transmission of 2 MeV He++ ions quantitative correlation study

Nuclear Instruments and Methods in Physics Research B 355 (2015) 324–327

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Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Backscattering/transmission of 2 MeV He++ ions quantitative correlation study V. Berec a,⇑, G. Germogli b, A. Mazzolari b, V. Guidi b, D. De Salvador c,d, L. Bacci d a

Institute of Nuclear Sciences Vinca, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia INFN Sezione di Ferrara and Dipartimento di Fisica e Scienze della Terra, Via Saragat 1, 44100 Ferrara, Italy c Dipartimento di Fisica, Università di Padova, Via Marzolo n.8, 35131 Padova, Italy d INFN Laboratori Nazionali di Legnaro, Viale Università 2, 35020 Legnaro, PD, Italy b

a r t i c l e

i n f o

Article history: Received 25 November 2014 Received in revised form 24 January 2015 Accepted 1 February 2015 Available online 20 February 2015 Keywords: Channeling Coherence effect Correlation Multiple scattering

a b s t r a c t In this work we report on detailed findings of planar channeling oscillations of 2 MeV He++ particles in (1 1 0) silicon crystal. The exact correlation and coherence mechanism between confined particles oscillating trajectories are analyzed theoretically and experimentally in backscattering/transmission geometry. Regular patterns of channeled He++ ion planar oscillations are shown to be dominated by the crystal harmonic-oscillator potential and multiple scattering effect. For the first time it was shown that under the planar channeling conditions trajectories of positively charged particles exhibit observable correlation dynamics, including the interference effect. Quantitative estimation of channeling efficiency is performed using path integral method. Ó 2015 Elsevier B.V. All rights reserved.

The effect of coherence arises during the interaction of charged particle with a crystal potential when the oscillations of the emitted/channeled particles and lattice atoms are superposed in the phase. Under the planar channeling condition [1,2], during transmission of energetic particles through crystal planes in the harmonic (oscillator potential) approximation, effect of coherence can be established under circumstances of phase summation, which depend on the periodical arrangement of the crystal atoms at selected beam parameters. Planar oscillations are induced by the interacting potential Upl (p), which is experienced by the energetic particle as a sum of the potentials generated by the strong electromagnetic field between crystal planes where the channeled oscillations occur. Phase summation between the oscillating energetic particles trajectories and the lattice atoms specific phase of the oscillation (parameterized by the thermal vibration) produces the effect of coherence in the channeling regime. Moreover, coherence being the central part of the interference phenomena [3], produces phase summing instantaneously identifying an interferential multiplier in the scattering cross-section, which leads to appearance of coherent peaks when the transferred momentum of the energetic particle coincides, i.e., when it is in resonance, with one of the reciprocal lattice vectors [4]. Experimental and theoretical studies [5] have revealed appearances of similar coherent peaks ⇑ Corresponding author. E-mail address: [email protected] (V. Berec). http://dx.doi.org/10.1016/j.nimb.2015.02.003 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.

in the orientation-dependant backscattering yield investigations and confirmed that collisions of energetic particles with crystal surfaces [6] can provide information about binary interactions involving outer-shell electrons [7]. By selecting specific backscattered particles whose energy losses are correlated with a particular surface atomic species, it is possible to determine binary collision produced trajectories, isolating in that way cases when the energetic particle is scattered off a single surface atom from those cases where multiple collisions or penetration into the solid occur. The result is variation in the amplitude and intensity of peaks which correspond to specific backscattering angles and incident energy of energetic particles, i.e., direct information about the binary particle-atom interactions. Former result in conjunction with coherent interaction mechanism of planar channeling [2] provides a direct quantitative description about effect of coherence and correlation, which are awaken by the crystal electromagnetic potential. Here we report on novel experimental results followed by detail theoretical description of 2 MeV He++ ions trajectory dynamics affected by the planar channeling oscillations between (1 1 0) atomic planes of a silicon crystal. Backscattering/transmission experiments with 2 MeV He++ ions were performed to study exact correlation between the phases of confined oscillating trajectories. Obtained regular patterns of channeled ion planar oscillations are shown to be under strong collective influence of the crystal harmonic-oscillator potential and multiple scattering effect. Quantitative estimation of channeling efficiency for 2 MeV He++ particles is

V. Berec et al. / Nuclear Instruments and Methods in Physics Research B 355 (2015) 324–327

performed in backscattering/transmission experimental setup, shown in Figs. 1 and 2. A silicon crystal of lateral sizes 20  20 mm2 and 0.5 mm thickness was obtained by dicing (Disco DAD3220) of a 4 diameter silicon wafer. The crystal surface is perpendicular to the h1 0 0i direction, and its (1 1 0) planes can be used for channelling. The crystal was mounted on goniometer with angular rotations resolution of 0.01 deg. Experiment is performed at Italian National Institute for Nuclear Physics, INFN-LNL facilities, where the crystal mounted in an ultra-high vacuum chamber and primarily oriented in order to achieve planar channeling between its (1 1 0) planes, was exposed to a 2 MeV He+ beam. Double ionization of helium was produced by impinging He+ ions onto a crystal entry face, subjecting 90% of bare 2 MeV He++ ions to a planar channeling regime. For this energy the estimated critical angle for planar channeling infers 0.17 deg, and the beam collimation was established with a divergence of 0.01 deg. Due to multiple-scattering effect on electrons and nuclei [8–10] the fraction of He++ ions becomes dechanneled [11,12]. Consequently, such particles are subjected to Rutherford backscattering [13] and collected by a pin-diode detector which can be moved along a circumference whose center coincides with the position of the sample, see Fig. 1. 1. Effect of coherence and correlation Effect of coherence, considering semi-classical framework [14], arises when the particles traverse a sufficient distance inside the crystal at constant velocity along a near-linear trajectory. In the presented case of 2 MeV He++ ions analysis is performed under backscattering/channeling geometry, where following types of correlations are considered: 1. Superposition of the Coulomb potential, waken by interaction of He++ ions with harmonic crystal potential, with the exchange– correlation potential generated by multiple scattering effect on electrons. The exchange–correlation potential becomes predominant as channeling trajectory increases its distance path from the entry face of the crystal. Depending on the selected parameters of beam energy, angles of incidence with respect to crystallographic planes, in this case two types of exchange– correlation can be established: static correlation which refers to situations where elastic multiple scattering effect occurs during correlated collisions, and the second is dynamical correlation which refers to capturing the effect of the instantaneous electron repulsion mainly between opposite-spin electrons, which is induced by prior collision of the energetic particle (channeled ion). Interaction (exchange) between those two

Fig. 1. Schematic representation of low energy channeling apparatus. 2 MeV He+ beam is focused to a spot size of 0.5  0.5 mm2 and collimated with divergence of 0.01 deg onto a 0.5 mm thick silicon crystal. The crystal is mounted on a goniometer and preferentially aligned in order to excite planar channeling between (1 1 0) planes.

325

Fig. 2. Study case experimental setup in backscattering/channeling geometry, a schematic representation. 2 MeV He+ beam was directed with 3/4 deg out of axis onto silicon (1 1 0) planes in order to avoid h1 0 0i axial channeling. Backscattering probability, as a function of depth and energy yield, is recorded respectively at three different angles: 108°, 115° and 170°.

correlation modes under influence of the crystal harmonic potential induces exchange–correlation potential. The exchange– correlation potential is usually separated into a sum of former exchange and static/dynamical correlation parts. 2. Superposition of coherent radiation of the crystal lattice atoms itself, induced by backscattered/channeled He++ ions, with the radiation field secondary emitted by projectiles (He++ ions). 3. Interference effects exhibited in neighboring (1 1 0) Si planar stacking sequences between coherent He++ channeled trajectories. It is well known that interference pattern can be produced by a classical force [15]. In terms of a classical electrodynamics, em waves are described as a continuous electromagnetic field; here used FLUX code [10] uses such description of the crystal potential. 2. Simulation model Taking in account the periodicity of (1 1 0) planar stacking sequences within the Bloch’s model [16,17] we express coherence effect, induced by correlation of the soft ion collisions over electron density of states by the waked Coulomb potential, in terms of a Fourier spectral decomposition:

/k ¼

X eikx /x ;

ð1Þ

x

where /k defines Fourier transformed components of Coulomb field. Simulation model of a cubic lattice consists of the parameter points x = d (n0, n1, n2, n3) where d is the lattice spacing and n1, n2, n3 are reciprocical lattice vectors where n0 is redundant parameter. Numerical simulation of channeled ion trajectories coherent in spacetime with a lattice, is performed applying the path integral formalism [18,19] across a system ((1 1 0) Si crystal plane) whose degrees of freedom are related to effective potential field variable /x at each lattice site x. In order to immerse into the simulation model the effective potential induced by the planar channeling oscillations, we applied the prescription for the path integral in the momentum space where particle’s Hamiltonian is H = p(t)2/ P 2m + 1/2mx2q2, (p, q e R3) and p(t)2 = jpj (t)2; incorporating the particle dynamics into the former Hamiltonian while summing over all possible paths and integrating over all possible momenta within specified time interval gives

U pl ðpðtÞÞ ¼ H;

Z

ðÞ ¼ H ¼

 _ 2  pðtÞ2 pðtÞ þ ; p_ ¼ Hq ¼ mx2 q ; 2m 2mx2

ð2Þ

326

V. Berec et al. / Nuclear Instruments and Methods in Physics Research B 355 (2015) 324–327

where Upl(p(t)) is the total potential in momentum space, which is experienced by the particle as a sum of the potentials of all the planes:

U pl ðpðtÞÞ ¼

X

V pl ðjz  zi jÞ  U min pl ¼

i

p_ ðtÞ2 pðt Þ2 þ ; 2m 2mx2

ð3Þ

U min pl

where is a constant subtracted in order to make the minimum value of U pl ðpÞ equal to zero, V pl ðpÞ is continuum planar crystal potential [20], and coordinates z and zi define points in a line normal to the set of planes where the total potential is measured.   Relations: U pl ðpðt ÞÞ ¼ H and H ¼ pðt Þ2 =2m þ 1=2 p_ ðt Þ2 mx2 are responsible for maintaining (keeping) harmonic potential approximation all the way during simulation. The continuum planar potential V pl ðpÞ at a distance q from the atoms in the plane is

V pl ¼ n

Z

1

 12 2pRV q2 þ R2 dR;

ð4Þ

0

where n denotes the atomic areal density in the plane [20]. Representation of a particle momentum change with time p_ ¼ Dp=Dt ¼ Hq ¼ mx2e q (during successive binary collisions) as a change of the particle momentum transverse component in impulse approximation Dp\ [10] increases its accuracy with convergence Dt ? d/v, i.e., as it approaches single collision time, as

p_ j Dt

¼

lim

Dt!dkj =

v

Dpj?ðbÞ ¼

Z

1

F ? dt 0

k

by distances

between atoms in the plane;

v is the particle velo-

city, and F ? ðqÞ ¼ dV pl =dq is the corresponding force transversal component derived from the potential Vpl(p). In next step we apply the initialization by setting the momentum pi at time interval ti, i = 1,. . .,n  1. Extending this time domain to all possible N sequences we get the following path integral relation which describes the dynamics of the channeled particle trajectories:

  lim R dp1 dðpi  p1 Þhðp1  pi Þdp2 dðp1  p2 Þhðp2  p1 Þ .. . G pf ; t; pi ¼ N!1   . .. dpn1 dðpn2  pn1 Þhðpn1  pn2 Þd pn1  pf ( ) N X   p_ j ðt Þ2 ; h pf  pn1 exp 2mx2 j¼1

ð6Þ

!) ðpn Þ2 ! ; m

ð7Þ

"

# pðt Þ2 dSðpðt ÞÞ ¼  U pl ðpðtÞÞ dt; 2m

ð8Þ

where the particle classical action, Sðpðt ÞÞ, is defined according to the total potential in momentum space that is experienced by the particle as a sum of the potentials of all the planes, U pl ðpðt ÞÞ:

Sðpðt ÞÞ ¼

Z

tf

ti

¼

Z

tf

ti

¼

transversal momentum of particle in limit dj =v , which is defined

e

where the particle momentum integral p1 at time t1 is   R dp1 dðpi  p1 Þhðt 1  t i Þ; GCH pf ; t f ; p1 ; t1 is the path integral function of initial momentum p1 and time t1 and final momentum pf and time tf of the channeled particle. N represents total numbers   of simulation intervals, e is the particle energy. dS pf ; t f ; p1 ; t 1 is the change of classical action for classical paths that extend from p1 (at time t1) to pn = pf (at time tn = tf corresponding to step   d pi  pf ; i ¼ 1; . . . ; n  1); it is given by the following relation

ð5Þ

where Dpj\(b) is impact parameter dependant function of the k dj

    lim GCH pf ;t f ; p1 ;t 1 ¼ Ad pi  pf N!1 (      h tf  t i exp N tf  ti

Z

ti

tf

!  2 1 dxðtÞ m  U pl ðpðt ÞÞ dt 2 dt !  2  2 dxðt Þ 1 dxðt Þ m  m  U pl ðpðtÞÞ dt dt 2 dt Z tf    pðt Þ p2   pðt Þdt  edt ¼ tf  t i  t f  t i e; m m ti

ð9Þ

where, first term, i.e., integrand along the path in momentum space 2

that the particles take, corresponds to a phase factor eip =mdt , and second term denotes relation of particle energy, e, to the total time which is needed to traverse the path, corresponding to a factor eiedt for each time step. Then, a phase eiSð~pðtÞÞ=h is assigned to each classi cal path and summed over the space of all paths p : ti ; t f ! Rn , giving rise to the field spectral distribution of Eq. (1). Applying the Fourier transformation on the Eq. (1) over the endpoints of total N simulation intervals gives the frequency spectra of the particle in momentum space. 3. Results Comparative analysis of simulation results, based on the FLUX code and the path integral calculations, have confirmed that effect of coherence, established over channeled He++ trajectories, becomes predominant at 200–600 nm distances from the entry

Fig. 3. Left: potential energy map during channeling He++ sequences in (1 1 0) Si, revealing the Coulomb potential superposition maxima in depth region 200–600 nm distant from the entry face of crystal (designated with colored circles). The intensity levels of effective potential, which influences coherent ion motion, are shown in scaled gray tones. Right: simulated trajectories of 2 MeV He++ ions channeled between (1 1 0) planes of a Si crystal display the interference effect induced in neighboring (1 1 0) Si planar stacking sequences between separately coupled He++ trajectories. Most pronounced minima (destructive interference) occur at depth points in vicinity of 300 nm and 400 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

V. Berec et al. / Nuclear Instruments and Methods in Physics Research B 355 (2015) 324–327

327

4. Conclusions

Fig. 4. Comparison between experimental RBS spectra obtained for 2 MeV He++ ions, and simulation. Silicon crystal is oriented along the (1 1 0) direction. Beam divergence is 0 degrees. Channeled He++ ion planar oscillations are extracted from the ratio of the total number of particles vs the channeled particles number, similarly to method presented in [21].

In summary, the planar oscillations of 2 MeV He++ ions in a straight (1 1 0) oriented Si crystal have been experimentally recorded and studied theoretically in conjunction to coherence and correlation mechanisms. Planar-channeled He++ trajectories are shown to receive contributions from correlation effect based on the superposition and interference, which are generated by the ordered structure of the crystal. The former, dominant in area of 200–600 nm from the first atomic layer is due to the effect of coherence and elastic multiple scattering during the interaction of charged particle with the harmonic Coulomb potential, while the latter, dominant after 600 nm involves dynamical exchange–correlation influenced by the omnipresence of singularities in instantaneous electron repulsion during multiple scattering and dechanneling. We have obtained new insights into phase space dynamics of 2 MeV He++ ions channeled between Si (1 1 0) atomic planes applying for the first time path integral quantitative and qualitative analyses for coherence and interference effect between neighboring planar stacking sequences of a silicon crystal, developing analytical and numerical tool to study dynamics of planar ion oscillations and exact measures of correlation arising due to multiple scattering effect on electrons. Acknowledgements

face of crystal, as shown in Fig. 3. Coherent motion of channeled He++ ions is precisely guided through (1 1 0) planar stacking sequence of silicon atoms using simultaneous correlation, i.e., superposition of the screened Coulomb potential caused by interaction of He++ ions with harmonic crystal potential, and the exchange–correlation potential due to multiple scattering effect on electrons (see Fig. 4). Keeping in mind that time interval, Dt, is defined by di/v via distances di ði ¼ 1; :::; nÞ between atoms in the plane, we have used   time term t f  ti and particle velocity v obtained from kinetic part of exponent argument of Eq. (7) in order to establish a clear connection between time–space dynamics of different paths of each channeled ion trajectory that exists between atomic planes of the crystal, analyzing them from the initial moment (ti) to the final moment of time (tf) through different length (thickness) of the crystal. Precisely, we get equation of motion in momentum space for each channeled particle between the crystal planes. Each interval in path integral relations (6) and (7) must have one d and one h factor in order to limit the possible paths and maintain causality. h factor attributes the specific electron loss [7,20] arising from the multiple scattering effect and collisions with valence electrons, as following

h¼

dE 4pZ 21 e4 2me v 2 ¼ ne ln ; 2 hxe dz me v

ð10Þ

where ne = DUth/4p = NZ is the density of the crystal’s electron gas averaged along the z axis, accounting Zloc  Zval = Z; D = oxx + oyy, DUth is the continuum potential of the crystal and xe = (4pe2ne/me)1/2 is the electron plasma frequency. Adding a h factor (noise) redistributes the ion paths in momentum space according to principle of least action.

We acknowledge Mr. Luca Bacci for the technical support and INFN ICE-RAD project for financial support. References [1] A. Mazzolari, E. Bagli, L. Bandiera, V. Guidi, H. Backe, W. Lauth, V. Tikhomirov, A. Berra, D. Lietti, M. Prest, E. Vallazza, D. De Salvador, Phys. Rev. Lett. 112 (2014) 135503. [2] J. Lindhard, Kongelige Danske Videnskabernes Selskab 34 (1965) 14. [3] G.L. Bochek, I.A. Grishaev, N.P. Kalashnikov, G.D. Kovalenko, V.L. Morokhovski, A.N. Fisun, Zh. Eksp. Teor. Fiz. 67 (1974) 808–815. [4] Kagan Yu, V. Kononets Yu., JETP 31(1) (1970) 124–132; Zh. Eksp. Teor. Fiz. 58(1) (1970) 226–244 (in Russian). [5] F. Abel, G. Amsel, M. Bruneaux, C. Cohen, A. L’Hoir, Phys. Rev. B 12 (11) (1975) 4617–4627. [6] E. Bogh, E. Uggerhoj, Phys. Lett. 17 (1966) 116–118. [7] V. Berec, Laser and Particle Beams, Cambridge University Press (2014), http:// dx.doi.org/10.1017/S0263034614000482. [8] J.A. Soininen, A.L. Ankudinov, J.J. Rehr, Phys. Rev. B 72 (4) (2005) 045136. [9] W. Scandale et al., Phys. Lett. B 680 (2009) 129. [10] P.J.M. Smolders, D.O. Boerma, Nucl. Instr. B 29 (1987) 471–489. [11] N.P. Kalashnikov, Coherent interactions of charged particles in single crystals: scattering and radiative processes in single crystals, (translated from the Russian by Amoretty S. J.) Harwood Academic Publishers, 1988. [12] A. Mazzolari, Manipulation of charged particle beams through coherent interactions with crystals, Annali Online Pubblicazioni dello IUSS 3 (1) (2009) 1–96. [13] L.C. Feldman, J.W. Mayer, S.T. Picraux, Materials Analysis by Ion Channeling, Academic Press, 1982. [14] J.M. Hansteen, O.M. Johnson, L. Kocbach, Atom. Data Nucl. Data Tables 15 (1975) 305. [15] T.H. Boyer, Phys. Rev. D 8 (1973) 1679. [16] E.A. Babakhanyan, V. Kononets Yu, Phys. Stat. Sol. 98 b(1) (1980) 59–77. [17] J.L. den Besten, D.N. Jamieson, L.J. Allen, Nucl. Instr. B 158 (1–4) (1999) 153– 158. [18] E. Cattaruzza, E. Gozzi, Francisco A. Neto, Phys. Rev. D 87 (2013) 06750. [19] J.D. Doll, R.D. Coalson, D.L. Freeman, Phys. Rev. Lett. 55 (1985) 1. [20] D.S. Gemmell, Rev. Mod. Phys. 46 (1974) 129. [21] M. Bianconi, G.G. Bentini, R. Lotti, R. Nipoti, Nucl. Instr. B 193 (2002) 66–70.