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28*12 RADIATION THEORETICAL Submitted A. UNIAXIAL
Delhart EMISSION
STUDY
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Physical
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CRYSTALS
III OF J.
Review
Derre ANISOTROPIC CHERENKOV
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Le DAPNIA (Departement d'Astrophysique, de physique des Particles, de physique Nude aire et de I'Instrumentation Assoriee)regroupe les activites du Service d'Astrophysique (SAp), du Departement de Physique des Particles Elementaires (DPhPE) et du Department de Physique Nucleaire (DPhN).
Adresse: DAPNIA, B aliment 141 CEA Saclay F -911 91 Gif-sur-Yvette Cedex Theoretical study of Cherenkov Radiation Emission in anisotropic uniaxial crystals
A. Delbart artd J. Derre CEA, DSM/DAPNIA, CE-Saclay, France
Abstract
We propose a theoretical review of the Cherenkov radiation emission in uniaxial crystals. The formalism is based on the previous work of C. Muzicar (1961) whose results in terms of energetic properties of the emitted waves are corrected. This formalism is used to simulate the Cherenkov radiation emission in a strongly birefringent sodium nitrate crystal (NaNO,) and to investigate the consequences of the slight anisotropy of sapphire (Al^OJ on the design of the Optical Trigger.
R&sum6
Nous presentons une etude theorique de 1’emission de radiation Cherenkov dans un milieu anisotrope uniaxe. Le formalisme est celui de C. Muzicar (1961) dent il corrige les rdsultats concernant les propriety energetiques des ondes emises. Ce formalisme est utilise pour simuler remission de la radiation Cherenkov clans un cristal tres birefringent de nitrate de sodium (NaNO,) et pour determiner ('influence de la faible anisotropic du saphir (A1,0J sur la conception du Trigger Optique. 1
1. Introduction
Our group built and studied a prototype of a new detector called the Optical Trigger [1], which uses the total reflexion of the Cherenkov light emitted by a particle in a LiF crystal shell [2] to detect short life particles. In order to improve the sensitivity of the device, we proposed and tested a second prototype [3,4] in which the Cherenkov radiator is a sapphire shell surrounded by a special liquid. Our simulations then had to take into account the influence of the slight anisotropy of sapphire on the Cherenkov radiation emission.
Cherenkov radiation in an isotropic medium is well understood and used in high energy physics detectors. It was discovered in 1934 by P. A. Cherenkov [5] who observed that pure sulfuric acid in a platinum crucible close to a radium source emitted a weak blue radiation. All the following observations revealed that a charged particle moving in a transparent medium of index of refraction n, with a velocity v - larger than the phase velocity of light in that medium, emits this new radiation in a well-defined direction which only depends on the product The radiation spectrum was found to be continuous and concentrated mainly in the blue-violet part of the spectrum. In 1937, 1. E. Tamm and I. M. Frank [6] developed a theory on the basis of classical electrodynamics which completely explained the properties of this radiation. A first theoretical study of the Cherenkov radiation in anisotropic media was due to V. L. Ginzburg [7] in 1940. A complete theory was then obtained in 1956 by V. E. Pafomov who extensively sudied the Cherenkov radiation. He investigated the case of the Cherenkov [8] emission in anisotropic ferrites and transposed his results for anisotropic dielectrics. He showed that, in the general case of a fast charged particle moving in an arbitrary direction in an uniaxial crystal, there should exist two noncircular conical surfaces, which correspond to the ordinary and extraordinary waves. The intensity distribution over these surfaces should not be uniform owing to the polarization features of the radiations. V. P. Zrelov experimentally confirmed in 1964 [9] the geometrical properties of the cones in the two special cases of a particle traveling along the optical axis of the crystal or perpendicularly to it. The only experimental verification of the energetic properties of the radiation was carried out in 1965 by D. Gfbller [10], who studied the Cherenkov radiation emitted by a ^P 1.71 MeV (3 source in a strongly anisotropic NaNOg plate. He found 15% less total emitted intensity for a particle moving perpendicularly to the optical axis than for a particle moving along this axis. This observation is well explained in Pafomov's theory, but we didn't manage to End how were obtained the corresponding formulae. Furthermore, we found that these formulae, applied to an isotropic medium, give an erroneous number of photons, except in the two previous special cases. Another approach was due to C. Muzicar [11] and led to the same conclusions concerning the geometrical properties of the two cones and the intensity of the ordinary wave. But his conclusions are different from those of V. E. Pafomov regarding the intensity distribution of the extraordinary wave. Moreover, he found the same total number of photons whatever the angle between the particle motion and the optical axis, not in agreement with the measurement of D. Gfbller.
We present the complete properties of these Cherenkov cones the same way as C. Muzicar did. The new result concerns the number of extraordinary photons, which is now in agreement with D. Gfbller and successfully applied to the isotropic case. 2
2. Basic formula and notations
Let a point charged particle move in an uniaxial crystal with a constant velocity F — c^ (c is the velocity of light in vacuum), and in a direction defined by a unit vector f. We choose a coordinate system with x, along the optical axis c of the crystal and %% in the (P,f) plane (/zgwe J). In this system, r is defined by % = cos(%) , % = sin(%) and % = 0. The dielectric tensor is diagonal with 6% = e, = and Eg = = &<,= n* , where n, and ^ are the ordinary and extraordinary refractive indices. The two eigen states of propagation [12], the ordinary and extraordinary waves, can be simultaneously emitted, and the total number of Cherenkov photons emitted d^Af by each wave, per path length df and per energy dE, is given by the following formula [7]:
d^Af oc' f.w du (1) dfdE
g- i where: (X = — « — is the fine structure constant; Ac 137 p (=1 in the following study) is the scalar magnetic permittivity of the medium; dw M an element of solid angle; ii is the unitary vector in the direction of the Cherenkov wave phase propagation; (ii is the normal to the wave front and is not necessarily the direction of the light ray) g is a vector in the direction of the electric field; 2 V,is the phase velocity of the emitted wave: v, = —g.d (2)
d is the unitary vector in the direction of the displacement vector: w.d = o ( 3 )
optical axis
Figure 1: Coordinate system and main notations 3
By definition, the two waves are orthogonally-polarized: (4) d and e are linked by the dielectric tensor: = (5)
We finally introduce in the principal plane another unitary vector & from which 3 is defined by its polar angle 8 and its azimutal angle (p:
w, = cos(8) -sin (8) cos((p) W2=&2cos(8) + tisin(8)cos((p) (6) w, = sin(8)sin((p)(6)
3. Ordinary waves
By definition, the displacement vector cM of the ordinary wave is perpendicular to the optical axis c. By using Eq.(3), and + = 1- we find and then g^by Eq. (5):
0 0 (8) and ^ (?) %3 (9) Vw «i
Since is constant, Eq. (1) then shows that the ordinary photons are emitted on a circular cone 8^) defined by f.w = cos(8(„) ) with:
cos(8(,)) = --^- (lo)
and the refractive index seen by the ordinary wave is n0
As in an isotropic medium, the symmetry axis of this cone is r and the pardcule must move faster than the critical speed (3% c with:
1 P («) " (11) ",
The integration of Eq. (1) over the angle 8 then gives:
d =_2L.^ (12)
where ii is defined by the relations (6) in which 8 = 8^ and = f. 4
We finally obtain, for an incident particle moving faster than the critical speed, the number of ordinary photons ^ emitted per azimutal angle d(p, energy dE and path length df:
_^L = _2L_sin:(8 )______sin:(%)sin\(p) ______dfdEdq) 2%Ac ^ l-(cos(8^)cos(%)-sin(%)cos((p)sin(8^)y (13)
V. E. Pafbmov [8] and V. L. Ginzburg [7] obtained a similar expression in another coordinate system.
We must now calculate the total number of ordinary photons by integrating Eq. (13) over the azimutal angle (p. The result of this integration depends on the sign of cos(%) —cos(8^) and we get two expressions, one when the optical axis is inside the ordinary cone (% 3 8^) and the other when it is outside:
-^-(l-cos(%)) optical axis ;%?fde the cone (14t) dME (14) optical axis oMtside the cone(14))
The polarization features of this wave are quite different from those of the Cherenkov radiation in an isotropic medium for which the electric field lies in the (F, 3) plane. By definition, the polarization of the ordinary wave lies in the plane perpendicular to the optical axis, i.e(x%,x,), &nd is of course perpendicular to the photon motion ii. This is confirmed by Eq. (7) and (8), which show that the electric field Z forms a fan in this plane as the normal runs around the cone (see figure 2).
The extreme values are obtained for electric fields located in the two planes tangent to the cone and passing through the optical axis. When the optical axis is inside the cone, these )lanes do not exist and the polarization vector fills the entire plane and draws a circle of radius g(">| = . When the optical axis goes outside the cone (%^8^), the electric field only draws a fan in this circle, whose vertex angle 2T| is found for Z^ minimum:
cos(n) = - ^cos^(8(,))-cosZ(%) (15) SM {%)
The vertex angle of the fan is minimum for a particle moving perpendicularly to the optical axis where it is equal to the ordinary Cherenkov angle 8^. Note that Z^ is along X; for a photon emitted in the (x, ,x%) plane (for (p = 0° and (p = 1800). 5
Figure 2: Ordinary electric field as the normal runs around the cone
4 Extraordinary waves
The displacement vector of the extraordinary wave is perpendicular to d ^ (Eq. 4).
Using Eq. (3) and Eq. (7) we obtain: 1-wf
1-w, e. 1 dW = (16) g(') = (17) Vi-«f e„ Ml": E.
v“=c4+bH and Eq. (3) ux (18)
The cone of normals of the extraordinary wave is defined by the argument of the Dirac v(" distribution in Eq. (1), f .w = —. We write this expression by means of Eq. (6): cP
Pcos* (8) + Q sin* (8)cos* (tp) -))sin (8) cos(8)cos((p) = (19)
^ with (20) f - (iAC - ^/P^ 0 = (tzr,-&,r%y-(l/n*-l/n*)^/P* 6
The interpretation of the cone's shape is simplified by choosing t to vanish the term - ^(t^ — &^) in (19). k is then the symmetry axis of the cone and, as a unitary vector, is given by:
(21)
k2 — sin(0 t) = —f= 1
The Cherenkov angle 8^ , which is relative to t is:
/;-6cos^( The critical speed is determined by 8^ =0 (or # = f): (23) PL V"! cos\%)+n^sin '(%) For a particle moving at the critical speed (3^ c, the Cherenkov radiation is emitted along a 2 direction 6* different from r since tan(0*) = —tan(%). Note that the two cones never intercept, except when the optical axis is tangent to the ordinary cone (% = 8(„, ) at (p = 180^: the two cones are then tangent to each other and to the optical axis (8^ =8* for (p = 1800). The optical axis is then simultaneously inside or outside the two cones. The index of refraction seen by an extraordinary photon now depends on |3,% and (p: (j_ r i 1 k + (24) 2 2 2 J J ", 7 Note that is equal to only in case of an isotropic medium (n, = n, ) or for an extraordinary wave emitted perpendicularly to the optical axis (u, = O). Furthermore, in case of a particle moving along the optical axis (% = O), the extraordinary cone is circular around the optical axis (8* = 0) and the Cherenkov wave has the same geometrical properties as the one emitted in an isotropic medium of refractive index independent of cos(8g)) = —(%2bis) 1 1 with 1 + - (24bis) We now have to determine the number of emitted photons. For that purpose, we use the properties of the Dirac distribution to find: f %.r8(cos(8)-cos(8(,))) 8 il.r ■■■■ J ^(T^Qcos^9))CP-^Gcos\^)T \ and we can write the scalar product on the cone of normals as: w.r F.gW = =[rfi\^ -w,] “i This leads, for an incident particle moving faster than the critical speed, to the number of extraordinary photons d^N ^ emitted per azimutal angle dtp, energy dE and path length dZ: — 1 t 1 w.r Lv1 *j + 2^ dW' _ a 1 1-^ (25) dZdEdtp ZrcAc V(E-Qcos'«p))(f-Ocos'(tp)) Although this result gives the same number of photons as the one given by V. E. Pafomov [8] in the two special cases ^ — 1 (r / /c) and r, = O (^77 E), it is different in all the other cases. We have to integrate Eq. (25) over the azimutal angle (p. After some calculations presented in Appendix, we find, like in the case of the ordinary wave, that this number depends on the relative position of the optical axis and the cone: z dW> a COS(%} Mk i/widc the cone (26a) dME Ac V p2 (26) dV q i p<-> owWide the cone (26b) d/dE he The polarization of the extraordinary wave lies in the plane defined by the photon and the optical axis, i.e (ii, c). The parametric equations of the cone of polarization are rather complicated and C. Muzicar [11] gives the following estimate: ® If the optical axis is outside the cone, the polarization vector dw is contained in a pyramid whose faces are the two tangent planes of the cone that pass through the optical axis and the two planes passing through ^ and perpendicular to the generators of the cone which lie in the principal plane (jq,^) ((p=0°and (p=l80°)(see/zgwe 3 ). optical axis Figure 3: Extraordinary displacement vector for a pardcule moving perpendicularly to the optical axis. 9 • If the optical axis is inside the cone, the polarization vector dM generates a cone of same axis & as the cone of normals (see/zgwre 4) co- 180° optical axis Figure 4: Extraordinary displacement vector for a particule moving along the optical axis 5. Total number of emitted photons The expression of the total number of photons emitted by a particle depends on the relative value of its speed (3 with respect to the two critical speeds, the ordinary one (3^(Eq. 11) and the extraordinary one (3^ (Eq. 23) which depends on the angle %: • If P is smaller than the two critical speeds, no photon is emitted. . If P is grea/er than the two critical speeds, the tiro comes are emitted and the total number of emitted photons is the sum of Eq. (14) and (26) which gives the same expression whether the optical axis is inside or outside the cone: (27) 10 . If P is AeAyeew the two critical speeds, only one of the two cones is emitted and the optical axis is always outside the cone. For a negative (positive) crystal, only the ordinary (extraordinary) cone is emitted and the total number of photons is given by the Eq. 14b (26b). For a given angle %, the total number of emitted photons increases with (5. Note that the distribution presents two kinks at the two critical speeds. But, for a given speed p , the variation of N' with % depends on the relative value of P with respect to and 1/n, . * If P > 1/n, and P> l/n@, P is greater than the two critical speeds and N' is thus given by Eq. (27). It decreases (increases) with % for a negative (positive) crystal from A/^ (% = 0°) down (up) to N; (%= 90°) given by: a AC = 1- (28a) Ac ZoZ a #1 = 1 — (28b) Ac Note that is equal to the number of photons emitted in an isotropic medium of refractive index /i,: A%,,/gV.mm = 37 sin '(8^) (29) . For p between 1/n, and 1/rig, a particular angle %o occurs when P[,, = P : i-iW sin(%o) (30) Vi-".2/-.1 11 For a negative crystal, ordinary photons are always emitted since P > P(„). Af' decreases with % down to and is given by the Eq. (27) since the two cones are emitted. Then, only ordinary photons are emitted (P c P^) and A/' is independent of % (Eq. (14b). For a positive crystal, ordinary photons are never emitted since P c p^. No photons are emitted for and then only extraordinary photons are emitted (P > P(,)) with increasing with % (Eq. (26b) since %>%o>8(«)). Our results are different from those of C, Muzicar who made a mistake in the integration of Eq. (25) which has also been misprinted. They are in agreement with the observations of D. Gfbller [10] who found Af^/Af,, = 85% for a P^p source of 1.71 MeV (P = 0.954) since Eq. (28a) and (28b) give TV}/N^ 85.2%. They are also successfully applied to the isotropic case (n, = n,) since Eq. (27) reduces, whatever the angle %, to the well-known formula (29) of the classical Cherenkov theory in an isotropic medium. 6. Cherenkov Radiation Emission in a NaNOg crystal As a numerical application, we choose the Cherenkov radiation emission in a highly negative birefringent NaNOg crystal whose refractive indices are n, = 1.61 and /%, = 1.34. J shows the dependence of the total number of emitted photons versus % and P. Note the location of the break angle line which runs from the point (x= 0°, P = P(o)) to the point (% = 90°, 1.3 = Pf.)). The two next figures illustrate the dependence of N(”), N(e), and N’ versus p . When p increases, the two cones open out and the optical axis can go inside them above the break point speed Po = Pw/cos(%). For cos(%) > P^ , Po For the dependence of TV^,N% and N' versus %, we expect two interesting figures: - For P > 1/n, , there is no kink in the TV' variation (/tgwne 7a for P = 1); - For 1/n, < P < 1/n, , there is a kink at % = %o (/zgwre 76 for P = 0.68). 12 These figures are cross sections of the two cones in a plane perpendicular to the direction of the particle r: - The cross section of the ordinary cone for which F is the symmetry axis, is the same circle whatever the angle %. - The cross section of the extraordinary cone is an ellipse whose center is slightly shifted from the symmetry axis t of this cone, except when the two cones have the same symmetry axis & = F for a particle moving along or perpendicularly to the optical axis. In a NaNOg crystal, the slight difference between the two symmetry axes is not greater than 5°. The energy of each cone is drawn as the thickness of the cross sections: for the ordinary cone, a number proportional to the amount of ordinary emitted photons is added to the circular cross section of the cone, whereas the corresponding number for the extraordinary cone is subtracted from its cross section. The features of the obtained intensities are well explained by polarization considerations. The Cherenkov energy is proportional to the square of the scalar product e.F (see Eq. (1)). No ordinary wave can thus be emitted for a particle moving along the optical axis and for photons emitted in the (7, E) plane. A special case occurs when the particle moves at an angle equal to the ordinary Cherenkov angle (/rgwre&z amf for % = 51.610). The two cones are then tangent to the optical axis and the ordinary photons can be emitted at 80 M-j-a d & l Figure 5: N' versus % and (3 in a NaNOg crystal. 13 7. Application to the Optical Trigger The Optical Trigger [1, 2] is a device which is devoted to fast triggering in fixed target experiments. It is based on the detection of the Cherenkov light produced in a thin spherical shell centered on the target. A particle which comes from the target produces Cherenkov light which is refracted at the outer interface of the device. Whereas part of the Cherenkov light produced by the particles which presents an impact parameter to the target is totally reflected and trapped in the spherical shell to give rise to a sizeable signal on the light detectors. Sapphire was the best choice for the Cherenkov radiator (see [3, 4]). For this slightly negative birefringent crystal (%,* 1.770 and n, * 1.762), the two cones can be separated up to 4 mr (-1/40), the same order of magnitude as the transition between total and non total reflexion needed for discrimination. However, for a particle moving along the optical axis, only a circular extraordinary cone is emitted, with an angle given by Eq. (22bis) and a total number of emitted photons given by Eq. (28a) with % = O, which then coincides with the number of photons emitted in an isotropic medium of index n, (Eq. 29). Taking the optical axis close to 0° is thus the best choice for the Optical Trigger device. Any other angle will degrade its discrimination power. 8. Conclusion The Cherenkov radiation emission in uniaxial crystals was extensively studied by V. E. Pafomov [8] in 1956 and C. Muzicar [11] in 1961. These two articles unfortunately lead to different results, especially in the most general case of a particle moving at an arbitrary angle to the optical axis of the crystal. We have developed a complete formalism and corrected their erroneous conclusions about the energetic properties. This formalism is in agreement with the experiments previously done in the 60's in the two special cases of a particle moving along or perpendicularly to the optical axis of the crystal: V. P. Zrelov [9] for the geometrical properties of the Cherenkov cones and D. GfOUer [10] for their energetic properties. Our results have now to be experimentally verified for particles moving at an arbitrary angle to the optical axis. Acknowledgements This study is a part of the PhD of A. Delbart who would like to thank R. Chipaux for his encouragements and help. The authors are also grateful to J. Haissinski, P. Micolon and M. Spiro for their constant support of the Optical Trigger project. 14 References [1] G. Charpak, Y. Giomatark and L. Lederman, Nucl. Instr. and Meth., A 306,439 (1991) [2] G. Charpak etal, Nucl. Instr. and Meth., A 332,439 (1993) [3] C.Kochowski, Nucl, Instr. and Meth., A351, 193 (1994) [4] M. Atac et al, NucL Instr. and Meth., A367,372 (1995) [5] P. A. Cherenkov, Doklady AN SSSR, 2,451 (1934) [6] LE. Tamm and L M. Frank, Doklady AN SSSR, 14,107 (1937) [7] V. L. Ginzburg, Zh. Eksperim. i teor. Fiz., 10,608 (1940) [8] V. E. Pafomov, J. Exptl.Theor.Phys. (USSR) 30,761, (1956) -Soviet Physics, JETP 3, n°4, 597 (November 1956) [9] V. P. Zrelov, J. Exptl.Thew. Phys. (USSR) 46,447, (1964) -Soviet Physics, JETP 19/% 301 (August 1964) [10] D. GfoHer, Zeitschrift fur Physik, 183,83, (1965) [11] C. Muzicar, J. Exptl.Theor.Phys. (USSR) 39, 163,(1960)-Soviet Physics, JETP 12/1,117 (January 1961) [12] A. Yariv, Optical waves in crystals (Wiley Interscience publications) 1984 15 0.6 0.65 ^ 0.7 0.75 0.8 0.85 0.95 1 Particle motion p=v/c 0.7 - - Particle motion p-v/c Figure 6a and 6b: Af ^, Af ^, and TV' versus P in a NaNO% crystal at % = 40" (a) and % = 90" (b) 16 Optical axis inside the cone Optical axis outside the cone Angle % betweentheopdcal axis and (he pardclemodon (in degrees) Angle % between the optical axis and the particle motion (in degrees) R^ureVa and 7b: TV^\ TV^'^, and TV' versus % in a NaNO^ crystal for (3 = 1 (a) and (3 = 0.68 (b) 17 Reure 8a: Ordinary (gray) and extraordinary (black) cone shapes for a relativistic particle moving at various angle inaNaNO^ crystal (see details in text) 18 Pi sure 8b: Ordinary (gray) and extraordinary (black) cone shapes for a relativistic particle moving at various angle in a NaNO; crystal (see details in text) 19 Appendix: Derivation of the total number of Extraordinary emitted photons We present here the main stages of the integration of Eq. (25) to derive the total number of extraordinary emitted photons (26). It is exactly the way of C. Muzicar, but his result for I; is erroneous (T is false). After some calculations, we can rewrite Eq. (25) to get: dW a mm ■)- d/dE 2%Ac 202 with: j ^ ...... 7 — r M.P * w.r^(R-Gcos\(p))(P -Gcos^((p)) J---1 + 2142^ dtp v v /y ' 7 l-wf)V(^-Geos'(tp))(P-Qcos" «p)) Using Eq. (10) to express w.r, and omitting terms odd in cos((p): r _ 1 fZ*rz* r'-'AU d dtp M.P ° +^2^)^ sin'((p)-Tcos^((p) with r = (t,r, +^2)^(6 -E) + (t2n-t/2)^(E-E) = - and therefore: I 2%m, ^cos^(%) + ^sin'(%) 20 For Ig we get: C/y* +z with PF = f , D = u zoz(*in + *z'2)^+^'2^2 [< rz*z 2/^D-(t,^ + ^r, _2ot (PM+^PF) 4MOf 2M The result of the integration is: az )slgn(.z, )-(ib-az2)sign( z,)] Z2 Z1 where iz^ and iz^ are the roots of (Wy^ -2iDy PF z^ is positive but the sign of z% depends on the relative position of the optical axis and the ordinary cone of normals, we finally get: cos(%) S cos(%) 2