- Email: [email protected]
New theoretical results in synchrotron radiation
Nuclear Instruments and Methods in Physics Research B 240 (2005) 638–645 www.elsevier.com/locate/nimb
New theoretical results in synchrotron radiation V.G. Bagrov
a,*
, D.M. Gitman c, V.B. Tlyachev b, A.T. Jarovoi
a
a Tomsk State University, Lenin Avenue 36, 634050 Tomsk, Russia Tomsk Institute of High Current Electronics, Akademicheskiy Avenue 4, Tomsk, Russia Instituto de Fı´sica, Universidade de Sa˜o Paulo, C.P. 66318, 05315-970 Sa˜o Paulo, SP, Brazil b
c
Received 17 December 2003; received in revised form 18 March 2005 Available online 21 September 2005
Abstract One of the remarkable features of the relativistic electron synchrotron radiation is its concentration in small angle D 1/c (here c-relativistic factor: c = E/mc2, E – energy, m – electron rest mass, c – light velocity) near rotation orbit plane [V.G. Bagrov, V.A. Bordovitsyn, V.G. Bulenok, V. Ya. Epp, Kinematical projection of pulsar synchrotron radiation profiles, in: Proceedings of IV ISTC Scientific Advisory Commitee Seminar on Basic Science in ISTC Aktivities, Akademgorodok, Novosibirsk, April 23–27, 2001, p. 293–300]. This theoretically predicted and experimentally confirmed feature is peculiar to total (spectrum summarized) radiating intensity. This angular distribution property has been supposed to be (at least qualitatively) conserved and for separate spectrum synchrotron radiation components. In the work of V.G. Bagrov, V.A. Bordovitsyn, V. Ch. Zhukovskii, Development of the theory of synchrotron radiation and related processes. Synchrotron source of JINR: the perspective of research, in: The Materials of the Second International Work Conference, Dubna, April 2–6, 2001, pp. 15–30 and in Angular dependence of synchrotron radiation intensity. http://lanl.arXiv.org/abs/physics/0209097, it is shown that the angular distribution of separate synchrotron radiation spectrum components demonstrates directly inverse tendency – the angular distribution deconcentration relatively the orbit plane takes place with electron energy growth. The present work is devoted to detailed investigation of this situation. For exact quantitative estimation of angular concentration degree of synchrotron radiation the definition of radiation effective angle and deviation angle is proposed. For different polarization components of radiation the dependence of introduced characteristics was investigated as a functions of electron energy and number of spectrum component. Ó 2005 Published by Elsevier B.V. PACS: 41.10; 41.70 Keywords: Relativistic electron; Classical theory of synchrotron radiation; Intensity angular distribution; Deconcentration
*
Corresponding author. Tel.: +7 3822 553 830. E-mail address: [email protected] (V.G. Bagrov).
0168-583X/$ - see front matter Ó 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.03.286
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
1. Introduction We assume, that a charged particle e moves on the plane z = 0 with a constant velocity v = cb, (0 < b < 1) along a circle of radius R. If this motion is realized by means of an external magnetic field H, then the following relations: ffi bm0 c2 bE m0 c2 pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ R¼ ¼ ð1Þ c 2 1; eH eH eH 1 b2 c ¼ ð1 b2 Þ1=2 ¼
E ; m 0 c2
c>1
ð2Þ
are satisfied [1], where m0 is the electron rest mass, c is the speed of light, E is the total energy and c is the Lorenz factor. Moreover, the angular velocity x0, which also determines the frequency of the first radiation harmonic, has the form: x0 ¼
cb ecH eH ¼ ¼ . R E m0 cc
Taking into account its polarization, the total of radiation intensity W can be represented as ce2 b4 e4 H 2 b2 ð1 b2 Þ W i ¼ W 0 Ui ðbÞ; W 0 ¼ ¼ ; m20 c3 R2 Z p Ui ðbÞ ¼ F i ðb; hÞ sin h dh; 0
F i ðb; hÞ ¼
1 X
fi ðm; b; hÞ;
z ¼ mb sin h; f2 ðm; b; hÞ ¼ f 3 ðm; b; hÞ ¼
ð6Þ m2 J 02 m ðzÞ; 2 2
m cos h 2 J m ðzÞ; b2 sin2 h
ð7Þ
where Jm(x) is a Bessel function of integer order. The functions f1, f2, f3 are characterized by the following apparent properties: k ¼ 0; 2; 3;
f 1 ðm; b; hÞ ¼ f1 ðm; b; p hÞ;
ð8Þ
which makes it possible to analyze restricting the functions fk(m, b; h) only in a closed angle interval 0 6 h 6 p/2(k = 0, 2, 3). Due to the special symmetry of functions f±1, it is enough to study only one of them (for instance f1). 2. Some of the features of the angular distribution of the total radiation intensity
ð4Þ
m¼1
where h is the angle between z-axis and the direction of radiation. The integer m is a number of the harmonic. The frequency x of the harmonic m is given by the relation x = mx0. An index i is a so-called polarization index. It represents several polarization types. For instance, the particular values i = 1, i = 1, i = 2 and i = 3 correspond to the circular left-polarized, circular right-polarized, linear ‘‘r’’-polarized component and linear ‘‘p’’-polarized component accordingly. A zero value of the polarization index i = 0 corresponds to the total integral radiation intensity (which is the sum of all the polarizations) f0 ðm; b; hÞ ¼ f1 ðm; b; hÞ þ f1 ðm; b; hÞ ¼ f2 ðm; b; hÞ þ f3 ðm; b; hÞ.
Detailed definition of the polarization components is given in [1–4]. The functions fi(m, b; h) give an exhaustive description of the spectral-angular distribution of the radiation intensity. According to the classical theory [1–4], these functions have the form: 2 m2 0 cos h J m ðzÞ ; J ðzÞ f1 ðm; b; hÞ ¼ b sin h 2 m
fk ðm; b; hÞ ¼ fk ðm; b; p hÞ; ð3Þ
639
ð5Þ
The exact analytic expressions for the functions Fk(b, h)(k = 0, 2, 3) are well known [1–4] F 2 ðb; hÞ ¼
7 3e ; 16e5=2
e ¼ 1 b2 sin2 h;
1 ðc2 e 1Þð5 eÞ 6 e < 1; F ðb; hÞ ¼ ; 3 c2 16ðc2 1Þe7=2 ð3 4c2 Þe2 þ 6ð2c2 1Þe 5 F 0 ðb; hÞ ¼ . 16ðc2 1Þe7=2
ð9Þ
For the functions F±1 we have the form: 1 F 1 ðb; hÞ ¼ F 0 ðb; hÞ Wðb sin hÞ cos h; 2 1 1X m2 J m ðmxÞJ 0m ðmxÞ WðxÞ ¼ x m¼1 1 1 d X mJ 2 ðmxÞ. ¼ 2x dx m¼1 m
ð10Þ
ð11Þ
640
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
However, no closed analytic expression for the function W(x) has yet been obtained. 3. Investigation of the integral characteristics of radiation It is well known that the overwhelming majority of the integral SR-intensity (spectral SR-intensity summed up over the spectrum) of the ultrarelativistic particle radiates in a small angular interval Dh 1/c near the orbit plane (i.e. neighborhood of h = p/2). In other words, the radiation is concentrated on the orbit plane in the ultrarelativistic case and the value of the concentration increases with increasing of the particle energy which results in the decrease of the angle Dh with increasing c. This assertion is right for any component of the radiation polarization. The same assertion was commonly considered to be true of the angular distribution of the separate spectral components of radiation. But detailed theoretical analysis of the angular distribution of the separate spectral components of the synchrotron radiation predicts a diametrically opposite tendency. In [5,6] it was shown that the angles corresponding to the maximum angular distribution of the separate spectral components SR move from the orbit plane (the point h = p/2) and for c ! 1 tend to the final values (different from p/2). These latter depend on the harmonic number m and on the state of polarization.
In this connection it is appropriate to introduce the quantitative characteristics which describe the degree of the angular concentration of radiation and to investigate the dependencies of these characteristics from the particle energy and state of polarization. We introduce the above-mentioned characteristics in the following way. For example, we consider the angular distribution of F1(b, h) for fixed b. The plot of this function for b = 0.6 is shown in Fig. 1. Let us take the points 0 6 h1 < h2 6 p to execute the condition Z
h2
h1
F 1 ðb; hÞ sin h dh ¼
1 2
Z
p
F 1 ðb; hÞ sin h dh. 0
ð12Þ This condition shows that the half of the right polarization SR-intensity radiates in the angular interval D1 ¼ h2 h1 . It is evident that the condition (12) does not determine the values h1 and h2 (correspondingly D1 ) but establishes a connection between them (or for example, the condition (12) establishes the functional dependence). Now let us select h1 so that D1 be minimum (i.e. weÕll find the minimum of function D1 ðh1 Þ). It is easy to the determine that if h2 does not coincide with the right-end of the interval (h25p) than the condition of minimum D1 is put down in form: F 1 ðb; h1 Þ sin h1 ¼ F 1 ðb; h2 Þ sin h2 .
. . . . . . . .
Fig. 1. On definition effective angle of radiation and deviation angle for function F1(b; h).
ð13Þ
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
From the system of equations (12) and (13) we find min unambiguously h1 and h2 and thereby D1 ¼ D1 will be the smallest. We shall name the D1 an effective angle radiation of right polarization component. The angle a1 ¼ ðp h1 h2 Þ=2
ð14Þ
characterizes the position of the interval [h1, h2] on the h-axes. The point a1 halves the interval D1. We shall call this angle a deviation angle. The introduced values are shown in Fig. 1. The plots of functions Fi(b, h) for i = 0, 2, 3 are symmetrical relative to vertical line h = p/2. For example, Fig. 2 show the plot of function F0(b, h) for b = 0.55. In this case it is appropriate use the described above definition for interval 0 < h < p/2 and otherwise define the deviation angle Z h2 Z 1 p=2 F i ðb; hÞ sin h dh ¼ F i ðb; hÞ sin h dh 2 0 h1 ð15Þ
0 6 h1 < h2 6 p=2; F i ðb; h1 Þ sin h1 ¼ F i ðb; h2 Þ sin h2 ;
i ¼ 0; 2; 3; ð16Þ
Di =2 ¼ h2 h1 ;
ai ¼ p=2 h2 .
ð17Þ
For ai > 0 the effective angle breaks up into two equal symmetrically located parts relative to the straight line h = p/2. These parts are divided by an interval just as it is shown in Fig. 2. In some cases (for example, for functions F2(b; h) for all
641
b and F0(b; h) for b near 1) the value h2 is equal to p/2 coinciding with the right end of the integration. To determine Di it is necessary to consider only Eq. (15), where h1 = (p D1)/2, h2 = p/2. For this case Eq. (16) does not hold and the deviation angle is equal to a1 = 0 as a result of which the two parts of the effective angle radiation get merged and the effective angle is located so that the straight line h = p/2 divides it in to two equal parts. It is absolutely evident that the effective angles and the deviation angles are the functions of particle energy (or value b) Di = Di(b), ai = ai(b). Analyzing the spectral-angular distribution of polarization components of radiation we likewise introduce the effective angle and the deviation angle Di = Di(m; b), ai = ai(m; b) for each harmonic m. The effective angle and the deviation angle are theoretically exact and descriptive-geometric characteristics of concentration degree and of the radiation geometry. One can see that finding the functions Di = Di(b), ai = ai(b), Di = Di(m; b), ai = ai(m; b) in a general case reduces to solution of the systems of two transcendental equations. It would still remain to find analytical expressions for these functions. Nevertheless there is no difficult in providing a numerical analysis of these functions. The results of this analysis make up the substance matter of present paper. Fig. 3 shows the plots of functions Di(b)(i = 0,1,2,3) and Fig. 4 shows the plots of functions ai(b) (i = 1,2,3;a2(b) = 0).
. . . . . . . . . . . . .
Fig. 2. On definition of effective angle of radiation and deviation angle for function F0(b; h).
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V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
In the ultrarelativistic case the following asymptotical formulas hold Di ðbÞ ai =c; c 1; a0 ¼ 0:8413; a1 ¼ 0:6765; a2 ¼ 0:7407; a3 ¼ 1:1950
.
.
.
.
.
.
ð19Þ
thus proving the statement about the concentration of total radiation in a narrow angular cone 1/c along the orbit particle plane. But there is one unexpected feature, namely the functions D1(b) and D3(b) which are not decreasing functions of b having the extremums (maximums) in the nonrelativistic case Dmax 1 ðb 0:17Þ 39 ;
Dmax 3 ðb 0:47Þ 54 .
Fig. 3. The dependence of effective angle on b for the polarization components. . . . . . . . . . . . . . . .
.
.
.
.
.
.
Fig. 5. Profiles of f0(m = 5; b; h) at different b.
Fig. 4. The dependence of deviation angle on b for the polarization components.
According to well known qualitative conclusions we obtain the following expressions for c ! 1 (b ! 1) lim Di ðbÞ ¼ 0; b!1
i ¼ 0; 1; 2; 3;
lim ai ðbÞ ¼ 0; i ¼ 1; 3; b!1
ð18Þ .
a2(b) 0 and while the function a0(b) = 0 for b > b0, where b0 0.295166 (c = 1.046632) (see Fig. 3).
.
.
.
.
.
Fig. 6. The dependence of deviation angle a0(m, b) for different harmonics m.
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
It is noteworthy that for b > b1 0.82(c 1.75) we have the inequality a1(b) < D1/2. This fact shows that part of radiation gets to the lower half-plane. But we have not yet found any plausible physical explanation of this fact. The situation is quite different in the case of spectral component of radiation considered separately. Fig. 5 shows the evolution of the function f0(m; b; h) for m = 5 with variable b. It is important to stress that all the function fi(m; b; h) are finite in the closed interval 0 6 b 6 1 (including the point b = 1). Fig. 5 shows the movement of maximums
.
.
.
.
.
of the function f0(5; b; h) from the plane h = p/2 with increasing b. Figs. 6, 8, 10, 12 show the plots of functions Di(m; b) (the dotted line representing the functions Di(b)). Figs. 7, 9, 11, 13 show the plots of functions ai(m; b) (the dotted line representing the functions ai(b)). Whereby the increasing particle energy brings about either an increase of the effective angle or the D3 for the pcomponent depending on b. There is always a finite limit for c ! 1 lim Di ðm; bÞ ¼ Dmi ¼ Di ðm; 1Þ > 0.
ð20Þ
b!1
.
Fig. 7. The dependence of effective angle D0(m, b) for different harmonics m.
643
.
.
.
.
.
.
Fig. 9. The dependence of effective angle D1(m, b) for different harmonics m.
. . . . . . . . . .
.
.
.
.
.
Fig. 8. The dependence of deviation angle a1(m, b) for different harmonics m.
.
.
.
.
.
.
.
.
.
Fig. 10. The dependence of deviation angle a2(m, b) for different harmonics m.
644
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
.
.
.
.
.
.
Fig. 11. The dependence of effective angle D2(m, b) for different harmonics m.
.
.
.
.
.
.
Fig. 13. The dependence of effective angle D3(m, b) for different harmonics m.
4. Conclusion
.
.
.
.
.
.
Fig. 12. The dependence of deviation angle a3(m, b) for different harmonics m.
The values Dmi decrease in a monotonic way with increasing m and for m 1 there are asymptotic formulas: 1=3 bi Dmi ; b0 ¼ 2:1267; m b1 ¼ 0:1271;
b2 ¼ 0:0491;
b3 ¼ 1:0616.
The general conclusion is that with an increase of particle energy each fixed spectral component (fixed harmonic number) of any radiation polarization tends to towards deconcentration of the angular distribution of radiation relative to the plane of particle orbit whereas the total (summed over spectrum) angular distribution concentrates unto this plane. There is a simple mathematical reason for such a striking contradiction – the series on harmonic m in formula (4) for b ! 1 has nonuniform convergence and therefore the behavior of the sum series and its separate terms may show opposite tendencies with changes in the parameters (b; h).
Acknowledgements This work was partially supported by RFBR Grant 03-02-17615 and President Grant of Russia.
ð21Þ
The functions ai(m; b) (i = 0, 1, 3) are a monotonically increasing functions of b. We have unexpectedly found that there is a range of b values where the functions a2(m; b) are non-zero for m = 1, 2, 3; if m > 3 then a2(m; b) 0.
References [1] A.A. Sokolov, I.M. Ternov, Synchrotron radiation, Academic Verlag/Pergamon Press, Berlin/New York, 1968. [2] A.A. Sokolov, I.M. Ternov, Radiation from relativistic electrons, American Institute of Physics, New York, 1986.
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645 [3] I.M. Ternov, V.V. Mihailin, Synchrotron radiation. Theory and experiment, Energoatomizdat, Moskow, 1986 [in Russian]. [4] Synchrotron Radiation Theory and its Development, World Scientific, Singapore, New Jersey, London, Hong Kong, 1999. [5] V.G. Bagrov, V.A. Bordovitsyn, V.G. Bulenok, V. Ya. Epp, Kinematical projection of pulsar synchrotron radiation
645
profiles, in: Proceedings of IV ISTC Scientific Advisory Commitee Seminar on Basic Science in ISTC Aktivities, Akademgorodok, Novosibirsk, April 23–27, 2001, p. 293. [6] V.G. Bagrov, V.A. Bordovitsyn, V. Ch. Zhukovskii, Development of the theory of synchrotron radiation and related processes, Synchrotron source of JINR: the perspective of research, in: The Materials of the Second International Work Conference, Dubna, April 2–6, 2001, p. 15.
New theoretical results in synchrotron radiation V.G. Bagrov
a,*
, D.M. Gitman c, V.B. Tlyachev b, A.T. Jarovoi
a
a Tomsk State University, Lenin Avenue 36, 634050 Tomsk, Russia Tomsk Institute of High Current Electronics, Akademicheskiy Avenue 4, Tomsk, Russia Instituto de Fı´sica, Universidade de Sa˜o Paulo, C.P. 66318, 05315-970 Sa˜o Paulo, SP, Brazil b
c
Received 17 December 2003; received in revised form 18 March 2005 Available online 21 September 2005
Abstract One of the remarkable features of the relativistic electron synchrotron radiation is its concentration in small angle D 1/c (here c-relativistic factor: c = E/mc2, E – energy, m – electron rest mass, c – light velocity) near rotation orbit plane [V.G. Bagrov, V.A. Bordovitsyn, V.G. Bulenok, V. Ya. Epp, Kinematical projection of pulsar synchrotron radiation profiles, in: Proceedings of IV ISTC Scientific Advisory Commitee Seminar on Basic Science in ISTC Aktivities, Akademgorodok, Novosibirsk, April 23–27, 2001, p. 293–300]. This theoretically predicted and experimentally confirmed feature is peculiar to total (spectrum summarized) radiating intensity. This angular distribution property has been supposed to be (at least qualitatively) conserved and for separate spectrum synchrotron radiation components. In the work of V.G. Bagrov, V.A. Bordovitsyn, V. Ch. Zhukovskii, Development of the theory of synchrotron radiation and related processes. Synchrotron source of JINR: the perspective of research, in: The Materials of the Second International Work Conference, Dubna, April 2–6, 2001, pp. 15–30 and in Angular dependence of synchrotron radiation intensity. http://lanl.arXiv.org/abs/physics/0209097, it is shown that the angular distribution of separate synchrotron radiation spectrum components demonstrates directly inverse tendency – the angular distribution deconcentration relatively the orbit plane takes place with electron energy growth. The present work is devoted to detailed investigation of this situation. For exact quantitative estimation of angular concentration degree of synchrotron radiation the definition of radiation effective angle and deviation angle is proposed. For different polarization components of radiation the dependence of introduced characteristics was investigated as a functions of electron energy and number of spectrum component. Ó 2005 Published by Elsevier B.V. PACS: 41.10; 41.70 Keywords: Relativistic electron; Classical theory of synchrotron radiation; Intensity angular distribution; Deconcentration
*
Corresponding author. Tel.: +7 3822 553 830. E-mail address: [email protected] (V.G. Bagrov).
0168-583X/$ - see front matter Ó 2005 Published by Elsevier B.V. doi:10.1016/j.nimb.2005.03.286
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
1. Introduction We assume, that a charged particle e moves on the plane z = 0 with a constant velocity v = cb, (0 < b < 1) along a circle of radius R. If this motion is realized by means of an external magnetic field H, then the following relations: ffi bm0 c2 bE m0 c2 pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ R¼ ¼ ð1Þ c 2 1; eH eH eH 1 b2 c ¼ ð1 b2 Þ1=2 ¼
E ; m 0 c2
c>1
ð2Þ
are satisfied [1], where m0 is the electron rest mass, c is the speed of light, E is the total energy and c is the Lorenz factor. Moreover, the angular velocity x0, which also determines the frequency of the first radiation harmonic, has the form: x0 ¼
cb ecH eH ¼ ¼ . R E m0 cc
Taking into account its polarization, the total of radiation intensity W can be represented as ce2 b4 e4 H 2 b2 ð1 b2 Þ W i ¼ W 0 Ui ðbÞ; W 0 ¼ ¼ ; m20 c3 R2 Z p Ui ðbÞ ¼ F i ðb; hÞ sin h dh; 0
F i ðb; hÞ ¼
1 X
fi ðm; b; hÞ;
z ¼ mb sin h; f2 ðm; b; hÞ ¼ f 3 ðm; b; hÞ ¼
ð6Þ m2 J 02 m ðzÞ; 2 2
m cos h 2 J m ðzÞ; b2 sin2 h
ð7Þ
where Jm(x) is a Bessel function of integer order. The functions f1, f2, f3 are characterized by the following apparent properties: k ¼ 0; 2; 3;
f 1 ðm; b; hÞ ¼ f1 ðm; b; p hÞ;
ð8Þ
which makes it possible to analyze restricting the functions fk(m, b; h) only in a closed angle interval 0 6 h 6 p/2(k = 0, 2, 3). Due to the special symmetry of functions f±1, it is enough to study only one of them (for instance f1). 2. Some of the features of the angular distribution of the total radiation intensity
ð4Þ
m¼1
where h is the angle between z-axis and the direction of radiation. The integer m is a number of the harmonic. The frequency x of the harmonic m is given by the relation x = mx0. An index i is a so-called polarization index. It represents several polarization types. For instance, the particular values i = 1, i = 1, i = 2 and i = 3 correspond to the circular left-polarized, circular right-polarized, linear ‘‘r’’-polarized component and linear ‘‘p’’-polarized component accordingly. A zero value of the polarization index i = 0 corresponds to the total integral radiation intensity (which is the sum of all the polarizations) f0 ðm; b; hÞ ¼ f1 ðm; b; hÞ þ f1 ðm; b; hÞ ¼ f2 ðm; b; hÞ þ f3 ðm; b; hÞ.
Detailed definition of the polarization components is given in [1–4]. The functions fi(m, b; h) give an exhaustive description of the spectral-angular distribution of the radiation intensity. According to the classical theory [1–4], these functions have the form: 2 m2 0 cos h J m ðzÞ ; J ðzÞ f1 ðm; b; hÞ ¼ b sin h 2 m
fk ðm; b; hÞ ¼ fk ðm; b; p hÞ; ð3Þ
639
ð5Þ
The exact analytic expressions for the functions Fk(b, h)(k = 0, 2, 3) are well known [1–4] F 2 ðb; hÞ ¼
7 3e ; 16e5=2
e ¼ 1 b2 sin2 h;
1 ðc2 e 1Þð5 eÞ 6 e < 1; F ðb; hÞ ¼ ; 3 c2 16ðc2 1Þe7=2 ð3 4c2 Þe2 þ 6ð2c2 1Þe 5 F 0 ðb; hÞ ¼ . 16ðc2 1Þe7=2
ð9Þ
For the functions F±1 we have the form: 1 F 1 ðb; hÞ ¼ F 0 ðb; hÞ Wðb sin hÞ cos h; 2 1 1X m2 J m ðmxÞJ 0m ðmxÞ WðxÞ ¼ x m¼1 1 1 d X mJ 2 ðmxÞ. ¼ 2x dx m¼1 m
ð10Þ
ð11Þ
640
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
However, no closed analytic expression for the function W(x) has yet been obtained. 3. Investigation of the integral characteristics of radiation It is well known that the overwhelming majority of the integral SR-intensity (spectral SR-intensity summed up over the spectrum) of the ultrarelativistic particle radiates in a small angular interval Dh 1/c near the orbit plane (i.e. neighborhood of h = p/2). In other words, the radiation is concentrated on the orbit plane in the ultrarelativistic case and the value of the concentration increases with increasing of the particle energy which results in the decrease of the angle Dh with increasing c. This assertion is right for any component of the radiation polarization. The same assertion was commonly considered to be true of the angular distribution of the separate spectral components of radiation. But detailed theoretical analysis of the angular distribution of the separate spectral components of the synchrotron radiation predicts a diametrically opposite tendency. In [5,6] it was shown that the angles corresponding to the maximum angular distribution of the separate spectral components SR move from the orbit plane (the point h = p/2) and for c ! 1 tend to the final values (different from p/2). These latter depend on the harmonic number m and on the state of polarization.
In this connection it is appropriate to introduce the quantitative characteristics which describe the degree of the angular concentration of radiation and to investigate the dependencies of these characteristics from the particle energy and state of polarization. We introduce the above-mentioned characteristics in the following way. For example, we consider the angular distribution of F1(b, h) for fixed b. The plot of this function for b = 0.6 is shown in Fig. 1. Let us take the points 0 6 h1 < h2 6 p to execute the condition Z
h2
h1
F 1 ðb; hÞ sin h dh ¼
1 2
Z
p
F 1 ðb; hÞ sin h dh. 0
ð12Þ This condition shows that the half of the right polarization SR-intensity radiates in the angular interval D1 ¼ h2 h1 . It is evident that the condition (12) does not determine the values h1 and h2 (correspondingly D1 ) but establishes a connection between them (or for example, the condition (12) establishes the functional dependence). Now let us select h1 so that D1 be minimum (i.e. weÕll find the minimum of function D1 ðh1 Þ). It is easy to the determine that if h2 does not coincide with the right-end of the interval (h25p) than the condition of minimum D1 is put down in form: F 1 ðb; h1 Þ sin h1 ¼ F 1 ðb; h2 Þ sin h2 .
. . . . . . . .
Fig. 1. On definition effective angle of radiation and deviation angle for function F1(b; h).
ð13Þ
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
From the system of equations (12) and (13) we find min unambiguously h1 and h2 and thereby D1 ¼ D1 will be the smallest. We shall name the D1 an effective angle radiation of right polarization component. The angle a1 ¼ ðp h1 h2 Þ=2
ð14Þ
characterizes the position of the interval [h1, h2] on the h-axes. The point a1 halves the interval D1. We shall call this angle a deviation angle. The introduced values are shown in Fig. 1. The plots of functions Fi(b, h) for i = 0, 2, 3 are symmetrical relative to vertical line h = p/2. For example, Fig. 2 show the plot of function F0(b, h) for b = 0.55. In this case it is appropriate use the described above definition for interval 0 < h < p/2 and otherwise define the deviation angle Z h2 Z 1 p=2 F i ðb; hÞ sin h dh ¼ F i ðb; hÞ sin h dh 2 0 h1 ð15Þ
0 6 h1 < h2 6 p=2; F i ðb; h1 Þ sin h1 ¼ F i ðb; h2 Þ sin h2 ;
i ¼ 0; 2; 3; ð16Þ
Di =2 ¼ h2 h1 ;
ai ¼ p=2 h2 .
ð17Þ
For ai > 0 the effective angle breaks up into two equal symmetrically located parts relative to the straight line h = p/2. These parts are divided by an interval just as it is shown in Fig. 2. In some cases (for example, for functions F2(b; h) for all
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b and F0(b; h) for b near 1) the value h2 is equal to p/2 coinciding with the right end of the integration. To determine Di it is necessary to consider only Eq. (15), where h1 = (p D1)/2, h2 = p/2. For this case Eq. (16) does not hold and the deviation angle is equal to a1 = 0 as a result of which the two parts of the effective angle radiation get merged and the effective angle is located so that the straight line h = p/2 divides it in to two equal parts. It is absolutely evident that the effective angles and the deviation angles are the functions of particle energy (or value b) Di = Di(b), ai = ai(b). Analyzing the spectral-angular distribution of polarization components of radiation we likewise introduce the effective angle and the deviation angle Di = Di(m; b), ai = ai(m; b) for each harmonic m. The effective angle and the deviation angle are theoretically exact and descriptive-geometric characteristics of concentration degree and of the radiation geometry. One can see that finding the functions Di = Di(b), ai = ai(b), Di = Di(m; b), ai = ai(m; b) in a general case reduces to solution of the systems of two transcendental equations. It would still remain to find analytical expressions for these functions. Nevertheless there is no difficult in providing a numerical analysis of these functions. The results of this analysis make up the substance matter of present paper. Fig. 3 shows the plots of functions Di(b)(i = 0,1,2,3) and Fig. 4 shows the plots of functions ai(b) (i = 1,2,3;a2(b) = 0).
. . . . . . . . . . . . .
Fig. 2. On definition of effective angle of radiation and deviation angle for function F0(b; h).
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In the ultrarelativistic case the following asymptotical formulas hold Di ðbÞ ai =c; c 1; a0 ¼ 0:8413; a1 ¼ 0:6765; a2 ¼ 0:7407; a3 ¼ 1:1950
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ð19Þ
thus proving the statement about the concentration of total radiation in a narrow angular cone 1/c along the orbit particle plane. But there is one unexpected feature, namely the functions D1(b) and D3(b) which are not decreasing functions of b having the extremums (maximums) in the nonrelativistic case Dmax 1 ðb 0:17Þ 39 ;
Dmax 3 ðb 0:47Þ 54 .
Fig. 3. The dependence of effective angle on b for the polarization components. . . . . . . . . . . . . . . .
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Fig. 5. Profiles of f0(m = 5; b; h) at different b.
Fig. 4. The dependence of deviation angle on b for the polarization components.
According to well known qualitative conclusions we obtain the following expressions for c ! 1 (b ! 1) lim Di ðbÞ ¼ 0; b!1
i ¼ 0; 1; 2; 3;
lim ai ðbÞ ¼ 0; i ¼ 1; 3; b!1
ð18Þ .
a2(b) 0 and while the function a0(b) = 0 for b > b0, where b0 0.295166 (c = 1.046632) (see Fig. 3).
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Fig. 6. The dependence of deviation angle a0(m, b) for different harmonics m.
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645
It is noteworthy that for b > b1 0.82(c 1.75) we have the inequality a1(b) < D1/2. This fact shows that part of radiation gets to the lower half-plane. But we have not yet found any plausible physical explanation of this fact. The situation is quite different in the case of spectral component of radiation considered separately. Fig. 5 shows the evolution of the function f0(m; b; h) for m = 5 with variable b. It is important to stress that all the function fi(m; b; h) are finite in the closed interval 0 6 b 6 1 (including the point b = 1). Fig. 5 shows the movement of maximums
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of the function f0(5; b; h) from the plane h = p/2 with increasing b. Figs. 6, 8, 10, 12 show the plots of functions Di(m; b) (the dotted line representing the functions Di(b)). Figs. 7, 9, 11, 13 show the plots of functions ai(m; b) (the dotted line representing the functions ai(b)). Whereby the increasing particle energy brings about either an increase of the effective angle or the D3 for the pcomponent depending on b. There is always a finite limit for c ! 1 lim Di ðm; bÞ ¼ Dmi ¼ Di ðm; 1Þ > 0.
ð20Þ
b!1
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Fig. 7. The dependence of effective angle D0(m, b) for different harmonics m.
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Fig. 9. The dependence of effective angle D1(m, b) for different harmonics m.
. . . . . . . . . .
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Fig. 8. The dependence of deviation angle a1(m, b) for different harmonics m.
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Fig. 10. The dependence of deviation angle a2(m, b) for different harmonics m.
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Fig. 11. The dependence of effective angle D2(m, b) for different harmonics m.
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Fig. 13. The dependence of effective angle D3(m, b) for different harmonics m.
4. Conclusion
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Fig. 12. The dependence of deviation angle a3(m, b) for different harmonics m.
The values Dmi decrease in a monotonic way with increasing m and for m 1 there are asymptotic formulas: 1=3 bi Dmi ; b0 ¼ 2:1267; m b1 ¼ 0:1271;
b2 ¼ 0:0491;
b3 ¼ 1:0616.
The general conclusion is that with an increase of particle energy each fixed spectral component (fixed harmonic number) of any radiation polarization tends to towards deconcentration of the angular distribution of radiation relative to the plane of particle orbit whereas the total (summed over spectrum) angular distribution concentrates unto this plane. There is a simple mathematical reason for such a striking contradiction – the series on harmonic m in formula (4) for b ! 1 has nonuniform convergence and therefore the behavior of the sum series and its separate terms may show opposite tendencies with changes in the parameters (b; h).
Acknowledgements This work was partially supported by RFBR Grant 03-02-17615 and President Grant of Russia.
ð21Þ
The functions ai(m; b) (i = 0, 1, 3) are a monotonically increasing functions of b. We have unexpectedly found that there is a range of b values where the functions a2(m; b) are non-zero for m = 1, 2, 3; if m > 3 then a2(m; b) 0.
References [1] A.A. Sokolov, I.M. Ternov, Synchrotron radiation, Academic Verlag/Pergamon Press, Berlin/New York, 1968. [2] A.A. Sokolov, I.M. Ternov, Radiation from relativistic electrons, American Institute of Physics, New York, 1986.
V.G. Bagrov et al. / Nucl. Instr. and Meth. in Phys. Res. B 240 (2005) 638–645 [3] I.M. Ternov, V.V. Mihailin, Synchrotron radiation. Theory and experiment, Energoatomizdat, Moskow, 1986 [in Russian]. [4] Synchrotron Radiation Theory and its Development, World Scientific, Singapore, New Jersey, London, Hong Kong, 1999. [5] V.G. Bagrov, V.A. Bordovitsyn, V.G. Bulenok, V. Ya. Epp, Kinematical projection of pulsar synchrotron radiation
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profiles, in: Proceedings of IV ISTC Scientific Advisory Commitee Seminar on Basic Science in ISTC Aktivities, Akademgorodok, Novosibirsk, April 23–27, 2001, p. 293. [6] V.G. Bagrov, V.A. Bordovitsyn, V. Ch. Zhukovskii, Development of the theory of synchrotron radiation and related processes, Synchrotron source of JINR: the perspective of research, in: The Materials of the Second International Work Conference, Dubna, April 2–6, 2001, p. 15.