PHYSICAL REVIEW A 83, 042508 (2011)

Blackbody-radiation correction to the polarizability of helium

M. Puchalski,1,2 U. D. Jentschura,1 and P. J. Mohr3

1Missouri University of Science and Technology, Rolla, Missouri 65409-0640, USA 2Faculty of Chemistry, Adam Mickiewicz University, Grunwaldzka 6, PL-60-780 Poznan, Poland  ́ 3National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8420, USA

(Received 14 February 2011; published 18 April 2011)

The correction to the polarizability of helium due to blackbody radiation is calculated near room temperature.

A precise theoretical determination of the blackbody radiation correction to the polarizability of helium is

essential for dielectric gas thermometry and for the determination of the Boltzmann constant. We find that the

correction, for not too high temperature, is roughly proportional to a modified hyperpolarizability (two-color

hyperpolarizability), which is different from the ordinary hyperpolarizability of helium. Our explicit calculations

provide a definite numerical result for the effect and indicate that the effect of blackbody radiation can be excluded

as a limiting factor for dielectric gas thermometry using helium or argon.

DOI: 10.1103/PhysRevA.83.042508 PACS number(s): 31.30.J−, 51.30.+i, 06.20.F−, 47.80.Fg

I. INTRODUCTION

The recent measurement [1] of the refractive index of

helium in a microwave cavity resonator has yielded the hitherto

most precise experimental value of the molar polarizability of

the helium atom,

A = αdNA

30

= 0.517 253 5(47)

cm3

mol . (1)

Here NA is Avogadro’s number, 0 is the vacuum permittivity

(also called the “electric constant”), and αd is the static electric

dipole polarizability of helium.We also recall that A is defined

in the limit of zero density. The techniques in this experiment

could lead to measurements of thermodynamic temperature

or to a determination of the value of the Boltzmann constant.

Each of these applications would take advantage of the fact

that αd has been accurately determined theoretically in a

series of calculations including complete leading relativistic

α2 and quantum electrodynamics (QED) α3 corrections in the

fine structure constant expansion [2–5] with an uncertainty of

0.2 ppm from the estimate of the α4 term. Here the magnitude

of the corrections is given in atomic units, that is, relative

to the Hartree energy scale. We note that for excitation by

low-energy radiation, as is relevant for the experiment [1],

the relativistic and radiative corrections to the polarizability

are unambiguously defined. However, for higher frequencies,

there may be additional field-configuration-dependent corrections (see Appendix E of Ref. [6]).

Blackbody radiation present in the cavity resonator will

lead to a temperature-dependent correction to the measured

value of the helium molar polarizability, which could affect the

interpretation of such measurements. However, the correction

due to blackbody radiation was assumed to be negligible

compared to the uncertainty of the measurement in Ref. [1].

(Heuristic arguments that support this assumption and indicate

that it is also valid for argon are presented in Sec. IV

below.) The experiment [1] determined the polarizability of

helium atoms through the interaction of microwaves with the

atoms in a cavity resonator. At the same time, blackbody

radiation is present in the resonator, and it also interacts with

the atoms. As a first approximation, the interaction of the

cavity microwaves with the atoms and the interaction of the

blackbody radiation with the atoms are independent processes

and do not affect each other. The dominant energy shifts due

to microwave radiation on the one hand, and due to blackbody

radiation on the other hand, are described by the corresponding

second-order ac Stark shifts [7–9] and are proportional, to

very good approximation, to the dynamic second-order dipole

polarizability at the microwave and blackbody frequencies.

The derivation of the theoretical second-order expressions in

both classical, time-ordered perturbation theory and in the

field-quantized framework are contrasted against each other in

Ref. [10]. From a QED point of view, both photon annihilation

as well as photon creation processes contribute to the dynamic

polarizability.

Besides the second-order polarizability, there is also a

fourth-order effect which is due to the exchange of four instead

of two photons with the radiation field(s). When the radiation

is monochromatic, the total fourth-order energy shift is proportional to the so-called hyperpolarizability of the atom [11].

However, when the atom is simultaneously interacting with

both microwave as well as blackbody radiation, the treatment

has to be modified because photon creation and annihilation

processes of one and the same field have to be “matched,” and

this excludes some intermediate, virtual states of the atom +

radiation field from the fourth-order expressions. Indeed, in

generalizing the fully quantized formalism to fourth order,

we find convenient expressions which describe the two-color

hyperpolarizability. The resulting fourth-order energy shift

finds a natural interpretation as a perturbation of the secondorder dynamic energy shift due to the microwave photons. The

latter is proportional to the dynamic polarizability. Therefore,

the fourth-order effect constitutes a correction to the dynamic

polarizability of the atom.

Note that some of the QED corrections to the polarizability of helium also involve fourth-order perturbation theory

[3,5], with blackbody photon interactions being replaced by

the interactions with the radiative photons. However, there is an

important difference. For example, for the QED corrections to

the Bethe logarithm, the atom only emits, then absorbs, virtual

radiative photons, while it emits then absorbs and absorbs

then emits photons with the probing electromagnetic waves.

In the evaluation of the blackbody radiation correction to the

dynamic polarizability, we have to take into account processes

1050-2947/2011/83(4)/042508(7) ©2011 American Physical Society 042508-1
M. PUCHALSKI, U. D. JENTSCHURA, AND P. J. MOHR PHYSICAL REVIEW A 83, 042508 (2011)

where the atom both emits then absorbs and absorbs then emits

blackbody and probing microwave photons.

Our paper is organized as follows. The theoretical foundations are recalled and derived in Sec. II. Numerical investigations are described in Sec. III. Conclusions are reserved for

Sec. IV.

II. THEORY

In order to formulate the problem, we need to take

into account the interaction of the helium atom with two

electromagnetic fields: (i) the microwave field used to probe

the electric dipole polarizability and (ii) the blackbody radiation field. We work with a second quantized radiation field

and with a first quantized theory for the atomic electrons.

Furthermore, we work in the Schrodinger picture. In SI units,  ̈

the Hamiltonian for the helium atom coupled to an external

source of microwaves and affected by blackbody radiation is

given as

H = H0 + hω ̄ Ma†

M aM +

B

hω ̄ Ba†

BaB + UM + UB. (2)

Here M and B are multi-indices defined as

M ≡ kMλM , B ≡ kBλB, (3)

where kM and kB are the wave vectors of the probing

microwave field and of the blackbody field, and λM,B denote

their polarizations. We sum over the modes of the blackbody

field and assume that the microwave radiation can be described

by a single mode. The other symbols used in Eq. (2) are as

follows. In the nonrecoil approximation, the helium atom is

described by the nonrelativistic Hamiltonian

H0 = 

a=1,2

p2

a

2m − Ze24π0ra



+ e2

4π0r12

, (4)

where r12 is the distance between the electrons, e is the electron

charge, Z = 2 the nuclear charge number, 0 is the vacuum

permittivity, and ra (a = 1,2) is the electron-nucleus distance.

We describe the interaction of the atom with the quantized

electromagnetic fields in the length gauge. The electric dipole

interactions of the atom with the microwave and blackbody

fields are as follows:

UM = −e



hω ̄ M

20VM

M · r(a†

M + aM ), (5)

UB = −e



B



hω ̄ B

20VB

B · r(a†

B + aB), (6)

where the photon creation and annihilation operators are a†

and a, respectively. We normalize the electric field operators

(see Ref. [12]) so that the energy density of the microwave

photon integrated over the volume VM is equal to  ̄hωM ,

and analogously for the blackbody photon. The effect of the

electromagnetic fields is assumed to be a small perturbation of

H0, so that a perturbative treatment of the dipole interaction of

the atom with the electromagnetic field becomes possible.

We first consider the second-order effect which gives the

main energy perturbation of the helium atom due to the ac Stark

effect. A single-mode microwave field probes the helium atom

in the ground state. The ground-state energy is denoted as E0

and its Schrodinger wave function is denoted as  ̈ ψ0. First-order

perturbation theory in UM gives a vanishing effect, and the

leading correction to the energy E0 is of second order. We

then average over the polarizations and propagation directions

of the microwave mode. We are interested in the classical limit

where the number of microwave photons is nM  1, the normalization volume is large (VM  1), but the ratio nM/VM remains finite and proportional to the intensity of the microwave

field. Using this formalism, one may easily rederive [10] the

dynamic ac Stark energy shift due to microwave photons,

E = e2

3

nMhω ̄ M

20V



±

ψ0|riR(±ω)ri|ψ0, (7)

where

R(±ω) = 1

E0 − H0 ∓ hω ̄ = 

m

|φmφm|

E0 − Em ∓ hω ̄ (8)

is the resolvent operator for the unperturbed helium atom,

and ri = ri

1 + ri2; that is, we are denoting the ith Cartesian

component of the sum r1 + r2 of the positions r1 and r2 of

both electrons simply by ri

. For repeated superscripts and

subscripts, we use the summation convention [an example

is given by the Cartesian superscripts i in Eq. (7)]. The sum

over ± is necessary because we treat both photon absorption

followed by emission as well as photon emission followed

by absorption. The intensity IM of the microwave field in our

normalization of the field operator is given as

IM = nMhω ̄ Mc

VM

. (9)

For our purposes (ground state of helium perturbed by a

microwave field), we may approximate to good accuracy the

microwave frequency by the static limit (ω = ωM → 0) in

Eq. (7). Then the well-known final result in second order is

rederived,

E = −e2IM

20c

αd , (10)

where the static dipole polarizability (divided by the square

of the elementary charge e2) in nonrelativistic limit for the

ground state is defined by

αd = −2

3 ψ0|ri

R(0)ri|ψ0. (11)

Here R

(0) = 1/(H0 − E0)

 is the reduced Green function,

where the reference state |ψ0 is excluded from the sum over

intermediate (virtual) states.

We now investigate the perturbation of the dipole polarizability αd due to the blackbody radiation. This is a

fourth-order process in the radiation field. The atom may emit

and absorb photons from the microwave field and also emit

and absorb photons from the blackbody field. Fourth-order

perturbation theory, with time-independent field operators in

the Schrodinger picture, can then be used in order to infer the  ̈

042508-2
BLACKBODY-RADIATION CORRECTION TO THE ... PHYSICAL REVIEW A 83, 042508 (2011)

energy shift. The result reads, when taking all combinations

of emission and absorption into account,

EB = e4

hω ̄ M

20VM



B

hω ̄ B

20VB



{2(	B a†

BR(ωM)	M a†M

×R(ωB + ωM )	B aBR(ωM )	MaM +	Ba†

BR(ωB)

×	Ma†

M R(ωB + ωM)	MaM R(ωB)	BaB+	Ma†M

×R(ωM)	Ba†

BR(ωB + ωM )	BaBR(ωM)	MaM 

+2 	Ba†

BR(ωB)	BaBR

(0)	Ma†

M R(ωM)	MaM 

−	Ba†

BR(ωB)	BaB	M a†M R2

(ω)	MaM 

−	Ba†

BR2

(ωB)	BaB(	Ma†

M R(ωM)	MaM }

+(a†

B,−ωB)↔(aB,ωB)+(a†M ,−ωM )↔(aM,ωM )

+(a†

B, − ωB)↔(aB,ωB); (a†M ,−ωM )↔(aM,ωM ),

(12)

where we use the shorthand notation

	B ≡ B · r, 	M ≡ M · r. (13)

Furthermore, we again use multi-indices M and B as defined

in Eq. (3). The replacement terms in Eq. (12) correspond to

the exchange of blackbody photon emission versus absorption, of microwave photon emission versus absorption, and

simultaneous exchange of both processes. Next, we consider

the “classical limit” of a high occupation number for both

fields, and the low-frequency limit for the microwave field,

ωM → 0 (this approximation is always valid for microwave

photons whose energy is low compared to the first available

atomic transition). Furthermore, we match the summation

over the blackbody modes B with an integration over the

frequency-dependent intensity of the blackbody radiation in

Eq. (12),



B

nBhω ̄ Bc

VB

→

 ∞

0

dωB u(ωB,T ), (14)

where Planck’s law gives

u(ωB,T ) dω = h ̄

4π3c2

ω3

BdωB

exp( ̄hωB/kBT ) − 1

. (15)

For room temperature, the blackbody spectrum has its maximum at frequencies much below typical atomic transition

frequencies. Therefore, in addition to approximating the

microwave frequency in Eq. (12) by zero, we may also

approximate the blackbody frequency by zero in the fourthorder polarizability defined in Eq. (12). Indeed, if we employ

the approximation ωB → 0 in the matrix element defined

in Eq. (12) (not in the prefactor which is proportional to

ωB), then we may even integrate over the blackbody photon

frequency analytically. This is explored in the following. For

now, we approach the problem by numerically integrating

over the blackbody photon frequency. The result can be

written as

EB = e2IM

20c

αdχ = Eχ , (16)

with the dimensionless factor

χ =

 ∞

0

dωBχ(ωB,T ). (17)

Thus, the product αdχ can be viewed as an effective static

dipole polarizability of helium in the presence of blackbody

radiation where αd is the (dipole) polarizability in the absence

of the blackbody radiation and χ is a multiplicative factor that

gives the change in the measured value due to the radiation.

The integrand χ(ωB,T ) involves the thermal distribution of

photons,

χ(ωB,T ) = 4

9

e2

αd 0c



±

g(±ωB)u(ωB,T ). (18)

The g function is obtained after summing over the photon

modes in Eq. (12) and after taking into account all photon

creation and annihilation processes and reads

g(ωB) ≡ −ψ0|r jR(ωB)rk

R(ωB)r jR(0)rk|ψ0

−ψ0|r jR(ωB)r jR

(0)rkR(0)rk|ψ0

−1

2 (ψ0|r jR(ωB)rkR(ωB)rkR(ωB)r j |ψ0

+ψ0|r jR

(0)rkR(ωB)rkR(0)r j |ψ0

−ψ0|r jR(ωB)r j |ψ0ψ0|rk

R2(0)rk|ψ0

−ψ0|r jR2(ωB)r j |ψ0ψ0|rkR(0)rk|ψ0). (19)

The dimensionless factor χ is the quantity we are looking for

as it gives the relative perturbation χ to the polarizability due

to blackbody radiation on αd according to Eq. (16).

To this point, we have kept SI mksA units in all formulas,

as practiced by the Committee on Data for Science and

Technology (CODATA). In calculations of atomic properties,

it is usually more convenient to use formulas in atomic units.

The SI mksA static polarizability αd , the angular frequency

ωB, and g(ωB) from Eq. (19) are related to their atomic unit

counterparts αd , ωB, and g(ωB) by

αd = αd

a2

0

Eh

, (20a)

ωB = Eh

h ̄ ωB, (20b)

g(ωB) = g(ωB)

a4

0

E3

h

, (20c)

where a0 is the Bohr radius and Eh is the Hartree energy. Then

χ as defined in Eq. (16) can be obtained as

χ =

 ∞

0

dωBχ(ωB,T )

= 4

9

α3

π2



±

 ∞

0

g(±ωB)

αd



expEh

kB T ωB



−1

−1

ω3

BdωB.

(21)

The atomic unit system is defined so that physical quantities

pertaining to atoms are of order one. We can thus conclude that

χ is an effect of order α3, which is additionally suppressed

by the Boltzmann factor. In the following, we use atomic

units (i.e., units with e = h ̄ = me = 1, and where the length is

measured in Bohr radii).

042508-3
M. PUCHALSKI, U. D. JENTSCHURA, AND P. J. MOHR PHYSICAL REVIEW A 83, 042508 (2011)

III. NUMERICAL CALCULATION

The crucial step in the evaluation is the calculation of the g

function defined in Eq. (19). According to Eq. (19), it contains

both manifestly fourth-order but also products of second-order

terms. Next, we integrate g over ωB in the interval (0,∞) with

a weight given by the Boltzmann distribution of the blackbody

radiation. The numerical integration is not completely trivial,

because we have to omit poles due to resonances given by

the virtual state in the denominators of the resolvent R(ωB).

By contrast, R(−ωB) has no poles. Bending the integration

contour into the complex plane around the resonances solves

the problem. Moreover, all intermediate discrete states with

a positive factor 1/(E0 − Em ± ωB) > 0 must be represented

very accurately as they define the position of the resonances.

In practice, we are interested in temperatures that do

not exceed the room temperature substantially (<400 K).

This simplifies the problem, because the weight given by

the Boltzmann factor is exponentially suppressed on the

scale of atomic transition frequencies. For T = 273 K, the

maximum of the blackbody radiation distribution lies at

ωBmax ≈ 0.002 44 a.u. (atomic units). If we are interested in

evaluating the total effect to a relative accuracy of 1%, we

may cut off the integration interval at ωBcut = 0.05 a.u., where

u(ωBcut,273 K)/u(ωBmax,273 K) ∼ 10−20, and the ratio is

even smaller for lower temperatures. The function g varies only

marginally in the frequency range relevant to the blackbody

radiation at 273 K, and ωBcut is still an order of magnitude less

than the energy (frequency) difference of the 1S ground state

of helium and the lowest excited states. On the integration path

from zero to ωBcut, we never approach the singular points in

the denominators of the resolvents in Eq. (19), and thus only

real values of ωB need to be considered.

The nonrelativistic wave function of the ground state ψ0

and its energy E0 in atomic units are determined based on

the Rayleigh-Ritz variational principle. We use a basis set of

explicitly exponentially correlated functions (see Refs. [13,14]

and also the Appendix):

ψ(r1,r2,r12) = 

NS

m=1

vm[e−air1−bir2−cir12 − (r1 ↔ r2)], (22)

where the parameters (a,b,c) for the ith function are randomly generated from an optimized box (A1,A2) × (B1,B2) ×

(C1,C2) under additional constraints ai + bi > ε, as well as

bi + ci > ε and ci + ai > ε, where ε =



2 (E+

0 − E0) and E+0

is the ground-state energy for He+. In atomic units, ε can

be interpreted as an approximate radial momentum of the

two-electron system that characterizes the radial exponential

falloff of the wave function. The minimal momentum ε must

be chosen to be large enough to be consistent with the binding

of the electrons to the helium nucleus. By requiring that all

combinations ai + bi, bi + ci, and ai + ci fulfill this criterion,

we ensure that the wave function falls off sufficiently rapidly

at large r1, r2, and r12. If a randomly generated orbital fails

to fulfill the requirements, we generate another one until

conditions are met. This method follows ideas outlined in

Refs. [13,14]. In order to fix ideas, we should reemphasize that

the six boundary parameters characterizing the box, that is, A1,

A2, B1, B2, C1, and C2, are subject to variational optimization,

not the random parameters ai, bi, and ci.

In order to obtain a more accurate representation of the wave

function, we use two boxes that model the short-range and

medium-range asymptotics of the helium wave functions. For

the calculation of the fourth-order effect which is the subject

of this paper, matrices with a moderate number of 2NS = 100,

200, 300, 400, and 600 basis functions are fully sufficient (we

use a prefactor 2 in order to clarify the—equal—distribution

of the basis functions onto the two boxes).

In this basis, all needed matrix elements can be represented

as linear combinations of the integrals (see the Appendix),

(a,b,c,n1,n2,n12)

=



d3

r1d3r2r

n1−1

1 r

n2−1

2 r

n12−1

12 e−ar1−br2−cr12 , (23)

with nonnegative integers n1, n2, and n12. Recurrence relations

for their computation are well known [14,15]. The result for

the ground-state energy extrapolated from 600 functions is

E0 = −2.903 724 377 034 118(3). The linear coefficients vi

in Eq. (22) are obtained from a solution of the generalized

eigenvalue problem. The numerical accuracy of the results

is estimated from the apparent numerical convergence of the

matrix elements as the size NS of the S state basis is increased.

In view of the above considerations, we can approximate

the blackbody frequency in Eq. (19) to good approximation by

ωB = 0 and evaluate g(0). This is instructive, because g(0) can

be broken down into distinct contributions, which allows us

to present them separately, possibly enabling an independent

verification of the calculations if needed. We thus calculate

first the quantity αP = ψ0|ri

R(0)ri|ψ0, which is directly

connected to the dipole polarizability by the relation αd =

−2αP /3 in the nonrelativistic approximation. The resolvent

R(0) can be replaced effectively by the sum over P states as

in

RP (0) = 

n

|φnP φnP |

E0 − En

, (24)

where the sum, for our calculation, is only over the singlet

states, and n is the principal quantum number. For the ground

state, all contributions from intermediate states fulfill E0 −

En < 0. In that case, the exact representation of the P state

component of the resolvent gives the lowest possible value for

the polarizability, thus leading to a variational principle for the

determination of the second-order polarizability.

For the calculation of the g function, we also need the

first-order correction to the wave function,

	

	δψi

P




= RP (0) ri

|ψ0

= 

NP

m=1

vP

m



ri

1e−air1−bir2−cir12 − (r1 ↔ r2)



, (25)

so that the dipole polarizability

αP = 

3

i=1

ψ0|ri

	

	δψi

P


 (26)

can be written in terms of the dipole matrix element of the

reference state wave function and of the perturbation |δψi

P .

042508-4
BLACKBODY-RADIATION CORRECTION TO THE ... PHYSICAL REVIEW A 83, 042508 (2011)

Variational parameters for δψP are generated just as for the

ground state, but the size NP = 3

2NS of the basis of P states

is chosen to be larger than NS used for the generation of the

ground state. With these results at hand, it is then easy to

calculate the other second-order element,

αP P = 

3

i=1

ψ0|ri R2

P (0)ri

|ψ0=δψP |δψP , (27)

needed for g(0).

The two fourth-order terms that enter g(0) can be expressed

as

αD = 


δψi

P

	

	r j

	

	δψij

D




, αS = 


δψi

P

	

	ri

|δψS, (28)

where the intermediate S and D states are represented in the

form

|δψS = R

S(0)ri

	

	δψi

P




=

N ̄ S

m=1

vS

m[e−air1−bir2−cir12 − (r1 ↔ r2)], (29a)

	

	δψjk

D




= RD(0)r j

	

	δψk

P




=

4



ND /5

m=1

vD

m

r

j

1 rk

1 − δjkr21



×e−air1−bir2−cir12 − (r1 ↔ r2)



+

N

D /5

m=1

vP P

m

×

r

j

12rk

12 − δjkr212

e−air1−bir2−cir12 − (r1 ↔ r2)



.

(29b)

The dominant effect due to intermediate D states comes from

the excitation of one of the electrons, and the second case

adds a mixture of single excitations of two electrons. For the

calculation, we choose 2N ̄S = 2ND = 4NS = 8

3 NP .

With the results for αP , αP P , αD, and αS as defined in

Eqs. (26), (27), and (28) (see also Table I), we can proceed to

the evaluation of

γ2 ≡ g(0) = −2

3 (5αS + 6αD − 3αP αP P ). (30)

The prefactors are determined by angular algebra [16]. Here

the term αP αP P is equal to the sum of the last two products

of terms on the right-hand side of Eq. (19), and αD and αS

correspond to the D- and S-state components of the sum of the

first four terms on the right-hand side of Eq. (19), in the limit

ωB → 0. Indeed, the two-color hyperpolarizability γ2 = g(0)

can be written in terms of just the matrix elements αD, αS, αP ,

and αP P because we set ωB → 0 in Eq. (19).

The two-color hyperpolarizability γ2 = g(0), calculated

here, is not to be confused with the static hyperpolarizability γ ,

which is used in order to describe the fourth-order perturbation

when all four photons are from the same field [4,17]. The latter

can be expressed in our parameters as

γ = − 8

15 (5αS + 6αD − 5αP αP P ). (31)

The numerical prefactors are different from the ones in

Eq. (30). Using our numerical algorithm, we may verify the

result for the static hyperpolarizability γ given in Ref. [4]

and add one more significant digit, γ = 43.104 227 7(1)

as opposed to γ = 43.104 227(1) from Ref. [4] (see also

Table I).

The above calculations illustrate the computational procedure for the static one-color hyperpolarizabilty γ and its

two-color generalization γ2. For finite excitation frequency,

there is a subtlety which deserves some extra considerations.

In the fourth-order matrix elements in Eq. (30), the outer S

state couples to odd-parity P states by dipole transitions. The

inner virtual states can be S or D states, but they can also

be even-parity P states. Let us suppose that the reference S

state has been coupled to a z-polarized odd-parity P state.

By a dipole transition, emitting or absorbing an y polarized

photon, the two-electron system may couple to an even-parity

P virtual state that is proportional to the x component of

the cross product r1 × r2. For ωB → 0, the contribution of

even-parity P states vanishes due to symmetry, but it gives a

finite effect for nonvanishing ωB. In order to fix the notation,

we define

αP E = ijk


δψi

P

	

	r j

	

	δψk

P E


, (32)

where the P even (P E) states are given by (we invoke the

summation convention over k and l)

	

	δψj

P E


= jkl RP E(0)rk

	

	δψl

P




= 

NP E

m=1

vP E

m



jklrk

1 rl2e−air1−bir2−cir12 − (r1 ↔ r2)



.

(33)

TABLE I. Various quantities of interest for the calculation of the two-color hyperpolarizability, expressed in atomic units. The definition of

αP , αP P , αD, and αS is given in Eqs. (26), (27), and (28). Numerical values are also indicated for the two-color hyperpolarizability γ2 ≡ g(ω = 0)

defined in Eq. (30), and for the hyperpolarizability γ defined in Eq. (31).

2NS αP αP P αS αD γ γ2 ≡ g(0)

100 −2.074 787 392 227 2 2.122 527 900 595 −11.261 152 615 −7.755 245 582 43.103 075 16 59.750 569 45

200 −2.074 788 259 836 5 2.122 530 424 544 −11.261 406 459 −7.755 398 231 43.104 221 68 59.752 012 04

300 −2.074 788 260 731 2 2.122 530 428 743 −11.261 407 099 −7.755 398 608 43.104 224 56 59.752 015 66

400 −2.074 788 261 670 8 2.122 530 432 055 −11.261 407 836 −7.755 399 056 43.104 227 94 59.752 019 89

600 −2.074 788 261 679 1 2.122 530 432 021 −11.261 407 802 −7.755 399 033 43.104 227 78 59.752 019 68

∞ −2.074 788 261 682(3) 2.122 530 432 01(2) −11.261 407 80(2) −7.755 399 03(2) 43.104 227 7(1) 59.752 019 7(1)

Literature −2.074 788 261 682 (3)a 43.104 227(1)b

a

This result was published in Ref. [3]. bSee Ref. [4] for this result.

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M. PUCHALSKI, U. D. JENTSCHURA, AND P. J. MOHR PHYSICAL REVIEW A 83, 042508 (2011)

We choose NP E = NS and recall the definition of the perturbation |δψi

P  due to the ith Cartesian component of the

position operator. The definition of αP E thus involves a total of

three resolvents of the helium atom. By numerical calculation,

using the same incremental values for the basis sets as those

indicated in Table I, we obtain αP E = −0.062 951 884 22(4).

This value should not be understood as the static value of a

dynamic fourth-order polarizability due to even-parity P states

(indeed, as outlined above, the contribution of even-parity P

states to the two-color hyperpolarizability vanishes). In our

definition of αP E, we have arranged for the  tensors to isolate

a nonvanishing contribution in the static limit. This approach

serves two purposes: (i) to implicitly define the variational

parameters used in the calculation of the resolvent RP E in both

the static as well as the dynamic regime, and (ii) to provide a

reference value for the contribution of even-parity P states to a

fourth-order matrix element that has a manifestly nonvanishing

static limit (but no direct physical interpretation).

For the integration in Eq. (21) over ωB, a 120-point GaussLegendre quadrature in the interval (0,ωBcut) is fully sufficient

to obtain

χ = 2.742 57(1) × 10−18 for T = 273 K. (34)

Under the approximation g(ωB) ≈ g(0) in Eq. (21), which

we can do because g varies slowly on the frequency scale of

the Boltzmann distribution at room temperature, the integral

over ωB can by done analytically. It has a character similar to

the Stefan-Boltzmann law with a T 4 temperature dependence.

For clarity, we now return to SI mksA units. With the

atomic unit quantities αd = 1.383 192 ... and g(0) = γ 2 =

59.752 019 ... defined as in Eq. (20), we obtain

χ ≈ 4α3

135

π2 γ 2

αd

kBT

Eh

4

≈ 4.9372 × 10−28 T

K

4

. (35)

This approximation gives χ = 2.7424 × 10−18 for T = 273 K

with an error of order 10−4 compared to Eq. (34). At room

temperature (T = 300 K), we obtain χ = 3.999 × 10−18.

Figures 1 and 2 illustrate the Boltzmann weight of the

integrand defining χ and the overall dependence of χ on the

temperature. As indicated in Eq. (16), χ is the multiplicative

factor that gives the temperature-dependent effective static

FIG. 1. (Color online) χ(ωB,T ) as a function of ωB at T = 273 K.

The frequency scale is in atomic units (a.u.).

FIG. 2. (Color online) A plot of the dimensionless quantity χ

against the temperature T measured in kelvin (K) is shown, with

special emphasis on the point T = 273 K.

polarizability of helium in the presence of the blackbody

radiation.

IV. CONCLUSIONS

We have investigated the effect of blackbody radiation on

the determination of the helium molar polarizability by fourthorder perturbation theory in the quantized electromagnetic

field of the probing microwave and the blackbody radiation

field. This shift of the molar polarizability is of interest because

it may affect a conceivable definition of thermodynamic temperature or a determination of the Boltzmann constant based on

the measurement [1]. Indeed, it has been suggested in Ref. [18]

that the relative correction to the polarizability amounts to

a correction as large as 2.1 × 10−10 K−2 T 2 = 16 × 10−6 at

T = 273 K, which would have been significant compared

to the 9.1 × 10−6 relative uncertainty of the measurement

[1]. However, our fourth-order calculations indicate that the

effect of blackbody radiation on the measurement of the

polarizability of helium reported in [1] is negligible. In fact,

the result of that measurement, (A,meas − A,theory)/A,theory =

(1.8 ± 9.1) × 10−6, supports this conclusion. The calculation

in [18] considers only the frequency dependence, discussed

in Ref. [19], for example, of the interaction of the blackbody

radiation with the atoms. In lowest order, this dependence does

not affect the interaction of the microwave radiation with the

atoms.

Here, we have presented a fully quantized approach to

the calculation of the blackbody radiation correction to the

polarizability of helium, and we have evaluated all expressions

numerically [see Eq. (34)], as well as within a semianalytic

approach [see Eq. (35)]. However, even without this formalism, an estimate for the effect of the blackbody radiation

on the polarizability of helium at room temperature could

have been performed based on known literature references,

as follows. We start from the ground-state blackbody shift

δν0 = 0.11 Hz given by Farley and Wing [20]. (This value

takes into account the 300 K to 273 K temperature difference

as compared to Ref. [20].) This amounts to a relative change in

the polarizability of order δν0/νHe ≈ 2 × 10−17, which affects

the virtual states in the defining expression for the ground-state

polarizability (we denote by νHe ≈ 5 × 1015 Hz a typical transition frequency in helium). Unlike the QED corrections, the

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BLACKBODY-RADIATION CORRECTION TO THE ... PHYSICAL REVIEW A 83, 042508 (2011)

blackbody radiation effect is largest for highly excited states,

so a possibly larger correction would come from the shift of the

excited-state energies due to the blackbody radiation. These

states enter the expression for the polarizability as virtual

states, and the relative frequency of the virtual transitions leads

to a proportional shift of the polarizability, as implied by the

fourth-order effect given in Eq. (19). Farley and Wing find

that the correction δνi for each excited state i approaches a

limiting shift of δν∞ → 2.0 kHz as i → ∞ [20]. For lower

states, this gives an overestimate because the blackbody

shifts for typical excited states are larger. Nevertheless, if

all excited-state energies were shifted by this amount, then

the polarizability would undergo a relative change of order

χ(T = 273 K) = δν∞/νHe ≈ 4 × 10−13, which needs to be

contrasted to our exact result (the latter is in the range of 10−18

and confirms the overestimation).

In this article, we have carried out a detailed analysis of

the effect and obtained the precise shift of the polarizability

in fourth-order perturbation theory, which can be expressed

as a relative perturbation of the second-order polarizability

effect proportional to a dimensionless factor χ. Our analytic

formula given in Eq. (35) and our numerical results obtained

by numerically integrating the blackbody spectrum confirm

this estimate under the proviso that the shift of the lowest

excited helium states, which is of the order of a few hertz, gives

the relevant contribution to the polarizability. In particular,

we find a shift of χ(T = 273 K) = 2.7424 × 10−18 for the

relative correction to the polarizability of the helium ground

state. Furthermore, according to Eq. (35), we find a T 4

dependence for the overall shift of the polarizability with an

analytic dependence of the form χ ≈ 4.9372 × 10−28(T /K)4.

Although we have carried out numerical calculations only up

to a temperature T = 400 K, an analytic estimate based on the

known form of the blackbody spectrum shows that the formula

Eq. (35) should be valid to better than 1% in a temperature

range to at least 2000 K.

Finally, we would also like to briefly discuss the size of

the effect for other noble gases like neon and argon, which

are of experimental interest. The two-color hyperpolarizability

scales as (Zeff α)

−10 where Zeff is the effective nuclear charge

seen by the electron in the valence shell. The final result for

the shift of the polarizability due to the blackbody radiation

interaction is obtained as an integral over the dynamic twocolor hyperpolarizability, and its value may thus additionally

depend on the overlap of the resonance frequencies with the

maximum of the temperature-dependent blackbody spectrum.

For temperatures not exceeding 400 K, however, the static

two-color hyperpolarizability of the noble gas in question

should provide a good estimate of the total effect, according to

Eq. (35). We thus expect that the result for χ in the temperature

range 0 <T < 400 K for neon and argon should not differ

from the result given in Eq. (35) by more than two orders

of magnitude and thus be negligible on the level of accuracy

reached in the experiment [1]. More accurate estimates require

an explicit calculation of γ2 for the atomic reference system

under investigation.

ACKNOWLEDGMENTS

The authors thank J. W. Schmidt and M. R. Moldover

for helpful discussions. This work has been supported by the

National Science Foundation and by a precision measurement

grant from the National Institute of Standards and Technology.

APPENDIX: TWO-ELECTRIC INTEGRALS

The two-electron integral  is defined by

(n1,n2,n3,α,β,γ )

≡

 d3r1

4 π

 d3r2

4 π e−α r1−β r2−γ r12 r

n1−1

1 r

n2−1

2 r

n3−1

12 . (A1)

This integral takes a very simple form when all ni = 0,

(0,0,0,α,β,γ ) = 1

(α + β)(α + γ )(β + γ )

. (A2)

The explicit form for ni > 0 can be obtained by differentiation

with respect to the corresponding parameter. For the actual

evaluation of , we use compact recurrence relations from

Refs. [13,15].

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