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Planar dechanneling of GeV particles in a bent crystal
Nuclear Instruments and Methods 189 (1981) 6 0 9 - 6 1 4 North-Holland Publishing Company
609
PLANAR DECHANNELING OF GeV PARTICLES IN A BENT CRYSTAL Hiroshi KUDO Institute of Applied Physics, University of Tsukuba, Ibaraki 305, Japan Received 15 April 1981
Planar dechanneling of GeV particles in a bent crystal has been analysed theoretically on the basis of a modified continuum model of dechanneling. Dechanneling probabilities have been calculated for low-index planes in bent Si and Ge crystals, and the efficiencies of the channeled steering of a GeV particle beam for typical bendings of crystals have been discussed.
1. Introduction The possibility of the channeled steering of a GeV particle beam by a bent crystal has been discussed first by Tsyganov [ 1] on a simple dechanneling model [2], in which the dechanneling is assumed to occur when the centrifugal force caused by the curvature of the channel is larger than the maximal electric field acting on channeled particles from crystal atoms. Recently, this effect has been observed at Dubna for an 8.4 GeV proton beam in a bent Si crystal [3]. It is suggested that the steering effect should provide some technical applications to high-energy experiments [4]. Therefore, the detailed analysis of the steering effect based on a ref'med dechanneling model, in which transverse motions of channeled particles are taken into account, is of practical interest. In the present paper, the planar dechanneling in a bent crystal is analysed on the basis of the modified continuum model of dechanneling in distorted crystals, which has been successfully applied to the dechanneling by dislocations in crystals [5].
where y(z) is the distance of the particle from the center of the planar channel at Z = z with the Z-axis being taken to be the running direction of the particle along the channel, IV(y) is the potential given below and u±(Z) is the displacement of atoms perpendicular to the channel at Z. The planar continuum potential V(yp) at a distance yp from a single atomic plane is given by, V(yp) = 2 ¢rnZ1Z2 e2 aT ~ ( y p/aT) ,
(2)
where n is the areal density of atoms in the atomic plane, e the electronic charge, Z1 the charge number of the incident particle, Z2 the atomic number of the crystal atom, aT=O.8853aoZ~ 1/3 is the ThomasFermi screening distance for the fully ionized particle with a0 being the Bohr radius, and ~(yp/aT) is obtained by integrating a screening function of Thomas-Fermi type [6]. The Moli6re approximation is available to calculate eq. (2). By using eq. (2), Iv(y) in eq. (1) can be generally written oo
Ivo,
[V(Kmdp[2 + y ) + V(Kmdp/2 - y) m=l
2. Dechanneling in a bent crystal - 2 V(K m dp/2)] , In the modified continuum model, the trajectory of a channeled particle in a distorted planar channel is described as a modified trajectory in the otherwise normal planar channel. For particles of relativistic momentum p and velocity o, the equation of trajectory is written
d2y_ dz 2
1 pv
Fd2uil dy
[_dZ 2 Jz=z
(1)
0029-554X/81/0000-0000/$02.50 © 1981 North-Holland
(3)
where dp is the interplanar distance of the channel, and the summation with respect to "m is taken over the pairs of atomic planes which are symmetrically located at y --- +-.Kmdp/2. Therefore, K1 = 1 and Km for m t> 2 depends o n the structure of the crystal lattice. For the case of a bending radius R, eq. (1)
610
H, Kudo / Planar dechanneling of Ge V particles
becomes d2y _
10 GeV H° ( p v = 10.86 GeV)
1
dz 2
d
pv dy
[W(y ) + (po/R )Yl •
(4)
Eq. (4) means that a channeled particle experiences the modified potential field W(y) + (pv[R)y in a bent crystal. The modified potential is chosen to be 0 at y = 0 for simplicity. Fig. 1 shows W(y), (pv[R)y, and the modified potential W(y) + (po]R)y for the case of 10 GeV H + in Si(110) which is bent with R = 10 cm. The critical distance of approach. Ye to the planes is assumed to be _1 Yc-~dp-a
"--~ Si (110)
f= cm
T .
(5)
The channeled trajectories only exist at -Ye < Y
(6)
and is also indicated in fig. 1. Ya is a function of pu/R. Fig. 2 shows examples of the modified trajectories of 10 GeV H ÷ in Si(110) where the region Z > 0 is bent with R = 10 cm. Dechanneling of particle occurs when y < -Yc after being modified its transverse motion in the bent region by the abrupt increase of the potential by (pv/R)y at Z = 0. Therefore, for the par-
0.4 0.2
/''",
o<
-0.2
_0, r ,o,
i
~ ;,"
Jt. .. . _ . Y':
/ •
_
,/"
i
0 li
l
-0.8| ..........-20'. . . . . . . . . . . . .-I'(3 ..
(~
~" .... I0
2~0
Fig. 2. Modified channeled trajectories o f 10 GeV H+ incident on a b e n t region at Z > 0 in S i ( l l 0 ) , where R = 10 em.
ticle of transverse energy E± and of position y =Yi at the entrance to the bent region, the dechanneling condition is written E± + (pv/R ) Yi > W(-Yc) - (pv/R ) Y e .
po/R > E c / y c .
20
(9)
Eq. (9) corresponds to the dechanneling criterion used in the simple treatment of the dechanneling [ 1 ]. Next, we consider the reverse process in which channeled particles enter the non-bent region from the bent region. In the bent region, the maximal transverse kinetic energy Ek of the channeled particle at y is written
lOom
~+ls t~
Ek = [ E c - ( p o / R ) Y c l - D D ' ) + ( P o / R ) Y ] . dp/2 (0.96)
-dp/2 -0.8
(8)
where e = EaJEc is the normalized transverse energy. For the case of particles running along the center of the channel, i.e., e =Yi = 0, eq. (8) reduces to
Si (110)
p v / R = 1.086 G e V / c m
I,~--
(7)
By using the critical energy for channeling E c = W(+yc), eq. (7) becomes e + (po/R)(Yi + Yc)/Ec > 1 ,
10 GeV H ' ( p v = 1 0 . 8 6 GeV) - -
l.- Z (iam)
Ya OJ. distance
0.8 y (~)
-tO
Fig. 1. Modified planar potential Wry) + (pv/R)y f o r 10 GeV I-I* in S i ( l l 0 ) which is b e n t with R = 10 cm.
(10)
At the entrance to the non-bent region, the maximal transverse energy Em of the particle at y becomes E m =E k + W(y) =E c - 0" +Yc)pO/R.
(11)
Consequently, dechanneling never occurs in this reverse process because E m < E c at lYl
11. Kudo / Planar dechanneling o f Ge V particles
3. Dechanneling probability In the calculation of dechanneling probabilities, the following two incident conditions have to be considered dependently on the bending conditions of the crystal.
3.1. Parallel incidence When the crystal is bent including its upstream end, the parallel particle beam is uniformly incident on the bent region. The bending of the crystal in forming an arc is an example in this case. In such parallel incidence, Yi and e are related by e = l¢(yi)/ E c in eq. (8) and the dechanneling probability Qp is simply written with the aid of fig. 1 :
Qp(po/R ) = (Yc - Ya)/2yc •
(12)
Qp becomes 50% at Ya = 0 which is equivalent to
po/R = Ec/y c from eq. (6). In addition, Qp becomes 100% at pv/R =(dW[dy)y=y c as understood from fig. 1.
3.2. Statistical equilibrium (SE) incidence When the crystal is bent excluding its upstream end, channeled particles run through the non-bent region before entering the bent region. It may be adequate to assume the statistical equilibrium of channeled particles [7] at the entrance to the bent region, so that the spatial distribution g(e,yi)dy i of particles of e is given by, g(e, Yi) dYi = C(e)[e - l¥(yi)/Ec]-1/2 d y i ,
(13)
where C(e) is a normalization constant. The dechanneling probability P(e, po/R) for particles of e is calculated from eqs. (8) and (13). The dechanneling probability Qs in such SE incidence is given by, 1
Qs(po/R ) = f P ( e , po/R )F(e) de,
(14)
o
where F(e)de is the distribution of particles with respect to e. F(e)de may be assumed to be the initial distribution function on the crystal surface, which is given by, F(e) de = (2yc) -1 dYs,
(15)
where e a n d y s are related by e = W(,Vs)/Ec. The Zl-scaling of Qp, P and Qs is obtainable. Since W(y) is proportional to Zl, the dechanneling condi-
611
tion of eq. (8) for the case of Z1 = N a n d pv/R can be regarded as that for the case of Z1 = 1 and pv/RN. Therefore, Qp(pv/R) for Z I = N is equal to Qp(pv/RN) for ZI---1. Taking into account that g(e,yi) and F(e) are independent of Z1, we can obtain the same Zx-scaling for P and Qs. It should be noted that the Zl-scaling has been derived when the particles are fully ionized. When the particles are partially ionized ions, aT should depend on Z1 [6] and W(.v) is no longer proportional to Z1.
4. Calculated results and discussion Qp and Qs have been calculated for (110), (100), and (111) planar channels in the static lattices of Si and Ge crystals. There are two interplanar distances for such diamond-type (111) channel, of which large and small ones are denoted by (111)-L and (111)-S, respectively. The summation in eq. (3) was taken over 4 pairs of nearest planes to the channel throughout the present calculation in order to obtain correct values of Qp and Qs within the accuracy of 0.1%. Fig. 3 shows Qp vs po/R curves for the planar channels, calculated for the case of Z1 = 1. Fig. 4 shows P vs pv/R curves for various e for the case of Z1 = 1 in Si(110). The oscillating amplitude s and e is related by e = I¢(s)[Ec. For e = 0, P becomes 100% at po/R > EcJyc (= 1.77) and otherwise it becomes 0. Fig. 5 shows Qs vs pv[R curves for the planar channels, calculated for the case of Z1 = 1. From eq. (9), a rough measure of the efficiency of channeled steering is the value ofEc/y c [1]. The values of Edyc of the planar channels in Si and Ge are summarized in table 1. At pu/R = Ec/Yc, Qv becomes 50% as pointed out in sec. 3 and Qs becomes about 65%. In fig. 3, Qp(po/R) for Si(lll)-S is larger than that for Si(111)-L, while Qp(pv/R) for Ge(111)-S is smaller than that for G e ( l l l ) - L in the region of ,po/R > 1.2 GeV/cm. The similar reversion o f Qs(PO/R) for those channels is also seen in fig. 5. This situation is consistent with the efficiencies of channeled steering estimated from the values of Ec/yc in table 1. From eqs. (2), (3) and (5), EcJyc is a function of dp[aT. Fig. 6 shows the universal curves of (2rrnZ1Z2e2)-lEcJy c vs D[aT for diamond-type (111)-L and (111)-S, where D is the interplanar distance of (111)-S. These two curves intersect at D[aT = 5.00 where the reversion of Ec]yc occurs for the two channels. The origin of the reversion of the dechanneling probabilities for those channels in figs. 3 and 5 may be clear because
612
H. Kudo / Planar dechanneling of Ge V particles
50 [',,
PLANAR DECHANNELING IN BENT Si AND Ge (parallel incidence)
,,,,@
100°/,
o~40
80
~L 60
20
40
20
I'
O0
i
i
i
2
3
4
i
50
pv, R (GeVlcm)
Fig. 3. Dechanneling probabilities Qp in bent Si and Ge crystals for parallel incidence, calculated for particles of charge e. This figure is available when the crystal is bent including its upstream end. For particles of charge Zle, the scale of the abscissa is replaced by po[RZ 1 (GeV/em).
the values o f D/aT for Si and Ge are 4.03 and 5.53, respectively as indicated in fig. 6. As a result of the Zl-scaling derived in sec. 3, figs. 3, 4 and 5 are applicable to the case o f Zx 4:1 b y replacing the abscissas pv/R by po/RZx. Fig. 7 shows the comparison between Qp and Qs for the case o f Si(110). A negligible difference is seen
s
0.96 ~
(dpl2)
~I00 (%.
.
.
.
between them at small po/R compared with Ec/y c. However, Qs exceeds Qp in the otherwise region. These trends are generally seen for all the planar channels in the present calculation. It may be concluded that the bending o f the upstream end o f the crystal should provide small dechanneling, i.e., high efficiency in the channeled steering o f a particle beam
5i (110) c , h a ~ / --.-7
~
.
Table 1 Ec/Y c of low-index planar channels in the static lattices of Si and Ge
,,~ 80 k a.
60
Channel
dp (A)
Yc (A)
E c (eV)
Ec/Y c (GeV/cm)
Si(110) Si(111)-L Si(1 ll)-S Si(100)
1.92 2.35 0.78 1.36
0.766 0.982 0.198 0.484
13.5 15.9 2.50 6.64
1.77 1.62 1.27 1.37
Ge(110) Ge(lll)-L Ge(lll)-S Ge(100)
2.01 2.46 0.82 1.42
0.855 1.08 0.262 0.562
27.6 29.1 7.84 14.8
3.22 2.70 2.99 2.64
z w z
~ zo z a
pvIR (GeVIcm)
Fig. 4. Calculated dechanneling probabilities P(e, po/R) as a function of pu/R in a bent Si(110) for particles of charge e. e is indicated together with the oscillating amplitude s.
H. Kudo / Planar dechanneling of Ge V particles PLANAR DECHANNELING IN BENT
613
$i AND Ge ( S E i n c i d e n c e ) 100%
50'
80
~ 4o _1
60
m < 30 0 (z: o. (9 z -.J 20
t,0
ply ~
Z -r u
20
~ lO
0
I
'
' pvl'R
0
(GeV/cm)
Fig. 5. Dechanneling probabilities Qs in bent Si and Ge crystals for SE incidence, calculated for particles of charge e. This figure is available when the crystal is bent excluding its upstream end. For particles of charge Zle, the scale of the abscissa is replaced by pv/RZ1 (GeV/cm).
except for the case of small pu/R compared with Ec/y c. In addition, the dechanneling o f channeled particles running in a bent planar channel occurs only when they are incident on the region where the bending radius becomes small, as discussed in sec. 2. Therefore, the most efficient channeled steering may be achieved by using the crystal which is bent in forming an arc.
(1i 1~-ID s (111)-L
In the experiments at Dubna [3], pv/R = 0 . 2 4 GeV/cm for the maximal bending of the downstream end o f Si(111). Taking into account the widths 2Yc o f Si(1 11)-L and Si(1 11)-S channels, we obtain the averaged value ~)s = 8.4% from fig. 5. This value reasonably explains the strong channeled steering observed at Dubna. The efficiency of the channeled steering is improved by a factor o f about 2 - 3 at the practical region of pv/R when Si is replaced b y Ge, as seen in figs. 3 and 5. Qs reduces to 3.0% when Ge(111) is used for the case at Dubna.
3D
0.2~
100[
........
I
o=
-
~" I " ' ~
5i (110)
.
,
_. ....
_(Ij 9_-s_..... ,.- 7
--
8
I0
~
'
Qp
'~ o.1,.o ,'
I i
o~-~,
2
~
6 D/a
T
p v l R (GeVlcm)
Fig. 6. Universal curves of (21rnZlZ2e2)-lEc/Yc vs D/a T calculated for diamond-type ( l l l ) - L and ( l l l ) - S channels, where D is the interplanar distance of the (111)-S channel.
Fig. 7. Comparison between Qp and Qs for the case of a bent Si(ll0) and Z 1 = 1.
614
H. Kudo / Planar dechanneling of Ge V particles
5. Conclusion
References
The analysis based, on the modified continuum model of dechanneling concludes that the dechanneling of channeled particles running in a bent crystal occurs only when they enter the region where the bending radius becomes small, and that the dechanneling probability is given as a function of po/RZ1. The best efficiency o f the channeled steering may be obtained for a given steering angle when the crystal is bent in forming an arc.
[1] E.N. Tsyganov, Fermilab TM-682, TM-684, Batavia (1976). [21 Y. Qu6r~, Phys. Stat. Sol. 30 (1968) 713. [3] A.F. Elishev et al. Phys. Lett. 88B (1979) 387. [4] Physics Today, Search and discovery (May, 1980). [5] H. Kudo, Nucl. Instr. and Meth. 170 (1980) 129, and references therein. [6] D.S. GemmeU, Rev. Mod. Phys. 46 (1974) 129. [7] J. Lindhard, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 34, no. 14 (1965).
I would like to express sincere thanks to Prof. M. Mannami of Kyoto University for his critical reading of the manuscript.
609
PLANAR DECHANNELING OF GeV PARTICLES IN A BENT CRYSTAL Hiroshi KUDO Institute of Applied Physics, University of Tsukuba, Ibaraki 305, Japan Received 15 April 1981
Planar dechanneling of GeV particles in a bent crystal has been analysed theoretically on the basis of a modified continuum model of dechanneling. Dechanneling probabilities have been calculated for low-index planes in bent Si and Ge crystals, and the efficiencies of the channeled steering of a GeV particle beam for typical bendings of crystals have been discussed.
1. Introduction The possibility of the channeled steering of a GeV particle beam by a bent crystal has been discussed first by Tsyganov [ 1] on a simple dechanneling model [2], in which the dechanneling is assumed to occur when the centrifugal force caused by the curvature of the channel is larger than the maximal electric field acting on channeled particles from crystal atoms. Recently, this effect has been observed at Dubna for an 8.4 GeV proton beam in a bent Si crystal [3]. It is suggested that the steering effect should provide some technical applications to high-energy experiments [4]. Therefore, the detailed analysis of the steering effect based on a ref'med dechanneling model, in which transverse motions of channeled particles are taken into account, is of practical interest. In the present paper, the planar dechanneling in a bent crystal is analysed on the basis of the modified continuum model of dechanneling in distorted crystals, which has been successfully applied to the dechanneling by dislocations in crystals [5].
where y(z) is the distance of the particle from the center of the planar channel at Z = z with the Z-axis being taken to be the running direction of the particle along the channel, IV(y) is the potential given below and u±(Z) is the displacement of atoms perpendicular to the channel at Z. The planar continuum potential V(yp) at a distance yp from a single atomic plane is given by, V(yp) = 2 ¢rnZ1Z2 e2 aT ~ ( y p/aT) ,
(2)
where n is the areal density of atoms in the atomic plane, e the electronic charge, Z1 the charge number of the incident particle, Z2 the atomic number of the crystal atom, aT=O.8853aoZ~ 1/3 is the ThomasFermi screening distance for the fully ionized particle with a0 being the Bohr radius, and ~(yp/aT) is obtained by integrating a screening function of Thomas-Fermi type [6]. The Moli6re approximation is available to calculate eq. (2). By using eq. (2), Iv(y) in eq. (1) can be generally written oo
Ivo,
[V(Kmdp[2 + y ) + V(Kmdp/2 - y) m=l
2. Dechanneling in a bent crystal - 2 V(K m dp/2)] , In the modified continuum model, the trajectory of a channeled particle in a distorted planar channel is described as a modified trajectory in the otherwise normal planar channel. For particles of relativistic momentum p and velocity o, the equation of trajectory is written
d2y_ dz 2
1 pv
Fd2uil dy
[_dZ 2 Jz=z
(1)
0029-554X/81/0000-0000/$02.50 © 1981 North-Holland
(3)
where dp is the interplanar distance of the channel, and the summation with respect to "m is taken over the pairs of atomic planes which are symmetrically located at y --- +-.Kmdp/2. Therefore, K1 = 1 and Km for m t> 2 depends o n the structure of the crystal lattice. For the case of a bending radius R, eq. (1)
610
H, Kudo / Planar dechanneling of Ge V particles
becomes d2y _
10 GeV H° ( p v = 10.86 GeV)
1
dz 2
d
pv dy
[W(y ) + (po/R )Yl •
(4)
Eq. (4) means that a channeled particle experiences the modified potential field W(y) + (pv[R)y in a bent crystal. The modified potential is chosen to be 0 at y = 0 for simplicity. Fig. 1 shows W(y), (pv[R)y, and the modified potential W(y) + (po]R)y for the case of 10 GeV H + in Si(110) which is bent with R = 10 cm. The critical distance of approach. Ye to the planes is assumed to be _1 Yc-~dp-a
"--~ Si (110)
f= cm
T .
(5)
The channeled trajectories only exist at -Ye < Y
(6)
and is also indicated in fig. 1. Ya is a function of pu/R. Fig. 2 shows examples of the modified trajectories of 10 GeV H ÷ in Si(110) where the region Z > 0 is bent with R = 10 cm. Dechanneling of particle occurs when y < -Yc after being modified its transverse motion in the bent region by the abrupt increase of the potential by (pv/R)y at Z = 0. Therefore, for the par-
0.4 0.2
/''",
o<
-0.2
_0, r ,o,
i
~ ;,"
Jt. .. . _ . Y':
/ •
_
,/"
i
0 li
l
-0.8| ..........-20'. . . . . . . . . . . . .-I'(3 ..
(~
~" .... I0
2~0
Fig. 2. Modified channeled trajectories o f 10 GeV H+ incident on a b e n t region at Z > 0 in S i ( l l 0 ) , where R = 10 em.
ticle of transverse energy E± and of position y =Yi at the entrance to the bent region, the dechanneling condition is written E± + (pv/R ) Yi > W(-Yc) - (pv/R ) Y e .
po/R > E c / y c .
20
(9)
Eq. (9) corresponds to the dechanneling criterion used in the simple treatment of the dechanneling [ 1 ]. Next, we consider the reverse process in which channeled particles enter the non-bent region from the bent region. In the bent region, the maximal transverse kinetic energy Ek of the channeled particle at y is written
lOom
~+ls t~
Ek = [ E c - ( p o / R ) Y c l - D D ' ) + ( P o / R ) Y ] . dp/2 (0.96)
-dp/2 -0.8
(8)
where e = EaJEc is the normalized transverse energy. For the case of particles running along the center of the channel, i.e., e =Yi = 0, eq. (8) reduces to
Si (110)
p v / R = 1.086 G e V / c m
I,~--
(7)
By using the critical energy for channeling E c = W(+yc), eq. (7) becomes e + (po/R)(Yi + Yc)/Ec > 1 ,
10 GeV H ' ( p v = 1 0 . 8 6 GeV) - -
l.- Z (iam)
Ya OJ. distance
0.8 y (~)
-tO
Fig. 1. Modified planar potential Wry) + (pv/R)y f o r 10 GeV I-I* in S i ( l l 0 ) which is b e n t with R = 10 cm.
(10)
At the entrance to the non-bent region, the maximal transverse energy Em of the particle at y becomes E m =E k + W(y) =E c - 0" +Yc)pO/R.
(11)
Consequently, dechanneling never occurs in this reverse process because E m < E c at lYl
11. Kudo / Planar dechanneling o f Ge V particles
3. Dechanneling probability In the calculation of dechanneling probabilities, the following two incident conditions have to be considered dependently on the bending conditions of the crystal.
3.1. Parallel incidence When the crystal is bent including its upstream end, the parallel particle beam is uniformly incident on the bent region. The bending of the crystal in forming an arc is an example in this case. In such parallel incidence, Yi and e are related by e = l¢(yi)/ E c in eq. (8) and the dechanneling probability Qp is simply written with the aid of fig. 1 :
Qp(po/R ) = (Yc - Ya)/2yc •
(12)
Qp becomes 50% at Ya = 0 which is equivalent to
po/R = Ec/y c from eq. (6). In addition, Qp becomes 100% at pv/R =(dW[dy)y=y c as understood from fig. 1.
3.2. Statistical equilibrium (SE) incidence When the crystal is bent excluding its upstream end, channeled particles run through the non-bent region before entering the bent region. It may be adequate to assume the statistical equilibrium of channeled particles [7] at the entrance to the bent region, so that the spatial distribution g(e,yi)dy i of particles of e is given by, g(e, Yi) dYi = C(e)[e - l¥(yi)/Ec]-1/2 d y i ,
(13)
where C(e) is a normalization constant. The dechanneling probability P(e, po/R) for particles of e is calculated from eqs. (8) and (13). The dechanneling probability Qs in such SE incidence is given by, 1
Qs(po/R ) = f P ( e , po/R )F(e) de,
(14)
o
where F(e)de is the distribution of particles with respect to e. F(e)de may be assumed to be the initial distribution function on the crystal surface, which is given by, F(e) de = (2yc) -1 dYs,
(15)
where e a n d y s are related by e = W(,Vs)/Ec. The Zl-scaling of Qp, P and Qs is obtainable. Since W(y) is proportional to Zl, the dechanneling condi-
611
tion of eq. (8) for the case of Z1 = N a n d pv/R can be regarded as that for the case of Z1 = 1 and pv/RN. Therefore, Qp(pv/R) for Z I = N is equal to Qp(pv/RN) for ZI---1. Taking into account that g(e,yi) and F(e) are independent of Z1, we can obtain the same Zx-scaling for P and Qs. It should be noted that the Zl-scaling has been derived when the particles are fully ionized. When the particles are partially ionized ions, aT should depend on Z1 [6] and W(.v) is no longer proportional to Z1.
4. Calculated results and discussion Qp and Qs have been calculated for (110), (100), and (111) planar channels in the static lattices of Si and Ge crystals. There are two interplanar distances for such diamond-type (111) channel, of which large and small ones are denoted by (111)-L and (111)-S, respectively. The summation in eq. (3) was taken over 4 pairs of nearest planes to the channel throughout the present calculation in order to obtain correct values of Qp and Qs within the accuracy of 0.1%. Fig. 3 shows Qp vs po/R curves for the planar channels, calculated for the case of Z1 = 1. Fig. 4 shows P vs pv/R curves for various e for the case of Z1 = 1 in Si(110). The oscillating amplitude s and e is related by e = I¢(s)[Ec. For e = 0, P becomes 100% at po/R > EcJyc (= 1.77) and otherwise it becomes 0. Fig. 5 shows Qs vs pv[R curves for the planar channels, calculated for the case of Z1 = 1. From eq. (9), a rough measure of the efficiency of channeled steering is the value ofEc/y c [1]. The values of Edyc of the planar channels in Si and Ge are summarized in table 1. At pu/R = Ec/Yc, Qv becomes 50% as pointed out in sec. 3 and Qs becomes about 65%. In fig. 3, Qp(po/R) for Si(lll)-S is larger than that for Si(111)-L, while Qp(pv/R) for Ge(111)-S is smaller than that for G e ( l l l ) - L in the region of ,po/R > 1.2 GeV/cm. The similar reversion o f Qs(PO/R) for those channels is also seen in fig. 5. This situation is consistent with the efficiencies of channeled steering estimated from the values of Ec/yc in table 1. From eqs. (2), (3) and (5), EcJyc is a function of dp[aT. Fig. 6 shows the universal curves of (2rrnZ1Z2e2)-lEcJy c vs D[aT for diamond-type (111)-L and (111)-S, where D is the interplanar distance of (111)-S. These two curves intersect at D[aT = 5.00 where the reversion of Ec]yc occurs for the two channels. The origin of the reversion of the dechanneling probabilities for those channels in figs. 3 and 5 may be clear because
612
H. Kudo / Planar dechanneling of Ge V particles
50 [',,
PLANAR DECHANNELING IN BENT Si AND Ge (parallel incidence)
,,,,@
100°/,
o~40
80
~L 60
20
40
20
I'
O0
i
i
i
2
3
4
i
50
pv, R (GeVlcm)
Fig. 3. Dechanneling probabilities Qp in bent Si and Ge crystals for parallel incidence, calculated for particles of charge e. This figure is available when the crystal is bent including its upstream end. For particles of charge Zle, the scale of the abscissa is replaced by po[RZ 1 (GeV/em).
the values o f D/aT for Si and Ge are 4.03 and 5.53, respectively as indicated in fig. 6. As a result of the Zl-scaling derived in sec. 3, figs. 3, 4 and 5 are applicable to the case o f Zx 4:1 b y replacing the abscissas pv/R by po/RZx. Fig. 7 shows the comparison between Qp and Qs for the case o f Si(110). A negligible difference is seen
s
0.96 ~
(dpl2)
~I00 (%.
.
.
.
between them at small po/R compared with Ec/y c. However, Qs exceeds Qp in the otherwise region. These trends are generally seen for all the planar channels in the present calculation. It may be concluded that the bending o f the upstream end o f the crystal should provide small dechanneling, i.e., high efficiency in the channeled steering o f a particle beam
5i (110) c , h a ~ / --.-7
~
.
Table 1 Ec/Y c of low-index planar channels in the static lattices of Si and Ge
,,~ 80 k a.
60
Channel
dp (A)
Yc (A)
E c (eV)
Ec/Y c (GeV/cm)
Si(110) Si(111)-L Si(1 ll)-S Si(100)
1.92 2.35 0.78 1.36
0.766 0.982 0.198 0.484
13.5 15.9 2.50 6.64
1.77 1.62 1.27 1.37
Ge(110) Ge(lll)-L Ge(lll)-S Ge(100)
2.01 2.46 0.82 1.42
0.855 1.08 0.262 0.562
27.6 29.1 7.84 14.8
3.22 2.70 2.99 2.64
z w z
~ zo z a
pvIR (GeVIcm)
Fig. 4. Calculated dechanneling probabilities P(e, po/R) as a function of pu/R in a bent Si(110) for particles of charge e. e is indicated together with the oscillating amplitude s.
H. Kudo / Planar dechanneling of Ge V particles PLANAR DECHANNELING IN BENT
613
$i AND Ge ( S E i n c i d e n c e ) 100%
50'
80
~ 4o _1
60
m < 30 0 (z: o. (9 z -.J 20
t,0
ply ~
Z -r u
20
~ lO
0
I
'
' pvl'R
0
(GeV/cm)
Fig. 5. Dechanneling probabilities Qs in bent Si and Ge crystals for SE incidence, calculated for particles of charge e. This figure is available when the crystal is bent excluding its upstream end. For particles of charge Zle, the scale of the abscissa is replaced by pv/RZ1 (GeV/cm).
except for the case of small pu/R compared with Ec/y c. In addition, the dechanneling o f channeled particles running in a bent planar channel occurs only when they are incident on the region where the bending radius becomes small, as discussed in sec. 2. Therefore, the most efficient channeled steering may be achieved by using the crystal which is bent in forming an arc.
(1i 1~-ID s (111)-L
In the experiments at Dubna [3], pv/R = 0 . 2 4 GeV/cm for the maximal bending of the downstream end o f Si(111). Taking into account the widths 2Yc o f Si(1 11)-L and Si(1 11)-S channels, we obtain the averaged value ~)s = 8.4% from fig. 5. This value reasonably explains the strong channeled steering observed at Dubna. The efficiency of the channeled steering is improved by a factor o f about 2 - 3 at the practical region of pv/R when Si is replaced b y Ge, as seen in figs. 3 and 5. Qs reduces to 3.0% when Ge(111) is used for the case at Dubna.
3D
0.2~
100[
........
I
o=
-
~" I " ' ~
5i (110)
.
,
_. ....
_(Ij 9_-s_..... ,.- 7
--
8
I0
~
'
Qp
'~ o.1,.o ,'
I i
o~-~,
2
~
6 D/a
T
p v l R (GeVlcm)
Fig. 6. Universal curves of (21rnZlZ2e2)-lEc/Yc vs D/a T calculated for diamond-type ( l l l ) - L and ( l l l ) - S channels, where D is the interplanar distance of the (111)-S channel.
Fig. 7. Comparison between Qp and Qs for the case of a bent Si(ll0) and Z 1 = 1.
614
H. Kudo / Planar dechanneling of Ge V particles
5. Conclusion
References
The analysis based, on the modified continuum model of dechanneling concludes that the dechanneling of channeled particles running in a bent crystal occurs only when they enter the region where the bending radius becomes small, and that the dechanneling probability is given as a function of po/RZ1. The best efficiency o f the channeled steering may be obtained for a given steering angle when the crystal is bent in forming an arc.
[1] E.N. Tsyganov, Fermilab TM-682, TM-684, Batavia (1976). [21 Y. Qu6r~, Phys. Stat. Sol. 30 (1968) 713. [3] A.F. Elishev et al. Phys. Lett. 88B (1979) 387. [4] Physics Today, Search and discovery (May, 1980). [5] H. Kudo, Nucl. Instr. and Meth. 170 (1980) 129, and references therein. [6] D.S. GemmeU, Rev. Mod. Phys. 46 (1974) 129. [7] J. Lindhard, Kgl. Dan. Vid. Selsk. Mat. Fys. Medd. 34, no. 14 (1965).
I would like to express sincere thanks to Prof. M. Mannami of Kyoto University for his critical reading of the manuscript.