Relativistic dynamical polarizability of hydrogen-like atoms

Transcription

1 J. Phys. B: At. Mol. Opt. Phys Printed in the UK Relativistic dynamical polarizability of hydrogen-like atoms Le Anh Thu, Le Van Hoang, L I Komarov and T S Romanova Department of Theoretical Physics, Belarussian State University, 4 Fr. Skariny av., Minsk 0050, Republic of Belarus Institute of Physico-Chemical Problems, Belarussian State University, 4 Leningradskaya str., Minsk 0080, Republic of Belarus Institute of Nuclear Problems, Belarussian State University, 4 Leningradskaya str., Minsk 0080, Republic of Belarus Received November 995, in final form 6 March 996 Abstract. Using the operator representation of the Dirac Coulomb Green function the analytical method in perturbation theory is employed in obtaining solutions of the Dirac equation for a hydrogen-like atom in a time-dependent electric field. The relativistic dynamical polarizability of hydrogen-like atoms is calculated and analysed.. Introduction In stationary perturbation theory as well as in the time-dependent one, the method of using the Coulomb Green function has found wide application in obtaining analytical solutions of the Schrödinger or Dirac equations. The main advantage of this method is the possibility of obtaining the final results in closed analytical form or reducing results to the summation of rapidly convergent series. Therefore, by using the above-mentioned method one can avoid a lot of complex numerical integrations see, for example, Makhanek and Korol kov 98, Zapryagaev et al 985 and references cited therein. On the basis of the application of the connection between the problem of the four-dimensional isotropic harmonic oscillator and that of a hydrogen-like atom in electromagnetic fields see Kustaanheimo and Stiefel 965, it has been proposed to establish a new representation of the Coulomb Green function in the form of a product of annihilation and creation operators this is called by us an operator representation. Operator representation of the Coulomb Green functions is very efficient in applications for the non-relativistic case Le Van Hoang et al 989 as well as for the relativistic one Le Anh Thu et al 994. The main elements of the algebraic method used with the aid of the operator representation of the Coulomb Green functions in Le Van Hoang et al 989 and in Le Anh Thu et al 994 are as follows. By using the above-mentioned connection, all operators of the algebra of the dynamical symmetry group SO4, can be found in the quadratic form of the annihilation and creation operators see, for example, Kleinert 968, Komarov and Romanova 98. Therefore, the calculation method, based on the use of the algebra of SO4,, leads only to the use of the simple commutation relations between the latter operators. The use of this method together with the operator representation of the Coulomb Green function, therefore essentially reduces the calculation process and provides for reducing rather complicated calculations of matrix elements with Coulomb wavefunctions to a purely algebraic procedure of transforming the product of the annihilation and creation operators to the normal form. The advantages of the /96/3897+0$9.50 c 996 IOP Publishing Ltd 897

2 898 Le Anh Thu et al proposed algebraic method are found not only in the simplicity of the calculation process but also in the possibility of obtaining the final results in the summation of rapidly convergent series. In fact, some clever results are obtained in Le Anh Thu et al 994 for the problem of calculation of the relativistic polarizability of hydrogen-like atoms. In this present paper, we consider the problem of calculating the relativistic polarizability of the ground state of hydrogen-like atoms on the basis of application of the operator representation of the Dirac Coulomb Green function established in Le Anh Thu et al 994. These calculations, besides their purely theoretical significance, are of great practical interest connected with recent developments in experimental investigations of multiply charged ions see, for example, Zapryagaev et al 985, Paratzacos and Mork 979. However, the majority of accurate calculations has been done only for the static polarizability of relativistic hydrogen-like atoms see also Drake and Goldman 98, 988, Johnson et al 988. In our calculations the radiation corrections are neglected, taking into account the fact that this effect is small compared with the external field effects. Our results are directly generalized from the nonrelativistic calculations Zapryagaev et al 985 and coincide with the results in the static limit Le Anh Thu et al Equation in two-dimensional complex space The Dirac equation for a hydrogen-like atom in the field of linearly polarized light can be written as follows h = m = c = : iα λ r + βr Ze + θ x λ erx 3e iνt + e iνt r,t=ir r,t t where α λ λ =,, 3 and β are the Dirac matrices; θ and ν are the amplitude and frequency of the external electric field respectively. Further on, we use the usual representation 0 σλ 0 = α λ = β = σ λ 0 0 where and are two-component spinors and σ λ λ =,, 3 are the Pauli matrices. The formal changes see Le Anh Thu et al 994 x λ σ λ st ξs ξ t r ξs ξ s r ˆp λ i σ λ st ξ t + ξ s ξ s ξt reduce equation to an equation describing the interaction between a particle with complex coordinates ξ s s =, and the external variable electric field. Here, in summation is indicated by means of repeated indices. The scalar product of wavefunctions in ξ-space is defined by the correlation ϕ = + + dξ dξ + + dξ dξ ξ,ξ,ξ,ξ ϕξ,ξ,ξ,ξ 3 where ξ s Reξ s, ξ s Imξ s. All operators appearing in are henceforth considered to conform with the formal changes. Thus, the operator on the left-hand side of equation is self-adjoint with respect to the scalar product of wavefunctions defined by 3. We will solve equation by using the method of perturbation theory assuming the external electric field to be small. Its solution can be found in the form r,t= 0 r,t+θ r,t 0 re iε0t +θure itε0 ν + θvre itε 0+ν 4

3 Relativistic dynamical polarizability of H-like atoms 899 where 0 r is a wavefunction in the zero-order approximation, i.e. a solution of the Dirac equation iα λ r + βr ε 0 r 0 r = Ze 0 r 5 x λ and ε 0 is the energy in the zero-order approximation. By substituting 4 into and taking into account 5, we obtain the equations iα λ r + βr Ze ur rε 0 νur = x erx 3 0 r 6 λ iα λ r + βr Ze vr rε 0 + νvr = x erx 3 0 r. 7 λ Noting that equations 6 and 7 have the same structure, we consider only equation 6 for the function ur; then by replacing ν by ν in ur we find the function vr. Let us now present ur and 0 in the form ϕ u = σ λ x λ /rϕ 0 ϕ 0 = σ λ x λ /rϕ 0. 8 The substitution of 8 into 6 leads to the following set of equations for ϕ and ϕ : i i x λ x λ + x λ + x λ ϕ iˆκϕ + r ε 0 + νϕ Ze ϕ = erx 3ϕ 0 9 ϕ + iˆκϕ r + ε 0 νϕ Ze ϕ = erx 3ϕ 0 0 where ˆκ = + σ λˆl λ ; ˆl is the orbital momentum operator. By using the transformations and ϕ 0 = i +ε 0 F 0 G 0 ϕ 0 = ε0 F 0 +G 0. ϕ = i +ε 0 νf G ϕ = ε0 +νf + G. we find x λ + + r Ze x λ ε 0 ν F + ˆκ + Ze x λ + r+ Ze x λ ε 0 ν G + ˆκ Ze where G = rx 3 Ã F 0 + Ã + G 0 F = rx 3 Ã + F 0 + Ã G 0 3 = ε 0 ν Ã ± = e 4 ε 0 ε 0 + ν ± + ε 0 + ε 0 ν. 4 After expanding the functions F and G in power series of the eigenfunctions of operators ˆL and ˆκ, the operator ˆκ appearing in and 3 becomes a c-number and, therefore, in order to solve equations and 3 we can employ the Green function method established in Le Anh Thu et al 994. In the next section we will show an example by calculating the dynamical polarizability of hydrogen-like atoms in the ground state.

4 900 Le Anh Thu et al 3. Relativistic dynamical polarizability in the ground state of hydrogen-like atoms The solution of equation 5 can be easily obtained by purely algebraic calculations. In particular, we can find for the ground state the solution see Komarov and Romanova 985: F 0 = 0 ε 0 Ɣε0 rε χ G 0 = 0 5 where 0 = Ze ; ε 0 = Ze ; χ are eigenvectors of operator σ 3 and 0 0 is the vacuum state defined by the equations a s =b s =0 s=,. Here the operators a s, b s are defined as follows a s = ξ s + b s = a + s = ξ s ξ s ξ s b + s = ξs + ξ s where is a positive parameter see Le Anh Thu et al 994. The perturbation term in equations and 3 thus has the form ±Ã rx 3 F 0 =±A r ε 0 Z, + Z, Here, we use the notation 60 ε 0 A ± = à ± Ɣε0 and Z l,κ are eigenvectors of i the square orbital momentum operator and ii the operator ˆκ. The structure of the perturbation term 6 prompts us to find the solution in the form F = F κ G = G κ 7 κ=, where F κ and G κ satisfy the equations x λ + + r Ze x λ ε 0 ν F κ + x λ + r + Ze x λ ε 0 ν From 9 it follows that F κ = A + κ κ Ze / rε Z κ 0 0 G κ + By substituting 0 into 8 we obtain: { x λ + + r Ze x λ ε 0 ν Ze +κ } G κ κ=, κ + Ze κ Ze κ Ze / x λ x λ ξ s ξ s G κ = A κ r ε 0 Z κ 0 0, 8 F κ = A + κ r ε 0 Z κ r + Ze x λ ε 0 ν G κ. + r + Ze x λ ε 0 ν 0

5 Relativistic dynamical polarizability of H-like atoms 90 = A κ κ Ze r ε 0 Z κ 0 0 +A + κ x λ + + r Ze x λ ε 0 ν r ε 0 Z κ 0 0. As mentioned above see, all operators in 0, though formally written for brevity via the usual coordinates x λ λ =,, 3 are understood in the sense of the formal changes. These changes are equivalent to the transformation from r-space to ξ-space Kustaanheimo-Stiefel transformation, see Kustaanheimo and Stiefel 965: { xλ = ξs σ λ st ξ t s, t =,. χ = argξ So we can rewrite the operators used in 0, via the annihilation and creation operators as follows: x λ = M M+ r = x λ M + M+ + N+ where the notations N = a s + a s + b s + b s M = a s b s M + = a s + b+ s. Therefore, equation has a more convenient form Ze {M + N + ε 0 ν M + Ze Ze + N + ε 0 ν + κ } G κ = A κ κ Ze r ε 0 Z κ 0 0 +A + κ M + N + Ze ε 0 ν r ε 0 Z κ 0 0. Here and henceforth, we omit for brevity the parameter in the expressions of the operators. Equation has the same structure as the equations appearing in the case of calculation of the static polarizability of hydrogen-like atoms see Le Anh Thu et al 994. The only difference is in the perturbation term on the left-hand side of these equations. Therefore, we can by analogy find the solutions of equation, using the Green function operator which, according to Le Anh Thu et al 994, can be established as follows: Ĝ lκ = ˆB lκ ˆB + N + M + M + lκ = d s N/M s 3 s=0 where d 0 n = 4 n+γ+ Ze ε/ d s n = s n + l +! γ + n + s + l +! Ɣn + + γ Ze ε/ Ɣn + s γ Ze ε/ γ = l + κ Z e 4 ε=ε 0 ν. Ɣn + s + γ Ze ε/ Ɣn + γ Ze ε/ s =,, 3... Here, we note that the Green function operator 3 acts on the basis of states with frequency. However, the wavefunctions in the zero-order approximation 5 have the frequency

6 90 Le Anh Thu et al 0 = Ze. Therefore, in order to use the algebraic calculation we have to transform the wavefunctions 5 from the frequency 0 to using the unitary transformation see Komarov and Romanova 98: 0 =Û 0,, where { Û 0,=exp ln } 0 M M+. This transformation can be reduced to the normal form Û 0,= exp 0 M + exp Nln exp 0 M Finally, by analogy with what was done in Le Anh Thu et al 994 for the same equation, we find the solution G κ =A κ κ Ze r γ Ĝ κ r ε0 γ Û 0,Z κ 0 0 +A + κ r γ Ĝ κ r γ N M+ + Ze ε 0 ν r ε 0 Û 0,Z κ The wavefunction F κ can be obtained by substituting 6 into 0. Let us now calculate the dynamical polarizability of the ground state of hydrogen-like atoms, the formula of which in ξ-space has the form aν = 0 erx 3. 0 r 0 This formula, after considering 8,, 5 and 7, can be written as follows aν = a κ ν where κ=, a κ ν = 4 0 κ 0 0 Z κ r ε 0 A + F κ + A G κ. 7 ε 0 By substituting the found solutions 0, 6 into 7 we find the expression for the positive frequency term of polarizability a κ +ν = 4 { 0 κ A + ε 0 B 0 rε A B+A +A δ+ A + δ B Here, we use the notations B = κ Ze δ = A + A + A + δ B H κ 00 H κ A + B H κ }. 8 Ze ε 0 ν + ε H κ nm = 0 Z κû+ 0,M n r ε 0+γ Ĝ κ r ε 0 γ M + m Û 0,Z κ 0. A similar expression for the negative frequency term can be obtained by replacing ν by ν in formulae 4, 8, 9.

7 Relativistic dynamical polarizability of H-like atoms 903 Figure. a The dependence of the dynamical polarizability of the hydrogen-like atom Z = 50 on the frequency ν of the external field one line A is shown in figure b with large scale. b The line A of figure a on a larger scale. By using the algebra of operators M +, M, N: M,M + = N + M,N + = M N +,M + =M + and the correlations NM + n Z lκ 0 =n + lm + n Z lκ 0 MM + n Z lκ 0 =nn + l + M + n Z lκ 0

8 904 Le Anh Thu et al Figure. The dependence of the dynamical polarizability of the hydrogen-like atom Z = 00 on the frequency ν of the external field. r ρ Z lκ 0 = Ɣρ + l + ρ Ɣ ρ s Ɣs ρ s! s + l +! M+ s Z lκ 0 we algebraically obtained the explicit form of the term Hnm κ as follows: H κ nm = ε s=0 p + 3! p=0 p q p Ɣq + ε0 γɣq +3+ε 0 γ q Ɣq + ε q=0 0 γ mq + 3! p s d sp s + p s! s=0 p s t p s t+q n m t + t=0 0 Ɣt + + ε 0 + γɣt +4+ε 0 +γ. 30 Ɣt + + ε 0 + γ nt + 3! Direct calculations show that all power series appearing in the term Hnm κ are rapidly convergent. The high convergency of these power series is directly related to the expansion 3 of the Dirac Coulomb Green function which, in fact, is established on the basis of harmonic oscillator wavefunctions. Moreover, the expression 8 for the polarizability contains only Hnm κ with n, m = 0,, the calculation of which needs only some of the first terms in the summation over p. For example, for frequency ν less than the fourth resonance frequency relative to the transition from the ground state to the 3P 3/ state the contribution of the terms with p = 0, inhnm κ n, m = 0, is about 98 99% for all Z 37.

9 Relativistic dynamical polarizability of H-like atoms 905 In figures a and b the dependence of relativistic polarizability on the external field frequency is given for hydrogen-like atoms with Z = 50. The dotted lines correspond to the non relativistic limit case. Figure gives the same for Z = 00. Let us now consider the non-relativistic limit, i.e. take into account only the first term in the expansion in the power series of Ze. For this case, we can effect summation over the variables t,q,s in the expression of Hnm κ and then have H nm κ in the form of hypergeometric functions. Consequently, we have µ 3 e { a non ±ν = Ze 4 µ + 0 /µ3 /µ F 5, µ, 4 µ µ, µ + 5µ + µ + 3 /µ F 6, 3 µ, 4 µ } µ, 3 µ + where µ = ± ν/ze. This expression coincides with the well known result obtained by Vetchinkin and Khristenko 968. By putting ν = 0 into 8, 30 and 3 we thus find the formula for the relativistic polarizability. It is easy to see that for = 0 the formula 8 contains only the term H00 κ. Taking into account the formula Prudnikov et al 98 n k n a+k = n a k m m n k=0 we can lead H00 κ to the form H00 κ = Ɣε 6 0 ε 0 γɣε 0 γ +3Ɣε 0 + γ + Ɣε 0 + γ p=0 p s=0 Ɣε 0 γ p d s p s + p s!p s + 3!Ɣε 0 + γ + p + s. 3 The substitution of 3 into 8 gives for ν = 0 the relativistic static polarizability a = e ε 0 + ε 0 + 4ε0 + 3ε e ε 0 Ɣε 0 + γ + 4Ɣε 0 γ Ɣε 0Ɣ ε 0 γɣ ε 0 +γ Ɣk ε 0 γɣk ε 0 +γ + γ + k!k + 3!k ε k=0 0 + γ + 3 Ɣq ε 0 + γ + Ɣq ε 0 γ + q + 3! Ɣq ε 0 + γ + 4 q= q s=0 Ɣs ε 0 γɣs ε 0 +γ +3 s!ɣs ε 0 γ + 3 where γ = + 4 Ze. This result absolutely coincides with the result obtained in Le Anh Thu et al 994 see also Barut and Nagel Conclusion In conclusion we would like to note that the magnetic field effects, as a rule, should be taken into account for a detailed investigation of the behaviour of a relativistic atom in the field of linearly polarized light. These effects can be neglected only in the non-relativistic limit.

10 906 Le Anh Thu et al The above method proposed with the use of the operator representation of the Coulomb Green function can also be employed, for example, in calculating magnetic polarizability. Consideration of magnetic field effects leads only to an enormous number of calculations, which are more complicated in comparison with the above calculations but could be done analogously by analytical methods. Moreover, we hope that our algebraic method would be helpful when considering the problem of an atom in a quantum field, in particular in calculating the Lamb shift of multiply charged ions, which is of great interest and has been widely investigated recently see, for example, Snyderman 99. Acknowledgments The authors would like to thank Professor A O Barut for useful discussions and for his interest in this work. One of the authors LVH would like to thank the Fundamental Research Foundation of the Republic of Belarus for the support rendered. References Barut A O and Nagel J 976 Phys. Rev. D Drake GWFandGoldman S P 98 Phys. Rev. A Adv. At. Mol. Phys Johnson W R, Blundell S A and Sapirstein J 988 Phys. Rev. A Kleinert H 968 Lectures in Theoretical Physics ed W E Brittin and A O Barut New York: Gordon and Breach p 47 Komarov L I and Romanova T S 98 Izv. Acad. Nauk BSSR, Ser. Fiz. Mat. Nauk J. Phys. B: At. Mol. Phys Kustaanheimo P and Stiefel E 965 J. Reine Angrew. Math Le Anh Thu, Le Van Hoang, Komarov L I and Romanova T S 994 J. Phys. B: At. Mol. Opt. Phys Le Van Hoang, Komarov L I and Romanova T S 989 J. Phys. A: Math. Gen. 543 Makhanek A G and Korol kov V C 98 Analytical Methods in the Quantum Perturbation Theory Minsk: Nauka i Tekcnika Paratzacos P and Mork K 979 Phys. Rep. C 8 Prudnikov A P, Brichkov Yu A and Marichev O I 98 Integrals and Series Moscow: Nauka Snyderman N J 99 J. Ann. Phys. 43 Vetchinkin S I and Khristenko S V 968 Opt. Spectrosc Zapryagaev S A, Manakov N L and Pal chikov V G 985 Ehtoery of Multiply Charged Ions with One or Two Electrons Moscow: Energoatomizdat Zon B A, Manakov N L and Rapoport L P 97 Yad. Phys