Interview Questions
Q1: What are the nuclear quantum effects (NQEs), and why and when are they important?
A1: One can categorise the NQEs into two groups: the so-called trivial [such as the zero-point energy (ZPE)] and non-trivial (such as nuclear quantum delocalisation and tunnelling).
The ZPE was historically the first effect, postulated as one of the consequences of the laws of quantum mechanics.
A classical nucleus (which can be approximated as a particle with infinite mass) is completely localised in space. If a classical nucleus is not in motion, the net external forces acting on this nucleus are zero. It is a point of local equilibrium, where not only the net forces are zero, but also the potential representing these net forces has a local minimum in space. As the particle is localised, its velocity, and in consequence, its kinetic energy, is equal to zero. Moreover, as we can always set the potential energy scale of a classical particle arbitrarily, we can set it to zero at the location in space corresponding to the minimum of the potential of the net external forces acting on a classical particle. Consequently, the total energy (sum of the potential and kinetic energies) can attain a value of zero. Physically, such a state of zero total energy of a classical particle corresponds to a classical particle at zero absolute temperature.
For a quantum nucleus, the lowest possible energy that it can have is always non-zero and is equal to the ZPE. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state, as described by the Heisenberg uncertainty principle. The uncertainty principle states that no object can have precise position and velocity values simultaneously. The uncertainty principle requires every quantum mechanical system to have the value of the zero-point energy greater than the minimum of its classical potential well. This results in a finite total energy value of a quantum nucleus even at absolute zero temperature.
The term ZPE is sometimes also referred to as 'zero-point motion'. However, one has to use the term 'motion' with a pinch of salt. In classical mechanics, motion is associated with the concept of a well-defined set of positions of a particle in time. In quantum mechanics, due to the Heisenberg uncertainty principle, particles are delocalised in space at any given time (nuclear quantum delocalisation). They can be described more as fuzzy objects with no sharp boundaries and exact positions in space. Mathematically, the quantum particles are described by distributions of positions and momenta. The Heisenberg uncertainty principle states that there is a mathematical correlation between the width of the position and the width of the momentum distribution of a quantum particle. This correlation exists at any given temperature and for any given value of the total energy of the quantum particle (not only at the absolute zero temperature). Thus in principle, by measuring the width of the particle momentum distribution using Vesuvio, one can infer the width of the position distribution and vice versa; by measuring the width of the position distribution (root mean squared displacement) using a diffractometer, one can infer the width of the momentum distribution of a quantum particle. The special case of such an experimental protocol is when it is conducted at the limit of absolute zero temperature. In such a case, one can speak about the measurement of the ZPE, either via the width of the position or the momentum distribution. In practice, due to the temperature dependence of these distributions, usually governed by the Boltzman statistics, it suffices to measure one of them at temperatures below 80K. When one does it in a diffraction experiment, the value of the root-mean squared displacement can be converted into the value of the potential energy. In the case of a Vesuvio experiment, one can convert the width of the momentum distribution into the value of the kinetic energy. As the most commonly used model to describe nuclei in chemical bonds in condensed matter systems and molecules is the quantum harmonic oscillator (QHO), one can use the theory of the QHO to calculate the value of the ZPE. The value of the ZPE for the QHO is simply twice the value of the potential energy or twice the value of the kinetic energy. (By virtue of the virial theorem as applied to the QHO, the value of the kinetic energy is equal to the value of the potential energy). For a single-dimensional QHO with frequency omega, the value of the ZPE is equal to ½ *hbar*omega. Thus, the value of the nuclear kinetic energy (and the value of the potential energy) corresponding to the ZPE is equal to ½*½ *hbar*omega = 1/4 *hbar*omega
For the QHO, the steeper e the walls of the confining potential, the higher the QHO frequency. In other words, the stronger the confinement, the bigger the value of the ZPE. Conversely, the flatter the potential curve of the QHO, the smaller is the value of the ZPE. Thus, in the limit of the vanishing potential of the QHE (a completely flat potential), the ZPE associated with the QHE also vanishes. Interestingly, this observation can be employed to set the lower conservative bound for the amount of the total (and kinetic) energy of a quantum particle. For a free particle, the potential energy is zero, and the entire energy is equal to the kinetic energy of the motion of the free particle. Now, the lower limit of the kinetic energy of a quantum particle can be set to a value of the kinetic energy of its classical counterpart (classical particle of the same mass) that is roaming free in space with no confinement whatsoever. In the classical gas theory, the value of the total energy is equal to 3/2kT in such a case. Thus, the mean kinetic energy of a quantum particle can be decomposed into a free-particle contribution proportional to the temperature and a non-classical excess kinetic-energy term proportional to the gradient of the mean force representing the confinement (which can be modeleddelled as the QHO) evaluated at zero value of displacement.
The last type of NQE to be discussed is nuclear quantum tunnelling. It can be either translational or rotational tunnelling. Tunnelling is a purely quantum phenomenon, which does not have any classical analogue. In this sense, it is 'non-trivial'. It amounts to a quantum particle penetrating the walls of the confining potential. As a result, parts of the wavefunction of the particle stretch into regions of space beyond the potential walls. The most iconic example is a particle penetrating into the other side of a barrier. A physical example of such behaviour is radioactive decay, where the state on the other side of the barrier corresponds to a state after the decay. Interestingly, for a QHO, the wavefunctions also stretch beyond the walls of the potential, and thus the quantum particle in a QHO also tunnels. A good model of tunnelling is the potential of a hydrogen bond, which is a function in space that has two minima separated by a local maximum (a barrier). For example, in water, protons may be initially localised in one of the minima representing a state in which protons are covalently bound to the donor oxygen atoms. Then they may tunnel into the other minimum representing the position closer to the acceptor oxygen of an adjacent water molecule. Depending on the strength of the hydrogen bond, the shape of the potential changes. For strong bonds, the potential has a well-pronounced maximum between the two minima. As the bond weakens, the barrier (the maximum) weakens, and for moderate hydrogen bonds, one reaches a flat-bottomed potential. Interestingly, even in this case, technically, one can speak of tunnelling as the proton wavefunction has a shape of a flat hat which reflects delocalisation around the centre of the potential, which is a signature of tunnelling.
Rotational tunnelling is a phenomenon whereby a quantum particle is likely to be simultaneously found in a set of positions related by rotation of 360 degrees/n around the origin of a binding potential with an n-fold rotational symmetry. Physically, it can be exhibited by protons in a methyl group or water molecules confined in potential with rotational symmetry.
Both rotational and translational tunnelling can be measured by neutron Compton scattering on Vesuvio, but it also can be measured using QENS and NMR. In the case of both QENS and NMR, the tunnelling manifests itself by the presence of additional peaks in the spectrum. In the case of neutron Compton scattering, the signature of tunnelling are distortions of the shapes of the nuclear momentum distributions beyond the usually observed Gaussian shape. In general, these distortions lead to a bi-modal nuclear momentum distribution, with a second, less-pronounced maximum lying on the shoulder of otherwise Gaussian distribution. The distortions of the shapes of momentum distributions lead to anomalously high values of nuclear kinetic energy (well above that predicted by QHO and Gaussian nuclear momentum distributions).
The nuclear quantum effects are always present, no matter which point (P, T) at a phase diagram is taken into account. However, in some cases, they are more visible and lead to more pronounced effects and consequences. The degree of influence of the ZPE and nuclear quantum delocalisation on the properties of materials is measured by the ratio of the de Broglie wavelength of a nucleus to a lattice constant of a crystal or a typical bond length. It depends on its mass and on the temperature. In the case of protons and other lightweight nuclei, sizeable effects are up to room temperature. However, in the case of nuclei as heavy as copper, they are still manifested around 10K. Also, the higher the pressure at which a given quantum system is measured, the more it becomes for nuclear quantum effects to be observed and for them to have an influence on material properties. The most iconic examples here are some very exotic phases of hydrogen predicted and observed to exist under very high pressure of hundreds of GPa. Generally, the higher the pressure, the higher is the degree of confinement of nuclei and, thus, the higher the degree of their quantumness.
Q2: Why are nuclear quantum effects important? They are important factors influencing many material properties.
A2: NQEs dictate the strength of hydrogen bonds. There is an interplay between the amount of nuclear delocalisation and the strength of the bending and stretching modes that leads to a net effect of strengthening or weakening of the hydrogen bonds.
ZPE, nuclear quantum delocalisation and tunnelling change the demarcation lines between different phases in phase diagrams of many materials. In general, the higher the ZPE and the higher the likelihood of tunnelling, the lower the effective barrier separating two adjacent phases.
NQEs can lead to anomalous mechanical properties, such as superfluidity or postulated supersolid behaviour of helium, whereby either liquid or solid (postulated) can move without internal friction.
NQES can influence the transport properties, especially of lightweight nuclei on surfaces and in confined geometries (hydrogen on metal surfaces, hydrogen and water in confined geometries).
NQEs influence a lot of thermodynamic properties, such as heat capacity, entropy, enthalpy, etc. Interestingly, the heat capacity of water would have been much higher than it is, had it not been for the fact that protons are quantum objects. Also, the pH of water with classical protons would have been ca. 9.5 instead of 7. This would probably destroy life as we know it or completely alter the kinetics of biochemical reactions and the physiology of plants and animals.
The difference of ZPEs of two different isotopes of a given nuclear species between two phases (for example, water and vapour) is directly linked to the isotope fractionation ratios of these two isotopes. These ratios are very important in the technological processes of isotope separation. Traditional methods of isotope separation are very costly and time-consuming, and not very environmentally friendly. The employment of ZPE and other NQEs in the separation of H2 and D2 can lead to much more efficient and less costly processes. Interestingly, the H/D isotopic ratio is different in living tissue and depends on the metabolic rate. For instance, H/D ratio is found different in leaving plant leaves compared to dead leaves. It has been postulated that, in principle, the H/D fractionation ratio could be used to detect signs of life as we know it on exoplanets.
Q3: How can the NQEs be measured?
A3: Historically, the ZPE was first proven to be a real effect in 1924 by Muliken during his analysis of isotope effects in electronic-vibrational spectra of molecules. When analysing the vibrational structure of the electronic spectra of the two molecules, 10B 16O and 11B 16O, Mulliken found out that the best description of emission spectra is possible only by taking into account the term 1/2 (ZPE) in formula E_n = (n+1/2)*h*omega, where omega is the frequency of vibration.
ZPE of atoms can also be inferred from the analysis of diffraction data. Laue was the first to derive formulas for X-ray scattering from crystals in 1912 on the assumption that the atoms are point-like and immobile. The influence of temperature on the intensity of X-ray reflection from the crystals was first considered theoretically by Debye in 1913; it was later modified by Waller in 1925. The heating of the crystal lattice induces irregular deflections of atoms from equilibrium. In order to take the thermal motion of the atoms into account, the crystal is considered to be an equilibrium thermodynamic ensemble whose members pass through all possible vibrational states during measurement. The measured intensity constitutes the average taken over all nuclear configurations. For this reason, X-ray scattering by such oscillating atoms is out of phase with that of an ideal lattice, and its intensity is lower. Evidently, the higher the temperature, the lower the intensity of the observed peaks corresponding to Bragg's reflection. Debye and Waller showed that the intensities of interference maxima should be multiplied by factor exp(-2W), where W is inversely proportional to the Debye temperature of the crystal and proportional to a function f(x)/(x+1/4) (Debye introduced zero-point energy into his formula in the form of factor ¼ as early as 1913). In the late 1920s, James and Firth measured X-ray scattering by rock salt (NaCl) crystals at different temperatures (down to a liquid-nitrogen temperature). Based on the results obtained, James, Waller, and Hartree published a
paper that turned out to be the first experimental confirmation of zero-point energy of atoms in a crystal lattice. The authors showed that the experimental findings agree with the computations only if the zero-point energy of atoms is taken into account, as is especially clear at large scattering angles. This seems natural because the greatest relative contribution of ZPE occurs at high-order reflections.
As far as the ZPE assessment is concerned, the situation with neutron diffraction by crystals is somewhat better than in the case of X-ray diffraction because neutrons are scattered from nuclei, and measurements of the temperature dependence of the Debye-Waller factor would give direct confirmation of ZPE since this factor cannot be unity even at absolute zero. However, absolute measurements of the Debye-Waller factor in the case of neutrons (as well as X-rays) are hampered by the weakening of the incident and diffracted beams (e.g., due to extinction) and the presence of the diffuse background. Moreover, a detecting system always possesses a finite resolving power that accounts for the broadening of the observed diffraction peaks and their decreased height.
The temperature dependence of the Debye-Waller factor was measured by many authors, but most of them were focused on studying the anharmonicity of atomic vibrations, i.e., the temperature dependence of the Debye temperature. The situation changed when C Wilson employed the single crystal neutron diffraction using SXD to study methyl group mean-square torsional amplitude (torsional RMSD) in paracetamol. RMSD obtained from TLS analysis of refinements of anisotropic displacement parameters was fitted by a linear function of temperature, and the zero-T intercept yielded the value of ZPE. Interestingly, in theory, from Rietveld refinement, one gets site-specific values of ZPEs, whereas using neutron Compton scattering at its present incarnation and precision level, as implemented on Vesuvio, the measurements of ZPEs are isotope specific but not site-specific.
Since the value of the Debye-Waller factor is practically determined at low temperatures by the ZPE alone, it opens up ample opportunities for studying the potential energy of the system of interest. In this respect, the experiment on the behaviour of water in nanotubes described by Kolesnikov et al. is very illustrative. Water in one-dimensional channels is of great interest to biology, geology, and materials science. Carbon nanotubes provide an excellent model for the investigation of such systems because the interaction between carbon and water molecules is very weak, and this concerns practically quasi-one-dimensional water. Kolesnikov et al. measured the elastic scattering of neutrons in such a system and compared the results with similar measurements in ice. When one plots temperature dependences of root-mean-square vibrations of water molecules, their amplitude in a nanotube is much higher than for ice. The authors concluded that water is present in the nanotube in a wide double-well potential. They estimated the delocalisation of water molecules at roughly 2 Angstroms.
The discovery of the Mossbauer effect in the early 1960s opened up the possibility of directly observing the ZPE of lattice atoms. Unlike nonresonant elastic scattering of X-rays by crystals, the Mossbauer effect comprises the resonant scattering of gamma rays by the nuclei of a crystal lattice. Because emission (scattering) is exercised by the lattice nuclei, phonons are also involved in the process. Therefore, the temperature dependence of the Mossbauer effect is directly related to atomic vibrations, including both thermal vibrations and ZPE. For a nucleus in a crystal lattice, the energy is consumed only for lattice excitations (phonon production or absorption). The lower the lattice temperature, the fewer phonons it contains and the higher the probability of phonons being uninvolved in the emission and absorption of a gamma quantum by the nucleus [phonons comprise Bose particles for which the probability of absorption (emission) is proportional to the number of particles present in the system]. The Mossbauer effect actually exhibits the phononless emission (absorption) of gamma quanta. In the Debye model of the spectrum of crystal vibration excitations, the relative intensity of the Mossbauer line, i.e., the fraction of the emission (absorption) processes proceeding without alteration of the vibrational state of the crystal, is actually determined by the Debye-Waller temperature factor, which contains ZPE. Results of Moesbauer experiments show that factor 2W at low temperatures depends on ZPE rather than thermal vibrations; in other words, the Debye-Waller factor always differs from unity (it is equal to ca. 0.8 below 80K).
Observation of quantum effects in macroscopic oscillators also allows for direct measurements of ZPE. In order to transfer a macroscopic oscillator to a quantum state, the mean number of thermal quanta (the Bose factor) must be close to unity (the oscillator is then close to the ground state). In other words, the vibrational quantum energy must be higher than the thermal energy, kbT. This means, for example, that for a pendulum swinging with an acoustic frequency of roughly 1 kHz, temperatures on the order of 50 nK are needed that are impossible to reach by standard cooling techniques. Thus, the oscillator frequency must be around 1 GHz at real temperatures on the order of 50 mK, attainable with a refrigerator. To this end, two studies published in 2005 and 2010, respectively, contributed to considerable progress in the creation and investigation of macroscopic oscillators.
Last but not least, the NQE's can be best observed on VESUVIO, using the techniques of neutron Compton scattering (NCS) (or, in other words, deep inelastic neutron scattering DINS), the resonant neutron absorption followed by gamma emission in the transmission mode (NRTA) , as well as the resonant neutron absorption followed by gamma emission in the scattering mode (NRCA).
The neutron scattering cross-section in the proximity of a resonance centred at the energy value E0 and experienced by a neutron of incident energy E can be expressed as the sum of the two contributions related to (i) the process where a neutron is absorbed, and a gamma ray is emitted, and (ii) the process where a neutron is absorbed and a neutron is emitted. The line shape of the resonance is well described by a Lorentzian function, according to the theory by Breit and Wigner. The width of the Breit-Wigner function is a consequence of the instability of the excited nucleus after the absorption of the incoming neutron. In addition to the Lorentzian broadening, the resonance line shape is furthermore widened by two additional effects that manifest themselves as a Gaussian broadening. The first term arises from the uncertainty in the time-of-flight measurement because of the finite size of the moderator. The second contribution to the Gaussian broadening of the resonance comes from the motion of the nucleus induced by its chemical bonding to surrounding atoms. Such Doppler broadening can be expressed as proportional to the atom's mean kinetic energy and, thus to its ZPE at low temperatures.
NRCA and NRTA have been successfully employed as an alternative to the NCS in studies of systems consisting of heavyweight isotopic species of similar masses. In such a situation in the NCS the recoil peaks would have overlapped too much to be analysed precisely. Instead, in NRCA or NRTA spectra, the peaks are only observed from those isotopic species that actually absorb neutrons, and those peaks are very well isolated in the time of flight spectra. However, experimentally, there are two drawbacks of the NRCA and NRTA techniques compared to the neutron Compton scattering. First, the resolution function can dominate the Doppler broadening component for samples of relatively small thicknesses, and in order to mitigate this problem, samples in the form of very thin (micrometre thick) foils or thin films need to be employed. Secondly, given the current resolution of the NRCA and NRTA techniques, detailed studies of shapes of nuclear momentum distributions (i.e., parameters of nuclear momentum distributions beyond the ZPE that can be inferred from the width of the distribution) are, at present, not possible.
The technique of neutron Compton scattering allows for precise measurements of the shapes of nuclear momentum distributions (NMDs) of lightweight nuclei (H, D) and the widths (proportional to the values of ZPEs) of nuclear momentum distributions of heavier nuclei. In the case of H and D, it allows extracting the shapes of the effective Born-Oppenheimer (interatomic) potentials by numerically inverting the nuclear Schroednigner equation (i.e., obtaining the shape of the potential from the known shape of the wave function of the nucleus, which can be expressed as the square root of its measured NMD, and the value of the nuclear kinetic energy measured in the experiment from the width of the NMD). In principle, no theoretical assumptions are needed for this numerical inversion problem, and thus it can be performed in a model-free manner. The shapes of NMDs are obtained in the process of NCS data reduction from the underlying shapes of the recoil peaks of individual isotopic species present in the time-of-flight NCS spectrum.
Interestingly, there exists another technique which is directly related to the NCS, the electron Compton scattering (ECS). The ECS experiments can be understood assuming that the electrons scatter from a moving target of vibrating atoms. The nuclear motion causes a Doppler shift in the energy of elastically scattered electrons. Electrons at low energy have insufficient momentum. This changes when the electron energy is increased in the keV range. At the same time, the interaction of fast electrons with the target becomes weaker and weaker. The mean free path for elastic and inelastic excitations is greater than or equal to 100 Å at energies above 20 keV. Thus it appears possible, at least in principle, to measure Compton profiles using energetic electrons that have been deflected over large angles by nuclei. The main obstacle that one has to overcome in order to use electrons for the study of lattice vibrations is the large mass difference between the electron and nuclei. The proton mass is 1836 times larger than that of an electron. The maximum transfer of momentum in the case of 180° deflection is twice the momentum of the incoming particle. This means that even for an electron backscattering from the proton, the maximum energy transfer is 0.2%. For heavier atoms, the transferred energy decreases even more, inversely proportional to the atomic mass. Thus we need a relatively monochromatic electron beam and a good quality energy analyser in order to see the effect of energy transfer and an even better one to resolve the effect of atomic vibrations on the elastic peak.
