- Email: [email protected]
Volume reflection efficiency for negative particles in bent crystals
Physics Letters B 765 (2017) 276–279
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Volume reflection efficiency for negative particles in bent crystals V.M. Biryukov Institute for High Energy Physics in National Research Center “Kurchatov Institute”, 142281 Protvino, Russia
a r t i c l e
i n f o
Article history: Received 11 September 2016 Received in revised form 23 November 2016 Accepted 14 December 2016 Available online 15 December 2016 Editor: M. Doser
a b s t r a c t We suggest a formula for the efficiency of a single volume reflection of negatively charged particles in bent crystal planes and compare it to recent experiments at SLAC, MAMI and CERN with electrons and negative pions in the energy range from 0.855 to 150 GeV in Si crystals. We show that Lindhard reversibility rule provides sufficient basis for quantitative understanding of these experiments. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
Keywords: Bent crystals Channelling Volume reflection
1. Introduction The increased interest to interaction of electrons with bent crystals has lead over the recent years to several important experiments on the so-called volume reflection [1–6]. While the full understanding of the process may be gained by the use of computer simulations taking into account both the electron motion in the field of bent crystal atomic planes and the scattering on the crystal constituents, it would be quite useful to obtain simple approximate formulas describing basic features of electron volume reflection in bent crystals. The volume reflection is caused by interaction of an incident particle with the potential of the bent crystal atomic planes, which give the particle an angular kick of the order of a critical channeling angle θC in the direction opposite to the crystal bending. Such a reflection is not 100%-efficient as some particles “stick” to the atomic planes (so called volume capture caused by scattering) instead of bouncing back. The “stuck” particles are trapped with the bent atomic planes in the channeled states and thus are steered away. The purpose of this paper is to understand the efficiency of volume reflection of electrons. We will show that the Lindhard reversibility rule [7] provides sufficient basis for its understanding. First, we follow the ideas of refs. [8,9] and repeat the derivation of the efficiency of particle reflection off the bent crystal plane. According to the Lindhard’ reversibility rule, the probability for a channeled particle to be scattered from a certain channeled state to certain unchanneled state equals the probability for the opposite
E-mail address: [email protected].
process. This rule was proven in the experiments with GeV-beams [10–12], where the state of each particle was detected by means of solid state detectors placed along the crystal length. In 1982 a PNPI experiment observed the effect of 1 GeV protons volume capture into channeling mode in the depth of the bent crystal [12], in the region where the particles’ trajectories are tangent to the crystallographic planes. 2. Formula for the reflection efficiency The reversibility rule allows to consider the feed-in (volume capture or rechanneling) and feed-out (dechanneling) processes from the unified point of view. The rate of particle transitions from the channeled to random states is set by the dechanneling length L D . Over the distance dz, the transition probability equals dz/ L D . According to the reversibility rule, the rate of the opposite transitions from random (over-barrier) to channeled (under-barrier) states is set by the same quantity L D . In a crystal bent with radius R, the non-channeled particle stays near the channel (on the phase plane) along the length of the order of R θC . Therefore, the probability of transition to channel over the all interaction time amounts to about R θC / L D . Respectively, the efficiency of volume reflection f VR is reduced by this value:
1 − f VR ≈
R θC LD
,
(1)
This formula, which is in fact the result of the reversibility rule, has agreed with the experiments where the volume-capture probabilities were measured at 1 GeV and 70 GeV [13]. Also, the 70-GeV experiment has confirmed the linear dependence of the probability on the crystal bending radius.
http://dx.doi.org/10.1016/j.physletb.2016.12.032 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
277
Table 1 The bending radius R, critical angle θC , measured dechanneling length L D , measured inefficiency of volume reflection, and the calculated efficiency, Eq. (1), for the experiments with electrons and negative pions at MAMI, SLAC and CERN.
2014 2015 2015 2015 2015 2015 2014 2009 2009
MAMI SLAC SLAC SLAC SLAC SLAC CERN CERN CERN
e− e− e− e− e− e− e−
π− π−
Si Si Si Si Si Si Si Si Si
(111) (111) (111) (111) (111) (111) (110) (110) (111)
E, GeV
R, cm
θC , μrad
Measured L D , μm
Measured 1 − f VR
R θC / L D
0.855 3.35 4.2 6.3 10.5 14 120 150 150
3.35 15 15 15 15 15 271 2279 1292
217 122 109 89 69 60 19 17 18
38 55.4 45.2 65.3 57.5 55.8 744∗ 930 930∗
23.3% 33% 27% 16% 16% 19.3% 4.5% 23.3% 17.26%
19% 31% 34% 19% 17% 15% 7%∗ 42% 25%∗
For protons, a typical probability of volume capture to stable channeled states is order of 0.2% at 70 GeV. For electrons, we expect the capture probability to be higher by two orders of magnitude because of stronger scattering. Channeled protons move (and scatter) mostly in electronic gas as they are trapped between atomic planes, while channeled electrons move across atomic plane (and hence scatter on nuclei). While R and θC are well known in each case, the dechanneling length L D for electrons is not so straightforward to obtain. Many experiments with electrons at low energies (MeV up to a few GeV) show rather flat dependence on energy, see refs. [1–4] and references therein, whereas a simple theory expects a linear dependence on energy [14]. This is why in the present paper we rely on measured L D values. Fig. 1. Inefficiency of a single volume reflection 1 − f VR in Si(111) as a function of energy. The SLAC data and theory, Eq. (1).
3. Comparison to the experiments Several experiments performed at MAMI, SLAC and CERN have measured both the efficiency and the dechanneling length of the particles in these crystals at the involved energies. These measurements were done in a broad energy range from sub GeV to 150 GeV with electrons and negative pions. Our idea in this paper is to use both measured quantities and check whether they agree with our formula (1). Table 1 shows the bending radius, critical angle, measured dechanneling length, measured efficiency of volume reflection and the calculated efficiency, Eq. (1), for the experiments with electrons and negative pions at SLAC [1,2], MAMI [3,4] and CERN [5, 6]. We take L D values from ref. [2] for SLAC, ref. [15] for CERN, and ref. [4] for MAMI. We use no free parameters at all, no fitting whatsoever. We just take the quantities R, θC , and L D reported in the papers and produce R θC / L D . This is compared then with the measured inefficiency of volume reflection 1 − f VR . Two cases are marked by (∗ ). The 120-GeV L D was not measured so we used the 150-GeV value scaled down linearly in proportion 120/150. For Si(111) at 150 GeV the L D value was not measured, so we tentatively assumed that is the same as for Si(110); actually it may differ somewhat. We see surprisingly good agreement between the measured inefficiencies of volume reflection and Eq. (1) in Table 1. On average, for the 9 experimental points of MAMI, SLAC and CERN experiments, the data are lower than prediction of Eq. (1) by a factor of 0.82:
1 − f VR ≈ 0.8 ·
R θC LD
(2)
For the SLAC data that is most complete, we can compare the experiment and the theory, R θC / L D , as a function of energy in Fig. 1. Notice that the comparison is with Eq. (1), not (2). Eq. (1) suggests a functional dependence on bending radius R. We believe, however, that real values should deviate from the linear R proportion of Eq. (1) both at too large R and at too small R.
Fig. 2. Inefficiency of volume reflection 1 − f VR as a function of the ratio R / R C . The SLAC data for electrons and theory, Eq. (1). Straight line is a guide to the eye.
At large R the efficiency cannot grow above 100% and thus the real values start to saturate below the values given by Eq. (1). At small R, that is R comparable to the critical radius R C , the effective potential well shrinks (due to centrifugal effect) thus starting to affect the L D value in Eq. (1) (and θC value to lesser extent). This should place the real values above the Eq. (1), at small R. There is no measured dependence on R for electrons. However, as the energy runs over a broad range with a constant R, the ratio R / R C spans over a broad range from small values where the potential well is distorted to large ones where the potential well is unchanged. This span for SLAC data can be seen in Fig. 2. The dependence of the volume reflection inefficiency, measured and calculated, on the R / R C ratio can be also plotted for CERN data, see Fig. 3. Notice some big differences between the experimental conditions of SLAC and MAMI versus CERN:
278
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
neling length L D behaviour is complicated. In the experimentally studied energy range from MeV up to a few GeV, L D is rather flat for a number of reasons. √ Then the reflection inefficiency in this range should follow 1/ E. At some high energy range, certainly 100 GeV and higher, L D should be a linear function of energy. Then the inefficiency should follow 1/ E 1.5 . Let us summarise this as
1 − f VR ∼ R / E 0.5 1 − f VR ∼ R / E
at low energies (MeV up to a few GeV)
1 .5
at higher energies (100 GeV domain)
One should remember that a typical R grows linearly with E. Only the ratio R / R C remains invariant. Therefore it is convenient to present the energy dependence with the use of nondimensional invariant parameter R / R C as Fig. 3. Inefficiency of volume reflection 1 − f VR as a function of the ratio R / R C . The CERN data for negative pions and electrons, and theory, Eq. (1). Straight line is a guide to the eye.
1 − f VR ∼ ( R / R C ) · 1 − f VR ∼ ( R / R C )/
√
√
E
at low energies (MeV up to a few GeV)
E
at higher energies (100 GeV domain) √
In the experiments considered above, the E factor varies as much as 13 as E goes from 0.855 to 150 GeV. The range of R / R C ratio has spanned from 3 to 100 in the variety of experiments on volume reflection. 4. Relation of Eq. (1) to a more general principle
Fig. 4. The ratio of MAMI, SLAC and CERN data to the predictions of Eq. (1) plotted as a function of energy.
(1) SLAC and MAMI data is for electrons, low energies (MeV up to a few GeV), and moderate R / R C = 14 to 27; (2) most of CERN data is for pions, 150 GeV, and high ratio of R / R C = 50 to 90. Different procedures for selection of reflected particles in the experiments may contribute further to the differences. Notice that the experimental errors in f VR and L D and some uncertainties in definition, especially for L D , are rather big in some cases. This adds to the scatter in the plot. Also, our model (1) is elementary and it would be too much to expect perfect agreement from it. Of course, a numerical coefficient (hopefully, of the order of magnitude of 1) would be natural in Eq. (1). As said above, on average the experimental data of MAMI, SLAC and CERN are lower than the prediction of Eq. (1) by a factor of 0.82. This ratio data-to-theory can be seen as a function of energy in Fig. 4. For low-GeV data of MAMI and SLAC the ratio is about 1.0. As the energy goes up by a factor of 200, from 0.855 to 150 GeV, this numerical factor in Eq. (2) decreases to 0.6 for CERN data on negative pions. The errors (mostly caused by the uncertainties in the L D measurements) are not shown here. The agreement between Eq. (1) and the data from three experimental groups means two things. One is the possibility to predict data, optimise applications, and understand the underlying physics. The second thing is that we demonstrate the consistency between three different experiments with the help of Eq. (1). The energy dependence of efficiency √is set by θC / L D term. While the critical angle θC behaviour ∼ 1/ E is clear, the dechan-
This agreement supports the basic idea that the Lindhard reversibility rule provides a good hint in understanding the efficiency of volume reflection. This link between the reversibility principle and the efficiency of volume reflection can be explored further. From Eq. (1) based on the reversibility rule, one can derive an interesting consequence for the efficiency of multiple volume reflection (MVR). If efficiency of a single reflection is f VR , then after N volume reflections the efficiency is ( f VR ) N . With Eq. (1) that makes
( f VR ) N = (1 − R θC / L D ) N = exp(− N R θC / L D ) where we approximate by exponent because f VR is close to 1. The angle of a single volume reflection is on the order of θC . After N reflections, the beam is bent by θ = N θC . That is, for beam bending a significant angle θ with MVR, the bending efficiency must decline with θ like
f MVR = ( f VR ) N = exp(− N R θC / L D ) = exp(− R θ/ L D ) The bending efficiency of bent crystal channeling (BCC) declines with bending angle θ in exactly the same way
f BCC = exp(− R θ/ L D ) = exp(− L / L D ) over distance L = R θ . This means that one cannot outsmart the nature with MVR and bend beam with efficiency higher than that of BCC. This argument can be treated vice versa, in a sense. From the postulate that the efficiency of the multiple volume reflection could not exceed the efficiency of bent crystal channeling at the same deflection angle, that is from the inequality
f MVR ≤ f BCC one derives that the efficiency of a single volume reflection f VR must be limited by Eq. (1). This is an alternative way to Eq. (1) from general postulate to the probability in specific physical process. We conclude that Eq. (1) based on reversibility rule helps to understand the quantitative results of volume reflection of negative particles in bent crystals.
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
References [1] U. Wienands, T.W. Markiewicz, J. Nelson, R.J. Noble, J.L. Turner, U.I. Uggerhøj, T.N. Wistisen, E. Bagli, L. Bandiera, G. Germogli, V. Guidi, A. Mazzolari, R. Holtzapple, M. Miller, Phys. Rev. Lett. 114 (2015) 074801. [2] T.N. Wistisen, U.I. Uggerhøj, U. Wienands, T.W. Markiewicz, R.J. Noble, B.C. Benson, T. Smith, E. Bagli, L. Bandiera, G. Germogli, V. Guidi, A. Mazzolari, R. Holtzapple, S. Tucker, Phys. Rev. Accel. Beams 19 (2016) 071001. [3] A. Mazzolari, E. Bagli, L. Bandiera, V. Guidi, H. Backe, W. Lauth, V. Tikhomirov, A. Berra, D. Lietti, M. Prest, E. Vallazza, D. De Salvador, Phys. Rev. Lett. 112 (2014) 135503. [4] H. Backe, W. Lauth, Nucl. Instrum. Methods B 355 (2015) 24. [5] L. Bandiera, et al., Nucl. Instrum. Methods B 309 (2013) 135. [6] W. Scandale, et al., Phys. Lett. B 719 (2013) 70.
[7] [8] [9] [10] [11] [12] [13] [14] [15]
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J. Lindhard, K. Dan, Vidensk. Selsk. Mat. Phys. Medd. 34 (1) (1965) 14. V.M. Biryukov, et al., Nucl. Instrum. Methods B 73 (1993) 153. V.M. Biryukov, Phys. Lett. A 205 (1995) 340. J.F. Bak, et al., Nucl. Phys. B 242 (1984) 1. J.S. Forster, in: R.A. Carrigan Jr., J. Ellison (Eds.), Relativistic Channeling, Plenum, NY, 1987, p. 39. V.M. Samsonov, in: R.A. Carrigan Jr., J. Ellison (Eds.), Relativistic Channeling, Plenum, NY, 1987, p. 129. V.M. Biryukov, Yu.A. Chesnokov, V.I. Kotov, Crystal Channeling and Its Application at High Energy Accelerators, Springer, Berlin, 1997. V.N. Baier, V.M. Katkov, V.M. Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, World Scientific, Singapore, 1998. W. Scandale, et al., Phys. Lett. B 681 (2009) 233.
Contents lists available at ScienceDirect
Physics Letters B www.elsevier.com/locate/physletb
Volume reflection efficiency for negative particles in bent crystals V.M. Biryukov Institute for High Energy Physics in National Research Center “Kurchatov Institute”, 142281 Protvino, Russia
a r t i c l e
i n f o
Article history: Received 11 September 2016 Received in revised form 23 November 2016 Accepted 14 December 2016 Available online 15 December 2016 Editor: M. Doser
a b s t r a c t We suggest a formula for the efficiency of a single volume reflection of negatively charged particles in bent crystal planes and compare it to recent experiments at SLAC, MAMI and CERN with electrons and negative pions in the energy range from 0.855 to 150 GeV in Si crystals. We show that Lindhard reversibility rule provides sufficient basis for quantitative understanding of these experiments. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
Keywords: Bent crystals Channelling Volume reflection
1. Introduction The increased interest to interaction of electrons with bent crystals has lead over the recent years to several important experiments on the so-called volume reflection [1–6]. While the full understanding of the process may be gained by the use of computer simulations taking into account both the electron motion in the field of bent crystal atomic planes and the scattering on the crystal constituents, it would be quite useful to obtain simple approximate formulas describing basic features of electron volume reflection in bent crystals. The volume reflection is caused by interaction of an incident particle with the potential of the bent crystal atomic planes, which give the particle an angular kick of the order of a critical channeling angle θC in the direction opposite to the crystal bending. Such a reflection is not 100%-efficient as some particles “stick” to the atomic planes (so called volume capture caused by scattering) instead of bouncing back. The “stuck” particles are trapped with the bent atomic planes in the channeled states and thus are steered away. The purpose of this paper is to understand the efficiency of volume reflection of electrons. We will show that the Lindhard reversibility rule [7] provides sufficient basis for its understanding. First, we follow the ideas of refs. [8,9] and repeat the derivation of the efficiency of particle reflection off the bent crystal plane. According to the Lindhard’ reversibility rule, the probability for a channeled particle to be scattered from a certain channeled state to certain unchanneled state equals the probability for the opposite
E-mail address: [email protected].
process. This rule was proven in the experiments with GeV-beams [10–12], where the state of each particle was detected by means of solid state detectors placed along the crystal length. In 1982 a PNPI experiment observed the effect of 1 GeV protons volume capture into channeling mode in the depth of the bent crystal [12], in the region where the particles’ trajectories are tangent to the crystallographic planes. 2. Formula for the reflection efficiency The reversibility rule allows to consider the feed-in (volume capture or rechanneling) and feed-out (dechanneling) processes from the unified point of view. The rate of particle transitions from the channeled to random states is set by the dechanneling length L D . Over the distance dz, the transition probability equals dz/ L D . According to the reversibility rule, the rate of the opposite transitions from random (over-barrier) to channeled (under-barrier) states is set by the same quantity L D . In a crystal bent with radius R, the non-channeled particle stays near the channel (on the phase plane) along the length of the order of R θC . Therefore, the probability of transition to channel over the all interaction time amounts to about R θC / L D . Respectively, the efficiency of volume reflection f VR is reduced by this value:
1 − f VR ≈
R θC LD
,
(1)
This formula, which is in fact the result of the reversibility rule, has agreed with the experiments where the volume-capture probabilities were measured at 1 GeV and 70 GeV [13]. Also, the 70-GeV experiment has confirmed the linear dependence of the probability on the crystal bending radius.
http://dx.doi.org/10.1016/j.physletb.2016.12.032 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
277
Table 1 The bending radius R, critical angle θC , measured dechanneling length L D , measured inefficiency of volume reflection, and the calculated efficiency, Eq. (1), for the experiments with electrons and negative pions at MAMI, SLAC and CERN.
2014 2015 2015 2015 2015 2015 2014 2009 2009
MAMI SLAC SLAC SLAC SLAC SLAC CERN CERN CERN
e− e− e− e− e− e− e−
π− π−
Si Si Si Si Si Si Si Si Si
(111) (111) (111) (111) (111) (111) (110) (110) (111)
E, GeV
R, cm
θC , μrad
Measured L D , μm
Measured 1 − f VR
R θC / L D
0.855 3.35 4.2 6.3 10.5 14 120 150 150
3.35 15 15 15 15 15 271 2279 1292
217 122 109 89 69 60 19 17 18
38 55.4 45.2 65.3 57.5 55.8 744∗ 930 930∗
23.3% 33% 27% 16% 16% 19.3% 4.5% 23.3% 17.26%
19% 31% 34% 19% 17% 15% 7%∗ 42% 25%∗
For protons, a typical probability of volume capture to stable channeled states is order of 0.2% at 70 GeV. For electrons, we expect the capture probability to be higher by two orders of magnitude because of stronger scattering. Channeled protons move (and scatter) mostly in electronic gas as they are trapped between atomic planes, while channeled electrons move across atomic plane (and hence scatter on nuclei). While R and θC are well known in each case, the dechanneling length L D for electrons is not so straightforward to obtain. Many experiments with electrons at low energies (MeV up to a few GeV) show rather flat dependence on energy, see refs. [1–4] and references therein, whereas a simple theory expects a linear dependence on energy [14]. This is why in the present paper we rely on measured L D values. Fig. 1. Inefficiency of a single volume reflection 1 − f VR in Si(111) as a function of energy. The SLAC data and theory, Eq. (1).
3. Comparison to the experiments Several experiments performed at MAMI, SLAC and CERN have measured both the efficiency and the dechanneling length of the particles in these crystals at the involved energies. These measurements were done in a broad energy range from sub GeV to 150 GeV with electrons and negative pions. Our idea in this paper is to use both measured quantities and check whether they agree with our formula (1). Table 1 shows the bending radius, critical angle, measured dechanneling length, measured efficiency of volume reflection and the calculated efficiency, Eq. (1), for the experiments with electrons and negative pions at SLAC [1,2], MAMI [3,4] and CERN [5, 6]. We take L D values from ref. [2] for SLAC, ref. [15] for CERN, and ref. [4] for MAMI. We use no free parameters at all, no fitting whatsoever. We just take the quantities R, θC , and L D reported in the papers and produce R θC / L D . This is compared then with the measured inefficiency of volume reflection 1 − f VR . Two cases are marked by (∗ ). The 120-GeV L D was not measured so we used the 150-GeV value scaled down linearly in proportion 120/150. For Si(111) at 150 GeV the L D value was not measured, so we tentatively assumed that is the same as for Si(110); actually it may differ somewhat. We see surprisingly good agreement between the measured inefficiencies of volume reflection and Eq. (1) in Table 1. On average, for the 9 experimental points of MAMI, SLAC and CERN experiments, the data are lower than prediction of Eq. (1) by a factor of 0.82:
1 − f VR ≈ 0.8 ·
R θC LD
(2)
For the SLAC data that is most complete, we can compare the experiment and the theory, R θC / L D , as a function of energy in Fig. 1. Notice that the comparison is with Eq. (1), not (2). Eq. (1) suggests a functional dependence on bending radius R. We believe, however, that real values should deviate from the linear R proportion of Eq. (1) both at too large R and at too small R.
Fig. 2. Inefficiency of volume reflection 1 − f VR as a function of the ratio R / R C . The SLAC data for electrons and theory, Eq. (1). Straight line is a guide to the eye.
At large R the efficiency cannot grow above 100% and thus the real values start to saturate below the values given by Eq. (1). At small R, that is R comparable to the critical radius R C , the effective potential well shrinks (due to centrifugal effect) thus starting to affect the L D value in Eq. (1) (and θC value to lesser extent). This should place the real values above the Eq. (1), at small R. There is no measured dependence on R for electrons. However, as the energy runs over a broad range with a constant R, the ratio R / R C spans over a broad range from small values where the potential well is distorted to large ones where the potential well is unchanged. This span for SLAC data can be seen in Fig. 2. The dependence of the volume reflection inefficiency, measured and calculated, on the R / R C ratio can be also plotted for CERN data, see Fig. 3. Notice some big differences between the experimental conditions of SLAC and MAMI versus CERN:
278
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
neling length L D behaviour is complicated. In the experimentally studied energy range from MeV up to a few GeV, L D is rather flat for a number of reasons. √ Then the reflection inefficiency in this range should follow 1/ E. At some high energy range, certainly 100 GeV and higher, L D should be a linear function of energy. Then the inefficiency should follow 1/ E 1.5 . Let us summarise this as
1 − f VR ∼ R / E 0.5 1 − f VR ∼ R / E
at low energies (MeV up to a few GeV)
1 .5
at higher energies (100 GeV domain)
One should remember that a typical R grows linearly with E. Only the ratio R / R C remains invariant. Therefore it is convenient to present the energy dependence with the use of nondimensional invariant parameter R / R C as Fig. 3. Inefficiency of volume reflection 1 − f VR as a function of the ratio R / R C . The CERN data for negative pions and electrons, and theory, Eq. (1). Straight line is a guide to the eye.
1 − f VR ∼ ( R / R C ) · 1 − f VR ∼ ( R / R C )/
√
√
E
at low energies (MeV up to a few GeV)
E
at higher energies (100 GeV domain) √
In the experiments considered above, the E factor varies as much as 13 as E goes from 0.855 to 150 GeV. The range of R / R C ratio has spanned from 3 to 100 in the variety of experiments on volume reflection. 4. Relation of Eq. (1) to a more general principle
Fig. 4. The ratio of MAMI, SLAC and CERN data to the predictions of Eq. (1) plotted as a function of energy.
(1) SLAC and MAMI data is for electrons, low energies (MeV up to a few GeV), and moderate R / R C = 14 to 27; (2) most of CERN data is for pions, 150 GeV, and high ratio of R / R C = 50 to 90. Different procedures for selection of reflected particles in the experiments may contribute further to the differences. Notice that the experimental errors in f VR and L D and some uncertainties in definition, especially for L D , are rather big in some cases. This adds to the scatter in the plot. Also, our model (1) is elementary and it would be too much to expect perfect agreement from it. Of course, a numerical coefficient (hopefully, of the order of magnitude of 1) would be natural in Eq. (1). As said above, on average the experimental data of MAMI, SLAC and CERN are lower than the prediction of Eq. (1) by a factor of 0.82. This ratio data-to-theory can be seen as a function of energy in Fig. 4. For low-GeV data of MAMI and SLAC the ratio is about 1.0. As the energy goes up by a factor of 200, from 0.855 to 150 GeV, this numerical factor in Eq. (2) decreases to 0.6 for CERN data on negative pions. The errors (mostly caused by the uncertainties in the L D measurements) are not shown here. The agreement between Eq. (1) and the data from three experimental groups means two things. One is the possibility to predict data, optimise applications, and understand the underlying physics. The second thing is that we demonstrate the consistency between three different experiments with the help of Eq. (1). The energy dependence of efficiency √is set by θC / L D term. While the critical angle θC behaviour ∼ 1/ E is clear, the dechan-
This agreement supports the basic idea that the Lindhard reversibility rule provides a good hint in understanding the efficiency of volume reflection. This link between the reversibility principle and the efficiency of volume reflection can be explored further. From Eq. (1) based on the reversibility rule, one can derive an interesting consequence for the efficiency of multiple volume reflection (MVR). If efficiency of a single reflection is f VR , then after N volume reflections the efficiency is ( f VR ) N . With Eq. (1) that makes
( f VR ) N = (1 − R θC / L D ) N = exp(− N R θC / L D ) where we approximate by exponent because f VR is close to 1. The angle of a single volume reflection is on the order of θC . After N reflections, the beam is bent by θ = N θC . That is, for beam bending a significant angle θ with MVR, the bending efficiency must decline with θ like
f MVR = ( f VR ) N = exp(− N R θC / L D ) = exp(− R θ/ L D ) The bending efficiency of bent crystal channeling (BCC) declines with bending angle θ in exactly the same way
f BCC = exp(− R θ/ L D ) = exp(− L / L D ) over distance L = R θ . This means that one cannot outsmart the nature with MVR and bend beam with efficiency higher than that of BCC. This argument can be treated vice versa, in a sense. From the postulate that the efficiency of the multiple volume reflection could not exceed the efficiency of bent crystal channeling at the same deflection angle, that is from the inequality
f MVR ≤ f BCC one derives that the efficiency of a single volume reflection f VR must be limited by Eq. (1). This is an alternative way to Eq. (1) from general postulate to the probability in specific physical process. We conclude that Eq. (1) based on reversibility rule helps to understand the quantitative results of volume reflection of negative particles in bent crystals.
V.M. Biryukov / Physics Letters B 765 (2017) 276–279
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