THE POSSIBILITY OF PRECISE MEASUREMENT OF ABSOLUTE ENERGY OF THE ELECTRON BEAM BY MEANS OF RESONANCE ABSORPTION METHOD R.A.

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1 THE POSSIBILITY OF PRECISE MEASUREMENT OF ABSOLUTE ENERGY OF THE ELECTRON BEAM BY MEANS OF RESONANCE ABSORPTION METHOD R.A. Melikian Yerevan Physics Institute, Yerevan, Armenia Abstract In this report we consider a new conception of the Resonant Absorption method for measurement of the absolute energy of the electron beam for a wide (up to few hundred GeV) range of energies, with accuracy. Detectors and lasers, necessary for the 4 measurement of the energy of electrons by this method, are not unique and can be chosen from existing industrial samples. The parameters of the electron beam will not worsen during the measurements of the energy and this feature of the method allows carrying out the continuous monitoring of the electron beam energy.. Determination of electrons energy on the basis of condition of resonant absorption As before[], we suppose that electrons and photons are injected in a magnetic field, under small angles ϕ and θ to the z-axis, as shown on Fig.. B r Fig. R.A. Melikian, YerPhI, 6.6.7

2 The energy of electrons can be determined using the condition of the resonant absorption of laser photons by electrons at transitions between quantum levels of energy in a magnetic field. For absorption of photons of optical or lower frequency hω ε << the resonant condition can be written as: where = ε mc is the relativistic factor of electron, Ω=ω ω, ω = eb c. c mc Equation () in relation to has the solution: For simplicity, we first consider the dependence of on in the case of values of Ω (Fig.). Then from () we have ε The dependence on ϕ for various values of Ω at ( ε ) ϕ= = GeV is shown on Fig.. ω cosϕ cosθ = Ω in approximation of Ω ± cosϕ cosθ Ω + (cosϕ cosθ ) = (cosϕ cosθ ) ϕ θ = Ω ± cosϕ Ω sinϕ = sinϕ. () for fixed () (3) Fig. R.A. Melikian, YerPhI, 6.6.7

3 From (3) it follows, that:. has a physical meaning if the following condition is satisfied :. At ϕ = the factor of electron has the minimal value: 3. At ϕ = the derivative of with respect to ϕ is equal to zero: sinϕ Ω Ω = + Ω. (4) (5). (6) From Fig. and (6) it is clear that the value of is less sensitive to changes of angle ϕ near to ϕ =. This behavior of dependence ε = ε (ϕ ) we shall use for exact measuring of absolute energy of electron. From (3) it follows, that the lower branch of curve (ϕ) at ϕ << Ω can approximately be described as: Ω Ω 4. (7). Precise measurement of electron beam absolute energy, in case of electrons spread over angles ϕ in the interval ϕ ϕ max. We assume that the electron beam has spread over energies ε ± Δε and over angles ϕ in the interval. Then, according to (7), the dependence ε (ϕ has the shape shown on Fig.3. ε Ω 8Ω 8 4Ω + ϕ [ ϕ] 3 ϕ ϕ max ) min d dϕ ϕ = = R.A. Melikian, YerPhI,

4 . From the Fig.3 it is clear, if Fig.3 Ω Ω then electrons with energies ε ε + Δε cannot give the contribution to absorption intensity of photons ( I = ). It is obvious that with growth of the absorption intensity gradually increases. When Ω = Ω, then into intensity of absorption I abs the contribution will gives electrons with energies in the interval ε and in ε ε + Δε the interval of angles ϕ ϕ. Finally, if Ω = Ω then into intensity of absorption will gives the contribution all electrons with energies in the interval ε and in the Δε ε ε + Δε interval of angles ϕ ϕ m. Then the absorption intensity will be maximal. abs Ω R.A. Melikian, YerPhI,

5 . Dependence of absorption intensity I on Ω has the shape, shown on Fig.4. abs Fig.4 Using the relations (7) and (5) we can estimate quantity ϕ and ϕm (Fig.3) approximately as: Δ Δ ϕ Ω, ϕ ϕ. m (8) If the spread over energies of electrons is equal to Δε ε = 3 then from (8) follows that: 3. (9) ϕ m Ω << Ω From the Fig.3 it is clear, that if the electron beam has angular distribution in the interval then in case of Ω=Ω into absorption intensity I will gives contribution electrons ϕ > ϕ m with the angles. In other words, the pseudo-collimation of the electron beam occurs. ϕ < ϕ m abs R.A. Melikian, YerPhI,

6 According to (5) and Fig.3 we have: 4 From () follows if Ω < then we can determine to the formula: At the same time the condition have Ω =, whence follows: 5 or.. () with accuracy δ < according () imposes a restriction on. Really, from () we Now let's find the width of Ω (Fig. 4). Taking into account that according to () + Δ = Ω we receive approximately:, Δ = and =, () 5 For example if = and = 3 8 Δ then Ω Ω =.5. From () follows the relation: According to (4) to find energy with accuracy with accuracy δω Ω <. 4 Ω Ω < 4 Ω Ω ( ) ϕ = = = ( + Ω ) min min = Ω ε 5.55 Ω Ω Ω 4 min MeV Δ Δ = δ = δω Ω Ω 4 <. (3) it is necessary to find δ Ω (4) R.A. Melikian, YerPhI,

7 The quantity of Ω with accuracy δω Ω < can be found using behavior of dependence I abs from Ω (Fig.4). It is obvious from the (Fig.3), that the greatest contribution to the intensity of absorption will be given by the electrons of distribution centre, with energy or when. From Fig.4 it is clear, that for points and the partial derivative =, while for point Ω I abs the private derivative is maximal. Using these properties of function (Ω) we Ω can find the behavior of dependence Ω I abs Ω 4 I Ω abs Ω on Ω (Fig.5). Ω = Ω I abs Using (3) and (4) we can find the ratio of Δ < 3, then: δ < 4 δω Ω Ω Fig.5 to Ω Ω, and its value in a case when δω δ = = Δ (5) R.A. Melikian, YerPhI,

8 Let's notice that real possibilities of measurement of B and ω allow us to find Ω with accuracy δω Ω < 4 because So, if spread over angles is ϕ ϕ max then energy of electron beam can be determined with accuracy δω Ω < 4 according to the formula: where the quantity of we can find over maximum of value of. Ω Dependence of absorption intensity δω Ω δb = ± m B use Nd:YAG and CO lasers has the shape, shown on Fig.6. δω ω I abs B = Ω, I abs (6) (7) from for ε = GeV and ε = GeV in case of Ω Fig.6 R.A. Melikian, YerPhI,

9 . Dependence of electron beam energy from magnetic field B in case of use Nd:YAG and CO lasers has the shape, shown on Fig.7. ε Fig.7 3. Measurement of absolute energy of electron beam in case of spread of electrons over spatial angles around of axis ϕ = in interval ϕ max ϕ ϕ max. Let s consider the version when the electron beam has spread over energies ε ± Δε, and directions of velocities of electrons are concentrated in narrow spatial angle around of axis ϕ = in interval: ϕ ϕ ϕ. max max R.A. Melikian, YerPhI,

10 . In this case of distribution over angles ϕ the dependence ε from ϕ will be described by the formula (3) and for ε = GeV has the form, shown on Fig.8. Fig.8 According to (3) (or (7)) for angles ϕ << Ω the dependence of ε (ϕ) has the form, shown on (Fig.9). Therefore for angles ϕ ϕ the dependence of max ϕ max (Ω) I abs will have the form similar shown on Fig.4. Energy of electron beam can be determined by means of the formula (7). Fig.9 R.A. Melikian, YerPhI, 6.6.7

11 4. Measurement of electron beam energy in case of spread of electrons over spatial angles in intervals ϕ max ϕ ϕ max and θ max θ θ max. 5 Dependence ε from ϕ and θ for constant Ω =.5 according to formula () has shape shown on Fig.. From () follows, that:. has physical sense if satisfies the condition: ϕ + θ. In a point = and, energy has the minimal value: ϕ θ = Fig.. (8) 3. In a point and, the derivative of with respect and are equal to zero: ϕ = θ = d dϕ ϕ = = Ω min ϕ θ Ω = + Ω (9), d. () dϕ θ = = R.A. Melikian, YerPhI, 6.6.7

12 From () follows, that the lower branch of curve ( ϕ, θ ) in case of ϕ << Ω and θ << Ω can be described approximately as: Using () and (9) we can estimate the quantity θ and θ m (similar to estimation of ϕ m according to (8)) approximately as: ( + Ω Ω ) ϕ + ( + Ω Ω ) θ + [ ϕ] [ θ Ω Ω 8Ω 8Ω ] 3 3 θ Ω Δ. () and,. () Energy of electron beam can be found in the form described above, by means of the formula (7). Δ θ m θ ϕ 5. Intensity of the laser necessary for absorption of photons by electrons We have shown that for determination of electrons energy (7), in case of constant laser ω B frequency it is necessary to measure only the magnetic field, which corresponds to absorption of photons by electrons of distribution centre over energy (Fig.3, Fig.6, Fig.9). In the formula (7) for determination of energy there is no explicit dependence on intensity of the laser I las, magnet length LM. However it is supposed that these parameters should satisfy to the certain conditions. R.A. Melikian, YerPhI, 6.6.7

13 . Intensity of the laser necessary for absorption of photons on the length l a we can find using the classical formula for a gain of energy of electrons in considered fields [,3]: where ξ ee / mωc (3) is parameter of laser intensity, is amplitude of electric field of wave. Let s take into account that is the length of formation of photon absorption with energy Δε = hω during time, relation (3) can be written as: d = ξω dt and V W, then the. (4) According to (4) dependence of laser intensity las a from energy of electrons of Nd:YAG and CO lasers, L = l = cm, has the shape, shown on Fig.. Ω, = E l a Δt = l a / Ve l a / c I las, a M a = Ω hω 9.4 = l a E = 9.4 cm I las, a cm I, ε in case Fig. R.A. Melikian, YerPhI,

14 6. Accompanying radiation of electrons. It is known that in fields considered above both processes are possible: acceleration (due to absorption of photons) and deceleration (and in consequence of that the radiation of photons) of electron in dependence of its initial phase[,3]. Let's analyze conditions of radiation of electrons for the given parameters, ω, LM, B, I las Earlier we have noted that frequency of radiation of electrons is determined by the classical formula [3]: ωr =, (5) l V V cosα c r where α is angle between electron velocity V r and direction of radiation, l r is the length of electron trajectory, onto which the radiation of photon is formed. On the other hand, using the discrete spectrum of electron energy in a magnetic field and the law of preservation of energy-momentum during radiation, for frequency of radiation we have [3]: ( n n ) ωc ωr = (6) V z cosψ c where ψ is angle between direction of magnetic field B r and direction of radiation, n n =,, 3,. R.A. Melikian, YerPhI,

15 In a case of >>, the main part of radiation intensity is concentrated in a narrow interval of angles α or α =ψ = the frequency ω r and for simplicity we shall consider for angles, which according to (5) and (6) will be: It is clear that radiation of a electron is possible if (7). (8) Intensity of the laser, necessary for radiation of electrons with frequency las r on the length we can find by means of similar to (4) formula: l r As radiation of electron is possible if then using relation (9) we receive a condition for radiation of electron: ψ c ω l r r ( n n ) ω ( n n ) ω c c l = l a ( n n ) ω r L M I, ω = ( n n ) ω l I las, r r l = hωr 9.4 = l r r L M ( n n ) hω L 9.4 I las, r M r (9) (3) Taking into account (4), the condition (3) can be written as: ( n n ) I las, a I las, r (3) R.A. Melikian, YerPhI,

16 If for absorption of electrons with energy the necessary parameters ω, L M, I las, a are already chosen, then radiation of electrons for the same parameters is possible if or if to use (3) las, a las, r las a. This condition can be carried out only if n n =. In case of n n =, using (8) we shall obtain a restriction on energy ( n n ) I I I, of electron: Thus radiation of electrons is possible, if their energy is limited by condition (3), while electrons with energy L M can absorb photons. ω c For example if L M cm and we use Nd:YAG and is possible if are satisfied accordingly the conditions: = CO L M ω c I las, r I las, a (3) lasers then radiation of electron MeV and ε 78. MeV. ε Influence of process of measurement on parameters of electron beam During the measurement of the electron beam energy his parameters will not worsen. a) As energy of the absorbed photons is much less than energy of electrons then the change of energy of electron beam because of absorption will be negligible. hω << ε b) Let s estimate change of direction of velocity of electron in magnetic field. For that we resolve onto components parallel ( B ) and perpendicular ( B ) to direction (Fig.). B r V r e V r e R.A. Melikian, YerPhI,

17 It is clearly that the trajectory of a electron can be bent only under influence of perpendicular component B ϕ B. Therefore angle of bend α on length LM of electron trajectory, we can estimate according to the formula: For example, if = 5, L M cm, = λ =. 6μk ϕ direction of electron velocity V r e is negligible. LMωc LMω ϕ = ϕ c c α. (33) then α 4. It means, that the change of 8. Experimental confirmations Acceleration of electrons in the electromagnetic wave and the magnetic fields, due to absorption of photons by electrons, has been experimentally confirmed [4,5]. The obtained results are in the good agreement with the theoretical predictions regarding validity of the resonant condition () and the electrons energy gain (3). It is known also about planned experiment in BNL: Laser Cyclotron Auto-Resonance Acceleration (LACARA), with use of the CO laser [6]. It seems this set-up is suitable for testing the considered conception. R.A. Melikian, YerPhI,

18 Laser Cyclotron Auto-Resonance Accelerator, (LACARA), AE5 Jay Hirshfield and Tom Marshall R.A. Melikian, YerPhI,

19 9. Summary. The considered conception of Resonant Absorption allows one to measure the absolute energy of the electron beam, for a wide range (up to few hundred GeV) energies, with accuracy 4.. The detectors and lasers required for measurement of the electron beam energy are not unique, and can be chosen from existing industrial samples. 3. In this conception, for measurement of high energy of electron beam, the required magnetic fields have relatively small value, equal a few hundred Gs. 4. The parameters of the electron beam will not worsen during the measurement of the energy. This feature of the method allows carrying out the continuous monitoring of the electron beam energy.. References. D.P. Barber, R.A. Melikian. Proc. 7th EPAC, Vienna,.. V.P. Milantiev. Uspekhi Fiz. Nauk, 67, (997) R.A. Melikian. Meeting on Beam Energy Measurement. Oct. 4-6, 6, Yerevan. 4. H.R. Jory and A.W. Trivelpiece, J. of Appl. Phys. 39, 353 (968). 5. M.A. LaPointe et al., Phys. Rev. Lett. 76, 78 (996). 6. Ilan Ben-Zvi, BNL, nd ORION Advanced Accel. and Beam Physics Workshop, 3. R.A. Melikian, YerPhI,