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7Бюллетень науки и практики /Bulletin of Science and Practice https://www.bulletennauki.com
Т. 7. №9. 2021 https://doi.org/10.33619/2414-2948/70
UDC 539.123
https://doi.org/10.33619/2414-2948/70/02
GENERAL FORMULA FOR IMPULSE LOSSES IN THE PROCESS OF EMISSION OF NEUTRINO PAIRS BY ELECTRONS IN A MAGNETIC FIELD
©Gajiyeva B., Ph.D., Nakhchivan State University, Nakhchivan, Azerbaijan
ОБЩАЯ ФОРМУЛА ИМПУЛЬСНЫХ ПОТЕРЬ ПРИ ИЗЛУЧЕНИИ ПАР НЕЙТРИНО ЭЛЕКТРОНАМИ В МАГНИТНОМ ПОЛЕ
©Гаджиева Б., канд. физ.-мат. наук, Нахичеванский государственный университет,
г. Нахичевань, Азербайджан
Abstract. Considered formula pulsed radiation losses pairs neutrinos electrons in a magnetic field. Gas consisting of polarized electrons in the direction of the magnetic field and spins composed of polarized electrons in the opposite direction of the magnetic field would receive a different impulse due to the asymmetric transmission of the impulse.
Аннотация. Рассматривается формула импульсных потерь при излучении пар нейтрино электронами в магнитном поле. Газ, состоящий из поляризованных электронов в направлении магнитного поля и спинов, состоящих из поляризованных электронов в направлении, противоположном магнитному полю, получит другой импульс из-за асимметричной передачи импульса.
Keywords: magnetic field, electrons, adiation pulse losses.
Ключевые слова: магнитное поле, электроны, излучение, импульсные потери.
An impulse is transmitted to the stellar conditions by the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field. Impulse transmitted to the scope of the conditions in a momentum unit (impulse losses) is generally calculated by the formula [1-5]:
§ = 7 in dnf П dnf (1 - ff )
К
(1)
Here in the dn and dnf scope element of the phase, respectively, the number of initial and final cases, the distribution functions of the particles in f — initial state, the distribution functions
of the particles in ff — final state, the momentum unit value of the modulus of |M/| /3 —
process matrix element, 3 — at the moment of the total interaction, in q — reaction it is an impulse transmitted to the conditions by a neutrino. The formulae for the impulse losses transmitted to a single scope of the stellar condition at a momentum unit due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field are correct.
2
Бюллетень науки и практики /Bulletin of Science and Practice https://www.bulletennauki.com
Т. 7. №9. 2021 https://doi.org/10.33619/2414-2948/70
dP_ 1 dt ~ ¥■
J dnjedne, (1 - fe, )dnv (1 - fv )dn~ (1 - f~)
M,,
3
(2)
L
L
dn =—dp —^ dp e 2ж Py 2ж d
(3)
L
L
dn' = —dp'—^dp' e 2ж p 2ж
(4)
dn„ =
dn~ =
Vdk'
(2Ж)3
Vdk
(Ж
(5)
(6)
By including here, we obtain the general expression for the impulse losses transmitted to a single scope of the stellar conditions at a momentum unit by the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field.
dP Gl LVL
dt (2
-l J Qq(i - f)(i - f~ f (i - f^ö(E E )x
ж) coco VJ
(7)
3(p'y + Чу - ?V )S(p'z + 4z - pz )dpVdpzdp'vdp'zdkdk'
In (7) and (1) formulae
fe = fe (E ) =
1
(8)
E-Me ' T + 1
Fermi-Dirac distribution of electrons in the initial state, Fermi-Dirac distribution of electrons in f' = fe (E') final state, chemical potential of /ue electrons, temperature of T — electron gas
fv= fM')=
(9)
С
+1
Fermi-Dirac distribution of neutrinos, chemical potential of ¡j.v — neutrinos, — temperature of Tv — neutrino gas:
1 (10)
f~ = f~ (c) =
Tv
e T +1
are the Fermi-Dirac distribution of antineutrinos. We use it in the unit system which is h = c = kB = 1. Here kB is the Bolasman constant. The Q quantity in formula (7) is determined as follows:
2
X
1
e
Бюллетень науки и практики / Bulletin of Science and Practice https://www.bulletennauki.com
Т. 7. №9. 2021 https://doi.org/10.33619/2414-2948/70
Q = NaeJ aJ
= Kk'p + Kkp - g((kk') +
a(3 *
^ a(3f.iv
(11) (12)
Here ga(3 is a metric tensor and saB is an antisymmetric tensor. (a, (, .,v = 0,1,2,3).
The following general expression is true for the R quantity for the arbitrary kinematics of the motion of neutrinos and antineutrons:
q = (cc +kk + kyk'y + kX 0 )2 + (cc+kxk'x - kyk'y - kzK J + (cc- kxk'x + kyk'y - kX )>
x| J 2|2 + (cC- kxk'x - kyk'y + kzk'z )( J3 )2 - 2 (®k'x + kx co' ) Re J0 J1 - 2 (ck'y + ky co' ) Re J0 J2 -
-2 (ak[ + kza ') J0 J3 + 2(kxk'y + kyk'x )Re JlJ2 + 2(kxk[ + kzk'x )Re + +2(kk'z + kzk' )Re J2 J3 - 2(kX - kX )Im J 0J1 - 2(kzkx - kXz )bn J J2 +
+2 (a 'it,-ak'y ) Im J1 J2 - 2(a 'kx-ak'x )Im J2 J3 + 2(aX-ak'y )Im J J3
(13)
Taking into account that p determines x coordinate of moving orbit of quantum number
electron
(14)
eH
and this changes in this coordinate part
L L
—x < x = xn <-*-2 0 2
It is possible to carry out integral for dpy in (7) expression.
J dPy = eHLx
Integrals for dp'y and dp'z are implemented by means of delta functions. JF(...)5(p'y + qy - py )dpy = F(...)
JF(..>(p; + qz - pz)dp'z = F(...)
(15)
(16)
(17)
(18)
Indicating the polar angle of antineutrinos (neutrinos) with $($' ) and the azimuthal angle with a(a')in the spherical coordinate system, the components of the impulses of antineutrinos (neutrinos) can be written as follows,
X = a sin 3 cos a, ky = a sin 3 sin a,
kz = a cos3, kX = a' sin3 x cosa
(19)
(20)
(21) (22)
Бюллетень науки и практики / Bulletin of Science and Practice Т. 7. №9. 2021
https://www.bulletennauki.com https://doi.org/10.33619/2414-2948/70
k'y =a' sin 3' sin c', (23)
k =a' cos S (24)
Here, non-dashed quantities refer to antineutrinos, and dashed quantities refer to neutrinos. Taking (19-24) expressions into account, it is possible to write (13) expression as:
Q = ®co'Q, (25)
The structure of the quantity depends on the polarization of the electrons in the initial and final cases, and we will give it later.
dk = 02 dadQ (26)
dk' = m' 2da'dQ.' (27)
Considering the angle elements of the body in the formula (7), the formula is obtained for the impulse losses transmitted to a single scope of the stellar conditions simultaneously due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation of electrons in a magnetic field:
^ = eH f 2 q(l - fv Xl - f )fe (1 - f W' + ф' + ф- E )dpzdada'dQdn ' (28) dt (2ж) J
As we view the effects of polarization, we do not average the spins of the electrons in the initial (final) state.
It is possible to write delta function simplifying the energy conservation as:
S(E' + + E ) = Y EE S (pz - Pz,) (29)
i\E, Pz, - EIPz,\
Here E , E satisfies energy conservation while the law of conservation of the third component of the impulse is satisfied by (pz = p\ + kz + k[). In this case, the impulse losses transmitted by electrons to a unit scope of the stellar conditions simultaneously due to the emission of neutrino pairs by electrons in the process of neutrino synchrotron radiation in a magnetic field are determined by the following formula:
^ = T^V eH E'El f.Q0a2v'2 q(l - f )(1 - f )fe (1 - f, ]doda'dQidQ.' (30)
dt (2k) j , \E ,pZI - E,Pz,\
Conclusion
During neutrino synchrotron radiation, these gases will be stimulated by the transmission of impulses as a result of the emission of neutrino pairs. However, a gas consisting of polarized electrons in the direction of the magnetic field and spins composed of polarized electrons in the opposite direction of the magnetic field would receive a different impulse due to the asymmetric transmission of the impulse.
Бюллетень науки и практики / Bulletin of Science and Practice Т. 7. №9. 2021
https://www.bulletennauki.com https://doi.org/10.33619/2414-2948/70
References:
1. Huseynov, V. A. (2012). Determination of spin and other quantum characteristics of neutral boson with mass around 126 GeV discovered in CMS and ATLAS experiments and its identification. arXivpreprint arXiv:1212.5830.
2. Borisov, A. V., Zhukovskii, V. Ch., & Lysov, B. A. (1983). Rozhdenie elektron-pozitronnoi pary neitrino v magnitnom pole. Izvestiya vuzov. Fizika, (8), 30-34. (in Russian).
3. Guseinov, V. A. (2003). Leptonnye protsessy vo vneshnem magnitnom pole: Dr. Diss.
Baku.
4. Kuznetsov, A. V., & Mikheev, N. V. (1999). Neutrino-electron processes in a strong magnetic field and plasma. Modern Physics Letters A, 14(36), 2531-2536. https://doi.org/10.1142/S0217732399002649
5. Chernodub, M. N. (2013). Electromagnetic superconductivity of vacuum induced by strong magnetic field. Strongly Interacting Matter in Magnetic Fields (pp. 143-180). Berlin, Heidelberg, Springer. https://doi.org/10.1007/978-3-642-37305-3_6
Список литературы:
1. Huseynov V. A. Determination of spin and other quantum characteristics of neutral boson with mass around 126 GeV discovered in CMS and ATLAS experiments and its identification. arXiv preprint arXiv:1212.5830. 2012.
2. Борисов А. В., Жуковский В. Ч., Лысов Б. А. Рождение электрон-позитронной пары нейтрино в магнитном поле // Известия вузов. Физика. 1983. №8. C. 30-34.
3. Гусейнов В. А. Лептонные процессы во внешнем магнитном поле: дисс. ... д-ра физ.-мат. наук. 2003. Баку, 211 с.
4. Kuznetsov A. V., Mikheev N. V. Neutrino-electron processes in a strong magnetic field and plasma // Modern Physics Letters A. 1999. V. 14. №36. P. 2531-2536. https://doi.org/10.1142/S0217732399002649
5. Chernodub M. N. Electromagnetic superconductivity of vacuum induced by strong magnetic field // Strongly Interacting Matter in Magnetic Fields. Berlin, Heidelberg: Springer, 2013. P. 143-180. https://doi.org/10.1007/978-3-642-37305-3_6
Работа поступила Принята к публикации
в редакцию 23.07.2021 г. 27.07.2021 г.
Ссылка для цитирования:
Gajiyeva B. General Formula for Impulse Losses in the Process of Emission of Neutrino Pairs by Electrons in a Magnetic Field // Бюллетень науки и практики. 2021. Т. 7. №9. С. 27-31. https://doi .org/10.33619/2414-2948/70/02
Cite as (APA):
Gajiyeva, B. (2021). General Formula for Impulse Losses in the Process of Emission of Neutrino Pairs by Electrons in a Magnetic Field. Bulletin of Science and Practice, 7(9), 27-31. https://doi.org/10.33619/2414-2948/70/02
