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Secondary electron emission induced by channeled relativistic electrons in a (1 1 0) Si crystal
Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Secondary electron emission induced by channeled relativistic electrons in a (1 1 0) Si crystal K.B. Korotchenko a,⇑, Yu P. Kunashenko a,b, T.A. Tukhfatullin a a b
Department of Theoretical and Experimental Physics, National Research Tomsk Polytechnic University, 30, Lenin Avenue, Tomsk 634050, Russia Department of Theoretical Physics, Tomsk State Pedagogical University, 60, Kievskaya Street, Tomsk 634041, Russia
a r t i c l e
i n f o
Article history: Received 23 November 2011 Available online 18 January 2012 Keywords: Channeling electrons Auger emission Quantum electrodynamics Bloch wave function
a b s t r a c t A new effect that accompanies electrons channeled in a crystal is considered. This phenomenon was previously predicted was called channeling secondary electron emission (CSEE). The exact CSEE crosssection on the basis of using the exact Bloch wave function of electron channeled in a crystal is obtained. The detailed investigation of CSEE cross-section is performed. It is shown that angular distribution of electrons emitted due to CSEE has a complex form. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction A new effect which accompanies electrons channeled in a crystal is channeling secondary electron emission (CSEE) [1]. CSEE still experimentally is not studied. This effect is similar to the Auger effect in atomic physics with the replacement of the primary atomic electron by a channeled electron and the secondary atomic electron is replaced by electron emitted by one of the crystal atoms. In comparison with the ordinary Auger effect, during CSEE the secondary electron can be produced by two types of transitions of channeled electrons (positrons). The first type takes place when the channeled electron remains in the same quantum state. It is a radiationless transition which is connected with the redistribution of channeled particles in a crystal. This effect is fairly well studied for non-relativistic and relativistic particles [2,3] and results in the appearance of orientation dependence of the K-ionization and characteristic X-ray radiation. A channeled electron changes its quantum state during the second type of transition, and emits a virtual photon. The virtual photon is absorbed by an electron of a target atom and an atom electron is emitted as a photoelectron. As a result one can suppose both the appearance of orientation dependence and an increase in the cross-section of K-ionization and characteristic X-ray radiation yield. Moreover the energy and angular distributions of the emitted electrons can be changed in comparison with distributions in amorphous target.
The CSEE, which corresponds to the second type of transition, was predicted in [1], where the qualitative estimation of CSEE was done. The goal of the present paper is a detailed investigation of CSEE cross-section on the basis of using the exact Bloch wave function of electrons (positrons) channeled in a crystal. Similar to CSEE, electron–positron trident pair production was considered in [4]. 2. CSEE cross-section The CSEE is a process of quantum electrodynamics which cannot be described in a framework of the ordinary perturbation theory. Therefore we should redefine Feynman rules for the channeled particles (see for example [4]). A new Feynman diagram for the channeling secondary electron emission in a first order over a (fine-structure constant) is presented in Fig. 1. The thick triple arrows show the initial Wi ð~ r1 Þ and final Wf ð~ r1 Þ channeled electron states, double arrow – crystal atom electron on a K-shell WK ð~ r 2 Þ, thin arrow – free electron emitted during CSEE process (CSEE electron) Wð~ r2 Þ and wavy line – photon propagator. According to the Feynman diagram, the channeling secondary electron emission cross-section is ( h ¼ c ¼ 1 units are used)
dr ¼ 2pjM if j2 dððEik þ ei? Þ ðEf k þ ef ? Þ E EK Þ
0168-583X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2012.01.006
ð1Þ
Here,
Mif ¼ e2
Z
ð~ f ð~ r2 W r1 Þ cl Wi ð~ r 1 Þ Dlm ð~ r1 ;~ r2 Þ W r 2 Þ cm WK ð~ r 2 Þ; d~ r 1 d~ ð2Þ
⇑ Corresponding author. Tel.: +7 3822 563 752. E-mail address: [email protected] (K.B. Korotchenko).
1 dq: J
is a matrix element of CSEE.
15
K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Wð~ r 2 Þ ¼ u exp½i~ p ~ r2 ; u ¼
Fig. 1. Feynman diagram for the channeling secondary electron emission in a first order over a (fine-structure constant).
In Eq. (1) and (2) indexes (l, m) are ones of three dimensional vectors, it implies a summation over the repeated Greek indexes, indexes (i, f) denote initial (i) and final (f) channeled electron states. Further: Wi ð~ r1 Þ and Wf ð~ r1 Þ are the wave functions (4-spinor) of channeled electron in initial and final states, WK ð~ r 2 Þ is the wave function of crystal atom bound electron and Wð~ r 2 Þ is the wave function of CSEE electron, ~ r1 and ~ r2 are corresponding radius vectors. Than Eik þ ei? ðEf k þ ef ? Þ are initial (final) electron total energy with Ei|| being longitudinal and ei? being transverse energies. Here and below the index || denotes longitudinal components of any vectors which parallel to the channeling plane of electron and index \ denotes transverse components of any vectors which are perpendicular to the channeling plane. Next, E is the emitted electron energy and EK the energy of a crystal atom bound electron, cl the Dirac matrix, J ¼ j~ pik j=ðEik dp L2 Þ the initial electron flux (~ pik the channeled electron longitudinal momentum in the initial state, dp a distance between neighbor planes and L2 the normalization area in the channeling plane, which will be equal to unit below), dq the phase volume connected with final electron state and emitted free electron:
dq ¼
d~ pf k
d~ p
ð2pÞ2 ð2pÞ3
ð3Þ
:
WK ð~ r 2 Þ ¼ uK UK ð~ r 2 Þ; uK ¼
1 2
x2if k i0
dlm
kl km
x
2 if
!
d~ k exp½i~ kð~ r2 ~ r 1 Þ : ð2pÞ3 ð4Þ
Here xif = (Ei|| + ei\) (Ef|| + ef\), and ~ k is the transferred momentum vector. We use the channeled electron wave function in an initial bound state i in a form [6–7]:
Wi ð~ r 1 ; tÞ ¼ Ui ð~ r 1 ÞexpðiEik tÞ;
Ui ð~ rÞ ¼
qffiffiffiffiffiffiffiffiffi
mþEik ui Eik
ui ðx1 Þ expði~ pik~ r 1k Þ; ui ¼
w ~ r~p^ mþEik
!
:
ð5Þ
Here /i(x1) is the transverse wave function of the channeled electron bound state; ~ r 1 ¼ ðx1 ;~ r 1k Þ and x1 is the channeled electron coordinate in the direction perpendicular to the crystal plane; the exponent expði~ pik~ r1k Þ describes free electron motion along the r1k are the projections of the channeled eleccrystal plane; ~ pik and ~ tron momentum and its radius vector onto crystal plane; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eik ¼ p2ik þ m2 is the channeled electron longitudinal energy, w is the two dimensional spinor normalized by the condition w+w = 1. Similarly notations are used for the final state f. We restrict our consideration to the case when the CSEE emitted electron is non-relativistic. In a common case its wave function should be a Bloch wave function. But for simplicity we take this wave function as a plane wave:
wþ w ¼ 1:
ð6Þ
w 0
;
wþ w ¼ 1:
ð7Þ
Therefore one has:
ð~ exp½i~ W r 2 Þ cm WK ð~ r2 Þ ¼ u p ~ r2 c0 cm uK UK ð~ r2 Þ p ~ r 2 a~ uK UK ð~ r2 Þ ¼ uþ exp½i~ ¼ uþ exp½i~ p ~ r 2 p^m uK UK ð~ r 2 Þ=m ¼ exp½i~ p ~ r 2 p^m UK ð~ r 2 Þ=m:
ð8Þ
Here we take into account that in the non-relativistic limit ^ is the electron velocity operator and u+u = w+w = 1. ~ a¼~ v^ ¼ ~p=m K After the substitution of Eqs. (4), (5), and (8) into (2) we arrive at
Mif ¼
Ce2 2p 2 m
Z
r 2 d~ k uf /f ðx1 Þcl ui /i ðx1 Þ dx1 d~ ! kl km 1 k~ r 2 k?x1 Þexp½i~ 2 dlm 2 exp½ið~ p ~ r2 xif k x2if Z þ1 p^m UK ð~ r2 Þ d~ r 1k exp½i~ r1k ð~ pik ~ pf k ~ kk Þ; m 1
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with C ¼ ðm þ Eik Þðm þ Ef k Þ=4Ef k Eik . The integration of Eq. (9) over ~ d~ r 1k dk result in:
Here ~ p is the momentum of CSEE emitted electron, d~ pf k the channeled electron longitudinal momentum in the final state f. The photon propagator Dlm ð~ r 1 ;~ r2 Þ is chosen in a form [5]:
Z
w ; 0
The wave function of the crystal atom bound electron is the electron wave function on the K-shell multiplied by the non relativistic spinor:
Mif ¼
Dlm ð~ r 1 ;~ r 2 Þ ¼ 4p
2ip 2 Ce mbif
dlm
Z
f /f ðx1 Þcl ui /i ðx1 Þexpðibif x1 Þ dx1 u
jl jm x2if
!Z
exp½ið~ p~ jÞ ~r2 p^m UK ð~r2 Þd~r2 :
ð10Þ
Here we introduce new vector ~ j ¼ fbif ; D~ pif k g, and following notations b2if ¼ x2if Dp2if k , D~ pik ~ pf k . pif k ¼ ~ For the electron on the K-shell the wave function does not depend on angle variables, i.e., UK ð~ r2 Þ ¼ UK ðjr2 jÞ=4p. For the further calculation the function UK(r2) is borrowed from the paper [8]:
UK ðr 2 Þ ¼
X
C snk ½ð2nks Þ!1=2 ð21ks Þ1=2þnks r2nks 1 expð1ks r 2 Þ:
ð11Þ
s
Here C snk , nks and 1ks are parameters defined in [8]. The integration over d~ r 2 in (10) can be performed analytically
JK ¼ ¼
Z
exp½ið~ p~ jÞ ~r2 UK ð~r2 Þd~r2
ffi X 21=2þnos pffiffiffiffiffiffi 10s C s10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð2n0s Þ! s Cð1 þ n0s Þ j~ p~ jj 1 þ nos sin arctg : 10s ½1 þ ðj~ j~ p~ jj p~ jj=10s Þ2 ð1þn0s Þ=2 ð12Þ
Here C(x) is Euler gamma function (for the K-shell n = 1 and k ¼ 0). f cui in (10) using the spinor (5) after For the matrix product u some algebra we obtain
f cl ui ¼ u
^f ~ ^i ~ p pf p pi : þ ; þ Ei þ m Ef þ m Ei þ m Ef þ m
ð13Þ k
~ It is convenient to introduce following vectors ~ I1 ¼ ðI? 1 ; I1 Þ and ? ~k ~ I2 ¼ ðI2 ; I2 Þ with components:
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Z
1 ^? uf ðx1 ÞÞui ðx1 Þexp½ibif x1 dx1 ; ðp Ef k þ m Z ~ pf k ~ Ik1 ¼ uf ðx1 Þui ðx1 Þexp½ibif x1 dx1 : Ef k þ m I?1 ¼
Z 1 uf ðx1 Þðp^? ui ðx1 ÞÞexp½ibif x1 dx1 ; Eik þ m Z ~ pik ~ uf ðx1 Þui ðx1 Þexp½ibif x1 dx1 : Ik2 ¼ Eik þ m
ð14Þ
To solve the problem (19) we make use of an earlier development by the authors [13–14] algorithm for MathematicaÓ 7.0. For describing the potential U(x1) we have used the formula which was obtained by Doyle and Turner [15] for the potential of a single atom
X 3=2 2 pffiffiffiffi U At ð~ rÞ ¼ ð2h = pme Þ aj cj expðr 2 =cj Þ;
I?2 ¼
ð20Þ
j
ð15Þ
Now finally we can write the matrix element of CSEE in a compact form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # 1 2p 2 ðm þ Eik Þðm þ Ef k Þ 1 e ð~ p~ Mif ¼ i pffiffiffi jÞ 2 ðð~ p~ jÞ~ jÞ~ j ð~I1 þ~I2 ÞJK : Ef k Eik xif 2 bif m ð16Þ
After substitution of the matrix element (16) in (1) we find exact formula for the cross-section of CSEE from the channeled electron transition from the bound state i to the state f.
where parameters aj and bj = 4p2cj are the coefficients, obtained by the fitting of the electron scattering form factor to the experimental values, r 2 ¼ x2 þ r2? . In order to have more correct approximation we use 6 point in comparison with standard Doyle–Turner procedure with 4 points. After integrating this potential over r|| = (y, z) we obtain the continuous potential for a single crystal plane. Next, we have summed the potentials of the single crystallographic planes, taking into account the symmetry of a particular crystal. Having an analytical expression for the continuous potential of the planes U(x1), it is easy to find Fourier components Um (details, see in Ref. [16]). 4. Numerical results
drif ¼
4p2 d b2if m2
e
4
ðm þ Eik Þðm þ Ef k Þ Ef k Eik
("
) # 2 1 ~ ~ ~ ~ ~ ~ ~ ~ ðp jÞ 2 ððp jÞjÞj I1 þ I2 J 2K
xif
dððEik þ ei? Þ ðEf k þ ef ? Þ E EK Þ
pf k d~ p Eik d~ : j~ pik j ð2pÞ5 ð17Þ
Recall that ~ j ¼ fbif ; D~ pif k g and x2if ¼ b2if þ Dp2if k . Further calculations are carried out using the package MathematicaÓ 7.0. 3. Electron planar channeling in a crystal: theoretical background For the stationary states of an electron its longitudinal motion along YZ–plane is described by the plane wave expði~ pk~ r k Þ. If E|| (e\ - U(x1)) and e\ is the energy of the electron transverse motion, than transverse motion of the electron is described by the transverse wave function /(x1) which is the solution of the Schrödinger-like equation with a relativistic mass cme [4,7]. The average potential of the system of crystal planes U(x1) is a periodic function over the coordinate x, therefore the transverse wave function /(x) and the energy of transverse motion of electron e\ are continuous functions of the electron wave vector component kx, which lie in the first Brillouin zone (e.g., see [9,10]). That means the wave function /(x1) is the Bloch one:
ui ðx1 ; k? Þ ¼
X
C i ðg ?m ; k? Þexpfiðk? þ g ?m Þx1 g:
We carry out the numerical calculation for the case of electrons channeled along the (1 1 0) plane in a Si crystal. The longitudinal electron velocity is parallel to the OZ axis. All calculations are performed taking into account all possible transitions of the channeled electron from the quantum state i to the state f. Fig. 2 shows the result of our calculation of the three dimensional angular distribution of CSEE electrons (differential crosssection dr/(dpdX)). The momentum of CSEE electron is equal to p = cpe = 15 keV (i.e., kinetic electron energy E p2e =2m 220 eV; here pe = mcbc is an electron momentum in the ordinary units). This momentum corresponds to maximal value of differential cross-section of CSEE (for the channeled electron relativistic factor is c = 100). Fig. 3 shows a sectional view of Fig. 2 along the plane XOZ. From Fig. 3 it follows that the cross-section has a maximum for a CSEE electron emission angle h near the values +76° and 71°. The angle h is the angle between the momentum of the secondary electron and the axis OZ. The asymmetry of the angular distribution of the secondary electrons with respect to the OZ axis arises from the asymmetry of the formula (17) by the vector ð~ p~ jÞ. The vector ~ pif k g does not depend on the angle h. j ¼ fbif ; D~ Fig. 4 plots the maximal value of the differential cross-section dr/(dpdX) (h = +76°) of CSEE as a function of CSEE electron momentum for the different relativistic factor c = 20...100 of the channeled electrons. From Fig. 4 one can see that for all values of the relativistic factor c the maximum of the cross-section appears
ð18Þ
m
Here g\m = g\m, and g\ is one dimension reciprocal lattice vector perpendicular to the crystal plane. Using the Fourier expansion of the periodic potential of the crystal plane system P U(x1) = mUm exp {i mg\x1}, we obtain the algebraic eigensystem problem [11,12]:
P m
Amn C i ðg ?m ; k? Þ ¼ ei? ðk? ÞC i ðg ?n ; k? Þ;
ð19Þ
2
Amn ¼ U mn þ dðm; nÞð h jng ? þ k? j2 =2me cÞ: Here Ci(gxm, kx) are the Fourier components of the electron wave function /i(x, kx) and Um the Fourier components of the periodic potential function U(x) which describes average potential of the system of crystal planes, d(a, b) is the Kronecker delta.
Fig. 2. Indicatrix of CSEE electrons with kinetic energy E 220 eV produced by planar channeled electrons with relativistic factor c = 100.
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Fig. 5. Maximal value of the differential cross-section dr/(dpdX) (h = +76, p = cpe = 15 keV) of CSEE as a function of relativistic factor c of the channeled electron. Fig. 3. Section of Fig. 2 along the plane XOZ. Table 2 The total cross-section CSEE.
c
r, barn
20 30 40 60 100
34 64 74 156 240
According to Figs. 4 and 5 we can estimate the total cross-section CSEE – the result is shown in Table 2. 5. Conclusion
Fig. 4. Maximal value of differential cross-section dr/(dpdX) (h = +76°) CSEE as a function of CSEE electron momentum for different relativistic factor c = 20...100 of the channeled electrons.
Table 1 Connection between the numerical values of the energy E and momentum p. Momentum p = cpe, keV
Kinetic energy E, eV
5 10 15 20 50
24, 46 97, 85 220, 2 391, 4 2446
at one point (marked on the axis) at p = cpe = 15 keV (i.e., kinetic electron energy E 220 eV). For reference, in Table 1 it is shown the corresponding numerical values of energy E and momentum p = cpe CSEE-electrons. Fig. 5 plots the maximal value of the differential cross-section dr/(dpdX) (h = +76°, p = cpe = 15 keV) of the CSEE-electrons as a function of relativistic factor c of the the channeled electron. The complex behavior of the cross-section dependence on the initial electron relativistic factor can be explained quite simply: at c = 39 and c = 98 the transitions (from new odd levels 5 and 7, respectively) should contribute to the cross-section. (The numerical calculation shows that contribution to the cross-section due to the transition from even levels is smaller than from odd ones).
Thus, within a QED formalism, the exact formula for the crosssection of new process – channeling secondary electron emission is found. The results of our calculations show the following. At planar channeling, CSEE electrons are emitted in the direction close to a direction perpendicular to channeling plane (at angles +76° and 71° with respect to the axis OZ – direction of the longitudinal motion of the channeled electron). The differential cross-section dr/(dpdX) of CSEE has a maximum at a rather small momentum of CSEE-electron p = cpe = 15 keV (i.e., kinetic electron energy E 220 eV). The position of maximum does not depend on relativistic factor c of initial electron. The CSEE cross-section has a maximal value at a fixed emission angle of the CSEE-electron. For all values of the relativistic factor c the maximum of the CSEE crosssection has same point at p = cpe = 15 keV (kinetic electron energy E 220 eV). This is explained by the fact that properties of the CSEE electron are mainly determined by the characteristics of the photoionization process by a virtual photon. Due to the fact that differential cross-section dr/(dpdX) does not depend neither on the energy of the channeled electron, nor on the CSEE-electron energy, one can obtain an approximate relation between the total and differential cross-sections of CSEE r(c) = dr(c)/(dpdX) G, here G 6 1014 sr eV. The contribution of CSEE in the appearance of the secondary electron in comparison with radiationless transition is approximately 15% (see for example [17]). But it is shown our estimation of its contribution should be a function of the initial angles of the channeled electrons. As is well known, the probability of electron capture into the channeling states is increased with decreasing electron emission angles with respect to the crystal plane. This phenomenon should result in the appearance of an orientation dependence of the CSEE phenomena. This orientation dependence will be the subject of our future investigation. Our
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
calculation shows that most of the CSEE electrons have a small kinetic energy E p2e =2m 220 eV. This fact did not allow the CSEE electrons to leave the crystal target. In a real experiment one can be conclude about existence of CSEE by the appearance of characteristic X-ray radiation. The properties of the characteristic X-ray radiation cross-section (such as dependence on channeled electron energy and orientation dependence) should be similar to the CSEE cross-section ones. Acknowledgments The authors are grateful to Prof. Yu. L. Pivovarov for useful discussions. The work is partially supported by RFBR, Grant 10-02-01386-a; by Russian Education Federal Agency program «Support of Scientific Schools», Contract 3558.2010.2 References [1] H. Nitta, Y.H. Ohtsuki, Phys. Rev. B 39 (1989) 2051–2053. [2] N.P. Kalashnikov, Coherent interactions of charged particles in single crystals: scattering and radiative processes in single crystals, Harwood Academic Publishers, New York, 1989.
[3] A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 372 (2008) 4451–4461. [4] J.C. Kimball, N. Cue, Phys. Rep. 125 (2) (1985) 69–101. [5] V.B. Berestetskii, T.M. Lifshiys, L.P. Pitaevskii, Quantum Electrodynamics, Pergamon, Oxford, 1975. [6] M.A. Rumarhov, Phys. Lett. Ser. A 57 (1976) 17. [7] V.A. Bazulev, N.K. Zhevago, Radiation of Fast Particles in Matter and External Fields, Nauka, Moscow, 1987 (in Russian). [8] E. Clementi, C. Roetti, Atom Data and Nuclear Tables 14 (1974) 177–478. [9] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1979. [10] C. Kittel, Quantum Theory of Solids, John Wiley & Sons, Inc., New York, London, 1963. [11] B.F. Buxtont, J.E. Loveluck, J. Phys. C: Solid State Phys. 10 (1977) 3941. [12] R.K. Klein, J.O. Kephart, R.H. Pantell, H. Park, B.L. Berman, R.L. Swent, S. Datz, R.W. Fearick, Phys. Rev. B 31 (1985) 68. [13] O.V. Bogdanov, K.B. Korotchenko, Yu.L. Pivovarov, JETP Lett. 85 (No. 11) (2007) 555. [14] K.B. Korotchenko, Yu.L. Pivovarov, T.A. Tukhfatullin, Nucl. Instr. Meth. B 266 (2008) 3753. [15] P.A. Doyle, P.S. Turner, Acta Cryst. A 24 (1967) 390. [16] K.B. Korotchenko, E.I. Fiks, Yu.L. Pivovarov, T.A. Tukhfatullin, J. Phys.: Conf. Ser. 236 (2010) 012016. [17] A.V. Schagin, V.V. Sotnikov, The Journal of Kharkiv National University, No.777, Physical Series ‘‘Nuclei, Particles, Fields’’, Issue 2/34/, Kharkov, 2007, p. 97–101.
Contents lists available at SciVerse ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Secondary electron emission induced by channeled relativistic electrons in a (1 1 0) Si crystal K.B. Korotchenko a,⇑, Yu P. Kunashenko a,b, T.A. Tukhfatullin a a b
Department of Theoretical and Experimental Physics, National Research Tomsk Polytechnic University, 30, Lenin Avenue, Tomsk 634050, Russia Department of Theoretical Physics, Tomsk State Pedagogical University, 60, Kievskaya Street, Tomsk 634041, Russia
a r t i c l e
i n f o
Article history: Received 23 November 2011 Available online 18 January 2012 Keywords: Channeling electrons Auger emission Quantum electrodynamics Bloch wave function
a b s t r a c t A new effect that accompanies electrons channeled in a crystal is considered. This phenomenon was previously predicted was called channeling secondary electron emission (CSEE). The exact CSEE crosssection on the basis of using the exact Bloch wave function of electron channeled in a crystal is obtained. The detailed investigation of CSEE cross-section is performed. It is shown that angular distribution of electrons emitted due to CSEE has a complex form. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction A new effect which accompanies electrons channeled in a crystal is channeling secondary electron emission (CSEE) [1]. CSEE still experimentally is not studied. This effect is similar to the Auger effect in atomic physics with the replacement of the primary atomic electron by a channeled electron and the secondary atomic electron is replaced by electron emitted by one of the crystal atoms. In comparison with the ordinary Auger effect, during CSEE the secondary electron can be produced by two types of transitions of channeled electrons (positrons). The first type takes place when the channeled electron remains in the same quantum state. It is a radiationless transition which is connected with the redistribution of channeled particles in a crystal. This effect is fairly well studied for non-relativistic and relativistic particles [2,3] and results in the appearance of orientation dependence of the K-ionization and characteristic X-ray radiation. A channeled electron changes its quantum state during the second type of transition, and emits a virtual photon. The virtual photon is absorbed by an electron of a target atom and an atom electron is emitted as a photoelectron. As a result one can suppose both the appearance of orientation dependence and an increase in the cross-section of K-ionization and characteristic X-ray radiation yield. Moreover the energy and angular distributions of the emitted electrons can be changed in comparison with distributions in amorphous target.
The CSEE, which corresponds to the second type of transition, was predicted in [1], where the qualitative estimation of CSEE was done. The goal of the present paper is a detailed investigation of CSEE cross-section on the basis of using the exact Bloch wave function of electrons (positrons) channeled in a crystal. Similar to CSEE, electron–positron trident pair production was considered in [4]. 2. CSEE cross-section The CSEE is a process of quantum electrodynamics which cannot be described in a framework of the ordinary perturbation theory. Therefore we should redefine Feynman rules for the channeled particles (see for example [4]). A new Feynman diagram for the channeling secondary electron emission in a first order over a (fine-structure constant) is presented in Fig. 1. The thick triple arrows show the initial Wi ð~ r1 Þ and final Wf ð~ r1 Þ channeled electron states, double arrow – crystal atom electron on a K-shell WK ð~ r 2 Þ, thin arrow – free electron emitted during CSEE process (CSEE electron) Wð~ r2 Þ and wavy line – photon propagator. According to the Feynman diagram, the channeling secondary electron emission cross-section is ( h ¼ c ¼ 1 units are used)
dr ¼ 2pjM if j2 dððEik þ ei? Þ ðEf k þ ef ? Þ E EK Þ
0168-583X/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2012.01.006
ð1Þ
Here,
Mif ¼ e2
Z
ð~ f ð~ r2 W r1 Þ cl Wi ð~ r 1 Þ Dlm ð~ r1 ;~ r2 Þ W r 2 Þ cm WK ð~ r 2 Þ; d~ r 1 d~ ð2Þ
⇑ Corresponding author. Tel.: +7 3822 563 752. E-mail address: [email protected] (K.B. Korotchenko).
1 dq: J
is a matrix element of CSEE.
15
K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Wð~ r 2 Þ ¼ u exp½i~ p ~ r2 ; u ¼
Fig. 1. Feynman diagram for the channeling secondary electron emission in a first order over a (fine-structure constant).
In Eq. (1) and (2) indexes (l, m) are ones of three dimensional vectors, it implies a summation over the repeated Greek indexes, indexes (i, f) denote initial (i) and final (f) channeled electron states. Further: Wi ð~ r1 Þ and Wf ð~ r1 Þ are the wave functions (4-spinor) of channeled electron in initial and final states, WK ð~ r 2 Þ is the wave function of crystal atom bound electron and Wð~ r 2 Þ is the wave function of CSEE electron, ~ r1 and ~ r2 are corresponding radius vectors. Than Eik þ ei? ðEf k þ ef ? Þ are initial (final) electron total energy with Ei|| being longitudinal and ei? being transverse energies. Here and below the index || denotes longitudinal components of any vectors which parallel to the channeling plane of electron and index \ denotes transverse components of any vectors which are perpendicular to the channeling plane. Next, E is the emitted electron energy and EK the energy of a crystal atom bound electron, cl the Dirac matrix, J ¼ j~ pik j=ðEik dp L2 Þ the initial electron flux (~ pik the channeled electron longitudinal momentum in the initial state, dp a distance between neighbor planes and L2 the normalization area in the channeling plane, which will be equal to unit below), dq the phase volume connected with final electron state and emitted free electron:
dq ¼
d~ pf k
d~ p
ð2pÞ2 ð2pÞ3
ð3Þ
:
WK ð~ r 2 Þ ¼ uK UK ð~ r 2 Þ; uK ¼
1 2
x2if k i0
dlm
kl km
x
2 if
!
d~ k exp½i~ kð~ r2 ~ r 1 Þ : ð2pÞ3 ð4Þ
Here xif = (Ei|| + ei\) (Ef|| + ef\), and ~ k is the transferred momentum vector. We use the channeled electron wave function in an initial bound state i in a form [6–7]:
Wi ð~ r 1 ; tÞ ¼ Ui ð~ r 1 ÞexpðiEik tÞ;
Ui ð~ rÞ ¼
qffiffiffiffiffiffiffiffiffi
mþEik ui Eik
ui ðx1 Þ expði~ pik~ r 1k Þ; ui ¼
w ~ r~p^ mþEik
!
:
ð5Þ
Here /i(x1) is the transverse wave function of the channeled electron bound state; ~ r 1 ¼ ðx1 ;~ r 1k Þ and x1 is the channeled electron coordinate in the direction perpendicular to the crystal plane; the exponent expði~ pik~ r1k Þ describes free electron motion along the r1k are the projections of the channeled eleccrystal plane; ~ pik and ~ tron momentum and its radius vector onto crystal plane; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eik ¼ p2ik þ m2 is the channeled electron longitudinal energy, w is the two dimensional spinor normalized by the condition w+w = 1. Similarly notations are used for the final state f. We restrict our consideration to the case when the CSEE emitted electron is non-relativistic. In a common case its wave function should be a Bloch wave function. But for simplicity we take this wave function as a plane wave:
wþ w ¼ 1:
ð6Þ
w 0
;
wþ w ¼ 1:
ð7Þ
Therefore one has:
ð~ exp½i~ W r 2 Þ cm WK ð~ r2 Þ ¼ u p ~ r2 c0 cm uK UK ð~ r2 Þ p ~ r 2 a~ uK UK ð~ r2 Þ ¼ uþ exp½i~ ¼ uþ exp½i~ p ~ r 2 p^m uK UK ð~ r 2 Þ=m ¼ exp½i~ p ~ r 2 p^m UK ð~ r 2 Þ=m:
ð8Þ
Here we take into account that in the non-relativistic limit ^ is the electron velocity operator and u+u = w+w = 1. ~ a¼~ v^ ¼ ~p=m K After the substitution of Eqs. (4), (5), and (8) into (2) we arrive at
Mif ¼
Ce2 2p 2 m
Z
r 2 d~ k uf /f ðx1 Þcl ui /i ðx1 Þ dx1 d~ ! kl km 1 k~ r 2 k?x1 Þexp½i~ 2 dlm 2 exp½ið~ p ~ r2 xif k x2if Z þ1 p^m UK ð~ r2 Þ d~ r 1k exp½i~ r1k ð~ pik ~ pf k ~ kk Þ; m 1
ð9Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with C ¼ ðm þ Eik Þðm þ Ef k Þ=4Ef k Eik . The integration of Eq. (9) over ~ d~ r 1k dk result in:
Here ~ p is the momentum of CSEE emitted electron, d~ pf k the channeled electron longitudinal momentum in the final state f. The photon propagator Dlm ð~ r 1 ;~ r2 Þ is chosen in a form [5]:
Z
w ; 0
The wave function of the crystal atom bound electron is the electron wave function on the K-shell multiplied by the non relativistic spinor:
Mif ¼
Dlm ð~ r 1 ;~ r 2 Þ ¼ 4p
2ip 2 Ce mbif
dlm
Z
f /f ðx1 Þcl ui /i ðx1 Þexpðibif x1 Þ dx1 u
jl jm x2if
!Z
exp½ið~ p~ jÞ ~r2 p^m UK ð~r2 Þd~r2 :
ð10Þ
Here we introduce new vector ~ j ¼ fbif ; D~ pif k g, and following notations b2if ¼ x2if Dp2if k , D~ pik ~ pf k . pif k ¼ ~ For the electron on the K-shell the wave function does not depend on angle variables, i.e., UK ð~ r2 Þ ¼ UK ðjr2 jÞ=4p. For the further calculation the function UK(r2) is borrowed from the paper [8]:
UK ðr 2 Þ ¼
X
C snk ½ð2nks Þ!1=2 ð21ks Þ1=2þnks r2nks 1 expð1ks r 2 Þ:
ð11Þ
s
Here C snk , nks and 1ks are parameters defined in [8]. The integration over d~ r 2 in (10) can be performed analytically
JK ¼ ¼
Z
exp½ið~ p~ jÞ ~r2 UK ð~r2 Þd~r2
ffi X 21=2þnos pffiffiffiffiffiffi 10s C s10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pð2n0s Þ! s Cð1 þ n0s Þ j~ p~ jj 1 þ nos sin arctg : 10s ½1 þ ðj~ j~ p~ jj p~ jj=10s Þ2 ð1þn0s Þ=2 ð12Þ
Here C(x) is Euler gamma function (for the K-shell n = 1 and k ¼ 0). f cui in (10) using the spinor (5) after For the matrix product u some algebra we obtain
f cl ui ¼ u
^f ~ ^i ~ p pf p pi : þ ; þ Ei þ m Ef þ m Ei þ m Ef þ m
ð13Þ k
~ It is convenient to introduce following vectors ~ I1 ¼ ðI? 1 ; I1 Þ and ? ~k ~ I2 ¼ ðI2 ; I2 Þ with components:
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Z
1 ^? uf ðx1 ÞÞui ðx1 Þexp½ibif x1 dx1 ; ðp Ef k þ m Z ~ pf k ~ Ik1 ¼ uf ðx1 Þui ðx1 Þexp½ibif x1 dx1 : Ef k þ m I?1 ¼
Z 1 uf ðx1 Þðp^? ui ðx1 ÞÞexp½ibif x1 dx1 ; Eik þ m Z ~ pik ~ uf ðx1 Þui ðx1 Þexp½ibif x1 dx1 : Ik2 ¼ Eik þ m
ð14Þ
To solve the problem (19) we make use of an earlier development by the authors [13–14] algorithm for MathematicaÓ 7.0. For describing the potential U(x1) we have used the formula which was obtained by Doyle and Turner [15] for the potential of a single atom
X 3=2 2 pffiffiffiffi U At ð~ rÞ ¼ ð2h = pme Þ aj cj expðr 2 =cj Þ;
I?2 ¼
ð20Þ
j
ð15Þ
Now finally we can write the matrix element of CSEE in a compact form: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" # 1 2p 2 ðm þ Eik Þðm þ Ef k Þ 1 e ð~ p~ Mif ¼ i pffiffiffi jÞ 2 ðð~ p~ jÞ~ jÞ~ j ð~I1 þ~I2 ÞJK : Ef k Eik xif 2 bif m ð16Þ
After substitution of the matrix element (16) in (1) we find exact formula for the cross-section of CSEE from the channeled electron transition from the bound state i to the state f.
where parameters aj and bj = 4p2cj are the coefficients, obtained by the fitting of the electron scattering form factor to the experimental values, r 2 ¼ x2 þ r2? . In order to have more correct approximation we use 6 point in comparison with standard Doyle–Turner procedure with 4 points. After integrating this potential over r|| = (y, z) we obtain the continuous potential for a single crystal plane. Next, we have summed the potentials of the single crystallographic planes, taking into account the symmetry of a particular crystal. Having an analytical expression for the continuous potential of the planes U(x1), it is easy to find Fourier components Um (details, see in Ref. [16]). 4. Numerical results
drif ¼
4p2 d b2if m2
e
4
ðm þ Eik Þðm þ Ef k Þ Ef k Eik
("
) # 2 1 ~ ~ ~ ~ ~ ~ ~ ~ ðp jÞ 2 ððp jÞjÞj I1 þ I2 J 2K
xif
dððEik þ ei? Þ ðEf k þ ef ? Þ E EK Þ
pf k d~ p Eik d~ : j~ pik j ð2pÞ5 ð17Þ
Recall that ~ j ¼ fbif ; D~ pif k g and x2if ¼ b2if þ Dp2if k . Further calculations are carried out using the package MathematicaÓ 7.0. 3. Electron planar channeling in a crystal: theoretical background For the stationary states of an electron its longitudinal motion along YZ–plane is described by the plane wave expði~ pk~ r k Þ. If E|| (e\ - U(x1)) and e\ is the energy of the electron transverse motion, than transverse motion of the electron is described by the transverse wave function /(x1) which is the solution of the Schrödinger-like equation with a relativistic mass cme [4,7]. The average potential of the system of crystal planes U(x1) is a periodic function over the coordinate x, therefore the transverse wave function /(x) and the energy of transverse motion of electron e\ are continuous functions of the electron wave vector component kx, which lie in the first Brillouin zone (e.g., see [9,10]). That means the wave function /(x1) is the Bloch one:
ui ðx1 ; k? Þ ¼
X
C i ðg ?m ; k? Þexpfiðk? þ g ?m Þx1 g:
We carry out the numerical calculation for the case of electrons channeled along the (1 1 0) plane in a Si crystal. The longitudinal electron velocity is parallel to the OZ axis. All calculations are performed taking into account all possible transitions of the channeled electron from the quantum state i to the state f. Fig. 2 shows the result of our calculation of the three dimensional angular distribution of CSEE electrons (differential crosssection dr/(dpdX)). The momentum of CSEE electron is equal to p = cpe = 15 keV (i.e., kinetic electron energy E p2e =2m 220 eV; here pe = mcbc is an electron momentum in the ordinary units). This momentum corresponds to maximal value of differential cross-section of CSEE (for the channeled electron relativistic factor is c = 100). Fig. 3 shows a sectional view of Fig. 2 along the plane XOZ. From Fig. 3 it follows that the cross-section has a maximum for a CSEE electron emission angle h near the values +76° and 71°. The angle h is the angle between the momentum of the secondary electron and the axis OZ. The asymmetry of the angular distribution of the secondary electrons with respect to the OZ axis arises from the asymmetry of the formula (17) by the vector ð~ p~ jÞ. The vector ~ pif k g does not depend on the angle h. j ¼ fbif ; D~ Fig. 4 plots the maximal value of the differential cross-section dr/(dpdX) (h = +76°) of CSEE as a function of CSEE electron momentum for the different relativistic factor c = 20...100 of the channeled electrons. From Fig. 4 one can see that for all values of the relativistic factor c the maximum of the cross-section appears
ð18Þ
m
Here g\m = g\m, and g\ is one dimension reciprocal lattice vector perpendicular to the crystal plane. Using the Fourier expansion of the periodic potential of the crystal plane system P U(x1) = mUm exp {i mg\x1}, we obtain the algebraic eigensystem problem [11,12]:
P m
Amn C i ðg ?m ; k? Þ ¼ ei? ðk? ÞC i ðg ?n ; k? Þ;
ð19Þ
2
Amn ¼ U mn þ dðm; nÞð h jng ? þ k? j2 =2me cÞ: Here Ci(gxm, kx) are the Fourier components of the electron wave function /i(x, kx) and Um the Fourier components of the periodic potential function U(x) which describes average potential of the system of crystal planes, d(a, b) is the Kronecker delta.
Fig. 2. Indicatrix of CSEE electrons with kinetic energy E 220 eV produced by planar channeled electrons with relativistic factor c = 100.
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
Fig. 5. Maximal value of the differential cross-section dr/(dpdX) (h = +76, p = cpe = 15 keV) of CSEE as a function of relativistic factor c of the channeled electron. Fig. 3. Section of Fig. 2 along the plane XOZ. Table 2 The total cross-section CSEE.
c
r, barn
20 30 40 60 100
34 64 74 156 240
According to Figs. 4 and 5 we can estimate the total cross-section CSEE – the result is shown in Table 2. 5. Conclusion
Fig. 4. Maximal value of differential cross-section dr/(dpdX) (h = +76°) CSEE as a function of CSEE electron momentum for different relativistic factor c = 20...100 of the channeled electrons.
Table 1 Connection between the numerical values of the energy E and momentum p. Momentum p = cpe, keV
Kinetic energy E, eV
5 10 15 20 50
24, 46 97, 85 220, 2 391, 4 2446
at one point (marked on the axis) at p = cpe = 15 keV (i.e., kinetic electron energy E 220 eV). For reference, in Table 1 it is shown the corresponding numerical values of energy E and momentum p = cpe CSEE-electrons. Fig. 5 plots the maximal value of the differential cross-section dr/(dpdX) (h = +76°, p = cpe = 15 keV) of the CSEE-electrons as a function of relativistic factor c of the the channeled electron. The complex behavior of the cross-section dependence on the initial electron relativistic factor can be explained quite simply: at c = 39 and c = 98 the transitions (from new odd levels 5 and 7, respectively) should contribute to the cross-section. (The numerical calculation shows that contribution to the cross-section due to the transition from even levels is smaller than from odd ones).
Thus, within a QED formalism, the exact formula for the crosssection of new process – channeling secondary electron emission is found. The results of our calculations show the following. At planar channeling, CSEE electrons are emitted in the direction close to a direction perpendicular to channeling plane (at angles +76° and 71° with respect to the axis OZ – direction of the longitudinal motion of the channeled electron). The differential cross-section dr/(dpdX) of CSEE has a maximum at a rather small momentum of CSEE-electron p = cpe = 15 keV (i.e., kinetic electron energy E 220 eV). The position of maximum does not depend on relativistic factor c of initial electron. The CSEE cross-section has a maximal value at a fixed emission angle of the CSEE-electron. For all values of the relativistic factor c the maximum of the CSEE crosssection has same point at p = cpe = 15 keV (kinetic electron energy E 220 eV). This is explained by the fact that properties of the CSEE electron are mainly determined by the characteristics of the photoionization process by a virtual photon. Due to the fact that differential cross-section dr/(dpdX) does not depend neither on the energy of the channeled electron, nor on the CSEE-electron energy, one can obtain an approximate relation between the total and differential cross-sections of CSEE r(c) = dr(c)/(dpdX) G, here G 6 1014 sr eV. The contribution of CSEE in the appearance of the secondary electron in comparison with radiationless transition is approximately 15% (see for example [17]). But it is shown our estimation of its contribution should be a function of the initial angles of the channeled electrons. As is well known, the probability of electron capture into the channeling states is increased with decreasing electron emission angles with respect to the crystal plane. This phenomenon should result in the appearance of an orientation dependence of the CSEE phenomena. This orientation dependence will be the subject of our future investigation. Our
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K.B. Korotchenko et al. / Nuclear Instruments and Methods in Physics Research B 276 (2012) 14–18
calculation shows that most of the CSEE electrons have a small kinetic energy E p2e =2m 220 eV. This fact did not allow the CSEE electrons to leave the crystal target. In a real experiment one can be conclude about existence of CSEE by the appearance of characteristic X-ray radiation. The properties of the characteristic X-ray radiation cross-section (such as dependence on channeled electron energy and orientation dependence) should be similar to the CSEE cross-section ones. Acknowledgments The authors are grateful to Prof. Yu. L. Pivovarov for useful discussions. The work is partially supported by RFBR, Grant 10-02-01386-a; by Russian Education Federal Agency program «Support of Scientific Schools», Contract 3558.2010.2 References [1] H. Nitta, Y.H. Ohtsuki, Phys. Rev. B 39 (1989) 2051–2053. [2] N.P. Kalashnikov, Coherent interactions of charged particles in single crystals: scattering and radiative processes in single crystals, Harwood Academic Publishers, New York, 1989.
[3] A.I. Mikhailov, A.V. Nefiodov, G. Plunien, Phys. Lett. A 372 (2008) 4451–4461. [4] J.C. Kimball, N. Cue, Phys. Rep. 125 (2) (1985) 69–101. [5] V.B. Berestetskii, T.M. Lifshiys, L.P. Pitaevskii, Quantum Electrodynamics, Pergamon, Oxford, 1975. [6] M.A. Rumarhov, Phys. Lett. Ser. A 57 (1976) 17. [7] V.A. Bazulev, N.K. Zhevago, Radiation of Fast Particles in Matter and External Fields, Nauka, Moscow, 1987 (in Russian). [8] E. Clementi, C. Roetti, Atom Data and Nuclear Tables 14 (1974) 177–478. [9] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1979. [10] C. Kittel, Quantum Theory of Solids, John Wiley & Sons, Inc., New York, London, 1963. [11] B.F. Buxtont, J.E. Loveluck, J. Phys. C: Solid State Phys. 10 (1977) 3941. [12] R.K. Klein, J.O. Kephart, R.H. Pantell, H. Park, B.L. Berman, R.L. Swent, S. Datz, R.W. Fearick, Phys. Rev. B 31 (1985) 68. [13] O.V. Bogdanov, K.B. Korotchenko, Yu.L. Pivovarov, JETP Lett. 85 (No. 11) (2007) 555. [14] K.B. Korotchenko, Yu.L. Pivovarov, T.A. Tukhfatullin, Nucl. Instr. Meth. B 266 (2008) 3753. [15] P.A. Doyle, P.S. Turner, Acta Cryst. A 24 (1967) 390. [16] K.B. Korotchenko, E.I. Fiks, Yu.L. Pivovarov, T.A. Tukhfatullin, J. Phys.: Conf. Ser. 236 (2010) 012016. [17] A.V. Schagin, V.V. Sotnikov, The Journal of Kharkiv National University, No.777, Physical Series ‘‘Nuclei, Particles, Fields’’, Issue 2/34/, Kharkov, 2007, p. 97–101.