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Page 1: Energy of neutrino radiation in beta decay in a quantizing magnetic field

ENERGY OF NEUTRINO RADIATION IN BETA DECAY IN A QUANTIZINGMAGNETIC FIELD

O. F. Dorofeev, V. N. Rodionov, S. G. Starcheus, and A. I. Studenikin

UDC 539.123.7

Beta decay in a quantizing magnetic field is studied, in conditions where the influ- ence of the temperature and density of the electron gas is significant. It is shown that the expressions for the probability W and power of neutrino emission Ev in con- ditions of degeneracy include a nonlinear.dependence on the parameter H/Hc; for field values at which quantum states of the electron with n # 0 are possible, they contain both monotonic and oscillating components. With increase in the field, the oscilla- tions of W and Ev vanish. In the limit of high temperatures T >> mc 2 the oscillat- ing corrections are considerably reduced.

The discovery of astrophysical objects of the type of pulsars with colossal magnetic fields even close to their surface has actively stimulated the theoretical investigation of physical phenomena taking account of the possibility of action of fields with an intensity comparable with the value H = m 2 c a / ( e h )---=--4,41.10 taG.

As is known, the important processes making a contribution to neutrino emission in the collapse of massive stellar nuclei are nuclear beta decay and cross-symmetric processes [i].

In the present work, further consequences (see, e.g., [2, 3]) of taking the action of an intense magnetic field on the course of 8 decay itself and on the power of the neutrino radia- tion into account in conditions when the temperature and density of the electron gas are also significant are studied.

It is known that the expression for the probability of 8 decay in a quantizing magnetic field takes the form [4]*

where

!

[NI S d W H ~(=-o--~) ~"

'" .=o~ % H~ V - " ~ - b ~ l + e x p d:,

(i)

~ (0~ n) 1 ~ ~,, cos O~ - - - - (1 + (~, + ~,,) cos < --r- a), " 2

tfl = 1 -4- 3=" ' m,. 2 3 , 1 + 3= 2 tea = 2 s n 1 -t- 3=" ' r m c = ,

W o = G ~" 1 + 3=._...._~ ~ N = I-I 8~ 4 ~ ( a ~ , - - 1 ) , b ~ = l + 2 n H , H e '

0~ is the angle of neutrino departure measured from the direction of the magnetic field; e and are the electron energy and the chemical potential, expressed in units of mca; ~ = IGA/GVI;

G A and G v are the axial and vector constants of the V--A model of interaction; Sn = _+i corre- sponds to the projection of the neutron spin on the direction of the magnetic field.

Using standard summation methods [5], after averaging over spin states of the neutron and integrating over the angle of neutrino departure for the probability of 8 decay [I] in the

*In obtaining~Eq. (i), the calculation of the probability of beta decay of a neutron at rest n + p + e- + 9 is taken as the basis, neglecting the motion of the proton; the energy-libera- tion parameter Eo is assumed to be arbitrary, however. This approach permits generalization of the results obtained to beta decay of the type of superpermitted transitions, for example, T+~+e-+~.

M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 84-88, April, 1989. Original article submitted July 9, 1986.

308 0038-5697/89/3204-0308512.50 �9 1989 Plenum Publishing Corporation

Page 2: Energy of neutrino radiation in beta decay in a quantizing magnetic field

expansion with respect to (H/H c), in conditions when the electron gas is degenerate (T § 0), it is found that

W ; ( 4 ~ We) = Fo (-Zo) - - F,, (,~) + (H/Hc)"- (F, (%) - - F , ( p ) ) q- (H/Hc)-" (F., ( - , , ) - -

- - P',_, (~)) + (2H/Ho)~- ' { ( - - 1/2, {N~} ) (% --,,.@"/2 + (2H'Ho)S2< (-- 3/2, (2) { N~}) (% - - ,,J.)/(3,?.) § ( 2 H / H J ~- [~, ( - - 5/2, { Ar:~}) .--, '?~:' - - ~ ( - - .5'2, A:) ~ ~ ]/15,

where r v) is a generalized zeta function; N u = H(U ~ -- l)/(2Hc) and

Fo (q) = (---, 4) [ t z lq + ]/q-' - - 1 [ -:- l / q ~ - - 1 (q~/5 - - %q:~. 2 + ~,q'-/3 -

- - q Z / 1 5 + ~ o q / 4 - - --~,13 - - 2 '15), ( 3 )

F, (q) = (~olnIq § U U - 11 + (1 q- ~ / 2 - ~ o q - q~/2);Vq- ' - 1) 6,

F~ (q) = (q'-' - - 1 )-~;'-' (2 - - 3-:~ - - 5q-" 4- I0 -oq '~ - - 4 %q~) /720 .

The energy of the neutrinos liberated in unit time in 13 decay, taking account of the ac- tion of an intense magnetic field, may be calculated using the following expression [6J:

E -- U.

dE~__ = W,, } (0,, n) , _,~

d ~ , t-/~ l ' , - ' - -b~ I + e x p ( - - - ~ - ' 1 /l=O hi,

Taking account of the first terms in the expansion of E v with respect to the parameter (H/H c) , the following result is obtained in the case of a degenerate electron gas

where

E , / ( 4 ~ We) = Go (-o) - Go (~) + (M,,',%) ~- (0, (-,) - o, (,?.)) + (t-//M~)' (O~ (~o) - -

--Gz (p)) + (2M/MY:~ ~ ( - - 1/2, { N ~ } ) ( % - F ) : V i + ( i M M~) s~2"-_ (--3 2, {N~})

(-~, - - ~)"-1(2~) + (2 t4114) ' " : ( - - Sii, {N.~}) ( ~ - - FDI( l0 v.") + -+- (2/-//k/o) 9n [~. ( - - 7~2, {N~}) (3so~/':f ' -- 1/,,~ 3) - - " (-- 7 _,o ,~- , , . ~ ' ~:~ o .%] 70,

(5)

Go (q) = ( 3 % / 8 + 1/16) In i q + t / q~ - - 1 I - - 1 /q~ - - 1 (q: ' 6 - - 3~.q~/5 q- , = 2 ' - - : ~ q U 3 " - '~ - - 4 - 3 ~ q 3 . 4 - - q : ~ / i H - - v . ~ , q / o - - - - 3 o o q 8 - - q ; 1 6 + : ~ ~, 3-4-, 2 - n 5 ) ,

G, (q) = [r~-~ ,9 , 3 ' 4 ) l n ] q + ' l , ' q ~ _ l l + (q : ' /4 - - " - o , o _ 3 % q ' j - 3-- ?,q/2 ( 6 ) ~ = ( ~ . , ' - "7- ,

- - 3q 4 § -~ ,2 + 3~o) ] / q - ' - - 11;:6, - - ' . . . . '~ § 2 % ) / 2 4 9 . O._, (q) = (q~ 1) -s '~ (q~ - - 2-?~q:' ~- 5zoiq :~ - - o%q- - - :-~

In Eqs. (2) and (5), the contributions not including the magnetic field coincide with the analogous literature data taking no account of the magnetic field [i].

The corrections for the magnetic field include both a monotonic component proportional to the integer powers of (H/H c) 2 and oscillatory contributions determining the generalized zeta function and proportional to (H/He)P/2, where p = 3, 5, 7... Thus, in the expressions for the B-decay probability and the intensity of the neutrino radiation summed and averaged over the polarizations of the particles participating in the reaction, the field corrections are significantly nonlinear. The appearance of linear contributions with respect to the field may be justified, for example, on taking account of the correlation of the neutron-spin ori- entation and the direction of the magnetic field [7].

The presence of oscillations is a very characteristic feature of the expressions describ- ing the processes in the presence of an intense magnetic field [8, 3]. Using the Hurwitz formula

0,

the order of magnitude of the oscillating terms may be estimated here.

It is also evident from Eq. (5) that the relation between the electron-gas density and the energy liberation in the decay may play a decisive role in determining the oscillating corrections.

In the limit of small - (g~ -- !) << 1 -- and large -- go >> i -- energy liberation in the decay, Eqs. (2) and (5) simplify still further. Retaining only the first terms in the expan- sion with respect to (H/Hc) , the expressions obtained are

309

Page 3: Energy of neutrino radiation in beta decay in a quantizing magnetic field

t05 12 ( - ~ - - I ) ~ ( ~ 1 ) / " 1

E j ( 4 r . Wo) 2 ( ~ - - 1)~'~ [ 21 (H/H~) ~" ' = 1 ~ ., )~ ' 3 1 5 4 (~.~ - - 1

: . . _ , 4; _ ( H H~) ~ , ~ _

, , _ ,~ _ ~ • - ( H / H A " " o- _ a__ ' ~~ >~1"

Note that, in the given cases, the true parameters of the expansions are (H/Hc)/(eo = --i) and H/(Hc~ ~) �9

At high temperatures, the expressions for the probability and power of neutrino radia- tion in 8 decay take the form

1 [Fo(~o) ~ ( H ' H J ' F ~ (~o) ~ (H."H~) ~ F., (~,>) - - W/'(4~. Wo) = ~ - . . . .

- - ( 2 H H~);r-~ (-- 5,2. {N} ) / ( lo=8 ) ] @ ~ - - [] --~, - - 1 (-z~,/30 - - ~ / 1 5 --?

+ 49%/120) - - A (~o~/8 -{- 1/16) - - pFo (-o) + ( H / H y - (51 ~,] - 1 - -

- - A (2--~ --? 3) - - 4~F~ (%)) /24 + (H /H~) ~ ((~.,] 2 4 - - --~ 6 4 - 5-~o/24 - -

- - %/12) (-~ - - 1) -~2 - - , ,F~ (%)/15) - - (2H."H~) 7,~ : ( - - 5:2, {N}) (% - - t,)/15l, ( 7 )

1 [6o 0o) + (H~f4)-G, (--o) + ( H , : H g ' G ~ ( ~ o ) - - E,"(4=Iro) -~

- ( 2 H / H , . p 2 : ( - - 7/2, {N)),~(35--30) + ~ [V' ,~ -- I 0~'70 -- 3 ~ / 7 0 +

-F421 ~ '840 -+- 16/15)12 - - A ( ~ / 8 + 3zo/16 ) - - ~Go (%) § (HiH#) ' ( (17%If12- -

- - ~ , /12 - - 4 1 3 ) / V ~ - 1 - A (~]/2 @ 9--o/4 - - t,G, ( % ) ) / 6 @ (H /Hc) ' X

_ r.o4 ' 1 1 6 ) - - : , G . > ( % ) ) - - X ((-% 1) =~m (.h]/24 - - 7=-~/24 + o-o, 8 - - 13~.~/24 --;- .

- - (2H;'He)gl2.: ( - - 7/2, {N}) (-% - - p), . ' (35-~)], ( 8 )

where A = i ~ [ 8 o + ~ 1 "

It is directly evident from Eqs. (7) and (8) that, in the limit of high temperatures, the monotonic field corrections, as before, increase quadratically with increase in the field, and the oscillating contributions due to the discreteness of the electron states in the magnetic field are significantly reduced: ~(H/Hc )7/'2 and ~(H/Hc)9/a for the probability and energy, respectively. Thus, in contrast to the above case of a degenerate electron gas, the quantiz- ing action of the magnetic field is significantly smoothed at high temperatures, as would be expected.

At magnetic fields

H > H c ( ~ - - 1)/2, H c ( ~ ~ - - 1)/2 ( 9 )

and above, the principal contribution to the probability and intensity of neutrino radiation is linear growth with increase in the field [2, 3]. The temperature expansions of Eqs. (i) and (4) in the limit r >> 1 when Eq. (9) holds are also given here:

W/(4~f 'o) = (H/He) L, 2 + (4(D) -~ (H/H~) (L._, -- DL,) --

- - 48(D~) - ' ) (H. H ~ ) I V ~ - I (.~[;/60 ~ - :~/20-- 181z.~/240 -3,,.-,. (~.,. 30 +

+ 11-=~i60 + 8,'15)) + A (3~,/16 + 5/32 + 9"oa, 8) -+. 3F~L,. - - FZL,], where

L, = ]..F- ,; - - 1 ( :0 .6 .~ _ . : , , ' 1/3) - - A--o/2.,. L~. = l/-~--~o 2 - 1 (~...24 - - 23%/48) + A (--0/4 + 3, '16),

E,/(4=W,0 = ( H t H ~ ) A4,,2 + (4 r ( H / H ~ ) (M.,. - - ,~NIJ - -

- - (48(I:);~) - ' (H'H~) []/-:~ - - 1 (z~/140 + ~ / 3 5 - 499%/560 - - 16/33 - - , , e , 15=o/32 ' - - 3:, (~-~/60 + 2-o~,/15 ~ 103%, 80 ) ) . - - r A ( 3 ~ / 1 6 +

--- 3~ ( 9 ~ / 1 6 + 5 32)) -+- 3p~M, - - ~;~AI,],

310

Page 4: Energy of neutrino radiation in beta decay in a quantizing magnetic field

where

M2 = l ~ - - 1 ( ~ / 4 0 - - 1 3 7 ~ / 2 4 0 - - 4 /18) + A (a~/4 -+- 9~o/16)'.

The presence of magnetic fields of the order of 10x2-10 x3 G close to pulsars is now al- most universally accepted. The possibility of fields of the order of i014-i0:5 G has been predicted on the basis of the analysis of data on the magnetic moments of x-ray pulsars. Thus, the results obtained may be interesting from the viewpoint of constructing an adequate pic- ture of the physical phenomena in the depths of collapsing stars and close to their surface. Note also that the given model of a homogeneous magnetic field may be a fairly good approxi- mation for the generally accepted model describing the stellar magnetic field as the field of a magnetic dipole.

It remains to thank Io M. Ternov, B. A. Lysov, and A. P. Krylov for useful discussions.

LITERATURE CITED

I. V. S. Imshennik, D. K. Nadezhin, and V. S. Pinaev, Astron~ Zh., 44, 768 (1967). 2. I.M. Ternov, B. A. Lysov, and L. I. Korovina, Vestn. Mosk. Univ., Ser. Fiz. Astron., No.

5, 58 (1965). 3. O. F. Dorofeev, V. N. Rodionov, and I. M. Ternov, Pis'ma Zh. Eksp. Teor. Fiz., 42, 222 (1985) o 4. Oo Fo Dorofeev, Vo N. Rodionov, and I. M. Ternov, Pis'ma Astron. Zho, II, 302 (1985). 5. M. A. Evgrafov, Asymptotic Estimates and Target Functions [in Russian], Nauka, Moscow

(1979) o 6. M. A. Ivanov and G. A. Shul'man, Astron. Zh., 57, 537 (1980). 7. Io M. Ternov, O. F. Dorofeev, and V. N. Rodionov, Pis'ma Zh. Eksp. Teor. Fiz., 39, 87

(1984) o 8o L. D. Landau, Collected Works [in Russian], Nauka, Moscow (1969), p. 317.

RADIATIONAL SELF-POLARIZATION OF ELECTRON--POSITRON BEAMS IN AN

AXIALLY SYMMETRIC FOCUSING ELECTRIC FIELD

V. G. Bagrov, V. V. Belov, I. M. Ternov, and B. V. Kholomai

UDC 530.12

Radiational self-polarization of an electron (positron) beam with spontaneous emis- sion in an axially symmetric focusing electric field with a potential of the form #(r) = UrU is considered. The analysis is based on the solutions of the Dirac equa- tions found in the approximation of small oscillations in the vicinity of the equi- librium radius of rotation. It is shown that the existence of self-polarization depends significantly on the structure of the field; in particular, the probabil- ity of electron transitions with spin reversal is zero when U = --i.

Spontaneous electromagnetic radiation in external fields may lead to the appearance of a predominant orientation of the particle spin (radiational self-polarization). This effect was first predicted for electrons in a constant magnetic field [I]. In [2], it was shown that ra- diational self-polarization of electrons (positrons) also occurs in the case of motion in an axially symmetric focusing electric field. This conclusion was based on direct calculation of the probability of electron radiation with spin reversal from the Dirac wave functions of a "rigid" cylindrical rotator. However, from a physical viewpoint, this model is very impre- cise for the description of actual motion. In the present work, radiational self-polarization in an axially symmetric electric field is analyzed on the basis of steady quasi-classical (h --~0) trajectory-coherent states of the electron [3], corresponding to motion along a spiral with small oscillations around the equilibrium orbital of rotation.

Institute of High-Current Electronics, Tomsk Branch, Siberian Section, Academy of Sci- ences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 88-92, April, 1989. Original article submitted July 16, 1986.

0038-5697/89/3204-0311512.50 �9 1989Plenum Publishing Corporation 311