For main chiral fields, one can also u~e the identification (7) with the substitution of the current ID by the covariant current IK. Doing that, one has to add to the Lagrangian (8) the term (6) with the substitution ID ~ I~. The final Lagrangian for the main chiral fields will have the form L = L0 + LI (cf. [18]). We can conclude by saying that nonlinear o-model Lagrangians can be obtained purely geometrically from Kaluza-Klein type theories with composition torsion. LITERATURE CITED 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12. 13. 14. 15. 16.

17. 18.

Geometric Ideas in Physics [in Russian], Mir, Moscow (1983). N. P. Konopleva and V. N. Popov, Gauge Fields [in Russian], Atomizdat, Moscow (1972). A. Khodos, Usp. Fiz. Nauk, !46, No. 4, 647 (1985). I. Ya. Aref'eva and I. V. Volovich, Usp. Fiz. Nauk, 146, No. 4, 655 (1985). A. I. Vainshtein, B. I. Zakharov, V. A. Novikov, et al., Elem. Chastis. Yad., 17, No. 3, 472 (1986). A. M. Poiyakov, Phys. Lett., B59, No. i, 79 (1975). E. Witten, Nucl. Phys., B223, No. 2, 422, 433 (1983). T. H. R. Skyrme, Proc. R. Soc. London, A260, 127 (1961). D. I. D'yakonov and M. I. Eides, Pis'ma Zh. Eksp. Teor. Fiz., 38, No. 7, 358 (1983). M.A. Semenov-Tyan-Shanskii and L. D. Fadeev, Vestn. LGU, No. 13, 81 (1977). P. Forgacs, Z. Horvath, and L. Palla, Z. Phys., C30, 251 (1986). D. D. Ivanenko, P. I. Pronin, and G. A. Sardanashvili, Gauge Theory of Gravitation [in Russian], Moscow State University, Moscow (1985). Yu. S. Vladimirov, Izv~ Vyssh. Uchebn. Zaved., Fiz~ No. 2, 133 (1963). T. L. Cutright and C. K. Zachos, Phys. Rev~ Lett., 53, No. 19, 1799 (1984); T. L. Curtright, L. Mezincescu, a n d C . K. Zachos, Phys. Lett., 161B, No. 1-3, 79 (1985). S. Mukhi, Phys. Lett., 162B, No. 4-5, 345 (1985). Yu. S. Vladimirov, in: Problems in Theory of Gravitation and Elementary Particles [in Russian], Energoatomizdat, Moscow (1986), No. 17, p. 66; Preprint of Physics Department of Moscow State University, No. 16/1985, Moscow (1985). A. Salam and J. Strathdee, Ann. Phys., 141, 316 (1982). L. D. Faddeev, Lett. Math. Phys., ~, 289 (1976).

RADIATION SPECTRUM OF A SINUSOIDAL UNDULATOR UDC 537.24:535.23

V. G. Bagrov, L. G. Glazov, Yu. N. Kargin, and G. K. Razina

The spectral radiation distribution is investigated for an electron moving along a sinusoid. It is shown that the radiation maximum is always at the first harmonic and no changes in the electron parameters will result in shifting the maximum to higher harmonics.

At the present time undulator radiation has been studied well theoretically and experimentally, and extensive scientific literature, including the survey [i], the monograph [2], and the textbook [3] has been devoted to it. At this time just one example of practical interest is known for an undulator that allows an exact theoretical analysis of all the radiation characteristic without utilization of simplifying assumptions within the framework of classical electrodynamics. This is the so-called sinusoidal undulator in which the electron motion can be given by the following equations: x=

c~,~t, y = c ~ •

z = O,

(1)

Institute of Heavy-Current Electronics, Tomsk Filial of the Siberian Branch of the Academy of Sciences of the USSR. V . V . Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 32-36, June, 1989. Original article submitted December 10, 1986.

0038-5697/89/3206-0435512.50

9 1989 Plenum Publishing Corporation

435

where c is the speed of light, m is the frequency of electron vibration, C~lI is the drift velocity along the x axis, and c81 is the maximal velocity along the y axis. The following condition should evidently be satisfied

where c8 is the maximum modulus of the velocity vector. Radiation of an electron moving in a sinusoidal undulator (for ~I! = 0) was studied by Schott (see [4], p. 120). However, Schott's results were apparently forgotten and obtained again in [5-13]. But in these papers (except [7]) approximations (that actually reduced to the constraint 8~ << 1 - 8~), were always utilized in analyzing the radiation properties in a sinusoidal undulator that did not permit detection of one property of the spectral radiation distribution that is important in principle, in our opinion, and that we here examine in detail. We note that a Lorentz coordinate system exists in which electron motion occurs without drift (811 = 0) and the charge performs just harmonic oscillations along the y' axis

x ' = O , y' = c x ~ T X s i n % t ' , z' = O, (3) where the Lorentz-invariant natural vibrationfrequency ~o and maximal velocity modulus c• (• < i) have the form %

=

=

(1

- -

~%)-lI2, x = ~ (1 -

~, )-'/~.

(4)

Evaluating the radiation field by standard classical electrodynamics methods [14], we obtain for the electrical intensity vector in the wave zone (e is the electron charge)

2e~j

~ . ~J, (~x) sin ~#,

.

E(r, t) =cr(1 -- ~t=sinOcos~)=sinOsin? ,_~--

(5) x=

[~• , ~---" to(t--r/cj l--[~,sinOcos? 1--[3,sinOcos~' j = (cos ? -- [~usin O) e~ + cos 0 sin ?e0.

Here r is the distance between the origin and the remote observation point, 8, ? are spherical angles yielding the direction of observation, ee, e= are the unit directions of the spherical coordinate system, Jr(x) is the Bessel function of integer order v. It follows from (5) that the radiation frequency m' of the harmonic v is [4-13]

~o' = ,,, (1 -- [3,,sin 8 cos ?)-1.

(6)

As follows from (5), the radiation is completely linearly polarized [5, 7, 13] and the electrical field vector is parallel to the constant vector J. The spectral-angular distribution of the power being emitted is found easily by means of the radiation field co W = e=~' (~ d ~ (1 -- ,8,,sin 8 cos ?)2 _ (1 -- 32 ) sin'e sin~? ~ ~2j~ (~x). Even Schott [4] knew this results (for ~ll ffi 0) and it is also presented

in

[2, 5-7, 13].

The angular radiation distribution, that was studied in [5-7], can be obtained by summation over: 6. Finally, by integrating with respect to the angles we obtain the total power being emitted [4-7] W = e ' ~ z 2 (4 -- 3x2) 12C (1 -- ~2)3'~ (8) Let us examine the spectral radiation distribution in greater detail. Integrating with respect to the angles in [7], we obtain after simple manipulation [4, 5, 7] ]

u7 = e2~,~ c

F (,, ~), F ( , , ~) = ,'

S

1x____ -x' 1 J; (~xx) dx.

(9)

The number N(x)of photons emitted per unit undulator length was investigated in [8-12]. Dividing the integrand of (7) by ~%~', where w' should be taken from (6), integrating with respect to the angles and dividing by c811, we obtain an expression for N(x), (~ ffi e2/~c)

N(x) = '.r~ N (~,, x), N('~, ~ ) = ato (c~,~.~)-'F(~,, x).

436

(t0)

The expression (10) for the number of photons emitted per unit undulator length is evidently valid under the assumption that the quantum corrections to the power being emitted are negligible as compared with the classical expression (8). Therefore, the nature of the radiation spectrum and the properties of the quantity N(z) are determined completely by the properties of the function P(v, x). In the approximation of small x (which again corresponds to the approximmtion 8~ << $~) we easily obtain from (9) (~x) 2~

F(~, ~)=

~<
22~ (~!) ~ (2~ -- 1 ) ' Hence, the evident deduction follows that the maximum in the radiation for • the first (v = I) harmonic. The known approximation [2, 3] of Bessel by Macdonald functions

A (~x)~'----r=--Kl.,3

1 -

(ll) arrives at

Kz/3(x)

~3!~ , e = 1 -- X~,

(12)

should be used to analyze the nature of the spectrum distribution over ~ for • close to i, where (12) is valid for (I--• and sufficiently large v. After simple manipulations, by replacing summation over ~) by integration, we obtain an approximation of the expression (9) in the form

dW,~, W~(y)dy, F (v, • ~,~.(y)/36, y : (v/3) (1--•

3/z,

(13)

where W should be determined in the form W =

9

e=~

,

(14)

12c(I -- x2)31~

which results from (8) for (i--•

The universal function ~(y)has the form OO

, (y) =--=,72 S (x __ y2mx,m)K~I~(x) dx.

( 15 )

Y

A. N. Matveev [7] first obtained the expressions (13)-(15); however, they were unfortunately not published in journals. Differentiating (15) with respect to y we find

~, ( y )

=

48

_ ~.~ y -

1i~ j

xII3KSa(x) dx,

(16)

Y

whence it follows that ~'(y)<0 (for y > 0) and ~(Y) of y. In particular we obtain from (15) and (16)

~(0) = 8 ] / 3 / = ~ 4 , The f u n c t i o n

~(y)

possesses

the easily

is a monotonically decreasing function

41063, ~'(0) = -- co.

verifiable

~ (y) dy

(17)

property =

I

(18)

0

and i n t e g r a t i o n o v e r t h e s p e c t r u m i n ( 1 3 ) r e s u l t s i n t h e t o t a l e m i t t e d power ( 1 4 ) . M a t v e e v d i d n o t e m p h a s i z e t h e f a c t t h a t t h e i n t e g r a l i n ( 1 5 ) can be e v a l u a t e d i n a n a l y t i c f o r m and t h e f o l l o w i n g e x a c t e x p r e s s i o n can be o b t a i n e d f o r ? (y) = 24y= -~ [KII3 (y) K:,,3 (y) + 3yK~/3 (y) - - 3yK~,3 (y)], (19) that was first presented in [15]. We tabulated this function and its table is presented in [15]. The maximal value of ~(y) is achieved for y = 0, and even as • the maximum in the spectral distribution in ~ can thereby not fit at v N(l--z=) -3/2 as would occur in synchrotron radiation. A numerical analysis of the function P(w, • is required for a singlevalued solution of the question of the maximum in the spectrum distribution over v. Matveev [7] gave such a very simple analysis (for v = i, 2, 5 and certain values of • ) and also did not publish it in a journal, whereupon it remained unknown to specialists. We performed electronic computer calculations of the function F(v, • for ~ = 1-20 and ~ = 0.I: 0.2, .... 0.9; i. These results are presented in [15]. The deduction that!the maximum

437

#

3 2~

tl l

!

Fig.

1

in the spectral distribution in 9 is always at the first harmonic (9 = i) for any values of (v = I) in the segment [0, i] follows single-valuedly from the analysis presented in [7, 15]. In our opinion, Hatveev [7] arrrived at this very important and interesting deduction. The detailed numerical computations we performed in [15] absolutely verify this deduction. It seems quite probable t o u s that Motz arrived at analogous deductions. This is indicated by the estimate- (70) on page 206 of the Russian translation of his paper [6] and the sentence following this estimate (see p. 533 in the original of [6]). However, all the computations in [6] were performed by using the assumption ~<
lira F(~, I) = 2V-3/gr.~0.12251, lira F(~, l ) / F ( l , I)~0.38599. 9 -a,-CO

(20)

~-bCO

Therefore, the spectrum of the undulator under consideration is not similar to the synchrotron radiation spectrum in the relativistic case for any values of the parameters. Considering the quantity

N(x)

in the approximation (l--x2)<
~o fY(Y) ay, CO

(21)

Y

36c,~ n Yo

where according to (13)

Y0 ~---(I/3)(I--•

(22)

and because of the divergence of the integral (21) it is impossible to set Y0 (21) we obtain the asymptotic expression [15] N (~) ~ -

(=~,,

K S ) zn ( I - ~), (I -

~) << I

= 0.

From (23)

We hence see that as the electron energy grows N(x) grows logarithmically. On the basis of numerical estimates, the deduction was made in [8-12] that N(• and N(v, x) are independent of the electron energy in the relativistic case. This deduction is absoutely valid for N(v, z); however,the results of [8-12] for N(• should be refined in this respect while the numerical estimates in these papers s out to be simply unresponsive to a very slow change in ln(l--• The examination performed here shows that the spectral distribution of undulator radiation can possess nontrivial singularities in specific cases.

438

LITERATURE CITED io 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12. 13. 14. 15.

I. M. Ternov, V. R. Khalilov, V. G. Bagrov, et al., Izv. Vyssh. Uchebn~ Zaved., Fiz., No. 2, 5 (1980). A. A. Sokolov and I. M. Ternov, Relativistic Electrons [in Russian], Nauka, Mocow (1983). I. M. Ternov, V. V. Mikhailin, and V. R. Khalilov, Synchrotron Radiation and Its Application [in Russian], Izd. Mosk. Gos. Univ. (1980). G. A. Schott, Electromagnetic Radiation, Oxford (1912). A. N. Matveev, Vestn. Mosk. Univ., Ser. Fiz., Astron., No. 10, 33 (1951). H. Motz, J. Appl. Phys., 22, No. 5, 527 (1951). A. N. Matveev, Candidate's Dissertation, Phys.-Mat. Sci. [in Russian], Mosk. Gos. Univ. (1954). N. A. Korkhmazyan, Izv. Akad. Nauk ArmSSR, Fiz., ~, No~ 4, 287 (1970)~ N. A. Korkhmazyan, Izv. Akad. Nauk ArmSSR, Fiz., 5, No. 6, 418 (1970). N. A. Korkhmazyan and S. S. ~ibakyan, Izv. Akad. Nauk ArmSSR, Fiz., 6, No. i, 7 (1971). N. A. Korkhmazyan, S. S. ~ibakyan, and A. M. Zverev, Nauchn. Soobshch. EFI 1-3 [in Russian], Erevan (1972). N. A. Korkhmazyan, Izv. Akad~ Nauk ArmSSR, Fiz., 8, No. 6, 405 (1973). A.S. Vshivtsev and V. Ch. Zhukovskii, Vestn. Mosk. Univ., Ser. Fiz. Astron., No. 6, 65~ (1975). L. D. Landau and E. M. Lifshits, Field Theory [in Russian], Nauka, Moscow (1973). V. G. Bagrov, L. G. Glazov, Yu. N. Kargin, and G. K. Razina, Preprint No. 35 [in Russian], Tomsk Filial, Siberian Branch, USSR Academy of Sciences, Tomsk (1986).

439