Extension of Lorentz Reciprocity and Poynting Theorems for Spatially Dispersive Media with Quadrupolar Responses
11 Extension of Lorentz Reciprocity and PoyntingTheorems for Spatially Dispersive Media withQuadrupolar Responses
Karim Achouri and Olivier J. F. Martin
Abstract —We provide a self-consistent extension of the Lorentzreciprocity theorem and the Poynting theorem for media pos-sessing electric and magnetic dipolar and quadrupolar responsesrelated to electric and magnetic fields and field gradients. Usingthese two theorems, we respectively deduce the conditions ofreciprocity and gainlessness and losslessness that apply to thevarious tensors mediating the interactions of these multipole mo-ments and the associated fields and field gradients. We expect thatthese conditions will play an essential role in developing advancedmetamaterial modeling techniques that include quadrupolar andspatially dispersive responses.
Index Terms —Lorentz reciprocity theorem, Poynting theorem,reciprocity, energy conservation, quadrupoles, metamaterials.
I. I
NTRODUCTION
The expansion of the electromagnetic fields induced bycharges and current distributions in terms of multipolar mo-ments plays an essential role in studying the interactionsof light with matter [1], [2]. It has been widely used, forinstance, in the electromagnetic characterizations of cubicand nonmagnetic crystals [3]–[5] and the studies of opticaleffects such as optical activity and circular birefringence inchiral media [6]. Historically, these studies have shown that,while dipolar approximations may be sufficient in some cases,it is often necessary to extend the multipolar expansion tohigher order terms, such as quadrupoles or even octupoles, inorder to properly assess the electromagnetic properties of theconsidered structures and model their optical response.In more recent years, we have witnessed the advances in thefield of metamaterials and metasurfaces research that have ledto a myriad of concepts and applications [7]–[12]. In the caseof metasurfaces, most of the design and modeling techniquesthat pertain to their synthesis and analysis are almost entirelybased on dipolar approximations, since the scattering particlesthat compose them are small enough compared to the wave-length that dipolar approximations are sufficient to model theirresponses [13]–[15]. However, the extreme number of degreesof freedom enabled by the ability to engineer the metasurfacescattering particles may be leveraged to induce non-negligiblehigher order multipolar components for additional field controlcapabilities, the necessity to control the angular scatteringresponse of metasurfaces for analog signal processing and thefact that dielectric metasurfaces are composed of dielectricresonators that are electrically large enough to exhibit at leastelectric quadrupolar responses prompt the need to improvethe existing modeling techniques by including higher ordermultipolar components [16]–[24]. However, the theoretical expressions derived in [1]–[5], [25]have been limited to rather few multipole moments and re-lated field interactions. Moreover, the permutation symmetries,which are associated to the multipolar components consideredin these studies and that correspond to reciprocity conditions,have been only derived based on quantum mechanical con-cepts.In this work, we shall overcome these limitations by provid-ing a multipolar expansion that includes electric and magneticdipolar and quadrupolar moments with components related tothe electric and magnetic fields as well as their first orderderivatives. Additionally, we also provide the permutationsymmetries that are associated to reciprocity and energy con-servation based on a purely electromagnetic derivation of theLorentz reciprocity theorem and the Poynting theorem.For self-consistency, we shall first review the fundamentalconcepts of the theory of multipoles and that of spatial disper-sion in Secs. II-A and II-B, respectively. Next, we derive theLorentz reciprocity theorem in Sec. III from which we derivethe associated conditions of reciprocity. Then, in Sec. IV, wederive the Poynting theorem and deduce the conditions ofgainlessness and losslessness, and finally conclude in Sec. V.II. T
HEORETICAL B ACKGROUND
A. Theory of Multipoles
The electric and magnetic responses of a material or amedium are conventionally expressed in terms of constitutiverelations [26], [27]. In this work, we will consider that theseconstitutive relations are given by D and B as functions of E and H . These relations are generally derived from a multipolarexpansion of the current density induced in the material [26],[28]. For completeness, we next provide a brief derivation ofthis multipolar expansion.The typical procedure to expand the electric current density J in terms of multipoles is to start from the definition of thevector potential A from which the magnetic field is definedas B = ∇ × A and that is given by [26], [28] A ( r ) = µ π (cid:90) J ( r (cid:48) ) e − jk | r − r (cid:48) | | r − r (cid:48) | dV (cid:48) , (1)where k = 2 π/λ and the time-harmonic e jωt is assumed butomitted throughout. Using a Taylor expansion , the free-space We use the following definition of the Taylor expansion: f ( r + a ) = (cid:80) ∞ n =0 1 n ! ( a · ∇ ) n f ( r ) [28]. a r X i v : . [ phy s i c s . op ti c s ] F e b Green function in (1) is expand into e − jk | r − r (cid:48) | | r − r (cid:48) | = ∞ (cid:88) n =0 n ! ( − r (cid:48) · ∇ ) n e − jkr r . (2)Substituting (2), truncated at n = 2 , into (1) yields A ( r ) = µ π (cid:20) (cid:18)(cid:90) J ( r (cid:48) ) dV (cid:48) (cid:19) − (cid:18)(cid:90) J ( r (cid:48) ) r (cid:48) dV (cid:48) (cid:19) · ∇ +12 (cid:18)(cid:90) J ( r (cid:48) ) r (cid:48) r (cid:48) dV (cid:48) (cid:19) : ∇∇ (cid:21) e − jkr r . (3)We next highlight the fact that the dyadic defined by J ( r (cid:48) ) r (cid:48) in (3) may be decomposed into symmetric and antisymmetricparts respectively as J ( r (cid:48) ) r (cid:48) = 12 [ J ( r (cid:48) ) r (cid:48) + r (cid:48) J ( r (cid:48) )] + 12 [ J ( r (cid:48) ) r (cid:48) − r (cid:48) J ( r (cid:48) )] , (4)which plays an important role in simplifying (3) for thefollowing steps.The integrals in brackets in (3) can now be individuallyassociated to a specific multipole moment. By using (4), wethus define the electric dipole P , the magnetic dipole M , theelectric quadrupole Q and the magnetic quadrupole S as [4] P i = 1 jω (cid:90) J i ( r (cid:48) ) dV (cid:48) , (5a) M i = 12 (cid:90) (cid:2) r (cid:48) j J k ( r (cid:48) ) − r (cid:48) k J j ( r (cid:48) ) (cid:3) dV (cid:48) , (5b) Q ij = 1 jω (cid:90) (cid:2) J i ( r (cid:48) ) r (cid:48) j + J j ( r (cid:48) ) r (cid:48) i (cid:3) dV (cid:48) , (5c) S ij = 23 (cid:90) (cid:2) r (cid:48) j J k ( r (cid:48) ) − r (cid:48) k J j ( r (cid:48) ) (cid:3) i r (cid:48) j dV (cid:48) . (5d)Note that the magnetic quadrupole corresponds to the anti-symmetric part of J ( r (cid:48) ) r (cid:48) r (cid:48) in (3), whereas its symmetric partis associated to the electric octupole, which is not shown here.Now, by association between (3) and (5), the electric currentdensity may be written as [25], [28] J = jω P + ∇ × M − jω Q · ∇ − ∇ × ( S · ∇ ) . (6)The constitutive relations may now be derived by consid-ering the frequency-domain Maxwell equations expressed interms of the fundamentals fields E and B given by [4], [26] ∇ × E = − jω B , (7a) ∇ × B = µ ( J + jω(cid:15) E ) . (7b)Substituting (6) into (7b) and re-arranging the terms, yields ∇ × (cid:18) µ − B − M + 12 S · ∇ (cid:19) = jω (cid:18) (cid:15) E + P − Q · ∇ (cid:19) . (8)Associating the first bracket in (8) to H and the second oneto D , we finally obtain D = (cid:15) E + P − Q · ∇ , (9a) B = µ (cid:18) H + M − S · ∇ (cid:19) . (9b) B. Spatial Dispersion
Spatial dispersion describes the spatially nonlocal nature ofelectromagnetic material responses [25], [29]. It implies, forinstance, that the electric current density J may be expressedas the convolution of an exciting electric field E with thecurrent response of the material K as [25], [29] J ( r ) = (cid:90) K ( r − r (cid:48) ) · E ( r (cid:48) ) dV (cid:48) . (10)We shall now approximate this expression by expanding theelectric field as E ( r (cid:48) ) = E ( r )+( r (cid:48) − r ) ·∇ E ( r )+ 12 [( r (cid:48) − r ) · ∇ ] E ( r ) , (11)which, upon insertion in (10), yields J i = b ij E j + b ijk ∇ k E j + b ijkl ∇ l ∇ k E j , (12)where b ij corresponds to (cid:82) K ( r − r (cid:48) ) , b ijk corresponds to (cid:82) K ( r − r (cid:48) )( r (cid:48) − r ) , and so on. Expression (12) shows how thecurrent is related to the electric fields and its derivatives, whichis a typical effect of spatial dispersion that implies that theinduced current, for instance in a uniform and homogeneousmedium, depends on the direction of wave propagation.It is particularly important to note that the terms related tothe derivatives of the electric field in (12) may be transformedto make the magnetic field H appear. Indeed, consider, forinstance, the second term on the right-hand side of (12), whichmay be split into symmetric and antisymmetric parts as b ijk = b sym ijk + b asym ijk . Then, considering that an antisymmetric third-rank tensor may be represented by a second-rank pseudotensorimplying the dual quantity b asym ijk ∝ ε ljk g il [30], where ε ijk isthe Levi-Civita symbol and g ij is a second-rank tensor, weobtain b ijk ∇ k E j = (cid:16) b sym ijk + b asym ijk (cid:17) ∇ k E j , = (cid:18) b sym ijk + jωµ ε ljk g il (cid:19) ∇ k E j , = b sym ijk ∇ k E j + g ij H j , (13)where we have used the fact that in vacuum (7a) may, using B = µ H , be written as ε ijk ∇ k E j = − jωµ H i . Relation (13)shows that the dependence of J on the magnetic field is onthe same order as its dependence on the first order derivativeof the electric field. Similarly, the last term in (12) may betransformed into two terms, one related to the first derivativeof H and one to the second derivative of E [25].From a general perspective, the convolution (10) and itsexpansion (12) may also be performed for any of the multipolemoments in (5) [25]. Accordingly, we now expand thesequantities in terms of the electric field, the magnetic field andtheir first order derivatives, which leads to P i M i Q il S il = χ · E j H j ∇ k E j ∇ k H j , (14) We use the three first terms of the Taylor expansion defined as f ( r (cid:48) ) = (cid:80) ∞ n =0 1 n ! [( r (cid:48) − r ) · ∇ ] n f ( r ) . where the hypersusceptibility tensor χ is given by (cid:15) χ ee ,ij c χ em ,ij (cid:15) k χ (cid:48) ee ,ijk c k χ (cid:48) em ,ijk η χ me ,ij χ mm ,ij η k χ (cid:48) me ,ijk k χ (cid:48) mm ,ijk(cid:15) k Q ee ,ilj c k Q em ,ilj (cid:15) k Q (cid:48) ee ,iljk c k Q (cid:48) em ,iljk η k S me ,ilj k S mm ,ilj η k S (cid:48) me ,iljk k S (cid:48) mm ,iljk . (15)Note that we have normalized each tensor in (15) so as to bedimensionless. As can be seen, we retrieve the conventionalbianisotropic susceptibility tensors χ ee , χ mm , χ em , and χ me ,and a plethora of other terms relating the fields and theirgradient to dipolar and quadrupolar responses.The hypersusceptibility tensor χ in (14) contains a totalnumber of 576 components. However, several of these compo-nents are not independent from each other. Indeed, inspectingthe definition of the electric quadrupole moment in (5c) revealsthat Q ij = Q ji , which directly implies that Q ee ,ijk = Q ee ,jik and Q em ,ijk = Q em ,jik . (16)Additionally, we know that the tensors in (15), which arerelated to the derivative of the electric field, are symmetric,as demonstrated in (13). This results in the permutability oftheir two last indices and thus implies that χ (cid:48) ee ,ijk = χ (cid:48) ee ,ikj and χ (cid:48) me ,ijk = χ (cid:48) me ,ikj . (17)The tensor Q (cid:48) ee ,ijkl combines both the symmetry of Q ij , whichaffects its two first indices, and the permutation symmetryof its two last indices due to the fact that it is related to aderivative of the electric field, we thus have that Q (cid:48) ee ,ijkl = Q (cid:48) ee ,jikl = Q (cid:48) ee ,ijlk = Q (cid:48) ee ,jilk . (18)Since there is no symmetry associated with the magneticquadrupole S or with the derivative of H , the tensors Q (cid:48) em ,ijkl and S (cid:48) me ,ijkl exhibit only the partial permutation symmetries Q (cid:48) em ,ijkl = Q (cid:48) em ,jikl and S (cid:48) me ,ijkl = S (cid:48) me ,ijlk , (19)whereas the tensors χ (cid:48) em ,ijk , χ (cid:48) mm ,ijk , S me ,ijk , S mm ,ijk and S (cid:48) mm ,ijkl exhibit no permutation symmetry at all. Taking intoaccount all of these permutation symmetries, the number ofindependent components in χ is reduced to 441.III. L ORENTZ R ECIPROCITY T HEOREM
We shall now investigate the reciprocal properties of thetensors χ and derive the associated reciprocity conditions thatapply to its various subtensors. For this purpose, we start fromthe definition of electromagnetic reaction given by [31] (cid:104) a , b (cid:105) = (cid:90) E a · J b dV, (20)which results from the interaction of a source a with acurrent distribution J a producing the field E a acting, withinthe volume V , on a source b with current distribution J b andelectric field E b , as illustrated in Fig. 1.The Lorentz reciprocity theorem then states that the mediumcontained in V is reciprocal if [27], [32], [33] (cid:104) a , b (cid:105) = (cid:104) b , a (cid:105) . (21) J a J b E a E b SV Medium
Fig. 1: Application of the Lorentz reciprocity theorem showingtwo regions of a medium, possessing dipolar and quadrupolarresponses, interact with each other.Substituting (20) into (21) and using J = ∇ × H − jω D fromMaxwell equations along with (7a), yields, after rearrangingthe terms, the equality [27], [32], [33] (cid:104) a , b (cid:105) − (cid:104) b , a (cid:105) = (cid:90) E a · J b dV − (cid:90) E b · J a dV = (cid:90) (cid:2) jω (cid:0) E b · D a − E a · D b + H a · B b − H b · B a (cid:1) + ∇ · (cid:0) H b × E a − H a × E b (cid:1)(cid:3) dV = 0 (22)This expression, combined with the constitutive relations (9),may now be used to derive the sought after reciprocityconditions. To do so, we next substitute (9) into (22) and pur-posefully ignore the trivial responses related to D = (cid:15) E and B = µ H and only concentrate on the dipolar and quadrupolarterms for convenience. We also make use of the followingidentities to simplify the quadrupolar expressions [18] E u · (cid:16) Q v · ∇ (cid:17) = ∇ · (cid:16) E u · Q v (cid:17) − (cid:88) i ∇ i E u · Q v · ˆu i , (23a) H u · (cid:16) S v · ∇ (cid:17) = ∇ · (cid:16) H u · S v (cid:17) − (cid:88) i ∇ i H u · S v · ˆu i . (23b)where u, v = { a , b } and ˆu = ˆx + ˆy + ˆz is a unit vector. Afterapplying the divergence theorem to the resulting expression and rearranging the terms, we obtain (cid:73) (cid:18) H b × E a − H a × E b + E a · Q b − E b · Q a − µ H a · S b + µ H b · S a (cid:19) · d S + jω (cid:90) (cid:32) E b · P a − (cid:88) i ∇ i E a · Q b · ˆu i − E a · P b + 12 (cid:88) i ∇ i E b · Q a · ˆu i + µ H a · M b − µ (cid:88) i ∇ i H b · S a · ˆu i − µ H b · M a + µ (cid:88) i ∇ i H a · S b · ˆu i (cid:33) dV = 0 . (24)Since the surface integral in (24) vanishes when taking thesurface S to infinity [18], [27], [32], [33], we next concentrateour attention only on the volume integral. Substituting (14)into the integrand of this volume integral, rearranging theterms and keeping only those that are not redundant forbriefness, we obtain (cid:90) (cid:104) (cid:15) E b i ( χ ee ,ij − χ ee ,ji ) E a j − µ k (cid:0) S (cid:48) mm ,klij − S (cid:48) mm ,ijkl (cid:1) ∇ l H a k ∇ j H b i + µ H a i ( χ mm ,ij − χ mm ,ji ) H b j − c k (cid:0) Q (cid:48) em ,klij + S (cid:48) me ,ijkl (cid:1) ∇ l E a k ∇ j H b i + (cid:15) k E b k (cid:0) χ (cid:48) ee ,kji − Q ee ,ijk (cid:1) ∇ i E a j − c k H b k (cid:0) χ (cid:48) me ,kji + Q em ,ijk (cid:1) ∇ i E a j +12 c k E b k (cid:0) χ (cid:48) em ,kij + S me ,ijk (cid:1) ∇ i H a j − µ k H b k (cid:0) χ (cid:48) mm ,kij − S mm ,ijk (cid:1) ∇ i H a j − (cid:15) k (cid:0) Q (cid:48) ee ,klij − Q (cid:48) ee ,ijkl (cid:1) ∇ l E a k ∇ j E b i +1 c E b i ( χ em ,ij + χ me ,ji ) H a j (cid:105) dV = 0 . (25)This equation allows us to straightforwardly deduce the reci-procity conditions that the subtensors in χ must satisfy. Indeed,since (25) must be zero far any field value, we directlyobtain the conventional reciprocity conditions for bianisotropicmedia [27], [32], [33] χ ee ,ij = χ ee ,ji , χ mm ,ij = χ mm ,ji , χ em ,ij = − χ me ,ji . (26)The reciprocity conditions relating the dipolar susceptibilitiesrelated to field gradients to the quadrupolar components relatedto the fields are [3], [25] χ (cid:48) ee ,kji = Q ee ,ijk , χ (cid:48) mm ,kij = S mm ,ijk ,χ (cid:48) em ,kij = − S me ,ijk , χ (cid:48) me ,kji = − Q em ,ijk , (27) and the those applying to the quadrupole components relatedto the field derivatives read [3], [25] Q (cid:48) ee ,klij = Q (cid:48) ee ,ijkl , Q (cid:48) em ,klij = − S (cid:48) me ,ijkl ,S (cid:48) mm ,klij = S (cid:48) mm ,ijkl . (28)Combining all permutation symmetries obtain in Sec. II-Bwith those from reciprocity reduces the number of independentcomponents in χ to 231.IV. P OYNTING T HEOREM
We shall now derive the conditions of gainlessness andlosslessness applying to the subtensors of χ and that canbe obtained from the Poynting theorem [32]. The Poyntingtheorem may be derived by starting from the time-domainMaxwell equation ∇ × E = − ∂ B ∂t , (29a) ∇ × H = J + ∂ D ∂t , (29b)and then subtracting (29b), pre-multiplied by E , to (29a), pre-multiplied by H , which yields ∇ · S = − E · ∂ D ∂t − H · ∂ B ∂t , (30)where S = E × H is the Poynting vector.We next develop the two terms on the right-hand sideof (30). For briefness, we concentrate our attention only onthe first term, which may be split into two equal parts andtransformed, using (9a), into E · ∂ D ∂t = 12 E · ∂ D ∂t + (cid:15) E · ∂ E ∂t + 12 E · ∂ P ∂t − E · ∂∂t (cid:16) Q · ∇ (cid:17) . (31)Adding the self-canceling terms ( E · ∂ P /∂t − E · ∂ P /∂t ) / and [( Q ·∇ ) · ∂ E /∂t − ( Q ·∇ ) · ∂ E /∂t ] / to (31) and combiningtogether terms that are similar to each other, we obtain E · ∂ D ∂t = 12 ∂∂t ( E · D ) + 12 (cid:18) E · ∂ P ∂t − P · ∂ E ∂t (cid:19) +14 (cid:20)(cid:16) Q · ∇ (cid:17) · ∂ E ∂t − E · ∂∂t (cid:16) Q · ∇ (cid:17)(cid:21) (32)By the same token, we transform the term H · ∂ B /∂t in (30),which we then substitute, along with (32), into (30) yielding ∂w∂t + ∇ · S = − (cid:18) E · ∂ P ∂t − P · ∂ E ∂t (cid:19) − (cid:20)(cid:16) Q · ∇ (cid:17) · ∂ E ∂t − E · ∂∂t (cid:16) Q · ∇ (cid:17)(cid:21) − µ (cid:18) H · ∂ M ∂t − M · ∂ H ∂t (cid:19) − µ (cid:20)(cid:16) S · ∇ (cid:17) · ∂ H ∂t − H · ∂∂t (cid:16) S · ∇ (cid:17)(cid:21) , (33)where w = 12 ( E · D + H · B ) . (34) We also use the identity ∇ · ( E × H ) = H · ( ∇ × E ) − E · ( ∇ × H ) . Relation (33) along with (34) constitute the instantaneousPoynting theorem in the presence of electric and magneticdipoles and quadrupoles.We next assume time-harmonic fields, which transforms thetime derivatives in (33) into ∂/∂t → jω , and take the time-average counterpart of (33) to obtain ∇ · (cid:104) S (cid:105) = ω (cid:20) E ∗ · P − P ∗ · E + (cid:16) Q · ∇ (cid:17) ∗ · E − E ∗ · (cid:16) Q · ∇ (cid:17) + 2 µ ( H ∗ · M − M ∗ · H ) + µ (cid:16) S · ∇ (cid:17) ∗ · H − µ H ∗ · (cid:16) S · ∇ (cid:17) (cid:21) , (35)where (cid:104) ∂w/∂t (cid:105) = 0 . Finally, we integrate (35) over avolume V and simplify the resulting expression using theidentities (23) with u = ∗ and v left blank and vice versaand, noting that ( A · ∇ ) ∗ = A ∗ · ∇ , with A = { S, Q } , weobtain (cid:73) (cid:104) S (cid:105) · d S = ω (cid:20) (cid:73) (cid:16) E · Q ∗ − E ∗ · Q + µ H · S ∗ − µ H ∗ · S (cid:17) · d S + (cid:90) (cid:18) E ∗ · P − (cid:88) i ∇ i E · Q ∗ · ˆu i − E · P ∗ + (cid:88) i ∇ i E ∗ · Q · ˆu i +2 µ H ∗ · M − µ (cid:88) i ∇ i H · S ∗ · ˆu i − µ H · M ∗ + µ (cid:88) i ∇ i H ∗ · S · ˆu i (cid:19) dV (cid:21) . (36)Now, for the same reason mentioned in Sec. III, the surfaceintegral on the right-hand side of (36) vanishes when S is takento infinity. The conditions of gainlessness and losslessnessare now obtained from the integrand of the volume integralin (36) knowing that it must be zero since, to satisfy energyconservation, all the energy entering the volume V must leaveit, i.e., (cid:72) (cid:104) S (cid:105) · d S = 0 , as illustrated in Fig. 2.After substituting (14) into (36) and selecting the termsthat must cancel each other as in (25), we obtain the desiredconditions for bianisotropic media as [29], [32] χ ee ,ij = χ ∗ ee ,ji , χ mm ,ij = χ ∗ mm ,ji , χ em ,ij = χ ∗ me ,ji . (37)The conditions applying to the third-rank tensors are thengiven by χ (cid:48) ee ,kji = Q ∗ ee ,ijk , χ (cid:48) mm ,kij = S ∗ mm ,ijk (38a) χ (cid:48) em ,kij = S ∗ me ,ijk , χ (cid:48) me ,kji = Q ∗ em ,ijk , (38b)whereas those for the fourth-rank tensors read Q (cid:48) ee ,klij = Q (cid:48)∗ ee ,ijkl , S (cid:48) mm ,klij = S (cid:48)∗ mm ,ijkl ,Q (cid:48) em ,klij = S (cid:48)∗ me ,ijkl . (39)Comparing relations (37), (38) and (39) to the reciprocityconditions (26), (27) and (28) reveals that these two setsof conditions are almost identical to each other. Indeed, therelations deduced from the Poynting theorem do not imply SV Medium
Fig. 2: Application of the Poynting theorem showing elec-tromagnetic waves impinging on and being scattered by amedium possessing dipolar and quadrupolar responses. Energyconservation requires that the energy entering V equals theenergy leaving it.additional permutation symmetries compared to those alreadyprovided by the reciprocity conditions. Instead, they requiresome tensors to be purely real and some others to be purelyimaginary for the medium to satisfy energy conservation.V. C ONCLUSIONS
We have provided a self-consistent and purely electro-magnetic derivation of the Lorentz and Poynting theorems,and have deduced the associated conditions of reciprocityand gainlessness and losslessness, in the presence of electricand magnetic quadrupolar responses expressed in terms offields and field gradients. We expect that these conditionswill be especially useful for the developments of advancedmetamaterial and metasurface modeling techniques requiringthe presence of quadrupolar responses.R
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