Study of the influence of spin on the angular distribution of synchrotron radiation for weakly excited particles This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys.: Conf. Ser. 295 012109 (http://iopscience.iop.org/1742-6596/295/1/012109) Download details: IP Address: 138.73.1.36 The article was downloaded on 26/04/2013 at 13:05 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
Transcript
Study of the influence of spin on the angular distribution of synchrotron radiation for weakly
excited particles
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
Abstract. In the framework of quantum theory, we obtain the precise analytical expressionsfor synchrotron radiation characteristics of first excited state charged particles. A detailedanalysis of the total radiation power angular distribution and the angular distribution ofpolarization components is given for particles with spin 0 and 1/2. For both these typesof particles one can calculate radiation characteristics and compare them demonstrating thedependance of the results on spin. It is demonstrated that the quantum effects (e.g. spinproperties) accounting leads us to the results which seriously differ from classical theorypredictions.
1. Introduction
The synchrotron radiation (SR) is essentially associated with a wide number of its practicalapplications. Nevertheless, during last 50 years the theory of SR stays a ’milker’ as well, mostlyfor the physicists. The basic properties of the phenomenon gained a detailed analysis in termsof classical theory [1, 2]. However, the contemporary accelerators have their parameters beingvery close to the region where the quantum corrections become significant. Guided by thetechnique development, one obtains a motive to fill the gaps in quantum description createdfor SR a few decades ago [3, 4]. We focus on the theoretical research of SR characteristics,thus necessarily studying the influence of quantum corrections on the basic expressions received.Without any possibility for an adequate classical interpretation, spin is a property of a purequantum nature. It is clear, that the investigation of spin properties can be carried out onlywith the use of quantum theory methods. To thoroughly study the influence of spin propertieson the SR characteristics, we consider a spinless (scalar) particle (a boson) and a spinor particle,namely, a particle of 1/2 spin (an electron). We consider first excited state particles as a specialcase with unique frequency being radiated. It seriously differs from classical theory where onedeals with a spectrum of radiated harmonics. Thus, this is a convenient way to observe quantumeffects, especially spin effects.
In the framework of classical SR theory one interprets the phenomenon as a radiation of aparticle moving around circular orbit in the plane straight perpendicular to the external magneticfield of strengths H > 0. Of course, the quantum picture differs completely, still we can considerthe motion in a similar magnetic field. In this case the energy of a first excited state boson and
19th International Spin Physics Symposium (SPIN2010) IOP PublishingJournal of Physics: Conference Series 295 (2011) 012109 doi:10.1088/1742-6596/295/1/012109
Published under licence by IOP Publishing Ltd 1
electron:
E = m0c2γ, where γ2 = (1−β2)−1 =
1 + 2B for an electron1 + 3B for a boson
, B =H
H0, H0 =
m20c
3
|e|~ . (1)
All the expressions received will be presented separately for a boson and an electron.We use index ’s’ to describe the polarization components as follows: if s = 2, 3 then it is σ-
or π- component of linear polarized radiation respectively, s = −1, 1 for left and right circularpolarization, s = 0 indicates the summed or so-called total radiation. Since, for an electron, wesuppose the solutions of motion equation being the eigenfunctions of transverse spin polarizationoperator, thus two following cases are under consideration. The initial state of an electron couldbe characterized with the use of spin quantum number ζ with its possible values ζ = 1 andζ = −1. Evidently, the transverse spin polarization means that the spin of a particle is collinearto the external magnetic field direction. To be more precise, it could be co-directional to thefield (we take ζ = 1 in this case), or have an opposite direction ( ζ = −1).
2. Boson
Let us introduce a parameter x0 and a variable x
x0 = x0(β) =
√3 −
√
3 − 2β2
√3 +
√
3 − 2β2; x = x(β, θ) =
√3 −
√
3 − 2β2 sin2 θ√3 +
√
3 − 2β2 sin2 θ; (2)
0 ≤ x0(β) ≤ 2 −√
3 ≈ 0, 26794919 ; 0 ≤ x(β, θ) ≤ x0(β).
Then for a boson we can write the angular distribution of radiated power as follows:
dW bs (β; θ)
dΩ=
Q0A(β)
54(1 + x)3e−x ϕb
s(β; θ); Q0 =e2m2
0c3
~2, A(β) =
β6
1 − β2=
(γ2 − 1)3
γ4;
ϕbg(β; θ) =
1
2ϕb
0(β; θ) + g(1 + x) cos θ, ϕb2(β; θ) = 1 − x, ϕb
3(β; θ) =(1 + x)2 cos2 θ
1 − x, (3)
ϕb0(β; θ) = ϕb
2(β; θ) + ϕb3(β; θ) = ϕb
−1(β; θ) + ϕb1(β; θ); dΩ = sin θdθ , g = ±1.
From (3) we obtain the expression for the total power radiated by the boson.
W b0 (β) =
4Q0A(β)
81f b(β), f b(β) =
3(1 + x0)2
8f b0(x0), f b
0(x) = f b2(x) + f b
3(x). (4)
Here the following designations are used
f b1(x) =
∫ 1
0(1 − x2y2) e−xy dy =
(1 + x)2e−x − 1
x, f b
1(0) = 1 ,
f b2(x) =
∫ 1
0
(1 + xy)(1 − xy)2√
(1 − y)(1 − x2y)e−xy dy , f b
2(0) = 2 , (5)
f b3(x) =
∫ 1
0(1 + xy)
√
(1 − y)(1 − x2y) e−xy dy , f b3(0) =
2
3.
One can easily rewrite (3)
dW bs (β; θ)
dΩ= W b
0 (β)pbs(β; θ), pb
s(β; θ) =(1 + x)3e−x ϕb
s(β; θ)
(1 + x0)2f b0(x0)
; (6)
pbs(β; θ) = pb
s(β;π − θ) (s = 0, 2, 3); pbg(β; θ) = pb
−g(β;π − θ),
in terms of pbs(β; θ) dΩ defining the contribution of power radiated by s-polarization component
in space angle dΩ near the direction giver by θ.
19th International Spin Physics Symposium (SPIN2010) IOP PublishingJournal of Physics: Conference Series 295 (2011) 012109 doi:10.1088/1742-6596/295/1/012109
2
3. Electron
Let us introduce a parameter x0 and a variable x
x0 = x0(β) =1 −
√
1 − β2
1 +√
1 − β2=
γ − 1
γ + 1; x = x(β, θ) =
1 −√
1 − β2 sin2 θ
1 +√
1 − β2 sin2 θ; (7)
0 6 x0(β) 6 1 ; 0 6 x(β, θ) 6 x0(β) .
As it was mentioned, our task includes the examination of the first excited state particles. Togo on with an electron, one should keep in mind that in the ground state its spin is opposite tothe external field. So, we can subdivide the transitions from the first excited state to the groundstate into the spin-flip transitions (ζ = 1) and the transitions without spin-flip (ζ = −1).
For an electron we obtain:
dW es (ζ;β; θ)
dΩ= d(ζ; β)
Q0A(β)
16(1 + x0)
(1 + x)3e−x
1 − xϕe
s(ζ;β; θ);
ϕe2(−1;β; θ) = ϕe
3(1;β; θ) = 1 − x0x ,
ϕe2(1;β; θ) = ϕe
3(−1;β; θ) =(1 + x)2 cos2 θ
1 − x0x, (8)
ϕe0(ζ;β; θ) = ϕe
0(β; θ) = ϕe2(−1;β; θ) + ϕe
3(−1;β; θ) = ϕe2(1;β; θ) + ϕe
3(1;β; θ),
ϕeg(ζ;β; θ) = ϕe
g(β; θ) =ϕe
0(β; θ)
2+ g(1 + x) cos θ , g = ±1.
Here we use the function
d(ζ; β) =1 − ζ + x0(1 + ζ)
2=
x0 at ζ = 1;1 at ζ = −1 .
(9)
Integrating over θ from 0 to π we obtain:
W e0 (ζ; β) = d(ζ; β)
Q0A(β)
6f e(β), f e(β) =
3(1 + x0)
8f e0 (x0), f e
0 (x) = f e2 (x) + f e
3 (x). (10)
Here
f e1 (x) =
2 − (2 + x) exp(−x)
x; 0 6 x 6 1;
f e2 (x) =
∫ 1
0(1 + xy) exp(−xy)
√
1 − x2y
1 − ydy,
f e3 (x) =
∫ 1
0(1 + xy) exp(−xy)
√
1 − y
1 − x2ydy.
We can create analogous functions pes(ζ; β; θ) for an electron
dW e
s(ζ;β;θ)dΩ = W e
0 (ζ; β)pes(ζ; β; θ), pe
s(ζ; β; θ) = (1+x)3e−x ϕe
s(ζ; β;θ)
(1+x0)2(1−x)fe
0(x0)
;
pe0(ζ;β; θ) = pe
0(β; θ), peg(ζ;β; θ) = pe
g(β; θ),
pe2(−1;β; θ) = pe
3(1;β; θ), pe2(1;β; θ) = pe
3(−1;β; θ);
pes(ζ;β; θ) = pe
s(ζ;β;π − θ) for s = 0, 2, 3; peg(β; θ) = pe
−g(β;π − θ).
(11)
19th International Spin Physics Symposium (SPIN2010) IOP PublishingJournal of Physics: Conference Series 295 (2011) 012109 doi:10.1088/1742-6596/295/1/012109
3
4. The comparative analysis of basic SR characteristics
According to the formulae presented, one can perfectly see that the electron with ζ = 1 radiatesx0 times less, than the electron with ζ = −1. This is an expected result, because intuitively weunderstand that the electron seems to loose energy while changing its spin direction during thetransition to the ground state. In ultrarelativistic case x0 → 1, thus providing the equiprobabilityof the transitions with and without spin-flip. Furthermore, we find an exciting fact: the angulardistributions of SR linear polarization components for an electron with spin ζ = −1 and anelectron with spin ζ = 1 are interchanging (subject to the scaling factor x0). Evidently, itoccurs so only for the first excited state electrons.
To compare the amount of radiation emitted by the boson and the electron we introduce thefunctions
k(ζ; β) =W e
0 (ζ; β)
W b0 (β)
; k(−1; β) =27
8
f e(β)
f b(β), k(1; β) = x0k(−1; β) 6 k(−1; β). (12)
With the help of these functions it becomes convenient to compare the power radiated by theelectron and the boson.
Table 1. Here the values of k-functions at different β are given to demonstrate the spin directioninfluence on the amount of radiated power
The data of the above table shows that the electron with ζ = −1 (left column) radiates morethan boson at any β, however, the electron of ζ = 1 (right column) starts radiating more thanboson only within the relativistic region of parameters, i.e. when β → 1.
5. The radiation polarization and spin
Now, let us consider the behavior of functions pe,bs in details. The first important thing about
the functions pbs and pe
s is their being finite at any β and θ (including β = 1, in contrast to theclassical theory). The evolution of the functions pb
2 and pe2 seems quite simple, both pb
2 and pe2 are
monotone increasing functions at 0 ≤ θ ≤ π/2 depending on θ (which qualitatively correspondswith classical theory). The functions pb
3 are steadily decreasing within [0;π/2] at any fixed valueof β, whereas pe
3 cease being monotone at β >√
3/2. And here we also find a remarkable fact:the behavior of the electronic functions is more similar to the classical theory in comparison tothe bosonic functions, though the spin itself has a pure quantum nature, and we deal exactly
with a case where its influence is expected. We observe the same situation for the functions pb,e0
and pb,e1 . The bosonic functions are monotone and decreasing on [0;π/2] ([0;π] for pb
1) at any β.
19th International Spin Physics Symposium (SPIN2010) IOP PublishingJournal of Physics: Conference Series 295 (2011) 012109 doi:10.1088/1742-6596/295/1/012109
4
Still, the electron radiation characteristics pe1 and pe
0 are loosing their monotony at β > 1/√
2.Finally, we see the function pe
1(β) tending to 0 when β → 1 within the interval π/2 ≤ θ ≤ π, thefact which makes us sure about right polarization extinction in the lower half plane.
As an example, we present the figures demonstrating the behavior of functions pb3 and pe
3
Figure 1. The graphs offunctions pb
3(β, θ)
Figure 2. The graphs offunctions pe
3(β, θ)
0,70
π_2
β
θ0 (β)max
θ1 (β)max
θ3 (β)max
Figure 3. Thegraphs of functionsθmax(β)
The above analysis provides a possibility to observe the absence of radiation powerconcentration near the orbit’s plane (θ = π/2 neighbourhood), which contradicts classical theorypredictions. For a more accurate analysis of polarization components contribution, we explorethe character of the functions θmax (Figure 3) - the maximum of pe
s depending on β. It couldbe defined parametrically as follows
β2 = 2(2−a3)(2−a)(2+a+a2)
,
cos2 θmax0 (β) = a4
(2−a3)(2−a);
β2 = 4(2−a3)2
(2−a)2(2+a+a2)(2+a−a3),
cos θmax1 (β) = a
2−a3 ;
β2 = 4(2−a3)(2+a2)(2−a)2(2+a+a2)2
,
cos2 θmax3 (β) = a2(1+a+a2)
(2−a3)(2+a2).
Concluding comments
The classical theory of SR provides a great number of adequate results. However, some propertiesof this phenomenon need to be described with the use of quantum theory methods. It turnedout, that for the first excited state particles the presence of spin makes SR characteristics behavemore similar to their classical analogues.
References[1] Sokolov A A and Ternov I M 1968 Synchrotron Radiation Berlin[2] Bagrov V G 2008 Russian Physics Journal, Special Features of the Angular Distribution of Synchrotron
Radiation 51, 335-352[3] Sokolov A A and Ternov I M 1986 Radiation from relativistic electrons New York[4] Ternov I M and Mikhailin V V 1986 Synchrotron Radiation. Theory and Experiment. Moskow
19th International Spin Physics Symposium (SPIN2010) IOP PublishingJournal of Physics: Conference Series 295 (2011) 012109 doi:10.1088/1742-6596/295/1/012109
