Magneto-Electric response functions for simple atomic systems
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Magneto-Electric response functions for simple atomic systems
J. Babington and B. A. van Tiggelen
Univ. Grenoble 1/CNRS,LPMMC UMR 5493,25 rue des Martyrs, Maison des Magist`eres,38042 Grenoble,France.e-mail: [email protected], [email protected].
Received: date / Revised version: date
Abstract.
We consider a simple atomic two-body bound state system that is overall charge neutral andplaced in a static electric and magnetic field, and calculate the magneto-electric response function as afunction of frequency. This is done from first principles using a two-particle Hamiltonian for both anharmonic oscillator and Coulomb binding potential. In the high frequency limit, the response function fallsoff as 1 /ω whilst at low frequencies it tends to a constant value. PACS.
Magneto-electric effects are by now a well established phe-nomena both theoretically and experimentally. The ten-dency has been to focus on relatively large molecular sys-tems where DFT calculations [1,2] apply and experimen-tal values have been measured [3]. In addition to beingassociated with important optical phenomena [2,4], theirexistence has played an important role in the Casimirphysics [5,6,7], in particular if it is possible in certaincircumstances to find a contribution from the quantumvacuum to a bodies momentum. For these reasons we con-sider it an interesting question to ask what are the simplestmodels that describe possible and display magneto-electriceffects.In this article we calculate the magneto-electric re-sponse function for the two simplest bound state systems -the harmonic oscillator and the hydrogen atom. The prin-cipal difference between these two atomic systems is thatthe harmonic oscillator is strongly bound whereas the hy-drogen atom with its Coulomb potential is weakly bound.This manifests itself in the accessibility of different energyeigenstates in the perturbation theoryThis article is organised as follows. In Section 2 we de-fine the atomic system and formulate the response for anarbitrary binding potential. As a tractable example thathas a closed form, the harmonic oscillator is chosen andits magneto-electric response function calculated explic-itly. In Section 3 the response function for hydrogen ispresented. It is then compared to the appropriate DFTresult and its relation to experimental values. Finally, in Section 4 we summarise our results and provide some com-ment on their validity and applicability.
We will now derive the ME response function for a two-body charge neutral composite system in static externalfields ( E , B ) using the Hamiltonian formalism (the bold-face used here is to indicate they are external fields). Onecan also use the path integral approach and the coupledclassical Lorentz force equations to obtain informationabout the response function. The path integral gives a cor-rect response function at high frequencies but it is not reli-able at low frequencies as well as suffering from the wronganalytic structure in the complex frequency plane [8]. Aconsideration of the coupled Lorentz force equations pro-duces similar difficulties and so we use the Hamiltonianmethod exclusively in this paper.The system we consider is illustrated in Figure 1. Twoequal but opposite electrical charges with coordinates ( q i , q i )and masses ( m , m ) interacting with a classical (c-number)gauge field ( φ, A i ) that are the combined contribution ofthe static external fields and the fluctuating source that isused to probe the system. A binding potential V ( q − q )holds the charges together giving a bound state that isoverall charge neutral. The Hamiltonian that describesthis system is given by H = 12 m ( p − eA ( q )) + eφ ( q )+ 12 m ( p + eA ( q )) − eφ ( q ) + V ( q − q ) . (1) J. Babington, B. A. van Tiggelen: Magneto-Electric response functions for simple atomic systems
Fig. 1.
The bound state two-body atomic system consistingof two equal but opposite electrical charges, with two differentmasses m and m and coordinates q i and q i (with respect tothe coordinate system Σ ). To deduce the response function forthis system we will change coordinates to the centre of masscoordinate X i and the separation coordinate x i . To calculate the response function of this body due to ahigh frequency electromagnetic field, it is necessary to passto a new set of coordinates that consists of both the centreof mass coordinate X i and the separation vector x i . Theseare shown in Figure 1. The new variables are then definedby X := ( m q + m q ) /M , x := ( q − q ), M := m + m , m := m m /M , and m ∆ = m − m . Correspondingly, wechange from the two particles momenta ( p , p ) to ( P, p )where P is the conjugate momenta of the centre of masscoordinate X and p is likewise the conjugate momenta tothe separation vector x . It is also necessary to implementthis change of coordinates on the gauge field; it can thenbe expanded about the centre of mass coordinate as A ( X + ( m /M ) x ) = A ( X ) + (cid:16) m M (cid:17) x i ∇ i A ( X )+ (cid:18) (cid:19) (cid:16) m M (cid:17) x i x j ∇ i ∇ j A ( X ) , (2) A ( X − ( m /M ) x ) = A ( X ) − (cid:16) m M (cid:17) x i ∇ i A ( X )+ (cid:18) (cid:19) (cid:16) m M (cid:17) x i x j ∇ i ∇ j A ( X ) , (3)and similarly for the scalar potential φ . We will work tofirst order in the spatial derivatives of the gauge potentialand thereby neglect the last terms in the above expansion.This corresponds to electric and magnetic fields that canvary in time, but that are spatially constant (so the ap-proach is restricted to wavevectors that are less than theinverse of the size of the atomic system).Using the Lagrangian as an intermediate step in per-forming the change of coordinates, we recognize here that E ( t, X ) = −∇ φ ( t, X ) − ∂ t A ( t, X ), whilst the derivative ofthe vector potential once projected with the Levi-Civitatensor will give the magnetic field. The Hamiltonian in the new coordinates after making this expansion reads H = 12 M P + 12 m p + V ( x ) − ex i · E i ( t, X ) − e (cid:16) m ∆ M (cid:17) P i ( x · ∇ ) A i ( t, X ) − e (cid:16) m ∆ M m (cid:17) p i ( x · ∇ ) A i ( t, X )+ e m (cid:16) m ∆ M (cid:17) x i x j ∇ i A k ( t, X ) ∇ j A k ( t, X )+ O ( ∇ A ) . (4)The ME activity results from a source magnetic field in-ducing electrical polarisation. We can now define the magneto-electric response of the bound state system by promotingall of the canonical degrees of freedom to operators. It isdefined by h eδ ˆ x i ( t ) i := Z dt ′ χ EBij ( t − t ′ ) δB j ( t ′ , X ) , (5)where the lhs is the standard expectation value of thefluctuating electric dipole moment induced on the rhs byan externally applied fluctuating magnetic field (i.e. a testsource) that is in general time and space dependent. Interms of correlation functions, it is given by the retardedtwo-point function χ EBij ( t − t ′ ) = − iθ ( t − t ′ ) h Ω | [ eδ ˆ x i ( t ) , δ ˆ O j ( t ′ )] | Ω i , (6)where the operator ˆ O j ( t ′ ) couples to the fluctuating mag-netic field δB j ( t ′ , X ), and is to be found from the micro-scopic theory given by the Hamiltonian Equation (4). Theground state | Ω i that we will use will be specified laterin this section. Taking the Fourier transform of this onefinds in frequency space χ EBij ( ω ) = h Ω | e ˆ x i H − E − ¯ hω ˆ O j | Ω i + h Ω | e ˆ x i H − E + ¯ hω ˆ O j | Ω i ∗ , (7)where E is the ground state energy of the system. To ob-tain the detailed form of the response function, we needto specify the form of the operator ˆ O j . One can see fromEquation (4) that the operator has two contributions. Oneis linear and the other is quadratic in the gauge poten-tial. To calculate the operator we make the specific gaugechoice for the gauge potential A = 12 ( B + δB ( t )) ∧ X. (8)With this choice the Hamiltonian is then a function of onlygauge invariant quantities. As a final step to fully specifythe Hamiltonian we include the static electric field by sub-stituting E i ( t, X ) = E i . The final form of the Hamiltonianthat will be used for subsequent calculations is given by H = 12 M ˆ P + 12 m ˆ p + V (ˆ x ) − e ˆ x i · E i . Babington, B. A. van Tiggelen: Magneto-Electric response functions for simple atomic systems 3 − e (cid:16) m ∆ M (cid:17) (ˆ x ∧ ˆ P ) · ( B + δB ( t )) − e (cid:16) m ∆ M m (cid:17) ˆ L · ( B + δB ( t ))+ e m (cid:16) m ∆ M (cid:17) ˆ x i ˆ x j B m δB n ( t )[ δ ij δ mn − δ im δ jn ] , (9)where L = x ∧ p is the orbital angular momentum aboutthe centre of mass origin. One can now just read off theoperator that couples to the fluctuating magnetic fieldˆ O i = − (cid:16) em ∆ mM (cid:17) ˆ L i + e m (cid:16) m ∆ M (cid:17) ˆ x a ˆ x b [ δ ab δ ij − δ ai δ bj ] B j . (10)There is also a contribution from the term linear in thecentre of mass momenta. However, this is zero when eval-uated between the ground states (zero momentum plane-wave eigenfunctions). This amounts to a choice of refer-ence frame (the centre of mass frame) and can be used todefine the components of the static external electromag-netic fields as well. Indeed, we have not specified so far thenature of the ground state | Ω i . For computational conve-nience we work with a perturbed ground state due to thepresence of static external electric field | Ω i = | i − e E i ∞ X n =1 E n − E | n ih n | ˆ x i | i . (11)This choice corresponds to the physical situation where weput the two particle system in the static external electricfield first , let the system settle down and then apply thestatic external magnetic field. From Equation (10) we seethere are two separate contributions to the response func-tion. The first is due to a coupling with the angular mo-mentum operator which we write as χ EBij ( ω, ˆ L ); the secondis due to a quadrupole moment coupling which we write as χ EBij ( ω, ˆ x ). Then the full response function is just givenby their sum χ EBij ( ω ) = χ EBij ( ω, ˆ L ) + χ EBij ( ω, ˆ x ). Firstconsider evaluating the contribution due to the angularmomentum operator χ EBij ( ω, ˆ L ) = − e ∆mmM h Ω | ˆ x i H − E − ¯ hω ˆ L j | Ω i− e ∆mmM h Ω | ˆ x i H − E + ¯ hω ˆ L j | Ω i ∗ . (12)The next step is to expand the denominators in terms ofthe static magnetic field. From Equation (9) the perturba-tion of the Hamiltonian is given by δH = − ( em ∆ /mM ) ˆ L i B i ,therefore we find (keeping only the terms linear in thestatic magnetic field) χ EBij ( ω, ˆ L ) = − (cid:18) e m ∆ mM (cid:19) (cid:16) em ∆ mM (cid:17) ×h Ω | ˆ x i H − E − ¯ hω ( ˆ L k B k ) 1ˆ H − E − ¯ hω ˆ L j | Ω i− (cid:18) e m ∆ mM (cid:19) (cid:16) em ∆ mM (cid:17) ×h Ω | ˆ x i H − E + ¯ hω ( ˆ L k B k ) 1ˆ H − E + ¯ hω ˆ L j | Ω i ∗ . (13)Inserting two complete sets of states gives χ EBij ( ω, ˆ L ) = − (cid:18) e m ∆ m M (cid:19) B k X m,n ( h Ω | ˆ x i | m ih m | ˆ L k | n ih n | ˆ L j | Ω i× E m − E − ¯ hω E n − E − ¯ hω +( h Ω | ˆ x i | m ih m | ˆ L k | n ih n | ˆ L j | Ω i ) ∗ × E m − E + ¯ hω E n − E + ¯ hω ) . (14)For the quadrupole moment contribution we have χ EBij ( ω, ˆ x ) = (cid:18) e m ∆ mM (cid:19) h Ω | ˆ x i (cid:18) H − E − ¯ hω (cid:19) × (ˆ x B j − B · ˆ x ˆ x j ) | Ω i + (cid:18) e m ∆ mM (cid:19) h Ω | ˆ x i (cid:18) H − E + ¯ hω (cid:19) × (ˆ x B j − B · ˆ x ˆ x j ) | Ω i ∗ . (15)Inserting a single complete set of states gives χ EBij ( ω, ˆ x ) = (cid:18) e m ∆ mM (cid:19) X n (cid:18) E n − E − ¯ hω (cid:19) ×h Ω | ˆ x i | n ih n | (ˆ x B j − B · ˆ x ˆ x j ) | Ω i + (cid:18) e m ∆ mM (cid:19) X n (cid:18) E n − E + ¯ hω (cid:19) × ( h Ω | ˆ x i | n ih n | (ˆ x B j − B · ˆ x ˆ x j ) | Ω i ) ∗ . (16)When the perturbed ground state given by Equation (11)is substituted into Equations (14) and (16) we arrive ata final expression for the magneto-electric response func-tion. To linear order in the static external fields Equa-tion (14) becomes χ EBij ( ω, ˆ L ) = (cid:18) e m ∆ m M (cid:19) B k E l X m,n X s =0 ( h | ˆ x i | m ih m | ˆ L k | n ih n | ˆ L j | s ih s | ˆ x l | i× E s − E E m − E − ¯ hω E n − E − ¯ hω )+( h | ˆ x i | m ih m | ˆ L k | n ih n | ˆ L j | s ih s | ˆ x l | i ) ∗ × E s − E E m − E + ¯ hω E n − E + ¯ hω ) . (17)The corresponding form that Equation (16) takes is χ EBij ( ω, ˆ x ) = − (cid:18) e m ∆ mM (cid:19) B k E l × X n X s =0 (cid:18) E s − E E n − E − ¯ hω (cid:19) J. Babington, B. A. van Tiggelen: Magneto-Electric response functions for simple atomic systems × (( h | ˆ x i | n ih n | (ˆ x δ kj − ˆ x k ˆ x j ) | s ih s | ˆ x l | i + h s | ˆ x i | n ih n | (ˆ x δ kj − ˆ x k ˆ x j ) | ih | ˆ x l | s i )( h | ˆ x i | n ih n | (ˆ x δ kj − ˆ x k ˆ x j ) | s ih s | ˆ x l | i + h s | ˆ x i | n ih n | (ˆ x δ kj − ˆ x k ˆ x j ) | ih | ˆ x l | s i ) ∗ ) . (18)To go further, it is necessary to specify a binding po-tential so that the energy eigenvalues and eigenfunctionscan be deduced. As the simplest example one might consider, the harmonicoscillator with V (ˆ x ) = mω ˆ x /
2. This model is both pos-sible to solve analytically and relevant phenomenologicallyin displaying the properties associated with real matter. Anotable feature here is that since the potential is stronglyconfining, the perturbed ground state due to the externalstatic electric field has only the first excited state surviv-ing in the summation. Using the operator algebra of theoscillator and the defining relationsˆ x i = r ¯ h mω (ˆ a † i + ˆ a i ) (19)ˆ L i = − i ¯ h/ ǫ ijk (ˆ a † j ˆ a k − ˆ a j ˆ a † k ) (20)= − i ¯ hǫ ijk ˆ a † j ˆ a k , (21)the matrix elements can be evaluated explicitly in Equa-tion (17) χ EBij ( ω, ˆ L ) = (cid:18) e m ∆ m M (cid:19) (cid:18) ¯ h e mω (cid:19) B k E l ǫ kib ǫ jbl × ( 1 E − E − ¯ hω E − E − ¯ hω + 1 E − E + ¯ hω E − E + ¯ hω ) . = (cid:18) e m ∆ ω m M (cid:19) ( E i B j − ( E · B ) δ ij ) × ω + ω ( ω − ω ) . (22)In an analogous use of the operator algebra the quadrupolecontribution Equation (18) can be similarly evaluated χ EBij ( ω, ˆ x ) = − (cid:18) e m ∆ ω m M (cid:19) (cid:18) ω − ω (cid:19) × (4 E i B j − E j B i − ( E · B ) δ ij ) . (23)Note here that both contributions have an anti-symmetricterm upon writing the tensor structure in the externalfields as a sum of symmetric and antisymmetric pieces,which will have a relevance to our later discussion. Boththe angular momentum and quadrupolar contribution areof the same order as can be seen from their multiplicative coefficients. The full response function then takes the finalform χ EBij ( ω ) = χ EBij ( ω, ˆ L ) + χ EBij ( ω, ˆ x )= − e m ∆ ω m M [ − ω + ω ( ω − ω ) ( E i B j − ( E · B ) δ ij )+ 1 ω − ω ( E i B j − E j B i −
14 ( E · B ) δ ij )] . (24) We now consider the second simplest system, namely thehydrogen atom. This is a weakly bound system with aCoulomb potential given by V ( r ) = − e / (4 πǫ r ) in spher-ical polar coordinates. Since the proton is very much moremassive than the electron, we can make the replacement m ∆ /M →
1, leaving just the reduced mass in all ex-pressions (which is just the electron mass). Equation (14)can be evaluated now using the hydrogenic eigenstates | n, L, m i and the energy spectrum E n = E /n . One finds χ EBij ( ω, ˆ L ) = − (cid:18) e m (cid:19) B k Y a =1 ∞ X n a =1 n a − X L a =0 L a X m a = − L a (cid:18) E n − E − ¯ hω E n − E − ¯ hω (cid:19) × ( h Ω | ˆ x i | n , L , m ih n , L , m | ˆ L k | n , L , m i×h n , L , m | ˆ L j | Ω i + (cid:18) E n − E + ¯ hω E n − E + ¯ hω (cid:19) ( h Ω | ˆ x i | n , L , m ih n , L , m | ˆ L k | n , L , m i×h n , L , m | ˆ L j | Ω i ) ∗ . (25)For the quadrupole contribution Equation (16) becomes χ EBij ( ω, ˆ x ) = + (cid:18) e m (cid:19) B k X n ,L ,m ( h Ω | ˆ x i | n , L , m i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x j | Ω i E n − E − ¯ hω +( h Ω | ˆ x i | n , L , m i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x j | Ω i ) ∗ E n − E + ¯ hω . (26)Because the electron is weakly bound the perturbed groundstate due to the external static electric field requires a fullsummation over the principal quantum number | Ω i = | , , i − e E l ∞ X n =2 1 X m = − E n − E | n, , m i×h n, , m | ˆ x l | , , i , (27) . Babington, B. A. van Tiggelen: Magneto-Electric response functions for simple atomic systems 5 in contrast to the strongly bound harmonic oscillator po-tential. In Equation (25) the only non-zero elements arewhen L = L = 1 (i.e. just standard selection rules oraddition of angular momentum) and thus reduces to χ EBij ( ω, ˆ L ) = (cid:18) e m (cid:19) B k E l X m = − X m = − ∞ X n =2 1 X m = − h , , | ˆ x i | n, , m ih n, , m | ˆ x l | , , ih , m | ˆ L k | , m i×h , m | ˆ L j | , m i E n − E − ¯ hω ) E n − E +( h , , | ˆ x i | n, , m ih n, , m | ˆ x l | , , ih , m | ˆ L k | , m i×h , m | ˆ L j | , m i ) ∗ E n − E + ¯ hω ) E n − E . (28)Turning now to the quadrupole case, Equation (26) isslightly more complicated as the expectation values of thequadrupole operator requires performing some additionof angular momenta. Indeed, the quadrupole operator canbe written as a linear superposition of spherical harmonics Y ,m and Y , . This implies non-trivial overlaps of matrixelements such that the summation over the L eigenvaluewill not completely reduce to a single value as in the pre-vious case but rather one finds χ EBij ( ω, ˆ x ) = − (cid:18) e m (cid:19) B k E l ∞ X n =2 ∞ X n =2 3 X L =1 2 X L =1 L X m = − L × L X m = − L (cid:18) E n − E E n − E − ¯ hω (cid:19) × ( h , , | ˆ x i | n , L , m ih n, L, m | ˆ x l | , , i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x i | n, L, m ih , , | ˆ x l | n, L, m ih n, L, m | ˆ x i | n , L , m i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x j | , , i )+ (cid:18) E n − E E n − E + ¯ hω (cid:19) × (( h , , | ˆ x i | n , L , m ih n, L, m | ˆ x l | , , i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x i | n, L, m i ) ∗ +( h , , | ˆ x l | n, L, m ih n, L, m | ˆ x i | n , L , m i×h n , L , m | ˆ x δ kj − ˆ x k ˆ x j | , , i ) ∗ ) . (29)Equations (28) and (29) are the main results of this pa-per. We see that like for the harmonic oscillator, thereis a 1 /ω behaviour at large frequencies. To avoid thesingularity when ¯ hω = E n − E , we supplement a phe-nomenological line width Γ so that any given excited en-ergy level can decay to a lower eigenstate by the replace-ment E n → E n + iΓ . We have evaluated Equations (28)and (29) analytically using Mathematica as an expansionin the principal quantum number. The real part of the sus-ceptibility is plotted in Figures 2 and 3, as a function offrequency both for the low and high frequency limits, andclose to the resonance respectively. To do this we have usednumerical data of typical field strengths of | B k | = 10 T , ´ ´ ´ ´ frequency H Hz L ´ - ´ - ´ - ´ - ´ - ´ - ´ - Re H Χ @ DL(cid:144)H Vc Ε L Fig. 2.
The real part of the (dimensionless) off-diagonalsusceptibility (the bi-anisotropic component χ with V =(4 / πa the atomic volume in the ground state) of a hydrogenatom with parameters E = (10 V m − , , B = (0 , T, ω = 10 Hz and Γ = 10 Hz . This plot shows the zero andhigh frequency limits. ´ ´ ´ ´ ´ ´ frequency H Hz L - - - - H Χ @ DL(cid:144)H Vc Ε L Fig. 3.
The real part of the (dimensionless) off-diagonalsusceptibility (the bi-anisotropic component χ with V =(4 / πa the atomic volume in the ground state) of a hydrogenatom with parameters E = (10 V m − , , B = (0 , T, ω = 10 Hz and Γ = 10 Hz . This plot shows the resonantstructure. | E k | = 10 V m − , a resonant frequency of ω = 10 Hz which corresponds to the energy difference between thefirst two levels in hydrogen, and a spontaneous decay rateof Γ ∼ s − . In Figure 4 the imaginary part of thesusceptibility is plotted close to the resonance. Of course,since the decay rate is very much smaller than the resonantfrequencies there is a huge enhancement in the suscepti-bility close to resonance.It is worth pointing out here an issue of the validity ofour calculation with respect to the photoelectric effect [9].Given that we are driving an atomic system with an elec-tromagnetic wave at some frequency we may wonder if itis valid at high frequencies. In this regime the associatedwave vector is also high and the photon can probe theshorter length scales and transfer more momentum to theelectron. However, if one considers the differential crosssection for photo-electric effect in hydrogen one knows athigh frequency it behaves as 1 /ω . Therefore at high fre-quency the atom will remain intact and the previous anal-ysis of the magneto-electric response should remain valid. J. Babington, B. A. van Tiggelen: Magneto-Electric response functions for simple atomic systems ´ ´ ´ ´ frequency H Hz L - - - H Χ @ DL(cid:144)H Vc Ε L Fig. 4.
The imaginary part of the (dimensionless) off-diagonalsusceptibility (the bi-anisotropic component χ with V =(4 / πa the atomic volume in the ground state) of a hydrogenatom with parameters E = (10 V m − , , B = (0 , T, ω = 10 Hz and Γ = 10 Hz . This plot shows the resonantstructure. It is instructive to give a numerical estimate of the size ofthe response function as compared to the standard electricsusceptibility. To do this, consider the harmonic oscillatorEquations (22) and (23) at zero frequency. Then up to nu-merical factors (i.e. simple dimensional analysis) we have | χ EBij ( ω ) / ( ǫ c ) | ∼ (cid:18) e ǫ mω (cid:19) (cid:18) e cm ω | B k || E l | (cid:19) ∼ (cid:18) e ǫ mω (cid:19) β (30)The last factor, β , can be seen too be a dimensionlessnumber, whilst the first is the standard static electric sus-ceptibility. Putting in the number leads to a factor of β ∼ − . This will also be approximately the same forthe hydrogen atom if the optical transition frequencies arechosen to coincide.We can compare this scale to experimental [3] andDFT values [1] found for the change in refractive ∆n ascompared to the absence of static external electromag-netic fields. For certain large complex molecules valuesof ∆n = ( N/V )( χ/ ( ǫ c )) ∼ − are found experimen-tally [3], where N/V is the number density of the sam-ple. For the helium atom a DFT calculation [1] gives arefractive index difference of ∆n helium ∼ − for a sam-ple with number density N/V ∼ m − evaluated at awavelength of λ = 632 . nm . The hydrogen estimate (forthe same number density) from our calculations takinginto account the number density scaling is ∆n hydrogen =( N/V )( χ/ ( ǫ c )) ∼ ( N/V )( e /ǫ mω ) β ∼ − . It is alsoworth mentioning that | χ EBij ( ω ) | , as for the harmonic oscil-lator, will scale as size of the system since it is proportionalto the standard polarisability. For the helium atom, thestatic polarisability is α helium (0) = 0 . × − Coulombmeter /Volt, which when divided by ǫ gives a volume of16 . a . For the hydrogen atom the corresponding volumeis 4 a from which we find a scaling factor of approximately four between the hydrogen and helium. The precise formhowever would have to be fitted empirically with the helpof DFT calculations in order to match on to experimentalvalues such as found in [3]. We have presented an exact quantum mechanical per-turbation theory calculation of the magneto-electric re-sponse function for atomic systems with the simplest bind-ing potentials, namely the harmonic oscillator and theCoulomb potential. We have deduced analytic forms forthe magneto-electric response tensor as a function of fre-quency that can be calculated exactly. A common featureis that at high frequency they have 1 /ω behaviour whilstat low frequency they tend to a constant value. It wouldbe interesting to try and apply the same method exactlyto the helium atom.There are interesting implications of this calculation.Concerning the Feigel effect [5], where a net momentumdensity for a medium (Equation (21) in [5]) with anmagneto-electric response function is developed, the re-sult found there is fourth power divergence in frequency,which is then simply cut-off. In fact it is only the anti-symmetric part of the magneto-electric response tensorthat contributes to the momentum. This neglected how-ever the dependence on frequency and was treated as aconstant. What we see now is that this divergence will besoftened to a quadratic divergence though it will not besimply washed away altogether. This shows that the as-sumption of a cut-off at high frequencies for ME is notjustified and that the divergence has to be resolved byother means, such as done recently in [10]. Acknowledgements
We would like to thank Geert Rikken for useful discussions.This work was supported by the ANR contract PHOTONIM-PULS ANR-09-BLAN-0088-01.
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