δcl (ε) − δcl (ε) .

In expression (63), the form factor fL2 max (ΩR ) (30) was replaced by the form factor f 2 (ω) (33). Performing integration with respect to the solid angle ω and making use of the relations (34), (13), and (44), we sJs s Js in (63) into the can then recast the quantity EM M form # sJs s Js (64) δM,Ks +m δM ,Ks bm EM M = Ks Js m Js +Js +2Ks

+ (−1)

as

calculating the multiparticle flux density jρ = j0ρ + ∆jρ in the direction of the hyperradius ρ, and performing integration with respect to the internal coordinates of the fission fragments and alpha particle and with respect to the Euler angles ω, we can find the first-order correction in the Coriolis interaction, BCor (ΩR , ΩrR , ε), because of the interference term Js M ∗ Js M Ψ ∆ΨK Ks s as

as

Js M ∗ Js M + ΨK ∆Ψ , Ks s as

as

B 0 (Ω

to the quantity R , ΩrR , ε) (36) appearing in the (n, f ) differential cross section (35). This correction can be represented by formulas (35)–(37) where the sJ s Js quantity AcKss M M in (39) is replaced by the quantity Cor sJ s Js , which has the form AcKss M M Cor % 2 & sJs s Js = − ¯ (62) AcKs M M 2J0 iδ iδ sJs s Js s Js sJs ∗ + e sJs s Js EM M , × e sJs s Js EM M where sJ s Js

s EM M

=

dωf 2 (ω)

m

Js∗ Js m × DM (Ks +m) (ω) DM Ks (ω) bKs Js Js + (−1)Js +Js +2Ks DM (−Ks +m) (ω)

J∗ × DMs −Ks (ω) bm −Ks Js

×

ll

l (l + 1)Ylm Ωr Pl θR

(ε) |dcl (ε)| × dCor Yl 0 Ωr Pl θR cl

(63)

×

$ δM,−Ks +m δM ,−Ks bm −Ks Js

l (l + 1)Ylm Ωr dCor (ε) cl

ll

Cor (ε) − δcl (ε) . × Yl 0 Ωr |dcl (ε)| exp i δcl

From expression (64), it follows that the function Cor sJ s Js in (62) is a nondiagonal matrix in AcKss M M

the indices M and M . This means that, upon the sJ s Js replacement of the function AcKss M M (39) by the Cor sJs s Js (62), expression (38) diffunction AcKs M M fers from zero only owing to the off-diagonal comJs Js σ in M and M of the spin denponent ρM M J J

sity matrix ρMs Ms (42), which is associated with the neutron polarization and which is determined by the second term in (42). If we consider that the matrix

J J

ρMs Ms

σ

changes sign under the interchange

of the indices Js M and Js M and that the matrix sJ s J DKss s (37) is invariant under the interchange of the indices sJs and s Js , we can perform the interchange of the indices sJs M and s Js M in the second term of expression (62) and replace the quantity iδsJs s J

e

s

s J sJs ∗

s EM M

in (62) by −iδ sJs s Js ∗ . −e sJs s Js EM M

(65)

If we substitute the first term of expression (62) into (38) and perform summation over M and M , taking into account (64) and employing the second term in expression (42) for the spin density matrix, the sJ s J Cor becomes corresponding quantity BcKss s % &% & 1 2 sJs s Js Cor = BcKs − ¯ (66) 16π 2 2J0 PHYSICS OF ATOMIC NUCLEI

Vol. 71 No. 11

2008


(2Js + 1) (2Js + 1) igKs Js Js pn (ε) dCor cl 2 (2I + 1) ll ! " × l (l + 1) Yl(−1) (Ωr ) − Yl1 (Ωr ) |dcl (ε)| Cor (ε) − δcl (ε) , × Yl 0 (Ωr ) exp i δsJs s Js + δcl

By analogy with the amplitude A0c (θr , ε) (47), we introduce the amplitude ACor c (Ωr , ε) as Cor (Ω , ε) = (ε) (71) ACor l (l + 1) d r c cl

×

l

" Cor × Yl(−1) (Ωr ) − Yl1 (Ωr ) eiδcl (ε) Cor iδc (θr ,ε) = ACor c (Ωr , ε) e !

where

(67)

gKs Js Js = A (Js , Js ) J K

s (Js + Ks ) (Js − Ks + 1)CJss1(K × s −1)1 Js Ks − (Js − Ks ) (Js + Ks + 1)CJs 1(Ks +1)−1 .

and define the quantity Fc as sJ s J Fc ≡ ΓcKs DKss s

Upon the substitution of the second term in (62) into (38) with allowance for (65) and performing summation over the indices M and M , there arises sJ s J Cor that is given by an a second term BcKss s expression differing from "(66) by the! substitution ! ∗ of − Yl(−1) (Ωr ) − Yl1 (Ωr ) and exp −i(δsJs s Js + " ! " Cor (ε) − δ (ε)) for Y δcl cl l(−1) (Ωr ) − Yl1 (Ωr ) and " ! Cor (ε) − δ (ε) , respectively. exp i δsJs s Js + δcl cl Since the spherical harmonics Yl(−1) (Ωr ) and Yl1 (Ωr ) are defined as [16] 2l + 1 (1) Pl (θr ) e−iϕr , (68) Yl(−1) (Ωr ) = 4π 2l + 1 (1) P (θr ) eiϕr , Yl1 (Ωr ) = − 4π l

×

(70)

ll

" × Yl(−1) (Ωr ) − Yl1 (Ωr ) |dcl (ε)| Yl 0 (Ωr ) Cor (ε) − δcl (ε) . × sin δsJs s Js + δcl !


No. 11

(2Js + 1) (2Js + 1) iδ gKs Js Js e sJs s Js 2 (2I + 1) = |Fc | eiδc ,

where the phase δc takes into account not only the effect of the phases δsJs s Js but also the randomness s (4) appearing in of the signs of the coefficients bJsK s sJ s J

expression (37) for DKss s , this randomness being associated with the stochastic character of the coeffis . In employing expressions (70)– cients aJKss and cJsK s Cor /dΩ dε to the (72) and (47), the correction d2 σnf r (n, f ) double-differential cross section (48) due to the Coriolis interaction effect can then be represented in the form Cor d2 σnf

pn 2 dΩr dε 2πkn2 2J¯0 0 (Ω , ε) × Ac (θr , ε) |Fc | ACor r c

(1)

sJ s Js Cor )

(72)

Ks sJs s Js

where Pl (θr )!is an associated Legendre " polynomial, the difference Yl(−1) (Ωr ) − Yl1 (Ωr ) is real-valued and can be represented in the form " ! (69) Yl(−1) (Ωr ) − Yl1 (Ωr ) 2l + 1 (1) cos ϕr Pl (θr ) . = π The total expression for the quantity (BcKss can then be reduced to % 2 & 1 sJs s Js Cor = 2 BcKs 8π 2J¯0 (2Js + 1) (2Js + 1) gKs Js Js pn × 2 (2I + 1) × (ε) l (l + 1) dCor cl

1899

=

(73)

c

× sin δc + δcCor (θr , ε) − δc (θr , ε) . In this case, the coefficient D of asymmetry in the angular distribution of products of ternary nuclear fission induced by polarized cold neutrons, % &+ % 2 &− ' % 2 &+ d σ d σ d2 σ − D= dΩr dε dΩr dε dΩr dε (74) % 2 &− d σ , + dΩr dε where the plus and minus signs correspond to choosing the neutron polarization vector pn to be directed along and against the y axis in the laboratory frame, is

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BUNAKOV et al.

D=

2 2J¯0

Cor A (Ωr , ε) sin δc¯ + δCor (θr , ε) − δc¯ (θr , ε) ( |Fc | c¯ c¯ c ( , 0 A0c¯ (θr , ε) cos δc¯0 |Fc |

(75)

c

0 where the complex-valued factor Fc0 = Fc0 eiδc is given by expression (72), where δJs ,Js is substituted for gKs Js Js and where c¯ is coincident with the channel index c for the most probable ternary-fission fragments. The expressions obtained above proved to be close in form to their counterparts derived previously in [6].

amplitudes can be expressed directly in terms of the aforementioned correlation functions as ACor (78) (e)c (Ωr , ε) Cor (θr , ε) , = (σn , [kα , kLF ]) (kα , kLF ) B(e)c Cor ACor (o)c (Ωr , ε) = (σn , [kα , kLF ]) B(o)c (θr , ε) ,

where Cor (θr , ε) = B(e)c

5. STRUCTURAL PROPERTIES OF THE ASYMMETRY COEFFICIENT

(79)

l=2,4,...

Upon choosing the z axis to be aligned with the light-fragment-emission direction, which is determined by the light-fragment unit wave vector kLF , and the y axis to be aligned with the neutron polarization vector σn , we can represent the spherical harmonics Yl 0 (Ωr ), which appear in the definition of the amplitude A0c (θr , ε) (47), in the form Yl 0 (Ωr ) = a0 + a1 (kα , kLF ) + . . .

(ε) l (l + 1) dCor cl

Cor × b2 + . . . + bl (kα , kLF )l−2 eiδcl (ε) Cor Cor iδ (θ ,ε) = B(e)c (θr , ε) e (e)c r , Cor Cor B(o)c (θr , ε) = dcl (ε) l (l + 1) Cor × b1 + . . . + bl (kα , kLF )l−1 eiδcl (ε) Cor Cor iδ (θ ,ε) = B(o)c (θr , ε) e (o)c r .

(76)

l

+ al (kα , kLF ) , where kα is the unit wave vector that specifies the direction of alpha-particle emission; the coefficients a0 , a2 , . . . for even powers of the (kα , kLF ) correlation functions are different from zero only for even values of the orbital angular momentum l ; and the coefficients a1 , a3 , . . . are nonzero only for odd values of l . In turn, the quantity in (69), which is different from zero for l 1, can be represented in the form 2l + 1 (1) cos ϕr Pl (θr ) = (σn , [kα , kLF ]) (77) π

× b1 + b2 (kα , kLF ) + . . . + bl (kα , kLF )l−1 , where the coefficients b1 , b3 , . . . at even powers of the (kα , kLF ) correlation functions are different from zero only for odd values of l, while the coefficients b2 , b4 , . . . at odd powers of the (kα , kLF ) correlation functions are nonzero only for even values of l. The amplitude ACor c (Ωr , ε) in (71) can then be represented as the Cor sum of the amplitudes ACor (e)c (Ωr , ε) and A(o)c (Ωr , ε), which are given by formula (71) upon performing in it summation only over even or only over odd values of l, respectively. Upon employing relation (77), the above

l=1,3,...

Cor /dΩ dε (73) In this case, we can represent d2 σnf r in the form Cor d2 σnf pn 2 = (80) dΩr dε 2πkn2 2J¯0 |Fc | A0c¯ (θr , ε) Hc¯ (Ωr , ε) , × c

where

(81) Hc¯ (Ωr , ε) = (σn , [kα , kLF ]) (kα , kLF ) Cor Cor (θ , ε) sin δ + δ (θ , ε) − δ (θ , ε) × B(e)¯ c¯ c¯ r c r (e)¯ c r Cor + (σn , [kα , kLF ]) B(o)¯ c (θr , ε)

Cor × sin δc¯ + δ(o)¯c (θr , ε) − δc¯ (θr , ε) ,

and the asymmetry coefficient D (75) in the form ( pn 2 (+) Hc¯ (Ωr , ε) c |Fc | 2J¯0 ( , (82) D = 0 |Fc0 | Ac¯ (θr , ε) cos δc¯0 c

(+)

where Hc¯ (Ωr , ε) is given by (81), with the vector σn being aligned along the y axis of the laboratory frame.


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If we now consider that, under time reflection, the quantity σn changes sign, while the [kα , kLF ] and (kα , kLF ) correlation functions remain invariCor /dΩ dε in (80) behaves as a pseudoscalar ant, d2 σnf r quantity—that is, its symmetry is T -odd. Under the inversion of the coordinate axes, σn and [kα , kLF ] behave as pseudovector quantities, while the (kα , kLF ) correlation function behaves as a true scalar quantity. Therefore, the correction to the double-differential cross section in (80) behaves as a scalar quantity— that is, it is a P -even quantity. At the same time, the inversion of the vector kα or the inversion of the vector kLF does not lead to sign reversal for the first term in the braced expression on the right-hand side of (81) (this term is associated with even values of l), but this does lead to a change in the sign of the second term there, which is associated with odd values of l. We can then conclude that the first term in the braced expression on the right-hand side of (81) corresponds to the experimental situation that was observed in [2] in studying 235 U fission induced by polarized neutrons and which was called a ROT effect, while the second term corresponds to the experimental situation observed for 233 U in [1] and called a TRI effect. In order to obtain deeper insight into the reasons behind the appearance of even and odd values of the alpha-particle orbital angular momentum in the amplitudes in (78), it is necessary to consider that the alpha particle leaves the fissile-nucleus neck because of the effect of superfluid nucleon–nucleon correlations whose orbital angular momentum is predominantly l = 0; as a result, the appearance of nonzero orbital angular momenta l is due entirely to the effect of nonspherical components of the potential of the alpha-particle interaction with ternaryfission fragments, VαF (R, r) (it includes the nuclear and Coulomb potentials [13]), since the Coriolis interaction only changes the projections Kl of the alpha-particle orbital angular momentum onto the symmetry axis of the nucleus undergoing fission without changing the orbital angular momentum l itself. Under the effect of the quadrupole component of this potential, VQ (R, r), the orbital angular momentum l = 2 of amplitude dc¯2 , which is dominant in absolute value among all l = 0 amplitudes dc¯l in the amplitude A0c¯ (θr , ε) (47) of the alpha-particle angular distribution not perturbed by the Coriolis interaction, is admixed to the alpha-particle orbital angular momentum l = 0. At the same time, the appearance of odd alpha-particle orbital angular momenta is caused by the effect of the dipole component VD (R, r) of the potential of alpha-particle interaction with ternary-fission fragments, this component being associated with the charge and mass asymmetries PHYSICS OF ATOMIC NUCLEI Vol. 71

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of fission fragments [13]. As the parameter of the mass and charge asymmetries of fission fragments, δ = (A1 − A2 )/(A1 + A2 ) ≈ (Z1 − Z2 )/(Z1 + Z2 ), tends to zero, the dipole component VD (R, r) of the potential VαF (R, r) disappears, along with odd values of the alpha-particle orbital angular momentum l. In this case, the TRI effect, which is due to odd values of l, must also disappear, with the result that only the ROT effect, which stems from even values of l, survives in expression (80). There remains the problem of clarifying the physical reason behind the emergence of such different T odd asymmetries in the angular distributions of products of the ternary fission of 234 U and 236 U compound nuclei, which are close in their fission properties and which arise upon cold-neutron capture by, respectively, 233 U and 235 U target nuclei. In order to solve this problem, we consider the limiting case where one neutron resonance characterized by fixed values of s and Js takes part in the formation of T -odd correlations. In this case, the phases δsJs s Js and δc vanish, and the cross section Cor /dΩ dε (80) assumes the form d2 σnf r pn (2Js + 1) 2 0 (83) (θ , ε) A r c ¯ 2 dΩr dε 2πkn 2 (2I + 1) 2J¯0 2 2 Js 0 Js hs bsKs ΓKs gKs Js Js , × Hc¯ (Ωr , ε)

Cor d2 σnf

=

Ks

where the function Hc¯0 (Ωr , ε) is given by the formula (84) Hc¯0 (Ωr , ε) = (σn , [kα , kLF ]) (kα , kLF ) Cor Cor × B(e)¯ c (θr , ε) sin δ(e)¯ c (θr , ε) − δc¯ (θr , ε) Cor (θ , ε) + (σn , [kα , kLF ]) B(o)¯ r c

Cor , × sin δ(o)¯ c (θr , ε) − δc¯ (θr , ε) which follows from (81); the factor gKs Js Js (67) has the form ⎧ Js (Js + 1) − Ks2 ⎪ ⎪ ⎪ ⎪ ⎪ Js ⎪ ⎪ ⎪ ⎨for Js = I + 1/2, (85) gKs Js Js = Js (Js + 1) − Ks2 ⎪ ⎪ ⎪− ⎪ ⎪ Js + 1 ⎪ ⎪ ⎪for J = I − 1/2; ⎩ s

( and the quantity ΓKs = c ΓcKs coincides with the total ternary-fission width of the fissile-nucleus prescission state introduced above, which is described by the wave function ΨJKssM . Since the main properties of the binary and ternary fission of 234 U

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and 236 U compound nuclei formed upon the capture of a polarized thermal neutron by, respectively, 233 U and 235 U target nuclei are close, it is natural to expect the of the structures of the amplitudes 0 proximity A (θr , ε) and H 0 (θr , ε) and, hence, the phases c¯ c¯ Cor (θ , ε), δ Cor (θ , ε), and δ (θ , ε) for the comδ(e)¯ c¯ r c r (o)¯ c r pound nuclei in question. Therefore, the coefficients of T -odd asymmetry, D in (82), for the (n, f ) ternaryfission reactions involving these compound nuclei should be identical, in contrast to what is observed experimentally. In order to obtain deeper insight into the situation around the structure of T -odd correlations in the case of a fission process involving only one swave neutron resonance, we consider the behavior of the asymptotic function in (54) for a prescission state that is characterized by fixed values of Js and Ks . The region of integration with respect to the multidimensional coordinate x in the expression for the function in (54) can be broken down into two subregions: x x0 and x0 x x, where the coordinate x0 is associated with a surface in configuration space of x . This surface corresponds to the situation where the alpha particle occurs in the vicinity of the Coulomb barrier maximum determined by the sum of the Coulomb and nuclear potentials of alpha-particle interaction with ternary-fission fragments and coincides with the surface at which initial conditions for calculations by the trajectory method are specified [3]. In the region x x0 , the condition r /R 0.15 then holds, and the dipole component VD (r , R ) of the potential of alpha-particle interaction with ternary-fission fragments, which includes the nuclear and Coulomb components, proves to be much greater in absolute value than the corresponding quadrupole component VQ (r , R ) of this potential [13]. It follows that, in this region, the orbital angular momentum of l = 1 is basically admixed to the leading alpha-particle orbital angular momentum of l = 0. Since the Coriolis interactions acts only on the alpha-particle orbital angular momentum l = 0, only the odd alpha-particle orbital angular momentum l = 1 is conserved at the boundary x = x0 of the region after the action of the Coriolis interaction. The quantity (Yl−1 (Ωr ) − Yl,1 (Ωr )) specified by Eqs. (69) and (77) at l = 1 is proportional to cos ϕr sin θr and has a broad maximum in the vicinity of the angle θr = π/2. In this case, initial conditions are generated at the surface x0 for calculations by the trajectory method in a form that is close to the form of standard initial conditions for such calculations without allowance for the Coriolis interaction. Thus, only in the region x x0 can one expect the appearance of the TRI effect upon taking into account the Coriolis

interaction; its amplitude ACor (o)¯ c (θr , ε) (78) as a function of θr and ε must be close to the unperturbed amplitude A0c¯ (θr , ε) (47), while the coefficient of T odd asymmetry, D in (82), must only slightly depend on the angle θr , as one observes experimentally for 233 U. But if the Coriolis Hamiltonian acts only in the region x x0 , where the ratio r /R proves to be rather large, reaching a value of r /R ≈ 1, the quadrupole component VQ (r , R ) generating, with a high probability, the even alpha-particle orbital angular momentum of l = 2 becomes the leading nonspherical component of the potential of alpha-particle interaction with fission fragments. In this case, the amplitude in (81) is dominated by even alpha-particle orbital angular momenta. This leads to the ROT effect and an asymmetry coefficient D [of the form in (82)] strongly dependent on the angle θr , as one did indeed observe experimentally for 235 U. Physically, it is quite clear that, because of its larger width, the region x x0 must make a greater contribution to the integral with respect to x in (54) than the region x x0 . Therefore, it is natural to expect that, in the case of taking into account only one isolated s-wave neutron resonance for the (n, f ) ternary-fission reactions induced by polarized cold neutrons in 233 U and 235 U target nuclei, there will arise the ROT effect exclusively. For the TRI effect to occur, it is necessary that a mechanism appear that would suppress the contribution of even values of the alpha-particle orbital angular momentum to the amplitude in (81) for the 234 U compound nucleus, which are responsible for the ROT effect, and that would not be operative for the amplitude in (81) for the 236 U compound nucleus, which is close to 234 U in global fission properties and where the ROT effect is observed. Since the complex-values amplitudes A0c¯ (θr , ε) (47) and ACor c¯ (θr , ε) (71) retain their properties upon going over from the 234 U to the 236 U compound nucleus, only the difference in the position and structure of the neutron resonances excited in 234 U and 236 U upon cold-neutron capture by the corresponding target nuclei and, hence, the difference in the phases δc¯ associated with the interference of these resonances may be the source of the above suppression. The existence of a strong effect of the interference between the above resonances on the structure of P odd and P -even correlations in the angular distributions of fragments originating from the binary fission induced by polarized thermal neutrons in analogous target nuclei was proven previously in [19, 20]. In order to explain the difference in the behavior of the asymmetry coefficient D for the (n, f ) reactions


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studied here for 233 U for 235 U target nuclei, it is necessary to take into account the interference between the sJs and s Js neutron resonances. Let us consider the coefficient D (82) in the vicinity of the most probable alpha-particle energies around ε ≈ εm . Under the assumption that, in 234 U, the phase δc¯ associated with the interference of neutron resonances has a Cor (θ , ε ) + δ (θ , ε ) in the vicinvalue of δc¯ ≈ −δ(e)¯ c¯ r m c r m (e)

ity of the angle θr corresponding to the maximum of the modulus of the amplitude ACor (e)¯ c (θr , εm ) for even values of l, this leads to the suppression of the contribution of even values of l to the amplitude in (81), which determines the coefficient D in (82), and to the occurrence of the TRI effect. In this case, the focusing effect of the Coulomb potential of alpha-particle interaction with ternaryfission fragments also manifests itself if only odd values of the alpha-particle orbital angular momentum l survive in the amplitude in (81). Even in the case where Coriolis interaction acts not only in the internal (x x0 ) but also in the external (x x0 ) region, this focusing effect leads to a predominant alphaparticle emission in the direction orthogonal to the direction along which ternary-fission fragments move apart. Concurrently, it is natural to expect that, in just the same way as in the case of where the Coriolis interaction acts only in the internal region, l = 1 is leading value of l in the amplitude in (81). At l = 1, the quantity [Y1−1 (Ωr ) − Y1+1 (Ωr )] ∼ sin θr cos ϕr , which appears in the amplitude ACor (o)¯ c (71), as a function of θr has a broad maximum in the vicinity of the angle θr = π/2, while the dependence of the amto the plitude Hc¯ (Ωr , ε) (81) on θr and ε is close analogous dependence of the amplitude A0c¯ (θr , ε) not perturbed by the Coriolis interaction. For the 236 U nucleus, the phase δc¯ must in turn differ significantly Cor (θ , ε ), from the phase difference δc¯ (θr , εm ) − δ(e)¯ c r m which is close to the analogous difference for the 234 U (e) compound in the vicinity of the angle θr , nucleus (e)

(e)

Cor θ , ε for sin δ(e)¯ r m − δc¯ /θr , εm − δc¯ not to c differ very strongly in absolute value from unity, in which case the ROT effect occurs.

6. COMPARISON OF THE QUANTUM-MECHANICAL THEORY OF T -ODD CORRELATIONS IN TERNARY FISSION WITH THE RESULTS OF CLASSICAL CALCULATIONS Classical calculations of angular and energy distributions of alpha particles in ternary fission by the PHYSICS OF ATOMIC NUCLEI Vol. 71

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trajectory method without allowance for the rotation of the system undergoing fission describe experimental data from [18] quite well. In [3], the ROT effect for the compound nucleus 236 U was reproduced fairly well by applying the method of classical trajectory calculations to the motion of the alpha particle in the external region Coul of two fission fragments in the Coulomb field VαF under the condition that the entire fissile system rotates classically as a discrete unit in the laboratory frame at an angular velocity Ω. Concurrently, one assumes that, to a high precision, escaping ternaryfission fragments move along or against the direction of the fissile-nucleus symmetry axis, which rotates in the laboratory frame at the angular velocity Ω, this being in accord with the above result that the Coriolis interaction does not affect the motion of these fragments in the body-frame system. At the same time, we note that, since the alpha particle emitted in ternary fission does not correlate tightly with fission fragments but interacts with them Coul , the in the external region through the potential VαF angle of alpha-particle rotation in a plane orthogonal to Ω differs from the angle of rotation of fission fragments in this plane. This leads to a change in the angular distribution of alpha particles with respect to the direction kLF of light-fragment emission in the laboratory frame and to the appearance of the ROT effect. Naturally, the interference between fission amplitudes of different sJs and s Js neutron resonances excited in the compound nucleus upon cold-neutron capture by the target nucleus could not be taken into account in [3] because of the use of the classical approach; therefore, the TRI effect could not be described there in principle fully in accord with the conclusions drawn in Section 5 of the present article. The angular velocity Ω of the classical rotation of a fissile system was calculated in [3] by the formula R σn . (86) Ω = P (J) J0 In this formula, P (J) stands for the polarization of the spin J of a spherical compound nucleus that arises upon the capture of a polarized neutron of polarization pn and unit polarization vector σn by an unpolarized target nucleus of spin I: ⎧J + 1 ⎪ pn for J = I + 1/2, ⎨ 3J (87) P (J) = ⎪ ⎩− 1 p for J = I − 1/2. n 3 The axisymmetric deformation of the fissile nucleus was taken into account by introducing in (86) the quantity R that was referred to in [3] as the angular momentum of the collective rotation of a deformed

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nucleus and which is related to its spin J and the spin projection K onto the symmetry axis of the nucleus by the equation R = J(J + 1) − K 2 . (88) In [3], the angular velocity Ω (86) was then represented in the form P (J) J(J + 1) − K 2 σn . (89) Ω= J0 The introduction of the vector R = Rσn in expressions (86) and (89), which is directed along the neutron-beam polarization vector σn , is questionable even within the classical approach. Indeed, the geometry of the experiment reported in [2] was chosen in such a way that the direction kLF of light-fragment emission (and of the fissile-nucleus symmetry axis) was orthogonal to the direction of the neutronbeam polarization vector σn . In the following, we denote these directions as, respectively, the z and the y axis, in which case R = Jy and K = Jz . In the case of K = 0, the nuclear spin J is due entirely to collective rotation; that is, R = J, which is at odds with relation (88). In the case of K = 0, the direction of the collective-rotation vector (and, hence, the direction of the angular-velocity vector Ω) cannot coincide with the direction of the fissile-nucleus total spin J, the latter being determined by the neutronpolarization vector σn , and this is inconsistent with expressions (86) and (88). It will now be shown that special features of the fission of a nonspherical nucleus possessing an axisymmetric deformation, the reaction of our prime interest, were taken into account incorrectly in expressions (86)–(88). The wave function for such a nucleus, ΨJM K (5), must be constructed with allowance for degeneracy in the sign of K and is expressed in terms of a linear superposition of the generalized J J spherical harmonics DM K (ω) and DM −K (ω), which depend not only on the projection M of the spin J onto the z axis of the laboratory frame but also on the projection K of the spin J onto the symmetry axis of the nucleus. The average spin J (K) of a polarized fissile nucleus is then dependent on the projection K and is given by - J , ρJM M ΨJM K |J|Ψ (90) J(K) = M K MM

- is the operator of the fissile-nucleus total where J spin in the laboratory frame and ρJM M is the spin density matrix defined in Eqs. (42) and (43). For our choice of laboratory-frame axes, the z axis coincides with the light-fragment-emission direction, which agrees to a high degree of precision with the direction of the fissile-nucleus symmetry axis, while

the y axis coincides with the direction of the neutronpolarization vector σn . Therefore, the axes of the body-frame system can be chosen to be coincident with the laboratory-frame axes, in which case the respective Euler angles ω are equal to zero. In expression (90), one can then replace the operators of the spin projection onto the laboratory-frame axes, Ji (i = x, y, z), by the corresponding operators of spin projection onto the axes of the body-frame system, Jα (α = x , y , z ), and make use of formulas that belong to the same type as (53). As might have been expected, only the spin projection onto the y axis is nonzero for our choice of coordinates; that is, gKJJ pn σn , (91) J(K) = 2 where the quantity gKJJ is given by expression (85) at Js = J and is dependent explicitly on the quantum number K. The polarization of the spin J of a deformed fissile nucleus, PK (J), | J (K) | , (92) PK (J) = J assumes the value pn gKJJ (93) PK (J) = 2J and differs substantially from the analogous polarization in (87) for a spherical nucleus. It can readily be proven that, upon averaging expression (93) over all possible values of K, one can arrive at the polarization P (J) given by (87): P (J) =

J K=−J

1 PK (J). (2J + 1)

(94)

Since a very strong mixing of quantum numbers K occurs in neutron resonances of a deformed compound nucleus owing to a dynamical enhancement of the Coriolis interaction [14, 15], the K-averaged expression (94), which coincides with expression (87) for the polarization for a spherical nucleus, is likely to be valid for the polarization of a nucleus for all channels of compound-nucleus decay, with the exception of fission. The point is that, in the fission channel, a nonuniform filtration of K values that is determined s in (4) occurs upon the passage by the amplitudes bJsK s of the saddle point, with the result that it is necessary to use formulas (91) and (93) for, respectively, the average spin J (K) and the polarization PK (J) of a deformed nucleus. In this case, the fissile system undergoes a classical rotation about the axis y, which is orthogonal to the symmetry axis of the nucleus, the angular velocity ¯ (J, K) being Ω ¯ (J, K) = J (K) = pn gKJJ σn . (95) Ω J0 2J0


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The ratio of the angular velocities in (95) and (89) is then given by ¯ (J, K) Ω 3 = J(J + 1) − K 2 . (96) Ω 2 In the case of J = 3 and 4 (that is, for the spins of the s-wave resonances in the 236 U nucleus, for which classical calculations by the trajectory method were performed), this ratio ranges between 1.3 and 0.5, depending on K. If fission proceeds through prescission fission modes characterized by various values of the quantum number K, then, as follows from (83), the expression for the effective angular velocity of classical rotation, Ωeff , has the form ¯ (J, K) α (J, K), Ω (97) Ωeff = K

where α (J, K) = ΓJK /ΓJ .

(98)

Here, ΓJ is the total ternary-fission width of a specific neutron resonance with a spin J, bJsK 2 ΓK , ΓJK = (99) ΓJ = K

K

while ΓK is the total width with respect to ternary fission from a prescission mode characterized by the wave function ΨJM K . The ratio α (J, K) (98) was evaluated in [21], where a detailed analysis of the angular anisotropy of fragments originating from the binary fission of aligned 235 U nuclei that was induced by neutrons of energy in the range between 0 and 20 eV was performed and where the contributions to the fission cross section from transition fission states characterized by various values of the quantum numbers Js and Ks were determined. In principle, expressions (97)– (99) could make it possible to find correct values of effective angular velocities for refining the results of classical calculations by the trajectory method. However, one of the basic conclusions drawn in [21] was that the use of the approximation of isolated resonances in quantitative calculations is illegitimate even in the region of resonance maxima, where the contribution of interference effects is minimal. Experimental measurements of the ROT effect [2] were performed with cold neutrons in the interresonance energy region, where the contributions from resonances of spin J = 4 and 3 to the cross section for the (n, f ) double-fission reaction were commensurate [21] (553 and 323 b, respectively). The contribution of interference effects is maximal in that case. For this reason, it is extremely difficult to estimate the accuracy of classical calculations PHYSICS OF ATOMIC NUCLEI Vol. 71

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by the trajectory method even in the case where such calculations are performed with refined angularvelocity values from (97). Allowance for the interference between resonances by formula (80) may change not only the magnitudes but also the signs of observed effects. As was shown above, the signs of the s amplitudes bJsK in (80) change from one resonance s to another at random. This is also suggested by the numerical analysis performed in [21]. Therefore, it is not surprising that a correlation between the signs of T -odd effects in neighboring uranium isotopes could not be found within the classical approach [2]. 7. CONCLUSIONS Allowance for Coriolis interaction within quantummechanical ternary-fission theory makes it possible to explain both of the observed T -odd effects and to show that the TRI effect disappears for fission symmetric in the fragment charges and masses. For fission that is strongly asymmetric in the fragment charges and masses, the effects being considered (TRI and ROT) always contribute to the angular distribution of products of ternary fission of unpolarized nuclei that is induced by polarized thermal and cold neutrons; only their ratio changes versus the target-nucleus mass and the projectile-neutron energy. (A preliminary analysis of experimental data is also indicative of this.) A change in the relative contributions of the TRI and ROT effects is due entirely to the interference between neutron resonances and cannot be explained within the classical approach. This interference affects both the magnitudes and the signs of the two effects (through the signs of the J s , and bssKs and of the signs quantities gKs Js Js , bJsK s dependent on the differences of phases). Therefore, the signs of the TRI effect do not correlate with the signs of the ROT effect, as is indicated by experimental data. The contributions of transition fission states characterized by various values of Js and Ks cannot change the TRI/ROT ratio, but they can only increase or decrease both effects simultaneously. It would be highly desirable to perform measurements of T -odd correlations at several strong s-wave resonances, where the contribution of interference effects is minimal. The analysis performed in Section 6 has revealed that, in that case, both the relative magnitudes of the TRI and ROT effects and their relative signs must be identical (even for strong resonances in different neighboring isotopes that have close global fission features). The theory could also be tested experimentally by changing the neutron-beam energy for the same

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target in such a way as to change substantially the character of the interference between neutron resonances and, hence, the contributions of the TRI and ROT effects and their ratio. REFERENCES 1. P. Jessinger, A. Kotzle, A. M. Gagarski, et al., Nucl. Instrum. Methods A 440, 618 (2000). 2. G. V. Valski ˘ı, A. S. Vorob’ev, A. M. Gagarski ˘ı, et al., Izv. Akad. Nauk, Ser. Fiz. 71, 368 (2007). 3. I. S. Guseva and Yu. I. Gusev, Izv. Akad. Nauk, Ser. Fiz. 71, 382 (2007). 4. S. G. Kadmensky, Yad. Fiz. 65, 1424, 1833 (2002) [Phys. At. Nucl. 65, 1390, 1785 (2002)]. 5. S. G. Kadmensky, Yad. Fiz. 66, 1739 (2003) [Phys. At. Nucl. 65, 1691 (2003)]; 67, 258 (2004) [67, 241 (2004)]. 6. V. E. Bunakov and S. G. Kadmensky, Yad. Fiz. 66, 1894 (2003) [Phys. At. Nucl. 66, 1846 (2003)]. 7. V. E. Bunakov, S. G. Kadmensky, and L. V. Rodionova, Izv. Akad. Nauk, Ser. Fiz. 69, 625 (2005); 70, 1611 (2006). 8. V. E. Bunakov and S. G. Kadmensky, Izv. Akad. Nauk, Ser. Fiz. 71, 364 (2007). 9. V. E. Bunakov and S. G. Kadmensky, Yad. Fiz. 71, 1227 (2008).

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Translated by A. Isaakyan


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