Annihilation of Relativistic Positrons in Single Crystal with production of One Photon

Transcription

1 Annihilation of Relativistic Positons in Single Cystal with poduction of One Photon Kalashnikov N.P.,Mazu E.A.,Olczak A.S. National Reseach Nuclea Univesity MEPhI (Moscow Engineeing Physics Institute), Moscow, Russia 1. Intoduction Nomally in the pocess of electon-positon annihilation when both leptons wee fee two photons ae poduced, as the annihilation pocess with poduction of one photon is fobidden by the consevation laws (fo example [1, 2]). But in case, when one o both leptons ae not fee (electon belongs to atom, fo example) this pocess may become possible [1, 3]. Anothe possibility fo this pocess opens in the case, when one of the leptons (e.g. high enegy elativistic positon) is moving in single cystal in channeling o quasi-channeling mode, when its motion is eithe bound o sufficiently uneven [4,5]. 2. Kinematics of the single photon annihilation of elativistic positon If the elativistic positon popagates along the axis ОZ with enegy >>m e c 2, the new poduced single photon must move also along the axis ОZ, what is evidently fobidden by momentum and enegy consevation laws: (enegy: + m e c 2 = ħω; momentum: P + = ħω /c. Consequence: P + c< < + m e c 2 = ħω =>ħω cannot be equal top + c). Momentum consevation law howeve can be satisfied in pojection upon axis ОZ, if the newly poduced photon popagates unde small angle θ to the axis ОZ: P + c= ħωcos θ ħω (1-θ 2 /2) <ħω (1) In this case tansvese momentum P x is not conseved, what is fobidden in vacuum. Howeve, it may be compensated by an inteaction with the cystal lattice, fo example if the positon moves in channeling o quasi-channeling mode, when its motion is eithe bound between neighboing cystal planes (channeling mode) o hoveing ove cystal planes (quasichanneling mode). Let us estimate fom consevation laws the possible angles θat which the photon may popagate in case of single-photon annihilation. When E + >>m e c 2, we may expect that θ<<1. Enegy consevation law looks like: E + + m e c 2 = ħω (2) Fo the longitudinal momentum P z the consevation law looks like: P z c= ħωcosθ ħω (1-θ 2 /2) (3) Enegy depends on momentum in elativistic case as: E + = (P 2 + с 2 + m 2 e c 4 ) 1/2 = (P 2 z с 2 + 2E + E x + m 2 e c 4 ) 1/2, (4) 1

2 wheee x - is the tansvese enegy of the positon, moving in aveaged continuous channeling potential [4,5].Compaing (2) and (4) the esult fo elativistic positon enegies will have the following fom: P z c(1 + E x /E + + m 2 e c 4 /2E 2 + ) +m e c 2 = ħω (5) Substituting P z c(3) we obtain: ħω - ħωθ 2 /2 + E x + m 2 e c 4 /2E + + m e c 2 = ħω (6) Taking into account, that E x <<m e c 2 [4,5] and m 2 e c 4 /2E + <<m e c 2 we get the esult: ħωθ 2 ~m e c 2 =>θ ~ (m e c 2 /ħω) 1/2 ~(m e c 2 /E + ) 1/2 >> m e c 2 /E + (7) 3. Some consideations fo plana channeling case Channeling phenomena occus when a chaged paticle entes a single cystal at a small angle with espect to the cystallogaphic axis o plane (fo example, [4,5,6] ). The entance angle θ must be smalle o compaable with the Lindhad angle[4,5,6], θ L ~ (2U 0 /E + ) 1/2 <<1, (8) wheeu 0 ~Ze 2 R at /d 2 is the effective depth of the continuous channeling potential (fo plana channeling), R at is the atom adius, d is the lattice constant (U 0 ~20-50 ev fo plana channeling in most of cystals [4]). The channeling lepton enegy E + must be much highe than the est enegy of the lepton E + >>m e c 2 = 511 kev. To poduce a photon with tansvese enegy ~ħωθ the initial positon should have the compaable value of tansvese momentum and enegy. (2E + E x ) 1/2 ~ħωθ~(m e c 2 E + ) 1/2 (9) Actually the tansvese enegy of positon in plana channeling potential is compaable to its height E + ~U 0 <<m e c 2. That means that in channeling mode the positon has much less tansvese enegy to satisfy the necessay condition (9), which stongly educes the possibility to poduce single photon in annihilation with fee electon, though this pocess is descibed by simple one vetex diagam Fig.1 Fig.1. Single-vetex Feynman diagam fo single-photon annihilation of channeled positon on fee electon. Howeve, the plana channeling motion povides yet one moe possibility to incease the coss section of single-photon annihilation. We will discuss it in the next section. 3. Specifics of positon motion in plana channeling mode 2

3 Motion of channeling paticles was thooughly studied in diffeent aspects by many eseaches in ies (see fo example [4,5,6]). The wave function of the positively chaged paticles which move in the plana channel can be pesented as ( = Σ k exp(ip z z)u k (x), (10) Fo the tansvesal component of the positon wave function u k (x) we may use the onedimensional Schödinge equation with elativistic mass E + [4, 5] and peiodical bode conditions u'' k (x) + 2E + (ε k U(x))u k (x) = 0 (11) u k (x+d) = u k (x) (12) wheeε k ae the allowed tansvesal enegies of the positon motion in the plana channel with aveaged potential U(x), numbeed by index k = 1,2,3 The tansvese enegies of a paticle in peiodical plana channeling potential fom a stuctue of enegy zones, which fo the deep unde-baie enegies can be consideed as discete levels. Calculation of zone stuctue with ealistic continuous potential by aveaging the potentials of atoms, constituting the cystal plane [4, 5,6] is a task fo numeical calculations.fig.2(a) shows numeical simulation of such zones fo 25 MeV positons in (110) plana channel in Si. Population of zones (Fig.2(b)) is depending on the incidence angle of the paticle elative to the cystal plane.fig.3 shows the numeical simulation of the aveaged wave function squae module (pobability to find a positon in this o that pat of a channel) fo diffeent zones.fo high enegy levels (whee quasi-classical appoach is applicable) the pobability density to find a paticle nea coodinate x is popotional to ~ (ε k U(x)) -1/2. That means that unde-baie paticles ae moving mostly close to the tuning points, whee ε k =U(x), and fo ove-baie paticles pobability density is maximal ove the baie. As a consequence - positons with close to the baie enegy levels (both ove and unde potential baie) ae hoveing ove the atomic planes, whee the electon density is ~d/r ~ 10 times highe, than aveage. Fo such paticles coss sections fo all the pocesses of inteactions with atomic electons, including the single-photon annihilation pocess on atomic electons, shall be multiplied by appoximately the same facto ~d/r. To achieve this it is necessay to populate by positons mostly close to the baie levels, what is possible when the entance angle of positon beam in elation to the atomic plane is close to the Lindhad angle [4,5]. As is known, the cystal potential V( ) fo cystals, such as silicon, gemanium, InSb, GaAs, should have the following fom if the point goup of the cystal contains the invesion V = V G G 3

4 , (13) ( ) = exp( ) = + 2 cos( ) V x V ig x V V G x Gx Gx x 0 Gx x. Gx> 0 Take concete lattice potential with the diamond stuctue in the definite diection of channeling leading to the equidistance channeling potential planes. Fo a significant pat of these cystals spatially inhomogeneous tems in the aveaged plana cystal potential (13) can be taken zeo, and the emaining tems ae expessed in tems of the Fouie components ( ) V4 n,0,0 n= 1,2,3... of the spatial lattice potential, so that V( x) = V + 2 V4 n,0,0 cos( 4 Gx). n= 1,2 Confining ouselves to the fist two tems in the sum (14), we can analytically conside the poblem of channeling of the chaged paticle. In this case, the equation (11) is educed to the known Mathieu equation [7] with the dimensionless coefficients (14) whee 2 U + 2 ( aɶ 2q cos 2S) U( S) = 0, (15) S EV0 EV aɶ = E / 4 ; G Għ c q= ; S = 2 Gx; G G min = 2 π / d; 2ħ c G 2ħ c G d is the lattice constant. In the case of channeled positively chaged paticles q> 0, sincev and V 400 ae lage than zeo. In the case of negatively chaged paticles V < 0 as V400. Afte the shift of the oigin of enegy q can be consideed positive and fo the negative paticles. The paamete q in both cases can be both smalle and geate than zeo, depending on the value of the channeling paticle (CP) tansvese enegy E ( P x ). Estimate the q value in the equation (16). At E= 28 MeV, V 15 ev (i.e., the depth of the potential wall is supposed to be equal to 30 ev, 10 G x = 10 1/ m), we obtain 11.2 E P x (16) q. The paamete q vaies depending on ( ), and can be both lage and small. The solutions of (15) have a zone Bloch chaacte. The boundaies of the enegy bands ae detemined by the well-known in the theoy of Mathieu functions [7] numbes, a and b + 1, whee = 0,1, 2..., and the allowed values of the CP tansvese enegy E ( P x ) should be detemined fom a E ( Px ) b+ 1 < < ( is the numbe of the allowed zone). It is clea that the efeencing of cetain aeas to the discete o continuous spectum is nominal and is detemined exclusively by the CP band width. Fig. 2 shows the plots of a and b + 1 on q 4

5 fo the most low-lying five bands [7]. As one can see, when q= 11.2 ( E 28 MeV), the fouth CP enegy band can not be consideed as a discete "level", while the fifth CP enegy band should emain a zone of the continuous spectum. Fig.2. (a) Enegy bands (levels) of tansvese motion of a fast oiented positons fo28 MeV in plana channeling along the plane (110) in single cystal Si in the appoximation of a sine cystal potentia l(with vetical line band spectum fo the enegy of the paticle E 25 MeV is maked, i.e.. q 0 = 10 ): quasi-momentum ( a, b, q in dime espect to the plane (110). - uppe bode of bands, b - lowe bode of bands, q - a ensionless units).(b) Population of enegy zones fo zeo entance angle of positon with Fig.3.The numeical simulation of the aveaged wave function squae module fo diffeent zones. Aveaged squae modules of odd and even wave functions fo 28 MeV positons in plana channeling along the plane (110) in single cystal Si fo diffeent enegy bands: (1) the 1-st deepest unde-baie level; (2) level 4 in the middle of the channel; (3) the highest unde- baie level; (4) the fist ove-baie band. Fom anothe hand, the paticles having low tansvese enegies ae moving between the atomic planes, whee the electon density is low and electons ae mostly covalent, having small binding enegy, which is not enough to open the possibility of one photon annihilation. 5

6 Thus the oientational dependence of the single photon annihilation of elativistic positons in single cystal would look like depicted on Fig.5 with the account to the elativistic positon matix elements behavio (Fig.4): Fig.4. Oientation dependence of the squaed positon even and odd matix elements on the positon entance angle θ measued in invese cystal lattice vectos. Fig.5. Oientation dependence of single-photon annihilation at the positon entance angle θ. 4. Conclusions As a esult of the quantum-mechanicaphoton annihilation of elativistic positons is a complicated function of the angle of incidence analysis we found that the dependence of the single- of the positon with espect to the cystallogaphic planes of the cystal. If the angle of incidence of the positon is smalle than the Lindhad angle, the pobability of the single-photon annihilation is depessed due to the low density of electons inthe channel between cystal 6

7 planes. At angles of incidence close to the Lindhad angle we see the incease in the pobability of single-photon annihilation due to hoveing of most of positons ove cystal planes with high concentation of atomic electons, as well as due to the behaviou of the quantum tansition matix elementsof the positon in the close to the baie states. Oientation behaviou of the single-photon annihilation of positons in a cystal is evidently ponounced and can easily be obseved in the expeiment. Refeences 1. Beestetskii V. B., Lifshitz E. M., Pitaevskii L.P. Quantum Electodynamics. Buttewoth-Heinemann, p. 2. Femi E., Uhlenbeck G.E. Phys. Rev., 1933, v.44, p Klepikov N. P. Zh. Eksp. Teo. Fiz., 1954.v.26, p Kalashnikov N.P. Coheent Inteactions of Chaged Paticles in Single Cystals. Hawood Academic Publishes. London-Pais-NewYok p. 5. Bayshevskii V.G. Channeling, Radiation and Reactions in Single Cystal at High Enegies. Minsk. Univesity. (in Russian) p. 6. Akhieze A.I., Shulga N.F. Electodynamics in Matte at High Enegies. Academic Publishes. Science. (in Russian) p. 7. Handbook of mathematical functions. Ed. M. Abamowitz, I.A.Stegun. National Bueau of Standads, Applied Mathematics Seies,