Mny-Body Clcultons of the Isotope Shft W. R. Johnson Mrch 11, 1 1 Introducton Atomc energy levels re commonly evluted ssumng tht the nucler mss s nfnte. In ths report, we consder correctons to tomc levels ssocted wth fnte nucler mss. These correctons re referred to s sotope shfts. We evlute sotope shfts usng mny-body perturbton theory (MBPT), followng the poneerng work by Mårtensson-Pendrll et l. [1,,3,4,5,6,7,8,9]. We frst gve bref dscusson of contrbutons to the sotope shft, then go on to specfc exmples. For smplcty, the exmples dscussed here re lmted to sotope shfts n toms wth sngle vlence electron. We consder nonreltvstc tom wth N electrons of mss m e t ( r 1, r, ) nd nucleus of mss M A t r. The Hmltonn for the N + 1 prtcle tom my be wrtten H( r, r 1, r,, p, p 1, p, )= p + M A m e + V e N ( r r )+ V e e ( r r j ). (1) j Let us trnsform to reltve coordntes: ρ = r r () R = M A r + m e r, (3) M T where M T = M A + Nm e. The generlzed moment conjugte to ρ nd R re: We fnd usng the four prevous equtons: π = 1 ρ (4) p P = 1 R (5) p = π + m e M T P (6) p = π + M A M T P (7) 1
The knetc energy term n the orgnl Hmltonn cn be rewrtten [ ] K. E. = π + m e π m e M P + Nm e T M P T ( ) + π M A π M A M P + M A T MT P = m e + M A π + π π j + 1 P (8) m e M A M A M T The wve functon for the tom s therefore fctorzble nto product of plne wve descrbng the center of mss moton nd n N-electron wve functon descrbng the moton reltve to the nucleus. The Hmltonn for the reltve moton s H( ρ 1, ρ,, π 1, π, ) = π µ + V e N ( ρ )+ V e e ( ρ ρ j ) j + π π j. (9) M A where the reduced mss µ s gven by j j 1.1 Norml nd Specfc Mss Shfts We wrte the Hmltonn s sum µ = M A m e M A + m e. (1) H( ρ 1, ρ,, π 1, π, ) = H µ + H (11) H µ ( ρ 1, ρ,, π 1, π, ) = µ + V e N ( ρ ) + V e e ( ρ ρ j ) (1) H( π 1, π, ) = j π π π j. (13) M A Norml Mss Shft The effect of the fnte nucler mss on the frst term s to scle the nfnte mss Rydberg constnt by the rto µ/m e = M A /(M A +m e ). The correspondng shft of the energy from the nfnte-mss vlue s s referred to s the norml mss shft. The vlue of the norml mss shft s ( ) MA δe NMS = E µ E me = 1 E me = m e E me m e E µ. M A + m e M A + m e M A (14) j
Here E me s the vlue of the energy n tomc unts (clculted wth the nfntemss Rydberg constnt). We my use the bove expresson wth E µ replced by the expermentl energy to evlute the norml mss shft to obtn n ccurte pproxmton to the norml mss shft. Specfc Mss Shft The correcton to the energy from H s referred to s the specfc mss shft. The vlue of the specfc mss shft s δe SMS = π π j. (15) M A The energy s proportonl to the mss (µ for H µ )or(m e for H me )nthe denomntor of the knetc energy. It follows tht lengths scle nversely wth mss nd tht knetc energy scles drectly s mss. The sclng of knetc energy mples tht momentum scles drectly wth mss. Wth the d of these sclng reltons, one my rewrte M A δe SMS = (M A + m e ) p p j, (16) n the center of mss system. The sclng s requred snce we evlute the SMS mtrx element usng nfnte nucler mss wve functons. The prescrpton s s follows: () Express ll nswers n terms of the R, the nfnte mss Rydberg constnt. (b) Multply the totl energy by m e /(M A + m e ) to obtn the the norml mss shft. (c) Multply the mtrx element of p p j by M A /(M A + m e ) to fnd the spectfc mss shft. Alterntvely, we my use expermentl energes for E µ nd evlute the norml mss shft s δe NMS = m e M A E expt. (17) 1. Feld Shft In ddton to the norml nd specfc mss shfts, we hve n ddtonl shft from the chnge n nucler sze s we shft from one sotope to the next. Ths shft s referred to s the feld shft nd s prmeterzed s j j δe = Fδ r, (18) where δ r s the chnge n the root-men-squre rdus f the nucleus. Assumng tht the nucleus cn be descrbed s unformly chrged bll of rdus 3
R, the nucler potentl s { [ (Z/R) 3 r /R ], r < R V (r, R) = Z/r, r R (19) The chnge n V (r, R) nduced by chnge δr n the rdus s δv = 3Z ] [1 R r R δr, r R. () Usng the the fct tht r =3R /5 for unform dstrbuton, one my rewrte the bove equton n the form δv = 5Z ] [1 4R 3 r R δ r, r R. (1) Wth ths result n mnd, we cn ntroduce the sngle-prtcle opertor h nuc (r) h nuc = 5Z ] [1 4R 3 r R, r R () nd fnd the feld-shft prmeter s MBPT Clcultons of SMS The opertor F = h nuc. (3) T = j p p j cn be expressed n second quntzton s the sum of one- nd two-prtcle opertors t jkl : j l k :+ t j : j :, j where jkl t jkl = j p 1 p kl (4) t j = p 1 p j p 1 p j. (5) The drect prt of the one-prtcle opertor t j vnshes by resons of symmetry. 4
.1 Angulr Decomposton The two-prtcle opertor t jkl my be decomposed n n ngulr-momentum bss s t jkl = ( 1) λ p λ k j p λ l, (6) λ whch, n turn, cn be expressed dgrmtclly s 1 t jkl = + j T 1 (jkl), (7) k where T 1 (jkl) = C 1 k j C 1 l P (c) P (bd). (8) In Eq. (8), the qunttes P (k) re rdl mtrx elements of the momentum opertor. We gve explct forms for these reduced mtrx elements n the followng subsecton.. Mtrx elements of momentum Let us dgress to gve explct forms for the mtrx elements of the momentum opertor. We frst consder the nonreltvstc cse...1 Nonreltvstc cse: We wrte p = 1 nd note tht n the nonreltvstc cse b = d 3 r 1 ( ) r P b(r) Yl 1 b m b r P (r) Y lm ( dp = dr P b (r) dr 1 ) r P (r) dω Yl b m b ˆrY lm + dr P b (r)p (r) dω Yl b m b Ylm (9) Note tht we cn rewrte the opertors on sphercl hrmoncs n terms of vector sphercl hrmoncs s l ˆrY lm (ˆr) = Y ( 1) lm (ˆr) (3) l(l +1) Y lm (ˆr) = Y (1) lm (ˆr) (31) r Usng the expnson of vector sphercl hrmoncs n terms of Y JLM (ˆr), we esly estblsh tht l b m b l m = { (l +1) l b m b ˆr l m for l b = l 1, l l b m b ˆr l m for l b = l +1. (3) 5
Wth the d of ths expresson, We fnd ( b dp = l b m b ˆr l m drp b (r) dr + l ) r P, l b = l 1, (33) nd b = l b m b ˆr l m We my therefore wrte, s n Eq. (8), ( dp drp b (r) dr l ) +1 P, l b = l +1. (34) r b p λ = l b m b C 1 λ l m P (b), where the rdl mtrx element P (b) s P (b) = 1 ( dp drp b (r) dr + l ) r P, l b = l 1, = 1 ( dp drp b (r) dr l ) +1 P, l b = l +1. (35) r.. Reltvstc cse I: It s smple to generlze the prevous nonreltvstc mtrx element to the reltvstc cse. We my wrte b p λ = κ b m b C 1 λ κ m P (b), where the reltvstc rdl mtrx elements P (b) s P (b) = 1 [ ( dg dr G b (r) dr + η ) ( r G df + F b (r) dr + ζ )] r F, (36) wth η = l or l 1, for l b = l 1orl b = l + 1, respectvely; nd ζ = l or l 1forl b = l 1orl b = l + 1, respectvely. Here l = l( κ). Ths s the proper form for the mtrx element of the momentum opertor...3 Reltvstc cse II: An lterntve form tht s equvlent to the bove n the nonreltvstc lmt s obtned by replcng p m e c α. Ths form leds to [ b p λ = m e c dr G b (r) F (r) κ b m b σ λ κ m ] F b (r) G (r) κ b m b σ λ κ m. (37) Wth the d of the dentty κ b m b σ λ κ m =(κ b + κ 1) κ b m b C 1 λ κ m, (38) 6
we my wrte the mtrx element n Eq. (37) s b p λ = m e c κ b m b C 1 λ κ m dr [ (κ b κ 1)G b (r) F (r) +(κ b κ +1)F b (r) G (r) ]. (39) From ths expresson, one obtns the lterntve expresson P (b) = m e c dr [ (κ b κ 1)G b (r) F (r)+(κ b κ +1)F b (r) G (r) ]. (4) for the rdl ntegrl of the momentum opertor. In the Pul pproxmton, we my replce F (r) 1 ( dg m e c dr + κ ) r G, ledng to P (b) = [ ( dg dr (κ b κ 1) G b (r) dr + κ ) r G ( dgb +(κ b κ +1)G (r) dr + κ )] b r G b = 1 ( dg dr G b (r) dr (κ ) b κ )(κ b + κ +1) G. (41) r By enumertng the sx possble combntons, one cn show tht { (κ l b κ )(κ b + κ +1) r l b = l 1 = r l, (4) +1 r l b = l +1 whch s just the nonreltvstc expresson for the rdl mtrx element. We wll use Eq. (4) n our clcultons becuse of ts reltve smplcty..3 Lowest-order clculton Consder n tom wth sngle vlence electron descrbed by n HF wve functon. The lowest-order mtrx element of T n stte v s gven by v T v (1) = t vv = t vv = v 1 + v T 1 (vv). We cn crry out the sum over mgnetc substtes usng stndrd grphcl rules to fnd: v T v (1) = 1 [v] v C1 P (v). (43) where we hve used the fct tht P (b) =P (b). 7
Tble 1: Lowest-order mtrx elements of the specfc-mss-shft opertor T for vlence sttes of L nd N. Lthum Z = 3 Sodum Z =3 Stte E HF v T v Stte E HF v T v s -.1963. 3s -.183 -.615 p 1/ -.1864 -.416 3p 1/ -.1949 -.31 p 3/ -.1864 -.416 3p 3/ -.194 -.3199 References [1] A. M. Mårtensson nd S. Slomonson, J. Phys. B15, 115 (198). [] E. Lndroth nd A. M. Mårtensson-Pendrll, Z. Phys. A39, 77 (1983). [3] S. Hörbäck, A. M. Pendrll, L. Pendrll, nd M. Pettersson, Z. Phys. A318, 85 (1984). [4] E. Lndroth, A. M. Mårtensson-Pendrll, nd S. Slomonson, Phys. Rev. A31, 58 (1985). [5] A. M. Mårtensson-Pendrll, L. Pendrll, S. Slomonson, A. Ynnermn, nd H. Wrston, J. Phys. B3, 1749 (199). [6] A. C. Hrtley nd A. M. Mårtensson-Pendrll, J. Phys. B4, 11193 (1991). [7] A. M. Mårtensson-Pendrll, A. Ynneremn, H. Wrston, L. Vermeeren, R. E. Slverns, A. Klen, R. Neugrt, C. Schulz, P. Levens, nd The ISOLDE Collborton, Phys. Rev. A45, 4675 (199). [8] A. M. Mårtensson-Pendrll, D. S. Gough, nd P. Hnnford, Phys. Rev. A49, 3351 (1994). [9] F. Kurth, T. Gudjons, R. Hlbert, T. Resnger, G. Werth, nd A. M. Mårtensson-Pendrll, Z. Phys. D34, 7 (1995). 8