QED calculations of three-photon transition probabilities in H-like ions with arbitrary nuclear charge

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by  T. Zalialiutdinov

QED calculations ofthree-photon transitionprobabilities inH-likeionswith arbitrary nuclear
charge
T. Zalialiutdinov1, D. Solovyev1 and L. Labzowsky1,2
1 Department of Physics, St.Petersburg State University,
Ulianovskaya 1, Petrodvorets, St.Petersburg 198504, Russia
2 Petersburg Nuclear Physics Institute, 188300, Gatchina, St. Petersburg, Russia
The quantum electrodynamical theory of the three-photon transitions in hydrogen-like ions is presented.
6 Emission probabilities of various three-photon decay channels for 2p3/2, 2p1/2 and 2s1/2 states are calcu-
1 latedforthenuclearchargeZ values1 6 Z 6 95. Theresultsaregivenintwodifferentgauges. Thefully
0 relativisticthree-photondecayratesofhydrogen-likeionswithhalf-integernuclearspinaregivenfortransitions
2 betweenfinestructurecomponents. TheresultscanbeappliedtothetestsoftheBose-Einsteinstatisticsforthe
multiphotonsystems.
n
a
J
I. INTRODUCTION
6
1
Duringlastyearsmultiphotonprocesseshaveattractedspecialattentionindifferentfieldsofphysicalscience. Multiphoton

] transitionsinatomicsystems,e.g.,two-photondecayandabsorptionhavebecomeausefulandevenastandardtoolforexper-
h
imentalstudiesof diversespectroscopiccharacteristicsin atoms: excitationof levelsof varioussystems, the determinationof
p
- physicalconstants, parity-violationphenomenaand etc. Since the primaryworksof Kramersand Heisenberg[1], Waller [2],
m
andGoeppert-Mayer[3],therehasbeenacontinuinginterestinaccuratecalculationsofmultiphotontransitionsinhydrogenlike
o ions. Two-andthree-photontransitionsarealsoofinterestinastrophysics[4],[5]. Moreover,multiphotontransitionshasbeen
t studied in connection with the test of Bose-Einstein statistics [6]-[10]. Recently the Spin-Statistics Selection Rules (SSSRs)
a
. whichpresentanextensionof theLandau-YangTheorem(LYT)to themultiphotonprocessesin atomswereestablished[12].
s
Examplesgivenin [12] concernedto the He-likeHighlyChargedIons(HCI) with photonfrequenciesinthe X-rayregion. In
c
i thepresentworkwesuggesttheexperimentsforthetestsofthe3-photonSSSRs onH-likeHCIwiththetransitionsinoptical
s
region.
y
h In present work the QED theory is applied to the study of spontaneousthree-photondecay processes. Calculations of the
p varioustransitionprobabilitiesforH-likeionsfornuclearchargeZ valueswithintheregion1 6 Z 6 95areperformed. Fur-
[ thermorefullyrelativisticcalculationsforthree-photontransitionsbetweenfinestructurecomponentswithaccountforhyperfine

structureof hydrogen-likeionsare presented. Thismeansthatwe have fixeda certainhyperfinesubstate of the fine structure
1
v component, not introducing explicitly the hyperfine level splitting. We performed calculations in two different gauges, what
8 allowsfortheaccuratecheckofthegaugeinvarianceoftheresults. ForthesummationoverthecompleteDirac spectrumthe
3 B-splinemethod[11]wasemployed.Relativisticunitsareusedthroughoutthispaper.
1 Ourpaperisorganizedasfollows. InsectionIIwepresentdetailedQEDderivationofthegeneralexpressionforspontaneous
4
three-photondecayrateinH-likeionforanarbitrarycombinationofelectricandmagneticmultipolesandinanarbitrarygauge
0
fortheelectromagneticpotentials. InsectionIIIweconsiderthree-photontransitionsbetweenfinestructurecomponentswith
.
1 accountforhyperfinestructure(asexplainedabove)ofanone-electronionwitharbitrarynuclearspinI. Numericalvaluesfor
0 thetransitionprobabilitiesconsideredinsectionsIIandIIIarepresentedinTablesI-III.InsectionIVwediscusstheapplication
6
of our results to the study of Bose-Einstein statistics and to the extension of LYT [12]-[14]. This section contains also the
1
concludingremarks.
:
v
i
X
II. QEDTHEORYOFTHREE-PHOTONTRANSITIONS
r
a
Wepresentacomputationallyconvenientfullyrelativisticformofageneralexpressionforthe3-photondecayrateinH-like
ionforanarbitrarycombinationofelectricandmagneticmultipolesandinanarbitrarygaugefortheelectromagneticpotentials.
Relativisticunits~=c=1areemployed.Inthissectionthehyperfinestructureofthelevelsisneglected.
TheS-matrixelementfortheemissionprocessi→f+3γ(iandf denotetheinitialandfinalstatesrespectively)reads[15],
[16],[17]
S(3) =(−ie)3 d4x d4x d4x ψ (x )γ A∗(k˜3˜e3)(x )S(x ,x )γ A∗(k˜2˜e2)(x )S(x ,x )γ A∗(k˜1˜e1)(x )ψ(x ), (1)
fi 3 2 1 f 3 µ3 µ3 3 3 2 µ2 µ2 2 2 1 µ1 µ1 1 i 1
Z
where
ψ (x)=ψ (~r)e−iEnt, (2)
n n
2
ψ (~r)isthesolutionoftheDiracequationfortheatomicelectron,E istheDiracenergy,ψ =ψ+γ istheDiracconjugated
n n n n 0
wavefunction,γ ≡ (γ ,~γ)aretheDiracmatricesandx ≡ (~r,it)arethespace-timecoordinates. InthispapertheEuclidean
µ 0
metric with an imaginary fourth vector componentis adopted. The photon wave function (electromagneticfield potential) is
describedby
A(~k,~e)(x)= 2πe eikµxµ =A(~k,~e)(~r)e−iωt, (3)
µ ω µ µ
r
wherek ≡ (~k,iω)isthephotonmomentum4-vector,~k isthephotonwavevector,ω = |~k|isthephotonfrequency,e arethe
µ
componentsofthephotonpolarization4-vector,~eisthe3-dimensionalpolarizationvectorforrealphotons,A(~k,~e) corresponds
µ
totheabsorbedphotonandA∗(~k,~e) representstheemittedphoton,respectively.
µ
Fortherealtransversephotons
2π 2π
A~(~k,~e)(x)= ~eei(k˜˜r−ωt) ≡ A~ e−iωt. (4)
ω ω ~e,~k
r r
The electron propagatorfor bound electronscan be presented in the form of the eigenmodedecompositionwith respect to
one-electroneigenstates[15],[16]
∞
1 ψ (~r )ψ (~r )
S(x ,x )= dωeiω(t1−t2) n 1 n 2 . (5)
1 2
2πi E (1−i0)+ω
−Z∞ Xn n
Here summation runs over entire Dirac spectrum for atomic electron. Insertion of the expressions (2)-(5) into Eq. (1) and
performingtheintegrationsovertimeandfrequencyvariablesyields
α˜A˜∗ α˜A˜∗ α˜A˜∗
S(3) =−2πie3δ(E −E −ω −ω −ω ) ˜e3,k˜3 fn′ ˜e2,k˜2 n′n ˜e1,k˜1 ni, (6)
fi i f 3 2 1 n′n (cid:16)(En′ −E(cid:17)f −(cid:16)ω3)(En−(cid:17)Ef −(cid:16) ω3−ω2(cid:17))
X
where α~ are the Dirac matrices, (...) denotesthe matrix elementwith Dirac wave functionψ , ψ . The amplitude of the
kn k n
emissionprocessU isrelatedtotheS-matrixelementvia
if
S =−2πiδ(E −E −ω −ω −ω )U . (7)
if i f 3 2 1 if
Thedifferentialprobability(transitionrate)oftheprocessisdefinedas
dWi3→γf =2πδ(E −E −ω −ω −ω ) U(3) 2 . (8)
dω dω dω i f 3 2 1 if
3 2 1
(cid:12) (cid:12)
(cid:12) (cid:12)
Thenthedifferentialtransitionrate in conjunctionwith theintegrationoverphoto(cid:12)ndir(cid:12)ections~ν = ~k/|~k| andsummationover
thephotonpolarizations~eofalltheemittedphotonswithallpermutationsofphotonsis
dWi→f(ω1,ω2) = ω3 ω2 ω1 d~ν d~ν d~ν × (9)
dω dω (2π)5 1 2 3
1 2 ~e3X,~e2,~e1Z
α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗
~e3,~k3 fn′ ~e2,~k2 n′n ~e1,~k1 ni + ~e3,~k3 fn′ ~e1,~k1 n′n ~e2,~k2 ni +
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:12)(cid:12)Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω2) Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω1)
(cid:12)
(cid:12)
(cid:12) α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗
(cid:12) ~e1,~k1 fn′ ~e3,~k3 n′n ~e2,~k2 ni + ~e1,~k1 fn′ ~e2,~k2 n′n ~e3,~k3 ni +
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
n′n (En′ −Ef −ω1)(En−Ef −ω1−ω3) n′n (En′ −Ef −ω1)(En−Ef −ω1−ω2)
X X
2
α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗ α~A~∗
~e2,~k2 fn′ ~e3,~k3 n′n ~e1,~k1 ni + ~e2,~k2 fm ~e1,~k1 n′n ~e3,~k3 ni ,
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12)
Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω3) Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω1) (cid:12)(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:12)
3
wherethefrequencyω isdefinesviaδfunctioninEq. (8). Thereforethetotaltransitionrateis
3
Wi→f = 1 1 dWi→f(ω1,ω2)dω1dω2 , (10)
3!2j +1 dω dω
ei 1 2
meXi,mef ZZ
wherej ,m ,j ,m aretheangularmomentaandtheirprojectionsfortheinitial(i)andfinal(f)electronstates.
ei ei ef ef
Expanding the plane waves into spherical waves in Eq. (9) we go over to the description of photons by the total angular
momentumj,itsprojectionmandparity(typeofthephoton).Thenwearriveat
dWi→f(ω1,ω2) = ω3 ω2 ω1 (11)
dω dω (2π)5
1 2
λ3Xλ2λ1jγ3Xjγ2jγ1mγ3mXγ2mγ1
Q(λ3) Q(λ2) Q(λ1) Q(λ3) Q(λ1) Q(λ2)
jγ3mγ3ω3 fn′ jγ2mγ2ω2 n′n jγ1mγ1ω1 ni jγ3mγ3ω3 fn′ jγ1mγ1ω1 n′n jγ2mγ2ω2 ni
+ +
(cid:12) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:12)(cid:12)Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω2) Xn′n (En′ −Ef −ω3)(En−Ef −ω3−ω1)
(cid:12)
(cid:12)
(cid:12) Q(λ1) Q(λ3) Q(λ2) Q(λ1) Q(λ2) Q(λ3)
(cid:12) jγ1mγ1ω1 fn′ jγ3mγ3ω3 n′n jγ2mγ2ω2 ni jγ1mγ1ω1 fn′ jγ2mγ2ω2 n′n jγ3mγ3ω3 ni
+ +
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
n′n (En′ −Ef −ω1)(En−Ef −ω1−ω3) n′n (En′ −Ef −ω1)(En−Ef −ω1−ω2)
X X
2
Q(λ2) Q(λ3) Q(λ1) Q(λ2) Q(λ1) Q(λ3)
jγ2mγ2ω2 fn′ jγ3mγ3ω3 n′n jγ1mγ1ω1 ni jγ2mγ2ω2 fn′ jγ1mγ1ω1 n′n jγ3mγ3ω3 ni
+ .
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:12)
Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω3) Xn′n (En′ −Ef −ω2)(En−Ef −ω2−ω1) (cid:12)(cid:12)
(cid:12)
(cid:12)
InEq. (11)weemploythereductionofthematrixelements(Q) totheradialintegralsdevelopedin[18],[19] (cid:12)
ab (cid:12)
Q(λ) =(−1)jeα−meα jeα jγ jeβ × (12)
(cid:16) jγmγω(cid:17)nαjeαleαmeα,nβjeβleβmeβ (cid:18)−meα mγ meβ(cid:19)
(−i)jγ+λ−1(−1)jeα−1/2 4π 1/2[(2j +1)(2j +1)]1/2 jeα jγ jeβ M(λ,jγ) (ω).
2j +1 eα eβ 1/2 0 −1/2 nαleαnβleβ
(cid:18) γ (cid:19) (cid:18) (cid:19)
Here j , m are the totalangularmomentumof the photonandits projection,λ characterizesthe type of the photon: λ = 1
γ γ
correspondsto electric and λ = 0 correspondsthe magneticphotons. The indicesn ,j ,l ,m presenta standard set of
α eα eα eα
one-electronDiracquantumnumbers.TheradialmatrixelementsM(λ,jγ) inEq. (12)areequalto
nαleαnβleβ
1/2 1/2
M(1,jγ) = jγ (κ −κ )I+ +(j +1)I− − jγ +1 (κ −κ )I+ −j I− (13)
nαlαnβlβ "(cid:18)jγ +1(cid:19) h α β jγ+1 γ jγ+1i (cid:18) jγ (cid:19) h α β jγ−1 γ jγ−1i#
−G (2j +1)J +(κ −κ ) I+ −I+ −j I− +(j +1)I− ,
γ jγ α β jγ+1 jγ−1 γ jγ−1 γ jγ+1
h (cid:16) (cid:17) i
M(0,jγ) = 2jγ +1 (κ +κ )I+, (14)
nαlαnβlβ [j (j +1)]1/2 α β jγ
γ γ
∞
±
I = (g f ±f g )j (ωr)dr, (15)
jγ α β α β jγ
Z
0
∞
J = (g g +f f )j (ωr)dr , (16)
jγ α β α β jγ
Z
0
whereg andf arethelargeandsmallcomponentsoftheradialDiracwavefunctionasdefinedin[18],κistheDiracangular
α α
number,ωisthephotonfrequency,j representsthesphericalBesselfunction,Gisthegaugeparameterfortheelectromagnetic
jγ
potentials. Inourcalculationsweemploythe”velocity”gauge(G=0)andthe”length” G= jγ+1 gaugeforthematrix
jγ
elementEq.(13)[20]. Notethatequation(4)correspondsto(G=0). (cid:16) q (cid:17)
4
The results can be further simplified by the summationsover projectionsof all the angular momenta. For this purposewe
definetheradialintegralpartforaparticularcombinationofmultipolesas
M(λi,jγi)(ω )M(λj,jγj)(ω )M(λk,jγk)(ω )
Sjn′jn(i,j,k)= f,n′ i n′,n j n,i k × (17)
lenX′,lennX,n′ En′jen′len′ −Enfjeflef −ωk−ωj Enjenlen −Enfjeflef −ωk
(cid:0) (cid:1)∆(cid:0)jen′,jen πflen′(i)πnle′n(j)πnlei(cid:1)(k),
where
1ifl +l+j +λ =odd
πl(t)= k γt t , (18)
k 0ifl +l+j +λ =even
(cid:26) k γt t
∆jen,jen′(i,j,k)= (4π)3/2[jef,jen′,jen,jei]1/2 jef jγi jen′ jen′ jγj jen jen jγk jei Θ(i,j,k) (19)
1/2 1/2 0 −1/2 1/2 0 −1/2 1/2 0 −1/2
jγi,jγj,jγk (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
and (cid:2) (cid:3)
Θ(i,j,k)=[jen′,jen]1/2menX′,men(−1)mei+mef+1(cid:18)−jmefef mjγγii mjeenn′′(cid:19)(cid:18)−jmene′n′ mjγγjj mjeenn(cid:19)(cid:18)−jmenen mjγγkk mjeeii(cid:19) .(20)
Theindicesi,j,kdenotetheserialnumberofthephotonwhichcantakethevalues1,2,3,thenotation[j,k,...]means(2j+
l)(2k+1)....
Finallyexpressionforthedecayratecanbewrittenintheform
2
dWi→f(ω1,ω2) = ω3 ω2 ω1 Sjen′jen(1,2,3)+(5permutations) . (21)
dω dω (2π)5 (cid:12) (cid:12)
1 2 λ1X,λ2,λ3jγ1,Xjγ2,jγ3mγ1,mXγ2,mγ3(cid:12)(cid:12)jenX,jen′ (cid:12)(cid:12)
(cid:12) (cid:12)
PermutationsinEq. (21)areunderstoodaspermutationsoftheindice(cid:12)s1,2,3. (cid:12)
(cid:12) (cid:12)
Thenumericalresultsforthetransitionrates2p → 1s +3γ(E1)and2s → 1s +3γ(E2)arepresentedforthe
1/2 1/2 1/2 1/2
H-likeionswith16Z 695inTablesI,IIrespectively.Summationoverthefullsetofone-electronstateswasperformedwithin
the B-spline approach[11]. The calculationswere carriedoutin two relativistic ”forms”, correspondingto the nonrelativistic
”length”and”velocity”forms[20];theresultscoincidewith3-6digits. Allvalueswascheckedforthestabilityandconvergence
for the differentlength of the spline basis set. In order to integrate over photon frequency, the 24 points of Gauss-Legendre
quadraturemethodwas employed. For the summationover Dirac spectrum set from40 B-spline basis states of order 9 were
used. ThefrequencydistributionsofsometransitionprobabilitiesarepresentedinFigs. 1-3.
III. THREE-PHOTONTRANSITIONSBETWEENFINESTRUCTURECOMPONENTS
Inthissectionwederivetheexpressionforthe3-photondecayrateinH-likeionforanucleiwithnonzerospinI andconsider
transitionsbetweendifferenthyperfinesublevelsofdifferentfinestructurelevels. Thematrixelementbetweenstateswithtotal
angularmomentumF andF (where|j −I|6F 6j +I)canbereducestotheform[21]:
α β e e
n F M Q(λ) n F M = (22)
α α α jγmγω β β β
D (cid:12) (cid:12) E
(−1)Fα−Mα+jeα+I [FαFβ]1/2 −FMαα mjγγ MFββ jFeαβ jIγ jFeβα (cid:12)(cid:12)nαjeα Q(cid:12)(cid:12)j(λγ)ω nβjeβ ,
(cid:18) (cid:19)(cid:26) (cid:27)D (cid:12)(cid:12) (cid:12)(cid:12) E
(cid:12)(cid:12) (cid:12)(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12)
1/2
n j Q(λ) n j =(−i)jγ+λ−1(−1)jeα−1/2 4π [j ,j ]1/2 jeα jγ jeβ M(λ,jγ)(ω). (23)
α eα jγω β eβ 2j +1 eα eβ 1/2 0 −1/2 αβ
D (cid:12)(cid:12) (cid:12)(cid:12) E (cid:18) γ (cid:19) (cid:18) (cid:19)
(cid:12)(cid:12) (cid:12)(cid:12)
Thenthedeca(cid:12)y(cid:12)rateis(cid:12)(cid:12)
2
dW ω ω ω
if = 3 2 1 SFn′Fnjen′jen(1,2,3)+(5permutations) , (24)
dω dω (2π)5 (cid:12) (cid:12)
2 1 λ1X,λ2,λ3jγ1,Xjγ2,jγ3mγ1,mXγ2,mγ3(cid:12)(cid:12)FenX′,FenjenX′,jen (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
5
where
SFn′Fnjen′jen(i,j,k)= Mf(λ,ni,′jγi)(ωi)Mn(λ′,jn,jγj)(ωj)Mn(λ,ik,jγk)(ωk) × (25)
lenX′,lennX,n′ En′jen′len′ −Enfjeflef −ωk−ωj Enjenlen −Enfjeflef −ωk
(cid:0) ∆Fn′,Fn,jen′,je(cid:1)n(cid:0)(i,j,k)πflen′(i)πnle′n(j)πnlei(cid:1)(k),
∆Fn′,Fn,jen′,jen(i,j,k)= (4π)3/2[jef,jen′,jen,jei]1/2 jef jγi jen′ jen′ jγj jen (26)
1/2 1/2 0 −1/2 1/2 0 −1/2
jγi,jγj,jγk (cid:18) (cid:19)(cid:18) (cid:19)
× 1je/n2 j0γk −j1e/i2 [jen′,jen]1/2[Fn′,Fn]1/2(cid:2) jFenf′ jIγi (cid:3)Fjenf′ Fjenn′ jIγj jFenn′ Fjeni jIγk jFeni Θ(i,j,k)
(cid:18) (cid:19) (cid:26) (cid:27)(cid:26) (cid:27)(cid:26) (cid:27)
and
Θ(i,j,k)=[Fn′,Fn]1/2MnX′,Mn(−1)Mf+Mn′+Mn(cid:18)−FMff mjγγii MFnn′′(cid:19)(cid:18)−FMn′n′ mjγγjj MFnn(cid:19) (27)
F j F
× n γk i .
−M m M
(cid:18) n γk i(cid:19)
In Eq. (24) we have neglectedthe energyshiftdueto the hyperfinesplitting, since it is small with respectto the total energy
difference. InTable III the numericalresultsfortransitionprobabilitiesof 2p (F = 0) → 2s (F = 2)+3γ(E1)decay
3/2 1/2
in H-like ions for the arbitrary nuclear charge Z and nuclear spin I = 3/2 are listed. In the two last columns of Table III
thehyperfinesplittingcoefficientsA(αZ) forthe levels2p and2s fordifferentZ valuesaregiven. Thesesplittingsare
3/2 1/2
evaluatedinafullyrelativistictheoryaccordingto[22]
F(F +1)−I(I +1)−j (j +1)
∆E =A(αZ) e e , (28)
µ
2Ij (j +1)(2l +1)
e e e
µ m (2l +1)κ[2κ(γ+n )−N]
A(αZ)=α(αZ)3 e e r . (29)
µ m N4γ(4γ2−1)
N p
Here I is the nuclear spin, κ = (−1)je+le+1/2(je +1/2) is a relativistic electron angular quantum number, nr is the radial
quantum number (n = n −|κ|), γ = κ2−(αZ)2, N = n2+2n γ+κ2 and µ is the magnetic moment of nucleus
r r r
expressed in units of nuclear magneton µ = |e|ℏ/(2m c), m and m is the electron and proton mass, respectively. The
N p e p
p p
valuesofI,µarealsogiveninTableIII.
These transitionscan be interestingin the view ofthe possible tests of Bose-Einsteinstatistics for the multiphotonsystems
[12].
IV. APPLICATIONFORTHETESTOFBOSE-EINSTEINSTATISTICS
In[12]theSSSRsforthemultiphotonatomictransitionswithequivalentphotonswhichpresentanextensionoftheLYT[13],
[14] wereformulated. These rulesconsistof: 1)SSSR-1: Two equivalentphotonsinvolvedin anyatomictransitioncanhave
onlyevenvaluesofthetotalangularmomentumJ,2)SSSR-2:Threeequivalentdipolephotonsinvolvedinanyatomictransition
canhaveonlyoddvaluesofthetotalangularmomentumJ = 1, 3,3)SSSR-3: Fourequivalentdipolephotonsinvolvedinany
atomictransitioncanhaveonlyevenvaluesofthetotalmomentumvaluesJ = 0, 2, 4. Itwasestablishedin[12]thatSSSR-2,
SSSR-3donothold,ingeneral,forthephotonmultipolarityj >1.
In[12]theexperimentsonthetestofSSSRswiththeabsorptionprocessesinHe-likeionofUraniumweresuggestedwhere
the lasers can be used as a source for the the equivalent photons. An advantage of the use of the laser source is that all the
photonswillhavethesamefrequency.IfwedividethisfrequencybyanintegernumberN andadjustthelaserfrequencyω to
γ l
thevalueoftransitionsfrequencyω ,ω =ω /N ,thenumberofphotonsN intheabsorptionprocesswillbefixed.
a l a γ γ
Using transitions with the initial total electron momentum J = 2 and the final momentum J = 0 we will fix the total
i f
momentum of the photon system J equal to J . Then, choosing N = 3 and J = 2 we will test SSSR-2. According to
i γ i
SSSR-2thevalueJ =2for3equalphotonsisforbidden,sothattheabsorptionofthelaserlightatthecorrespondingfrequency
ω = ω /3shouldbeabsent. InthesamewaySSSR-3canbetested. ThenumericalexampleswiththehighlychargedHe-like
l a
ionsweregivenin[12]. ThephotonfrequenciesinthiscaseareintheX-rayregion.
6
TheexperimentsofthistypecanbeextendedtotheH-likeionswiththehalf-integernuclearspinI. Theadvantageofsuch
experimentsconsistsinthepossibilitytousetheopticalrangelasers. Intherecentexperimentswithheavyatomsandions[23]
itispossibletomeasurethefrequencydistributionforthetransitionsrates. Inthiscasethevalueofthetotalangularmomentum
F forN -photonsystemcanbefixedbychoosingtheappropriatevaluesF andF fortheinitial(lower)andfinal(upper)levels
γ i f
inthetransitionprocess.
Inlaserbeamallthepossiblemultipolaritiesofphotonarepresented.Thusitshouldproduceallthetransitionswiththesame
total parity: E1E1E1, E1M1E2, E1E1M2 etc. However the processes with the photons of higher multipolarities are usually
stronglysuppressedinatoms. DuetothissuppressiontheE1E1E1transitionwillbedominant. Measuringtheabsorptionrate
attheω = ω /N frequencyonecanestablishthevalidityornon-validityoftheparticularSSSRs: theatomicvapourshould
l a γ
betransparentforthelaser lightatthefrequencyω = ω /N . Notealso thatunlikethespontaneousemissionwhichisvery
l a γ
weak for multiphoton transitions, the multiphoton absorption depends on the laser intensity and can be well observed in the
experiments.
Inourexamplesweconsideredtransitionsbetweenfinestructurecomponentswithfixationofacertainfinestructuresubcom-
ponents.VaryingthenuclearchargeZ wecanfindthesituationwheneachphotonwillbeinopticalrange(0.857eV-3.27eV).
For example, to test the SSSRs the H-like ions with Z = 16, 17, 19 can be examined. In this case transitionsbetween fine
structurecomponents2s (F = 2) → 2p (F = 0)+3γ(E1)canbechosen. Theenergyintervalsbetween2p (F = 0)
1/2 3/2 3/2
and2s (F = 2)statesforZ = 16, 17, 19arelistedinthe8thcolumnofTableIIIandcorrespondingω forN = 3isin
1/2 l γ
theopticalregion. Thenucleioftheseionsarestable[24]. ItisimportantthatforZ = 16, 17, 19thehyperfinesplittingboth
for2p and2s areresolvable(seeTableIII)i.e. thetransition2s (F = 2) → 2p (F = 0)canbewellseparatedout.
3/2 1/2 1/2 3/2
The3E1transitionratevalueis1013timessmallerthentheone-photonM2transitionrate. Howevertuningthelaserfrequency
to the one third of transition frequenciesexcludesone-photonabsorption. Thusthe frequencydistribution depicted in Fig. 1
(and its two-dimensionalsectional cutFig. 2) should be observablein experimentsof such type. The same picturearises for
1s (F = 0)→2p (F =0)+3γ(E1)transitioninneutralhydrogenatom(Z =1)withI =1/2andµ =2.793(seeFig.
1/2 3/2
3). Inthiscaseω = 3.40148eV.Forthistransitionthevirtualstatesns (F = 1)andnp (F = 1)inthesuminEq. (24)
l 1/2 1/2
forthe n = 2 lie betweenthe initial2p (F = 2)andfinal1s (F = 0)states whichleadsto the resonance(thesituation
3/2 1/2
when the energy denominatorturns to zero). The presence of the cascade-producingstates in the sum over the intermediate
statesinthetransitionamplitudeleadstothearrivalofthehigh,butnarrow”ridge”inthefrequencydistribution dW(ω1,ω2) [12].
dω1dω2
The ”ridge”does not influence the SSSR-2: it does not correspondto the case of three equivalentphotons. This is a general
situationforallthepossiblecascadetransitions. While in[6,7]itwasdemonstratedthattwophotonsbehaveliketwobosons
theexperimentssuggestedabovewoulddemonstratethatthreephotonsalsoobeytheBose-Einsteinstatistics.
Acknowledgments
TheworkwassupportedbyRFBR(grantsNo. 14-02-00188).T.Z.,D.S.andL.L.acknowledgethesupportbySt.-Petersburg
State University with a research grant 11.38.227.2014. The work of T. Z. was supported also by the nonprofit foundation
”Dynasty”(Moscow).
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7
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TABLEI:Transitionprobabilitiesfor2p →1s +3γ(E1)and2p →1s +γ(E1)decayratesins−1fordifferentZ.Thenumber
1/2 1/2 1/2 1/2
inparenthesesindicatesthepoweroften.TransitionenergiesineVarelistedinthelastcolumn.
Z W3γ W3γ W1γ ∆E
vel. len.
1 1.168620(−8) 1.168632(−8) 6.268(8) 10.204393
5 4.561173(−3) 4.561219(−3) 6.918(11) 2.551846(2)
10 1.164659 1.164670 6.272(12) 1.021675(3)
20 2.950782(2) 2.950754(2) 1.005(14) 4.101827(3)
30 7.430658(3) 7.430721(3) 5.106(14) 9.287031(3)
40 7.237156(4) 7.237208(4) 1.621(15) 1.665912(4)
50 4.170711(5) 4.170735(5) 3.980(15) 2.634226(4)
60 1.717285(6) 1.717293(6) 8.310(15) 3.851493(4)
70 5.579235(6) 5.579255(6) 1.552(16) 5.342837(4)
80 1.514879(7) 1.514883(7) 2.671(16) 7.144008(4)
90 3.554091(7) 3.554098(7) 4.316(16) 9.305821(4)
95 5.1821200(7) 51821200(7) 5.380(16) 1.054361(5)
TABLEII: Transition probabilitiesfor 2s → 1s +3γ(E2) and 2s → 1s +2γ(E1) in s−1 for different Z. The number in
1/2 1/2 1/2 1/2
parenthesesindicatesthepoweroften.TransitionenergiesineVarelistedinthelastcolumn.
Z W3γ W3γ W2γ ∆E
vel. len.
1 1.284270(−27) 1.284254(−27) 8.229063 10.204393
5 7.833773(−18) 7.833681(−18) 1.284705(5) 2.551846(3)
10 1.281043(−13) 1.284925(−13) 8.200646(6) 1.021674(3)
20 2.082775(−9) 2.087035(−9) 5.195127(8) 4.101829(3)
30 6.001701(−7) 6.008097(−7) 5.821090(9) 9.287046(3)
40 3.306380(−5) 3.307976(−5) 3.198616(10) 1.665919(4)
50 7.335633(−4) 7.337452(−4) 1.186615(11) 2.634252(4)
60 9.129899(−3) 9.131310(−3) 3.426453(11) 3.851581(4)
70 7.601921(−2) 7.602718(−2) 8.305995(11) 5.343122(4)
80 0.470047 0.470080 1.767273(12) 7.144853(4)
90 2.304994 2.305110 3.393473(12) 9.308396(4)
95 4.744875 4.745083 4.551134(12) 1.054819(5)
TABLEIII:Transitionprobabilitiesfor2p (F =0)→2s (F =2)+3γ(E1)and2p (F =0)→2s (F =2)+γ(M2)ins−1for
3/2 1/2 3/2 1/2
differentH-likeions.NuclearspinI =3/2.Thenumberinparenthesesindicatesthepoweroften.Transitionenergiesandhyperfinesplitting
constantsA(Eq.(29))ineVarelistedinthreelastcolumns.
Ion Abundance% µ I Wv3eγl. Wl3eγn. W1γ ∆E A2p3/2 A2s1/2
33S15+ 0.75 0.64382 3 8.944609×10−20 8.944608×10−20 6.406678×10−6 2.992524 2.61179×10−4 2.67850×10−4
2
35Cl16+ 93.3 0.82187 3 3.404860×10−19 3.404860×10−19 1.917035×10−5 3.817932 4.00118×10−4 4.11687×10−4
2
39K18+ 75.8 0.39147 3 3.963697×10−18 3.963697×10−18 1.434485×10−4 5.971640 5.59230×10−4 5.79554×10−4
2
8
FIG.1: (ColorOnline)3-dimensionalplotforfrequenciesdistributionofthetransitionrate2p (F = 0) → 2s (F = 2)+3γ(E1)in
3/2 1/2
H-likeionwithZ = 19. NuclearspinI = 3/2. Ontheverticalaxisthetransitionrate dW ins−1 isplotted;onthehorizontalaxesthe
photonfrequenciesareplottedinunitsω1/∆,ω2/∆where∆denotestheenergydifferednωce1d∆ω2= E(2p3/2)−E(2s1/2). Thelowest(zero)
pointisthepointwithcoordinatesω1/∆=ω2/∆=1/3atthebottomofthe”pit”inthefrequencydistributionforthetransitionratewhich
arisesduetoSSSR-2.
FIG.2:(ColorOnline)Two-dimensionalsectionalcutofFig.1atthefrequencypointω2/∆=1/3.AlldetailsarethesameasinFig.1.The
lowest(zero)pointisthepointwithcoordinatesω1/∆=1/3arisesduetoSSSR-2.
FIG.3: (Color Online) 3-dimensional plot for frequencies distributionof the transitionrate 2p (F = 2) → 1s (F = 0)+3γ(E1)
3/2 1/2
in the neutral hydrogen atom (Z = 1). The energy difference between 2p (F = 2) and 1s (F = 0) states is 10.204435 eV and
3/2 1/2
ω =3.40148eV.AlldetailsarethesameasinFig.1.Here∆=E(2p )−E(1s ).Thelowest(zero)pointisthepointwithcoordinates
l 3/2 1/2
ω1/∆=ω2/∆=1/3atthebottomofthe”pit”inthefrequencydistributionforthetransitionratewhicharisesduetoSSSR-2.