Hidden momentum of electrons, nuclei, atoms and molecules
HHidden momentum of electrons, nuclei, atoms and molecules
Robert P. Cameron ∗ SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, U.K.
J. P. Cotter
Centre for Cold Matter, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2AZ, U.K. (Dated: September 26, 2018)We consider the positions and velocities of electrons and spinning nuclei and demonstrate thatthese particles harbour hidden momentum when located in an electromagnetic field. This hiddenmomentum is present in all atoms and molecules, however it is ultimately cancelled by the momen-tum of the electromagnetic field. We point out that an electron vortex in an electric field mightharbour a comparatively large hidden momentum and recognise the phenomenon of hidden hidden momentum.
I. INTRODUCTION
A loop of electric current I and magnetic-dipole mo-ment m at rest in a static electric field E has ‘hiddenmomentum’ p hidden = m × E /c , even though the loopis not moving [1–4]. This system is illustrated in FIG. 1.The hidden momentum results from the different chargecarriers in the loop having different speeds, due to a mod-ification of their usual motion around the loop by E [2–4]. It is cancelled by the momentum − m × E /c ofthe electromagnetic field [2, 3, 5–7]. The phenomenon ofhidden momentum is not unique to this system, nor is itunique to electrodynamics [1–3, 8]. FIG. 1. A loop of electric current in a static electric field.Different charge carriers in the loop have different speeds,as indicated here by their colour. The imbalance of theirmomenta is the loop’s hidden momentum [1–4].
This paper was motivated by a question posed recentlyby Filho and Saldanha: “does an electron with a mag-netic moment resulting from its spin in the presence ofan applied electric field have hidden momentum” [4]? In § II, we consider a free electron, as described by the (first-quantised) Dirac equation. We highlight subtleties asso-ciated with ‘the’ position and velocity of the electron, an ∗ understanding of which is necessary for the analysis thatfollows. In § III, we introduce an external electromag-netic field and demonstrate that the electron harbourshidden momentum associated with its spin, thus provid-ing an affirmative answer to the question above. In § IV,we consider an isolated atom or molecule and reaffirmthat its constituent electrons, as well as any spinning nu-clei present, harbour hidden momentum individually. Wealso show that the sum total of this hidden momentum iscancelled by the momentum of the electromagnetic field,as it should be. In § V, we point out that an electronvortex in an electric field might harbour a comparativelylarge hidden momentum and recognise the hitherto ne-glected phenomenon of hidden hidden momentum. Ourwork is timely, given the recent surge of interest in rela-tivistic electron vortices [9–16].In what follows, ‘hats’ are used to indicate physi-cal quantities whereas the mathematical operators usedto express these quantities in different representationsdo not have hats – we alternate between the Diracrepresentation (primed) [17] and the Foldy-Wouthuysenrepresentation (unprimed) [18], defined in appendix A.These distinctions are important. Consider, for example,ˆ r (cid:48) q = ˆ r M = r . Here, two different physical quantities(ˆ r (cid:48) q and ˆ r M ) are expressed in two different representa-tions (primed and unprimed) by the same mathematicaloperator ( r ). II. FREE ELECTRON
Let us consider first a free electron . In the Dirac rep-resentation, the electron obeysi¯ h ˙ ψ (cid:48) = ˆ H (cid:48) ψ (cid:48) , (1)with ψ (cid:48) = ψ (cid:48) ( r , t ) the electron’s spinor andˆ H (cid:48) = cααα · p + βmc (2)the free Dirac Hamiltonian [17]. Here, ααα = (cid:18) σσσσσσ (cid:19) , p = − i¯ h ∇∇∇ , β = (cid:18) − (cid:19) , (3) a r X i v : . [ phy s i c s . a t m - c l u s ] M a y and m is the rest mass of the electron. The momentum ofthe electron can be identified unambiguously as ˆ p (cid:48) = ˆ p = p . However, ‘the’ position and velocity of the electronare not unique [8, 9, 16, 18–26]. For the purposes of thispaper, we find it necessary to identify and distinguishbetween the instantaneous position ˆ r (cid:48) q of the electron’selectric charge, the kinetic position ˆ¯ r (cid:48) q of the electron andthe so-called mean position ˆ r (cid:48) M of the electron. As theelectron is free, these positions can be defined as follows. A. Positions and velocities
The position of charge takes on a simple form in theDirac representation [17],ˆ r (cid:48) q = r . (4)The interpretation of ˆ r (cid:48) q as the position of charge [22, 24]will be made apparent in the next section, where we im-pose an electromagnetic field.The kinetic position is [19, 27, 28]ˆ¯ r (cid:48) q = 14 (cid:20) H (cid:48) (cid:16) ˆ H (cid:48) ˆ r (cid:48) q + ˆ r (cid:48) q ˆ H (cid:48) (cid:17) + (cid:16) ˆ H (cid:48) ˆ r (cid:48) q + ˆ r (cid:48) q ˆ H (cid:48) (cid:17) H (cid:48) (cid:21) . (5)This coincides with the centre of the electron’s electriccharge, as evidenced by the result that (cid:104) ˆ¯ r (cid:48) q (cid:105) = (cid:104) ˆ r (cid:48) q (cid:105) fora state with energy of definite sign [16]. ˆ¯ r (cid:48) q is sometimesreferred to as the ‘observable’ part of the position ˆ r (cid:48) q ofcharge [21], being the projection of ˆ r (cid:48) q onto positive andnegative energy subspaces [9, 16]. ˆ¯ r (cid:48) q is not the electron’scentre of energy [16], in spite of its suggestive form.The mean position takes on a simple form in the Foldy-Wouthuysen representation [18, 20],ˆ r M = r . (6)Loosely speaking, ˆ r M can be thought of as the kinetic po-sition in the electron’s rest frame, actively boosted withappropriate velocity [16, 21, 29, 30]. It is ˆ r M that is usu-ally regarded as being ‘the’ position of the electron inlow-energy studies [18, 31], although one can argue thatthe kinetic position ˆ¯ r q is closer to the classical notion ofposition for a particle like the electron [23]. The ‘mean’terminology introduced in [18] for ˆ r M and other quanti-ties is something of a misnomer – it is ˆ¯ r q rather than ˆ r M that embodies the electron’s ‘average’ position [24].The components of the velocity ˆ v (cid:48) q = dˆ r (cid:48) q / d t = cααα ofcharge [32, 33] support discrete eigenvalues of ± c whilstthe kinetic velocity ˆ¯ v q = dˆ¯ r q / d t = βc p /E p [23] andmean velocity ˆ v M = dˆ r M / d t = βc p /E p [18] vary con-tinuously with p and are equal. Here, E p = (cid:112) m c + c p . (7) The above can be summarised as follows:quantity definitionposition of charge ˆ r (cid:48) q = r kinetic position ˆ¯ r (cid:48) q = (cid:110) H (cid:48) , (cid:110) ˆ H (cid:48) , ˆ r (cid:48) q (cid:111)(cid:111) mean position ˆ r M = r velocity of charge ˆ v (cid:48) q = cααα kinetic velocity ˆ¯ v q = βc p /E p mean velocity ˆ v M = βc p /E p momentum ˆ p (cid:48) = ˆ p = p (8)where we have used curly brackets to indicate anti-commutators. B. Zitterbewegung
In the Heisenberg picture, the positions evolve as [24,27, 28] ˆ r q ( t ) = ˆ¯ r q ( t ) + ˆ ξξξ ( t ) , (9)ˆ¯ r q ( t ) = ˆ r M ( t ) + ˆ δδδ, (10)ˆ r M ( t ) = ˆ r M (0) + ˆ v M t, (11)with ˆ ξξξ ( t ) = i¯ hc (cid:20) ˆ v q ( t ) − c ˆ p ˆ H (cid:21) e −
2i ˆ
Ht/ ¯ h ˆ H , (12)ˆ δδδ = c p × s E p ( E p + mc )= p × s m c + O( c ) . (13)Here, s = ¯ h (cid:32) σσσ σσσ (cid:33) . (14)The position difference ˆ ξξξ ( t ) executes a complicated oscil-latory motion with amplitude comparable to the Comp-ton wavelength 2 π ¯ h/mc and the resulting motion of theposition ˆ r q ( t ) of charge is referred to as the electron’s Zitterbewegung [18, 24, 27, 28, 34]. Meanwhile, the ki-netic position ˆ¯ r q ( t ) and the mean position ˆ r M ( t ) trans-late uniformly, with ˆ¯ r q ( t ) offset from ˆ r M ( t ) by the posi-tion difference ˆ δδδ . The equality ˆ¯ v q = dˆ¯ r q ( t ) / d t = ˆ v M =dˆ r M ( t ) / d t = βc p /E p holds as ˆ δδδ is constant. C. Relativistic Hall effect
The position difference ˆ δδδ might be regarded as a mani-festation of the relativistic Hall effect [25]. It embodies adistortion of the trajectory of the electron’s charge, dueto the electron’s spin and translation. A rotating, trans-lating wheel serves as an instructive (classical) analogy –different elements on the rim (ˆ r q ) have different speedsand are Lorentz contracted by different amounts, givinga shift (ˆ δδδ ) of the element-weighted centre (ˆ¯ r q ) away fromthe axle (ˆ r M ) [25, 35]. Due to the spin dependence ofˆ δδδ , the components of ˆ¯ r q do not commute: [ˆ¯ r qα , ˆ¯ r qβ ] = − i¯ hc (cid:15) αβγ ˆ¯ s γ / ˆ H [19]. Here, ˆ¯ s = ˆ j − ˆ¯ r q × ˆ p [27, 28], withˆ j the total angular momentum of the electron [17]. Itseems that ˆ¯ s is the electronic analogue [9, 16] of the spinof freely propagating light [36–41] – each is conserved andboth have similar commutations relations. ˆ r q ( t ), ˆ¯ r q ( t )and ˆ r M ( t ) are depicted schematically in FIG. 2.Position differences like ˆ δδδ are well known for electronsin the solid state and can be regarded as Berry connec-tions in momentum space [9, 16, 42–47]. FIG. 2. There is a sense in which a free electron resem-bles an electric current ‘loop’ – the electron’s
Zitterbewegung sees the position of charge circulate, in spite of there be-ing no obvious external fields. It can be argued that thisis the origin of the electron’s spin and magnetic-dipole mo-ment [18, 24, 27, 28, 34]. It seems natural, therefore, to an-ticipate that an electron in an electric field harbours hiddenmomentum associated with its spin, due to a modification ofits
Zitterbewegung by the field.
III. ELECTRON IN AN EXTERNALELECTROMAGNETIC FIELD
To demonstrate that an electron can harbour hiddenmomentum associated with its spin, let us consider nowan electron in an external electromagnetic field , withscalar potential Φ = Φ( r , t ) and magnetic vector poten-tial A = A ( r , t ) in the Coulomb gauge [48]. We work toorder 1 /c and assume that the leading-order contribu-tion to A is of order 1 /c . According to the principleof minimal coupling, the Hamiltonian in the Dirac rep- resentation becomes [17]ˆ H (cid:48) = cααα · ( p − q A ) + βmc + q Φ , (15)where q is the electron’s electric charge. It is importantnow to distinguish between the canonical momentumˆ p (cid:48) = p = ˆ p + O(1 /c ) and the total kinetic momentumˆ πππ (cid:48) = p − q A = ˆ πππ + O(1 /c ) of the electron. It is ˆ πππ (cid:48) ratherthan ˆ p (cid:48) that obeys the Lorentz force law [32, 33, 49],dˆ πππ (cid:48) d t = q ( E + ααα × B ) . (16)Here, E = E ( r , t ) = −∇∇∇ Φ − ˙ A is the electric field and B = B ( r , t ) = ∇∇∇ × A is the magnetic field. The ab-sence of explicit magnetic-dipole moment terms in (16)agrees with the view that the magnetic-dipole moment ofthe electron is an emergent feature, due to the electron’s Zitterbewegung [18, 24, 27, 28, 34].
A. Positions and velocities
We continue to identify and distinguish between theposition ˆ r (cid:48) q of charge, the kinetic position ˆ¯ r q and the meanposition ˆ r M – as the electron is in the presence of anelectromagnetic field, we now define these asˆ r (cid:48) q = r , ˆ¯ r q = ˆ r M + ˆ δδδ, & ˆ r M = r , (17)with ˆ δδδ = p × s m c + O( c ) . (18)Note that the potentials in (15) are evaluated at r , inaccord with the interpretation of ˆ r (cid:48) q as the position ofcharge [22, 24]. Unlike the case for a free electron, theposition difference ˆ δδδ is not necessarily constant – theelectromagnetic field can alter p × s to leading order,as dˆ p / d t = − q ∇∇∇ Φ + O(1 /c ) and dˆ s M / d t = 0 + O(1 /c ),with ˆ s M = s the mean spin of the electron [18]. It followsthat the kinetic velocity ˆ¯ v q = dˆ¯ r q / d t no longer equalsthe mean velocity ˆ v M = d ˆ r M / d t , asˆ¯ v q − ˆ v M = dˆ δδδ d t = q s × ∇∇∇ Φ2 m c . (19)This subtlety will prove important below.Velocity contributions like dˆ δδδ/ d t are also known forelectrons in the solid state and are sometimes referred toas being ‘anomalous’ [42–47]. B. Hidden momentum
Explicit calculation of i[ ˆ H, ˆ r M ] / ¯ h reveals that themean velocity isˆ v M = β p m − βp p m c − βq A + q s × ∇∇∇ Φ2 m c + O( c ) . (20)Multiplying this by β and rearranging reveals that thecanonical momentum isˆ p = βm ˆ v M + ˆ p ˆ p m c + q A − βq ( s × ∇∇∇ Φ)2 mc + O( c ) . (21)It is tempting to identify the first and second termshere with the relativistically corrected kinetic momen-tum, and the third term with the electromagnetic mo-mentum. However, this leaves the fourth term unac-counted for. To proceed, we must recognise that thekinetic momentum should be cast in terms of the kineticvelocity ˆ¯ v q , rather than the mean velocity ˆ v M . This leadsus to recast the spin-dependent term in (21) as − βq ( s × ∇∇∇ Φ)2 mc = βq ( s × ∇∇∇ Φ)2 mc − βq ( s × ∇∇∇ Φ) mc = βm (ˆ¯ v q − ˆ v M ) + ˆ m × ˆ E c + O( c ) , (22)with ˆ m = βq s /m the magnetic-dipole moment of theelectron. Here, we have made use of (19) and E = −∇∇∇ Φ+O(1 /c ). Substituting (22) into (21) givesˆ p = βm ˆ¯ v q + ˆ p ˆ p m c + q A + ˆ m × E c + O( c ) . (23)Thus, ˆ p is comprised of relativistically corrected kineticmomentum terms ( βm ˆ¯ v q + ˆ p ˆ p / m c ), an electromag-netic momentum term ( q A ) and, pleasingly, a hiddenmomentum term ( ˆ m × E /c ) with the prototypical formdescribed in the introduction. We attribute this hiddenmomentum to a modification of the electron’s Zitterbewe-gung by the electric field E . For the special case in which E is due to a “test particle”, a complementary result wasderived in [8]. A similar result was derived in [23], butwith no explicit recognition of the hidden momentum.Note that the total kinetic momentum ˆ πππ includes thehidden momentum ˆ m × E /c .The hidden momentum ˆ m × E /c is small, its expec-tation value being < ∼ − q ¯ h | E | / mc = | E | × − A . s inmagnitude. IV. ISOLATED ATOM OR MOLECULE
The formalism employed in the previous section doesnot allow us to confirm that the hidden momentumˆ m × E /c is cancelled by the momentum of the elec-tromagnetic field, as the field is externally imposed. Letus conclude, therefore, by considering an isolated atomor molecule – a closed system. Our description is ef-fectively truncated at order 1 /c and we therefore ne-glect terms of order 1 /c or smaller. The subscripted‘ q ’ and ‘ M ’ notation used above is henceforth dropped,for the sake of clarity. Let us focus our discussion upona molecule (an atom being a special case with one nu-cleus). We regard the molecule as being an electricallyneutral collection of electrons (subscript i ) and spin 0 or 1 / j ), bound together by electromag-netic interactions in the absence of external influences.We refer to the electrons and nuclei collectively as ‘theparticles’ (subscript k ) and treat the k th particle as apoint-like object of rest mass m k , mean position ˆ r k = r k ,canonical momentum ˆ p k = − i¯ h ∇∇∇ k , electric charge q k andmagnetic-dipole moment ˆ m k = γ k ˆ s k , with γ k the gyro-magnetic ratio and ˆ s k = ¯ hσσσ k / m k /γ k = ˆ s k = 0 for spin 0 nuclei.Let R i = √ h/ m i c account for the effective finite sizesof the electrons [50, 51], R j account for the finite size ofthe j th nucleus [52, 53] and f k = (1 − q k / m k γ k ) be theusual spin-orbit factor [54–57] for the k th particle. Weregard the R k as being of order 1 /c and take the Hamil-tonian governing our molecule to be [33, 51–53, 58–61]ˆ H = (cid:88) k ˆ p k m k + (cid:88) k q k ˆΦ k − (cid:88) k ˆ p k m k c + (cid:88) k q k R k ∇ k ˆΦ qk − (cid:88) k f k ˆ m k · (ˆ p k × ∇∇∇ k ˆΦ qk ) m k c − (cid:88) k q k ˆ p k · ˆ A k m k − (cid:88) k
12 ˆ m k · ( ∇∇∇ k × ˆ A k ) , (24)with ˆΦ k = ˆΦ qk + ˆΦ Rk (25)the intramolecular scalar potential seen by the k th par-ticle at ˆ r k and ˆ A k = ˆ A m k + ˆ A v k (26)the intramolecular magnetic vector potential, whereˆΦ qk = (cid:88) k (cid:48) (cid:54) = k q k (cid:48) π(cid:15) ˆ r kk (cid:48) (27)ˆΦ Rk = − (cid:88) k (cid:48) (cid:54) = k q k (cid:48) R k (cid:48) δ (ˆ r kk (cid:48) )6 (cid:15) (28)account for the electric charges and finite sizes of theother particles andˆ A m k = (cid:88) k (cid:48) (cid:54) = k µ ˆ m k (cid:48) × ˆ r kk (cid:48) π ˆ r kk (cid:48) (29)ˆ A v k = (cid:88) k (cid:48) (cid:54) = k µ q k (cid:48) πm k (cid:48) (cid:34) r kk (cid:48) ˆ p k (cid:48) + ˆ p k (cid:48) r kk (cid:48) + ˆ r kk (cid:48) r kk (cid:48) (ˆ r kk (cid:48) · ˆ p k (cid:48) ) + (ˆ p k (cid:48) · ˆ r kk (cid:48) ) 1ˆ r kk (cid:48) ˆ r kk (cid:48) (cid:35) (30)account for the intrinsic magnetic moments and orbitalmotions. A. Hidden momentum of the electrons and nucleiindividually
Defining [62] ˆ δδδ k = ˆ¯ r k − ˆ r k = ˆ p k × ˆ s k m k c , (31)ˆ¯ v k = dˆ¯ r k d t , (32)ˆ v k = dˆ r k d t , (33)a calculation analogous to that outlined in the previoussection reveals that the canonical momentum of the k thparticle isˆ p k = m k ˆ¯ v k + ˆ p k ˆ p k m k c + q k ˆ A k − ˆ m k × ∇∇∇ k ˆΦ qk c + O( c ) . (34)Thus, each electron and spinning nucleus in the moleculeharbours a hidden momentum − ˆ m k × ∇∇∇ k ˆΦ qk /c .A basic estimate suggests that the hidden momentumof an electron in a hydrogen atom corresponds to a no-tional electronic speed of only < ∼ × m . s − . Signifi-cantly stronger electric fields can be found in heavy atomsand molecules [63], in which case the hidden momentummight be significantly larger.In the calculation leading to (34), the emergence of thehidden momentum can be traced to the ‘1’ in the spin-orbit factor f k (a translating magnetic-dipole momentresembles an electric-dipole moment [23, 35, 54]) whilstthe emergence of the momentum difference m k (ˆ¯ v k − ˆ v k )can be traced to the ‘ − q k / m k γ k ’ (Thomas preces-sion [23, 35, 55–57]). This seems natural, as the posi-tion difference ˆ δδδ k is intimately associated with Thomasprecession [23, 35].For a more detailed discussion of the energy, linearmomentum, angular momentum and boost momentumof a molecule to order 1 /c see [64]. B. Total hidden momentum and its cancellation
We recognise ˆ P = (cid:80) k ˆ p k as being the total momentumof the molecule. ˆ P is conserved and generates (Carte-sian [65]) translations of the molecule in space [66, 67].The hidden contribution to ˆ P is countered by an equaland opposite contribution due to the magnetic-dipole mo-ments of the particles, − (cid:88) k ˆ m k × ∇∇∇ k ˆΦ qk c + (cid:88) k q k ˆ A m k = 0 . (35)Thus, the total hidden momentum of the molecule is can-celled by the momentum of the intramolecular electro-magnetic field, as one might expect [2, 3, 5–7]. V. OUTLOOK
An electron vortex [9–16] in an electric field E mightharbour a hidden momentum due to a modification of theelectron’s orbital motion by E , in addition to the spin-based hidden momentum identified in this paper. Theorbital-based hidden momentum should take the form m (cid:96) × E /c , with m (cid:96) the orbital magnetic-dipole mo-ment of the electron. Assuming that | m (cid:96) | ∼ − q ¯ h | (cid:96) | / m ,this is < ∼ − q ¯ h | (cid:96) || E | / mc = | (cid:96) || E | × − A . s in mag-nitude. The orbital-based hidden momentum could besignificantly larger than the spin-based hidden momen-tum (expectation value < ∼ − q ¯ h | E | / mc in magnitude),as the orbital angular momentum quantum number (cid:96) ∈{ , ± , . . . } is unbounded.Inferring the existence of hidden momentum in the lab-oratory is an interesting problem. One might endeavourto measure the associated angular momentum, which isnot necessarily cancelled by the angular momentum ofthe field that gives rise to the hidden momentum - unlikethe total linear momentum, the total angular momentumof a system ‘at rest’ need not vanish [2]. An electron vor-tex with a large orbital angular momentum, perturbedby an electric field, might prove particularly suitable forthis purpose.The hidden momentum of a system like the one de-scribed in the introduction might be referred to moredescriptively as a hidden kinetic momentum, to empha-sise that it is an imbalance of the kinetic momenta ofthe system’s constituent particles: ‘ (cid:80) γm v (cid:54) = 0’ [1–4].In this paper we have established that even a single par-ticle like the electron can harbour a hidden momentumassociated with its spin. We can now conceive, therefore,of systems containing such particles in which there is no imbalance of the kinetic momenta of the particles and yetthe total hidden momentum of the particles is non-zero:‘ (cid:80) γm v = 0’ but ‘ (cid:80) m × E /c (cid:54) = 0’. One might say thatsuch a system harbours hidden hidden momentum, in dis-tinction to hidden kinetic momentum. A loop of electriccurrent (driven through a resistive element by a battery)encircling the tip of a (long) magnetised needle is onesuch system. To appreciate this, consider a simple modelof such a system in which the loop is circular and lies inthe x − y plane whilst the tip of the needle coincides withthe centre of the loop, at the origin. If we imagine thatthe magnetic-dipole moment ‘ m ’ of each charge carrieris aligned radially due to the magnetic field of the needlewhilst the electric field ‘ E ’ driving the current aroundthe loop is aligned azimuthally, then the hidden momen-tum ‘ m × E /c ’ of each charge carrier is aligned axially.Thus, the system harbours a hidden hidden momentum‘ (cid:80) m × E /c (cid:54) = 0’, with no hidden kinetic momentum tospeak of: ‘ (cid:80) γm v = 0’. Hall effects [68, 69] have beenneglected in our argument. We do not expect these todramatically alter the underlying physics, however. VI. ACKNOWLEDGEMENTS
This work was supported by the EPSRC(EP/M004694/1) and The Leverhulme Trust (RPG-2017-048). We thank Gergely Ferenczi for his adviceand encouragement.
Appendix A: Representations
The Foldy-Wouthuysen representation was introducedin [18] by Foldy and Wouthuysen to establish a correspon-dence between Dirac’s fully relativistic theory expressedin the Dirac representation [17] and the low-energy Paulidescription of spin 1 / r (cid:48) q = r of charge have simple op-erator representatives in the Dirac representation whilstothers such as the mean position ˆ r M = r have simpleoperator representatives in the Foldy-Wouthuysen rep-resentation instead. The following is a summary of keyresults from [18].For a free electron, the Foldy-Wouthuysen representa-tion is related to the Dirac representation by the unitaryoperator U = exp (cid:26) i (cid:20) − i βααα · p p tan − (cid:16) pmc (cid:17)(cid:21)(cid:27) . 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