CC BY
38
UDC 530.1; 539.1
New results in the fundamental theory of synchrotron radiation: the evolution of
spectral maximum
V. Bagrov1 2, A. Burimova3, D. Guitman3
1 Department of Quantum Field Theory, Tomsk State University, Tomsk, 634050, Russia;
2 Tomsk Institute of High Current Electronics, SB RAS, Tomsk, 634034, Russia;
3institute of Physics, University of Sao Paulo, Brazil.
E-mail: [email protected]
New results referred to the fundamental theory of synchrotron radiation (SR) are to be presented. We thoroughly analyze the relation between the amount of radiation emitted by a scalar particle at the transitions to the first excited state and to the ground state. Generalizing basic expressions we can follow the evolution of spectral maximum. It turns out there is a condition for radiation maximum to stay at highest harmonic.
Keywords: synchrotron radiation; spectral maximum; quantum transitions
1 Introduction
The motion of a scalar particle (boson) in a constant uniform magnetic field of intensity H = (0,0, H) can be described by Klein-Gordon equation from which follows that the spectrum of particle’s energy is discrete (see [ 1—3] for details)
E
2
Ymgc'
moc2
1
v/T—в2 1 - в2
1 + (2n + 1)b, (1)
„2„3
0=
________= Qh
jeojft e2|eo| ’
„2™2„3
Q=
h2
Here 7 is the relativistic factor; m0 is the rest mass of the particle; c is the speed of light; ft = v/c, where v is the speed of the particle in classical theory; h is the Plank constant and e0 is the charge of the particle. H0 may be interpreted as the Schwinger field.
Energy levels are numbered with n = 0, 1, 2, 3, etc., n = 0 for the ground state. The concept of quantum spectrum of SR bases on the quantum transitions between energy levels, i.e. at the transition from initial level n to final level s the harmonic v = n — s of quantum spectrum is radiated. Obviously, quantum spectrum contains a finite number of harmonics v < n in contrast to classical one which, as we know, is infinite whatever the radiation parameters are.
It is well-known that for non-relativistic limit classical theory of SR predicts the maximum of radiation to be found at the first harmonic of classical spectrum and quantum theory does not contradict this prediction. Indeed, at low values of ft (ft « 0) only the first harmonic is radiated (in classical theory the radiation associated with other harmonics is vanishingly small),
so there is no other possible position for radiation maximum. Increasing ft we get to relativistic domain where the shift of radiation maximum to higher harmonics is expected. In [4] L. A. Artsimovich and I. Ya. Pomer-anchuk demonstrated that according to classical theory of SR the number of harmonic vmax associated with the maximum amount of emitted radiation is proportional to y3, vmax ~ y3. It means that the number vmax can be increased infinitely with ft. This result can not be repeated in quantum theory at least because v < n, as mentioned above. However, in quantum theory one can ask about the condition for vmax = n, i.e. for the radiation maximum to lie on the highest possible harmonic.
2 Basic definitions and notations
Let us consider the transitions from initial quantum state n to the first excited state s = 1 (v = n — 1)
s = 0 v = n of unpolarized radiation may be interpreted in terms of ft Mid 9 (the angle 9 gives the direction of photon emission) [11
W b(n,ft) =
Qft6 £ Fb(n,v,ft)
V=1_____________
2(2n +1)3(1 - в2):
Fb(n,v,ft) = y f b(n, v, ft, 0) sin 0d0,
(2)
f b(n,n - 1,ft,0) =
(n — 1 + x)
3n—2
e x ([n(n — 1) —
n!(n — 1 — x)
— (2n + 1)x + x2]2 + (n — x)2(n — 1 + x)2 cos2 0) v = n — 1, n > 2,
2
o
f b(n, n, ft, 9) = (n + x) x ) e x [(n — x)2+ n!(n — x)
(n + x)2 cos2 9] , v = n ^ 1.
J i 12 ■
1J0 ■
0.8 ■
0.6
04 ■
02 ■
Figure 1: The functions K(n,/3) at n = 5, 10, 20, 50, 100.
n
20000 40000 60000 80000 100000
Figure 2: The y2 = 2fc0n line with the (n, Y2)~roots of K(n,^)=l.
3 The evolution of spectral maximum
Now we introduce the functions K(n, ft)
K(n ft) = Fb(n,n,ft)
K (n,ft) Fb(n,n - 1,ft)'
The graphics of these functions are shown in Fig. 1.
K( n, ft)
amount of radiation emitted at v = n and v = n — 1 transitions. It is easy to see that if K(n, ft) > 1, then the maximum of radiation stays at highest harmonic. vmax = n and if K(n, ft) < 1, then the maximum does not shift to highest harmonic. Finally, the condition
K (n,ft ) = 1 (3)
at fixed n defines such ft = ftn and, therefore, such Y = Yn = (1 — ftn)-1/2 that the maximum in radiation spectrum shifts from harmonic v = n — 1 to the high-v=n
n
Y2 ~ 2bn, n ^ 1. (4)
So, as the solutions of (3) can be rewritten in the form
(n,Y^ )> according to (4) they must lie on the ’line’ Y2 = 2k0n (Fig. 2).
The coefficient k0 was calculated numerically, k0 « 1,146128792697. It is importât to emphasize that k0 does not depend on n. On the other hand, y^ = 2k0n = 2b0n, i.e. k0 = b0, where b0 is associated with a certain value of external magnetic field. Thus, whatever
the initial energy level, the radiation maximum shifts to the highest harmonic only if the external field is more or equal to a certain critical value, b > b0. This can be formulated differently - no matter how much we increase the energy of a particle or change other conditions, the shift of maximum to vmax = n can only occur when b > b0
if b < b0, vmax < n — 1, if b > b0, vmax < n.
Thus, we assume that there exists an ordered set of numbers b0 > b1 > b2 > ... > bs such that if b < bs then the spectral maximum lies on vmax < n — s and it shifts to higher harmonics with n. If bm < b < bm-1 (1 < m < s) then for any initial state n > m the spectral maximum stays at vmax = n — s. Finally, if b > b0 then at any n we have vmax = n.
4 Conclusion
In the framework of quantum theory we consider the evolution of SR spectral maximum. We found the condition for the maximum to lie at the highest harmonic of the spectrum, and, therefore, increasing the energy, any other shift of the maximum for higher har-
monics does not occur. The condition of its shift to the highest harmonic is that the external magnetic field intensity is higher than certain critical value. If the intensity of external field is less than this critical value, whatever the other parameters are, the spectral maximum does not move to the highest harmonic. Thus, we found the restrictions for the rule concerning the evolution of spectral maximum in classical theory.
Acknowledgement
This work was supported by FAPESP (Sao Paulo, Brazil) and the Ministry of Education of Russia, contract numbers N 14.B37.21.0911 and N P789.
References
[1] Bagrov V. G., Bisnovatyi-Kogan G. S., Bordovitsyn V. A. et al. Synchrotron radiation theory and its development. Singapore: World Scientific. 1999.
[2] Sokolov A. A., Ternov I. M. Radiation from relativistic electrons. NY: American Institute of Physics. 1986.
[3] Sokolov A. A., Ternov I. M. Synchrotron Radiation. Berlin: Akademie Verlag. 1968.
[4] Artsimovich L. A., Pomeranchuk I. Ya. JETP. 1946. V. 16. P. 379.
Received 01.10.2012
В. Г. Багров, A. H. Вуримова, Д. М. Гитман,
НОВЫЕ РЕЗУЛЬТАТЫ В ФУНДАМЕНТАЛЬНОЙ ТЕОРИИ СИНХРОТРОННОГО ИЗЛУЧЕНИЯ: ЭВОЛЮЦИЯ СПЕКТРАЛЬНОГО МАКСИМУМА
Представлены новые результаты, относящиеся к фундаментальной теории синхротронного излучения (СИ). Исследовано отношение количества испускаемого излучения при переходах в основное состояние и первое возбужденное. Обобщение основных выражений позволило изучить вопрос об эволюции спектрального максимума. Обнаружено условие смещения максимума на последнюю гармонику.
Ключевые слова: синхротронное излучение, максимум спектра излучения, квантовые переходы,
Багров В.Г., доктор физико-математических наук, профессор.
Томский государственный университет.
Пр. Ленина, 36, Томск, Россия, 634050.
Институт сильноточной электроники СО РАН.
Пр. Академический, 2/3, Томск, Россия, 634055.
E-mail: [email protected]
Вуримова A.H., аспирант.
~Университет Сан-Паулу, Институт Физики, Факультет ядерной физики.
Rúa do Matao Travesea 187, СЕР 05508-090, Sao Paulo, Brazil.
E-mail: [email protected]
Гитман Д.М., доктор физико-математических наук, профессор.
~Университет Сан-Паулу, Институт Физики, Факультет ядерной физики.
Rúa do Matao Travesea 187, СЕР 05508-090, Sao Paulo, Brazil.
E-mail: [email protected]
