Relativistic corrections to the thermal interaction of bound particles
RRelativistic corrections to the thermal interaction of bound particles
D. Solovyev, ∗ T. Zalialiutdinov, and A. Anikin
Department of Physics, St.Petersburg State University, St.Petersburg, 198504, Russia
This paper discusses relativistic corrections to the thermal Coulomb potential for simple atomic systems. Thetheoretical description of the revealed thermal corrections is carried out within the framework of relativisticquantum electrodymamics (QED). As a result, thermal corrections to the fine and hyperfine strucutres of atomiclevels are introduced. The theory presented in this paper is based on the assumption that the atom is placed in athermal environment created by the blackbody radiation (BBR). The numerical results allow us to expect hteirsignificance for modern experiments and testing the fundamental interactions.
I. INTRODUCTION
Since the early days of quantum mechanics (QM), the studyof characteristics of atoms, such as the energy of bound states,has played a key role in the development of modern quan-tum field theory and its practical application in various fieldsof physics. Further development of quantum mechanics ledto the creation of a quantum electrodynamical description(QED) of atoms and the evaluation of the corresponding rela-tivistic QED corrections to bound energies [1–7]. Subsequentexperimental observations and their growing accuracy haverequired taking into accounting more complex effects, such as,for example, a multitude of radiative QED corrections mutu-ally providing a versatile verification of fundamental physics.Much effort has been put into this type of theoretical research,see, for example, [8–10].Nowadays, the most accurate atomic experiments can beattributed to three directions corresponding to measurementsof the transition frequencies in hydrogen [11, 12] with a rel-ative uncertainty of . × − , in helium [13, 14], wherethe experimental accuracy reaches the level of several partsin − , and in atomic clocks possessing an accuracy about − for time scaling [15, 16]. Such precise experiments re-quired theoretical calculations of various QED effects at the α m /M and α m levels, see [17] and references therein,where α is the fine structure constant, m and M are the elec-tron and nuclear masses, respectively. Apart from accuratetheoretical calculations of the binding energies in the hydro-gen atom, the fine structure, and the isotope shift of the low-lying states of helium tend to serve as an independent tool fortesting of fundamental interactions. Similar to well-studiedone-electron atomic systems the measured transition frequen-cies should be compared with the theoretical calculations pur-suing the search of possible discrepancy [18].Such extraordinary calculations however pay attention tothe effects of the other type: corrections induced by the ex-ternal blackbody radiation (BBR) field. The influence of theBBR field is well-known in atomic physics, it is manifestedin the existence of a Stark shift of bound states. The the-ory and corresponding calculations for one- and few-electronatoms were presented in [19] in the framework of the QMapproach. These calculations were continued for the case of ∗ [email protected] atomic clocks (many-particle systems) in [20–22] and are thesubject of theoretical investigations in present days. Not longago, the QED derivation of the Stark shift induced by the BBRfield was performed in [23] and, subsequently, applied to cal-culations in the helium atom [24].Recently in [25] a QED description of the effects inducedby the BBR field for one-electron atomic systems was pre-sented. Although this theory is based on Thermal Quan-tum Electrodynamics (TQED) pioneered in [26–28], thermalCoulomb interaction was first rigorously introduced in [25].In particular, it was found that the energy shift associatedto this interaction can exceed the corresponding Stark shift.The effect of thermal one-photon exchange between the boundelectrons and nucleus was investigated in [29] for the heliumatom. From the results obtained in this work it follows that intwo-electron atomic system the thermal shift reaches the levelof the experimental accuracy [30] at room temperature.In view of the close attention to the verification of fun-damental physical interactions in such experiments and thesearch for new constraints on dark matter [31], the derivationof thermal effects leading to fine and hyperfine splitting oflevels is of considerable interest. This problem can be solvedusing the formalism presented in [25]. Then, the representa-tion of the thermal photon propagator in the thermal Coulombgauge admits an explicit analogy with the case of zero vac-uum, see [4], i.e. arising relativistic corrections are easily ex-tended to thermal ones. It can be expected that thermal cor-rections of these types can serve for further testing of funda-mental interactions on atomic systems.In this paper, the relativistic thermal corrections arisingfrom the scalar and transversal part of the thermal photonpropagator are evaluated. Then their contributions for one-and two-electron atoms are estimated. All the derivationsare performed within the framework of the rigorous quantumelectrodynamics at finite temperatures and are applicable toH-like ions. For clarity, the mass of particles and speed oflight c are written out explicitly in the basic formulas. II. THERMAL COULOMB INTERACTION:RELATIVISTIC CORRECTIONS
Starting with the description of the interaction of twocharges, one can use the relation from textbooks (see, for ex-ample, [2]) connecting the nuclear current, j ν ( x (cid:48) ) , with the a r X i v : . [ phy s i c s . a t o m - ph ] D ec field, A µ ( x ) , it creates: A µ ( x ) = (cid:90) d xD µν ( x, x (cid:48) ) j ν ( x (cid:48) ) , (1)where x = ( t, (cid:126)r ) represents the four-dimensional coordi-nate vector ( t represents time and (cid:126)r denotes a space vector), D µν ( x, x (cid:48) ) is the Green’s function of the photon, and µ , ν arethe indices running the values , , , . Then, the zero com-ponent of A µ ( x ) corresponds to the Coulomb interaction, andthe components , , are the transversal part, which givesthe interaction of retardation and advance. According to [26–28], the photon Green’s function (photon propagator) is rep-resented by the sum of two contributions, which are the resultof expectation value on the states of zero and heated vacuum, D µν ( x, x (cid:48) ) = D µν ( x, x (cid:48) ) + D βµν ( x, x (cid:48) ) , respectively.Thermal interaction can be introduced by analogy, see [25],when the ’ordinary’ photon Green’s function is replaced bythe thermal one, D βµν ( x , x ) , [26–28]. In [25] is was estab-lished that the thermal part of photon propagator D βµν ( x, x (cid:48) ) isgiven by the Hadamard propagation function [2, 7] and, there-fore, admits a different (equivalent) form: D βµν ( x, x (cid:48) ) = − πg µν (cid:90) C d k (2 π ) e ik ( x − x (cid:48) ) k n β ( | (cid:126)k | ) , (2)where g µ ν is the metric tensor, k = k − (cid:126)k and n β is thePlanck’s distribution function. The contour of integration in k -plane for Eq. (2) is given in Fig. 1. FIG. 1. Integration contour C in k plane of Eq. (2).Arrows on thecontour define the pole-bypass rule. The poles ± ω k are denoted with × marks. Thermal photon propagator in the form Eq. (2) has the ad-vantage of allowing the introduction of gauges in completeanalogy with the ’ordinary’ QED theory, see [25]. Then in theCoulomb gauge the function D βµν ( x , x ) recasts into D β ( x, x (cid:48) ) = 4 πi (cid:90) C d k (2 π ) e ik ( x − x (cid:48) ) (cid:126)k n β ( ω ) , (3) D βij ( x, x (cid:48) ) = 4 πi (cid:90) C d k (2 π ) e ik ( x − x (cid:48) ) k n β ( ω ) (cid:18) δ ij − k i k j (cid:126)k (cid:19) . Concentrating first on the thermal Coulomb interaction, weconsider the zero component of the thermal photon propaga-tor, D β ( x, x (cid:48) ) . Substituting this into Eq. (1) the thermalpotential for the point-like nucleus in the static limit can befound. For the conciseness we omit the discussion and em-ploy the procedure proposed in [25], where the appropriate analytical calculations of the integral over κ was obtained as V β ( r ) = − e π − γβ + i r ln Γ (cid:16) irβ (cid:17) Γ (cid:16) − irβ (cid:17) , (4)where β ≡ / ( k B T ) ( k B is the Boltzmann constant and T is the temperature in kelvin), r is the modulus (length) of thecorresponding radius vector for the interpartcle distance, Γ isthe gamma function and γ is the Euler-Mascheroni constant, γ (cid:39) . .The potential (4) was used to find thermal corrections to theenergy of a bound electron in the hydrogen atom [25] and he-lium [29], where the thermal correction tuned out to be of theorder of the experimental accuracy [30]. As it should be, inthe lowest order the heat bath environment removes the orbitalmomentum degeneracy. To find other corrections (to the fineand hyperfine structure), one should turn to the Pauli approx-imation or determine the relativistic corrections proportionalto /c , where c is the speed of light, see [2, 4, 6, 7]. Withinthe second order approximation, the particle interaction oper-ator (in the case of zero vacuum) in momentum representationand Coulomb gauge has the form [4]: U ( (cid:126)p , (cid:126)p , (cid:126)k ) = 4 πe (cid:20) (cid:126)k − m c − m c + ( (cid:126)k(cid:126)p )( (cid:126)k(cid:126)p ) m m c (cid:126)k − (cid:126)p (cid:126)p m m c (cid:126)k + i(cid:126)σ [ (cid:126)k × (cid:126)p ]4 m c (cid:126)k − i(cid:126)σ [ (cid:126)k × (cid:126)p ]4 m c (cid:126)k (5) − i(cid:126)σ [ (cid:126)k × (cid:126)p ]2 m m c (cid:126)k + i(cid:126)σ [ (cid:126)k × (cid:126)p ]2 m m c (cid:126)k + ( (cid:126)σ (cid:126)k )( (cid:126)σ (cid:126)k )4 m m c (cid:126)k − (cid:126)σ (cid:126)σ m m c (cid:35) . Here (cid:126)p is the electron momentum operator, (cid:126)σ is the Pauli ma-trix, m is the particle mass and the index 1 or 2 refers to thecorresponding particle.The Fourier component Eq. (5) contains relativistic correc-tions arising from the scalar and transverse parts of the pho-ton propagator D ( x, x (cid:48) ) and D ij ( x, x (cid:48) ) , respectively. Thelatter also corresponds to the Breit-Pauli interaction. Ap-plying the Fourier transform, (cid:82) d k (2 π ) e i(cid:126)k(cid:126)r , to the scatteringamplitude U ( (cid:126)p , (cid:126)p , (cid:126)k ) , the coordinate representation can befound. Then the operators (cid:126)p and (cid:126)p should be replaced by the (cid:126)p = − i ∇ and (cid:126)p = − i ∇ . In the thermal case, however,the Fourier transform to coordinate representation is given as (cid:82) d k (2 π ) n β ( | (cid:126)k | ) e i(cid:126)k(cid:126)r , where the factor 2 occurs in a result ofintegration along the contour C .The result of such a Fourier transform for the first term inEq. (5) results in the expression (4), where the regularizationof divergent contribution at | (cid:126)k | → was performed by intro-ducing a coincidence limit [25]. The same can be easily givenfor the second and third terms in Eq. (5). After the integrationover angles one can find (2) + (3) → − e π (cid:20) m c + 1 m c (cid:21) ∞ (cid:90) dκ n β ( κ ) κ sin κr r . (6)Then the series expansion in small values of r reveals a con-stant contribution proportional to ∞ (cid:82) dκ κ n β ( κ ) . In princi-ple, this contribution is state independent and, therefore, van-ishes for the difference between the energy states of the atom.But, it can be found immediately that the coincidence limit[25] regularizes such contributions along with divergences. Inother words, subtraction of the limit r → in Eq. (6) givesthe regular expression, which is (2) + (3) → e π ζ (5) β r (cid:20) m c + 1 m c (cid:21) . (7)Here we have integrated over κ ≡ | (cid:126)k | and ζ ( s ) gives the Rie-mann zeta function.In the fourth term in Eq. (5) first note that (cid:126)k in the numera-tor can be obtained by the gradient action on the exponent: πe m m c (cid:90) d k (2 π ) n β ( κ ) e i(cid:126)k(cid:126)r ( (cid:126)k(cid:126)p )( (cid:126)k(cid:126)p ) (cid:126)k = (8) πe m m c (cid:90) d k (2 π ) n β ( κ ) ( (cid:126) ∇ (cid:126)p )( (cid:126) ∇ (cid:126)p ) (cid:126)k e i(cid:126)k(cid:126)r . Then, integrating over angles in the expression (8) and actingby the gradient operators, the formula (8) reduces to e πm m c ∞ (cid:90) dκ n β ( κ ) κ (cid:20) cos κr r − sin κr κr (cid:21) ( (cid:126)p (cid:126)p ) (9) − e πm m c ∞ (cid:90) dκ n β ( κ ) κ (cid:20) κr r + 3 sin κr κr − κ sin κr r (cid:21) ( (cid:126)r (cid:126)p )( (cid:126)r (cid:126)p ) . The coincidence limit r → in this case is − e πm m c ∞ (cid:90) dκ n β ( κ ) , (10)which cancels the divergence in Eq. (9). Finally, the Fouriertransform of fourth term in Eq. (5) reduces to πe ( (cid:126)k(cid:126)p )( (cid:126)k(cid:126)p ) m m c (cid:126)k → ζ (3) e πβ m m c × (11) (cid:2) r ( (cid:126)p (cid:126)p ) + 2( (cid:126)r (cid:126)p )( (cid:126)r (cid:126)p ) (cid:3) . Evaluation of the fifth contribution in Eq. (5) repeats thecalculation of the first one. The result of the lowest order is − (cid:126)p (cid:126)p m m c (cid:126)k → ζ (3) e πβ m m c r ( (cid:126)p (cid:126)p ) . (12)The Fourier transform for the next four terms can be per-formed using the substitution (cid:126)k → − i(cid:126) ∇ . Acting by the gra-dient operator on the expression arising after angular integra- tion, we find i(cid:126)σ [ (cid:126)k × (cid:126)p ]4 m c (cid:126)k → − ζ (3) e πβ m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) , − i(cid:126)σ [ (cid:126)k × (cid:126)p ]4 m c (cid:126)k → ζ (3) e πβ m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) , (13) − i(cid:126)σ [ (cid:126)k × (cid:126)p ]2 m m c (cid:126)k → ζ (3) e πβ m m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) ,i(cid:126)σ [ (cid:126)k × (cid:126)p ]2 m m c (cid:126)k → − ζ (3) e πβ m m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) . Similar calculations for the last two contributions leads to ( (cid:126)σ (cid:126)k )( (cid:126)σ (cid:126)k )4 m m c (cid:126)k → − ζ (5) e πβ m m c × (cid:2) r ( (cid:126)σ (cid:126)σ ) + 2( (cid:126)r (cid:126)σ )( (cid:126)r (cid:126)σ (cid:3) (14) − (cid:126)σ (cid:126)σ m m c → ζ (5) e πβ m m c r ( (cid:126)σ (cid:126)σ ) . We emphasize that the replacement (cid:126)k → − i(cid:126) ∇ assumes firstthe action of the gradient operator and then going to the coin-cidence limit.Finally, the total contribution in the lowest order in temper-ature can be obtained in the form: U ( (cid:126)p , (cid:126)p , (cid:126)r ) = − ζ (3) e r πβ + 8 ζ (3) e πβ m m c r ( (cid:126)p (cid:126)p )+ 8 ζ (3) e πβ m m c (cid:126)r ( (cid:126)r (cid:126)p ) (cid:126)p − ζ (3) e πβ m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ])+ 2 ζ (3) e πβ m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) + 4 ζ (3) e πβ m m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) − ζ (3) e πβ m m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) . (15) III. HYDROGEN ATOM
The expressions (4), (7) and (11)-(14) allow significant re-duction for the hydrogen atom, when the approximation of theinfinite nucleus mass is considered. Then the thermal correc-tions of lowest order in temperature and α (the fine structureconstant) arises from Eq. (15) as U ( (cid:126)p , (cid:126)r ) = − ζ (3) e r πβ − ζ (3) e πβ m c ( (cid:126)σ [ (cid:126)r × (cid:126)p ]) . (16)To the latter the approximation of the point-like nucleus can beapplied, when (cid:126)r can be replaced by the vector (cid:126)r representingthe distance between the bound electron and the nucleus, m is the electron mass, and we can take into account that (cid:126)σ = (cid:126)s corresponds to the electron spin. Noticing that the operator [ (cid:126)r × (cid:126)p ] ≡ (cid:126)l is the orbital momentum, one can find U ( (cid:126)p, (cid:126)r ) = − ζ (3) Ze πβ r − ζ (3) Ze πβ m c ( (cid:126)s (cid:126)l ) , (17)where Z denotes the nuclear charge.The parametric estimation in relativistic units for the ex-pression (17) arises with r ∼ / ( mαZ ) , β ∼ mα ( k B T ) , c = 1 . Then the first term is proportional to mα ( k B T ) /Z and the second term to mα Z ( k B T ) . To get an estimate inatomic units, this results must be divided by a factor mα .The evaluation of the ( (cid:126)s(cid:126)l ) operator can be done using therelation (cid:126)j = ( (cid:126)l + (cid:126)s ) , which results to the average value (cid:104) ( (cid:126)s(cid:126)l ) (cid:105) = [ j ( j + 1) − l ( l + 1) − s ( s + 1)] . Whereasthe average value r for the state a in the hydrogen atom is n a (5 n a + 1 − l a ( l a + 1)) , the energy shift taking into ac-count the fine structure in the lowest order in atomic units is ∆ E a = − ζ (3)3 πβ α n a [5 n a + 1 − l a ( l a + 1)] (18) − ζ (3)3 πβ α [ j a ( j a + 1) − l a ( l a + 1) − s a ( s a + 1)] . As a next step one can take into account the effect of thefinite nuclear mass in the lowest order. For this, the nuclearmomentum (cid:126)p is replaced by the − (cid:126)p in the center-of-mass system. Then in approximation of the point-like nucleus, theresult is U ( (cid:126)p, (cid:126)r ) = − ζ (3) Ze πβ r + 4 ζ (5) Ze πβ r + 2 ζ (5) Ze πβ m c r (19) + 2 ζ (5) Ze πβ M c r − ζ (3) Ze πβ mM c r p − ζ (3) Ze πβ mM c (cid:126)r ( (cid:126)r(cid:126)p ) (cid:126)p − ζ (3) Ze πβ m c (cid:20) mM (cid:21) ( (cid:126)s(cid:126)l ) − ζ (3) Ze πβ mM c (cid:104) m M (cid:105) ( (cid:126)I p (cid:126)l )+ 64 ζ (5) Ze πβ mM c r ( (cid:126)s(cid:126)I p ) − ζ (5) Ze πβ mM c ( (cid:126)r(cid:126)s )( (cid:126)r(cid:126)I p ) , where (cid:126)I p denotes the nuclear spin operator and M ≡ m isthe nuclear mass. The meaning of these terms can be foundin textbooks on quantum electrodynamics, see also [32], withthe additional notation ’thermal’.In a hydrogen atom at room temperature, corrections (19)can be neglected except for corrections (18). For clarity, thevalues of the thermal corrections Eq. (19) are shown in Table Ifor specific low-lying states at room temperature. TABLE I. The values of thermal corrections Eq. (19) in Hz. The first column gives the thermal correction (Th. corr.). The following columnsshow the values obtained for the specific states. The dependence on the nuclear charge Z is left to determine the behavior of corrections forhydrogen-like ions.Th. corr. s s p / p / − ζ (3) Ze πβ r − . Z . Z . Z . Z ζ (5) Ze πβ r . · −
10 1 Z . · − Z . · − Z . · − Z ζ (5) Ze πβ c r (cid:2) m + M (cid:3) . · −
14 1 Z . · −
13 1 Z . · −
13 1 Z . · −
13 1 Z − ζ (3) Ze πβ mMc r p − . · − Z − . · − Z − . · − Z − . · − Z − ζ (3) Ze πβ mMc (cid:126)r ( (cid:126)r(cid:126)p ) (cid:126)p − . · − Z − . · − Z − . · − Z − . · − Z − ζ (3) Ze πβ m c (cid:2) mM (cid:3) ( (cid:126)s(cid:126)l ) 0 0 1 . · − Z − . · − Z − ζ (3) Ze πβ mMc (cid:2) m M (cid:3) ( (cid:126)I p (cid:126)l ) 0 ( F = 0 ) ( F = 0 ) . · − Z ( F = 0 ) . · − Z ( F = 0 ) ( F = 1 ) ( F = 1 ) − . · − Z ( F = 1 ) − . · − Z ( F = 1 ) ζ (5) Ze πβ mMc r ( (cid:126)s(cid:126)I p ) − . · −
17 1 Z ( F = 0 ) − . · −
16 1 Z ( F = 0 ) − . · −
17 1 Z ( F = 0 ) − . · −
17 1 Z ( F = 1 ) . · −
18 1 Z ( F = 1 ) . · −
17 1 Z ( F = 1 ) . · −
17 1 Z ( F = 1 ) . · −
17 1 Z ( F = 2 ) − ζ (5) Ze πβ mMc ( (cid:126)r(cid:126)s )( (cid:126)r(cid:126)I p ) 4 . · −
18 1 Z ( F = 0 ) . · −
17 1 Z ( F = 0 ) . · −
17 1 Z ( F = 0) 1 . · −
17 1 Z ( F = 1) − . · −
18 1 Z ( F = 1 ) − . · −
17 1 Z ( F = 1 ) − . · −
17 1 Z ( F = 1 ) − . · −
18 1 Z ( F = 2) In particular, from Table I it follows that the relativistic ther-mal corrections in the hydrogen atom are negligible at roomtemperature. Nevertheless, there are corrections that increasewith increasing nuclear charge Z . The most interesting in thissense is the correction corresponding to the thermal shift ofthe fine sublevel and proportional to (cid:126)s(cid:126)l . Note also that in thenonrelativistic limit the thermal correction for the motion ofthe nucleus can be found in the lowest order, see [1], by re- placing the electron mass with the reduced one. Then, for theresults listed in the first row of Table I, one can find that thethermal correction on the finite mass of the nucleus is about − Hz (multiplication factor is / . ), representing,thus, a leading order correction with respect to relativistic cor-rections. Having made rough estimates for highly chargedhydrogen-like ions, it can be found that these corrections stillgo beyond the accuracy of laboratory experiments. However,assuming the astrophysical applications, one can expect a cu-bic increase of the thermal corrections with a rise in tempera-ture. IV. HELIUM ATOM
Operator Eq. (15) admits the evaluation of the relativisticthermal corrections for the helium atom. For this, it is conve-nient to generalize the formula (15) to the case of an arbitrarynumber of electron-electron and electron-nuclear interactions: U = − ζ (3) e πβ N +1 (cid:88) i TABLE III. Expectation values of U (2) A = ( (cid:126)s (cid:126)l ) + ( (cid:126)s (cid:126)l ) , U (2) B = Z ([ (cid:126)r × (cid:126)p ]( (cid:126)s + 2 (cid:126)s ) − [ (cid:126)r × (cid:126)p ]( (cid:126)s + 2 (cid:126)s )) and U (2) C = 3 r ( (cid:126)p (cid:126)p ) + (cid:126)r ( (cid:126)r (cid:126)p ) (cid:126)p operators (in a.u.) and the cor-responding total energy shift ∆ E β ≡ (cid:104) U (2) (cid:105) , see Eq. (22), at roomtemperature ( T = 300 K) in Hz for the He( M = ∞ ) atom.State (cid:104) U (2) A (cid:105) in a.u. (cid:104) U (2) B (cid:105) in a.u. (cid:104) U (2) C (cid:105) in a.u. ∆ E β in Hz S . . · − S . . · − S . . · − P . . · − P − . . . . · − P − . . . . · − P . − . . . · − S . . · − S . . · − P . . · − P − . . . . · − P − . . . . · − P . − . . . · − The total contribution of thermal corrections Eq. (22),listed in Table III for different states in helium, does not ex-ceed several parts of − . With that, the thermal shift arisingdue to the finite mass of the nucleus (dividing the results ofTable II by the nuclear mass) may turn out to be significantfor highly excited states. For example, for the P state, it isabout . Hz, whereas for the S it reaches . Hz. Themost problem in this context arises with the method of calcu-lating the binding energies of highly excited states in the he-lium atom. Moreover, in [37] it can be found that the calcula-tions of the lowest order thermal shift for highly excited statesmust be performed with a closed form of thermal potential,Eq. (4). Still, the calculations presented here can serve to pickout the trend of thermal shifts (21) and (22) in this case. First,as can be found analytically, the energy shift correspondingto the thermal spin-orbit interaction does not depend on theprincipal quantum number (almost the same). Second, the en-ergy shift, Eq. (21), has the order of a thermal Stark shift andalways remains negative as opposed to a Stark shift, see [19]. V. CONCLUSIONS AND DISCUSSION In this article, we examined relativistic corrections due tothermal interaction. As was found in [25], applying the rig-orous QED theory to the description of the interaction of twocharges placed in a heat bath reveals effects that do not arisein the framework of the quantum mechanical approach. Asa consequence, the thermal potential Eq. (4) was obtained in[25], where the leading-order radiative corrections were alsodescribed.To complete the QED description of thermal effects, it isnecessary to take into account the relativistic corrections tothe interaction potential. The procedure is greatly simplifiedby virtue of the form of thermal photon propagator Eq. (2).First of all, it admits a simple introduction of thermal gauges,see [25]. Then, by choosing the thermal Coulomb gauge, therelativistic corrections to the thermal interaction can be eas-ily written by analogy with the formula (5) in the momen-tum space, see [4]. The difference from the ordinary (zerovacuum) case is the presence of contour integration and thePlanck’s distribution function. Then, by performing the se-quential calculations, the expression (15) can be found. In ahydrogen atom, it reduces to Eq. (19) and then can be general-ized to the expression (20) in the case of an arbitrary numberof charges.The expression (20) allows parametric estimation in the finestructure constant. For the first term, it reads α /Z · ( k B T ) in atomic units. Then the leading-order relativistic correctionsare α times less (as usual) and correspond to the spin-orbitthermal interaction. The found estimate by the nuclear chargenumber Z shows a decrease of thermal correction Eq. (17)for the hydrogen-like ions. The dependence on the nuclearcharge Z is left in Table I to demonstrate inference. In thecase, Z = 1 , one of the possible applications of the presentedtheory is the study of the proton form factor effects involv-ing the spin-spin and spin-orbit thermal interaction [32]. An-other conclusion following from the expression (19) is that thethermal relativistic corrections corresponding to the spin-orbit interaction do not depend on the principal quantum numberand, more interestingly, are proportional to the nuclear chargenumber Z . The results listed in Table I, however, demonstratethat these corrections are still outside the experimental accu-racy for arbitrary Z .Nonetheless, the dependence only on angular momenta al-lows an approximate estimate of the correction Eq. (18) forfine sublevels in other atoms. Consider, for example, a pre-cision measurement of atomic isotope shifts in Sr + and Sr + based on S / ↔ D / transition [38]. The ther-mal shift proportional to ( (cid:126)s(cid:126)l ) for the d / state in hydro-gen is − . Hz at room temperature and an approxi-mate estimate in Sr + is − . Hz, while the declared er-ror is about . Hz. Another example can be given forthe neutral Sr atom. Performing the same estimates usingthe results in Table III, a thermal correction of the order of − . Hz can be found, resulting in a relative value of about − . × − (which exceeds the dc-Stark effect on the or-der) for the measured frequency between the ground state S and the metastable state P [39]. Especially, in [39] it wasassumed that the planned controlling the BBR shift would in-crease the accuracy to × − or below in a near future,making such thermal corrections potentially important.Moreover, since the thermal Stark shift can change signwith a varying in temperature, it was recently predicted in[40] that the rubidium atomic clock could be operated withzero net BBR shift at the temperature of about K. In turn,the thermal corrections Eq. (20) would increase by about . times when raised to this temperature. Despite such a roughanalysis, this estimate discloses the need for a separate de-scription of thermal effects in many-electron atomic systems.It should be noted that in this case the dominant contribution − ζ (3) Ze πβ r was not taken into account, which can only becalculated using specialized numerical methods.Along with that, one should separately focus on the ten-dency of recent years towards the search for ’new physics’and verification of fundamental interactions within the frame-work of atomic physics. The achieved experimental accuracyat the level of − of operating atomic clocks with singleatomic systems [16, 41] makes them the most accurate toolfor searching for dark matter setting constraints on its mass[42]. The study of new physics using the Rydberg states ofatomic hydrogen was proposed in [43]. All spectroscopic ex-periments with this level of accuracy require stabilization ofthe temperature environment and the corresponding consid-eration of the thermal Stark shift. The results obtained in thispaper show the need to take into account additional thermal ef-fects in such an analysis. In contrast to the traditional relativis-tic QM description of the thermal interaction determined bythe multipolar method, [22, 44] or, for example, the study ofthe attractive force induced by black body radiation betweenthe atom and the heated cavity [45], the description given inthis work reveals the fundamental nature of thermal interac-tions of another type. In addition, astrophysical prolongationof the discovered effects is evident, when the temperature canreach much higher values. ACKNOWLEDGEMENTS This work was supported by Russian Foundation for BasicResearch (grant 20-02-00111). [1] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Springer Berlin Heidelberg, 1957).[2] A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynam-ics (Wiley-Interscience, New York, 1965).[3] I. I. Sobel’man, Introduction to the Theory of Atomic Spectra (Pergamon, 1972).[4] V. Berestetskii, E. Lifshits, and L. Pitaevskii, Quantum Elec-trodynamics (Oxford Butterworth-Heinemann, 1982).[5] I. Lindgren and J. Morrison, Atomic many-body theory (Springer, 1986).[6] L. Labzowsky, G. Klimchitskaya, and Y. Dmitriev, RelativisticEffects in the Spectra of Atomic Systems (Institute of PhysicsPublishing, 1993).[7] W. Greiner and J. Reinhardt, Quantum Electrodynamics ,Physics and astronomy online library (Springer, 2003).[8] O. Y. Andreev, L. N. Labzowsky, G. Plunien, and D. A.Solovyev, Phys. Rep. , 135 (2008).[9] V. Shabaev, Phys. Rep. , 119 (2002).[10] P. Indelicato, Journal of Physics B: Atomic, Molecular and Op-tical Physics , 232001 (2019).[11] C. G. Parthey and et al., Phys. Rev. Lett. , 203001 (2011).[12] A. Matveev and et al., Phys. Rev. Lett. , 230801 (2013).[13] R. van Rooij, J. S. Borbely, J. Simonet, M. D. Hoogerland,K. S. E. Eikema, R. A. Rozendaal, and W. Vassen, Science , 196 (2011).[14] X. Zheng, Y. R. Sun, J.-J. Chen, W. Jiang, K. Pachucki, andS.-M. Hu, Phys. Rev. Lett. , 063001 (2017).[15] F. Levi, D. Calonico, C. E. Calosso, A. Godone, S. Micalizio,and G. A. Costanzo, Metrologia , 270 (2014).[16] T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti,B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S.Safronova, G. F. Strouse, W. L. Tew, and J. Ye, Nat. Commun. , 6896 (2015).[17] P. J. Mohr, D. B. Newell, and B. N. Taylor, J. Phys. Chem. Ref.Data , 043102 (2016).[18] K. Pachucki, V. Patk´oˇs, and V. A. Yerokhin, Phys. Rev. A ,062510 (2017).[19] J. W. Farley and W. H. Wing, Phys. Rev. A , 2397 (1981).[20] M. S. Safronova, D. Jiang, B. Arora, C. W. Clark, M. G. Ko-zlov, U. I. Safronova, and W. R. Johnson, IEEE Transactionson Ultrasonics, Ferroelectrics, and Frequency Control , 94(2010).[21] M. S. Safronova, M. G. Kozlov, and C. W. Clark, Phys. Rev.Lett. , 143006 (2011).[22] S. G. Porsev and A. Derevianko, Phys. Rev. A , 020502(R)(2006).[23] D. Solovyev, L. Labzowsky, and G. Plunien, Phys. Rev. A ,022508 (2015).[24] T. Zalialiutdinov, D. Solovyev, and L. Labzowsky, J. Phys. B:At. Mol. Opt. Phys. , 015003 (2017).[25] D. Solovyev, Annals of Physics , 168128 (2020).[26] L. Dolan and R. Jackiw, Phys. Rev. D , 3320 (1974).[27] J. F. Donoghue and B. R. Holstein, Phys. Rev. D , 340 (1983).[28] J. F. Donoghue, B. R. Holstein, and R. Robinett, Annals ofPhysics , 233 (1985). [29] D. Solovyev, T. Zalialiutdinov, and A. Anikin, Phys. Rev. A , 052501 (2020).[30] K. Kato, T. D. G. Skinner, and E. A. Hessels, Phys. Rev. Lett. , 143002 (2018).[31] C. J. Kennedy, E. Oelker, J. M. Robinson, T. Bothwell,D. Kedar, W. R. Milner, G. E. Marti, A. Derevianko, and J. Ye,Phys. Rev. Lett. , 201302 (2020).[32] F. G. Daza, N. G. Kelkar, and M. Nowakowski, Journal ofPhysics G: Nuclear and Particle Physics , 035103 (2012).[33] V. I. Korobov, D. Bakalov, and H. J. Monkhorst, Phys. Rev. A , R919 (1999).[34] V. I. Korobov, Phys. Rev. A , 064503 (2000).[35] G. W. F. Drake, Atomic, Molecular and Optical Physics Hand-book (Springer, New York, NY, 1996).[36] A. M. Frolov, Phys. Rev. A , 2436 (1998).[37] D. Solovyev, T. Zalialiutdinov, and A. Anikin, Journal ofPhysics B: Atomic, Molecular and Optical Physics (2020).[38] T. Manovitz, R. Shaniv, Y. Shapira, R. Ozeri, and N. Akerman,Phys. Rev. Lett. , 203001 (2019).[39] R. Le Targat, L. Lorini, Y. Le Coq, M. Zawada, J. Gu´ena,M. Abgrall, M. Gurov, P. Rosenbusch, D. Rovera, B. Nag´orny,R. Gartman, P. Westergaard, M. Tobar, M. Lours, G. Santarelli,A. Clairon, S. Bize, P. Laurent, P. Lemonde, and J. Lodewyck,Nature Communications , 8pp (2013).[40] K. W. Martin, B. Stuhl, J. Eugenio, M. S. Safronova, G. Phelps,J. H. Burke, and N. D. Lemke, Phys. Rev. A , 023417(2019).[41] N. Huntemann, C. Sanner, B. Lipphardt, C. Tamm, and E. Peik,Phys. Rev. Lett. , 063001 (2016).[42] P. Wcisło, P. Morzy´nski, M. Bober, A. Cygan, D. Lisak,R. Ciuryło, and M. Zawada, Nature Astronomy , 0009 (2016),arXiv:1605.05763 [physics.atom-ph].[43] M. P. A. Jones, R. M. Potvliege, and M. Spannowsky, Phys.Rev. Research , 013244 (2020).[44] B. K. Sahoo, “Relativistic calculations of atomic clock,” in Handbook of Relativistic Quantum Chemistry , edited by W. Liu(Springer Berlin Heidelberg, Berlin, Heidelberg, 2016) pp. 1–44.[45] P. Haslinger, M. Jaffe, V. Xu, O. Schwartz, M. Sonnleitner,M. Ritsch-Marte, H. Ritsch, and H. M¨uller, Nature Phys. ,257 (2018), arXiv:1704.03577 [physics.atom-ph].[46] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Sin-gapore, 1988). APPENDIX: REDUCTION OF MATRIX ELEMENTS In this Appendix a most general formulas for the reductionof matrix elements for operators (cid:126)s i [ (cid:126)r i × (cid:126)p i ] , ( (cid:126)s i +2 (cid:126)s j )[ (cid:126)r ij × (cid:126)p j ] and r ij ( (cid:126)p i (cid:126)p j ) + (cid:126)r ij ( (cid:126)r ij (cid:126)p j ) (cid:126)p i in the LSJM coupling scheme( L is the total orbital momentum, S is the total spin, J, M are the total angular momentum and its projection) are given.Since the spin operator and orbital momentum operator acton different subsystems, the expectation value of (cid:126)s i (cid:126)l i operatorcan be written, see [46], as (cid:104) n (cid:48) L (cid:48) S (cid:48) J (cid:48) M (cid:48) | (cid:126)s i [ (cid:126)r i × (cid:126)p i ] | nLSJM (cid:105) = δ J (cid:48) J δ M (cid:48) M × (A1) ( − J + L + S (cid:48) (cid:26) J S (cid:48) L (cid:48) L S (cid:27) (cid:104) n (cid:48) L (cid:48) || r i × p i || nL (cid:105)(cid:104) S (cid:48) || s i || S (cid:105) , where (cid:104) S (cid:48) || s || S (cid:105) = ( − S ( S (cid:48) ) (cid:112) (2 S (cid:48) + 1)(2 S + 1) (A2) × (cid:26) / S (cid:48) / S / (cid:27) (cid:112) / and (cid:104) n (cid:48) L (cid:48) || . . . || nL (cid:105) is the reduced matrix element.Similar equations can be written for expectation values ofoperator ( (cid:126)s i + 2 (cid:126)s j ) (cid:104) n (cid:48) L (cid:48) S (cid:48) J (cid:48) M (cid:48) | ( (cid:126)s i + 2 (cid:126)s j )[ (cid:126)r ij × (cid:126)p j ] | nLSJM (cid:105) (A3) = δ J (cid:48) J δ M (cid:48) M ( − J + L + S (cid:48) (cid:26) J S (cid:48) L (cid:48) L S (cid:27) ×(cid:104) n (cid:48) L (cid:48) || r ij × p j || nL (cid:105)(cid:104) S (cid:48) || s i + 2 s j || S (cid:105) . The latter operator r ij ( (cid:126)p i (cid:126)p j ) + (cid:126)r ij ( (cid:126)r ij (cid:126)p j ) (cid:126)p i is scalar anddo not acts on spin variables. Therefore, (cid:104) n (cid:48) L (cid:48) S (cid:48) J (cid:48) M (cid:48) | r ij ( (cid:126)p i (cid:126)p j ) + (cid:126)r ij ( (cid:126)r ij (cid:126)p j ) (cid:126)p i | nLSJM (cid:105) (A4) = δ S (cid:48) S ( − J + L (cid:48) + S √ J + 1 C J (cid:48) M (cid:48) JM (cid:26) L S JJ (cid:48) L (cid:48) (cid:27) ×(cid:104) n (cid:48) L (cid:48) || r ij ( p i p j ) + r ij ( r ij p j ) p i || nL (cid:105) . Here C J (cid:48) M (cid:48) JM = δ J (cid:48) J δ M (cid:48) M is the Clebsch-Gordan coefficientand (cid:26) j j j j j j (cid:27)(cid:27)