Generalized electromagnetic theorems for non-local plasmonics
GGeneralized electromagnetic theorems for non-local plasmonics ´Emilie Sakat, ∗ Antoine Moreau, and Jean-Paul Hugonin Universit´e Paris Saclay, Center for Nanoscience and Nanotechnology,C2N UMR9001, CNRS, 91120 Palaiseau, France Universit´e Clermont Auvergne, CNRS, Institut Pascal, F-63000 Clermont-Ferrand, France Universit´e Paris-Saclay, Institut d’Optique Graduate School,CNRS, Laboratoire Charles Fabry, 91127 Palaiseau, France
The ultraconfined light of plasmonic modes put their effective wavelength close to the mean freepath of electrons inside the metal electron gas. The Drude model, which can not take the repulsiveinteractions of electrons into account, then clearly begins to show its limits. In an intermediate lengthscale where a full quantum treatment is computationally prohibitive, the semiclassical hydrodynamicmodel, instrinsically nonlocal, has proven successful. Here we generalize the expression for theabsorption volume density and the reciprocity theorem in the framework of this hydrodynamicmodel. We validate numerically these generalized theorems and show that using classical expressionsinstead leads to large discrepancies.
Plasmonic mode volumes, which are several ordersof magnitude smaller than the cubic wavelength, offerunique opportunities to produce local heating [1], to en-hance chemical reactions at a precise location [2, 3] orto increase the spontaneous emission rate of a quan-tum emitter associated to a cavity [4–6]. These deeplysubwavelength volumes occur because light is noticeablyslowed down when it propagates in the vicinity of metals,leading to slow guided modes [7]. At these ultraconfinedscales, the effective wavelength of the plasmonic guidedmodes gets close to the mean free path of free carriersinside the metal electron gas, so that the Drude model,which can not take such interactions into account, clearlybegins to show its limits [8, 9].A precise modeling of the light-matter interactionsin these systems requires an accurate description ofboth quantum effects and far-field radiation. Time-dependent density functional theory (TD-DFT) [10] orsingle-band theory in the Random-Phase Approxima-tion (RPA) [11] can provide full first-principles quantumtreatment. However, they become computationally pro-hibitive for sizes that exceed a few nanometers and forhigh density of carriers. At the intermediate length scale,the semiclassical hydrodynamic Drude model (HDM) hasproven successful to describe experimental results [8, 12–14]. In the framework of this model, the optical responseof metals cannot be described simply by a local permit-tivity any more so that they are called spatially dispersiveor nonlocal.Losses are both sought after for many applications ofplasmonics like sensing, heating or even electro-opticalmodulation [15], while being often responsible for the lim-itations of ultimately miniaturized plasmonic devices [16]– in which case minimizing losses is crucial. It is thus ofparamount importance to be able to compute accuratelythe local absorption inside nanostructures for which spa-tial dispersion (non-local effects) has to be taken into ac- ∗ [email protected] count. The standard expression for the local absorptionin the harmonic regime α abs ( r , ω ) = ω Im( ε ( ω )) | E ( r , ω ) | , (1)where ω is the frequency, ε the dielectric permittivity ofthe material and E the total electric field, is obviouslynot valid anymore.The lack of a correct formula has led many authors tolook for a work-around or to rely only on far-field quanti-ties, as it hinders from computing the actual absorptioncross-section by integrating directly the absorption den-sity [9, 13, 14, 17–23]. Furthermore, given the fact thatthe electric field in the framework of the hydrodynamicmodel contains a longitudinal component, the regular ex-pression of the reciprocity theorem can not be relied oneither. This is a source of concern, as many other prop-erties or laws are proven using this theorem (e.g. theKirchhoff law), and it is routinely used to check the ac-curacy of numerical methods [24].In this work, we first derive a formula for the absorp-tion volume density that is valid even in a non-localmedium. Then, we derive a generalized version of thereciprocity theorem valid also when one of the sourcesis located inside a non-local medium. The expression weobtain corresponds to the classical expression for the the-orem except only the transverse component of the elec-tric field should be considered. Finally, our theoreticalresults are validated numerically, using simulation meth-ods which integrate the hydrodynamic model rigorously.These results show that different expressions (like Eq.(1))for the absorption volume density may lead to a largeerror when computing the absorption cross-section of aspherical nanoparticle. I. HARD-WALL HYDRODYNAMIC MODEL
Drude’s model is first based on the idea that the den-sity of volume currents can be integrated as an effec- a r X i v : . [ phy s i c s . op ti c s ] F e b tive polarization P f of the medium, using the relation j = ∂ P f ∂t . A metal can thus always be described withsuch a polarization. The following assumption is thatthe movement of electrons is dependent only on the lo-cal electric field and that the repulsion between electronsinside the metal can be neglected. Such an hypothesisholds only if the typical scale of the field variation islarge in comparison to the mean free path of electrons.However, in plasmonics guided modes typically tend topresent a high effective index and thus very short effec-tive wavelength. Therefore, the local hypothesis does nothold any more.The hydrodynamic model [19] is the most simple pos-sible way to take into account the repulsion between elec-trons, by integrating a pressure term in the descriptionof the electron gas response, which leads to the follow-ing relation between the electric field and the effectivepolarization: ∂ P f ∂t + γ ∂ P f ∂t − β ∇ ( ∇ . P f ) = ε ω p E . (2)The electron pressure term β ∇ ( ∇ . P f ) obviously in-cludes spatial derivatives of the polarization, making thedescription non-local. However, even in the framework ofthe hydrodynamic model, the quantity χ f = − ω p ω + iγω ,which represents the local susceptibility of the Drudemodel plays an important role in the following.The parameter β quantifies the non-local effects andis called the hydrodynamic parameter. It can be definedas β = (cid:112) / v F with v F = (cid:126) m ∗ (3 π ) / n / , the Fermivelocity [19].The bound electrons contribute to the response ofthe metal, however, their response can be considered aspurely local and the corresponding polarization written P b = ε χ b E ( r ). We can thus write the nullity of thedivergence of the displacement field: ∇ . D = ε ∇ . E + ∇ . P f + ∇ . P b = 0 (3)in order to express and inject ∇ . P f in Eq.2. In the har-monic regime, the following non-local expression for P f is thus obtained: P f = − ε ω p ω + iγω (cid:20) E − β (1 + χ b ) ω p ∇ ( ∇ . E ) (cid:21) . (4)Then, Amp`ere’s circuital law ∇ × H = − iω D = − iω ( ε E + P f + P b ) yields ∇× H = − iωε (cid:20) (1 + χ b ) E + P f ε (cid:21) = − iωε ε [ E − α ∇ ( ∇ . E )](5)with α = χ f ( ε − χ f ) β εω p and ε =1 + χ f + χ b .According to Helmholtz’s theorem, any sufficientlysmooth, rapidly decaying vector field in three dimensionscan be resolved into the sum of an irrotational vector field (the longitudinal component) and a divergence-free vec-tor field (the transverse component).In a non-local medium, E can always be written as: E = E + eH = H ( E , H ) corresponding to a transverse wave and e to alongitudinal wave. By definition ∇ × H = − iωε ε E and ∇ . E =0. Thus, by using the second term of Eq.5, we canwrite in the non-local media: ∇ × ( H − H ) = 0 = − iωε ε [( E − E ) − α ∇ ( ∇ . E )]0 = − iωε ε [ e − α ∇ ( ∇ . e )]If we now define ρ such as ρ = α ∇ . E , the above equationgives e = ∇ ρ (6)and given that ∇ . E = ∇ . E + ∇ . e , another HDM funda-mental relation is obtained:∆ ρ = ρα (7) n ε + ε Non-local medium E + =E E - =E - + ∇ ρ V e S(V e ) Figure 1. Non-local medium of permittivity ε inserted in alocal background medium of permittivity ε + which is not nec-essarily spatially uniform. With these definitions, and in particular e = α ∇ ( ∇ . e ),Eq. 5 leads to the following expression: P f ε = χ f E − ε e = χ f E − ( ε − χ f ) e (8)In the usual implementation of the HDM, electron tun-neling or electron density spill-out effects are neglected.This leads to the hard-wall boundary condition, statingthat P f . n =0 [9]. In this paper, we will limit ourselvesto this hypothesis even if few works have proposed to gobeyond [10, 20]. Within this framework, the continuityconditions for the problem illustrated on Fig. 1 are:1. the tangential components of E and H are continuous,2. the normal component of ε E is continuous,3. the normal component of ( ε − χ f ) E is continuous.These hard-wall boundary conditions along with Eq.8and Eq.6 imply one last fundamental relation of the HDMmodel at the non-local/local interface: ∇ ρ. n = χ f ( ε − χ f ) E − . n (9) II. NON-LOCAL ABSORPTION VOLUMEDENSITY
In order to describe the absorption power in the non-local medium with the hard-wall HDM model describedabove, we can write the surface integral of the Poyntingvector flux around the non-local medium: P abs ( ω ) = − ‹ S ( V e ) Re( E ∗ × H ) . n . (10)Given the continuity of the tangential components ofthe electromagnetic fields ( E , H ), this integral can be cal-culated indifferently on the inner or on the outer contourof the non-local media. On the inner contour of the in-tegral, Eq. 10 gives P abs ( ω ) = − ‹ S ( V e ) Re( E ∗ × H ) . n − ‹ S ( V e ) Re( ∇ ρ ∗ × H ) . n (11)The first term of this equation is straightforward to ex-press since E and H satisfy the classical Maxwell equa-tion: − ‹ S ( V e ) Re( E ∗ × H ) . n = ωε ˆ V e Im( ε ( ω )) | E ( r , ω ) | d r (12)Now let us consider the second term in Eq. 11. ∇ . ( ∇ ρ ∗ × H ) = −∇ ρ ∗ . ( ∇ × H ) + ( ∇ × ∇ ρ ∗ ) . H (13)By definition ∇ × ∇ ρ ∗ =0. Thus Eq. 13 reduces to: ∇ . ( ∇ ρ ∗ × H ) = ∇ ρ ∗ . i ωε ε E (14)If we now apply the Ostrogradski theorem to expressthe second term of Eq. 11, we obtain: − ‹ S ( V e ) Re( ∇ ρ ∗ × H ) . n = − ˆ V e Re ( ∇ ρ ∗ .iωε ε E )(15)Eq. 11 can thus be rewritten as: P abs ( ω ) = ωε (cid:20) ˆ V e Im( ε ) | E | − ˆ V e Re [ iε ∇ ρ ∗ .E ] (cid:21) = ωε (cid:20) ˆ V e ε E ∗ . E + ˆ V e ε ∇ ρ ∗ . E (cid:21) = ωε (cid:20) ˆ V e ε E ∗ . E (cid:21) (16)giving a non-local expression for the absorption volumedensity: α abs ( r , ω ) = ωε ε E ∗ . E ) (17) This generalized expression is different from the localone given Eq. 1. But if V e contains only local media,no longitudinal component exists, E = E and the two ex-pressions are equivalents. One important point to noticewith this generalized expression of the absorption poweris that unlike Eq. 1, it can be negative. To understandthe necessary conditions on the material properties tomaintain the integral positive, this generalized expres-sion can be rewritten differently (see demonstration inAppendix A). P abs ( ω ) = ωε ˆ V e (cid:32) Im( ε − χ f ) | E | + Im( χ f ) (cid:12)(cid:12)(cid:12)(cid:12) P f ε χ f (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) (18)This second expression highlights clearly that ifIm( ε − χ f ) and Im( χ f ) are positive, which is in theoverwhelming majority of cases true for metals, thisgeneralized expression for the absorption power remainspositive. One has to be careful with this second formulathough. Integrands of Eq. 16 and Eq. 18 are indeednot equivalents. The demonstration of the second oneincludes the condition P f . n =0, which is true only on thenon-local/local interface, while the demonstration of Eq.16 can be done on any closed volume inside the non-localmedia. Thus, the two integrals are equals only for theclosed volume surrounded by the contour where P f . n =0.For other closed volumes inside the non-local media, thisis not the case. In fact, only the integrand of Eq. 16 isan absorption volume density: it is the unique expressiongiving an integral equal to the Poynting flux surfaceintegral on any closed volume inside the non-local media.Now, we illustrate the difference between the classicalformula (1) and the non-local expression (17) on anexample. Figure 2 compares absorption volume densitymaps calculated with these two formula for a 5-nm z ( n m ) a) Local, β =0 ε E * . E ε )| E | z ( n m ) b) 50-550-5 x(nm) z ( n m ) c)010302040 ε )| E | β =1.15. 10 m.s -1 β =1.15. 10 m.s -1 Figure 2. Absorption volume density α abs ( r ) maps for a5-nm radius nanosphere embedded in a background of re-fractive index n=1.5 at λ =2.4 µ m. The HDM parametersfor the nanosphere’s permittivity are: ε ∞ = 1 + χ b =4, ω p =1 . . rad.s − , γ = 1 . . rad.s − . a) classicalMaxwell formula in the local case, i.e. β = 0; b) classicalMaxwell formula but the field E = E + e is derived withinthe HDM framework, β =1 . . m.s − ; c) formula (17), β =1 . . m.s − . The absorption volume density is normal-ized by the norm of the Poynting vector of the incident planewave. The sphere is illuminated by a plane wave polarizedalong the x -axis and propagating at normal incidence alongthe z axis. radius nanosphere embedded in a background mediumof refractive index n = 1 .
5. The permittivity chosenfor the nanosphere is realistic and corresponds to ahighly doped n-ITO nanocrystal. Figure 2a is calculatedusing Eq. 1 with the local Drude model. Figure 2b iscalculated using Eq. 1, but within the HDM framework,i.e. | E | = | E + ∇ ρ | . Figure 2c is calculated withEq. 17. Figure 2 clearly shows that while, in the localframework, electromagnetic field would be evaluated asquasi constant everywhere in the sphere (Fig. 2a), inthe HDM framework on the contrary it is not the case,and the correct expression show that absorption occurscloser to the surface of the metal (Fig. 2c) than theclassical formula would lead to expect (Fig. 2b).Figure 3 represents the losses computed for the same5-nm nanosphere by integrating different expressions forthe absorption volume density α abs , as a function of thewavelength. Only expression (17) allows to retrieve thecorrect value for the absorption cross-section computedusing the flux of the Poynting vector. Assuming α abs = Im( ε ) | E | yields a σ abs value more than twice as lowas the actual value; choosing α abs = Im( ε ) | E | gives avalue almost six times too large. Wavelength (µm)
042 10 -5 ε ) |E| ε ) |E | ε E * . E )Poynting Flux ext.Poynting Flux int. σ a b s ( µ m ) Figure 3. Absorption cross-section as a function of the wave-length for the same non-local nanosphere as Fig.2, assumingdifferent expressions for the absorption volume density. Onlythe value obtained using Eq. 17 matches the surface integralof the Poynting vector flux on the inner or outer contourssurrounding the non-local medium.
This new expression is thus the only way to estimatelocally and accurately the losses in a non-local media.Given the importance of understanding where losses oc-cur in any deeply subwavelength plasmonic structure, wethink such an expression could prove very useful in thefuture by providing a more accurate physical picture ofthe phenomenon at play, like quenching [25].
III. RECIPROCITY THEOREM WITHNON-LOCAL MEDIA
The reciprocity theorem allows to connect the electro-magnetic fields generated by two different and arbitrarypoint sources. By assuming time-harmonic fields in lin-ear and local media in which the tensors ε and µ aresymmetric (reciprocal materials), the reciprocity theo-rem between two punctual time-harmonic source currents( j , j m ) and ( j , j m ) located at positions r and r andwhich emit respectively the fields ( E , H ) and ( E , H )can be written: j E ( r ) + j m H ( r ) = j E ( r ) + j m H ( r ) . (19) PML ( j ,j m1 ) p e ( E ,H )( r ) n V e S + n V e ( E ,H )( r ) S - S + S - b) PML r r a) S ∞ S ∞ ( j ,j m2 ) E - =E - + ∇ ρ E - =E - + ∇ ρ E + =E + E + =E + Figure 4. A punctual source current ( j , j m ) located at a po-sition r radiates, illuminating the non-local medium. It cre-ates an electromagnetic field ( E , H ). (b) A punctual sourcecurrent ( j , j m ) located at a position r inside the non-localmedium radiates an electromagnetic field ( E , H ). In the case where a non-local media is introduced, thereciprocity theorem can still be written but not underthe same form. To find the right expression for the non-local reciprocity theorem, we consider the two problemsdescribed in Fig. 4, where a source is located inside thenon-local medium while the other is placed in the sur-rounding (local) medium. The Perfect Matched Layers(PML) around the object shown Fig. 4 are here only fornumerical purposes, in order to take into account the ra-diation boundary conditions at infinity. Two solutions ofMaxwell’s equations in the HDM framework can thus beconsidered. In both case inside V e : E = E + ∇ ρ E = E + ∇ ρ H = H H = H The Lorentz reciprocity formula, which relates twotime-harmonic solutions of Maxwell’s equations can stillbe written on the transverse component E (a generalform of Lorentz reciprocity formula and its derivationcan be found for instance in Annex 3 of Ref. [26]). Herewe use this Lorentz reciprocity formula for the two trans-verse components E at the same frequency ω , (i) on theclosed surface surrounding the volume outside the non-local medium and located in between the contour S ∞ located at infinity and the contour S + corresponding to x(nm) z ( n m ) a) b) x(nm) z ( n m ) n=4n=2n=1.5 n=3.4n-ITO( j ,j m1 ) n=4n=2n=1.5 n=3.4n-ITO( j ,j m2 ) ( j ,j m1 )( j ,j m2 ) Figure 5. Calculations performed with a multipole method[27]. The electromagnetic field maps are plotted in log scaleat λ =2.4 µ m. Non-local reciprocity theorem verification witha complex electromagnetic environment composed of a 20-nmradius core-shell with a core of the same non-local medium (re-alistic permittivity of n-ITO) as in Fig. 2 and a ligands shellof refractive index n=2 and 5-nm thickness. This core-shellis deposited on a silicon substrate (n=3.4) and is embeddedin a background of n=1.5. In the vicinity of the core-shell isplaced another nanosphere of refractive index n=4. In thiscomplex environment, two punctual source currents ( j , j m )and ( j , j m ) are inserted randomly. The reciprocity theoremof Eq.21 is verified with 16 digits of precision. the outer contour of the non-local medium, and (ii) onthe closed surface S − surrounding the non-local volume V e (see Fig. 4). The surface integral on S ∞ being nullby construction, this gives: − ‹ S + ( E × H − E × H ) .dS n = j E ( r ) + j m H ( r ) ‹ S − ( E × H − E × H ) .dS n = − [ j E ( r )+ j m H ( r )]We can now use the field continuity conditions to relatethese two surface equations. It is in particular possibleto show that (see Appendix B): ‹ S + ( E × H − E × H ) .dS n = ‹ S − ( E × H − E × H ) .dS n (20)which immediately leads to the non-local reciprocity the-orem: j E ( r ) + j m H ( r ) = j E ( r ) + j m H ( r ) (21)Note that in the non-local medium, unlike Eq. 19,the reciprocity theorem is valid only for the transversecomponent E of the field and not for the total field E .As a consequence, the Green tensor is symmetrical for E but not for the total field E .This derivation has been done for the case where onesource is outside and one source is inside the non-localmedium but it is still valid if the two sources are in thesame medium, whether it is local or not. We underline that placing a punctual source infinitely far is a straight-forward way to generalize this non-local reciprocity the-orem for a plane wave and a punctual source. Finally, itshould be stressed that the reciprocity theorem we havederived gives the same expression as the classical theo-rem Eq. 19 when the two points considered are locatedin the surrounding local medium, which is not necessarilyobvious in presence of nonlocal media.We have numerically checked that this non-localreciprocity theorem is satisfied for a complex setupincluding a non-local core-shell nanostructure (see Fig.5). We place the sources randomly, one inside thenonlocal medium, and the other one outside. The elec-tromagnetic field maps created by these sources are thencomputed using a multipole method [27]. The reciprocitytheorem of Eq.21 is verified with 16 digits of precision re-gardless of the position or polarization of the two sources. IV. CONCLUSION
We have derived here rigorous formulas which allowto generalize the classical expressions for the absorptionvolume density and the reciprocity theorem beyond theDrude model, to the case where non-local media de-scribed by the hydrodynamic model are present. Ournumerical simulations show that the absorption volumedensity can significantly differ from classical predictionswhen nonlocality is taken into account. We underlinealso that our results can prove particularly useful tocheck that numerical methods based on the hydrody-namic model [18, 28] are indeed accurate, as done in thepresent work.For a long time, nonlocality has been expected to playa role only in the tiniest metallic nanoparticles [29]. Re-cent results point towards a larger influence of nonlocal-ity than previously expected, for much larger structures[22, 23], and for semiconductors even more than for met-als [21, 30]. Moreover, the recent trend towards minia-turized plasmonic devices [16, 31] means that nonlocalitywill have, increasingly often, to be taken into account.Given the large number of situations in which local ab-sorption plays a crucial role (photothermal therapy, localchemical reaction catalysis, HAMR,... etc.) we think ourwork could simply allow to better understand and finallyto help design plasmonic nanostructures.
ACKNOWLEDGMENTS
This work was supported by French state funds man-aged by the ANR within the Investissements d’Avenirprogrammes under reference ANR-20-CE24-00XX MO-SAIC and 16-IDEX-0001 CAP 20-25. The authorsthanks also Philippe Lalanne for the fruitful discussionsat the beginning of this project. Antoine Moreau is anAcademy CAP 20-25 chair holder.
Appendix A: Absorption
In this section, we show the derivation allowing to ob-tain Eq. 18 of the main text. By using Eq. 8, Eq. 16 ofthe main text can be written as a function of P f : P abs ( ω ) = ωε (cid:20) ˆ V e ε | E | − ε E ∗ e (cid:21) = ωε (cid:20) ˆ V e ( ε − χ f ) | E | + E ∗ P f ε (cid:21) = ωε (cid:34) ˆ V e ( ε − χ f ) | E | + 1 χ ∗ f (cid:12)(cid:12)(cid:12)(cid:12) P f ε (cid:12)(cid:12)(cid:12)(cid:12) + ε ∗ e ∗ χ ∗ f P f ε (cid:35) The third term of this last equation can be expressedthanks to the Ostrogradski theorem and the hard-wallboundary conditions: ‹ S ( V e ) ρ ∗ ( P f ε . n ) =0 = ˆ V e ∇ . ( ρ ∗ P f ε ) (A1)0 = ˆ V e e ∗ P f ε + ρ ∗ ∇ . P f ε (A2)Then by using Eq. 8 along with Eq. 7 and by express-ing α : ∇ . P f ε = χ f ∇ . E − ( ε − χ f ) ∇ . e = − ( ε − χ f )∆ .ρ = − ( ε − χ f ) ρα = − εχ f ω p β ρ which leads to the final expression of the absorptionpower inside V e corresponding to Eq. 18 of the maintext: P abs ( ω ) = ωε (cid:34) ˆ V e ( ε − χ f ) | E | + 1 χ ∗ f (cid:12)(cid:12)(cid:12)(cid:12) P f ε (cid:12)(cid:12)(cid:12)(cid:12) + ε χ f ω p β | ρ | (cid:35) = ωε (cid:34) ˆ V e ( ε − χ f ) | E | + 1 χ ∗ f (cid:12)(cid:12)(cid:12)(cid:12) P f ε (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) = ωε ˆ V e (cid:32) Im( ε − χ f ) | E | + Im( χ f ) (cid:12)(cid:12)(cid:12)(cid:12) P f ε χ f (cid:12)(cid:12)(cid:12)(cid:12) (cid:33) Appendix B: Reciprocity
Eq. (20) relates two surface integrals, one outside andone inside the non-local medium. For the sake of com-pleteness, we give here the derivation leading to this re-lation. In particular, thanks to the continuity of the tan-gential components of E and H , it is possible to writethe following expression: ‹ S + ( E × H − E × H ) .dS n = ‹ S − ( E × H − E × H ) .dS n (B1)By definition of the field inside the non-local medium,the second term of this equation can also be written as: ‹ S − ( E × H − E × H ) .dS n + ‹ S − ( ∇ ρ × H −∇ ρ × H ) .dS n (B2)To establish (20), we demonstrate first that the secondterm of Eq. (B2) is null, by noticing that ∇ ρ × H = ∇ × ( ρ H ) − ρ . ∇ × H = ∇ × ( ρ H ) − ρ . ( − iωε ε E + j δ ( r − r )) . Symmetrically, we can write ∇ ρ × H = ∇ × ( ρ H ) − ρ . ( − iωε ε E + j δ ( r − r )) . Using the Ostrogradski theorem then yields ‹ S − ∇ × ( ρ i H j ) .dS n = ˆ V e ∇ . ( ∇ × ( ρ i H j )) dV = 0 , so that the second term in Eq. 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