- Email: [email protected]
Radiation from channeling electrons, stimulated by laser beam
Nuclear Instruments and Methods in Physics Research B 309 (2013) 67–69
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Radiation from channeling electrons, stimulated by laser beam N.P. Kalashnikov, E.V. Khangulyan, A.S. Olchak ⇑ National Research Nuclear University ‘‘MEPhI’’, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 26 November 2012 Received in revised form 18 January 2013 Accepted 18 January 2013 Available online 21 March 2013 Keywords: Quantum electrodynamics Channeling Fast electrons Accelerators Laser Single crystal
a b s t r a c t Fast electrons, moving in a channel formed by crystal planes, may occupy only discrete transverse energy quantum levels. Transition between levels can be accompanied with emitting high energy photons. This radiation can be additionally and resonantly stimulated by external photons having energy equal to the distance between certain channeling electron transverse energy levels in the accompanying system. Due to the Doppler effect, the optical frequency laser beam can provoke intensive c-radiation from channeling electrons. The effect can be also used for precise resonance measuring of transverse energies of channeling electrons. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. c-Radiation from planar channeling electrons
In 1970–80s predicted theoretically possibility to get a new type of radiation sources of ‘‘nearly monochromatic’’ high energy photons was one of the strongest challenges, which boosted interest towards the effect of channeling of fast charged particles between crystal planes or axises (see, for example, [1–4]). X-ray and c-radiation from channeling electrons were very soon discovered experimentally (see, for example [5,6]). However, it proved to be not so intensive and not so monochromatic as has been expected. During the following years the first results were many times confirmed by different researchers [7]. The curiosity about radiation from channeling electrons was stimulated by the fact that their transverse motion between crystal planes or along some crystal axis is finite. As the result the particle has a discrete spectrum of transverse energies in the channel. Transition between the energy levels results in emitting photons the same way as it happens in atoms. The difference is that the longitudinal motion of electron is not finite and can be very fast (relativistic). The energy of emitted photons depends on both interval between transverse energy levels and angle, under which they are emitted. Due to the Doppler effect, photons emitted in straight forward direction or close to it, may obtain energies, compared to the energy of electron itself.
From here on we will consider only planar channeling of electrons – the case, which promises more chances to get monochromatic c-quants. The typical structure of transverse energy quantum levels of electron in planar channel is shown in Fig. 1 [4]. This calculation was performed to compare with experimental data [5], where quite distinctive radiation spectral lines were detected (Fig. 2). However, for higher electron energies (E > 1 GeV) the number of quantum energy levels is growing as E1/2 and all spectrum lines are merging together in one broad spectrum (see, for example [8], Fig. 2). Spectral characteristics of radiation from channeling electrons strongly depend on electron energy as well as on averaged planar channel potential, which, in turn, depends on the type of crystal (Si, Ge, diamond, etc.), plane orientation, interplanar distance etc. To be free of particularities we will take only approximate estimations of basic parameters [4,9]: 1. Planar channel potential width and depth in laboratory system:
U 0 20eV;d 0; 2—0; 4A23 ðSiÞ;
ð1Þ
2. Planar channel potential depth in accompanying system (longitudinally moving synchronously with electrons):
U ¼ U 0 ðE=mc2 Þ ðm is the electron mass; c is the light velocityÞ ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A.S. Olchak). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.01.060
ð2Þ
68
N.P. Kalashnikov et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 67–69
Fig. 3. Stimulation of radiation from channeling electron by the laser beam.
Fig. 1. The structure of transverse energy quantum levels of 54 MeV electrons in Si planar channel (1 1 0) (reproduced from the book [4]).
3. Number of quantum transverse energy levels:
N ðEU 0 Þ1=2 d=hc
ð3Þ
4. Distance between levels in accompanying system:
DE U=N ðEU 0 Þ
1=2
ðh=mcdÞ
x ¼ x0 ð4Þ
5. Radiated photon energy (with respect to the Doppler effect) in the forward direction (angle to straight forward direction w << 1):
hx ¼ 2DE=ðmc2 =E þ Ew2 =mc2 Þ < 2DEE=mc2 2ðE=mc2 Þ2 ðU 0 =EÞ1=2 hc=d
Another way to increase radiation flux from channeling electrons is to use some external stimulation. Such a stimulus can be provided by electromagnetic laser wave. Due to the Doppler effect when the laser beam is propagating through the crystal in direction nearly opposite to that of electron beam, the energy of photons may become comparable or equal to energy distances between electron energy levels in the accompanying system. Fig. 3 illustrates the proposed experiment geometry. In the accompanying system the energy of photon due to the Doppler effect will be much higher than in laboratory system
ð5Þ
3. Stimulated c-radiation from channeling electrons
c-Radiation from channeling electrons can be additionally stimulated by some external periodical force. In [9], for example, the case was studied when radiation from channeling electrons was stimulated by periodical fluctuations of planar channeling potential, hence any crystal plane consists of crystal axises and any crystal axis consists of discrete atoms. This radiation may have high energy but still is not intensive.
qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 vc2 1 þ vc cos h
¼ 2x0 =ðmc2 =E þ Ew2 =mc2 Þ < 2x0 E=mc2
ð6Þ
here x0 is the photon frequency in laboratory system. In accompanying system with the increase of electron energy E the energy of photon grows (⁄x E), and grows faster than the distance between transverse energy levels (DE E1/2) (4). For high enough electron energies the photon energy ⁄x (6) in accompanying system may reach the same value as DE (4) or even exceed it.
hðEU 0 Þ1=2 =mdc < 2hx0 E=mc2
ð7Þ
if to substitute x0 with 2pc/k0, where k0 is the laser wavelength, condition (7) can be rewritten more explicitly
E=U 0 > ðk0 =4pdÞ2 ;
ð8Þ
where U0 is the depth of channeling potential and d is the distance between crystal planes. For the typical crystal parameters (1) the effect of resonance stimulation of c-radiation from channeling electrons can be observed for electrons with energies exceeding 108 eV. We will need also high intensity optical laser, directing the beam nearly
Fig. 2. Experimental radiation spectra from channeling electrons in Si at different energies (three spectra reproduced from [5] relate to 54 MeV electrons moving along different crystal planes and one reproduced from [8] for 855 MeV electrons).
N.P. Kalashnikov et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 67–69
anti-parallel to the electron beam. Such a facilities are available at LNF INFN [10]. Important is that accurately directing laser beam (changing angle w at Fig. 3) it is possible to change the photon energy (6) so, that to bring it into precise resonance with this or that transverse energy transition!
2hx0 =ðmc2 =E þ Ew2 =mc2 Þ ¼ DEij
ð9Þ
Changing angle further may result in resonance with another transition and so on. It means that proposed effect may serve as direct measurement of transverse energy spectrum. Also it is important that the energy of emitted by channeling electron–photons (5) due to the Doppler effect will be much higher than the energy of initial laser photon in accompanying system (6). For 1 GeV electrons it is >1 MeV [8]. It means: we may convert the energy of the electron beam into the energy of emitted c-quants with high energies. 4. Preliminary estimations of stimulated c-radiation probability Accurate calculation of the cross-section of the resonance effect under consideration and of the consequent increase of c-radiation flux require quite complicated quantum analyses with regard also to dechanneling and electrons scattering effects. Such an analyses is in progress now and we hope to publish the results in nearest future. Currently we may just present some preliminary estimations. For channeling electron moving in the electromagnetic field of a laser beam with flux P [Wt/m2] the effective radiation length (for one stimulated c-quant emission) may be estimated as
L ðhxc=PrcÞðC=DEÞ;
ð10Þ
here r is an effective electron–photon scattering cross-section; c is the Lorenz-factor, accounting for the increase of photon density in accompanying system, U is the width of transverse energy levels due to dechanneling effect. In zero-approximation the cross-section of electron–photon scattering equals 1029 m2 [11]. If to estimate the resonance ratio C/DE as 0.01 (just for example) the effective radiation length will be L 0.01 m for 1 GeV electrons and optical (k0 500 nm) laser with flux P 1015 Wt/m2. 5. Conclusions 1. High energy electrons, moving in a channel along crystal plane or axis, may occupy only certain discrete transverse energy quantum levels. Transitions between levels may be accompanied with emitting photons (like in regular atoms). Due to the Doppler effect, the energy of radiated photons can be very high.
69
2. This radiation can be additionally and resonantly stimulated by external photons, having energy equal to the distance between certain channeling electron transverse energy levels in the accompanying system (longitudinally moving synchronously with electrons). 3. Due to the Doppler effect optical frequency laser photons directed nearly opposite to electron beam will have energies in the accompanying system, reaching the required resonance conditions. 4. Such photons will resonantly provoke c-radiation from channeling electrons. It means: we may convert the energy of the accelerated electron beam into the energy of emitted c-quants with much higher energies, than that of the provoking optical energy photons. 5. Resonance with certain transverse energy transition can be reached just by correctly orienting laser beam around the close to opposite to electrons direction. 6. The effect can be observed only if the electrons energy E is high enough:
E=U 0 > ðk0 =4pdÞ;2
ð11Þ
where U0 is the depth of the planar channeling potential; k0 is the provoking laser wavelength; d is distance between crystal planes. 7. The effect can be also used as the method of precise resonance measuring of the transverse energy spectrum of channeling electrons. References [1] N.P. Kalashnikov, V.S. Remizovich, M.I. Ryazanov, Collisions of Fast Charged Particles in Solids, Gordon and Breach Science Publishers, London-NY, 1985. [2] M.A. Kumakhov, Phys. Lett. A57 (1976) 17. [3] R. Wedell, Phys. Status Solidi B 99 (1) (1980) 11–45. [4] N.P. Kalashnikov, Coherent Interactions of Charged Particles in Single Crystals. (Scattering and Radiative Processes in Single Crystals), Harwood Academic Publishers, London-NY, 1988. [5] R.L. Swent et al., Phys. Rev. Lett. 43 (1979) 1723. [6] Y.N. Adishchev et al., Phys. Lett. A75 (1980) 316. [7] P. Rullhusen, X. Artru, P. Dhez, Novel Radiation Sources Using Relativistic Electrons, World Scientific, Singapore, 1998. [8] H. Backe, et al. In: ‘‘Channeling 2012, International Conference’’, Alghero, Italy, 2012. [9] N.P. Kalashnikov, A.S. Olchak, Nuovo Cimento 1D (2) (1982) 257–279. [10] N.P. Kalashnikov, A.S. Olchak, Commun. INFN Frascatti (private communication), 2012. [11] V.B. Berestetskii, E.M. Lifshits, L.P. Pitajevskii, Quantum Electrodynamics (in Russian), Nauka, Moscow, 1980 (Chapter X).
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Radiation from channeling electrons, stimulated by laser beam N.P. Kalashnikov, E.V. Khangulyan, A.S. Olchak ⇑ National Research Nuclear University ‘‘MEPhI’’, Moscow, Russia
a r t i c l e
i n f o
Article history: Received 26 November 2012 Received in revised form 18 January 2013 Accepted 18 January 2013 Available online 21 March 2013 Keywords: Quantum electrodynamics Channeling Fast electrons Accelerators Laser Single crystal
a b s t r a c t Fast electrons, moving in a channel formed by crystal planes, may occupy only discrete transverse energy quantum levels. Transition between levels can be accompanied with emitting high energy photons. This radiation can be additionally and resonantly stimulated by external photons having energy equal to the distance between certain channeling electron transverse energy levels in the accompanying system. Due to the Doppler effect, the optical frequency laser beam can provoke intensive c-radiation from channeling electrons. The effect can be also used for precise resonance measuring of transverse energies of channeling electrons. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
2. c-Radiation from planar channeling electrons
In 1970–80s predicted theoretically possibility to get a new type of radiation sources of ‘‘nearly monochromatic’’ high energy photons was one of the strongest challenges, which boosted interest towards the effect of channeling of fast charged particles between crystal planes or axises (see, for example, [1–4]). X-ray and c-radiation from channeling electrons were very soon discovered experimentally (see, for example [5,6]). However, it proved to be not so intensive and not so monochromatic as has been expected. During the following years the first results were many times confirmed by different researchers [7]. The curiosity about radiation from channeling electrons was stimulated by the fact that their transverse motion between crystal planes or along some crystal axis is finite. As the result the particle has a discrete spectrum of transverse energies in the channel. Transition between the energy levels results in emitting photons the same way as it happens in atoms. The difference is that the longitudinal motion of electron is not finite and can be very fast (relativistic). The energy of emitted photons depends on both interval between transverse energy levels and angle, under which they are emitted. Due to the Doppler effect, photons emitted in straight forward direction or close to it, may obtain energies, compared to the energy of electron itself.
From here on we will consider only planar channeling of electrons – the case, which promises more chances to get monochromatic c-quants. The typical structure of transverse energy quantum levels of electron in planar channel is shown in Fig. 1 [4]. This calculation was performed to compare with experimental data [5], where quite distinctive radiation spectral lines were detected (Fig. 2). However, for higher electron energies (E > 1 GeV) the number of quantum energy levels is growing as E1/2 and all spectrum lines are merging together in one broad spectrum (see, for example [8], Fig. 2). Spectral characteristics of radiation from channeling electrons strongly depend on electron energy as well as on averaged planar channel potential, which, in turn, depends on the type of crystal (Si, Ge, diamond, etc.), plane orientation, interplanar distance etc. To be free of particularities we will take only approximate estimations of basic parameters [4,9]: 1. Planar channel potential width and depth in laboratory system:
U 0 20eV;d 0; 2—0; 4A23 ðSiÞ;
ð1Þ
2. Planar channel potential depth in accompanying system (longitudinally moving synchronously with electrons):
U ¼ U 0 ðE=mc2 Þ ðm is the electron mass; c is the light velocityÞ ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A.S. Olchak). 0168-583X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nimb.2013.01.060
ð2Þ
68
N.P. Kalashnikov et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 67–69
Fig. 3. Stimulation of radiation from channeling electron by the laser beam.
Fig. 1. The structure of transverse energy quantum levels of 54 MeV electrons in Si planar channel (1 1 0) (reproduced from the book [4]).
3. Number of quantum transverse energy levels:
N ðEU 0 Þ1=2 d=hc
ð3Þ
4. Distance between levels in accompanying system:
DE U=N ðEU 0 Þ
1=2
ðh=mcdÞ
x ¼ x0 ð4Þ
5. Radiated photon energy (with respect to the Doppler effect) in the forward direction (angle to straight forward direction w << 1):
hx ¼ 2DE=ðmc2 =E þ Ew2 =mc2 Þ < 2DEE=mc2 2ðE=mc2 Þ2 ðU 0 =EÞ1=2 hc=d
Another way to increase radiation flux from channeling electrons is to use some external stimulation. Such a stimulus can be provided by electromagnetic laser wave. Due to the Doppler effect when the laser beam is propagating through the crystal in direction nearly opposite to that of electron beam, the energy of photons may become comparable or equal to energy distances between electron energy levels in the accompanying system. Fig. 3 illustrates the proposed experiment geometry. In the accompanying system the energy of photon due to the Doppler effect will be much higher than in laboratory system
ð5Þ
3. Stimulated c-radiation from channeling electrons
c-Radiation from channeling electrons can be additionally stimulated by some external periodical force. In [9], for example, the case was studied when radiation from channeling electrons was stimulated by periodical fluctuations of planar channeling potential, hence any crystal plane consists of crystal axises and any crystal axis consists of discrete atoms. This radiation may have high energy but still is not intensive.
qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 vc2 1 þ vc cos h
¼ 2x0 =ðmc2 =E þ Ew2 =mc2 Þ < 2x0 E=mc2
ð6Þ
here x0 is the photon frequency in laboratory system. In accompanying system with the increase of electron energy E the energy of photon grows (⁄x E), and grows faster than the distance between transverse energy levels (DE E1/2) (4). For high enough electron energies the photon energy ⁄x (6) in accompanying system may reach the same value as DE (4) or even exceed it.
hðEU 0 Þ1=2 =mdc < 2hx0 E=mc2
ð7Þ
if to substitute x0 with 2pc/k0, where k0 is the laser wavelength, condition (7) can be rewritten more explicitly
E=U 0 > ðk0 =4pdÞ2 ;
ð8Þ
where U0 is the depth of channeling potential and d is the distance between crystal planes. For the typical crystal parameters (1) the effect of resonance stimulation of c-radiation from channeling electrons can be observed for electrons with energies exceeding 108 eV. We will need also high intensity optical laser, directing the beam nearly
Fig. 2. Experimental radiation spectra from channeling electrons in Si at different energies (three spectra reproduced from [5] relate to 54 MeV electrons moving along different crystal planes and one reproduced from [8] for 855 MeV electrons).
N.P. Kalashnikov et al. / Nuclear Instruments and Methods in Physics Research B 309 (2013) 67–69
anti-parallel to the electron beam. Such a facilities are available at LNF INFN [10]. Important is that accurately directing laser beam (changing angle w at Fig. 3) it is possible to change the photon energy (6) so, that to bring it into precise resonance with this or that transverse energy transition!
2hx0 =ðmc2 =E þ Ew2 =mc2 Þ ¼ DEij
ð9Þ
Changing angle further may result in resonance with another transition and so on. It means that proposed effect may serve as direct measurement of transverse energy spectrum. Also it is important that the energy of emitted by channeling electron–photons (5) due to the Doppler effect will be much higher than the energy of initial laser photon in accompanying system (6). For 1 GeV electrons it is >1 MeV [8]. It means: we may convert the energy of the electron beam into the energy of emitted c-quants with high energies. 4. Preliminary estimations of stimulated c-radiation probability Accurate calculation of the cross-section of the resonance effect under consideration and of the consequent increase of c-radiation flux require quite complicated quantum analyses with regard also to dechanneling and electrons scattering effects. Such an analyses is in progress now and we hope to publish the results in nearest future. Currently we may just present some preliminary estimations. For channeling electron moving in the electromagnetic field of a laser beam with flux P [Wt/m2] the effective radiation length (for one stimulated c-quant emission) may be estimated as
L ðhxc=PrcÞðC=DEÞ;
ð10Þ
here r is an effective electron–photon scattering cross-section; c is the Lorenz-factor, accounting for the increase of photon density in accompanying system, U is the width of transverse energy levels due to dechanneling effect. In zero-approximation the cross-section of electron–photon scattering equals 1029 m2 [11]. If to estimate the resonance ratio C/DE as 0.01 (just for example) the effective radiation length will be L 0.01 m for 1 GeV electrons and optical (k0 500 nm) laser with flux P 1015 Wt/m2. 5. Conclusions 1. High energy electrons, moving in a channel along crystal plane or axis, may occupy only certain discrete transverse energy quantum levels. Transitions between levels may be accompanied with emitting photons (like in regular atoms). Due to the Doppler effect, the energy of radiated photons can be very high.
69
2. This radiation can be additionally and resonantly stimulated by external photons, having energy equal to the distance between certain channeling electron transverse energy levels in the accompanying system (longitudinally moving synchronously with electrons). 3. Due to the Doppler effect optical frequency laser photons directed nearly opposite to electron beam will have energies in the accompanying system, reaching the required resonance conditions. 4. Such photons will resonantly provoke c-radiation from channeling electrons. It means: we may convert the energy of the accelerated electron beam into the energy of emitted c-quants with much higher energies, than that of the provoking optical energy photons. 5. Resonance with certain transverse energy transition can be reached just by correctly orienting laser beam around the close to opposite to electrons direction. 6. The effect can be observed only if the electrons energy E is high enough:
E=U 0 > ðk0 =4pdÞ;2
ð11Þ
where U0 is the depth of the planar channeling potential; k0 is the provoking laser wavelength; d is distance between crystal planes. 7. The effect can be also used as the method of precise resonance measuring of the transverse energy spectrum of channeling electrons. References [1] N.P. Kalashnikov, V.S. Remizovich, M.I. Ryazanov, Collisions of Fast Charged Particles in Solids, Gordon and Breach Science Publishers, London-NY, 1985. [2] M.A. Kumakhov, Phys. Lett. A57 (1976) 17. [3] R. Wedell, Phys. Status Solidi B 99 (1) (1980) 11–45. [4] N.P. Kalashnikov, Coherent Interactions of Charged Particles in Single Crystals. (Scattering and Radiative Processes in Single Crystals), Harwood Academic Publishers, London-NY, 1988. [5] R.L. Swent et al., Phys. Rev. Lett. 43 (1979) 1723. [6] Y.N. Adishchev et al., Phys. Lett. A75 (1980) 316. [7] P. Rullhusen, X. Artru, P. Dhez, Novel Radiation Sources Using Relativistic Electrons, World Scientific, Singapore, 1998. [8] H. Backe, et al. In: ‘‘Channeling 2012, International Conference’’, Alghero, Italy, 2012. [9] N.P. Kalashnikov, A.S. Olchak, Nuovo Cimento 1D (2) (1982) 257–279. [10] N.P. Kalashnikov, A.S. Olchak, Commun. INFN Frascatti (private communication), 2012. [11] V.B. Berestetskii, E.M. Lifshits, L.P. Pitajevskii, Quantum Electrodynamics (in Russian), Nauka, Moscow, 1980 (Chapter X).