Appendix 1 Dynamic Polarizability of an Atom

Appendix 1 Dynamic Polarizability of an Atom Definition of Dynamic Polarizability The dynamic (dipole) polarizability aðoÞ is an important spectroscopic characteristic of atoms and nanoobjects describing the response to external electromagnetic disturbance in the case that the disturbing field strength is much less than the atomic ꢀelectric field strength E<<Ea ¼ me2 e5 ꢀh4 ffi 5:14 Á 109 V=cm, and the electromagnetic wave length is much more than the atom size. From the mathematical point of view dynamic polarizability in the general case is the tensor of the second order aij connecting the dipole moment d induced in the electron core of a particle and the strength of the external electric field E (at the frequency o): X<ul style="display: flex;"><li style="flex:1">diðoÞ ¼ </li><li style="flex:1">aijðoÞ EjðoÞ: </li></ul>(A.1) jFor spherically symmetrical systems the polarizability aij is reduced to a scalar: aijðoÞ ¼ aðoÞ dij; (A.2) where dij is the Kronnecker symbol equal to one if the indices have the same values and to zero if not. Then the Eq. A.1 takes the simple form: dðoÞ ¼ aðoÞ EðoÞ: (A.3) The polarizability of atoms defines the dielectric permittivity of a medium eðoÞ according to the Clausius-Mossotti equation: eðoÞ À 1 43¼p na aðoÞ; (A.4) 347 eðoÞ þ 2 V. Astapenko, Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures and Solids, Springer Series on Atomic, Optical, and Plasma Physics 72, #<ul style="display: flex;"><li style="flex:1">DOI 10.1007/978-3-642-34082-6, </li><li style="flex:1">Springer-Verlag Berlin Heidelberg 2013 </li></ul><ul style="display: flex;"><li style="flex:1">348 </li><li style="flex:1">Appendix 1 Dynamic Polarizability of an Atom </li></ul>where na is the concentration of substance atoms. For simplicity it is assumed in Eq. A.4 that a medium consists of atoms of the same kind. It should be noted that the basis for a number of experimental procedures of determination of the dynamic polarizability aðoÞ is its connection with the refractive index of a substance (that for a transparent nonmagnetic medium is pffiffiffiffiffiffiffiffiffi determined by the equation nðoÞ ¼ eðoÞ). Dynamic polarizability defines the shift of the atomic level energy DEn in an external electric field. In the second order of the perturbation theory for the nonresonant external field E and a spherically symmetric electronic state a corresponding correction to energy looks like 1DEðn2Þ ¼ À anðoÞ E2: (A.5) 2The formula (A.5) describes the quadratic Stark effect. In the case that the external field frequency coincides with the eigenfrequency of an atom, the energy shift is found to be linear in electric field intensity – the linear Stark effect. The linear Stark effect is realized also in case of an orbitally degenerate atomic state as it takes place for a hydrogen atom and hydrogen-like ions. Static polarizability, that is, polarizability at the zero frequencyaðo ¼ 0Þdefines the level shift in a constant electric field and, besides, the interatomic interaction potential at long distances (the Van der Waals interaction potential). Since static polarizability is a positive value (see below), from the Eq. A.5 it follows that the energy of a nondegenerate atomic state decreases in the presence of a constant electric field. The potential of interaction of a neutral atom with a slow charged particle at long distances is also defined by its static polarizability: að0Þ VpolðrÞ ¼ Àe20 ;(A.6) 2 r4 where e0 is the particle charge. With the use of Eq. A.6 it is possible to obtain the following expression for the cross-section of elastic collision of a charged particle with an atom in case of applicability of the classical approximation for description of motion of an incident particle with the energy E: rffiffiffiffiffiffiffiffiffi að0Þ el scat sðEÞ ¼ 2 p e0 :(A.7) 2 E It should be noted that the Eq. A.7 follows (accurate to the factor equal to 2) from Eq. A.6 if the effective scattering radius rE is determined with the use of the equation ꢁꢁ ꢁꢁE ¼ VpolðrEÞ : (A.8) <ul style="display: flex;"><li style="flex:1">Appendix 1 Dynamic Polarizability of an Atom </li><li style="flex:1">349 </li></ul>Thus the knowledge of dynamic polarizability is very important for description of a whole number of elementary processes. Expression for the Dynamic Polarizability of an Atom Let us calculate the dipole moment of an atom d in the monochromatic field EðtÞ ¼ 2RefEo expðÀi o tÞg that by definition is dðtÞ ¼ 2RefaðoÞ Eo expðÀi o tÞg: The Fourier component of the dipole moment is given by the expression do ¼ aðoÞ Eo: (A.9) (A.10) In the formulas (A.9) and (A.10) Eo is the complex electric field vector in monochromatic radiation being a Fourier component of EðtÞ. The dipole moment of an atom in the absence of external fields is equal to zero in view of spherical symmetry, so the value of an induced dipole moment can serve as a measure of disturbance of an atom by an external action. The linear dependence dðtÞon electric field intensity (A.9) is true in case of smallness of the field strengthE from the standpoint of fulfilment of the inequations E<<Ea. Thus for low enough field strengths the response of an atom to electromagnetic disturbance can be characterized by its polarizability aðoÞ. For description of the electromagnetic response of an atom – a quantum system – within the framework of classical physics, it is convenient to use the spectroscopic conformity principle. It can be formulated as follows: an atom in interaction with an electromagnetic field behaves as a set of classical oscillators (transition oscillators) with eigenfrequencies equal to the frequencies of transitions between atomic energy levels. This means that each transition between the atomic states jji and jni is assigned an oscillator with the eigenfrequency ojn and the damping constant djn< <ojn. The contribution of the transition oscillators to the response of an atom to electromagnetic action is proportional to a dimensionless quantity called oscillator strength – fjn , the more is the oscillator strength, the stronger is a corresponding transition. Transitions with the oscillator strength equal to zero are called forbidden transitions. According to the spectroscopic conformity principle, the change of an atomic state is made up of the change of motion of transition oscillators. The inequation E<<Ea means the smallness of perturbation of an atomic electron state as a result of action of the electromagnetic field. Thus it is possible to consider the deviations of the transition oscillators from the equilibrium position under the action of the field EðtÞ small, so for a nth oscillator the equation of motion in the harmonic approximation is true: <ul style="display: flex;"><li style="flex:1">350 </li><li style="flex:1">Appendix 1 Dynamic Polarizability of an Atom </li></ul>e2<ul style="display: flex;"><li style="flex:1">€</li><li style="flex:1">_</li></ul>rn þ dn0 rn þ on0 rn ¼ fn0 EðtÞ; (A.11) mwherern is the radius vector corresponding to the deviation of the transition oscillator from the equilibrium position; dn0, on0, fn0 are the damping constant, eigenfrequency and the oscillator strength. For simplicity we consider a one-electron atom in the ground state, the dipole moment of which is equal to d ¼ e r . (In case of a multielectron atom the dipole moment is equal to the sum of dipole moments of atomic electrons.) In view of the conformity principle the induced dipole moment of an atom is made up of induced dipole moments of the oscillators of transitions to <ul style="display: flex;"><li style="flex:1">P</li><li style="flex:1">P</li></ul>the nth state dn: d ¼ dn ¼ e we have rn. Going in this equation to Fourier components, <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>Xdo ¼ e rno ;(A.12) nwhere rno is the Fourier transform of the radius vector of the transition oscillator deviation from the equilibrium position. The expression for this value follows from the equation of motion (A.11): efn0 rno ¼Eo: (A.13) m o2n0 À o2 À i o dn0 Substituting the formula (A.13) in the Eq. A.12 and using the definition of polarizability (A.10), we find for it the following expression: e2 mfn0 o2n0 À o2 À i o dn0 XaðoÞ ¼ :(A.14) nHence it follows that the dynamic polarizability of an atom represents, generally speaking, a complex value with a dimensionality of volume. The imaginary part of polarizability is proportional to the damping constants of the transition oscillators. The sum on the right of the Eq. A.14 includes both summation over the discrete energy spectrum and integration with respect to the continuous energy spectrum. The imaginary part of polarizability is responsible for absorption of radiation, and the real part defines the refraction of an electromagnetic wave in a medium. The expression (A.14) describes not only a one-electron atom, but also a multielectron atom. The multielectron nature of an atom is taken into account by the fact that in definition of the oscillator strength the dipole moment of an atom is equal to the sum of dipole moments of its electrons. From the Eq. A.14 several important limiting cases can be obtained. For example, if the frequency of the external field is equal to zero, the formula (A.14) gives the expression for the static polarizability of an atom: <ul style="display: flex;"><li style="flex:1">Appendix 1 Dynamic Polarizability of an Atom </li><li style="flex:1">351 </li></ul>e2 mfn0 o2n0 Xa0 ꢀ aðo ¼ 0Þ ¼ :(A.15) nHence it is seen that static polarizability is a real and positive value. It has a large numerical value if in the atomic spectrum there are transitions with high oscillator strength and low eigenfrequency as it is, for example, for alkaline-earth atoms. In the opposite high-frequency limit, when ꢀh o>>IP ( IP is the ionization potential of atom) and the eigenfrequencies in the denominators of Eq. A.14 can be neglected, from the formula (A.14) in view of the golden rule of sums, according to which the sum of oscillator strengths is equal to the number of electrons in an atom Na, we obtain e2 Na m o2 a ðoÞ ¼ À 1:(A.16) Hence it is seen that the high-frequency polarizability of an atom is a real and negative value that decreases quadratically with growing frequency of the external field. If the external field frequency is close to one of eigenfrequencies of the transition oscillators, so that the resonance condition jo À on0j ꢁ dn0 (A.17) is satisfied and one resonant summand in the sum (A.14) can be retained, then from Eq. A.14 the expression for resonant polarizability follows: <ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><ul style="display: flex;"><li style="flex:1">e2 </li><li style="flex:1">fn0 </li></ul>aresðoÞ ¼ :(A.18) 2 m on0 on0 À o À i dn0=2 In derivation of Eq. A.18 from Eq. A.14 in nonresonance combinations the distinction of the external field frequency from the transition eigenfrequency was neglected. Resonant polarizability is a complex value, the real part of which can be both positive and negative. The Eq. A.10 determining dynamic polarizability after the Fourier transformation can be rewritten as 1ðdðtÞ ¼ aðtÞ Eðt À tÞ dt; (A.19) À1 <ul style="display: flex;"><li style="flex:1">352 </li><li style="flex:1">Appendix 1 Dynamic Polarizability of an Atom </li></ul>where aðtÞ is the real time function, the Fourier transform of which is equal to the dynamic polarizability aðoÞ. The most simple expression for aðtÞ follows from the formula (A.18): e2 fn0 <ul style="display: flex;"><li style="flex:1">aresðtÞ ¼ </li><li style="flex:1">ðÀiÞ yðtÞ expðÀi on0 t À dn0 t=2Þ; </li></ul>(A.20) 2 m on0 where yðtÞ is the Heaviside step function. The time dependence of the induced dipole moment dðtÞ coincides with the time dependence of the right side of the Eq. A.20 for the delta pulse of the field: EðtÞ ¼ E0 dðtÞ, where dðtÞ is the Dirac delta function. In the general case the expression for bðtÞ can be obtained by replacement qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2of the frequency on0 !o2n0 À ðdn0=2Þ and summation over all transition oscillators. It should be noted that the decrease of the eigenfrequency of oscillations in view of damping following from the said replacement is quite natural since friction (the analog of damping) reduces the rate of motion. Concerned above was dipole polarizability that describes the response of an atom to a spatially uniform electric field. In the case that the characteristic size of the spatial nonuniformity of a field is less than the size of an atom, dipole polarizability should be replaced by the generalized polarizability of an atom aðo; qÞ depending on the momentum ꢀh q transferred as a result of interaction. The spatial scale of the field nonuniformity l is inversely proportional to the value of the wave vector l ꢂ 1=q. With the use of the generalized polarizability of an atom aðo; qÞ the formula (A.3) is modified to the form ðdq DðoÞ ¼ aðo; qÞ Eðo; qÞ ;(A.21) 3ð2 pÞ where Eðo; qÞ is the spatio-temporal Fourier transform of the electric field vector. For the spatially uniform field Eðo; qÞ ¼ EðoÞ dðqÞ the Eq. A.21 (in case of a spherically symmetric atomic state) goes to Eq. A.1 in view of the fact that aðoÞ ¼ aðo; q ¼ 0Þ. Appendix 2 Methods of Description of the Electron Core of Multielectron Atoms and Ions Slater Approximation For definition of the effective field and the concentration of a atomic core, for simple estimations, and in a number of applications, in which the behavior of wave functions of atomic electrons at long distances is essential, nodeless Slater functions of the following form are used: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mþ1 ð2bÞ PgðrÞ ¼ rmeÀb r ;(A.22) Gð2m þ 1Þ whereg ¼ ðnlÞis the set of quantum numbers characterizing an electronic state, b; m are the Slater parameters. The wave functions (A.22) are normalized, have correct asymptotics at long distances. The main advantage of the functions (A.22) consists in their simplicity. To determine the parameters b; m, Slater proposed empirical rules that for shells more than half-filled look like Z À Sg bg ¼ ;(A.23) mg where Z is the atomic nucleus charge, Sg is the screening constant, the values of which, together with the parameter mg and the number of electrons Ng for different electron shells, are given in Table A.1. For shells that are half-filled or less than half-filled, the best results are given by another rule: .pffiffiffiffiffiffiffiffi m ¼ half-integer value nearest to Z 2jEj; b ¼ m 2jEj=Z; (A.24) 353 V. Astapenko, Polarization Bremsstrahlung on Atoms, Plasmas, Nanostructures and Solids, Springer Series on Atomic, Optical, and Plasma Physics 72, #<ul style="display: flex;"><li style="flex:1">DOI 10.1007/978-3-642-34082-6, </li><li style="flex:1">Springer-Verlag Berlin Heidelberg 2013 </li></ul>354 Appendix 2 Methods of Description of the Electron Core of Multielectron Atoms and Ions Table A.1 Slater parameters of atomic shells Shell g ¼ ðnlÞ Sg 0.30 mg Ng <ul style="display: flex;"><li style="flex:1">1s2 </li><li style="flex:1">1</li><li style="flex:1">2</li></ul><ul style="display: flex;"><li style="flex:1">8</li><li style="flex:1">2(sp)8 </li><li style="flex:1">4.15 </li></ul>11.25 21.15 27.75 39.15 45.75 44.05 57.65 71.15 2<ul style="display: flex;"><li style="flex:1">3(sp)8 </li><li style="flex:1">3</li><li style="flex:1">8</li></ul>3d10 <ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">10 </li></ul><ul style="display: flex;"><li style="flex:1">8</li><li style="flex:1">4(sp)8 </li><li style="flex:1">3.5 </li></ul>3.5 44d10 10 8 5(sp)8 без 4f 4(df)24 5(sp)8 с 4f 5d10 c 4f 3.5 4 24 8<ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">10 </li></ul>where E is the electronic state energy in atomic units. With the use of the functions (A.22) the radial distribution of the electron density of an atom in the Slater approximation can be obtained as XrðrÞ ¼ Ng P2gðrÞ: (A.25) (A.26) (A.27) gThe atomic (Slater) potential corresponding to this electron density is zSðrÞ USðrÞ ¼ À ;rwhere zSðrÞ is the effective charge of the core: r1<ul style="display: flex;"><li style="flex:1">ð</li><li style="flex:1">ð</li></ul>rðr0Þ zSðrÞ ¼ Z À rðr0Þdr0 À r dr0: r0 r0It is possible to make sure that the potential of Eqs. A.26 and A.27 satisfies the electrostatic Poisson equation with the boundary conditions: zSð0Þ ¼ Z; zSð1Þ ¼ Zi; (A.28) where Zi is the charge of an ion that is equal, naturally, to zero for a neutral atom. Substituting in Eq. A.27 the formulas (A.22), (A.25) and performing integration, we find <ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul>2mÀ1 k<ul style="display: flex;"><li style="flex:1">X</li><li style="flex:1">X</li></ul>2m À k ð2b rÞ 2m k! <ul style="display: flex;"><li style="flex:1">zSðrÞ ¼ Z À </li><li style="flex:1">Ng 1 À eÀ2br </li></ul>

Appendix 1 Dynamic Polarizability of an Atom

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support