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Nonlinear Compton scattering of strong laser radiation on channeled particles in a crystal
8 July 2002
Physics Letters A 299 (2002) 331–336 www.elsevier.com/locate/pla
Nonlinear Compton scattering of strong laser radiation on channeled particles in a crystal A.K. Avetissian, K.Z. Hatsagortsian, G.F. Mkrtchian ∗ , Kh.V. Sedrakian Department of Theoretical Physics, Plasma Physics Laboratory, Yerevan State University, 1 A. Manukian, Yerevan 375049, Armenia Received 29 November 2001; received in revised form 11 May 2002; accepted 14 May 2002 Communicated by P.R. Holland
Abstract A version for intense γ -ray radiation based on the multiphoton scattering of strong laser radiation on relativistic particle beam channeled in a crystal is proposed. The incident laser beam and charged particles beam are counter-propagating and the laser radiation is resonant to the energy levels of transversal motion of channeled particles. 2002 Elsevier Science B.V. All rights reserved. PACS: 61.80.Fe; 07.85.Fv; 41.75.Ht; 52.25.Os Keywords: Radiation; Crystal; Nonlinear; Compton
1. Introduction As is known channeling occurs if a charged particle enters a crystal at an angle to a crystallographic axis √ or plane smaller than Lindhard angle θL = 2U0 /E, where U0 is the depth of a transverse potential well, and E is the particle’s energy [1]. The spontaneous radiation of channeled particles [2,3] has some significant properties that open possibilities for implementation of short-wave radiation sources since due to their large Doppler shift and the high oscillation frequency in the channel (Ω ∼ 1014 –1016 s−1 for particles energies E ∼ 10 GeV–10 MeV) particles emit quanta mainly in the X-ray and γ -ray domain with an intensity much higher than the intensity * Corresponding author.
E-mail address: [email protected] (G.F. Mkrtchian).
of other types of radiation (see [4] and references therein). The spontaneous radiation of channeled particles has been comprehensively studied both theoretically and experimentally (see, e.g., Refs. [1,4–6]), while induced channeling radiation, that is radiation in the presence of external electromagnetic wave (EMW), has been studied mainly in the linear regime of interaction [7–14]. To achieve a considerable amplification in the single pass nonlinear regime of X-ray amplification has been studied in [15], but these investigations show that implementation of short-wave coherent radiation sources due to stimulated channeling radiation is far from being realized yet. One reason is that because of short lifetime of a particle transverse-motion levels the length of coherent interaction of a channeled particle with EMW is quite short (e.g., of the order of one micrometer at positron energies ∼10 MeV) compared to the interaction length in other versions of Free
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 5 6 - 4
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A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
Electron Laser (such as the undulator and Cherenkov lasers). There is an another problem connected with the controlling of the channeled particles overpopulation [16]. Two component laser assisted schemes for radiation enhancement have also been studied. One of those is based on stimulated photon scattering by channeled particles [13]. Since absorption of a photon by a channeled particle is a resonant process, the cross section of photon scattering by a channeled particle is 104 times larger than the free electron Compton scattering cross section [12]. Nevertheless, the gain of a EMW in stimulated Compton scattering on channeled particles does not exceed the gain of stimulated emission in the channel from the initially inverse populated states. The second scheme concerns the quantum mode of interaction, that is the coherent radiation of quantum modulated beam at the frequency of the stimulating wave [17] and its harmonics [18]. In this Letter we investigate multiphoton scattering of strong laser radiation on relativistic particle beam channeled in a crystal which can serve as a possible scheme for γ -ray generation. The scheme is considered when averaged potential for a plane channeled particles is good enough described by the harmonic potential. Then it is assumed that the incident laser beam and charged particles beam are counterpropagating and the laser radiation is resonant to the energy levels of transversal motion of channeled particles. Due to the resonant interaction with given pump laser the equidistant transverse levels are excited and as a consequence spontaneous transitions from the upper levels take place with the emission of a hard quanta. In the following within the scope of quantum electrodynamics the spectral intensity of multiphoton Compton scattering in a strong laser field is obtained. The first-order Feynman diagram, where the electron/positron lines correspond to the wave functions in the strong laser field is calculated and the resonant case of interaction is discussed.
enough described by the harmonic potential x2 . 2 For plane channeled positron
U (x) = κ
(1)
8U0 (2) , d2 where U0 is the transverse potential hole depth, and d is the interspace distance [4]. For plane channeled electrons the approximate potential is actually not harmonic, but for the high energies it can be approximated by (1). As it is known, for the channeled particles the depth of the potential hole is U0 ∼ 10–100 eV [4] and U0 E, where E is the particle energy. The spin interaction which is ∼ ∇U (x) is again less than E. For this reason the transverse motion is described by the Schrödinger equation [4,5] with effective mass mef = E (the natural units h¯ = c = 1 will be used throughκ=
out this Letter), where E = p2 + m2 is the energy of longitudinal motion, m is the particle mass and p is the longitudinal momentum. On the other hand, the spin interaction can play a role in the spontaneous radiation process if the radiated photons energy is ω ∼ E. But if the particle energy is not enough high, i.e., E
m2 , E⊥
(3)
where E⊥ = E − E is the energy of transverse motion, then ω E and the spin effects are not substantial. In addition we restrict the total energy change of the channeled particles with the external EMW ∆E E,
(4)
so that we may ignore the spin interaction and instead of Dirac equation use the Klein–Gordon equation 2 2 ∂ i − U (x) Ψ = pˆ − eA + m2 Ψ, (5) ∂t
2. Wave function of a plane channeled particle in the field of transverse electromagnetic wave
where e is the particle charge and A = A0 cos ω0 (t + nz), 0, 0 ,
We will consider the case when averaged potential of the crystal for a plane channeled particle is good
is the vector potential of the plane EMW with amplitude A0 and frequency ω0 . Here n is the crystal refracting index on ω0 . The channeled particle initial
(6)
A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
motion (before the interaction with EMW) is separated into longitudinal (Y, Z) and transverse (X) degrees of freedom. For longitudinal motion we assume initial state with momentum p = {py , pz }, while for transverse motion we assume the state {s}, where by s we indicate the energy levels in the harmonic potential (1). As the external EMW field depends only on the τ = t + nz then raising from the problem symmetry, the wave function can be found in the following form: Ψ (r, t) = f (x, τ ) exp[ipy y + ipz z − iEt].
(7)
Taking into account (4) we can consider f (x, τ ) as a slowly varying function of τ and neglect the second derivative compared with the first order. So, for f (x, τ ) we will have the following equation:
˜ τ ∂xx + 2E E⊥ − U (x) + 2i p∂ 2 2 − 2iA(τ )∂x − e A (τ ) f = 0, (8) where p˜ = E + npz .
(9)
In Eq. (8) transverse and longitudinal motions are not separated. But after a certain unitary transformation in the equation for the transformed function the variables are separated [18] and for wave function we obtain Ψ=√
1 2Π
exp{iΠy y + iΠz z − iΠt}
ω( ˜ ω˜ 2 + Ω 2 ) 2 2 e A0 sin 2ω0 τ × exp −i 8E ∆2
Ω2 eA0 cos ω0 τ − ix ∆ ω˜ eA0 sin ω0 τ , × Vs x + E ∆
Πz = pz − n
ω˜ 2 2 2 e A0 . Π =E+ 4p∆ ˜
Here ω˜ =
p˜ ω0 , E
and Vs (x) = χ=
1 π 1/4
E Ω,
∆ = ω˜ 2 − Ω 2 ,
(12)
χ 2x2 χ exp − Hs (χx), 2s s! 2 κ Ω= , E
(13)
is the wave function of the harmonic oscillator with Hermit polynomial Hs (χx).
3. Compton scattering on channeled particles in a crystal As we saw in Section 2 Πy , Πz and s are the quantum numbers (neglecting spin interaction) describing the state of a particle moving in the fields (1), (6). It is clear that between them there will be a spontaneous transitions which causes spontaneous radiation. The spontaneous radiation may be considered by the theory of perturbation. In this case first order Feynman diagram describes spontaneous radiation where wave functions (10) correspond to electron/positron lines. The probability amplitude of transition from the state {Π0y , Π0z , s0 } to the state {Πy , Πz , s} with emission of a photon with the frequency ω and momentum k will be [20]: µ∗ √ e0ν = e 4π j √ Ms(ν) , µ 0s 2ω
(14)
µ∗
(10)
where the state of the particle in fields (1), (6) is characterized by the average energy and momentum (“quasimomentum”) defining via free particle energymomentum by the following equations Πy = py ,
333
ω˜ 2 2 2 e A0 , 4p∆ ˜ (11)
where e0ν is the four-dimensional polarization vector, index ν corresponds to the emitted photons of two possible polarizations (ν = 1, 2). Here µ j µ = d 4 x jf,i (r, t)ei(ωt −kr) , (15) µ
where jf,i (r, t) is the four-dimensional transition current. As is known the polarization vector may always be µ chosen in a way that e0 = (0, e0 ); e0 k = 0 (three dimension gage). The differential probability calculated in unit volume and unit time will be dWs(ν) = 0s
(ν) |Ms0 s |2 dk dΠy dΠz , Ly Lz T (2π)3
(16)
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A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
where Ly , Lz are quantization length and T is the interaction time. If we are not interested in the dependence of process on the photons polarization then the probabilities must be summed by all possible polarizations 2 (2) 2 + M . |Ms0 s |2 = Ms(1) (17) s s0 s 0 As long as in our case ω E and the condition (4) must be satisfied then the transition currents may by calculated using the solution of the Klein–Gordon equation. The transition currents will be jf,i = i Ψi ∇Ψf∗ − Ψf∗ ∇Ψi − 2eAΨi Ψf∗ . (18) Using this expression for transition currents and taking into account (15), (16) and (17) we will arrive to the following expression for the differential probability dWs0 s =
e2 8πωΠΠ0 ∞ × Ws(/) δ(Πz − Π0z + kz + /nω) 0s /=−∞
× δ(Πy − Π0y + ky ) × δ(Π − Π0 + ω − /ω) × dk dΠy dΠz .
(19)
In Eq. (19) Π0 , Π are the initial and final quasienergies which are defined via free particle energy and momentum by Eq. (11). In general the expression of the partial probability is very complicated and we will not bring it here. The δ functions in expression (19) for differential probability express the quasimomentum and quasienergy conservation laws in the given process. Different / correspond to different parare tial processes with fixed photon numbers and Ws(/) 0s the partial probabilities. Let us find the energy of emitted photons rising from the conservation laws. Taking into account (4) and ω E we will have the following expression for the frequency 1 + nv˜0z
/ω0 + Ω (s0 − s) , ω= (20) k˜v0 1− ω where Ω =
Ω , 1 + nv˜0z
(21)
Π , Π0
(22)
and v˜ 0 =
Fig. 1. Schematic illustration of considering process for the particular transition. The equidistant transverse levels of moving particle are shown which are excited by the external counter propagating EMW with frequency ω0 (solid arrow). Due to the resonant interaction (Doppler shifted frequency is resonant to the energy levels) the transverse levels are excited (solid vertical arrow) and as a consequence spontaneous transition from the upper levels (dotted vertical arrow) takes place and a photon is emitted with Doppler upshifted frequency ω (dashed arrow).
is the mean longitudinal velocity, v˜0z is the (Z) component of the mean longitudinal velocity. In Fig. 1 the schematic illustration of considering process for the particular transition is given. It follows from (20) that / > 0 corresponds to the multiphoton absorption and / < 0 to the multiphoton emission of a wave quanta. It is noteworthy to mention that in the nonlinear Compton process on free electrons [21] only the case of / > 0 , i.e., the multiphoton absorption process in the strong EMW takes place. In contrary, at the scattering of a strong wave on channeling particles the multiphoton emission (/ < 0) of quanta of EMW takes place. When Ω = 0 and n = 1 formula (20) is reduced to the well-known Compton effect one for the scattered frequency (neglected quantum recoil). Let us consider the resonance case which is of more interest and expression for the differential probability may be simplified. We will consider the case when |ω˜ 0 − Ω| 1, Ω
|ω˜ − Ω| 1, Ω
(23)
A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
where ω˜ 0 and ω˜ are initial and final Doppler shifted frequencies (12). Besides we will assume that ω˜ 2 − Ω 2 eA0 (24) |δ|, δ = 0 2 . m Ω Here ξ is the relativistic invariant parameter of the wave intensity. Considering (23) it is possible that despite ξ 1 but the condition (24) may be satisfied. To obtain the total cross section of nonlinear scattering the expression (19) must be summed by all discrete states of transverse motion in the channel. After integrating by Πy and Πz , then summing by /, using the δ functions, and taking into account (23), (24) for differential cross section we will have
ξ≡
dW =
m2 e 2 2πωω0 ΠΠ0 2 ξ 2 Λ1 (N, α, β) × −Λ20 (N, α, β) + δ − Λ0 (N, α, β)Λ2 (N, α, β) dk. (25)
Here Λr (N, α, β) = (2π)
−1
π dθ cosr θ
−π
× exp i(α sin θ − β sin 2θ − Nθ ) , (26) are known functions [21] and represent nonlinear processes in the field of linear polarized wave (multiphoton Compton effect, pair production, etc.). The arguments α and β are defined by the relations α=ξ
2mkx ω˜ 0 ω˜ 0 ω˜ − Ω 2 , E ∆∆0
β = ξ2
m2 (ω˜ 0 − ω) ˜ ω˜ 0 ω˜ − Ω 2 , 8E ∆∆0
(27)
1 + nv˜0z 1−
k˜v0 ω
Nω0 .
spectral intensity of an one-photon emission (if product to ω) in the crystal at simultaneously nonlinear “Compton” scattering of a strong EMW on the channeled particle at the resonance. Instead of nonlinearity parameter ξ 2 in the Compton effect on free electrons the effective nonlinearity in the channeling process is determined by the resonance parameter (ξ/δ)2 , increasing the cross section of the multiphoton “Compton” scattering. For the actual cases δ ∼ 10−2 –10−1 [19], and consequently the parameter of nonlinearity increases ∼103 times. By the analogy with nonlinear Compton scattering on free particles [21] the multiphoton processes become essential for ξ/|δ| 1 and the cutoff number of harmonics Nc ∼ (ξ/|δ|)3 . Typically, the depth of the channeling potential well is of order of U0 ∼ 10–100 eV, the interspace distance d ∼ 10−8 cm, so for positrons with energy E ∼ 50 MeV, the resonance can be achieved by the optical lasers as h¯ ω0 h¯ Ω/2 ∼ 1 eV (see (21)). The number of absorbed photons should be restricted by condition Nc ω0 U0 to avoid the dechanneling effects, so for the forward radiation of a positrons with the energy E ∼ 50 MeV, Nc ∼ 10 maximum of emitted quanta energies up to h¯ ω ∼ 1 MeV are achievable. In conclusion, we have presented the theoretical treatment of the multiphoton scattering of a strong laser radiation on relativistic particle beam channeled in a crystal. The discussed scheme has several advantages in respect to the known ones. First of all, the cross section of the process is resonantly enhanced with respect to the Compton scattering process. At the second, the multiphoton processes arise at the much lower laser intensities than in the case of the Compton scattering. Besides, the scheme enables to use the forward channeling radiation arising due to transitions from short living, high excited states of a particle to the ground state, that could not be achievable in the process of spontaneous channeling radiation.
(28)
and N is fixed by the conservation law, which in the resonant case is ω=
335
(29)
In expressions (27), (28) ∆0 and ∆ are initial and final resonance widths (12). Formula (25) defines the
Acknowledgements We would like to thank Prof. H.K. Avetissian for valuable discussions during the work under the present Letter. This work was supported by International Science and Technology Center (ISTC) Project No. A353.
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References [1] A. Kumakhov, Emission by Channeled Particles in Crystals, Energoatomizdat, Moscow, 1986 (in Russian). [2] R.L. Walker, Phys. Rev. Lett. 25 (1970) 5. [3] A. Kumakhov, Phys. Lett. A 57 (1976) 17. [4] V.A. Bazylev, I.K. Zhevago, Radiation of High Energy Particles in a Medium and External Fields, Nauka, Moscow, 1987 (in Russian). [5] V.V. Beloshicky, F.F. Komarov, Phys. Rep. 93 (1982) 117. [6] G. Kurizki, J.K. McIver, Phys. Rev. B 32 (1985) 4358. [7] V.V. Beloshicky, M.A. Kumakhov, Zh. Eksp. Teor. Fiz. 74 (1978) 1244, Sov. Phys. JETP 47 (1978) 652 (in English). [8] R.H. Pantell, M.J. Alguard, J. Appl. Phys. 50 (1979) 798. [9] A.V. Andreev et al., Zh. Eksp. Teor. Fiz. 84 (1983) 1743, Sov. Phys. JETP 57 (1983) 1017 (in English). [10] A.V. Tulupov, Zh. Eksp. Teor. Fiz. 86 (1984) 1365, Sov. Phys. JETP 59 (1984) 797 (in English).
[11] I.M. Ternov et al., Zh. Eksp. Teor. Fiz. 88 (1985) 329, Sov. Phys. JETP 61 (1985) 192 (in English). [12] A.V. Tulupov, Pis’ma Zh. Tekh. Fiz. 7 (1981) 460, Sov. Tech. Phys. Lett. 7 (1981) 196 (in English). [13] V.A. Bazylev, I.K. Zhevago, Phys. Status Solidi B 97 (1980) 63. [14] P. Kalman, Phys. Rev. A 48 (1993) R42. [15] H.K. Avetissian et al., Phys. Rev. A 56 (1997) 4121. [16] A.V. Tulupov, Zh. Eksp. Teor. Fiz. 81 (1981) 1639, Sov. Phys. LETP 54 (1981) 872 (in English). [17] H.K. Avetissian et al., Zh. Eksp. Teor. Fiz. 109 (1996) 1159, Rus. Phys. LETP 82 (1996) 622 (in English). [18] H.K. Avetissian et al., Phys. Lett. A 206 (1995) 141. [19] V.A. Bazylev, I.K. Zhevago, Usp. Fiz. Nauk 160 (1990) 47. [20] A.E. Akhiezer, V.B. Berestetzki, Quantum Electrodynamics, Nauka, Moscow, 1969. [21] V.I. Ritus, Trudy Fiz. Inst. Akad. Nauk 111 (1979) 141.
Physics Letters A 299 (2002) 331–336 www.elsevier.com/locate/pla
Nonlinear Compton scattering of strong laser radiation on channeled particles in a crystal A.K. Avetissian, K.Z. Hatsagortsian, G.F. Mkrtchian ∗ , Kh.V. Sedrakian Department of Theoretical Physics, Plasma Physics Laboratory, Yerevan State University, 1 A. Manukian, Yerevan 375049, Armenia Received 29 November 2001; received in revised form 11 May 2002; accepted 14 May 2002 Communicated by P.R. Holland
Abstract A version for intense γ -ray radiation based on the multiphoton scattering of strong laser radiation on relativistic particle beam channeled in a crystal is proposed. The incident laser beam and charged particles beam are counter-propagating and the laser radiation is resonant to the energy levels of transversal motion of channeled particles. 2002 Elsevier Science B.V. All rights reserved. PACS: 61.80.Fe; 07.85.Fv; 41.75.Ht; 52.25.Os Keywords: Radiation; Crystal; Nonlinear; Compton
1. Introduction As is known channeling occurs if a charged particle enters a crystal at an angle to a crystallographic axis √ or plane smaller than Lindhard angle θL = 2U0 /E, where U0 is the depth of a transverse potential well, and E is the particle’s energy [1]. The spontaneous radiation of channeled particles [2,3] has some significant properties that open possibilities for implementation of short-wave radiation sources since due to their large Doppler shift and the high oscillation frequency in the channel (Ω ∼ 1014 –1016 s−1 for particles energies E ∼ 10 GeV–10 MeV) particles emit quanta mainly in the X-ray and γ -ray domain with an intensity much higher than the intensity * Corresponding author.
E-mail address: [email protected] (G.F. Mkrtchian).
of other types of radiation (see [4] and references therein). The spontaneous radiation of channeled particles has been comprehensively studied both theoretically and experimentally (see, e.g., Refs. [1,4–6]), while induced channeling radiation, that is radiation in the presence of external electromagnetic wave (EMW), has been studied mainly in the linear regime of interaction [7–14]. To achieve a considerable amplification in the single pass nonlinear regime of X-ray amplification has been studied in [15], but these investigations show that implementation of short-wave coherent radiation sources due to stimulated channeling radiation is far from being realized yet. One reason is that because of short lifetime of a particle transverse-motion levels the length of coherent interaction of a channeled particle with EMW is quite short (e.g., of the order of one micrometer at positron energies ∼10 MeV) compared to the interaction length in other versions of Free
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 5 6 - 4
332
A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
Electron Laser (such as the undulator and Cherenkov lasers). There is an another problem connected with the controlling of the channeled particles overpopulation [16]. Two component laser assisted schemes for radiation enhancement have also been studied. One of those is based on stimulated photon scattering by channeled particles [13]. Since absorption of a photon by a channeled particle is a resonant process, the cross section of photon scattering by a channeled particle is 104 times larger than the free electron Compton scattering cross section [12]. Nevertheless, the gain of a EMW in stimulated Compton scattering on channeled particles does not exceed the gain of stimulated emission in the channel from the initially inverse populated states. The second scheme concerns the quantum mode of interaction, that is the coherent radiation of quantum modulated beam at the frequency of the stimulating wave [17] and its harmonics [18]. In this Letter we investigate multiphoton scattering of strong laser radiation on relativistic particle beam channeled in a crystal which can serve as a possible scheme for γ -ray generation. The scheme is considered when averaged potential for a plane channeled particles is good enough described by the harmonic potential. Then it is assumed that the incident laser beam and charged particles beam are counterpropagating and the laser radiation is resonant to the energy levels of transversal motion of channeled particles. Due to the resonant interaction with given pump laser the equidistant transverse levels are excited and as a consequence spontaneous transitions from the upper levels take place with the emission of a hard quanta. In the following within the scope of quantum electrodynamics the spectral intensity of multiphoton Compton scattering in a strong laser field is obtained. The first-order Feynman diagram, where the electron/positron lines correspond to the wave functions in the strong laser field is calculated and the resonant case of interaction is discussed.
enough described by the harmonic potential x2 . 2 For plane channeled positron
U (x) = κ
(1)
8U0 (2) , d2 where U0 is the transverse potential hole depth, and d is the interspace distance [4]. For plane channeled electrons the approximate potential is actually not harmonic, but for the high energies it can be approximated by (1). As it is known, for the channeled particles the depth of the potential hole is U0 ∼ 10–100 eV [4] and U0 E, where E is the particle energy. The spin interaction which is ∼ ∇U (x) is again less than E. For this reason the transverse motion is described by the Schrödinger equation [4,5] with effective mass mef = E (the natural units h¯ = c = 1 will be used throughκ=
out this Letter), where E = p2 + m2 is the energy of longitudinal motion, m is the particle mass and p is the longitudinal momentum. On the other hand, the spin interaction can play a role in the spontaneous radiation process if the radiated photons energy is ω ∼ E. But if the particle energy is not enough high, i.e., E
m2 , E⊥
(3)
where E⊥ = E − E is the energy of transverse motion, then ω E and the spin effects are not substantial. In addition we restrict the total energy change of the channeled particles with the external EMW ∆E E,
(4)
so that we may ignore the spin interaction and instead of Dirac equation use the Klein–Gordon equation 2 2 ∂ i − U (x) Ψ = pˆ − eA + m2 Ψ, (5) ∂t
2. Wave function of a plane channeled particle in the field of transverse electromagnetic wave
where e is the particle charge and A = A0 cos ω0 (t + nz), 0, 0 ,
We will consider the case when averaged potential of the crystal for a plane channeled particle is good
is the vector potential of the plane EMW with amplitude A0 and frequency ω0 . Here n is the crystal refracting index on ω0 . The channeled particle initial
(6)
A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
motion (before the interaction with EMW) is separated into longitudinal (Y, Z) and transverse (X) degrees of freedom. For longitudinal motion we assume initial state with momentum p = {py , pz }, while for transverse motion we assume the state {s}, where by s we indicate the energy levels in the harmonic potential (1). As the external EMW field depends only on the τ = t + nz then raising from the problem symmetry, the wave function can be found in the following form: Ψ (r, t) = f (x, τ ) exp[ipy y + ipz z − iEt].
(7)
Taking into account (4) we can consider f (x, τ ) as a slowly varying function of τ and neglect the second derivative compared with the first order. So, for f (x, τ ) we will have the following equation:
˜ τ ∂xx + 2E E⊥ − U (x) + 2i p∂ 2 2 − 2iA(τ )∂x − e A (τ ) f = 0, (8) where p˜ = E + npz .
(9)
In Eq. (8) transverse and longitudinal motions are not separated. But after a certain unitary transformation in the equation for the transformed function the variables are separated [18] and for wave function we obtain Ψ=√
1 2Π
exp{iΠy y + iΠz z − iΠt}
ω( ˜ ω˜ 2 + Ω 2 ) 2 2 e A0 sin 2ω0 τ × exp −i 8E ∆2
Ω2 eA0 cos ω0 τ − ix ∆ ω˜ eA0 sin ω0 τ , × Vs x + E ∆
Πz = pz − n
ω˜ 2 2 2 e A0 . Π =E+ 4p∆ ˜
Here ω˜ =
p˜ ω0 , E
and Vs (x) = χ=
1 π 1/4
E Ω,
∆ = ω˜ 2 − Ω 2 ,
(12)
χ 2x2 χ exp − Hs (χx), 2s s! 2 κ Ω= , E
(13)
is the wave function of the harmonic oscillator with Hermit polynomial Hs (χx).
3. Compton scattering on channeled particles in a crystal As we saw in Section 2 Πy , Πz and s are the quantum numbers (neglecting spin interaction) describing the state of a particle moving in the fields (1), (6). It is clear that between them there will be a spontaneous transitions which causes spontaneous radiation. The spontaneous radiation may be considered by the theory of perturbation. In this case first order Feynman diagram describes spontaneous radiation where wave functions (10) correspond to electron/positron lines. The probability amplitude of transition from the state {Π0y , Π0z , s0 } to the state {Πy , Πz , s} with emission of a photon with the frequency ω and momentum k will be [20]: µ∗ √ e0ν = e 4π j √ Ms(ν) , µ 0s 2ω
(14)
µ∗
(10)
where the state of the particle in fields (1), (6) is characterized by the average energy and momentum (“quasimomentum”) defining via free particle energymomentum by the following equations Πy = py ,
333
ω˜ 2 2 2 e A0 , 4p∆ ˜ (11)
where e0ν is the four-dimensional polarization vector, index ν corresponds to the emitted photons of two possible polarizations (ν = 1, 2). Here µ j µ = d 4 x jf,i (r, t)ei(ωt −kr) , (15) µ
where jf,i (r, t) is the four-dimensional transition current. As is known the polarization vector may always be µ chosen in a way that e0 = (0, e0 ); e0 k = 0 (three dimension gage). The differential probability calculated in unit volume and unit time will be dWs(ν) = 0s
(ν) |Ms0 s |2 dk dΠy dΠz , Ly Lz T (2π)3
(16)
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A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
where Ly , Lz are quantization length and T is the interaction time. If we are not interested in the dependence of process on the photons polarization then the probabilities must be summed by all possible polarizations 2 (2) 2 + M . |Ms0 s |2 = Ms(1) (17) s s0 s 0 As long as in our case ω E and the condition (4) must be satisfied then the transition currents may by calculated using the solution of the Klein–Gordon equation. The transition currents will be jf,i = i Ψi ∇Ψf∗ − Ψf∗ ∇Ψi − 2eAΨi Ψf∗ . (18) Using this expression for transition currents and taking into account (15), (16) and (17) we will arrive to the following expression for the differential probability dWs0 s =
e2 8πωΠΠ0 ∞ × Ws(/) δ(Πz − Π0z + kz + /nω) 0s /=−∞
× δ(Πy − Π0y + ky ) × δ(Π − Π0 + ω − /ω) × dk dΠy dΠz .
(19)
In Eq. (19) Π0 , Π are the initial and final quasienergies which are defined via free particle energy and momentum by Eq. (11). In general the expression of the partial probability is very complicated and we will not bring it here. The δ functions in expression (19) for differential probability express the quasimomentum and quasienergy conservation laws in the given process. Different / correspond to different parare tial processes with fixed photon numbers and Ws(/) 0s the partial probabilities. Let us find the energy of emitted photons rising from the conservation laws. Taking into account (4) and ω E we will have the following expression for the frequency 1 + nv˜0z
/ω0 + Ω (s0 − s) , ω= (20) k˜v0 1− ω where Ω =
Ω , 1 + nv˜0z
(21)
Π , Π0
(22)
and v˜ 0 =
Fig. 1. Schematic illustration of considering process for the particular transition. The equidistant transverse levels of moving particle are shown which are excited by the external counter propagating EMW with frequency ω0 (solid arrow). Due to the resonant interaction (Doppler shifted frequency is resonant to the energy levels) the transverse levels are excited (solid vertical arrow) and as a consequence spontaneous transition from the upper levels (dotted vertical arrow) takes place and a photon is emitted with Doppler upshifted frequency ω (dashed arrow).
is the mean longitudinal velocity, v˜0z is the (Z) component of the mean longitudinal velocity. In Fig. 1 the schematic illustration of considering process for the particular transition is given. It follows from (20) that / > 0 corresponds to the multiphoton absorption and / < 0 to the multiphoton emission of a wave quanta. It is noteworthy to mention that in the nonlinear Compton process on free electrons [21] only the case of / > 0 , i.e., the multiphoton absorption process in the strong EMW takes place. In contrary, at the scattering of a strong wave on channeling particles the multiphoton emission (/ < 0) of quanta of EMW takes place. When Ω = 0 and n = 1 formula (20) is reduced to the well-known Compton effect one for the scattered frequency (neglected quantum recoil). Let us consider the resonance case which is of more interest and expression for the differential probability may be simplified. We will consider the case when |ω˜ 0 − Ω| 1, Ω
|ω˜ − Ω| 1, Ω
(23)
A.K. Avetissian et al. / Physics Letters A 299 (2002) 331–336
where ω˜ 0 and ω˜ are initial and final Doppler shifted frequencies (12). Besides we will assume that ω˜ 2 − Ω 2 eA0 (24) |δ|, δ = 0 2 . m Ω Here ξ is the relativistic invariant parameter of the wave intensity. Considering (23) it is possible that despite ξ 1 but the condition (24) may be satisfied. To obtain the total cross section of nonlinear scattering the expression (19) must be summed by all discrete states of transverse motion in the channel. After integrating by Πy and Πz , then summing by /, using the δ functions, and taking into account (23), (24) for differential cross section we will have
ξ≡
dW =
m2 e 2 2πωω0 ΠΠ0 2 ξ 2 Λ1 (N, α, β) × −Λ20 (N, α, β) + δ − Λ0 (N, α, β)Λ2 (N, α, β) dk. (25)
Here Λr (N, α, β) = (2π)
−1
π dθ cosr θ
−π
× exp i(α sin θ − β sin 2θ − Nθ ) , (26) are known functions [21] and represent nonlinear processes in the field of linear polarized wave (multiphoton Compton effect, pair production, etc.). The arguments α and β are defined by the relations α=ξ
2mkx ω˜ 0 ω˜ 0 ω˜ − Ω 2 , E ∆∆0
β = ξ2
m2 (ω˜ 0 − ω) ˜ ω˜ 0 ω˜ − Ω 2 , 8E ∆∆0
(27)
1 + nv˜0z 1−
k˜v0 ω
Nω0 .
spectral intensity of an one-photon emission (if product to ω) in the crystal at simultaneously nonlinear “Compton” scattering of a strong EMW on the channeled particle at the resonance. Instead of nonlinearity parameter ξ 2 in the Compton effect on free electrons the effective nonlinearity in the channeling process is determined by the resonance parameter (ξ/δ)2 , increasing the cross section of the multiphoton “Compton” scattering. For the actual cases δ ∼ 10−2 –10−1 [19], and consequently the parameter of nonlinearity increases ∼103 times. By the analogy with nonlinear Compton scattering on free particles [21] the multiphoton processes become essential for ξ/|δ| 1 and the cutoff number of harmonics Nc ∼ (ξ/|δ|)3 . Typically, the depth of the channeling potential well is of order of U0 ∼ 10–100 eV, the interspace distance d ∼ 10−8 cm, so for positrons with energy E ∼ 50 MeV, the resonance can be achieved by the optical lasers as h¯ ω0 h¯ Ω/2 ∼ 1 eV (see (21)). The number of absorbed photons should be restricted by condition Nc ω0 U0 to avoid the dechanneling effects, so for the forward radiation of a positrons with the energy E ∼ 50 MeV, Nc ∼ 10 maximum of emitted quanta energies up to h¯ ω ∼ 1 MeV are achievable. In conclusion, we have presented the theoretical treatment of the multiphoton scattering of a strong laser radiation on relativistic particle beam channeled in a crystal. The discussed scheme has several advantages in respect to the known ones. First of all, the cross section of the process is resonantly enhanced with respect to the Compton scattering process. At the second, the multiphoton processes arise at the much lower laser intensities than in the case of the Compton scattering. Besides, the scheme enables to use the forward channeling radiation arising due to transitions from short living, high excited states of a particle to the ground state, that could not be achievable in the process of spontaneous channeling radiation.
(28)
and N is fixed by the conservation law, which in the resonant case is ω=
335
(29)
In expressions (27), (28) ∆0 and ∆ are initial and final resonance widths (12). Formula (25) defines the
Acknowledgements We would like to thank Prof. H.K. Avetissian for valuable discussions during the work under the present Letter. This work was supported by International Science and Technology Center (ISTC) Project No. A353.
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