- Email: [email protected]
Ultrarelativistic electron and positron radiation in planar channeling
Nuclear Instruments and Methods 170 (1980) 2 7 - 2 8 ©North-Holland Publishing Company
27
ULTRARELATIVISTIC ELECTRON AND POSITRON RADIATION IN PLANAR CHANNELING
N.P. KALASHNIKOV and A.S. OLCHACK Moscow Physical Engineering Institute, Moscow, U.S.S.R. The coherent electromagnetic radiation from channeling electrons and positrons is given by similar expressions. However for the channeling positrons the close collisions are suppressed due to the fact that the positron wave function is exponentially small near the atoms of the crystal lattice. It follows that the coherent bremsstrahlung decreases for the channeling positrons. We have investigated the ultrarelativistic channeling electron and positron radiations, connected with the electromagnetic transitions from the continuum spectrum states to the quasi-bound spectrum states and between the different quasi-bound spectrum states. The radiation probabilities are calculated by using the model continuum planar potential. It is shown that the radiation from the channeling electrons is several orders of magnitude larger than the positron radiation, while the electron and positron radiation have similar characteristics such as frequency limitation and angular distribution of the radiation.
When the angle between the incoming particle momentum and the crystallographic plane is small ( 0 o ' ~ 0 L with 0 L the critical Lindhard channeling angle [1]), then the new radiation mechanism, connected with the transitions of the charged particles between the continuum free state and the channeling bound state or between two various channeling bound states, first studied in refs. 3, 4 and then in refs. 5 - 1 2 and some other works, is possible. The electromagnetic radiation probability of a charged particle, transiting from the initial state 1 to the final state 2 is given by [21
_ e 2 d03 d ~ dW12
871"
(1
g21
2
+ me + 2 E I E I ± 2E~ - cos 0 3,
cos 0.y E l zS~wl+ 03L'2j~] 03E1 ]
X {E 2 sin20v I:dxF~ (x) exp(-ikxx ) F 1(x)I "2
e2
dW12 = ~n le f f f ; 6 t f f l e x p ( - i k - r) d3rl 2 dak ts(g 1 - E 2 - 03) I-[ dP2iLi 03 i 2rt '
transverse energies. If we consider planar channeling, then for the majority of single crystals (,-)max ~ E l , when E1 < 10 GeV. Averaging over the photon and charged particle polarizations, we obtain (considering 03 ~ E1 for the planar channeling)
,
X --
exp(-ikxx)
(1 - s i : O ,
(1)
where e is the photon polarization vector, h = c = 1, e ~ = 1/137, i = (z,y) if planar channeling is treated, or i = z if axial channeling is treated, if2 and ffl are the final and the initial bispinor wave functions that are the solutions of the Dirac equation, 6t is the Dirac matrix. The energy conservation law can be satisfied only if 2 AEIE~ 03 ~ 03max me2 + 2 AE±EI (2) AEz is the difference between the initial and the final
ax
'
(3)
where Fl(X) and F2(x) are the initial and final wave functions of the transverse motion, that are the solutions of the one-dimensional equation
~ 5 + 2E(E± - U(x)) F(x) = 0 ,
(4)
where U(x) is the model planar channeling potential. Eq. (3) is the general result for planar channeling. For electrons channeling states coincide with the I. COHERENT RADIATION
N.P. Kalashnikov, A.S. Olchack / Ultrarelativisticradiation
28
negative values of transverse energy E± < 0 and their transition into the channeling state is accompanied by electromagnetic radiation, called quasi-characteristic radiation, which is the main mechanism of electron radiation in thin single crystals [10,11]. The electron quasi-characteristic radiation probability for the model atomic plane potential is given by [ 11 ] 2e 2 dco( dW(e-) = -~- V0 --o3 1
co) ~ax
'
(5)
where Vo is the potential depth and COmax= 2Vo E~/ Me2. At co > COmax quasi-characteristic radiation does not exist, but at co < COmax it is more intensive than coherent bremsstrahlung. There exists a strict connection between the emission angle and the radiation frequency of the electron 0.~ = m e E--~I
- 1 .
(6)
After the quasi-characteristic radiation process the electron falls into a channeling bound state. That is why the quasi-characteristic radiation o f electrons takes place only up to a length L C H ~ [W(e-)] -1 1 0 3 ~ A . The coherent bremsstrahlung probability increases with the thickness o f the single crystal, and it becomes comparable with the quasi-characteristic radiation probability at L ~ 1 cm. Positron radiation is connected with the transitions between various channeling bound states. The most probable are transitions between neighbouring levels [ 6 - 8 ] . The radiation intensity connected with these transitions has a maximum, sufficiently exceeding the amorphous medium bremsstrahlung intensity, in the region of the upper limit of the frequency spectrum COpeak= 2 (~kE±)neig EA
Z 1/3 (E~t3/: ~ me (~ea)2 \rn--~e!
(7) High frequency radiation (co >> Ca)peak) can arise at the transitions between far levels. If the initial positrons are uniformly distributed over the channeling levels, then the high frequency positron emission probability, calculated for the model atomic plane
potential [II] is equal to
dW(e.)=e2vodW (1 ~-
to ) ~ ) W-~ax m2 a
,
(8)
where a is the atomic lattice constant. In thin single crystals (L '~ 1 mm) high frequency positron radiation is weaker than electron quasi-characteristic radiation with co >> (")peak. The total intensity of channeling electron and positron radiation in a broad frequency region can be estimated according to [ 6 - 8 , 1 0 - 1 1 ] ~max
l =f o
dW
L -d~ w dw ;
I(e -)
2
l(e ÷)
7r: x/~-QoF 1 "mea
me (9)
Thus one can see that for thin targets electron radiation is more intense than positron radiation.
References
[1] J. Lindhard, Mat. Fys. Med. Dan. Vid. Selsk. 34, No. 1 (1965). [2] E.M. Lffshits and L.P. Pitaevskij, Quantum relativistic field theory, Part 1 (Nauka, Moscow, 1968). [3] N.P. Kalashnikov, E.A. Koptelov and M.I. Riazanov, Fiz. Tverd. Tela 14 (1972) 1211. [4] N.P. Kalashnikov, E.A. Koptelov and M.I. Riazanov, in Atomic collisions in solids, ed. S. Datz (Plenum Press, New York, 1975) p. 559. [5] A.A. Vorobiev, V.V. Kaplin and S.A. Vorobiev, Nucl. Instr. and Meth. 127 (1975) 265. [6] M.A. Kumakhov, Phys. Lett. A57 (1976) 17. [7] M.A. Kumakhov, Phys. Stat. Sol. B84 (1977) 41. [8] M.A. Kumakhov and R. WedeU, Phys. Stat. Sol. B84 (1977) 581. [91 V.G. Baryshevskij and I.Ja. Dubovskaja, Phys. Stat. Sol. B82 (1977) 403. [10] E.A. Koptelov and N.P. Kalashnikov, "Characteristic" ultrarelativistic electrons bremsstrahlung in single crystal, Preprint Inst. Nucl. Res., Acad. Sci. USSR, N-0054 (Moscow, 1977). [11] N.P. Kalashnikov and A,S. Olchack, Interactions of nuclear emissions with single crystals (Moscow Phys. Engin. Inst., Moscow, 1979). [12] A.A. Vorobiev, V.V. Kaplin and S.A. Vorobiev, ZhETP. Pis. Red. (JETP Lett.) 4 (22) (1978) 1340.
27
ULTRARELATIVISTIC ELECTRON AND POSITRON RADIATION IN PLANAR CHANNELING
N.P. KALASHNIKOV and A.S. OLCHACK Moscow Physical Engineering Institute, Moscow, U.S.S.R. The coherent electromagnetic radiation from channeling electrons and positrons is given by similar expressions. However for the channeling positrons the close collisions are suppressed due to the fact that the positron wave function is exponentially small near the atoms of the crystal lattice. It follows that the coherent bremsstrahlung decreases for the channeling positrons. We have investigated the ultrarelativistic channeling electron and positron radiations, connected with the electromagnetic transitions from the continuum spectrum states to the quasi-bound spectrum states and between the different quasi-bound spectrum states. The radiation probabilities are calculated by using the model continuum planar potential. It is shown that the radiation from the channeling electrons is several orders of magnitude larger than the positron radiation, while the electron and positron radiation have similar characteristics such as frequency limitation and angular distribution of the radiation.
When the angle between the incoming particle momentum and the crystallographic plane is small ( 0 o ' ~ 0 L with 0 L the critical Lindhard channeling angle [1]), then the new radiation mechanism, connected with the transitions of the charged particles between the continuum free state and the channeling bound state or between two various channeling bound states, first studied in refs. 3, 4 and then in refs. 5 - 1 2 and some other works, is possible. The electromagnetic radiation probability of a charged particle, transiting from the initial state 1 to the final state 2 is given by [21
_ e 2 d03 d ~ dW12
871"
(1
g21
2
+ me + 2 E I E I ± 2E~ - cos 0 3,
cos 0.y E l zS~wl+ 03L'2j~] 03E1 ]
X {E 2 sin20v I:dxF~ (x) exp(-ikxx ) F 1(x)I "2
e2
dW12 = ~n le f f f ; 6 t f f l e x p ( - i k - r) d3rl 2 dak ts(g 1 - E 2 - 03) I-[ dP2iLi 03 i 2rt '
transverse energies. If we consider planar channeling, then for the majority of single crystals (,-)max ~ E l , when E1 < 10 GeV. Averaging over the photon and charged particle polarizations, we obtain (considering 03 ~ E1 for the planar channeling)
,
X --
exp(-ikxx)
(1 - s i : O ,
(1)
where e is the photon polarization vector, h = c = 1, e ~ = 1/137, i = (z,y) if planar channeling is treated, or i = z if axial channeling is treated, if2 and ffl are the final and the initial bispinor wave functions that are the solutions of the Dirac equation, 6t is the Dirac matrix. The energy conservation law can be satisfied only if 2 AEIE~ 03 ~ 03max me2 + 2 AE±EI (2) AEz is the difference between the initial and the final
ax
'
(3)
where Fl(X) and F2(x) are the initial and final wave functions of the transverse motion, that are the solutions of the one-dimensional equation
~ 5 + 2E(E± - U(x)) F(x) = 0 ,
(4)
where U(x) is the model planar channeling potential. Eq. (3) is the general result for planar channeling. For electrons channeling states coincide with the I. COHERENT RADIATION
N.P. Kalashnikov, A.S. Olchack / Ultrarelativisticradiation
28
negative values of transverse energy E± < 0 and their transition into the channeling state is accompanied by electromagnetic radiation, called quasi-characteristic radiation, which is the main mechanism of electron radiation in thin single crystals [10,11]. The electron quasi-characteristic radiation probability for the model atomic plane potential is given by [ 11 ] 2e 2 dco( dW(e-) = -~- V0 --o3 1
co) ~ax
'
(5)
where Vo is the potential depth and COmax= 2Vo E~/ Me2. At co > COmax quasi-characteristic radiation does not exist, but at co < COmax it is more intensive than coherent bremsstrahlung. There exists a strict connection between the emission angle and the radiation frequency of the electron 0.~ = m e E--~I
- 1 .
(6)
After the quasi-characteristic radiation process the electron falls into a channeling bound state. That is why the quasi-characteristic radiation o f electrons takes place only up to a length L C H ~ [W(e-)] -1 1 0 3 ~ A . The coherent bremsstrahlung probability increases with the thickness o f the single crystal, and it becomes comparable with the quasi-characteristic radiation probability at L ~ 1 cm. Positron radiation is connected with the transitions between various channeling bound states. The most probable are transitions between neighbouring levels [ 6 - 8 ] . The radiation intensity connected with these transitions has a maximum, sufficiently exceeding the amorphous medium bremsstrahlung intensity, in the region of the upper limit of the frequency spectrum COpeak= 2 (~kE±)neig EA
Z 1/3 (E~t3/: ~ me (~ea)2 \rn--~e!
(7) High frequency radiation (co >> Ca)peak) can arise at the transitions between far levels. If the initial positrons are uniformly distributed over the channeling levels, then the high frequency positron emission probability, calculated for the model atomic plane
potential [II] is equal to
dW(e.)=e2vodW (1 ~-
to ) ~ ) W-~ax m2 a
,
(8)
where a is the atomic lattice constant. In thin single crystals (L '~ 1 mm) high frequency positron radiation is weaker than electron quasi-characteristic radiation with co >> (")peak. The total intensity of channeling electron and positron radiation in a broad frequency region can be estimated according to [ 6 - 8 , 1 0 - 1 1 ] ~max
l =f o
dW
L -d~ w dw ;
I(e -)
2
l(e ÷)
7r: x/~-QoF 1 "mea
me (9)
Thus one can see that for thin targets electron radiation is more intense than positron radiation.
References
[1] J. Lindhard, Mat. Fys. Med. Dan. Vid. Selsk. 34, No. 1 (1965). [2] E.M. Lffshits and L.P. Pitaevskij, Quantum relativistic field theory, Part 1 (Nauka, Moscow, 1968). [3] N.P. Kalashnikov, E.A. Koptelov and M.I. Riazanov, Fiz. Tverd. Tela 14 (1972) 1211. [4] N.P. Kalashnikov, E.A. Koptelov and M.I. Riazanov, in Atomic collisions in solids, ed. S. Datz (Plenum Press, New York, 1975) p. 559. [5] A.A. Vorobiev, V.V. Kaplin and S.A. Vorobiev, Nucl. Instr. and Meth. 127 (1975) 265. [6] M.A. Kumakhov, Phys. Lett. A57 (1976) 17. [7] M.A. Kumakhov, Phys. Stat. Sol. B84 (1977) 41. [8] M.A. Kumakhov and R. WedeU, Phys. Stat. Sol. B84 (1977) 581. [91 V.G. Baryshevskij and I.Ja. Dubovskaja, Phys. Stat. Sol. B82 (1977) 403. [10] E.A. Koptelov and N.P. Kalashnikov, "Characteristic" ultrarelativistic electrons bremsstrahlung in single crystal, Preprint Inst. Nucl. Res., Acad. Sci. USSR, N-0054 (Moscow, 1977). [11] N.P. Kalashnikov and A,S. Olchack, Interactions of nuclear emissions with single crystals (Moscow Phys. Engin. Inst., Moscow, 1979). [12] A.A. Vorobiev, V.V. Kaplin and S.A. Vorobiev, ZhETP. Pis. Red. (JETP Lett.) 4 (22) (1978) 1340.