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“Volume reflection” of high-energy charged particles in quasi-channeling states in bent crystals
Volume 119, number 8
PHYSICS LETTERS A
12 January 1987
“VOLUME REFLECTION” OF HIGH-ENERGY CHARGED PARTICLES IN QUASI-CHANNELING STATES IN BENT CRYSTALS A.M. TARATIN’ Institute ofPhysics, University ofAarhus, DK-8000 Aarhus C, Denmark
and S.A. YOROBIEV Institute of Nuclear Physics. Tomsk Polytechnical Institute, 634050 Tomsk, USSR Received 27 October 1986; accepted for publication 13 November 1986
Planar quasi-channeling ofhigh-energy charged particles in bent crystal is considered. In the computer analysis ofparticle trajectories it is discovered that at a crystal bend far from the critical one quasi-channeled particles are deflected to the side opposite to the bend, i.e. a “volume reflection” of particles (positive and negative) by the field ofbent atomic planes takes place.
The deflection of high-energy charged particles in channeling states in bent crystals has been investigated in the series of experimental and theoretical works (see the review [1]). In the present paper the deflection of particles passed through a bent crystal in the planar quasi-channeling regime is considered. In the computer analysis it is discovered that quasichanneled particles are deflected by a bent crystal to the side opposite the crystal bend when are from the critical bend,toFc’~Fcr, where F~ is thewe centrifugal force acting on a particle and F~.is the critical F~for channeling (Fe. is determined by the maximal strength of the averaged field of atomic planes Fcr = eEmax).
Let the crystal be uniformly bent with radius
2~R
-.2:~ 2
/
/ // LLU
Z
x
a)
R
2~
b)
0
along a set of parallel atomic planes and let particles enter the crystal at sufficiently small angles tothe bent planes, O~.<1, that the approximation of the continuum potential is valid. The equation for the particle trajectory in the averaged field of the system of bent atomic planes, which has axial symmetry with respect to the with bending (see in polar coordinates (r,ço) the axis origin at fig. the 1), intersection of the axis
Fig. 1. Schematicpicture ofthe deflection of high-energycharged particles by the averaged field of atomic planes in a crystal bent with a radius R0 for the case ofsmall crystal bend, FclcFcr. (a) 0~=O,particles enter a crystalparallel to bent planes, (b) 00> O~, where 8~is the critical channeling angle. Here, (1) are channeled particles captured at the entrance (a) or in the crystal volume (a,b). 0R (2)for are (a)quasi-channeled and 26R for (b).particles, the deflection angle equals
Permanent address: Institute of Nuclear Physics, Tomsk Polytechnical Institute, 634050 Tomsk, USSR.
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
with the orbital plane has the form 425
Volume 119, number 8
PHYSICS LETTERS A
2{c2[E_
U(r)]2
~(r) =M$ dr r —
M21r2 — m 2c2 }
—
1/2 +
12 January 1987 a>
30 .Or
(1)
~
~ ~
where U( r) is the potential energy in the field, E and Mare the total energy and the component of angular momentum along the bending axis, m is the particle mass, c is the velocity of light, ~ is a constant. The values of the integrals of motion (E,M) are obtained from the initial conditions (r 0,v0) at the crystal surface, E=Eko+ U( r0), M= my0 I r~>
\~
/ I
V
ii!
/ 0 .00 0.00
I .00
2.00
‘~0.0 b)
A/ \1 \
~
20.0
—
/
/
.~
i \.~
...i
,~‘
/
A ‘~! I
,./
I .00
a..
2.00
_________________________
w
j
\
I00~
I
ro
+ (2E*/R0)(r_ro) —
[U(r)
—
U(ro)]} - 1/2,
(2)
50.0
where E’~’=Eko/3~I2. Characterizing the bend of a crystal by the centrifugal force F~= 2E*/R0 acting on a particle and measuring angles in2, units of the where U critical channeling angle O~=(U0IE*)h/ 0 is the depth ofthe planar potential, we obtain the equation of the trajectory on a form which is independent of particle energy ~‘(r,F~)=2~,2
-
E
•~ ‘
0.00
/
I
~
XR
I .00
2.00
250. dl I
,.•~
125.
2, (3) +Ueff(ro,Fc)—Ueff(r,Fc)]” where ço’=ço/O~, O~=O~/O~,Ueff(r,Fc)=U(r) —F~r+U~ 0(R0),U~0is a constant. Fig. 2 shows the
426
~‘
0
Jdr[UoO~2
effective potential Ueff(r,Fc) for positive and negative particles in a Si crystal bent along (110) planes at T= 293 K. The potential U( r) was calculated in the Moliere approximation for an atomic potential with thermal vibrations of the crystal atoms included [2]. Let us consider the case when the beam orientational angle 00=0 (fig. la). The quasi-channeled states correspond to the particles entering the channd at a distance larger than XR from the outer channel wall (see fig. 2), the potential of which defines the critical transverse energy for channeling in bent
-
0.00 0.00 01 STANCE
1.00
2.00
(x / d~)
Fig. 2. The effective potential in a Si crystal at T= 293 K bent along (110) planes with different bends F~,GeV/cm: (a) 0.1, (b)
0.5, (c) 2, (d) 6. The full (dashed) line is for positive (negative) particles. d0 is the interplanar distance (d0 =0.192 nm for (110) Si). crystals E~~(R0). In the radially periodic field of a bent crystal particles execute oscillatory motion. With increasing depth of penetration into the crystal the oscillations in v decrease and the average velocity v, becomes stable. Fig. 3 shows the calculated angular
Volume 119, number 8
PHYSICS LETTERS A
-1.00
0.00
1.00
2.00
-1.00
0.00
1.00
2.00
12 January 1987
—1.00
0.00
1.00
2.00
1.00
0.00
1.00
2.00
1.00
2.00
1.00
2.00
bI .~
c)
~
—1.00
0.00
1.00
__
—1.00
,
2.00
.n
________________
—1.00
0.00 ~d)
0.00 ANGLE ~
1.00
2.00
Fig. 3. Angular distributions of high-energy positive particles scattered in quasi-channeled states by the axially symmetric field of (110) bent planes of a Si crystal, T= 293 K, O~ = 0, at different crystal bends F 0, 0eV/cm: (a) 0.01, (b) 0.5, (c) 2, (d) 6.
distributions of high-energy positive particles scattered in quasi-channeled states by the field of (110) bent planes of a Si crystal at different bends F0, GeV/cm: (a) 0.01, (b) 0.5, (c) 2, (d) 6. For the small bend (fig. 3a) the 0R~ angular distribution presentsarea 0.800, i.e. particles narrow peak at to the crystal bend. Whenthe beam deflected opposite enters a crystal at O~> O~(fig. lb), the narrow peak is removed at an angle 20R from the initial direction, i.e. the deflection is doubled. In addition all particles are in quasi-channeled states and are deflected through this angle. The doubling of the angle of deflection is caused by the presence ofthe two trajectory branches which are symmetric with respect to the turning point in the effective potential and are distinguished by the sign of the radial velocity. We can consider the observed deflection as the “volume reflection” of particles by bent planes. The broadening of the angular distributions with increasing crystal bend (figs. 3b—3d) is explained by the increase of —
-1.00
0.00 AWGLE ~
Fig. 4. The same as in fig. 3 for high-energy negative particles.
the “reflecting” part of the effective potential. The angular distributions stretch to the bend side because the particles appear which have smaller radial velocities and spend longer times near the turning points. Fig. 4 shows the angular distributions for highenergy negative particles scattered in quasi-channeled being states by the fieldthe of bent crystal, the 3. conditions otherwise sameSi as for fig. The “volume reflection” at a small crystal bend takes place for negative particles also. However, the reflection angle 0R is only approximately half the value corresponding to positive particles. This is caused by the difference in Ueff( r,F~),its “reflecting” top for negative particles is more flat and they accomplish a longer path near turning points than positive partides. With increasing crystal bend the angular distributions stretch to the bend side also (figs. 4c, 4d), but the centre of gravity remains within the “reflected” part at 0,, <0, due to the wide “reflecting” top of the effectivepotential. The “volume reflection” angle °R is determined by the depth of the planar potential and its behaviour 427
Volume 119, number 8
PHYSICS LETTERS A
near the maximum. The dependence of °R on the crystal parameters is similar as for the critical channeling angle O~.The intense multiple scattering by crystal electrons and nuclei for quasi-channeled partides leads to a considerable broadening of the beam. In addition in a bent crystal near the turning points of particles in the effective potential the conditions exist for capture into the channeling regime through multiple scattering [3,41.However, as our calculations have been shown, the “volume reflection” at a small crystal bend is still observed through the maximum displacement in the angular distribution to the side opposite to the crystal bend. So, for positive partides at a beam orientation 0~>0,, the displacement angle is about 20,,.
428
12 January 1987
The authors wish to thank Professor J. Lindhard and also E. Bonderup, A.H. Sorensen, J.F. Bak, S.P. Møller, H.E. Schiøtt and E. Uggerhøj for their support and useful discussions on the topic.
References [1] R.A. Carrigan Jr. and W.M. Gibson, in: Coherent radiation sources (Springer, Berlin, 1985). [2]Yu.L. Pivovarov 256 (1981) 837. and S.A. Vorobiev, Doki. Akad. Nauk SSSR [3] V.A. Andreev et al., Soy. Phys. JETP Lett. 36 (1982) 415. [4] A.M. Taratin and S.A. Vorobiev, Phys. Lett. A 115 (1986) 398.
PHYSICS LETTERS A
12 January 1987
“VOLUME REFLECTION” OF HIGH-ENERGY CHARGED PARTICLES IN QUASI-CHANNELING STATES IN BENT CRYSTALS A.M. TARATIN’ Institute ofPhysics, University ofAarhus, DK-8000 Aarhus C, Denmark
and S.A. YOROBIEV Institute of Nuclear Physics. Tomsk Polytechnical Institute, 634050 Tomsk, USSR Received 27 October 1986; accepted for publication 13 November 1986
Planar quasi-channeling ofhigh-energy charged particles in bent crystal is considered. In the computer analysis ofparticle trajectories it is discovered that at a crystal bend far from the critical one quasi-channeled particles are deflected to the side opposite to the bend, i.e. a “volume reflection” of particles (positive and negative) by the field ofbent atomic planes takes place.
The deflection of high-energy charged particles in channeling states in bent crystals has been investigated in the series of experimental and theoretical works (see the review [1]). In the present paper the deflection of particles passed through a bent crystal in the planar quasi-channeling regime is considered. In the computer analysis it is discovered that quasichanneled particles are deflected by a bent crystal to the side opposite the crystal bend when are from the critical bend,toFc’~Fcr, where F~ is thewe centrifugal force acting on a particle and F~.is the critical F~for channeling (Fe. is determined by the maximal strength of the averaged field of atomic planes Fcr = eEmax).
Let the crystal be uniformly bent with radius
2~R
-.2:~ 2
/
/ // LLU
Z
x
a)
R
2~
b)
0
along a set of parallel atomic planes and let particles enter the crystal at sufficiently small angles tothe bent planes, O~.<1, that the approximation of the continuum potential is valid. The equation for the particle trajectory in the averaged field of the system of bent atomic planes, which has axial symmetry with respect to the with bending (see in polar coordinates (r,ço) the axis origin at fig. the 1), intersection of the axis
Fig. 1. Schematicpicture ofthe deflection of high-energycharged particles by the averaged field of atomic planes in a crystal bent with a radius R0 for the case ofsmall crystal bend, FclcFcr. (a) 0~=O,particles enter a crystalparallel to bent planes, (b) 00> O~, where 8~is the critical channeling angle. Here, (1) are channeled particles captured at the entrance (a) or in the crystal volume (a,b). 0R (2)for are (a)quasi-channeled and 26R for (b).particles, the deflection angle equals
Permanent address: Institute of Nuclear Physics, Tomsk Polytechnical Institute, 634050 Tomsk, USSR.
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
with the orbital plane has the form 425
Volume 119, number 8
PHYSICS LETTERS A
2{c2[E_
U(r)]2
~(r) =M$ dr r —
M21r2 — m 2c2 }
—
1/2 +
12 January 1987 a>
30 .Or
(1)
~
~ ~
where U( r) is the potential energy in the field, E and Mare the total energy and the component of angular momentum along the bending axis, m is the particle mass, c is the velocity of light, ~ is a constant. The values of the integrals of motion (E,M) are obtained from the initial conditions (r 0,v0) at the crystal surface, E=Eko+ U( r0), M= my0 I r~>
\~
/ I
V
ii!
/ 0 .00 0.00
I .00
2.00
‘~0.0 b)
A/ \1 \
~
20.0
—
/
/
.~
i \.~
...i
,~‘
/
A ‘~! I
,./
I .00
a..
2.00
_________________________
w
j
\
I00~
I
ro
+ (2E*/R0)(r_ro) —
[U(r)
—
U(ro)]} - 1/2,
(2)
50.0
where E’~’=Eko/3~I2. Characterizing the bend of a crystal by the centrifugal force F~= 2E*/R0 acting on a particle and measuring angles in2, units of the where U critical channeling angle O~=(U0IE*)h/ 0 is the depth ofthe planar potential, we obtain the equation of the trajectory on a form which is independent of particle energy ~‘(r,F~)=2~,2
-
E
•~ ‘
0.00
/
I
~
XR
I .00
2.00
250. dl I
,.•~
125.
2, (3) +Ueff(ro,Fc)—Ueff(r,Fc)]” where ço’=ço/O~, O~=O~/O~,Ueff(r,Fc)=U(r) —F~r+U~ 0(R0),U~0is a constant. Fig. 2 shows the
426
~‘
0
Jdr[UoO~2
effective potential Ueff(r,Fc) for positive and negative particles in a Si crystal bent along (110) planes at T= 293 K. The potential U( r) was calculated in the Moliere approximation for an atomic potential with thermal vibrations of the crystal atoms included [2]. Let us consider the case when the beam orientational angle 00=0 (fig. la). The quasi-channeled states correspond to the particles entering the channd at a distance larger than XR from the outer channel wall (see fig. 2), the potential of which defines the critical transverse energy for channeling in bent
-
0.00 0.00 01 STANCE
1.00
2.00
(x / d~)
Fig. 2. The effective potential in a Si crystal at T= 293 K bent along (110) planes with different bends F~,GeV/cm: (a) 0.1, (b)
0.5, (c) 2, (d) 6. The full (dashed) line is for positive (negative) particles. d0 is the interplanar distance (d0 =0.192 nm for (110) Si). crystals E~~(R0). In the radially periodic field of a bent crystal particles execute oscillatory motion. With increasing depth of penetration into the crystal the oscillations in v decrease and the average velocity v, becomes stable. Fig. 3 shows the calculated angular
Volume 119, number 8
PHYSICS LETTERS A
-1.00
0.00
1.00
2.00
-1.00
0.00
1.00
2.00
12 January 1987
—1.00
0.00
1.00
2.00
1.00
0.00
1.00
2.00
1.00
2.00
1.00
2.00
bI .~
c)
~
—1.00
0.00
1.00
__
—1.00
,
2.00
.n
________________
—1.00
0.00 ~d)
0.00 ANGLE ~
1.00
2.00
Fig. 3. Angular distributions of high-energy positive particles scattered in quasi-channeled states by the axially symmetric field of (110) bent planes of a Si crystal, T= 293 K, O~ = 0, at different crystal bends F 0, 0eV/cm: (a) 0.01, (b) 0.5, (c) 2, (d) 6.
distributions of high-energy positive particles scattered in quasi-channeled states by the field of (110) bent planes of a Si crystal at different bends F0, GeV/cm: (a) 0.01, (b) 0.5, (c) 2, (d) 6. For the small bend (fig. 3a) the 0R~ angular distribution presentsarea 0.800, i.e. particles narrow peak at to the crystal bend. Whenthe beam deflected opposite enters a crystal at O~> O~(fig. lb), the narrow peak is removed at an angle 20R from the initial direction, i.e. the deflection is doubled. In addition all particles are in quasi-channeled states and are deflected through this angle. The doubling of the angle of deflection is caused by the presence ofthe two trajectory branches which are symmetric with respect to the turning point in the effective potential and are distinguished by the sign of the radial velocity. We can consider the observed deflection as the “volume reflection” of particles by bent planes. The broadening of the angular distributions with increasing crystal bend (figs. 3b—3d) is explained by the increase of —
-1.00
0.00 AWGLE ~
Fig. 4. The same as in fig. 3 for high-energy negative particles.
the “reflecting” part of the effective potential. The angular distributions stretch to the bend side because the particles appear which have smaller radial velocities and spend longer times near the turning points. Fig. 4 shows the angular distributions for highenergy negative particles scattered in quasi-channeled being states by the fieldthe of bent crystal, the 3. conditions otherwise sameSi as for fig. The “volume reflection” at a small crystal bend takes place for negative particles also. However, the reflection angle 0R is only approximately half the value corresponding to positive particles. This is caused by the difference in Ueff( r,F~),its “reflecting” top for negative particles is more flat and they accomplish a longer path near turning points than positive partides. With increasing crystal bend the angular distributions stretch to the bend side also (figs. 4c, 4d), but the centre of gravity remains within the “reflected” part at 0,, <0, due to the wide “reflecting” top of the effectivepotential. The “volume reflection” angle °R is determined by the depth of the planar potential and its behaviour 427
Volume 119, number 8
PHYSICS LETTERS A
near the maximum. The dependence of °R on the crystal parameters is similar as for the critical channeling angle O~.The intense multiple scattering by crystal electrons and nuclei for quasi-channeled partides leads to a considerable broadening of the beam. In addition in a bent crystal near the turning points of particles in the effective potential the conditions exist for capture into the channeling regime through multiple scattering [3,41.However, as our calculations have been shown, the “volume reflection” at a small crystal bend is still observed through the maximum displacement in the angular distribution to the side opposite to the crystal bend. So, for positive partides at a beam orientation 0~>0,, the displacement angle is about 20,,.
428
12 January 1987
The authors wish to thank Professor J. Lindhard and also E. Bonderup, A.H. Sorensen, J.F. Bak, S.P. Møller, H.E. Schiøtt and E. Uggerhøj for their support and useful discussions on the topic.
References [1] R.A. Carrigan Jr. and W.M. Gibson, in: Coherent radiation sources (Springer, Berlin, 1985). [2]Yu.L. Pivovarov 256 (1981) 837. and S.A. Vorobiev, Doki. Akad. Nauk SSSR [3] V.A. Andreev et al., Soy. Phys. JETP Lett. 36 (1982) 415. [4] A.M. Taratin and S.A. Vorobiev, Phys. Lett. A 115 (1986) 398.