Inner-shell Annihilation of Positrons in Argon, Iron and Copper Atoms
IInner-shell Annihilation of Positrons in Argon, Ironand Copper Atoms
M. A. Abdel-Raouf (1) , M. M. Abdel-Mageed (2) and S. Y. El-Bakry (2) (1)
Physics Department, Faculty of Science, UAEU, Al Ain 17557, United Arab Emirates (2)
Physics Department, Faculty of Science, Ain Shams University, Cairo, Egypt
Abstract
The annihilation parameters of positrons with electrons in different shellsof Argon, Iron and Copper atoms are calculated below the positronium (Ps)formation thresholds. Quite accurate ab initio calculations of the boundstate wavefunctions of Argon, Iron and Copper orbitals are obtained fromCowan computer code. A least-squares variational method (LSVM) is usedfor determining the wavefunction of the positrons. The program isemployed for calculating the s-wave partial cross sections of positronsscattered by Iron and Copper atoms. Our results of the effective charge arecompared with available experimental and theoretical ones.------------------------------------------------------------------
PACS: 34.85, 78.70.B [email protected]
Key Words: Positron Annihilation, Inner-Shell Ionizations, Positron Scattering byAtoms
Positrons, since their prediction by Dirac [1], and their observation by Anderson [2] andBlackett and Occhialini [3], have been extensively used as probes in different branches ofphysics. Studies of positron interactions with gases, atoms and molecules with and withoutpositronium formation have been carried out over the years. Much interest, however, has beendevoted in the last decade to the interaction of slow positrons (below the positroniumformation thresholds) with one, two and three dimensional macromolecular structures, (e.g.large molecules, chains, surfaces, crystals and bulks). In these cases, positron annihilation isemployed as an effective nondestructive tool for the investigation of electronic structures aswell as defects of materials. Particularly, positrons may also annihilate with inner-shellelectrons creating holes which consequently induce the emission of highly efficient Augerelectrons with extremely low background secondary electrons as the ultimate parallel tools(see e.g. Weiss et al [4] and Kim et al [5]) for the investigation of surface metals.The aim of the present paper is to use a least-squares variational method (LSVM) fordetermining the wavefunctions of slow positrons interacting with inner-shell electrons ofdifferent atoms below the positronium (Ps) formation thresholds. In order to illustrate thestrength of our algorithm, we study the annihilation of positrons with one of the noble gases,namely Argon. (Previous interesting works on positron collisions with noble gases werecarried out by Montgomery and LaBahn [6] and McEachran et al [7]). Our main goal,however, will be the annihilation of positrons in two, from the industrial point of view,extremely important metals, namely iron and copper. Each wavefunction is used to calculatethe effective charge (annihilation parameter) Z eff , which stands for the effective number ofelectrons at the positron position at a given subshell of the target atom. In this case thecalculation of annihilation rates and cross sections are directly related to the average densityof electrons at the position of the positron. An elaborate version of Cowan computer code([8], program RCN32) is used to calculate quite accurate ab initio orbital wavefunctions of thetarget atoms.
In non-relativistic time-independent quantum mechanics, Schrödinger’s equation is written as EH , (1)where H and E are the total Hamiltonian and energy, respectively, of a quantum mechanicalsystem described by the wavefunction . The boundary conditions of characterize variousquantum mechanical systems, e.g. bound-state system, scattering process, etc. In the collisionof positrons (e + ) with target atoms (A), the positrons annihilation is subjected to the emissionof 2 or 3 photons according to one of the following processes:A + + 2 (or 3 ) ( Direct annihilation )e + + A Ps + A + A + + 2 ( or 3 ) (2)[e + , A] A + + 2 ( or 3 )Ps and A + stand for the positronium and the residual ion. In the first process which is calleddirect annhilation, the incident positron annihilates ( below the Ps formation threshold ) withone of the atomic electrons of the neutral target atom A and the annihilation rate is calculatedusing the electron charge density ( Z eff ) at the positron position. In the second process theincident positron (above the Ps formation threshold) picks up an electron to form positroniumand after that annihilates. The positron in the third process is captured to the atom to form [e + ,A] bound system and the photon annihilation then occurs within the positron-many-electroncomplex system.In the present work, we concentrate ourselves on the first (i.e. the direct annihilation) process.In positron-atom scattering, , the rate of annihilation of an incoming positron and an atomicelectron with the emission of two gamma rays, is given by the expression (Ferrell [9] andFraser [10]) )( kZcr ffe , (3)where r is the classical radius of the electron, c is the velocity of light. is the density ofelectrons per atom available to the positron for annihilation and k is the positron wavenumber. )( kZ ffe is defined in general as the effective number of electrons per atom availableto the positron for annihilation. (In our case it stands for the effective number of electronsoccurring at the positron position at a given subshell of the target atom). It depends onspecific properties of the e + - Atom system under consideration and is equal to Z , the numberof atomic electrons, if the interaction potential between the positron and the atom is set to bezero. The annihilation parameter )( kZ ffe can be calculated using the scattering wavefunctionobtained via the least-squares variational method. Remembering that the annihilationparameter is related to the probability of an electron and a positron to be found in the sameposition, we can write krxxrkrxkZ Ni iffe ;,)(;,)( , (4)where krx ;, is the full scattering wavefunction, including all partial waves, for thesystem made up of the incident positron with wave vector k and the target atom. x and r standfor the position vector of the positron and the target (composed of N electrons), respectively.For s-wave scattering process, the variational treatment (Abdel-Raouf [11]) starts by defininga trial wavefunction );,( krx nt . It consists of two multiplicative wavefunctions );,( krx nt = )( r T );( kx ncS (5)where )( r T represents the target in its ground-state and );( kx ncS is the positronscattering wavefunction which is composed of the angular part ( Y ) multiplied bythe radial part );( kx nP . Thus, we have ,)();( ˆ );( ˆ );( xdkxCbkxSakx ini innnP (6a)n refers to the dimension of the square integrable part of the trial wavefunction representingall possible virtual states of quantum mechanical system composed of the positron and thetarget. );( ˆ kxS and );( ˆ kxC specify the regular and irregular parts of the wavefunction,respectively. Usually, the latter is accompanied with a cut-off function for avoiding thesingularity at the origin. This cut-off function will tend to zero at the origin and to unity atinfinity. );( kx nP has to satisfy the boundary conditions:);0( k nP = 0);( kx nP nnx bkxSa );( ˆ );( ˆ kxC (6b)The function )( x i appearing at eq. (6a) is a square integrable wavefunction. inn dandba , are variational parameters. In this case the reactance matrix K contains a single element whichis identical with the tangent of the s-wave scattering phase shift ( ) and is calculated by nn abK tan . (7)The s-wave elastic scattering trial wavefunction for the system may be written in abbreviatedform as: nt = S + K C + n (8)where S is the regular part ; )(.sin41)(. ˆ rcrSS TT , (9)( kwherexkc )(,sinsin is the momentum of the incident positron). Thefunction C consists of a cut-off function and the irregular part, i.e. )( ˆ )1( rCeC Tx (10) )()cos()1(41 rce Tx , (11)where andc coscos is an adjustable (free) parameter which is selected from thevalues that give a plateau of K (see ref. [11], P.73). )( r T is the target ground statewavefunction (see Appendix). The square integrable ),( rx n possesses the form ni iiini iTn dxdrrx )(.)(),( (12)where xii ex and Tii . (13)The next step in the variational treatment is to select a proper test-wave function S anddefine the functional VEH ntS (14)The linear variational parameters K and i d are chosen according to the following variationalprinciple: V (15)Thus, they are chosen following a least-squares variational principle in which the squaredmodulus of the projection of the vector nt EH )( in S is minimal. The test wavefunction S is constructed [11] by: ......,2,1;,, njCS jS (16)In this case we have the system of projections VSdCSKSS ni ii VCdCCKSC ni ii (17) njVdCKS jni ijijj ,....2,1; .The LSVM implies:
21 2 nj j V . (18)This means that the sum of squared moduli of the projections of nt EH on the testfunction space s is minimum.The minimization of
21 2 nj j V guarantees that the vector nt EH has a minimumlength. The variational parameters are obtained by applying this variational principle (18).The total Hamiltonian (in Rydberg units) of positron-target atom system has the form: ),( xrVHH tnixT (19)where T H is the Hamiltonian of the target atom, x is the kinetic energy operator for theincident positron, ),( xrV tni stands for the interaction potential between the positron and thetarget and r is used to represent the assemblage coordinate for Z atomic electrons.The total energy E of the system may be written, in Rydberg, as kEE T , (20)where kandE T are the energy of the target and the kinetic energy of the incidentpositrons, respectively. ),( xrV tni is the interaction potential between the incident positron andthe target and is given by Ni itni rxxZxrV nt can be expressed (see Appendix) interms of wavefunction determinants as nMCKnMSn C nS nnt : ......: .....:1 : 111: 1 (2 ffe Z can be determined experimentally to a high degree of accuracy and thus, the calculationof this parameter is a criterion for the goodness of the employed wavefunction t whichrepresents the system of a low energy positron moving in the field of the atom andapproximated by eq.(8). Using equations defined for t , ffe Z can be written as .)( drxrkZ tffe (23)Therefore, we have );(41)( jj njPffe RkrdrkZ (24)0or jj njni irirffe RrederkcKrkcdrkZ (25)where jj n K denotes the electron radial wavefunctions and the summation is over the electronstates in the atomic level defined by quantum numbers jj andn ( see Appendix ).The s-wave elastic scattering cross section (in a units) is related to the phase shift by thefollowing relation sin4 k le . (26) The computation of the annihilation rates was started by calculating the orbital wavefunctionsand energies of the target atoms using Cowan computer code (program RCN32). Thesewavefunctions were used for calculating the positron-atom potentials (eq. A5b). After theconstruction of the matrix elements SS , CS , i S , SC , CC , i C , S j , C j , and ij , we employed the LSVM program (at certain starting valuesof the free non linear parameter and n (the number of the square integrable functionsincluded in the trial wavefunction) in order to test the validity of the whole program. Later on,we changed and increased n until we reached convergence and stability in the resultsof K . This was achieved at = 0.3 and n = 7.Our computational process of the annihilation rates started with calculations of this quantityargon atoms. These results were compared with already existing experimental data, seeFig.(1). This figure shows that our theoretical calculations have a good agreement with theexperimental data and show similar behavior as the results of Mitroy and Ivanov [12]1developed using a two-parameter semiempirical theory of positron scattering and annihilation.Obviously, our results lie higher than the value Z eff = 13.6 obtained by Dzuba et al [13]. Theimprovements demonstrated by our calculations are attributed to the accurate forms of thebound state wavefunctions and the positron wavefunctions obtained, respectively, via ab initioand least-squares variational techniques.Since the ionization energies of the argon, iron and copper atoms are approximately 15.76 eV,7.87 eV and 7.726 eV, respectively, the energy of the incident positron k must lie below theso called Ore gap, i.e. it must be less than the difference between the values of the ionizationenergies and the binding energy of the Ps (- 6.8 eV). This means that k should be less thanthe Ps formation thresholds 8.96 eV, 1.07 eV and 0.926 eV, respectively, where the onlypossible channels are the elastic scattering and the direct annihilation. Therefore, we havecalculated the effective charge )( ffe Z at each subshell in the collision of positrons with Ar,Fe and Cu atoms through the energy ranges below 8.96 eV, 1.07 eV and 0.926 eV,respectively.The contribution of each subshell to the total effective charge for the collision of positronswith argon, iron and copper atoms are shown in figures (2), (3) and (4), respectively. Theannihilation parameters for iron and copper atoms are plotted in figures (5) and (6). Fig.6contains also a comparison with the results of Mitroy and Ivanov [12] . The s-wave elasticscattering cross-sections )( aofunitsin le of positrons by iron and copper atoms aredrawn in figures (7) and (8). These figures demonstrate the monotonic decrease of le as theenergy of the incident positron increases.2 Fig.1: Comparison between present ( — ) annihilation parameter (Z eff ) of e + - Ar scattering andthe results of Mitroy and Ivanov ( - - - ) [12] Canter and Roellig ( ■ ) [14], and Paul ( ♦ ) [15]. Fig.2: The annihilation factor (Z eff ) as a function of the incident positron energy (k ) fordifferent subshells of positron-argon scattering. Fig.3. Energy dependence of the annihilation parameter (Z eff ) for different subshells ofpositron-iron scattering.Fig.4.The energy dependence of the annihilation parameter (Z eff ) for different subshellsof positron-copper scattering. e + -Fe Z e ff k (eV) Fig. 5: Total effective charge (Z eff ) as a function of the incident positron energy (k ) forpositron-iron scattering. e + -Cu Present workMitroy & Ivanov Z e ff k (eV) Fig.6: The total effective charge (Z eff ) as a function of the incidentpositron energy (k ) for positron-copper scattering. e + - Fe e l ( a o ) k ( eV ) Fig.7: The s-wave elastic cross-sections for positron-iron scattering. e + - Cu e l ( a o ) K ( eV ) Fig.8: The s-wave elastic cross-sections for positron-copper scattering. Appendix
This appendix contains a brief discussion of the potential of the positron in the target field andthe matrix elements needed to specify the matrices Q and q required for calculating thevariational parameters using the least-squares variational method (LSVM) program.The target ground state wavefunction can be expressed as a Slater determinant of mutuallyorthonormal one-electron wavefunctions i u in the form: )(.....)()()(det!1)( zzT rurururuzr (A1)In equation (A1) z denotes the total number of electrons. According to the central field model(Heyland et al [16]), )( ii ru can be expressed as ,)() ˆ (1)( imniii rYRrru iiii (A2)where ii n R is the radial wavefunction, ) ˆ ( im rY ii are the usual spherical harmonics and )( stands for the spin vector of the orbit i such that iii mandn , are the correspondingprincipal, orbital and magnetic quantum numbers, respectively. The energy of the target isgiven by drrHrE TTTT )()( (A3)In order to calculate the matrix elements, the positron potential has to be determined. Thepotential )( xU of the positron in the target field is defined as )(),()()( rxrVrxU TtniT . (A4)7In the original work of Madison and Shelton [17] )( xU takes the form: ),(max22)( ininii i rxRdrRdrxZwxU iiii (A5a)It can be written (Cowan 1981) as: inx inx inii i rRdrRdrxRdrxZwxU iiiiii (A5b)where i w denotes the occupation number of electrons in ii n atomic orbit and ii n , are theprincipal and orbital quantum numbers of an orbit i , respectively. The orbital radialwavefunction ii n R is the solution of the equation: iiii niniiiiii RRrVrdrd )()1( (A6)where )( ii rV is the assumed potential energy function for the field in which the atomicelectron i moves. These functions are generated from Cowan program, which is based on thedescription of Herman and Skillman [18] with Hartree plus statistical exchange approximatedpotential.The system of (n + 2) equations, i.e. eqs.(17),can be reduced in matrix representation to theform: VqdQ (A7a)where the matrices Q , d , q and V are defined below. In other words, the least-squaresprinciple is equivalent to the minimization of the norm of the vector qdQ , which leavesus with8 qQQQd qQdQQ (A7b) nC TCTS M MCC MCSQ , n ddRd , S M SC SSq and n VVVV (A8)where TCTS
MandM are the transpose of the column vector CS MandM ,respectively, whichare defined as
SSSM nS , CCCM nC and nnn nn (A9)The closed form of the matrix elements required for the employment of the LSVM,namely SS , CS , i S , SC , CC , i C , S j , C j , and ij ,are needed. These matrix elements have the general form: dxfHgxddfHEgfg ˆ sin (A10)where x is the position vector of the positron, is the angle between x and the Z-axis and is the azimuthal angle. The operator H ˆ possesses the form9 , ˆ HEH (A11)which can be written in several different ways depending on the particular form ofwavefunction on which it operates.The effects of i andCSonH , ˆ are given by: ˆ cxrVrSH tniT (A12a) cecekcexrVrCH xxxtniT cossin2cos14`1),()( ˆ (A12b) xixixixitniTi exiiexiexkexxrVrH ˆ (A12c)(Remember that )(,sinsin xkc and coscos c ).Therefore, the matrix elements appearing in the matrices Q and q have the following finalforms: dxxUkSS sin)(1 (A13) sincos1)1(cossin)(1 ALAkdxexUkCS x (A14) dxexUkSC x )1(cossin)(1 (A15)0 .coscossin sin)1(cos)(1 BLALBM AMdxexUkCC x (A16) )sin())1(sin(2))2((sin!)1(sin)(1 iF iiDkidxexxUkS ixii . (A17)The elements of TC M are given by )cos())1((cos2))2cos((!)1( )cos())1(cos(2))2(cos(!)1()1(cos)(1
212 2120 )1( iD iiGDGki iD iiDkidxexxUkC i ixii (A18) dxxexUkS jxj sin)(1 (A19) ))2cos(())2((sin2!)1(cos)1()(1 jkjGjdxeexxUkC jxxjj (A20) ijiiijiijDdxxexU ijxij (A21)where .2,4,, ,44,4,2arctan,arctan,2arctan kMandkLkGkFkD kBkAkkk The final form of the trial expansion space nt can be expressed in terms of vectordeterminants as1 nMCKnMSn C nS nnt : ......: ......:1 : 111: (A22)where
SSSM nS ::: , CCCM nC ::: and nnnn nn n ::: ::: ::::
21 22212 12111 (A23)The elements of the Hermitean matrix QQ can be abbreviated by gf : . The matrixelement gf : is defined by nk k gfgf : . The positron wavefunction nP canbe also expressed in terms of vector determinants as nMCKnMSn C nS nnP :..... ˆ :..... ˆ :1 : 111: 1 (A24)In the final analysis, K is given in the following form CCnSCn K ::11 (A25)where the determinants gfn : are defined as ng Tfgfn M Mgf ):( : (A26)where Tf M is the transpose of f M and gf CorS .2 References [1] P. A. M. Dirac,
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