Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering

An economic technique for calculation of polarized bremsstrahlung process is proposed, assuming typical atomic momentum transfer q << m. The adopted approach is based on the natural reduction of the matrix element to the form Vαγα + A₅γ⁵. Polarization distribution in the fully differential cro...

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Дата:	2009
Автор:	Bondarenco, M.V.
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Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2009
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Теми:	Электродинамика
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Цитувати:	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering / M.V. Bondarenco // Вопросы атомной науки и техники. — 2009. — № 3. — С. 89-94. — Бібліогр.: 6 назв. — англ.

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spelling	irk-123456789-1112622017-01-09T12:43:18Z Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering Bondarenco, M.V. Электродинамика An economic technique for calculation of polarized bremsstrahlung process is proposed, assuming typical atomic momentum transfer q << m. The adopted approach is based on the natural reduction of the matrix element to the form Vαγα + A₅γ⁵. Polarization distribution in the fully differential cross-section is analyzed. It is found that at a given momentum transfer to the atom polarization in the plane of small radiation angles is oriented along circles passing through two common points. It is shown that with angular selection of radiated photons carried out, even with momentum transfers to the atom being integrated over, for particular radiation angles polarization may stay as high as 100%. Without angular selection of photons, only by control of the recoil, it is impossible to gain radiation polarization above 50%. Запропоновано спрощений спосіб обчислення диференціального перерізу поляризованого гальмівного випромінення за типової атомної передачі импульсу q << m. Використовуваний підхід грунтується на природній редукції спінового матричного елементу процесу до форми Vαγα + A₅γ⁵. Аналізується розподіл поляризації у повністю диференціальному перерізі. Показано, що якщо здійснювати кутову селекцію випромінених фотонів, то навіть після інтегрування по импульсах переданих атому, для певних кутів випромінення поляризація може зберігатися до 100%. Без кутової селекції фотонів, тільки за рахунок контролю віддачі, неможливо отримати поляризацію випромінення вищу ніж 50%. Предложен упрощенный способ вычисления дифференциального сечения поляризованного тормозного излучения при типичной атомной передаче импульса q << m. Используемый подход основывается на естественной редукции спинового матричного элемента процесса к форме Vαγα + A₅γ⁵. Анализируется распределение поляризации в полностью дифференциальном сечении. Показано, что если производить угловую селекцию излученных фотонов, то даже после интегрировании по импульсам, переданным атому, для определенных углов излучения поляризация может сохраняться до 100%. Без угловой селекции фотонов, только за счет контроля отдачи, нельзя получить поляризацию излучения выше 50%. 2009 Article Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering / M.V. Bondarenco // Вопросы атомной науки и техники. — 2009. — № 3. — С. 89-94. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 41.60.-m, 61.80.Az, 61.80.Ed, 78.70.-g, 95.30.Gv http://dspace.nbuv.gov.ua/handle/123456789/111262 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Электродинамика Электродинамика
spellingShingle	Электродинамика Электродинамика Bondarenco, M.V. Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering Вопросы атомной науки и техники
description	An economic technique for calculation of polarized bremsstrahlung process is proposed, assuming typical atomic momentum transfer q << m. The adopted approach is based on the natural reduction of the matrix element to the form Vαγα + A₅γ⁵. Polarization distribution in the fully differential cross-section is analyzed. It is found that at a given momentum transfer to the atom polarization in the plane of small radiation angles is oriented along circles passing through two common points. It is shown that with angular selection of radiated photons carried out, even with momentum transfers to the atom being integrated over, for particular radiation angles polarization may stay as high as 100%. Without angular selection of photons, only by control of the recoil, it is impossible to gain radiation polarization above 50%.
format	Article
author	Bondarenco, M.V.
author_facet	Bondarenco, M.V.
author_sort	Bondarenco, M.V.
title	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
title_short	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
title_full	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
title_fullStr	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
title_full_unstemmed	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
title_sort	computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2009
topic_facet	Электродинамика
url	http://dspace.nbuv.gov.ua/handle/123456789/111262
citation_txt	Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering / M.V. Bondarenco // Вопросы атомной науки и техники. — 2009. — № 3. — С. 89-94. — Бібліогр.: 6 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT bondarencomv computationandanalysisofthepolarizationdegreeforbremsstrahlungatperipheralscattering
first_indexed	2025-07-08T01:52:31Z
last_indexed	2025-07-08T01:52:31Z
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fulltext	ELECTRODYNAMICS COMPUTATION AND ANALYSIS OF THE POLARIZATION DEGREE FOR BREMSSTRAHLUNG AT PERIPHERAL SCATTERING M.V. Bondarenco∗ National Science Center ”Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine (Received December 12, 2008) An economic technique for calculation of polarized bremsstrahlung process is proposed, assuming typical atomic momentum transfer q ¿ m. The adopted approach is based on the natural reduction of the matrix element to the form V αγα + A5γ 5. Polarization distribution in the fully differential cross-section is analyzed. It is found that at a given momentum transfer to the atom polarization in the plane of small radiation angles is oriented along circles passing through two common points. It is shown that with angular selection of radiated photons carried out, even with momentum transfers to the atom being integrated over, for particular radiation angles polarization may stay as high as 100%. Without angular selection of photons, only by control of the recoil, it is impossible to gain radiation polarization above 50%. PACS: 41.60.-m, 61.80.Az, 61.80.Ed, 78.70.-g, 95.30.Gv 1. INTRODUCTION Radiation from ultra-relativistic electrons is natu- rally collimated along the forward direction, with the opening angle being inversely proportional to the electron energy, but up to energies ∼ 10 GeV, corre- sponding to typical radiation angles ∼ 10−4 rad, an- gular distribution of radiation can be resolved. If that is done in practice, photon polarization effects neces- sarily come into play. In particular, the connection between the polarization degree and the angular dis- tribution is important for purposes of preparation of polarized photon beams. The issue of polarization ac- count may arise also at investigation of spatial evolu- tion of electromagnetic showers. Calculations of dif- ferential cross-section for the bremsstrahlung process were conducted in various frameworks [1], [2], but acquired reputation of rather cumbersome a subject. Presentation of the final results in the literature does not help gaining detailed intuition. The conditions of radiation from ultra-relativistic electrons in matter are such that small momentum transfers to atoms dominate (q ∼ r−1 B ¿ m). Un- der such conditions the expression for bremsstrahlung cross-section, called dipole approximation, is similar to Compton scattering cross-section from the stand- point of the equivalent photon method. As a matter of fact, the latter method is usually used to derive characteristics of bremsstrahlung, averaged over di- rections of emission and over polarizations of final, and also (through azimuthal integration in momen- tum transfers) over polarizations of initial quanta. Nevertheless, it can be extended to the polarized case, provided one traces correspondence of polarizations (that is traditionally achieved via a transition be- tween reference frames [1]). From the technical side, the case of Compton scat- tering has the advantage that photon polarization vectors in it might be chosen orthogonal simultane- ously to two momenta in the problem (out of three momenta not bound by 4-momentum conservation). That is expected to substantially simplify the pro- cedure of calculation with the account of polariza- tions and, in the conventional technology of cross- section calculation via computation of a spur from the squared matrix amplitude [3], this is indeed the case. Despite the simplifications gained, calculations for this process still are of formidable complexity. This is in mark contrast with the concise final result, and should be blamed on nothing but poor efficiency of the method of straightforward spurring. As was discussed in [4], a more adequate method of calculation of QED-processes, also offering access to fermion polarization observables, is the evaluation of spin amplitudes in a matrix basis chosen by some convenience reasons. In the present paper, firstly, it is desired to ap- ply the spin amplitude approach to the specific case of Compton scattering. It appears that in this par- ticular case the most convenient spin matrix choice is different from what may be appropriate in other cases. With the choice adopted in the present article, the whole procedure of differential cross-section cal- culation with the account of initial and final photon polarizations becomes fairly elementary. Issuing from the representation for the differential cross-section for ∗Corresponding author E-mail address: bon@kipt.kharkov.ua PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2009, N3. Series: Nuclear Physics Investigations (51), p.89-94. 89 Compton scattering in a gauge- and Lorenz-invariant form, it is straightforward further on to pass to the differential cross-section of bremsstrahlung in labo- ratory frame, corresponding to conditions of periph- eral scattering, i.e. dipole approximation. This pro- vides an alternative to explicit implementation of the equivalent photon method. The second objective of the present work is to discuss the features of polarization distribution as a function of the radiation angle at fixed transverse di- rection of momentum transfer to the atom. The char- acteristic feature is that the polarization is aligned along circles, passing through two knot points in the space of radiation angles. One of the circles has its centre coinciding with the origin - and that has an important consequence: upon averaging over momen- tum transfers to the atom at radiation direction fixed, only contributions with identical direction of polar- ization are summed up. And since at small radiation frequencies the polarization in the doubly differential cross-section is close to 100%, after the averaging it remains near so. 2. CALCULATION OF AMPLITUDES AND CROSS-SECTION FOR COMPTON SCATTERING We are considering the process of electron scattering on an atom1, accompanied by emission of a single photon. Denoting by p and p′ 4-momenta of the ini- tial and the final electrons, q and q′ - the momentum transfer to the atom and the momentum of the emit- ted photon, the 4-momentum conservation and mass shell conditions for them read as: p + q = p′ + q′, p2 = p′2 = m2, q′2 = 0. For q2 there is no strict condition, but q2 ∼ r−2 B ∼ e4m2 ¿ m2, (1) rB being Bohr radius. This estimate relates mainly to q components orthogonal to p . The component qz parallel to p is small, as long as typical denom- inators emerging in Feynman diagrams are of order pq ' Eqz ∼ m2, and E is large. Then qzrB ¿ 1, owing to which condition the matrix element for the whole process factorizes into Born-level radiation ma- trix element Mrad(q⊥, q′) and the exact elastic scat- tering amplitude Ascat(q⊥): Tfi = Ascat (q⊥) √ 4πeMrad (q⊥, q′) , dσscat = \|Ascat\|2 d2q⊥ (2π)2 , Mrad = ūp′ ( ê′∗(p̂ +q̂+m)ê 2pq − ê(p̂−q̂′+m)ê′∗ 2pq′ ) up , dσrad = 1 2E \|Tfi\|2 d2q⊥ (2π)22E′ d3q′ (2π)32q′0 , (2) (the elastic scattering amplitude through small an- gles can be regarded as independent from the elec- tron polarization). We are interested in Mrad(q⊥, q′) modulus squared and averaged over initial electron and summed over final electron polarizations. By the straightforward Feynman’s method, this quantity is to be computed as 〈 \|Mrad\|2 〉 = 1 2 Sp (p̂′+m) ( ê′∗(p̂ + q̂ + m)ê 2pq − ê(p̂− q̂′ + m)ê′∗ 2pq′ ) (p̂ +m) ( ê∗(p̂ + q̂ + m)ê′ 2pq − ê′(p̂− q̂′ + m)ê∗ 2pq′ ) . (3) However, at that one needs to calculate a spur from a polynomial of 8th degree in γ-matrices. Typically, even relying on properties of photon crossing sym- metry and advantages of the gauge choice (see (4) below), that entails calculations along 2-3 pages (cf. [3]). A more efficient approach would be prior re- duction of the spin matrix to some ”minimal” form, embarking on the condition of initial and final elec- tron bispinors belonging to the mass shell. 2.1. Deduction of basic amplitudes As usual, the computations of Compton scatter- ing are strongly facilitated in the gauge ep = eq = 0 = e′p = e′q′. (4) To start with, commute in the matrix element (2) p̂ through ê , ê′ to a position neighboring to up: Mrad = ūp′ ( ê′∗(p̂ + q̂ + m)ê 2pq − ê(p̂− q̂′ + m)ê′∗ 2pq′ ) up ∼= ūp′ ( ê′∗q̂ê 2pq + êq̂′ê′∗ 2pq′ ) up. (5) Feynman in [3] started squaring from this modified representation, but it is still 8-th order in γ-matrices, and we shall proceed a little further. With the appli- cation of the standard formula γαγβγγ = gαβγγ − gαγγβ + gβγγα + iεναβγγνγ5, (5) naturally reduces to a basic-matrix form: Mrad = ūp′ ( V αγα + A5γ 5 ) up. (6) The coefficients in the decomposition are V α = eαqe′∗ − qαee′∗ 2pq + e′α∗q′e− q′αee′∗ 2pq′ , A5 = −iεµαβγ pµ m eαe′β∗Gγ , ( Gγ = qγ 2pq − q′γ 2pq′ ) and it was utilized that vector G, as well as e, e′, is orthogonal to p, Gp = 0, 1A composite and lower-dimensional scattering system will be addressed in Sec. 3.1. 90 so, εναβγeαe′β∗Gγ = pν m · εµαβγ pµ m eαe′β∗Gγ , whereas action of pνγνγ5/m on up gives −γ5. One can still add to the vector V α an arbitrary vector, proportional to (p − p′)α. It is advantageous to tune it so that V be orthogonal to momentum p: V α → V α p = eαqe′∗ − qαee′∗ 2pq + (p− p′)α ee′∗ (p− p′)2 + e′α∗q′e− q′αee′∗ 2pq′ + (p− p′)α ee′∗ (p− p′)2 = eαe′∗G− e′α∗eG−Gαee′∗ pq + pq′ m2 − pp′ . Now V α p and A5 together have 4 independent components, as it should be for parametrization of a matrix describing transition between two spin-1/2 on-shell states. 2.2. Computation of the differential cross-section, averaged over fermion polarizations Substituting (6) to (3), the spur is calculated easily: 〈 \|Mrad\|2 〉 = 1 2 Sp (p̂′ + m) ( V α p γα + A5γ 5 ) (p̂ + m) ( V α∗ p γα + A∗5γ 5 ) = 2(m2 − pp′) { \|Vp\|2 − \|A5\|2 } . The entries thereat are evaluated to be \|Vp\|2 = \|eαe′∗G− e′α∗eG\|2 + G2 \|ee′∗\|2 ( pq + pq′ m2 − pp′ )2 , − \|A5\|2 = ∣∣∣∣∣∣ \|e\|2 ee′ eG (ee′)∗ \|e′\|2 e′∗G e∗G e′G G2 ∣∣∣∣∣∣ = \|e\|2 \|e′\|2 G2 + 2Re ee′ · e′∗G · e∗G− \|e\|2 \|e′G\|2 − \|e′\|2 \|eG\|2 −G2 \|ee′\|2 = − \|eαe′∗G− e′α∗eG\|2 + \|e\|2 \|e′\|2 G2 −G2 \|ee′\|2 . In sum, after cancellation of terms ± \|eαe′∗G− e′α∗eG\|2, \|Vp\|2 − \|A5\|2 = \|e\|2 \|e′\|2 G2 −G2 \|ee′\|2 + G2 \|ee′∗\|2 ( pq + pq′ m2 − pp′ )2 , G2 = m2 − pp′ 2pq · pq′ , 〈 \|Mrad\|2 〉 = 1 pq · pq′ {( \|ep\|2 ∣∣e′p ∣∣2 − ∣∣epe ′ p ∣∣2 ) (pq − pq′)2 + ∣∣epe ′∗ p ∣∣2 (pq + pq′)2 } . (7) (In the final formula an explicit subscript p at polarization vectors is introduced emphasizing the employed gauge). In what follows, we will be mainly interested in the case of linearly polarized initial photons. Then, the final photon polarization is also linear. For those conditions one can set ∣∣epe ′∗ p ∣∣ = ∣∣epe ′ p ∣∣ = ( epe ′ p ) and add in (7) the two like terms: 〈 \|Mrad\|2 〉 = e2 pe ′2 p (pq − pq′)2 pq · pq′ + 4(epe ′ p) 2. (8) This is the renowned Klein-Nishina’s formula for linearly polarized initial and final photons [3], [5]. To apply formula (8) to bremsstrahlung in laboratory frame, where the scatterer atom is at rest, it should first be rendered a gauge-invariant appearance. To this end, substitute for ep, e′p expressions ep = e− q ep pq , e′p = e′ − q′ e ′p pq′ , where e, e′ are polarization vectors in an arbitrary gauge: 〈 \|Mrad\|2 〉 = ( e− q ep pq )2 ( e′ − q′ e′p pq′ )2 (pq − pq′)2 pq · pq′ + 4 {( e− q ep pq )( e′ − q′ e′p pq′ )}2 . (9) Thereupon, this formula can be applied in the laboratory frame. 3. APPLICATION TO THE BREMSSTRAHLUNG IN LABORATORY FRAME For bremsstrahlung in the laboratory frame e = (1,0), e′ = (0, e′), q = (0,−q), q′ = (ω,k). At that, combinations ep, e′p in components equal e− q ep pq ' ( 1, 1, q⊥ qz ) , (10) 91 e′−q′ e′p pq′ ' ( e′zω E E′qz , e′z ( 1+ω E E′qz ) , e′⊥+k⊥ e′p E′qz ) . (11) In terms of orthogonal components, the combination e′p entering (11) equals e′p ' E ω (e′k − e′⊥k⊥) = −E ω e′⊥k⊥. Apparently, in scalar products present in (9), the temporal and the longitudinal spatial components of vectors (10-11) do not essentially contribute, ex- cept in e′2p , which is easier calculated in the Lorenz- invariant fashion, with the use of e′q′ = 0, q′2 = 0: e′2p = e′2 = −1. So, e2 p ' −q2 ⊥ q2 z , e′2p = −1, epe ′ p ' \|q⊥\| qz ( −nq⊥ + ( k⊥nq⊥ ) E′qz E ω k⊥ ) e′ , with nq⊥ = q⊥ \|q⊥\| . Next, the kinematical combinations required are (pq − pq′)2 pq · pq′ ' (qq′)2 pq · pq′ ' (ωq′z) 2 Eqz · E′qz = ω2 EE′ , E′qz ' p′q ' pq′ = Eω − pzkz ' Eω − ( E − m2 2E )( ω − k2 ⊥ 2ω ) ' ω 2E m2+ E 2ω k2 ⊥, or, introducing the radiation angle θk = k⊥/ω and ratios γ = E/m À 1, xω = ω/E, 0 ≤ xω ≤ 1, qz = mxω 2(1− xω) ( 1 γ + γθ2 k ) . Substituting all the ingredients into (9), one ar- rives at the expression 〈 \|Mrad\|2 〉 =4q2 ⊥ (1− xω) m2 ( 1 γ +γθ2 k )2 { 1 + 4(1− xω) x2 ω \|νe′\|2 } . (12) Here ν is a vector ν = −nq⊥ + 2 γ−2 + θ2 k (θknq⊥)θk, along which the radiation polarization orients itself2. The absolute value of the polarization amounts P = ν2 x2 ω 2(1−xω) + ν2 , with ν2 = 1− 4γ−2 (γ−2 + θ2 k)2 (θknq⊥)2 ≡ (γθk + nq⊥)2(γθk − nq⊥)2 (1 + γ2θ2 k)2 . It is easy to show by solving the differential equation dθy/dθx = νy (θx, θy) /νx (θx, θy), that the curves, in every point tangential to the direction of vector ν, are circles passing through two specific points: θk = ±nq⊥/γ (see Fig. 1). In those two points the polarization turns to zero. In vicinities of those points ν2 ' (γθk ∓ nq⊥)2 . At distances from them much greater then xω 2γ √ 1−xω polarization is close to 100%. -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 Fig.1.Curves in γθk plane, directing polarization of radiation. All displayed curves are circles. nq⊥ is a unit vector along the vertical axis The doubly differential cross-section in itself is usu- ally not measured in experiment, as long as recoiling atoms are not detected. So, the picture described above serves mainly for intuition purposes. Below we shall discuss two most important cases of Ascat dependence on nq⊥ . In both those cases there is a factorization of azimuthal and radial dependences Ascat(q⊥) = Φ(nq⊥)A(\|q⊥\|), the latter being irrele- vant for polarization effects in view of homogeneity of Mrad dependence on q2 ⊥. Hence, it suffices to discuss averaging of 〈 \|Mrad\|2 〉 over nq⊥ . 3.1. Planar geometry In the planar geometry Ascat(q⊥) exhibits a sharp peak along some particular direction nq⊥ .3 If one selects photons away from knot directions ±nq⊥/γ, where polarization is maximal - say, emitted in the middle band of angles ∣∣θknq⊥ ∣∣ < 1/2γ (ref. to Fig. 1), the dependence of polarization on xω may be es- timated from formula (12) upon substitution in it 2It may be worth pointing out, that except overall proportionality to q2 ⊥, D \|Mrad\|2 E is also dependent on nq⊥ despite q2 ⊥ ¿ m2 - unless it is integrated over directions of θk and summed over e′. That circumstance is often missed in presentations of the equivalent photon method, including treatises [5], [6], though is taken care in [1]. 3Physically, it may correspond either to electron passage through a thin crystal close to a strong crystalline plane, or to passage through gap of a magnet deflecting electrons to small angles. It should be minded that in those cases actual dimensions of the scattering system exceed the cross-section of electron beam in both transverse directions, so the concept of differential cross-section looses direct physical sense. Nevertheless, values of polarization extracted from it do not depend on Ascat and are correct. 92 k⊥ ⊥ q⊥: 〈 \|Mrad\|2 〉 = 4q2 ⊥(1− xω) m2 ( 1 γ + γθ2 k )2 { 1+ 4(1− xω) x2 ω ∣∣nq⊥e′ ∣∣2 } . Then, P ' P (xω) = 1 1 + x2 ω/2(1− xω) , ( ∣∣θknq⊥ ∣∣ ¿ 1/γ) independently on θk × nq⊥ . As Fig. 2 displays, po- larization stays higher then 90% for xω < 0.35. 0.2 0.4 0.6 0.8 1 xΩ 0.2 0.4 0.6 0.8 1 P Fig.2. Polarization of bremsstrahlung at planar scattering and orthogonal radiation ∣∣θknq⊥ ∣∣ ¿ 1/γ On the other hand, if angular separation is not at- tempted (which might be technically challenging at γ > 104), and only the natural collimation due to emission from an ultra-relativistic particle is used, one needs to integrate over radiation angles, or equiv- alently, photon transverse momenta. Evaluation of the integral of (12) over d2k⊥ yields: ∫ 〈 \|Mrad\|2 〉 d2k⊥ (2π)2 = ∫ 〈 \|Mrad\|2 〉 ω2d2θk (2π)2 =q2 ⊥γ2 1− xω π { x2 ω + 2 3 (1− xω) [ 1 + 2(nq⊥e′)2 ]} , P ' P (xω) = 1− xω 3 2x2 ω + 2(1− xω) . Thus, without separation in θk, by means of only keeping nq⊥ fixed, polarization can not be obtained higher then 50% (see Fig. 3). On the other hand, we are going to show below, that with separation in θk performed, it is possible to achieve a 100% po- larization even at a spherically-symmetric scatterer. 0.2 0.4 0.6 0.8 1 xΩ 0.1 0.2 0.3 0.4 0.5 P Fig.3. Polarization of bremsstrahlung at planar scattering, averaged over θk angles 3.2. Centrally-symmetric scatterer, az- imuthal integration over q⊥ The averaging in (12) over directions of q⊥ is achieved through the substitution (nq⊥)i(nq⊥)k → 1 2δik. One gets 〈 \|Mrad\|2 〉 → 8q2 ⊥ m2 (1− xω)2 (γ−1 + γθ2 k)2 · { x2 ω 2(1− xω) + 1− 4γ−2(θke′)2 (γ−2 + θk)2 } , P (xω, γ \|θk\|) = 2γ2θ2 k x2 ω 2(1−xω) (1 + γ2θ2 k)2 + 1 + γ4θ4 k . 1 2 3 4 ΓΘk 0.2 0.4 0.6 0.8 1 P Fig.4. Polarization of bremsstrahlung on a centrally-symmetric scatterer for xω = 0; 0.3; 0.6 (from top to bottom) At xω → 0, \|θk\| = 1/γ polarization reaches 100%, in spite of the angular averaging. That could hardly be expected based on very general reasons only. (The corresponding result was displayed in Fig. 4 of [1] but with no explanation supplied for the backing of such possibility). From our Fig.1 it is apparent that 100% magnitude of polarization may exist because at \|θk\| = 1/γ polarization is oriented along a circle, cen- tered at the origin of the plane and invariant under rotations of nq⊥ , corresponding to angular averaging. Thus, observation at some radiation angles of po- larization close to 100% does not yet signal existence in the target of some special collective fields, guiding electron motion. Such an effect is also possible in an amorphous medium 4. 4. SUMMARY The present work has offered formulation of a method for evaluation of differential cross-section of Comp- ton scattering for polarized initial and final pho- tons, based on advanced reduction of the matrix el- ement. As was demonstrated, after the full reduc- tion, the calculation leading to Lorentz-invariant for- mula (7) consumes only a few lines. To compare with, the derivation starting with squaring of the ini- tial matrix element or expression (5), occupies a few pages. The transition to differential cross-section of 4As usual, at that in order to be able to neglect multiple scattering effects as compared to deflection by emitting radiation, one needs fulfillment of the Landau-Pomeranchuk’s type condition L ¿ e2Lrad (see, e.g., [5]), where L is target thickness and Lrad the radiation length (centimeters to decimeters for solid targets). Moreover, at xω rather small, photon emission angle which is of main interest for us exceeds electron deflection angle by a factor 1−xω xω , so the true condition may be L ¿ � 1−xω xω �2 e2Lrad. 93 bremsstrahlung in dipole approximation was based on specification of covariant expressions in terms of vector components in laboratory frame, which lifts the necessity to manually implement the equivalent photon method and transit between different refer- ence frames - not very trivial task when dealing with polarization of the emitted photon. In application to bremsstrahlung, a previously overlooked feature which seems to be worth empha- sizing is that polarization as a function of (small) radiation angles at fixed nq⊥ orients itself along per- fect circles, including one centered at the origin. If averaging over nq⊥ is performed, at the radius of the latter circle \|θk\| = 1/γ only polarizations with the same orientation add up. Moreover, at small polar- ization values along the circle are close to 100%, and so the averaged polarization can be close to 100%, too. Actually, polarization stays in excess of 80% in the interval 0.8 < γ \|θk\| < 1.3, xω < 0.3. The practical conclusions reached were as follows. If a beam of energetic and polarized γ-quanta needs to be prepared, it is beneficiary to use bremsstrahlung in an amorphous medium at \|θk\| ' 1/γ (not just ∼ 1/γ). The polarization is orthogonal to the ra- diation plane - as had long been established. The radiation recoil influence is always depolarizing, but for xω < 0.3 rather weak. The method is most con- venient to use for ω ≤ 10 GeV. There is still an interesting question, whether po- larization must necessarily be taken into account at computations of electromagnetic shower spatial de- velopment in matter. For showers polarizations of γ-quanta are often neglected even at calculations of the spatial picture subsequently used for evaluation of radioemission from the shower as a whole. Taking into account that polarization of bremsstrahlung on a single atom/nucleus is orthogonal to the direction of photon deflection from the shower axis, at each step the development of the shower roughly makes an azimuthal turn to 90◦. After a few steps the shower shall become axially symmetric. Therefore, neglect of polarization may prove acceptable in a thick (as com- pared to the radiation length) target, but not when the shower is short. References 1. M.L. Ter-Mikaelian. High-Energy Electromag- netic Processes in Condensed Media. N.Y.: Wiley-Interscience, 1972, 458 p. 2. V.N. Baier, V.M. Katkov, V.S. Fadin. Radiation of relativistic electrons. Moscow: ”Atomizdat”. 1973, 374 p. 3. R.P. Feynman. The Theory of Fundamental Processes. New York: ”Benjamin”, 1961. 172 p. 4. M.V. Bondarenco. Covariant amplitude decom- position in relativistic fermion scattering prob- lems // Probl. Atom. Sci.and Techn. 2007, v.3 (1), p. 104-110. 5. V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii. Quantum Electrodynamics. Oxford: ”Pergamon-Press”. 1982. 652p. 6. A.I. Akhiezer, V.B. Berestetskii. Quantum Elec- trodynamics. Moscow: ”Nauka”. 1981, 432p. РАСЧЕТ И АНАЛИЗ ВЕЛИЧИНЫ ПОЛЯРИЗАЦИИ ДЛЯ ТОРМОЗНОГО ИЗЛУЧЕНИЯ ПРИ ПЕРИФЕРИЧЕСКОМ РАССЕЯНИИ Н.В. Бондаренко Предложен упрощенный способ вычисления дифференциального сечения поляризованного тормозно- го излучения при типичной атомной передаче импульса q ¿ m. Используемый подход основывается на естественной редукции спинового матричного элемента процесса к форме V αγα + A5γ 5. Анали- зируется распределение поляризации в полностью дифференциальном сечении. Показано, что если производить угловую селекцию излученных фотонов, то даже после интегрировании по импульсам, переданным атому, для определенных углов излучения поляризация может сохраняться до 100%. Без угловой селекции фотонов, только за счет контроля отдачи, нельзя получить поляризацию излучения выше 50%. РОЗРАХУНОК ТА АНАЛIЗ СТУПЕНЮ ПОЛЯРИЗАЦIЇ ДЛЯ ГАЛЬМIВНОГО ВИПРОМIНЮВАННЯ В УМОВАХ ПЕРИФЕРИЧНОГО РОЗСIЯННЯ М.В. Бондаренко Запропоновано спрощений спосiб обчислення диференцiального перерiзу поляризованого гальмiвно- го випромiнення за типової атомної передачi импульсу q ¿ m. Використовуваний пiдхiд грунтується на природнiй редукцiї спiнового матричного елементу процесу до форми V αγα + A5γ 5. Аналiзується розподiл поляризацiї у повнiстю диференцiальному перерiзi. Показано, що якщо здiйснювати кутову селекцiю випромiнених фотонiв, то навiть пiсля iнтегрування по импульсах переданих атому, для пев- них кутiв випромiнення поляризацiя може зберiгатися до 100%. Без кутової селекцiї фотонiв, тiльки за рахунок контролю вiддачi, неможливо отримати поляризацiю випромiнення вищу, нiж 50%. 94

Computation and analysis of the polarization degree for bremsstrahlung at peripheral scattering

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