RADIATIONAL CAPTURE OF NEUTRONS BY DEUTERONS n + d + aH + AND P-ODD NUCLEAR FORCES
UDC 539.12
M. P. Rekalo
P-odd polarizational phenomena are considered in the radiative capture of neutrons by deuterons: n + d + 3H + ~. It is shown that, in the general case, 18 different P-odd asymmetries of the angular distribution of ~ quanta arise in the collision of arbitrarily polarized neutrons with a deuteron target characterized by vector and tensor polarizations. The P-odd contribution to the density matrix of the quanta formed in the capture of polarized neutrons by unpolarized deuterons is determined by eight real structural functions, while the P-odd dependence of the Stokes parameters of the photons on the tensor polarization of the deuterons is characterized, in the general case, by ten structural functions. In slow-neutron capture, the number of P-odd correlations is reduced.
i. The suppression of the P-even transition in the radiative capture of thermal neutrons by deuterons [1-5] n + d § 3H + y makes it attractive for the search for P-odd effects [6-9]. As well as P-odd asymmetry of the angular distribution of the u quanta formed in the capture of longitudinally polarized neutrons [I0], other P-odd characteristics may also be measured in the reaction n + d + BH + 7, such as the degree of circular polarization of the ~ quanta formed in the interaction of unpolarized particles. The present work constitutes a systematic investigation of P-odd polarizational characteristics in the reation n + d + 3H + ~. Without specifying the mechanism of the P-odd and P-even amplitudes, the dependence of the P-odd contribution to the differential cross section of the process n + d + 3H + y on the polarizational states of the incident neutrons and on the vector and tensor polarization of the deuteron target is studied. The P-odd contributions to the Stokes parameters of the y quanta which form are also determined, as a function of the polarizations n and d. Conducting the analysis in terms of structural functions allows the general structure of the polarizational P-odd characteristics valid for any incidentneutron energy and angle of y-quantum formation to be established. The spin structure of the P-even and P-odd amplitudes is significantly simplified at the threshold, but nevertheless the choice of possible experiments to search for N-odd correlations at the threshold is sufficiently broad that there are various methods of determining those P-odd multipole threshold amplitudes which determine the P-odd effects in n + d + 3H + 7. 2. The P-odd contribution to the differential cross section of the process n + d § ~H + summed over the polarizations of ~ and 3H in the final state depends on the polarizations of n and d in the initial state as follows
d•
A
dg.
A
A
A
--~.ma,( s, t)+~.sca2(s, t)+ s.ma3(s, t)+s.Ka~(s, t ) + A
A
A A
A
A
A
+ (%maUl) a.~ (s, t) q- (sJ%rzo) a~ (s, t) -~- ~.ms. na~ (,~, t) -.4A
A
A
A
A
A
+ ~,Ics.1~as (s, t) + ~.ns.ma9 (s, t) + ~.ns.lCa,o(S, t) + A
A
A
A
A
A A
A
A A
+ ~.Ir [(sa~m~mb) a . (s, t) + ( s ~ # b ) A
A A
(i)
A A
A A
a,,~ (s, t) + (s~bm,,~) a,~, (~, t)] +
+ ~.n [(s.brn,,~) a . (s, 0 + ( s ~
A
A
a,~ (s, 0],
Khar'kov Physicotechnical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 23-26, March, 1987. Original article submitted November 30, 1984.
0038-5697/87/3003-0201512.50
9 1987 Plenum Publishing Corporation
201
where ~ (s) is the polarization vector of the incident neutrons
(deuteron target), A
quadrupole-polarization A
tensor of the deuteron
(s~o = sba, S~a ~ 0 ) ,
Sab is the
A
A
~ = ~/I ~[, ~ = ~ ~ q / [ ~ ~ ql, m
A
= ~ ~ ~, ~ and q are the 3-momenta of the 7 quantum and neutron (in the center-of-inertia system) in the reaction n + d § 3H + 7. The real structural function a~(s, t) depends on two Invariant variables s = (q + p)= and t = (k - q)=; k, q, and p are the 4-momenta of 7, n, and d. Thus, in the general case, 18 independent P-odd asymmetries (according to the number of corresponding structural functions) arise in the collision of polarized neutrons with a deuteron target. 3. The polarized states of the 7 quanta which form may be characterized by the tensor Hij, which is related to the differential cross section of formation of 7 quanta with polarization vector e in the reaction n + d § 3H + 7 as follows: do/d~ = Hijeie* j. The P-odd part of the tensor Hij depends as follows on the polarization of the incident neutron
~i](~ )
{Am., A}ijl, l ,
=
A
A
A A A + i [m, nluh., + ~ . m
A
AA
A
AA
A
A
A A A A (h~mLm~ + h~njz~) +
A
A
A
+ ~.tr (hsm~m~ + h~n~n~) + ~.n ({m, n~h~ + i [m, n]~h~), A
A
{m, n}U = rlzr eters
A
A
A
A
A
(2)
A A
m#~, [m, rtli~ = m # # - - m#~.
The polarizational states of the photon are characterized by the standard Stokes param[ii]
d~ % dO,
h~ + ~.nhT,
2% ~
-2 ~-~
= h2 + ~.nhs,
(~)
= ~-m (h:~ - ~ ) + ~. ~ ( ~ - h~).
It is evident from Eq. (3) that the P-odd contributions to ~i and $2 are nonzero even in the case when unpolarized particles are encountered in the initial state. In the absorption of unpolarized neutrons by a deuteron target with vector polarization, the structure of Hij(s) is the same as in Eq. (2), but ~ must be replaced by S. If the target has tensor polarization, the following general structure may be written for Hij(Sab) (it is valid in the absorption of unpolarized neutrons) A A
A
A
A
A
/4U (S,,~) = (s,~omjno) Ih~ {m, rt}i.i + ih,o ira, nlul § A A
A
A
A
A
A
A
A
A
A A
AA
A A
A
A
A
A
A
A
AA
§ (s,~m~n~)(hjn~mj § h~nj) § (S~b~nb) (h,~mpzj§ h~n#j), a total of ten real structural functions. The P-odd contributions to the Stokes parameters depend as follows on the tensor-polarization components of the deuteron target
dg
A
A
AA
AA
(~,/~
A
A
A A
A
"
n
~ ~ : h~o (s~bm~mo) + h,2 (sjcjc~) + h~ (SubtC~mb), d~
A A
A A
2~:, ~-~ = (h~.~ - h~o) (s,,~m~no) + (h~ -- h ~ ( s ~ , n ~ ) . In the capture of thermal neutrons in the process n + d + 3H + y, the structure of the P-odd effects is simplified, and the number of independent structural functions is reduced: more precisely, the structural functions degenerate to constants here. The P-even amplitude of the process n + d + aH + y with nonzero neutron momentum corresponds to the formation of M1 and E2 y quanta; the total angular momentum of the system n + d may be 1/2 or 3/2 here, and as a result three transitions must be present in n + d + 3H + y: 1/2 § MI, 3/2 + MI, and 3/2 + E2. The P-odd amplitude of the process n + d § 3H + 7 as q + 0 is also characterized by three multipole amplitudes: 1/2 § El, 3/2 ~ El, and 3/2 § M2. 202
As a result, the following expression may be written for the P-even and P-odd amplitudes of the process n + d + SH + A
A
A
e. U ~< ic -+f~ (Z~+=. tc~,) e.U +f3 (~+=.el,) U.K,
F = i/~ (•
~
A
A
=f, (X~Z,)e, U - v i?, (X+e.e >( UX, ) + if3 (X+ e ' e P; gX,) U.~r
where Xi (X2) i s t h e t w o - c o m p o n e n t s p i n o r o f t h e n e u t r o n polarization; U is the 3-vector of deuteron polarization. In the capture is reduced to five
of thermal neutrons d~
(3H); e i s t h e 3 - v e c t o r
( w i t h z e r o momentum q ) ,
A /'. A A = ~.Kx~ + s.^'x, " i~.~ • sx~ + ~.Ir
A A
of v-quantum
t h e number o f P-odd a s y m m e t r i e s
A x~,+ (~u~s~) x~.
The P-odd asymmetry because of x n is simultaneously also T-odd; therefore, at the threshold x 3 = 0, since the relative phases of the threshold P-even and P-odd amplitudes are 0 or ~. For the threshold tensor Hij determining the polarizational characteristics of the quantum as a function of the polarization of the deuteron target, the following general structure may be written A
A
A
A A
A
A
A
H~j = i~iitxlx~ + ~Us. ~r + i~ijt~ l (s~o~mb) x8 + i~i~s~b%x9 ~- (si..~x~sj~ @ ~.~Jcmsm) x,~. In view of the T-invariance xi0 = 0 at the threshold, are real.
since the threshold multipole amplitudes
Regardless of the specific form of the coefficients xi, the ratio x 2 = 2x 7 holds. In terms of the threshold amplitudes fi and fi, the following amplitudes may be obtained for x i
~,
= ~ (L/. § J~:, + f,L ~-S;J,) ~. = 4 (f#, + L > L - LA0,
., =
r(/,+ y,)Z + (/.+n) (--7 + Z
., = ~
f---(/, §
4
7
+i)J,
(L + 7-,,)+ 1, o>, +/,il, ~
-
.,, = ~[{,g, + f,f,, -f,I. --f,);,l. ., = - Jr.4, .,-,. . . . .
2 r,Z +17, +,/j,).
Since the reaction mechanism was not given specific form anywhere above, all the formulas obtained here are also valid for the radiative capture of protons p + d § 3H + u LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii.
L. Kaplan, C. R. Ringo, and K. E. Wiezbach, Phys. Rev., 87, 785 (1952). S. Merritt, J. G. Taylor, anbd A. W. Boud, Nucl. Sci. Eng., 34, 195 (1968). E. T. Jurney, P. J. Bendt, and J. C. Browne, Phys. Rev., C25, 2811 (1982). Ao E. Cox, A. R. Wynchank, and C. H. Collie, Nucl. Phys., 74, 500 (1968). L. I. Schiff, Phys. Ray., 52, 242 (1937). R. J. Blin-Stoyle and F. Feschbach, Nucl. Phys., 27, 395 (1961). A. N. Moskalev, Yad. Fiz., 8, 1156 (1968). A. N. Moskalev, Yad. Fiz., 9, 163 (1969). E. Hadjimichael, E. Harms, and V. Newton, Nucl. Phys., A228, No. 1 (1974). Yu. G. Abov and P. A. Krupchitskii, Usp. Fiz. Nauk, 118, 141 (1976). V. B. Berestetskii, E. M. Lifshits, and L. P. Pitaevskii, Relativistic Quantum Theory [in Russian], Nauka, Moscow (1968), Part I, pp. 39-45.
203
