between electron energy levels. Using the spectrum of electron energy (1) and the law of energy-momentum conservation for photon absorption:

Transcription

1 ENERGY MEASUREMENT O RELATIVISTIC ELECTRON BEAMS USING RESONANCE ABSORPTION O LASER LIGHT BY ELECTRONS IN A MAGNETIC IELD R.A. Melikian Yerevan Physics Institute, Yerevan ABSTRACT The possibility o a precise measurement o electron beam energy using a resonant orption o laser light by electrons in а static homogeneous magnetic ield is considered. The method permits to measure the average electron beam energy or wide range up to ew hundred GeV energies. The relative accuracy o the electron beam energy measurement can reach 1-4. The inluence o a laser light diraction and spread in angles o electrons on a measurement accuracy o electron beam energy is considered. The power o the required laser is estimated. 1. THE CONDITION O RESONANT ABSORPTION or a precise measurement o electron beam energy we suppose to use a resonant orption o laser photons by electrons in a static and homogeneous magnetic ield. Let's assume, that the magnetic ield B directed along z axis (ig.1). It is supposed, that the electrons and photons are injected in a magnetic ield under small angles and θ to the z-axis, accordingly. In a magnetic ield the electrons have a discrete spectrum o energy [4]: 1 ε n, ζ = [ m + Pz + eb(n + 1+ ζ )], (1) where n=,1,, labels the electron energy levels, P z is the z-component o the electron momentum, ζ = ± 1 is the projection o electron spin on a direction o B. We use units or which =c=1. rom (1) ollows, that the electrons in a magnetic ield can be on equidistant energy levels (ig.) and photons o requency ω can be resonantly orbed at transitions ε n, ζ ε n, ζ between electron energy levels. Using the spectrum o electron energy (1) and the law o energy-momentum conservation or photon orption: ε n, ζ + ω = ε n, ζ, P z, + ω = Pz, () we can ind the condition or resonant electron transition (see also [5], [6]): ω (sinθ ) ω (1 Vz, cosθ ) + = ω c ( n n), (3) m where ε n, ζ = ε and Pz, are the energy and z-component o electron momentum beore o photon orption, ω c = eb/m is the cyclotron requency o electron, = ε /m is the electron relativistic actor, V z, = V cos is the z-component o electron velocity beore orption. We consider transitions without o change o electrons spin direction, because the probability o spin-lip transitions is negligible. Besides we consider only transitions on the main harmonic n' - n = 1, because the probability o transitions or higher harmonic is negligible. 1

2 or orption o photons o optical and more lower requencies o interest to us in approximation o ω/ε <<1 the resonance condition can be written as (see also [5,6]): ω c cos cosθ 1 =. (4) ω In relation to the relation (4) has a solution: ± cos cosθ 1+ (cos cosθ ) =, (5) 1 (cos cosθ ) where = ω c ω. The dependence o the -actor rom or some ixed angles and θ is represented schematically on ig.3. In case o one electron, i the angles and θ are known, then having measured a resonant requency, due to a unique dependence between and, we can calculate -actor o electron according to the ormula (5). Because in reality the electrons o beam always have some spread on angles in limits i and laser light always diverges in limits o diraction angles, thereore the precise measurement o olute value o immediately according to (5) is diicult. Below we consider the possibility o determination o electron bunch energy using relation (5), taking in account the spread in angles o electrons and inluence o light diraction.. DETERMNATION O ELECTRON BEAM ENERGY TAKING IN ACCOUNT THE SPREAD IN ANGLES O ELECTRONS AND LIGHT DIRACTION It is known, that the light beam o diameter D, because o diraction diverges in limits o angle θ d λ D around o direction o wave vector k o incident wave. Here k = ω c, λ and ω are the length and requency o wave. At diraction o light the wave vector changes only direction, so that k = k, where k is wave vector ater diraction. The distribution o light intensity depending on angle o diraction θ is determined by expression (see or example [7], [8]): J1( α) I ( θ ) = I, (6) α and has the shape schematically represented on ig.4. Here I is the intensity o incident wave under angle θ =, J 1( α) is the cylindrical unction o the irst order and α = Dkθ /. rom (6) ollows, that the main part o radiation intensity is concentrated in interval o angles θ λ D. rom (5) ollows that in a case o >> 1 and i >> 1 (cos cosθ ) the electron energy can be approximately determined by ormula:. (7) + θ

3 rom (7) it is clear, that or ixed values o and we have + θ = = const. (8) On the graph with axes (, θ ) the relation (8) represents the equation o a circle with radius (ig.5). rom (8) ollows, that the intensity o a resonant orption I is minimum or electrons with angles = i and θ = at min = i = i, because the number o electrons orbing photons resonantly, is minimum. rom ig.6 it is obvious, that i < i then I =. With growth o the intensity o orption I is increase, because in process o orption will be included more electrons with angles in limits i and θ θ. Intensity reaches up to maximum at = and = =. At the urther growth o the intensity o orption I decreases because the interval o angles θ and thereore the intensity o light interacting with electrons decrease. Thus the intensity o orption I as a unction rom has a shape, which is schematically shown on ig.7. Note, that i the electron beam interacts with light (see ig.5) in limits o angles < θ < θ, then the diraction divergence o this part o light will be smaller by actor o θ θ d << 1. It is clear rom ig.7, that on axis the point = is singled out and can be used or determination o electrons energy. The irst method: Determination o a relative energy o electrons. I the energy o electron bunch 1 is known, then having measured the resonant requency, 1, which corresponds to the maximum o resonance orption intensity I ( ) 1 or this energy (ig.8), we can calculate according to the ormula (7) the, angle: =,1 1. urther we suppose, that or an unknown energy o electrons the angular distribution o electrons o beam does not vary and the maximum o resonant orption intensity I ) is all on the same angle (, =,. Then having measured the requency, calculate the value o unknown :, = 1., (see ig.6), we can (9) It is obvious, that the accuracy o will be determined by accuracy o 1,, 1 and,. The second method: Determination o an olute energy o electrons. Let's assume, that we can turn the electron beam on some small angle δ =, known or us (ig.9). Then having measured resonant requencies and (ig.1), which correspond to a maximum o resonance orptions intensity at angles 3

4 = and =, we can calculate the olute value o actor o electrons: 1 = (1) δ In particular it is possible to use a bending magnet (BM) or turn o electron beam on some small angle δ = (ig.9), which is determined by the ormula: eh l δ =, (11) mc c where H is the value o the magnetic ield, and l is the length o bending magnet. Having measured resonant requencies and, which correspond to a maximum o resonance orptions intensity at angles (ig.1, ig.11, ig1), we have the relation: = and = ( ) eh δ = = = mc rom here we can calculate the olute value o actor o electron: l c. (1) 1 1 eh l =. mc c ( ) (13) In this case the accuracy o will be determined by accuracy o parameters, H and l. At both methods the precise measurement o a resonant value, is diicult because o some width o resonance o orption intensity I () (ig.7). However, prove to be, that by tuning o the laser intensity parameter ξ = ee mωc (or length o a magnet L ) it is possible to increase the accuracy o determination and to increase a spectral resolving power o measuring method. 3. ESTIMATION O INTENSITY O A RESONANT ABSORPTION Let s estimate the parameter o the laser intensity ξ at orption by electron o ν laser photons on length o interaction L and or angle o incidence o electrons (ig.13). It is clear rom ig.13, that under action o a component E = sine E o electric vector o light, the electron will be accelerated in a direction o velocity V e o electron and the growth o electron energy will be eel. I at acceleration the electron orbs ν photons, then we have the relation: ω ν ω = eel = mc ξl, (14) c whence we have 1 mc L[ cm]1 ν = ξl ξ. (15) 38,61 4

5 In order to the electron would orb even 1 photon, it is necessary to satisy the requirement: mc ξl 1. (16) or ixed length o a magnet L, the dependence ξ rom have the shape, schematically shown on ig.14. It is clear rom ig.14, that or example in a case o ξ < ξ 1 the electrons with angles in an interval i < < 1 will not orb photons, while the electrons with angles > 1, will orb. It means, that or < 1 there is not resonant orption o photons (see ig.14, ig.15). Analogously, in cases o ξ < ξ and ξ < ξ (ig.14) the electrons will be orb photons only at realization o requirements > and >, accordingly (ig.17, ig.18 and ig.19, ig.). Thus it is obvious (ig.18 and ig.), that at ξ < ξ and ξ < ξ the intensity o orption have sharp let-hand edge and this property o dependence o I () allows increasing the accuracy o determination o. It is clear, that in case o determination o energy o electrons according to the irst and second method described above, we suppose that ξ > ξ i. Let's remark, that the orption o photons does not increase noticeably the spread o electron beam energy ε b because o relation: ω << ε b. Let's estimate number o photons N orbed by electron beam o length l b, i l b >> L and taking into account, that each electron transits the region o interaction L only once.. To observe the resonant orption we suppose, that in (15) the parameters ξ, and L are chosen so that ν 1. I the number o electrons in the beam is N e, then N = N e. I the parameters o the laser ξ and ω are known, then we have ee = ξmωc and the minimum power o the laser can be ound by a relation: [ ] [ ] E V cm W cm W. (17) 19,4 Actually, because only the part o laser light interacts with electron beam, the necessary minimum o a laser power (intensity) should be more (see ig.5). Now, using the known value o W we can ind the number N l, ph o laser photons, incident on a surace S during time Tb = lb c : W N l, ph = STb. (18) ω Resonance orption can be observed by measuring the ratio o the orbed photons N to total number o photons, incident on area o interaction N l, ph or ( I I tot ). 5 Let's consider a numerical example or = 1, λ = 1, 6 micrometer ( CO laser) 4 and = 1, θ =. Then according to (7) the ield strength is B =. kgs. or 9 L = 5 cm and T b =. 64 msec (TESLA) using (15) and (17) we have: ξ and W.36W cm. I N e = 4 1 e/pulse and S =.5cm, then using (18) we 15 ind N l, ph 3 1 photon/pulse and N N l, ph

6 Thus, the numerical estimations show, that near to a resonance, the ratio has a reasonable value or measuring. N N l, ph REERENCES [1] Brinkmam R. EPAC98, Stockholm, [] Boge et al. EPAC98, Stockholm, [3] Bravin et al. EPAC98, Stockholm, [4] Berestetski V.B., Lishitz E.M., Pitaevski L.P. Quantum Electrodinamics, Pergamon (198), (Nauka, Moscow, 198). [5] Barber D.P. and Melikian R.M. EPAC, Vienna,. [6] Milantev V.P. Uspekhi iz. Nauk, 167, 1997, 3. [7] Landau L.D. and Lishits E.M. Classical Theory o ields. Nauka, Moscow, [8] Demtoder W. Laser Spectroscopy. Basic Concepts and Instumentations. Springer- Verlag, Berlin, Heidelberg, New York,

7 igure Captions ig. 1. Conceptual sketch o the set-up. e - - electron beam, L laser beam, D detector, M magnet. ig.. Equidistant energy levels o electrons in magnetic ield. ig. 3. Depedence o electron -actor rom or ixed angles and θ. ig. 4. Distribution o laser light intensity I (θ) rom angle o diraction θ. ig. 5. The dependence o angle rom angle θ. ig. 6. Dependence o -actor o electrons rom or dierent angles I, 1/,. The intensity is minimum or = i and = i. I reaches to maximum at = and =. At > the intensity I decreases. ig. 7. Intensity I o photon orption by electrons as a unction rom. ig. 8. Determination o the unknown -actor o electrons relative to the known 1 -actor. ig. 9. Sketch o the arrangement or determination o the olute energy o electrons. BM bending magnet, L the length o the main magnet. ig. 1, ig. 11, ig.1. Measuring o requencies and relevant to the maximum o resonant orption at angles and. ig. 13. Laser beam intersects the electron beam under angle. Electrons can accelerated in direction o electron velocity V e under action o a component E = E sin. ig. 14. The dependence laser intensity parameter o ξ rom or ixed length o magnet L. ig. 15. In case o ξ< ξ 1/ electrons with angles in interval << 1/ orb photons, while electrons with angles > 1/ will orb. ig. 16. In case o ξ< ξ 1/ or values o < 1/ there is not resonant orption o photons. ig. 17, ig. 18. In case o ξ< ξ electrons will be orbs photons only at. Intensity o orption have sharp let-hand edge at =. ig. 19, ig. In case o ξ ξ electrons can orb photons only at. Intensity o orption have sharp let-hand edge at =. 7

8 B e - beam D Laser beam θ ig. 1 z ħω ħω ig. ħω n+ n+1 n n-1 (;θ) ig. 3 I(θ) I θ d θ ig. 4 7

9 1/ i θ 1/ θ θ θ d ig. 5 θ i 1/ I I 1 I 1/ i 1/ i d ig.6 ig.7 i 1,1, ig. 8 8

10 e - beam BM B δ Z Laser beam e - L ig. 9 e - i θ θ θ d θ ig. 1 ig. 11 I I 1 ig. 1 9

11 e - beam Laser Beam E E V e Z ig. 13 ξ ξ i ξ 1/ 1/ ξ ξ i i 1/ θ 1/ θ θ d θ ig. 14 ig.15. ξ < ξ 1 I I 1 I 1/ 1/ i i 1/ θ θ d ξ ξ 1 ig.17. ξ ξ ig.16. θ 1

12 I I 1 i θ θ θ d θ ig.18. ξ ξ. ig.19. ξ ξ I I 1 ig.. ξ ξ 11