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Gamma-quanta propagation in single crystals

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γ-quanta propagation in single crystals V.A.Maisheev ?Institute for High Energy Physics,142284,Protvino,Russia Abstract Propagation of γ-quanta in the single crystals,oriented in a region of the coher-ent pair production is considered.The qualitative description of the process is also discussed.The theory of γ-quanta propagation in the anisotropic medium is illus-trated by the help of the particular calculations of such a process in silicon single crystals.It is shown that the single crystals are sensitive to the initial circular polarization of γ-beam despite the fact that the cross section of absorption is independent of it.The reason is that the normal electromagnetic waves (an eigenfunctions of the problem)are elliptically polarized.The speeds of absorption and motion of both the normal waves are di?erent and as a result the process of γ-quanta propagation depend on the initial polarization state.The calculated value of asymmetry is about 20%for 25GeV γ-quanta propagating in 100cm silicon single crystal.The obtained results are useful in creating of polarimeters for high energy elec-trons and γ-quanta.1Introduction It is well known that the optical properties of a medium may be described with the use of a permittivity tensor [1,2].The anisotropic medium referred as such a medium in which the permittivity tensor is a symmetric tensor [1,2].In the general case the components

of this tensor are complex values and,because of this,the complex permittivity tensor does not reduce to principal axes,except in speci?c cases.The simplest example of such an anisotropic medium of the general type is a combination of the two linearly polarized monochromatic laser waves with di?erent frequencies (dichromatic wave)moving in the same direction.In general the directions of linear polarization of these monochromatic waves are di?erent.Another more complicated example is a single crystal oriented in the region of coherent e ?e +-pair production [3,4,5].

The permittivity tensor was found for both these examples in papers[6,7].Besides,in [6]the theory of γ-quanta propagation in the anisotropic medium of a general type is also considered.One of the main results of these papers is the prediction of sensitivity of the anisotropic medium to the initial circular polarization of propagating γ-quanta.It is important for creating of devices measuring the circular polarization of γ-quanta or longitudinal polarization of electrons.In the last case the electron beam is transformed into bremsstrahlung γ-beam by the use of amorphous target [8].The circular polariza-tion of the end of bremsstrahlung spectrum is approximately equal to the longitudinal polarization of a primary electron beam.In this paper we consider the theoretical basis for creating of the γ-beam polarimeters.

2Qualitative consideration of process

As already noted,the theoretical description ofγ-quanta propagation in the anisotropic medium is contained in paper[6].The permittivity tensor in single crystals oriented in the region of coherent e?e+-pair production is found in[7].In the case of high energy γ-quanta the process of propagation is determined primarily by the transverse part of the permittivity tensor,while the longitudinal components of the tensor are higher-order in?nitesimals in the interaction constantα[9,10].In this way the permittivity tensor in anisotropic medium is a symmetric tensor of rank two.

Now we make an attempt of the qualitative consideration of theγ-beam propagation in the anisotropic medium.As is well known that the imaginary components of permittivity tensor in anisotropic medium describe the absorption ofγ-quanta.The cross sectionσe?e+ depends on the linear polarization ofγ-quanta

σe?e+=A+B(eτ)2,(1) where the unit vectors e andτdetermine the polarization plane of theγ-quanta and some de?nite plane in single crystal(the wave vector of theγ-quanta lies in both planes).The formula(1)essentially determines a symmetric tensor of rank two,whose components we denote asσkl(k,l=1,2).Then the imaginary parts of the permittivity tensor components are

Nσkl c

ε”kl=

P ∞0xε′′ij(x)dx

of the symmetry plane for real permittivity tensor di?er from the similar position for imaginary part of this tensor.In other words,the axes of both tensors are not parallel or perpendicular in between.Hence we get that in general the preferred two planes exist in a space for propagatingγ-quanta of a?xed energy.

As is well known thatγ-beam in a medium present the superposition of two states. These states referred as normal electromagnetic waves[2].The normal waves are the eigen-functions of the problem and they have determinate polarization characteristics,which depend on the optical properties of a medium.So,theγ-quanta,having the polarization characteristics identical to one of normal waves,are conserve their at the propagation in a medium.In general the speeds of absorption of normal waves are di?erent.

It is evidently that both normal waves are linearly polarized in case illustrated in Fig.1a.These polarizations are perpendicular in between and their directions coincide with the principal axes of ellipses.In other case(see Fig.1b),in general the principal axes of both ellipses are not parallel,and it is believed that normal wave are elliptically polarized.It is also follows from paper[7].In the case,when the absorption in medium is absent,the all imaginary components of permittivity tensor are equal to zero and because of this the normal waves are always linearly polarized.

In the general case the initial polarization state ofγ-quanta changes at propagation in a medium.The intensity ofγ-beam on thickness x in the anisotropic medium can calculate by the help of follows relation[6]

Jγ(x)=A(x)+B(x)ξ1+C(x)ξ2+D(x)ξ3(4) whereξi,(i=1,3)are the initial Stokes parameters ofγ-quanta,A,B,C,D are simple functions of x including also some parameters as,for example,refractive indices of normal waves.When the normal waves have only linear polarization the function C(x)is equal to zero on any thickness x.In the case,when the normal waves have nonzero component of circular polarization the function C(x)is equal to zero at x=0and in general it is nonzero at x>0.Besides,the?rst derivative of the C(x)-function is equal to zero at x=0.It means that the cross section of the absorption process independent of the circular polarization of theγ-quanta.Here we employ the well-known formula dJ=?J(0)Nσdx for thin targets.

In this way the relation(4)show that intensity ofγ-quanta propagating in the anisotro-pic medium depends on the initial polarization despite the fact that the cross section of absorption is independent of it.The reason is that the normal waves are elliptically polarized.The speeds of absorption and motion of both the normal waves are di?erent due to the processγ-quanta propagation depends on the initial polarization state.On the other hand the initially unpolarizedγ-beam became elliptically polarized[6].This is an essential prerequisite for creating of theγ-polarimeter.To do this requires computations of theγ-quanta propagation in single crystals.

3γ-quanta propagation in single crystals

In this section we present some results of the calculations ofγ-beam propagation in silicon single crystals.The investigated process is determined with the help of such parameters as refractive indices and polarization states of normal electromagnetic waves.The imaginary parts of refractive indices are responsible for the speed of normal wave absorption.The real parts and polarization states of normal waves are also in term C(x)(see equation (4)).Let us de?ne the polarization state of one normal wave with the help of Stokes parameters X1,X2,X3(X21+X22+X23=1).Then the polarization state of another

wave is correspondingly equal to?X1,X2,?X3.Intuition suggest that the parameter X2should be signi?cant for circular polarization detection.As it is follows from[7] parameter X2is signi?cant whenγ-beam moving under no large angle with respect to any strong crystallographic axis.We select the<001>axis in silicon for calculations. We also select the Cartesian coordinate system with one axis along<001>axis and two another axes along<110>and<1?10>axes.Direction ofγ-quanta motion we can determine with the use of angleθwith respect to<001>axis and azimuth angle αaround of this axis(α=0,when theγ-quanta momentum lies in the(110)-plane). However another angles are more convenient to use?H=θcosα,?V=θsinα.

The characteristics of process have the same numerical values at di?erentγ-quanta energies Eγwhen the following invariant parameters are used:W H=EγG2?H/(2mc2), W V=EγG3?V/(2mc2).Here m is the electron mass,G2,G3are the reciprocal lattice constants(in our case G2=G3=0.01264in units of the inverse electron Compton length). In the case being considered we can write these parameters as W H=12.366Eγ?H,W V= 12.366Eγ?V[GeV radian].The calculations are carried out for Eγ=25GeV.The Figs.2-6illustrate these results of calculations.In the calculations the Moliere form factor was employed[10].

As Fig.3illustrates,the regions with high circular polarization X2exist in single crystals.The absolute value of X2is near1at?H=3.008mrad and?V=3.558mrad. We can see that the high quantity of X2is observed in region≈±0.1with respect to central?H,?V-values.The direction,in which the circular polarization X2is equal to1,is the direction of the so-called singular axis,described in the crystal optic of the visible light [2].We can see in Fig.2that refractive indices are approximately equal to one another near this direction.Notice,that the value Im(n)?εA/2,as a function of invariant W H,W V-parameters,is independent of theγ-quanta energy.TheεA-value is determined in[7],and for the silicon single crystal is numerically equal to1.3210?15/Eγ.Fig.5illustrates the variation of polarization state of the propagatingγ-quanta.We can see that the initially unpolarized beam obtain in general the linear and circular polarization.Notice,that equations of paper[6]break down for X2=1(see also[2]).However they are true in the neighborhood of this point and our calculations are made for X2=0.995.

Figs.4and6illustrate the absolute and relative losses ofγ-quanta intensity.One can see that these losses depend on the initial polarization of theγ-quanta.We select the value A s=|I p?I0|/I0as the degree of relative losses of intensity,where I0is the losses of initially unpolarized beam and I p is the losses of completely circularly polarized beam.

Notice,that the initial polarization ofγ-quanta is not needed to change for measuring of the asymmetry A s.So,letγ-beam is completely circularly polarized at the entry of the single crystal which have the orientation angles are equal to?H1and?V1.After measuring the intensity losses we change the single crystal orientation so,that new angles are equal to?H2=?V1and?V2=?H1.Then we get that|I1?I2|/0.5(I1+I2)=|I1?I2|/I0=2A s.

As already noted,the symmetry of problem have in general the dynamic character. It is means that our knowledge of such characteristics as refractive indices and X i-values depends on the possible of uncertainty in the initial data for calculations.We think that main uncertainty is connected with the electric?elds of single crystals.We carry out calculations of X2-value with the use of experimental silicon form factors[11,12],instead Moliere ones.The results of these calculations agree closely with each other.However,a little displacement of the singular axis direction(when X2=1)take a place(in this case W H=0.975,W V=1.075,instead W H=0.93,W V=1.10for Moliere potential).

4Discussion and Summary

The results of calculations in the silicon single crystal oriented near<001>axis show detectable sensitivity to the circular polarization of propagatingγ-quanta.Below we discuss some questions of the limitations and optimization.

Our consideration of the process are based on the traditional theory of the coherent e+e?-pair production in single crystals[3,4].However,this process is transformed into the analogous process in a”strong?eld”of the crystallographic axis[13]and the mathematical description of the both processes is di?erent.The process in the”strong?eld”take a place at anglesφB≤V a/mc2with respect to axis,where V a is the potential of this axis.In the case of<001>silicon axis this angle is equal to≈0.15mrad.It is means that the upper bound for our consideration(<001>,Si)is～500GeV.

We believe that the asymmetry of process A s on the same thickness is growing with theγ-quanta energy increasing.It is true because of the increasing of the cross section of theγ-beam absorption.As Fig.6illustrates,the detectable value A s take a place for orientations with not high quantity of the X2-value.This orientation region is more wide and?at then region near singular axis.It is important for measuring polarization of the γ-beam with a large angle divergence.

It is well-known that the<011>and<111>axes in silicon single crystals have more strong electric?elds,then<100>axis.Because of this,it is expected more high A s-values for these axes.

We believe that the theoretical prerequisites exist for creatingγ,e-polarimeters on the base of the considered here phenomenon.However,a lot of optimization calculations need to be done before for it.

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Figure1:The geometrical interpretation of permittivity tensor in anisotropic medium.Further explanations are given in the text.

Figure2:Real minus unit(1′,2′)and imaginary(1,2)parts of the refractive indices near of<001>-axis in silicon single crystal as a function of the parameter W H. Eγ=25GeV,W V=1.10.Temperature of single crystal T Si=300?K.

Figure3:Absolute quantity of normal wave circular polarization X2near of< 001>-axis in silicon single crystal as a function of the parameter W H.W V=1.10, 1.13,1.07for1,2,3-curves.T Si=300?K.

Figure4:Intensity of25GeVγ-quanta as a function of the single crystal thickness. Central curve is the intensity of initially unpolarized beam,upper and bottom curves are the intensities of beam with the corresponding initial circular polarization equal to-1and+1.W H=0.93,W V=1.10,T Si=300?K.

Figure5:Stokes parameter variations of25GeVγ-quanta as a function of the single crystal thickness.The?rst number in the parenthesis is initial circular polarization (1,0,-1),the second number is number of Stokes parameter i=1-3.W H=0.93,W V= 1.10,T Si=300?K.

Figure6:Asymmetry of the process as a function of the silicon single crystal thickness.Curve1is for W H=0.93,W V=1.10and curve2is for W H=2.0,W V= 1.10.Eγ=25GeV,T Si=300?K.