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Volume capture of ultrarelativistic electrons in the axial channeling regime
Volume
115, number
PHYSICS
8
LETTERS
5 May 1986
A
VOLUME CAPTURE OF ULTRARELATIVISTIC ELECTRONS IN THE AXIAL CHANNELING REGIME A.M. TARATIN and S.A. VOROBIEV Nuclear Physics Institute, Received
1 August
Tomsk Polytechnical
1985; revised manuscript
Institute, received
634050 Tomsk, 30 December
USSR
1985; accepted
for publication
13 February
1986
Using computer simulation the volume capture effect of ultrarelativistic electrons in the axial channeling regime in crystals is investigated. The thickness, temperature and orientational dependences of the beam fraction captured in the channeling regime in the volume of the crystal are obtained.
The first experimental studies of the radiation of ultrarelativistic electrons in single crystals have shown a large contribution from the channeling radiation of electrons bound to the atomic rows [ 11. The electrons initially captured in the bound states at the incidence in a crystal are rapidly dechanneled due to the multiple scattering by the crystal electrons and nuclei. They cannot provide the observed large effect in the rather thick crystals in experiments [ 11. So one may suppose that the additional fraction of the electron beam may be captured in the channeling regime in the crystal volume. This process results from the multiple scattering of electrons in a crystal. A basic condition for capture of electrons in the channeling regime in a crystal volume is the existence of an essential fraction of particles with transverse energies close to a critical one Elc . This paper shows the results of a computer experiment for the investigation of the volume capture of electrons in the axial channeling regime. The calculations have been done with a model of a lattice of atomic rows [2]. We used a real atomic potential and the appropriate electron density obtained from the amplitude of atomic scattering [3]. The beam fraction of particles undergoing a volume capture has been calculated, and its thickness, temperature and orientational dependences have been obtained for the first time. The system of equations, describing the electron motion along the atomic rows in a crystal, has the form 0.375-9601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
my d2xldt2 = -au(p)/i3x
7
my d2y/dt2 = -au(p)/ay , my d2z/dt2 = 0,
(1)
where m is the electron rest mass, y is the Lorentz factor, p = {x, y } and z are the tranverse and longitudinal coordinates of a particle, respectively, and u(p) is the crystal potential averaged along the crystallograph. ic axes, 2.5
‘) ,2 24 4 us(r) = - - c ai exp(-B--), a0 4 i=l~~ t uf
(2)
where u&r) is the averaged potential of a single atomic row, r, are the two-dimensional vectors of the atomic rows, the summation was taken over a maximum of 25 rows, a0 is the Bohr radius, d, is the distance between the atoms of a row, uI = flu1 is the twodimensional mean-square displacement of the crystal atoms from the row axis during thermal motion, Bi = hi/(2X)2, ai and bi are the expansion coefficients of the electron scattering factor. The electron trajectories in the crystal have been computed by numerical solution of system (1) with the potential (2). Multiple scattering of particles by crystal electrons and nuclei 401
Volume 115, number 8
PHYSICS LETTERS A
has been taken into account by the method described in ref. [2]. Transmission of a 1 GeV electron beam through a silicon crystalwith thickness 40 pm has been considered. The particle beam is incident at an angle \k 5 \k, relative to the (100) crystallographic direction, where \k, is a critical channeling angle (\k, = 0.418 mrad for (100) Si at T = 300 K). The particle state (the transverse coordinates (x ,y ) and velocity (uX , u,,)) is recorded in the end of every crystal layer of AZ = 4 I.tm. Also the transverse energy of particles El = i my (~1x2 + II;) t u(x ,y) is estimated. Fig. la shows the dependence on the orientation angle \k of the beam fraction of the electrons being captured in the bound states with El < 0 at the incidence at the crystal (z = 0, curve 1). The particles with EL < 0 are attributed to the states of bound motion with a single atomic row - the channeled electrons. It is seen that the electron capture decreases with increasing orientational angle \k, and at \k = 9, it is equal to zero. When a beam penetrates into the crystal, the number of channeled electrons de&eases due to multiple scattering by the crystal electrons and nuclei, i.e. particles are dechanneled and transfer into the states with El > 0 (quasichanneled electrons). However, as was noted by us [2], multiple scattering also leads to an inverse process; the capture of particles from the states with EL > 0 into the channeling regime takes place in the crystal volume. Fig. lb shows the z-dependences of the fraction of the channeled elec-
0
0.2
0,4 0 UI(mrad)-
trons captured in the channeling regime in the crystal volume. Results are shown for four values of the incident angle \k. The volume capture is maximum at a crystal depth which increases with increasing ?Ir. The maximum number of captured electrons is also changed, achieving a maximum value of 16.5% at an orientational angle \k = 0.12 mrad. The volume capture occurs from the over-barrier states with transverse energy EL z 0 near the potential barrier. The initial population of these states is maximum at orientation angle \k = (0.2-0.3)\kc, which stipulates a maximum of the obtained orientational dependence. At large orientational angles, \k 5 qc, the initial population of the near-barrier states with EL a 0 is small. After penetrating into the crystal, a definite fraction of the particle beam finds itself in a near-barrier region due to the diffusion process. This effect explains the obtained orientational dependence (curve 2 in fig. la). We have also considered the temperature dependence of the volume capture. Fig. 2 shows the z-dependences of the fraction of channeled electrons at two crystal temperatures: T = 0 K (curves 1,l ‘) and T = 540 K (curves 2,2’). Curves 1,2 are for the electrons captured in a crystal volume, and curves 1’,2’ are for the electrons captured in the channeling regime at the incidence into the crystal. The number of particles captured in the channeling regime at the incidence in a crystal reduces quickly, i.e. the electrons are dechanneled. The dechanneling grows with increasing temperature because the transverse dimension of the nuclear scattering region is increased. The volume capture of electrons slightly increases with temperature rise (curves 1,2). This is a consequence of increased nuclear scattering also.
20t oJrn,40
Fig. 1. (a), Dependence of the fraction of channeled electrons (in the states with ,!Z’*< 0) in ( 100) Si on the orientation angle for a 1 GeV electron beam: (1) particles captured at incidence at the crystal (z = 0), (2) particles captured in the crystal volume (in maximum of the depth dependence, see (b)). (b) Dependence of the fraction of channeled electrons in (100) Si captured in the crystal volume, on the crystal depth at different orientation angles, (1) q = 0.12-0.2 mrad; (3) * = 0.3 mrad; (4) q = 0.4 mrad. Crystal temperature T = 300 K.
402
5 May 1986
Fig. 2. Dependence of the fraction of channeled electrons on the crystal depth at different temperatures, (1 ,l’) T = 0 K, (2,2’) T= 540 K. Curve (1,2) electrons are captured in a crystal volume, curve (2,2’) electrons are captured at incidence in the crystal. The incident angle q = 0.1 mrad.
Volume 115, number 8
PHYSICS LETTERS A
This computer experiment shows that multiple scattering of particles by the crystal electrons and nuclei cause both dechanneling of relativistic electrons from the bound states with the atomic rows, and the inverse effect of volume capture of electrons into the bound states from the quasi-channeled fraction of the beam. Moreover, the electrons captured in the crystal volume constitute the main fraction of channeled particles for crystal thickness larger than 10 pm. These electrons are responsible for the observed channeling radiation in experiments [ 11. The contribution of the electrons captured at the incidence in a crystal is important only for thin crystals. The volume capture ef-
5 May 1986
feet increases the effective crystal thickness at which bound state electrons contribute to the different processes accompanying the electron beam penetration into the crystal. References [ 1] B.N. Kalinin, V.V. Kaplin, A.P. Potylitsin and S.A. Vorobiev, Phys. Lett. A 70 (1979) 447. [ 21 A.M. Taratin and S.A. Vorobiev, Phys. Stat. Sol. (b) 124 (1984) 641. [ 31 P.A. Doyle and P.S. Turner, Acta Crystallogr. A 24 (1968) 390.
403
115, number
PHYSICS
8
LETTERS
5 May 1986
A
VOLUME CAPTURE OF ULTRARELATIVISTIC ELECTRONS IN THE AXIAL CHANNELING REGIME A.M. TARATIN and S.A. VOROBIEV Nuclear Physics Institute, Received
1 August
Tomsk Polytechnical
1985; revised manuscript
Institute, received
634050 Tomsk, 30 December
USSR
1985; accepted
for publication
13 February
1986
Using computer simulation the volume capture effect of ultrarelativistic electrons in the axial channeling regime in crystals is investigated. The thickness, temperature and orientational dependences of the beam fraction captured in the channeling regime in the volume of the crystal are obtained.
The first experimental studies of the radiation of ultrarelativistic electrons in single crystals have shown a large contribution from the channeling radiation of electrons bound to the atomic rows [ 11. The electrons initially captured in the bound states at the incidence in a crystal are rapidly dechanneled due to the multiple scattering by the crystal electrons and nuclei. They cannot provide the observed large effect in the rather thick crystals in experiments [ 11. So one may suppose that the additional fraction of the electron beam may be captured in the channeling regime in the crystal volume. This process results from the multiple scattering of electrons in a crystal. A basic condition for capture of electrons in the channeling regime in a crystal volume is the existence of an essential fraction of particles with transverse energies close to a critical one Elc . This paper shows the results of a computer experiment for the investigation of the volume capture of electrons in the axial channeling regime. The calculations have been done with a model of a lattice of atomic rows [2]. We used a real atomic potential and the appropriate electron density obtained from the amplitude of atomic scattering [3]. The beam fraction of particles undergoing a volume capture has been calculated, and its thickness, temperature and orientational dependences have been obtained for the first time. The system of equations, describing the electron motion along the atomic rows in a crystal, has the form 0.375-9601/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
my d2xldt2 = -au(p)/i3x
7
my d2y/dt2 = -au(p)/ay , my d2z/dt2 = 0,
(1)
where m is the electron rest mass, y is the Lorentz factor, p = {x, y } and z are the tranverse and longitudinal coordinates of a particle, respectively, and u(p) is the crystal potential averaged along the crystallograph. ic axes, 2.5
‘) ,2 24 4 us(r) = - - c ai exp(-B--), a0 4 i=l~~ t uf
(2)
where u&r) is the averaged potential of a single atomic row, r, are the two-dimensional vectors of the atomic rows, the summation was taken over a maximum of 25 rows, a0 is the Bohr radius, d, is the distance between the atoms of a row, uI = flu1 is the twodimensional mean-square displacement of the crystal atoms from the row axis during thermal motion, Bi = hi/(2X)2, ai and bi are the expansion coefficients of the electron scattering factor. The electron trajectories in the crystal have been computed by numerical solution of system (1) with the potential (2). Multiple scattering of particles by crystal electrons and nuclei 401
Volume 115, number 8
PHYSICS LETTERS A
has been taken into account by the method described in ref. [2]. Transmission of a 1 GeV electron beam through a silicon crystalwith thickness 40 pm has been considered. The particle beam is incident at an angle \k 5 \k, relative to the (100) crystallographic direction, where \k, is a critical channeling angle (\k, = 0.418 mrad for (100) Si at T = 300 K). The particle state (the transverse coordinates (x ,y ) and velocity (uX , u,,)) is recorded in the end of every crystal layer of AZ = 4 I.tm. Also the transverse energy of particles El = i my (~1x2 + II;) t u(x ,y) is estimated. Fig. la shows the dependence on the orientation angle \k of the beam fraction of the electrons being captured in the bound states with El < 0 at the incidence at the crystal (z = 0, curve 1). The particles with EL < 0 are attributed to the states of bound motion with a single atomic row - the channeled electrons. It is seen that the electron capture decreases with increasing orientational angle \k, and at \k = 9, it is equal to zero. When a beam penetrates into the crystal, the number of channeled electrons de&eases due to multiple scattering by the crystal electrons and nuclei, i.e. particles are dechanneled and transfer into the states with El > 0 (quasichanneled electrons). However, as was noted by us [2], multiple scattering also leads to an inverse process; the capture of particles from the states with EL > 0 into the channeling regime takes place in the crystal volume. Fig. lb shows the z-dependences of the fraction of the channeled elec-
0
0.2
0,4 0 UI(mrad)-
trons captured in the channeling regime in the crystal volume. Results are shown for four values of the incident angle \k. The volume capture is maximum at a crystal depth which increases with increasing ?Ir. The maximum number of captured electrons is also changed, achieving a maximum value of 16.5% at an orientational angle \k = 0.12 mrad. The volume capture occurs from the over-barrier states with transverse energy EL z 0 near the potential barrier. The initial population of these states is maximum at orientation angle \k = (0.2-0.3)\kc, which stipulates a maximum of the obtained orientational dependence. At large orientational angles, \k 5 qc, the initial population of the near-barrier states with EL a 0 is small. After penetrating into the crystal, a definite fraction of the particle beam finds itself in a near-barrier region due to the diffusion process. This effect explains the obtained orientational dependence (curve 2 in fig. la). We have also considered the temperature dependence of the volume capture. Fig. 2 shows the z-dependences of the fraction of channeled electrons at two crystal temperatures: T = 0 K (curves 1,l ‘) and T = 540 K (curves 2,2’). Curves 1,2 are for the electrons captured in a crystal volume, and curves 1’,2’ are for the electrons captured in the channeling regime at the incidence into the crystal. The number of particles captured in the channeling regime at the incidence in a crystal reduces quickly, i.e. the electrons are dechanneled. The dechanneling grows with increasing temperature because the transverse dimension of the nuclear scattering region is increased. The volume capture of electrons slightly increases with temperature rise (curves 1,2). This is a consequence of increased nuclear scattering also.
20t oJrn,40
Fig. 1. (a), Dependence of the fraction of channeled electrons (in the states with ,!Z’*< 0) in ( 100) Si on the orientation angle for a 1 GeV electron beam: (1) particles captured at incidence at the crystal (z = 0), (2) particles captured in the crystal volume (in maximum of the depth dependence, see (b)). (b) Dependence of the fraction of channeled electrons in (100) Si captured in the crystal volume, on the crystal depth at different orientation angles, (1) q = 0.12-0.2 mrad; (3) * = 0.3 mrad; (4) q = 0.4 mrad. Crystal temperature T = 300 K.
402
5 May 1986
Fig. 2. Dependence of the fraction of channeled electrons on the crystal depth at different temperatures, (1 ,l’) T = 0 K, (2,2’) T= 540 K. Curve (1,2) electrons are captured in a crystal volume, curve (2,2’) electrons are captured at incidence in the crystal. The incident angle q = 0.1 mrad.
Volume 115, number 8
PHYSICS LETTERS A
This computer experiment shows that multiple scattering of particles by the crystal electrons and nuclei cause both dechanneling of relativistic electrons from the bound states with the atomic rows, and the inverse effect of volume capture of electrons into the bound states from the quasi-channeled fraction of the beam. Moreover, the electrons captured in the crystal volume constitute the main fraction of channeled particles for crystal thickness larger than 10 pm. These electrons are responsible for the observed channeling radiation in experiments [ 11. The contribution of the electrons captured at the incidence in a crystal is important only for thin crystals. The volume capture ef-
5 May 1986
feet increases the effective crystal thickness at which bound state electrons contribute to the different processes accompanying the electron beam penetration into the crystal. References [ 1] B.N. Kalinin, V.V. Kaplin, A.P. Potylitsin and S.A. Vorobiev, Phys. Lett. A 70 (1979) 447. [ 21 A.M. Taratin and S.A. Vorobiev, Phys. Stat. Sol. (b) 124 (1984) 641. [ 31 P.A. Doyle and P.S. Turner, Acta Crystallogr. A 24 (1968) 390.
403