Evidence of macroscopically entangled protons in a 0 mixed isotope crystal of KH D CO p 1 p 3 1 − 0 2 Fran¸cois Fillaux n a J § LADIR-CNRS, UMR 7075, Universit´e P. et M. Curie, 2 rue Henry Dunant, 94320 3 Thiais, France 1 ] r Alain Cousson e h Laboratoire L´eon Brillouin (CEA-CNRS), C.E. Saclay, 91191 Gif-sur-Yvette, cedex, t o France . t a m Matthias J Gutmann - d ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK n o c Abstract. We examine whether protons and deuterons in the crystal of [ KH0.76D0.24CO3 at 300 K are particles or matter waves. The neutron scattering 1 function measured over a broad range of reciprocal space reveals the enhanced v diffraction pattern anticipated for antisymmetrized macroscopic states for protons 0 (fermions). These features exclude a statistical distribution of protons and deuterons. 5 1 Raman spectra are consistent with a mixture of KHCO3 and KDCO3 sublattices 2 whose isomorphous structures are independent of the isotope content. We propose . 1 a theoretical framework for decoherence-free proton and deuteron states. 0 0 1 : PACS numbers: 61.12.-9,03.65.Ud, 63.10.+a,82.30.Hk. v Keywords: Quantum entanglement, Single crystal, Neutron scattering, Raman i X scattering, Hydrogen bonding r a To whom correspondence should be addressed: § fi[email protected] Tel: +33149781283 Fax: +33149781218 (fi[email protected]; http://www.ladir.cnrs.fr/pagefillaux eng.htm) Macroscopically entangled protons 2 1. Introduction A defect-free crystal is a macroscopic quantum system with spatial periodicity. In principle, only nonlocal observables are probed by plane waves, eg. with diffraction or spectroscopy techniques. This is attested by the marked localization in reciprocal space of Bragg peaks as well as discrete phonon states. However, since the seminal work of Slater [1], the nonlocal nature of protons is commonly overlooked in crystals containing bistable O–H O hydrogen bonds, when the coexistence of two configurations, say ··· “L” for O1 H O2 and “R” for O1 H O2, is conceived of as classical disorder. − ··· ··· − This leads to a conflict of interpretation consisting in that a statistical distribution of protons should destroy the translation periodicity of the crystal, whereas infrared and Raman spectra evidence extended states at the Brillouin-zone center (BZC) obeying the symmetry-related selection rules of the crystal. This conflict has been tolerated because the nonlocal nature of protons cannot be evidenced by x-ray diffraction. Even with neutrons, a Bragg peak analysis is not conclusive, for statics and dynamics cannot be unravelled effectively from the unit-cell structure factor [2]. The same conflict of interpretation arises for crystals containing hydrogen bonded centrosymmetric dimers, such as carboxylic acids [3, 4], or potassium hydrogen carbonate (or bicarbonate, KHCO ) [5, 6, 7, 8]. The thermal interconversion 3 between tautomeric configurations, LL and RR, is supposed to be correlated with a reorganization of the single and double CO bonds, along a reaction path across a multidimensional potential. This is commonly rationalized with rigid proton-pairs in an asymmetric double-well coupled to an incoherent thermal bath mimicking a liquid- like environment. It is supposed that “phonon-induced incoherent tunnelling” should prevail at low temperatures, whereas stochastic jumps should take the lead at elevated temperatures. By contrast, vibrational spectra are consistent with nonlocal observables represented by normal coordinates. There is no evidence of any symmetry breaking due tolocaldisorderandthedensity-of-statesremainsdiscreteatanytemperature[9,10,11]. Nowadays, the local or nonlocal behaviour of bistable hydrogen bonds can be distinguished at advanced pulsed neutron sources, with a time-of-flight Laue diffractometer measuring a much broader range of reciprocal space than any reactor- based four-circle instrument. The scattering function measured at large momentum transfer values (Q = k k , where k and k are the initial and final wave vectors, i f i f − respectively) contains information on the scale of spacial correlations between L and R sites ( 0.5 ˚A). Moreover, such an instrument is unique to observing diffuse scattering, ∼ because signals in addition to Bragg peaks are effectively measured across the whole rangeof reciprocal space foreach neutron pulse. It isthus possible to determine whether or not coherent scattering is affected by the coexistence of L and R configurations at various temperatures. This has been exemplified with the crystal of KHCO made of hydrogen bonded 3 − dimers, (HCO ) [12, 13, 14, 15]. Below 150 K, only those proton sites (L), 3 2 corresponding to a unique configuration for dimers (LL), are occupied. The neutron Macroscopically entangled protons 3 scattering function reveals enhanced Bragg peaks exclusively at those particular Q- values corresponding to the reciprocal sublattice of protons. Furthermore, the enhanced peaks are superposed to a prominent continuum of incoherent scattering, cigar-like shaped along Q-vectors perpendicular to dimer planes. By contrast, there isno evidence of similar enhanced features for the isomorphous derivative KDCO . To the best 3 of our knowledge, the dramatic increase of both coherent and incoherent scattering intensities for the sublattice of protons, not observed for deuterons, cannot be explained with any conventional model based upon Slater’s localized particles. Alternatively, a quantum theory leads to different macroscopic quantum correlations (entanglement) for indistinguishable protons (fermions) or deuterons (bosons). The enhanced features can be thus explained by the spin symmetry of the “super-rigid” fermion states for which phonons are forbidden. By contrast, deuteron states are represented by phonons with no spin-symmetry. Above 150 K, the less favored sites along the hydrogen bonds (R), at ∆ 0.6 x ≈ ˚A from the L sites, are progressively populated, while the space group symmetry and the unit cell parameters remain unchanged. At 300 K, the population ratio is 18(RR) : 82(LL). The enhanced features observed continuously from cryogenic to ≈ room temperatures show that the quantum correlations are temperature independent. ThecoexistenceofLLandRRconfigurationsisconsistentwithaquantumsuperposition of proton states perfectly insulated from decoherence by the other degrees of freedom [13, 14]. Then, scattering functions, vibrational spectra, and quasi-elastic neutron scattering (QENS) data, canbereinterpreted consistently withnonlocal observables and quantum interference arising from the superposition of macroscopic tunnelling states. Structural transitions of second order (P2 /a C2/m) occur at T 318 K 1 cH ←→ ≈ (KHCO ) or T 353 K (KDCO ). Above T , dimers split into two centrosymmetric 3 cD 3 c ≈ entities, symmetric with respect to the plane (a,c). It is quite remarkable that the enhanced features of the scattering function, as well as proton dynamics and quantum interference, do not show any discontinuity at T [14]. There is no evidence of c classical disorder above T . It transpires that quantum correlations arise from the space c periodicity of centrosymmetric dimers, irrespective of the space group symmetry. Here, we examine whether the enhanced features survive, or not, in a mixed isotope crystal of KHpD1−pCO3. In this case, static or dynamical disorder are clearly differentiated and the existence of distinct crystallographic domains of KHCO or 3 KDCO is excluded. A static distribution of mutually exclusive particles should destroy 3 the quantum entanglement and the enhanced features should vanish. Alternatively, if isotopes merge into separable macroscopic states, such that quantum correlations across the sublattice of protons survive, the enhanced features should be observed, although with lower contrast compared to KHCO . To the best of our knowledge, such a nonlocal 3 approach has never been examined in depth, in this context. In a preliminary report [16], the structure of KH D CO at 300 K was refined 0.76 0.24 3 from Bragg peaks measured with the four-circle diffractometer 5C4 at the Orph´ee reactor of the Laboratoire L´eon Brillouin [17]. There was no evidence of any symmetry Macroscopically entangled protons 4 breaking: the space group (P2 /a) is identical to that of KHCO or KDCO at 1 3 3 the same temperature (Fig. 1). A structural model consistent with the proposed quantum theory was represented by four sublattices, H , H , D , D , with weights LL RR LL RR p q ,p q ,p q ,p q , respectively: H LLH H RRH D LLD D RRD p [q H +q H ]+p [q D +q D ]. (1) H LLH LL RRH RR D LLD LL RRD RR ¯ ¯ The bound coherent scattering lengths are b = 3.741 fm or b = 6.671 fm. The H D − refined positional and thermal parameters (R 3%) were identical for protons and ≈ deuterons whose sites are indiscernible, to within experimental precision and thermal ellipsoids. The estimated concentrations were p = 0.76, p = 0.24, and the relative H D weights were such as S = p q +p q +p q +p q = 0.99(4). There was no H LLH H RRH D LLD D RRD evidence ofanysignificant decreaseofBraggpeakintensities intoLauemonotonicdiffuse scattering due to a statistical distribution of isotopes. Alternatively, for a sublattice of paired up nonseparable protons and deuterons, the averaged scattering length should be: ¯ ¯ ¯ b = 0.76b +0.24b 1.242 fm, (2) H D ≈ − The model is more artificial because p is no longer a free parameter and it is less satisfactory (R 4%). However, it cannot be definitely excluded. Consequently, the ≈ question as to whether the sublattices of protons and deuterons are separable entities remains largely open. The conclusions of the present work are based upon the neutron scattering function of the mixed isotope crystal presented in Sec. 2. The enhanced features show that the quantum correlations of the pure crystal are not destroyed by partial deuteration and Raman spectra are consistent with separable proton and deuteron states. In Sec. 3, the quantum theory [14, 15] applied to the mixed crystal accounts for macroscopic proton and deuteron states, each of which being a superposition of LL and RR states. A detailed interpretation of the enhanced features is proposed. In Sec. 4, we show that alternative explanations are inappropriate. 2. Experimental 2.1. The neutron scattering function The mixed isotope crystal ( 30 mm3) measured in the previous diffraction work [16] ≈ was cut from a twinning-free parent crystal ( 20 15 3 mm3) obtained from a water ≈ × × solution( 75%H O,25%D O).Thisparent crystal wasloadedinthevacuumchamber 2 2 ≈ of the time-of-flight Laue diffractometer SXD, at the ISIS pulsed neutron source [18, 19]. The temperature was (300 1) K. We measured 8 orientations, each for 8 hours, with ± ≈ ∗ (b) (b ) either parallel or perpendicular to the equatorial plane of the detector bank. k The reduced data were concatenated in order to compute the scattering function (Fig. 2). Previous studies of crystals with similar sizes have shown that multiple-scattering effects are not important. However, Bragg peaks show severe extinction. A standard Macroscopically entangled protons 5 analysis is consistent with the structure and isotope contents previously determined, but the modest precision (R 10%) shows that measurements conducted at different ≈ sources are complementary. The interpretation of the scattering function is based upon the crystal structure made of centrosymmetric dimers lying in (103) planes with O O bonds parallel to each ··· other. Proton/deuteron coordinates are x,y,z, (Fig. 1), Q ,Q ,Q , are projections of x y z Q, U ,U ,U , are the dominant temperature factors. Dotted lines in Fig. 1 enhance xx yy zz the network of double lines of proton/deuteron sites, either parallel to (b), separated by ∆ 0.6 ˚A, or parallel to x, with ∆ 2.2 ˚A. The spatial periodicity of these grating- x y ≈ ≈ like structures is D a/cos42◦ 20.39 ˚A or D = b. The distance between (x,y) x y ≈ ≈ planes is D 3.28 ˚A. A previous analysis of rigid body motions has shown that there z ≈ is no short-range correlation between protons and carbonate entities [12] and inelastic neutron scattering (INS) spectra show that proton dynamics are separated from heavy atoms and virtually dispersion-free [20, 21]. ∗ ∗ In Fig. 2, cuts of the scattering function parallel to (a ,c ) show spots of intensity, corresponding to Bragg peaks, and, at large Q-values, enhanced streaks along Q z ∗ perpendicular to the trace of dimer planes parallel to (Q ,Q ), with Q (b ). These x y y || streaks are not an artefact of the instrument because: (i) the detectors were calibrated with vanadium; (ii) the streaks are measured by different detectors for each of the 8 orientations of the crystal; (iii) spurious signals eventually due to an anomalous detector should appear along different trajectories. The maps in Fig. 2 are similar to those of KHCO at the same temperature [13], 3 apart from the weaker intensity of diffuse scattering due to the smaller concentration of protons. On the other hand, they are markedly different from those of KDCO 3 for which enhanced features are totally extinguished [12]. The enhanced diffraction peaks correspond to nodes of the reciprocal sublattice of protons/deuterons and there is no obvious alternative sublattice with similar parameters. They occur when k = ∗ ∗ Q /b = 2πn /(b ∆ ) n 2.57. For k = 0, in-phase scattering by double y y y y ± ≈ ± × lines of sites parallel to x gives peaks at Q = 2π/∆ = (10.00 0.25) ˚A−1 and x x ± ± ± Q n 1.92 ˚A−1. At k = 3, n 1, anti-phase scattering along Q and Q give z z y y x ≈ × ± ≈ peaks at Q = (5.00 0.25) ˚A−1, Q n 1.92 ˚A−1. For k = 5 (n 2) and x z z y ± ± ≈ × ± ≈ k = 8 (n 3), the patterns of enhanced peaks are similar to those at k = 0 and y ± ≈ k = 3, respectively. ± A cut along the streak at Q = (10.00 0.25) ˚A−1 in the plane k = 0 (Fig. 3) x ± shows enhanced Bragg peaks on the top of a continuum along Q . Because there is no z evidence of disorder along z, this continuum is essentially due to incoherent scattering [12]. It compares favorably with the Gaussian profile I exp Q2U , with U 0.04 10 − z zz zz ≈ ˚A2 [16]. Compared to the intensity I at Q = 0, Q = 0, the ratio I /I 2, 0 x z 0 10 ≈ with substantial uncertainty, is much greater than expected from conventional models accordingtowhichincoherentscatteringbyprotonsshouldbeproportionaltotheDebye- Waller factor DWF = exp (Q2U + Q2U + Q2U ), while incoherent scattering by − x xx y yy z zz other nuclei should be negligible. With U 0.035 ˚A2 [16], the expected intensity ratio xx ≈ Macroscopically entangled protons 6 ′ should be I /I 30. The observed intensity (I ) suggests that U is dramatically 0 10 10 xx ≈ reduced for these Q ,Q -values. x y ∗ Fig. 4 (top) shows a cut at k = 0, parallel to a , through the reciprocal plane at (5.0 0.2) ˚A−1 off-a∗. The enhanced Bragg peak observed at (8.5 0.2) ˚A−1 − ± − ± (spotted by an arrow) hides the underlying incoherent scattering signal. Alternatively, a cut at (4.2 0.2) ˚A−1 off-a∗ is free of Bragg peaks (Fig. 4, bottom). It reveals − ± the sharp profile of incoherent scattering centered at (9.0 0.2) ˚A−1 (spotted by an − ± arrow). Compared to the enhanced Bragg peak, the shift in position by 0.5 ˚A−1 is ≈ − consistent with the slope of the Qz axis. The half-width at half-maximum, ∆a∗ 0.4 ≈ ˚A−1, gives the half-width along Qx, ∆Qx ∆a∗/cos42◦ 0.3 ˚A−1, similar to that of the ≈ ≈ Bragg peak ( 0.2 ˚A−1). Consequently, U vanishes exclusively if Q matches a node of xx x ≈ the reciprocal sublattice of protons (see below). Otherwise, incoherent scattering is not observed, presumably because it is dramatically depressed as exp (Q2U ). Similarly, − x xx cuts around k = 0 (not shown) suggest that U vanishes sharply outside the plane. yy 2.2. Separation of proton and deuteron states: Raman scattering In Fig. 5, the Raman profile of the stretching mode νOD for the pure crystal is red- shiftedby 600cm−1 withrespecttotheνOHprofileofKHCO andthespectrumofthe 3 ≈ isotope mixture is a mere superposition of these profiles. There is no visible frequency shift for each profile and, therefore, there is no significant dynamical correlation between isotopes. By contrast, if protons and deuterons were paired up, a single νO(HpD1−p) profile should appear at an intermediate frequency, shifting smoothly from νOH (p = 1) to νOD (p = 0). This is clearly excluded. The spectra accord with separable proton and deuteron extended states consistent with the mixture of sublattices (1). In addition, the O O bond lengths are significantly different for KHCO (R = 3 H ··· 2.58 ˚A) or KDCO (R = 2.61 ˚A) and R = 2.59 ˚A for KH D CO is seemingly 3 D 0.76 0.76 0.24 3 a weighted average [16]. Isomorphous sublattices of protons and deuterons exclude a statistical distribution of O O distances. Otherwise, the νOH frequency should ··· vary significantly with p, for ∆νOH/∆R 5000 cm−1˚A−1 [22]. The absence of p ≈ frequency shift, or band broadening, suggests that a mixed crystal can berepresented by isomorphous sublattices p[KHCO ] and (1 p)[KDCO ], whose structures and dynamics 3 3 − are independent of p. These sublattices are not resolved by diffraction. 3. Theory Admittedly the enhanced features are anevidence of quantum correlations, but different mechanisms have been envisaged: symmetry related entanglement [12, 13, 14, 15]; quantum exchange [23]; spin-spin interaction [24]. They will be compared. The Raman spectra suggest that the Hamiltonian for a mixed crystal can be Macroscopically entangled protons 7 partitioned as = p( + + ) Hp HH HKCO3H CH,KCO3H (3) + (1 p)( + + )+ . − HD HKCO3D CD,KCO3D Hep , , represent the sublattices of H+, D+; , , those of heavy nuclei; HH HD HKCO3H HKCO3D , ,couplelightandheavynuclei. Thesublatticesareboundthroughthe CH,KCO3H CD,KCO3D electronic term . There is no dynamical coupling between the KHCO and KDCO ep 3 3 H subsystems which are supposed to be separable: any state vector of is a tensor p H product of state vectors of the subsystems. Furthermore, the adiabatic separation of and , or and , leads to factorable state vectors for the subsystem HH HKCO3H HD HKCO3D of bare protons (fermions), or deuterons (bosons) [23, 24, 25]. The separation of proton and deuteron states is, therefore, consistent with the “super selection rule” for fermions and bosons [26]. The mixed crystal is described with N , N , N ( = N N N ) unit cells labelled a b c a b c N j,k,l, alongcrystal axes(a),(b),(c), respectively. Thesublattices aredescribed with the sameunitcellcontaining4pKHCO and4(1 p)KDCO entities. Proton/deuteronsites 3 3 − ′ ′ are organized in two dimer entities indexed as j,k,l and j ,k,l, respectively, with j = j . For each dimer, dynamics are represented by normal coordinates, either antisymmetric α or symmetric α (r = 1,2,3), which are linear combinations of displacements ar sr { } { } ′ along x,y,z, respectively. The coordinates of the centers of symmetry, α , α , { 0r} { 0r} ′ ′ ′ ′ are (x ,y ,z ), (x ,y ,z ). For planar dimers, z = z = 0. Configurations LL and 0 0 0 0 0 0 0 0 ′ RR correspond to opposite signs for x and x . Although proton and deuteron sites 0 0 may be not strictly identical, their dynamics are formally represented by the same set of normal coordinates. The wave functions in the ground state are represented by gaussian functions Ψa (α ) and Ψs (α √2α ), with C = “LL” or “RR”. 0jkl ar 0jklC sr − 0rC A key issue of the theory is to clearly differentiate sites and states. Sites refer to local positional parameters, whereas each single state is extended over all indistinguishable sites. Conversely, the probability density (occupancy) at any site comprises contributions from every single state. The nonlocal observables (normal coordinates and conjugated momentums) are not bound to any particular unit cell. We should emphasize that nonlocality due to the lattice periodicity is not related to any physical interaction. It is energy free. This is totally different in nature from the long-range quantum exchange in a supersolid, like the crystal of He [27]. Obviousy, the proposal by Keen and Lovesey [23] that nonlocality could arise from non vanishing exchange integrals is incorrect, even for the nearest sites separated by 2.2 ˚A [28]. ≈ 3.1. Macroscopic states It iscustomary to represent latticedynamics withplane waves (phonons). Fordeuterons in the ground state, this can be written as: 1 Nc Nb Na |D+kςiC = √ XXX|Ξ0DjklςCiexp(ik·Ljkl); (4) N l=1 k=1 j=1 Macroscopically entangled protons 8 where k is the wave vector; L = ja+kb+lc is a lattice vector; a, b, c, are the unit jkl cell vectors. The wave functions for a unit cell, Ξ0DjklςC = Θ0DjklC+ςΘ0Dj′klC with ς = “+” or “ ”, is a linear combination of the wave functions for centrosymmetric sites: − Θ0DjklC({α0r}) = QΨa0Djkl(αar)Ψs0DjklC(αsr−√2α0rC) and likewise for Θ0Dj′klC({α0′r}). r For centrosymmetric dimers, there is no permanent dipole, so long-range correlations through inter-dimer interaction are negligible. Phonons are virtually dispersion-free [20, 21]. D+ is a single-deuteron state with an occupancy number (4 )−1 per site. This | kςiC N state cannot be factored with eigen states for local coordinates (x ,x ...). It is 1jkl 2jkl entangled in position. (By definition, a pure state of an arbitrary number of particles is said to be entangled when, because of its indivisible nature, it cannot be described by treating each particle separately.) Likewise, the state in momentum space is also entangled. In addition, this state meets the symmetry postulate for bosons stipulating ′′ ′′ ′′ that the sign should not change when any two dimer sites, say j,k,l, and j ,k ,l , or any two sites within dimers, are interchanged [29]. Alternatively, eigen states for fermions should be antisymmetric. An interchange ′′ ′′ ′′ of lattice sites j,k,l, and j ,k ,l , should transform a plane wave into the anti-phase wave and superposition of these waves should annihilate the amplitude. Hence, phonons are forbidden, except those matching the lattice periodicity, for which any interchange is neutral. Such states represent in-phase dynamics in each unit cell, namely ς = “+”, and in-phase dynamics for all unit cells, namely k L 0 modulo 2π, where L is H H · ≡ any vector of the sublattice of protons. This macroscopic quantum effect excluding elastic distortions has been named “super-rigidity” [13]. In addition, eigen states must be antisymmetrized with respect to interchange of proton sites within dimers, what is equivalent to changing the signs of the coordinates for the center of symmetry, say α α . To this end, the wave function of a dimer, supposed to benon planar 0r 0r { } −→ −{ } for the ease of notation, is rewritten as a linear combination of the wave functions for permuted sites as: Θ′ = 2−1/2[Θ ( α )+τΘ ( α )] (and likewise 0HjklCτ 0HjklC { 0r} 0HjklC −{ 0r} ′ forj ),withτ =“+”or“ ”. Then, antisymmetrizationrequiresaspin-symmetry, either − singlet-like (S for τ = +) or triplet-like (T for τ = ). Finally, with the translation − ′ ′ ′ invariant wave function of the unit cell, Ξ = Θ +Θ , protons states 0Hjkl+Cτ 0HjklCτ 0Hj′klCτ can be written as: 1 Nc Nb Na + ′ |H+iC = √ XXX|Ξ0Hjkl+C+i|Si; N l=1 k=1 j=1 (5) 1 Nc Nb Na + ′ |H−iC = √ XXX|Ξ0Hjkl+C−i|Ti. N l=1 k=1 j=1 For each configuration C, the ground states are entangled in position, momentum, and spin. The singlet-like and triplet-like states are degenerate because the spin symmetry is energy-free. The proposal by Sugimoto et al. [24] that the spin symmetry could be due to spin-spin interaction is incorrect because the actual coupling between nearest spin sites ( 104 Hz 10−6 K) is far too weak to give rise to long-lived correlations. ∼ ∼ Macroscopically entangled protons 9 For an isolated crystal, symmetry-related quantum-correlations are decoherence- free. Thanks to the adiabatic separation, there is no phonon-induced decoherence and, therefore, no transition to the classical regime, before melting or decomposition. In practice, some states can be transitorily disentangled by external perturbations, such as thermal radiations, but this is counterbalanced by spontaneous re-entanglement via decay to the ground state. In a standard gaseous atmosphere, the density of the surroundings is 1000-fold smaller than that of the crystal, so the impact ∼ of environment-induced decoherence to measurements is invisible. A source of disentanglement is the thermal population of excited proton states (for example γ a OH and γ OH at 1000 cm−1), as the a s splitting destroys the spin-symmetry, but s ≈ − this is marginal. Because the spin symmetry holds for both LL and RR ground states, for which there is no a s splitting, the sublattice at 300 K can be represented by a mixture − comprising a coherent superposition of super-rigid states for protons (5), on the one hand, and, on the other, a coherent superposition of phonon states for deuterons (4): H+(T) = [q′ (T) H+ +q′ (T) H+ ]; | i P LLH | τ iLL RRH | τ iRR τ (6) + ′ + ′ + D (T) = [q (T) D +q (T) D ]; | k i P LLD | kςiLL RRD | kςiRR ς with q′ (T) 2 = q (T), etc, in accordance with (1). | LLH | LLH 3.2. Neutron scattering The partial differential nuclear cross-section can be written as [30] d2σ k = f Xpi kf f Vˆ(r) i ki 2δ(~ω +Ei Ef). (7) dΩdE k |h |h | | i| i| − i i,f The summation runs over initial states i , E , weight p , and final states f , E . The i i f | i | i energy transfer is ~ω and dΩ is an element of solid angle. The interaction potential between the incident neutron and the target sample, Vˆ(r), comprises the interaction potential at every site. For diffraction, i f , E E , ~ω 0. Neutrons are i f | i ≡ | i ≡ ≡ diffracted by each single state. Because of the adiabatic separation, neutrons diffracted by proton or deuteron states are free of any contribution from heavy atoms. Moreover, neutrons diffracted by proton states probe exclusively the interaction potential at the proton sites and likewise for deuteron states, irrespective of whether or not these sites are identical. There is no Laue monotonic diffuse scattering. In general, Vˆ(r) does not have the periodicity of the lattice, for the potential will depend upon the nuclear position and spin orientation. However, this is not the case for the super-rigid states which match exactly the periodicity of the crystal. Upon momentum transfer to the target proton states (6), the final state is, in general, such that k L = 0 modulo 2π, so elastic distortions are generated. They f H · 6 are represented by ellipsoids at H/D sites in Fig. 1. The super-rigid states are probed without induced distortions when Q L = 0 modulo 2π. (8) H · Macroscopically entangled protons 10 Then, neutrons probe the spin-symmetry and the coherent cross-section is the total nuclear cross-section, σ 82.0 b, that is 45 times greater than the coherent cross- H ≈ ≈ section for regular Bragg peaks, σ 1.8 b, or 15 times greater than σ 5.59 Hc Dc ≈ ≈ ≈ b, or 3-fold greater than σ 27.7 b for heavy atoms. These scattering events ≈ KCO3c ≈ emerge from the diffraction pattern with spectacular contrast, even at large Q -values | | as they are not attenuated by the Debye-Waller factor. Besides, they are temperature independent. Forthe P2 /astructure, z isnot parallel to anaxis ofthesublattice ofprotons (Fig. 1 1), sothematching condition(8)isnever realized alongQ . Thesuper-rigidityalongz is z necessarily destroyed by measurements and intensities are proportional to exp U Q2. − zz z ′ On the other hand, since z = z = 0 for planar dimers, the spin-symmetry holds only in 0 0 the (x,y) planes, so the differential cross-section for enhanced diffraction can be written as [14] dσc (cid:12)Na′ Nb Nc (cid:12) expiQ lD [expiQ (kD ∆ /2) dΩ ∝ PP(cid:12)P P P z z { y y − y τi τf (cid:12)j=1k=1l=1 (cid:12) + τ τ expiQ (kD +∆ /2)] i f y y y (9) 2 2(cid:12) [expiQ (jD ∆ /2)+τ τ expiQ (jD +∆ /2)] x x x i f x x x (cid:12) × − } (cid:12) exp( U Q2) δ(ω), (cid:12) × − zz z × where 2 takes into account indistinguishable scattering events, either in-phase {···} ( S S and T T ), or anti-phase ( S T and T S ). Enhanced | i −→ | i | i −→ | i | i −→ | i | i −→ | i diffraction by (x,y) planes is anticipated at Q = 2πn /∆ , Q = 2πn /∆ for Hy y y Hx x x ± ± n even (in-phase scattering) and Q = 2π(2n + 1)/∆ , for n odd (anti-phase y Hx x x y ± scattering). Then, enhanced Bragg peaks occur at Q = 2πn /D , with intensities z z z proportional to exp U Q2. − zz z Otherwise, if Q = 2πn /D , the super-rigid (x,y) lattice is probed without z z z 6 distortion for the same (Q ,Q )-values, but the spin-symmetry is destroyed. Hx Hy Incoherent scattering is observed and the intensity is proportional to 2 σ exp (Q U )δ(Q Q +Q Q ), (10) Hi − z zz x − Hx y − Hy hence the cigar-like shape. Compared to a nonrigid lattice, the enhancement is merely due to U = U = 0. The observed intensity ratio I /I 2 is at variance with the xx yy 0 10 ≈ expected ratio of 1.5, but this discrepancy could be representative of errors in the ≈ background correction. Figures 2-4 can be thus rationalized with three different kinds of scattering events: (i) regular Bragg diffraction by the crystal lattice; (ii) enhanced diffraction by the fully entangled sublattice of protons; (iii) incoherent scattering by the super-rigid planes of protons.