Parametric X-ray radiation from polarized electrons

Nuclear Instruments and Methods in Physics Research B 173 (2001) 27±29

www.elsevier.nl/locate/nimb

Parametric X-ray radiation from polarized electrons A.P. Potylitsyn a, V.A. Serdyutsky

a,*

, A.V. Mazunin a, M.N. Strikhanov

b

a

b

Tomsk Polytechnic University, Lenin Ave. 30, 634050 Tomsk, Russia Moscow Engineering Physical Institute, Kashirskoe Shosse 31, 115 409 Moscow, Russia Received 7 December 1999

Abstract Characteristics of parametric X-ray radiation (PXR) from polarized charged fermions based on the quantum theory have been investigated. Spin-dependent part of PXR cross-section on spin of the incident particle has been obtained near the K-edge of crystal target. Estimation of coecient of PXR process asymmetry was made. The comparison with quantum e€ects in transition radiation has been carried out. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Parametric X-ray radiation (PXR) in crystals

By the present time, a number of successful experiments on the polarized beams of electrons, muons and protons has been carried out. One of the main problems of polarization experiments is the analysis of the initial particle beam polarization. The technique for determination of non-relativistic particle polarization characteristics is investigated rather in details, whereas the analysis of polarization states of ultrarelativistic particles is a highly actual task. It is obvious that for that purpose usage of the processes described by quantum electrodynamics has undoubted advantages: the calculation of the analyzing power without using any models and, as a rule, a high process cross-section. For example, in papers [1,2]

*

Corresponding author. Tel.: +7-3822-423992; fax: +73822-423934. E-mail address: [email protected] (V.A. Serdyutsky).

there has been considered the electromagnetic radiation of polarized protons in undulator and in bent crystal, where a principle possibility of using the noticed processes for proton polarization has been shown. In the present paper, another kind of electromagnetic radiation of polarized particles (parametric X-ray radiation (PXR) in crystals) has been investigated. On the base of quantum-mechanic PXR theory developed in the paper of Nitta [3] we have calculated a coecient of PXR process asymmetry, Aˆ

…dr" =dX† ÿ …dr# =dX† : …dr" =dX† ‡ …dr# =dX†

…1†

In (1) dr";# =dX is the PXR cross-section calculated in the vicinity to the Bragg direction, for the particles polarized in the opposite directions. For being correct we shall consider the PXR from

0168-583X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 0 7 9 - 3

28

A.P. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 173 (2001) 27±29

polarized electron. Let us write the matrix element of the process as follows [3]:     e~ha
~ a
u~ps Y ~ k; a ˆ P ~ k; a
u~‡ p0 s0
~   ‡Q ~ k; a
u~‡ …2† ps : p0 s0
u~ Here ~ u~ps is the four-dimensional spinor describing an electron with the momentum ~ p, in spin state of S, ~ k the PXR photon momentum, ~ e~ha …a ˆ 1; 2† the polarization orts of the virtual photon with the k ‡~ h and ~ h is the crystal recoil momentum ~ k~h ˆ ~ (reciprocal lattice vector). Functions P …~ k; a† and Q…~ k; a† were determined in [3]. Further we shall present the calculations in lab frame where the two-component spinor presentation is more convenient:   1 …3† u~ps ˆ ~r~p
ms ; e‡1

 ~ aˆ

 0 ~ r ; ~ r 0

…4†

where ~ p; e are the electron momentum and energy, respectively, and ms is the two-component spinor. Here and further the system of units  hˆm ˆ c ˆ 1 is used. In two-component form the matrix element (2) is written in the following form: n       ~ r~ Bp Y ~ k; a ˆ m‡ s0
P k; a
Ap ‡ i~    o …5† ‡Q ~ k; a
Aq ‡ i~ r~ Bq
ms : In (5), there are the following notations: ~ ~ e~ ~ p e~ ~ p0 Ap ˆ ha ‡ 0 ha ; e‡1 e ‡1 # " 0 ~ ~ p p ~ e~ha ; Bp ˆ ~ ; ÿ e ‡ 1 e0 ‡ 1 ~ p
~ p0 ; …e ‡ 1†
…e0 ‡ 1† h i ~ p p0 ;~ ~ : Bq ˆ 0 …e ‡ 1†
…e ‡ 1† Aq ˆ 1 ‡

tron polarization ~ n will be described by the density matrix  1 qˆ 1 ‡~ n~ r : …7† 2 The squared matrix element will be calculated without averaging over the initial electron polarizationÕs:   i 1 nh  2 r~ Bp ‡ Q Aq ‡ i~ r~ Bq jY j ˆ Sp P Ap ‡ i~ 2 h    io r~ Bp ‡ Q Aq ÿ i~ r~ Bq
…1 ‡ ~ n~ r†
P  Ap ÿ i~ 1 ˆ …Sp1 ‡ Sp2 †; 2

   2 2 2 2 2 ~ ~ Sp1 ˆ j P j
Ap ‡ Bp ‡ jQj
Aq ‡ Bq 2



  Bq ; Bp ~ ‡ 2Re…PQ †
Ap Aq ‡ ~

In formulas (6) ~ p0 ; e0 are the ®nal electron momentum and energy, respectively. The initial elec-

…9†

n     h io n~ B q ÿ Aq ~ n~ Bp ‡ ~ Bq n ~ Bp ; ~ Sp2 ˆ 2Im…PQ †
Ap ~   n : …10† ˆ 2Im…PQ †
K ~ The aim of the paper is to estimate the asymmetry value (1) for relativistic case …e; e0  1†, therefore, the exact taking into account of the polarization photon states in (9) and (10) will not be performed (the ``a'' index is missed). Putting the energy and momentum for ®nal electron from the conservation laws, ~ p ÿ~ k ÿ~ h ˆ~ p ÿ~ k~h ; p0 ˆ ~ 0 e ˆ e ÿ x;

…6†

…8†

…11†

and leaving the main terms in the expansion by degrees of 1=e, instead of (6) we have   b ; e~ha~ Ap ˆ 2 ~ Aq ˆ 2; 1 h ~i ~ e~ ; b ; Bp ˆ ~ e ha 1h ~ i ~ b; k~h ; Bq ˆ ~ e

…12†

A.P. Potylitsyn et al. / Nucl. Instr. and Meth. in Phys. Res. B 173 (2001) 27±29

where    1 x ~ b: b 1ÿ ‡ ~ bˆ ~ k~h ÿ x~ e e In formulas (12) ~ b is the initial particle velocity and x is the PXR photon energy. Using the derived expressions let us estimate the ``kinematic'' multiplier K…~ n† in the spin-depending part of the cross-section (10),     h h i i 1 b
~ n
~ b; ~ k~h ÿ 2
~ 2
~ e~ha~ b n
~ e~ha ; ~ K…~ n† ˆ e hh ii i h 1 n
~ e~ha ; ~ b ; ~ b; ~ k~h ‡
~ e  h h io i 2 n ~ n
~ b; ~ k~h ÿ ~ e~ha
~ b ~ b ;  n
~ e~ha ; ~ e …13† because the last term can be neglected. It is clear that the kinematic multiplier is determined by the experiment geometry (Bragg angle) only, and it has the order of x=e. Further, one can notice that in (13) the ®rst term depends on transverse elecb) only, whereas the sectron polarization (~ n? ? ~ ond one depends on both longitudinal and transverse polarization components. In (10) the multiplier Im…PQ † can be considered as a ``dynamic'' one, which is determined by the crystal susceptibility 2 ˆ 1 ‡ v00 ‡ i
v000 :

…14†

For real 2 Im…PQ † ˆ 0, whereas in a range, where v000 6ˆ 0 (for example, in the vicinity to the K-edge of the absorption) Im…PQ † ˆ ÿP0 Q0

v000 2

:

…15†

In (15), P0 and Q0 are the functions P ; Q calculated for 2 ˆ 1 ‡ v00 . Thus, using (13) and (15) one can estimate the asymmetry of (1),

Aˆ

h    i 2Im…PQ †
K ~ n ÿ K ÿ~ n Sp1

29

 v000


x : e …16†

For instance, in the range of anomalous absorption in crystal (x  10 KeV; v000  10ÿ3 †; for electrons with energy e ˆ 5 MeV the asymmetry will have the value of 10ÿ5 ±10ÿ6 . It is obvious that for muons the e€ect will be two orders less. One should expect that in case of PXR from polarized protons there could appear peculiarities for some crystal orientations, because of anomalous magnetic momentum existence, however, in general case increasing of initial particle mass on three orders leads to the analogous decreasing of asymmetry. In conclusion, letÕs estimate the p. It asymmetry for a hypothetical case, when ~ p0 jj~ can easily be shown that in this case i x h ~ ~ e~ha ; ~ Bp ˆ
~ b : …17† Bq ˆ 0; ce Comparing the obtained values with (12) one can notice that for considering case spin-dependent part of cross-section, that is connected with pure quantum e€ects, is determined by the value of x=ce (c is the Lorentz factor). The obtained result is in good agreement with the estimation of quantum corrections in transition radiation [4], p). where the particle moves rectilinearly (~ p0 jj~ References [1] A. Luccio, in: Proceedings of the 1993 Part. Accel. Conf. (1995) 2560. [2] A.P. Potylitsyn, in: Proceedings of the Seventh Workshop on High Energy Spin Physics, E2-97-413, Dubna, 1997, p. 229. [3] H. Nitta, Phys. Rev. B 45 (1992) 7621. [4] V.N. Baier, V.M. Katkov. Preprint BINP 98-91, Novosibirsk, 1998.