Electric and magnetic multipoles

Transcription

1 Electric and magnetic multipoles Trond Saue Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

2 Multipole expansions In multipolar gauge the expectation value of the interaction Hamiltonian takes the form H int = [ρ(r)φ(r) j(r) A(r)] d 3 r = Q [0] φ [0] n=1 1 n! Q[n] j 1j 2...j n 1 E [n 1] j 1...j n 1 n=1 where appears electric multipoles Q [n] j 1...j n = r j1 r j2... r jn ρ(r)d 3 r 1 n! m[n] j 1j 2...j n 1 B [n 1] j 1...j n 1 and magnetic multipoles m [n] j 1...j n 1 = n n + 1 r j1 r j2... r jn 1 [r j(r)] d 3 r Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

3 Electric and magnetic multipoles Examples Electric dipole: Electric quadrupole: Magnetic dipole: Magnetic quadrupole: µ i = Q [1] i = Q [2] ij = m [1] i = 1 2 m [2] ij = 2 3 r i ρ(r)d 3 r r i r j ρ(r)d 3 r (r j(r)) i d 3 r r j (r j(r)) i d 3 r Many molecular properties are defined in terms of (induced) electric and magnetic multipoles. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

4 Origin dependence Origin independence generally holds true only for lowest order non-zero multipole. Consider a change of origin: r r A: Electric dipole moment: = (r A) ρ(r)d 3 r = Q [1] A rρ(r)d 3 r A ρ(r)d 3 r = Q [1] 0 AQ[0] Magnetic dipole moment: m [1] A = 1 {(r A) j(r)} d 3 r = m [1] 0 2 A 1 2 j(r)d 3 r Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

5 Traceless electric multipoles Let us consider the nth order electric multipole interaction H En = 1 n! Q[n] j 1j 2...j n E [n 1] j 1;j 2...j n The nth order electric multipole is symmetric in all indices Q [n] j 1...j n = r j1 r j2... r jn ρ(r)d 3 r The (n 1)th derivative of the electric field is symmetric in all but one index E [n 1] j 1;j 2...j n = n 1 E j1 r j2... r jn...except in the static case where it is symmetric in all indices since it is then given by E = φ E [n 1] j 1;j 2...j n = φ [n] j 1j 2...j n 0 Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

6 Traceless electric multipoles We generally have In the static case we have and E = ρ/ε 0 ; E [1] i;i = ρ [0] /ε 0 E [1] i;i = φ [2] ii = ρ [0] /ε 0 H En = 1 n! Q[n] j 1...j n φ [n] j 1...j n When the expansion point of the multipole expansion is outside the charge distribution generating the scalar potential (so not the molecular one!), we have ρ [0] = 0. This restriction can be built into the electric multipoles by introducing traceless multipoles, that is tracing over any two indices gives zero. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

7 Traceless electric quadrupoles Let us focus on the electric quadrupole interaction in the static case H E2 = 1 2 Q[2] ij φ [2] ij = 1 { } Q xx [2] φ [2] xx + Q yy [2] φ [2] yy + Q zz [2] φ [2] zz + Q xy [2] φ [2] xy }{{} (no summation) We introduce the traceless quadrupole moment Θ [2] ij = 3 ( Q [2] ij 1 ) 2 3 Q[2] kk δ ij which gives H E2 = 1 3 Θ[2] ij φ [2] ij 1 6 Q[2] kk φ[2] ii = 1 3 Θ[2] ij φ [2] ij 1 6 Q[2] kk ρ[0] /ε 0 }{{} Poisson term Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

8 Traceless electric multipoles Traceless electric multipoles can be generated from the formula Θ [n] j 1...j n = ρ(r) ( 1)n r 2n+1 {Π n n! k=1 jk } 1 r d 3 r where we temporarily assume ρ [0] = 0. We obtain Θ [2] ij = 1 ( ) 3Q [2] ij Q [2] kk 2 δ ij Θ [3] ijk = 1 2 Θ [4] ijkl = 1 8 ( 5Q [3] ijk Q[3] ill δ jk Q [3] jll δ ik Q [3] kll δ ij ( 35Q [4] ijkl 5δ ij Q [4] klmm 5δ ikq [4] jlmm 5δ ilq jkmm 5δ jk Q [4] ilmm 5δ jlq [4] ikmm 5δ klq [4] ilmm + {δ ijδ kl + δ ik δ jl + δ il δ jk } Q [4] mmnn The number of traceless and non-traceless electric multipoles is (2n + 1) and 1 2 (n + 2) (n + 1), respectively. ) ) Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

9 Traceless electric multipoles The set of electric multipoles of order n is Q [n] = x i y j z k ρ(r)d 3 r; i + j + k = n For instance, for n = 2 (0, 0, 2) (0, 1, 1) (0, 2, 0) (1, 0, 1) (1, 1, 0) (2, 0, 0) Generally, the number of triples (i, j, k) is 1 2 (n + 1)(n + 2). When switching to traceless multipoles we introduce constraints for each index pair, that is 1 2n(n 1) constraints. The effective number of traceless multipoles is 2n + 1. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

10 Spherical electric multipoles The number of traceless and general electric multipoles of order n correspond exactly to the number of Cartesian and spherical Gaussian-type orbitals, respectively, of order l Indeed, in the static case the electric multipole expansion can be carried out in spherical harmonics using the addition theorem 1 r r = +l l=0 m= l 4π 2l + 1 r< l r> l+1 Y lm ( r ) ( ) r Y lm r where r > = max (r, r ) and r < = min (r, r ). Inserted into the interaction energy (Coulomb gauge) we obtain (assuming r < = r) H int = with q lm = ρ (r) φ (r) d 3 r = 1 4πε 0 ρ(r)ρ(r ) r r d 3 r = r l Ylm (θ, φ) d 3 r; E lm = 1 4π 4πε 0 2l + 1 r l l=0 m= l q lm E lm r (l+1) Y lm (θ, φ ) d 3 r Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

11 Limit on possible electric multipoles For atoms (and other systems) with well-defined angular moment, the state function may be expressed as ηlm Such functions form a basis for the spherical multipole operator q lm ηlm =...such that the integral η L M q lm ηlm = is non-zero only for l+l l = l L l+l l = l L C l L = l L,..., l + L; C l ηl (M + m) η L M ηl (M + m) M = M + m This results implies that the state ηlm has maximally an electric multipole moment of order 2L. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

12 Range of validity of traceless electric multipoles In the static case traceless (and spherical) electric multipoles are perfectly equivalent to the general ones. In the general case: H E2 = 1 2 Q[2] E [1] ij i;j = 1 3 Θ ij E [1] 1 i;j 6 Q[2] kk E [1] i;i H E3 = 1 6 Q[3] ijk E [2] i;jk = 1 15 Θ[3] ijk E [2] i;jk 1 15 Q[3] E [2] ill i;jj 1 30 Q[3] kll E [2] i;ik H E4 = 1 24 Q[4] ijkl E [3] = 1 ijkl 105 Θ[4] ijkl E [3] i;jkl 1 56 Q[4] ilmm E [3] 1 i;jjl 56 Q[4] klmm E [3] + 1 i;ikl 280 Q[4] mmnn E [4] i;ikk Conclusion: If one wants to introduce electric multipoles that can be applied in both static and dynamic situations, then the general definition is to be preferred over the traceless forms. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

13 Nuclear electric moments Nuclear wave functions have well-defined parity [Ĥ nuc, P] = 0 Pψ nuc (r) = ±ψ nuc (r); ρ nuc ( r) = ρ nuc (r) Integrand of electric multipole Q [n] (r) = r j1 r j2... r jn ρ(r) Q [n] ( r) = ( 1) n r j1 r j2... r jn ρ( r) = ( 1) n r j1 r j2... r jn ρ(r) For nuclei (and atoms) all odd electric multipoles are zero. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

14 Mössbauer spectroscopy is high-energy spectroscopy Nuclear isomers are metastable nuclear excited states (half-life 1 ns or longer) Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

15 Mössbauer spectroscopy in a nutshell Recoil-free, resonant absorption and emission of gamma rays in solids The energy E γ of the nuclear γ-transition is modified by the local chemical environment Isomer shift (mm/s): δ = c ( E γ E a γ Eγ s ) E γ : change in the electron- nucleus interaction due to the change in nuclear size between the nuclear excited and ground state Further structure by quadrupole splitting and nuclear Zeeman effect Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

16 Models for the nuclear charge distribution Point nucleus: φ point (r; Z) = Z r Homogeneously charged sphere: φ H (r; Z, R) = { Z 2R Z ; r Gaussian charge distribution: φ G (r; Z, η) = Z r erf ( r η ) ( 3 r 2 R 2 ) ; r < R r > R We impose the same root mean square radius for all finite charge distributions based on the empirical formula 1/2 [ ] rn 2 = 0.836A 1/ fm Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

17 A nuclear model-independent expression for the electron-nucleus interaction The potential of point and finite nuclei are identical outside some critical radius r c: E en = ρ e (r e) φ point n (r e) d 3 r e + E en [ ] E en = ρ e (r e) φ n φ point n (r e) d 3 r e r e <r c Using the first mean value theorem for definite integrals: [ ] E en = ρ e φ n φ point n (r e) d 3 r e r e <r c introduces the effective electronic charge density ρ e = ρ e( r); r 0, r c Further manipulation gives: E en = ρ e (r e) φ point n (r e) d 3 Ze ρe r e rn 2 6ε 0 and shows that the mean square radius of the nuclear charge distribution is the appropriate measure of nuclear size Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

18 Mössbauer spectroscopy Isomer shift Modification of nuclear γ-transition by electrostatic interaction between electronic and nuclear charge distributions E γ = E en ( rn 2 ) EeN ( rn 2 The change δ rn 2 = r 2 n r 2 n so we may use a first-order Taylor expansion E γ = de en d rn 2 δ r 2 n 0 0 ) 0 is small compared to r 2 n rn 2 = Ze ρe δ rn 2 6ε 0 0, Mössbauer isomer shift δ = α ( ρ a e ρ s e) ; α = Zec δ r 2 n 6ε 0 E γ Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

19 Mössbauer spectroscopy Effective density vs. contact density The effective density is usually approximated by the contact density, that is, the electron density at the nucleus ρ e ρ e (0) For heavy nuclei this will lead to (systematic) errors Electron (number) density in the nuclear region of the mercury atom Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

20 57 Fe isomer shifts The Mössbauer isomer shift probes the electron density in the close vicinity of the nucleus and is sensitive to the chemical environment P. Gütlich and C. Schröder, Mössbauer Spectroscopy, Bunsenmagazin, 12 (2010) 4 Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

21 Mössbauer spectroscopy Electric quadrupole splittings Nuclei of spin I > 1 2 may possess a quadrupole moment which then interacts with the electronic electric field gradient at the position of the nucleus H E2 = 1 2 Q[2] ij φ [2] ij = 1 3 Θ ije [1] i;j 1 6 Q[2] kk E [1] i;i = 2 m= 2 Q 2mE [2] lm 1 Q [2] kk 6ε ρ[0] e 0 Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

22 Nuclear electric quadrupole moments The nuclear spherical electric quadrupole moments reads Q 20 = 1 ρ n (3z 2 r 2) d 3 r n 2 3 Q 2,±1 = ρ nz n(x n ± iy n)d 3 r n 2 3 Q 2,±2 = ρ n(x n ± iy n) 2 d 3 r n 8 ) The operator (3Î z 2 I 2 has the same angular dependence with respect to nuclear orientation as Q 20 which implies that [ ] 3Î Im I Q 20 ImI = C Im 2 I z I 2 ImI = C 3mI 2 I (I + 1) (Wigner-Eckart theorem) We define eq = II Q 20 II = C [I (2I 1)] ; C = eq I (2I 1) Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22

23 Electric quadrupole splitting For a system with axial symmetry E Q (I, m I ) = [1] eqe zz { 3m 2 4I (2I 1) I I (I + 1) } The electric quadrupole splitting is sensitive to deviations from spherical symmetry. Trond Saue (LCPQ, Toulouse) Electric and magnetic multipoles Virginia Tech / 22