Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms

Transcription

1 NSTTUTE OF PHYSCS PUBLSHNG JOURNAL OF PHYSCS B: ATOMC, MOLECULAR AND OPTCAL PHYSCS J. Phys. B: At. Mol. Opt. Phys. 39 ) 7 85 doi:.88/953-75/39/8/ Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess of hydrogen-like atoms Chang-qi Cao, Xiao-wei Fu and Hui Cao 3 Department of Physis, Peking University, Beijing 87, People s Republi of China Department of Physis, Tsinghua University, Beijing 8, People s Republi of China 3 Department of Physis and Astronomy, Northwestern University, Evanston, L 8-3, USA qao@pku.edu.n and h-ao@northwestern.edu Reeived Otober 5, in final form Marh Published April Online at staks.iop.org/jphysb/39/7 Abstrat A relativisti and non-markovian study is arried out on the magneti-dipole emission of a hydrogen-like atom of large atomi number Z without perturbation approximation. The orrelation spetra are derived analytially and the orresponding orrelation funtions are alulated numerially. The Markovian approximation is seen to be satisfied with more auray as ompared with the ase of eletri-dipole emission while the relativisti orretions are quite larger. n transition S / to S /, the relativisti values even beome several times of the non-relativisti values. As a omparison with the magneti-dipole emission, the eletri-quadrupole emission in the ase of the D 3/ to S / transition is also studied.. ntrodution n a previous paper [], we investigated the possible relativisti and non-markovian orretions to the eletri-dipole emission. Now we make a similar onsideration of the magneti-dipole spontaneous emissions. For quite a long time the interest in magneti-dipole emission has been limited in astrophysis [], where the ollision probability between the atoms is so small that the magneti-dipole emission may have the hane to take plae. Subsequently it also beame of interest for solar physis, in onnetion with the solar orona observation. Only sine 97 has the development of experimental tehniques made it possible to be studied in laboratories both for hydrogen-like and helium-like atoms, leading to a relatively high interest in their theoretial investigation. n a low Zatomi number) region, the two-photon eletri-dipole emission will overwhelm the magneti-dipole emission, but for large Z the situation beomes different; magneti-dipole emission may turn out to be the dominant mode //87+5$3. OP Publishing Ltd Printed in the UK 7

2 7 C-Q Cao et al Usually the magneti-dipole transitions are divided into two types. n the ase where the non-relativisti atomi radial wavefuntions of the initial and final states are the same, suh as in the proess from P 3/ to P /, the orresponding transition is alled unhindered []. n ontrast, if the non-relativisti initial and final atomi radial funtions are orthogonal to eah other, suh as in the proess from S / to S /, the orresponding transition is alled hindered, sine its transition rate will be zero in the non-relativisti theory provided the atom-size effet is negleted, namely the fator e ik x is taken as within the range of atoms. The earlier theories just alulate the emission rate, and mainly give the results in the lowest order of perturbation; some even just give an estimate of the order of magnitude [ ]. Lin and Feinberg [7] alulated the radiative orretion to the lowest order in the proess S / S / + γ deay in hydrogen-like atoms. Barut and Salamin [8], on the other hand, derived the rates of spontaneous emission based on the alulation of eletron self-energy formulation; their result for the S / S / + γ deay rate is the same as the known lowest perturbation value. n [] we used the multi-pole photon field formulation to study the whole proess of eletri-dipole emission without perturbation approximation. Now we still use the same formulation to study the magneti-dipole emission proess, whih says the magneti emission orresponds to radiating a photon with a total angular momentum J = and with a parity P = +. For omparison, the eletri quadrupole emission is also studied whih orresponds to radiating a photon with a total angular momentum J = and with a parity P = +. The general multipole photon field formulation has been desribed in [], in whih the multipole vetor potential A kjmp is expressed by the spherial Bessel funtion g L kr) and vetor spherial funtion Y JLM θ, ϕ). f only the two atomi levels onerned need to be onsidered, the interation Hamiltonian in rotating-wave approximation is given by Ĥ int = i h kjmp [ gkjmp ˆσ + â kjmp e iω ωr) )t gkjmp ˆσ â kjmp eiω ωr) )t ], ) where ˆσ + and ˆσ are the atom level upward and downward hange operators, ω R) is the relativisti value of h E E ), ω = k, g kjmp is the orresponding oupling onstant given by g kjmp = ē d 3 xψ h x)γ ψ x) A kjmp x), ) in whih ψ x) and ψ x) are the upper level and lower level atomi wavefuntions, respetively; they are now Dira spinors. The state of our system is expressed by t =C t) ψ ; + C,kJ MP t) ψ ; kjmp 3) kjmp with ψ ; denoting the state in whih the atom is in its upper level and no photon exists, ψ ; kjmp denoting the state in whih the atom is in its lower level with one photon in the mode kj MP ). After eliminating C,kJ MP t) from the oupled equation as we did in [], one easily gets the integro-differential equation for C t) as d t dt C t) = Ut t )C t ) dt. ) The funtion Ut t ), the so-alled orrelation funtion, is given by Ut t ) = Rω) e iω ωr) )t t ) dω 5)

3 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 73 with R Rω) = π g kjmp, ω = k, ) JMP in whih R is the radius of the normalized sphere of the photon field. The funtion Rω) is alled the orrelation spetrum, and an be alulated by the relevant Dira wavefuntions and orresponding multipole vetor potential of the photon. Having got Ut t ), equation ) may be solved numerially by a simple reurrene formula [9], n C t n+ ) = C t n ) Ut m )C t n m ) t). 7) m=. The magneti-dipole emission in atomi transition S / to S / and 3D 3/ to S / First, let us onsider the atomi transition S / to S /, whih is also onsidered in [7, 8]. The only possible one-photon emission is magneti-dipole emission, as an be seen from the angular momentum onservation and parity onservation. By these onservation laws, the quantum number set J, P ) of the emitted photon should be, +). However, in the nonrelativisti theory, the orresponding magneti transition moment m is zero; even the spin ontribution is taken into aount. The reason is that m = e h ϕ + m x)ˆl + σ)ϕ x) d 3 x 8) while both integrals ϕ + x)ˆlϕ x) d 3 x and ϕ + x)σ ϕ x) d 3 x are zero. t is evident that the first integral is equal to zero, beause the orbital angular momenta of both initial and final states are zero. The seond integral is also equal to zero by the reason that the radial wavefuntions of S and S are orthogonal. Thus, only when the finite size orretion of higher order of ka) and relativisti orretion are taken into aount, the one-photon emission for this transition beomes possible. t is evident that under rotating-wave approximation this spontaneous emission is just a two-level proess, and no other atomi level an be involved in this proess as an intermediate state; hene the formulation of the last setion an be applied. We take, by will, the m of the initial state S / as /; so its wavefuntion is given by ψ x) = r G r) r F r) θ, ϕ) = B θ, ϕ) +B ρ)ρ s e ρ ) A + A ρ)ρs e ρ r, 9) in whih s = Z α, the same as that in ψ desribed in [], and A Zα = +s, A = s +s), B Zα+s) = Zα + s), Zα+s) ρ = Z s) r, E = m Zα +s), ) ) Z s) B s+ [ +A ) = + B Zα Ɣ+s) + A A )+s) + ] B + A ) +s)+s).

4 7 C-Q Cao et al The final atomi wavefuntion ψ x) is the same as in []. ω o R) is also the same as that of the P / to S / transition, namely ω R) = m h +s) s). The third omponents of angular momenta of photon and final-state atom M, m) now have two possible ombinations:, ) and, ). The orresponding two oupling onstants g kjmp are given by πω g k+) = e ψ hr x)γ ψ x) g kr)y θ, ϕ) d 3 x, a) πω g k+) = e ψ hr x)γ ψ x) g kr)y θ, ϕ) d 3 x. b) As desribed above, the quantum number m of ψ is taken as / in equation a), and taken as /) in equation b). After arrying out the integration over spatial oordinates, we get the orresponding spetrum of orrelation funtion as Rω) = R π g k+) + g k+) ) = B B α 5+ π 3 ω S ω) + S ω) a) in whih B = Z s +s Ɣ+s), b) B is given by equation ) and S ω) = Za k [ A + S ω) = k with + ) s Za Za Zα Za +s Za [ B a = ) s { A ] ) A ) s Za /Z ) ] + s)/ Zα Ɣs ) ) s Ɣs) ) s A ) s { Ɣs) ) s B Za ) } Ɣs +) +.., ) s+ 3a) Ɣs ) ) s Za ) Ɣs +) +.. ) s+ }, 3b), ) and denotes Bohr s radius. We note in passing that S ω) and S ω) ome from the integrals and respetively. g kr)g r)f r) dr g kr)f r)g r) dr,

5 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 75 f we just keep the leading order term in Zα, we will get the orresponding non-relativisti orrelation spetrum: Rω) = )Z α 3 ω ω ) a 3 7 ) π + ω a, 5) in whih a is redued to a = 3Z. ) We see that in equation 5) the leading term of the last fator is proportional to ) ωa, ontrasting with the eletri-dipole result: ) ωa f equation a) of[]). The reason is that the magneti-dipole moment has an extra fator v ω a as ompared with the eletridipole moment, and Rω ) is proportional to the square of the relevant moment hene in the unhindered ase the leading term will be ω ) a ). Now in the transition S/ to S / the magneti-dipole moment is zero, whih is refleted in the above relativisti alulation that the leading terms from S ω) and S ω) anel eah other. The next terms in non-relativisti S ω) and S ω) are proportional to ) ωa 3. Sine Rω)/ω S ω) + S ω), an additional ) will appear in the leading term of Rω) as ompared with eletri-dipole results. fator ωa n sum, the Rω) of equation 5) has taken into aount the finite size effet of magnetidipole, and in equation ) further orretion of relativisti effet is inluded. Next we onsider the single photon transition from 3D 3/ to S /. n this ase both photons with J =,P = +) and J =,P = ) may be emitted. The former orresponds to magneti-dipole emission similarly as in the above disussion; the latter orresponds to eletri-quadrupole emission whih will be disussed in the next setion for omparison. The magneti-dipole transition is also a hindered one with zero non-relativisti magneti moment, as an be seen from the fator δ l l in the following formula: j l m ˆL + σ) j l m d = δ l l ) m m n m m ) / s= / C j,m l,m s;,s l l +)C j,m,m s l,m s;,scl l,m s;,m m + 3C j,m C,m m +s ) l,m s;,m m +s. 7),s;,m m This situation will make the eletri-quadrupole emission dominate in the 3D 3/ to S / transition, as shown in setion 3. n the relativisti theory, we take, by will, m = 3/ for the initial state 3D 3/ ;sothe initial wavefuntion is with ψ x) = r G r) 3 3 r F r) 3 3 θ, ϕ) = B θ, ϕ) +B ρ)ρ s e ρ ) 3 3 A +, 8a) A ρ)ρs e ρ 3 3 s = Z α, Z ρ = r, 5+s 8b) A = Zα, s 8)

6 7 C-Q Cao et al A = Zα9+s 5+s ) +s ) s 3s +3s ) ), 8d) 5+s B = 5+s +s 3, E = m +s 8e) 5+s 5+s and ) Z B +s = 5+s Ɣ+s ) [ ) +A + B + A A )+s ) ]. ) + 8f ) B + A +s )+s ) The emission frequeny of this transition is ω R) = m h +s 5+s s ), 9) where s = s. n the magneti-dipole emission the only allowed value of the third omponents of angular momenta for photon and for final atomi state, namely M, m), is just, /); hene only one oupling onstant need be alulated: πω g k+) = e ψ hr x)γ ψ x) g kr)y θ, ϕ) d 3 x ) in whih the value m for ψ takes /. After arrying out the integration over spatial oordinates, we get the relevant orrelation spetrum in the two-level approximation as Rω) = R π g k+) = B B α ω π S ω) + S ω). ) B is still given by equation b). As before, S ω) and S ω) ome from the radial integrals and respetively, and are given by S ω) = [ + k S ω) = k g kr)g r)f r) dr g kr)f r)g r) dr 5+s ] s [ + 5+s ] s { A Ɣs + s ) A Ɣs + s ) ) s +s ) s +s Zα +s [ + B 5+s ] s [ + 5+s ] s { Ɣs + s ) ) s +s + + A + 5+s }, A + Ɣs + s +) 5+s ) +.. s +s + Ɣs + s ) ) s +s + a) + Ɣs + s ) Ɣs + s ) 5+s ) s +s ) s +s B } + Ɣs + s +) 5+s ) +.., b) s +s +

7 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 77 R Z ab Figure. The relativisti orrelation spetra of magneti-dipole emission for atomi transition S / to S /. : Z = 9; : Z = 5; dashed lines the orresponding non-relativisti results. R Z ab Figure. The relativisti orrelation spetra of magneti-dipole emission for atomi transition 3D 3/ to S /.:Z = 9; : Z = 5; dashed lines the orresponding non-relativisti results. in whih ase a = /Z. 3) + 5+s Similarly, we may just keep the term of leading order in Zα to get the orresponding non-relativisti spetrum, the result is Rω) = 3 3 Z α 3 ω ω ) a 3 [ ] π + ω a 8. ) n this ase, the leading term in S ω) also anels with that in S ω), hene this non-relativisti Rω) has the same form of leading term as that in equation 5), but the numerial oeffiient in equation )ismuhsmaller. Now the value of a is redued to its non-relativisti value a = 3 Z. 5) We plot in figure and figure the orrelation spetrum of magneti-dipole emission in atomi transition S / to S / and 3D 3/ to S / respetively.

8 78 C-Q Cao et al R Z ab R Z ab Figure 3. The small peak of relativisti orrelation spetra of magneti-dipole emission in a low frequeny region for the transition S / to S /. : Z = 9; : Z = 5; dashed lines are the orresponding non-relativisti Rω). The absissa is taken as ω/ω, where the ω s denote the orresponding non-relativisti value. For S / to S /, ω is given by ω = 3 8 Z α, a) and for 3D 3/ to S / ω = 9 Z α. b) t is seen that, in the transition S / to S /,Rω)spread over a range somewhat larger than the range of those of eletri-dipole emission. n the transition 3D 3/ to S /, the range is smaller than that of S / to S / for the same Z. We note that beyond the main peak of relativisti Rω), there is a very small peak in the low frequeny region whih is invisible in figures and. We plot it separately in figures 3 and. These small peaks are very important, sine the key values ω = ω R) are loated in these small peaks. The numerial values of ω R) alulated up to three signifiant digits are given by { ω R).3ω, for Z = 9, =.3ω, for Z = 5 for transition S / to S /, and ω R) = {.ω, for Z = 9,.ω, for Z = 5 7a) 7b) for transition 3D 3/ to S /. These small peaks do not appear in the non-relativisti spetrum see the doted line of figure 3 and ), hene their appearane is a harateristi of relativisti theory. We also define the orrespondent deay oeffiient γ for magneti-dipole emission by γ = πr ω R) ). 8)

9 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 79 3 R Z ab 3 8 R Z ab Figure. The small peak of relativisti orrelation spetra of magneti-dipole emission in a low frequeny region for the transition 3D 3/ to S /. : Z = 9; : Z = 5; dashed lines are the orresponding non-relativisti Rω). For transition S / to S /, alulated up to three signifiant digits { 5. 3 γ A, for Z = 9, γ = a) γ A, for Z = 5. γ A is the Einstein A oeffiient, haraterizing the order of magnitude of the deay rate of eletri-dipole emission. Equation 9a) may be ompared with the non-relativisti values alulated by equation 5)forω = ω : {.9 γ A, for Z = 9, γ =.9 5 9b) γ A, for Z = 5. We see that the relativisti values are about 8. times and.9 times of non-relativisti values. The differenes are quite large. For transition 3D 3/ to S /, the relativisti results alulated by equation 8)are { 7.5 γ A, for Z = 9, γ = a) γ A, for Z = 5, while the orresponding non-relativisti values alulated by use of equation ) with ω = ω )aregivenby {.8 5 γ A, for Z = 9, γ =. 7 3b) γ A, for Z = 5. We see that, in this ase, the relativisti values are quite smaller than non-relativisti values, ontrary to the ase S / to S /, in whih the relativisti values are muh larger. So one annot generally say that magneti-dipole emission is mainly aused by relativisti effet. Having the orrelation spetrum, the orrespondent orrelation funtion an be alulated by the numerial method as desribed by equation 7). The absolute value of Uτ)/U) are shown in figure 5 for both transitions with Z = 5 and Z = 9. The widths τ W at half the height of Uτ) /U) for Z = 9 and 5 in the transition of S / to S / are given by τ W =.87/ω and τ W =.9/ω, respetively, while in the

10 8 C-Q Cao et al.8 a).8 b) U U.. U U.... U U ).8 d) U U Figure 5. The relativisti orrelation funtion Uτ) of magneti dipole emission: the absolute value. a) For transition S / to S /,Z = 5,U) =.98 3 ω for the relativisti ase and U) =.7 3 ω for the non-relativisti ase. b) For transition S / to S /,Z = 9,U) = ω for the relativisti ase and U) =.7 3 ω for the non-relativisti ase. ) For transition 3D 3/ to S /,Z = 5,U) =. 7 ω for the relativisti ase and U) = 7. 7 ω for the non-relativisti ase. d) For transition 3D 3/ to S /,Z = 9,U) = ω for the relativisti ase and U) = 7. 7 ω for the non-relativisti ase. transition 3D 3/ to S /, the relevant values are τ W =.7/ω and τ W =.38/ω.Wesee the values of τ W for S / to S / transition are muh smaller than /ω. Figure presents the orrespondent arguments of Uτ) versus τ. They show a similar behaviour as those of eletri-dipole emission, but they drop muh deeper in the peaked region of Uτ), whih will redue the magnitude of γ onsiderably. We have noted that the small peak of Rω) shown in figures 3 and is important for the deay behaviour, sine the ruial point ω = ω R) is loated in this small peak. As to the large peak at higher frequeny, it may greatly affet the shape of orrelation funtion. f we just retain the small peak by removing the large peak, the urve of Uτ) will extend to a quite larger region than that in figure 5, as shown in figure 7. These results are aording to expetation, sine the small peak has a muh narrower ω, so its Fourier transformation will have a muh wider extension τ. Now turn to the deay behaviour of the upper level population N. We have seen that the orrelation interval τ in the above disussed magneti-dipole emissions are omparable with or smaller than those in the eletri-dipole emissions, and /γ ), whih haraterize the deay times, are muh longer than /γ A see equations 9) and 3)); hene the Markovian approximation of C t) must be valid with more auray, resulting N t) = e γt 3) with γ given by equation 8). This result shows that the deay oeffiient γ atually represents the deay rate.

11 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 8 a) b) U ) d) U Figure. The relativisti orrelation funtion Uτ) magneti-dipole emission: θ U the argument of U). a) For transition S / to S /,Z = 5. b) For transition S / to S /,Z = 9. ) For transition 3D 3/ to S /,Z = 5. d) For transition 3D 3/ to S /,Z = Figure 7. The absolute value of relativisti orrelation funtion Uτ)of magneti-dipole emission if only the small peak in the low frequeny region is retained. : for transition 3D 3/ to S /,Z = 9,U) =. 9 ω ; : for transition S / to S /,Z = 9,U) =.39 ω. We note in passing that despite the fat that orrelation funtion hanges a lot when the large peak of spetrum is removed and hene the orrelation interval τ is inreased by several times, the deay of N alulated by this modified orrelation funtion is still almost the same as the original. The reason is that in the present ase γ is extremely small, leading to the ratio of modified τ to /γ still muh smaller than those for eletri-dipole emission disussed in []. 3. The eletri-quadrupole emission in atomi transition 3D 3/ to S / As a omparison we finally onsider the main emission way in the transition of a hydrogen-like atom from 3D 3/ to S / the eletri-quadrupole emission.

12 8 C-Q Cao et al The allowed values of M, m), the third omponents of the total angular momentum of the photon and final state atom, are ), and, ). The orresponding oupling onstants are given by [ ] πω 3 g km+) = e ψ 5 hr x)γ ψ x) g kr)y M θ, ϕ) g 3 kr)y 3M θ, ϕ) d 3 x, 3) in whih M takes the values and, and the orresponding m for ψ x) should take / and / respetively. After arrying out the integration over spae oordinates, we finally get the orrelation spetrum for eletri-quadrupole emission as Rω) = R π g k+) + g k+) ) = B B 3α ω 5 π 5S ω) + S ω) S 3 ω), 33) in whih S ω) and S ω) are the same as those given by equations ), while S 3 ω) omes from the integral g 3 kr)f r)g r) dr, with the expression [ ] Zα s [ { ] s 5i s 5+s S 3 ω) = k +s + 5 k a + i ka Ɣs + s ) ) s +s B + 5+s + i ka ) Ɣs + s ) ) 3 s +s ka B Ɣs + s 3) k 3 a 3 ) s +s 3 i 5 B ) ka + 5+s ) Ɣs + s ) + 5+s ) s +s B } + + Ɣs + s +) 5+s ) +.., 3) s +s + in whih a is still given by equation 3). f we just keep the term of leading order in Zα, the result will transit to the non-relativisti ase. By this way we get the non-relativisti orrelation spetrum for eletri-quadrupole emission in the atomi 3D 3/ to S / transition as ) +3 ωa ) ) ωa Rω) = 9 ) Z α 3 ω [ 5 π + ωa ) ] 8 35) where a is redued to that given by equation 5). We see that the leading term of non-relativisti Rω)/α 3 ω is proportional to ) ωa ompared with ) ωa in the eletri-dipole emission. This differene reflets the fat that quadrupole moment has an additional spae dimension fator a as ompared with dipole moment. On the other hand, this leading term is muh larger than the leading term ) ωa in the non-relativisti magneti-dipole emission of the same atomi transition, whih will lead to the dominane of eletri-quadrupole emission over the orresponding magneti-dipole emission. The orrelation spetra of eletri-quadrupole emission are plotted in figure 8, the absissa being still taken as ω/ω with ω given by equation ). The range of spread of Rω) is somewhat smaller than that of eletri-dipole emission. The relevant orrelation funtion is shown in figure 9. The widths τ W of Uτ) /U) are.8/ω for Z = 5, and.3/ω for Z = 9, both of order /ω. The urves of

13 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess Figure 8. The relativisti orrelation spetra Rω) of eletri-quadrupole emission for transition 3D 3/ to S /.:Z = 9; : Z = 5; dashed lines are the orresponding non-relativisti Rω) Figure 9. The relativisti orrelation funtion Uτ) of eletri-quadrupole emission for transition 3D 3/ to S / : the absolute value. : Z = 9,U) =.8 5 ω ; : Z = 5,U) =.8 5 ω Figure. The relativisti orrelation funtion Uτ)of eletri-quadrupole emission for transition 3D 3/ to S / : θ U the argument of U). : Z = 9; : Z = 5. θ U τ), the argument of Uτ) versus τ are plotted in figure. We see θ U τ) first drop down, but not so deep as those in figure, then rise up linearly when ω τ is larger than.5 and..

14 8 C-Q Cao et al n the present investigation, the values of γ are given by γ = πr { ω R) ).7 3 γ A, for Z = 5 =.7 3 3) γ A, for Z = 9 alulated up to three signifiant digits, where ω R) is given by equation 7b). We see that these γ are muh larger than those of orresponding magneti-dipole emission. As a omparison, we list the non-relativisti γ alulated by equation 35): {.5 3 γ A, for Z = 5, γ = ) γ A, for Z = 9. The differene between equations 3) and 37) gives the relativisti orretion of deay oeffiients. Their values are about 9% for Z = 5 and % for Z = 9 with positive sign. n the present ase the ratio γ/ω, whih roughly measures the ratio of orrelation internal τ to the deay time /γ, is muh smaller than the orresponding value for eletridipole emission, indiating the Markovian approximation should be a good approximation. A numerial alulation onfirms this assertion. Hene equation 3) holds with γ given by equation 3).. Brief summary The multi-pole em field formulation is used to study the relativisti orretion and non- Markovian orretion of magneti-dipole and eletri-quadrupole emissions as we did in [] for eletri-dipole emission. n this formulation, the magneti-dipole emission orresponds to the emission of a photon with J = and P = +, and the eletri-quadrupole emission orresponds to the emission of a photon with J = and P = +. The eletron spin urrent ontribution is now fully inluded, even in the eletri-quadrupole emission ase. The relativisti orrelation spetra Rω) are first analytially derived, then the orresponding orrelation funtions Uτ) are numerially alulated. The absolute value of Uτ)has a width smaller than / ω R), espeially in the transition of S / to S /. Outside the peaked region Uτ) also osillates with the frequeny ω R). The general feature is similar to the ase of eletri-dipole emission. The Markovian approximation is well satisfied over the main period of deay up to an atom of Z = 9, with the deay rate γ defined by the relativisti orrelation spetrum Rω) as γ = πr ω R) ). But the relativisti orretions to the deay rates are quite large. For S / S / magneti-dipole emission and 3D / S / eletri-quadrupole emission, the relativisti orretions are positive, while for 3D 3/ S / magneti-dipole emission the relativisti orretion on the ontrary, is negative. n our studied ases S / to S / and 3D 3/ to S /, the magneti-dipole emission rates are very small. This situation reflets the fat that in the non-relativisti theory the orrespondent magneti-dipole moments are zero, namely the magneti-dipole emission is hindered. A new feature in the relativisti orrelation spetra of the investigated magnetidipole emission is the appearane of a small peak around ω R), whih makes the relativisti deay rate muh different from the lassial value. The eletri-quadrupole emission is dominant in the atomi transition 3D 3/ to S /, as an be understood by the nonzero of non-relativisti eletri-quadrupole moment. As to the relativisti orretion, in this emission it is muh larger than that for eletri-dipole emission.

15 Non-Markovian study of the relativisti magneti-dipole spontaneous emission proess 85 Lastly, the ondition for the validity of Markovian approximation holds muh better in these two kinds of emissions than in eletri-dipole emission, sine the orresponding γ are muh smaller than that of eletri-dipole emission while the orrelation time τ s are still of order / ω R) or smaller). Aknowledgments This work is supported by the National Siene Foundation of USADMR-9399). Referenes [] Cao C-Q, Fu X-W and Cao H 5 J. Opt. B: Quantum Semilass. Opt. 7 3 [] Suher J 978 Rep. Prog. Phys. 78 Marrus R and Mohr P J 978 Adv. At. Mol. Phys. 8 [3] Breit G and Teller E 9 Astrophys. J. 9 5 [] Shapiro J and Breit G 959 Phys. Rev [5] Marrus R and Shmieder P W 97 Phys. Rev. A 5 [] Johnson W 97 Phys. Rev. Lett. 9 3 [7] Lin Dong L and Feinberg G 97 Phys. Rev. A 5 [8] Barut A O and Salamin Y 988 Phys. Rev. A 37 8 [9] Cao C-Q, Tian J-R and Cao H Phys. Lett. A 33 38