Wave Vector Transfer Research Articles

Vibrational modes of hemoglobin in red blood cells

Equine red blood cells were washed in saline heavy water (2H2O) to exchange the hydrogen atoms of the non-hemoglobin components with deuterons. This led to novel neutron scattering measurements of protein vibrations within a cellular system and permitted a comparison with inelastic neutron scattering measurements on purified horse hemoglobin, either dry or wetted with 2H2O. As a function of wavevector transfer Q and the frequency transfer v the neutron response typified by the dynamic structure factor S(Q, v) was found to be similar for extracted and cellular hemoglobin at low and high temperatures. At 77 K, in the cells, a peak in S(Q, v) due to the protein was found near 0.7 THz, approximately half the frequency of a strong peak in the aqueous medium. Measurements at higher temperatures (170 and 230 K) indicated similar small shifts downwards in the peak frequencies of both components. At 260 K the low frequency component became predominantly quasielastic, but a significant inelastic component could still be ascribed to the aqueous scattering. Near 295 K the frequency responses of both components were similar and centered near zero. When scattering due to water is taken into account it appears that the protein neutron response in, or out of, red blood cells is little affected by hydration in the low frequency regime where Van der Waals forces are thought to be effective.

Quasi-Elastic Light Scattering in Hopping Conductors -Effects of Sublattice-

An investigation of quasi-elastic light scattering by hopping particles is made on the basis of the relaxation mode theory. The prediction by Klein that the scattering cross section gives a nonzero value at zero wave-vector transfer when the polarizability depends on lattice site is confirmed in a sublattice system. The depolarized scattering, found in some superionic conductors, is also discussed in terms of the polarizability fluctuation of pairs of mobile and adjacent deformed immobile ions.

Neutron Scattering Studies of Surfaces and Interfaces

During the past decade, scientific and technological interest in the properties of surfaces and interfaces has grown at an astounding rate. On first thought, one might not consider a neutron or even an x-ray photon to be a particularly sensitive surface probe given their relatively weak interactions with matter compared to that of a low-energy electron or atom. Indeed, low-energy electron diffraction and atomic beam scattering techniques have contributed significantly to our understanding of surface phenomena. Nonetheless, the very fact that electrons and atoms are so strongly interacting makes quantitative analysis of their scattering data difficult. The interaction of neutrons or x-rays with matter, on the other hand, is weak enough that the potential can be characterized by a relatively simple scattering amplitude. Presently attainable neutron intensities, though not yet comparable to those of x-ray synchrotron sources, are still of sufficient strength to permit a variety of surface or near surface reflectivity and grazing angle diffraction experiments. Because neutrons can distinguish between different isotopes of the same element, most notably hydrogen and deuterium, as well as couple to atomic magnetic moments via a dipolar interaction, they can be indispensable and complementary probes.More conventional neutron diffraction techniques can also be applied to the study of interfacial phenomena and the effects of reduced dimensionality and compositional modulation in super-lattice structures grown by a variety of thin film deposition methods. In this article we will differentiate between reflectivity and diffraction measurements as follows: if the scattering occurs at a wavevector transfer low enough that the scattering medium appears as a continuum, so that amorphous and crystalline states are indistinguishable, then it will be considered to be in the reflectivity regime whereas diffraction will be taken to correspond to higher wavevector transfer where the precise arrangement of atoms is discernible.

Vibrational properties of the modulated spring chain: neutron cross section and dual spectrum

The inelastic neutron cross section is calculated for a modulated spring chain, which models an incommensurate composite system. The authors obtain a difference equation for the neutron cross section which has an exact and analytic solution for a periodic modulation. The structure of the neutron spectrum is studied as functions of wavevector and energy transfers for a wide range of model parameters. In general the spectra consist of bands of intensity, except at special wavevectors where bands collapse to a singular structure characteristic of normal modes of vibration.

A neutron scattering study of the impulse approximation in single-crystal beryllium

The inelastic scattering of high-energy neutrons from a single crystal of beryllium has been measured in order to test the validity of the impulse approximation. Separate measurements were performed, all with the sample at room temperature, with the wave-vector transfer along (100), (110) and (001) using the time-of-flight spectrometer HET at ISIS. For wave-vector transfers >or approximately=35 AA-1, corresponding to an incident neutron energy of approximately=1 eV, the authors find that the scattering is centred at the recoil energy given by the impulse approximation. Also, in this regime, the width is independent of the orientation of the wave-vector transfer with respect to the crystal axes and has a value consistent with that expected from a harmonic model of beryllium. They show that their experimental criterion for the validity of the impulse approximation is in agreement with a numerical calculation of the asymptotic behaviour of the exact incoherent neutron dynamic structure factor.

The vibrational motions in vitreous silica at high temperatures

Vitreous silica was studied at the temperatures 293 K, 673 K and 1073 K using inelastic neutron scattering. The zeroeth moment of the dynamic scattering function S( Q, E) agrees well with neutron diffraction results. It was found that the structure factors for elastic and inelastic scattering change with temperature in a Q region which reveals mostly long-range atomic interactions rather than those from nearest and next-nearest neighbours. The temperature variation was interpreted as if the structural and dynamical correlations between neighbouring SiO 4 polyhedra decrease when the temperature increases. Effective Debye-Waller factors were derived and found to increase linearly with temperature. It was also found that the vibrational amplitudes of the atoms in silica, within the experimental errors, are the same as those in high cristobalite. Through extrapolation to zero wave vector transfer of the spectrum of the longitudinal current-current correlation function, effective frequency distribution functions were obtained as well as generalized structure factors for inelastic scattering.

Collective intersubband resonances of inversion electrons on GaAs.

Collective intersubband resonance of inversion electrons in gated AlAs-GaAs single heterojunctions is studied with Fourier-transform spectroscopy using grating coupler techniques. Large grating periods are chosen to test the interface potential at nearly zero wave-vector transfer $q$ parallel to the sample plane. With highly efficient grating couplers up to three intersubband transitions are excited simultaneously. A coupling of the collective intersubband resonance to confined AlAs optical phonons is observed.

Two-dimensional quantum Heisenberg antiferromagnet at low temperatures.

It is argued that the long-wavelength, low-temperature behavior of a two-dimensional quantum Heisenberg antiferromagnet can be described by a quantum nonlinear $\ensuremath{\sigma}$ model in two space plus one time dimension, at least in the range of parameters where the model has long-range order at zero temperature. The properties of the quantum nonlinear $\ensuremath{\sigma}$ model are analyzed approximately using the one-loop renormalization-group method. When the model has long-range order at $T=0$, the long-wavelength behavior at finite temperatures can be described by a purely classical model, with parameters renormalized by the quantum fluctuations. The low-temperature behavior of the correlation length and the static and dynamic staggered-spin-correlation functions for the quantum antiferromagnet can be predicted, in principle, with no adjustable parameters, from the results of simulations of the classical model on a lattice, combined with a two-loop renormalization-group analysis of the classical nonlinear $\ensuremath{\sigma}$ model, a calculation of the zero-temperature spin-wave stiffness constant and uniform susceptibility of the quantum antiferromagnet, and a one-loop analysis of the conversion from a lattice cutoff to the wave-vector cutoff introduced by quantum mechanics when the spin-wave frequency exceeds $\frac{T}{\ensuremath{\hbar}}$. Applying this approach to the spin-\textonehalf{} Heisenberg model on a square lattice, with nearest-neighbor interactions only, we obtain a result for the correlation length which is in good agreement with the data of Endoh et al. on ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$, if the spin-wave velocity is assumed to be 0.67 eV $\frac{\AA{}}{\ensuremath{\hbar}}$. We also argue that the data on ${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$ cannot be easily explained by any model in which an isolated Cu${\mathrm{O}}_{2}$ layer would not have long-range antiferromagnetic order at $T=0$. Our theory also predicts a quasielastic peak of a few meV width at 300 K when $k\ensuremath{\xi}\ensuremath{\ll}1$ (where $k$ is wave-vector transfer and $\ensuremath{\xi}$ is the correlation length). The extent to which this dynamical prediction agrees with experiments remains to be seen. In an appendix, we discuss the effect of introducing a frustrating second-nearest-neighbor coupling for the antiferromagnet on the square lattice.

Neutron scattering as a probe of superconducting correlations in the new high T c superconductors

We suggest that a direct experimental estimation of the superconducting gap and the coherence length may be possible by neutron scattering experiments in the new high T c superconductors. We show that both orbital and spin currents associated with the Cooper pairs make equally important contributions to the inelastic neutron scattering, the former dominating the latter in the region of small wave vector transfer.

Large-scale oscillatory motions in myoglobin

Elastic and inelastic components of neutron scattering from lyophilized myoglobin were measured at wavevector transfers, Q of 1.0, 1.4, 1.85, and 3.0 A-1 and at temperatures of 77 and 298 K. The elastic scattering determines an overall effective mean square atomic displacement which was employed in a simple harmonic Langevin formalism to obtain effective force constants describing molecular deformations. These force constants and the inelastic data are compatible with alpha helices as a vibrating species at room temperature, and with amino acid residues as the vibrating entities at 77 K. The present measurements are consistent with recent molecular dynamics calculations.

Low frequency dynamics of chymotrypsin by neutron spectroscopy

The low frequency vibrational modes of enzymes have large amplitudes of vibration which should be related to conformational changes that occur during enzyme action. In the present paper we present inelastic neutron scattering measurements for α-chymotrypsin that show a peak in the low frequency spectrum. This peak is well defined at 77 K. Gaussian fits yield values of 0.93±0.05, 0.86±0.04, 0.81±0.05, and 0.87±0.06 THz for the peak position at wave vector transfers (Q) of 1.00, 1.40, 1.85, and 3.00 A-1, respectively. The full widths at half maximum are all greater than the resolution (0.2 THz) by at least a factor of two. At 298 K a weak peak at about 0.6 THz was observed for Q values of 1.0, 1.4 and 1.85 A-1. The data are interpreted in terms of the allowed oscillations of a large globular protein treated as an elastic sphere. Assuming a Raman active oscillation at 0.9 THz it is shown that a peak in the neutron scattering response at 0.6 THz may arise from a rotational shear mode of the chymotrypsin molecule.

Dynamic form factor of liquid4He at intermediate momentum transfer

Measurements of the inelastic neutron scattering from liquid4He atT=1.2 K in the range of wavevector transfer 3≤Q≤12 A−1 have been performed to verify the oscillations in the peak position and the width of the scattering function withQ observed by Cowley and Woods and by Martelet al. We confirm these oscillations and extend the measurements to show that the oscillations in the width continue up toQ≈12 A−1 while oscillations in the peak position, which have a much smaller amplitude, could be observed with confidence up toQ≈8 A−1 only. We also present a straightforward RPA calculation of the dynamic form factorS(Q, ω), beginning from the pair interatomic potential, to see how well the observedS(Q, ω) can be described and to investigate the origin of the oscillations. The observedS(Q, ω) is quite well reproduced and the oscillations in the width and peak position ofS(Q, ω) are seen to originate from oscillations in the interaction in the RPA which we have approximated as the4He-4He scattering amplitude calculated fromv(r). While the present calculation is quite different from the model proposed previously by Martelet al., the physical origin of the oscillations inS(Q, ω) is the same, namely oscillations in the4He-4He atom scattering amplitude.

On multiple small-angle scattering

An analytical approach has been taken for analyzing the multiple scattering effects in small angle scattering (SAS) from both monodisperse and polydisperse systems. Two limiting regions, viz the Guinier region and the Porod region have been studied. A modified form of Guinier law has been deduced for the scattered intensity distribution in the region of small wave vector transfer, q. In the regionq(=|q|)→∞, it is shown that the effect of multiple scattering does not alter the Porod (q −4) law. In the case of polydisperse systems, a correlation has been established between the size distribution of the inhomogeneities and the experimentally extractable parameters. The validity of the formalism has been examined by reinterpretation of the multiple SANS data (Hardman-Rhyne and Berk 1985) on polydisperse Al2O3 samples. This formalism is useful in characterizing the inhomogeneities from SAS measurements, particularly when thick samples are used.

Analysis of small angle neutron scattering data from strongly interacting polydisperse ionic micellar solutions

AbstractSmall angle neutron scattering (SANS) intensity distributions from ionic micellar solutions without added salt generally show a prominent interaction peak at finite Q (magnitude of the wave vector transfer) related to a certain inter‐micellar correlation distance. Analysis of this type of data requires, aside from the structural model of the micelle itself, a statistical mechanical theory for dealing with the inter‐micellar correlations. We present a method for a consistent analysis of SANS data taking into account the surfactant chain packing, the aggregation number polydispersity, and the effective micellar charge. The micelle is modelled as a two‐region spheroidal particle and the inter‐micellar correlations calculated according to a generalized one‐component macroion (GOCM) theory. GOCM uses an effective inter‐micellar pair potential which is a finite concentration extension of the well‐known Derjaguin‐Landau‐Verway‐Overbeek (DLVO) double layer interaction potential. Two micellar solutions are treated as examples, namely that of sodium dodecyl sulfate (SDS) and sodium 1,2‐bis(2‐ethylhexyloxycarbonyl)ethanesulfonate (AOT). The effects of polydispersity are appreciable immediately above the critical micellar concentrations (CMC). Both the structural parameters of the micelle and the free energies of micellization and micellar growth can be extracted from SANS data.

Dynamics of quantum liquids at high momentum.

The dynamic form factor S(Q,\ensuremath{\omega}) in liquid $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ is evaluated in the wave-vector transfer range 3\ensuremath{\le}Q\ensuremath{\le}15 A${\r{}}^{\mathrm{\ensuremath{-}}1}$. The input is the pair interatomic potential, developed by Aziz et al. The S(Q,\ensuremath{\omega}) is calculated within the random-phase approximation (RPA) which becomes valid when \ensuremath{\Elzxh}Q is much larger than the average momentum in the liquid. A T-matrix approximation represents the interaction appearing in the RPA. The aim is to explore how well S(Q,\ensuremath{\omega}) can be described for 3\ensuremath{\le}Q\ensuremath{\le}15 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$ from first principles. In $^{3}\mathrm{He}$, we find S(Q,\ensuremath{\omega}) is a broad, nearly Gaussian function, centered just below the recoil frequency having a width and shape that agrees well with experiment. It does, however, have tails at high frequency which make important contributions to its moments. In $^{4}\mathrm{He}$, S(Q,\ensuremath{\omega}) is a more sharply peaked function which also agrees quite well with experiment. We are able to reproduce the oscillations in the peak position and in the width of S(Q,\ensuremath{\omega}) with Q in liquid $^{4}\mathrm{He}$ observed by Martel et al. In the present model, these oscillations originate from oscillations in the magnitude of the T-matrix interaction with Q. The corresponding oscillations are predicted to be very small and probably unobservable in liquid $^{3}\mathrm{He}$.

Electronic screening at the surface of an sp-bonded metal

A review is given of recent work on the screening response of the conduction electrons at a simple metal surface. Particular emphasis is placed on the density response function at wave vector transfers typical of electronic excitations involved in, e.g., coupling to an adsorbate. This means that q|| (the wave vector transfer parallel to the surface) can range from a small fraction of kF to q|| > kF, where kF is the Fermi wave vector. This necessitates an entirely microscopic description of the screening process. The solution of the self-consistent problem for the ground state is considered first (for the jellium model). The response problem is then addressed within self-consistent field response theory. The physical model used in the evaluation of the effective potential in the ground state determines the dynamical model for the response function. Several response theories are discussed in unison according to the procedure used to introduce (or entirely drop, as in the case of the random-phase approximation) the many-body effects of exchange and correlation. Applications to dynamic and static screening processes are considered. Examples of dynamical processes are the damping of a vibration of an adsorbed molecule due to decay into electron-hole pairs, and the excitation of collective modes in an atomically-thin overlayer of alkali-metal atoms adsorbed on a metal surface. In the static limit, recent work on lattice relaxation at Al(110) is discussed. In each case contact is made with the relevant experimental evidence.

Coupled interface plasmons of the Ag-Si(111) system as investigated with high-resolution electron energy-loss spectroscopy

Using the high-resolution electron energy-loss spectroscopy technique (HREELS), we have probed ultra-thin films of silver deposited under UHV conditions onto a clean silicon (111) substrate displaying a (7 × 7) superstructure. Coupled interface plasmons of the Ag layer with its adjacent media (vacuum and Si) have been clearly observed. The dispersion of these interface plasmons has been plotted versus q| D where q| where q| is the wavevector transfer and D is the Ag layer thickness. The experimentally found dispersion behaviour can be understood within the dielectric theory applied to a two interfaces-bounded layer.

Scattering of X-rays from crystal surfaces

X-ray measurements performed on a variety of materials demonstrate that it is possible to observe diffuse scattering that originates in the abrupt change of density at a crystal surface. Such a discontinuity gives rise, in general, to rods of scattering in reciprocal space which are most intense close to the Bragg peaks tau and are well defined for sufficiently smooth surfaces. For wave-vector transfer Q= tau +q the q-dependence of the intensity of scattering gives information on the topographic structure of the crystal surface. Experimental results on crystals of GaAs and KTaO3, with surfaces prepared in various ways, were obtained using conventional X-ray techniques with a rotating anode source and can be described by a continuum model of the surface. There are discrepancies between the predictions of the models and the experimental results and these suggest that further experiments are needed to achieve a more complete understanding.

Quasielastic light scattering from diamond.

Strong quasielastic features have been found in light scattering from diamond in the frequency range 0 to \ifmmode\pm\else\textpm\fi{} 200 GHz and have been investigated as a function of wave-vector transfer ($Q$) and temperature ($T$). The evolution of a relatively narrow Lorentzian component (\ensuremath{\sim} 10 GHz) demonstrates for the first time the manner in which a two-phonon difference spectrum in the collision-free regime (low $T$ and large $Q$) develops into entropy-fluctuation scattering in the collision-dominated regime (high $T$ and low $Q$).

Quasi-elastic neutron-scattering studies of intercalated molecules in charge-deficient layer silicates. Part 2.—High-resolution measurements of the diffusion of water in montmorillonite and vermiculite

The relative influence of water–surface and water–cation interactions in hydrates of the charge-deficient expanding-lattice layer silicates montmorillonite and vermiculite have been studied using high-resolution quasi-elastic neutron-scattering (QENS) measurements made with the back-scattering spectrometer at ILL, Grenoble. Detailed measurements have been made Ca2+–montmorillonite equilibrated at seven different water-vapour partial-pressure (p/p0) values and for Ca2+, Mg2+ and Na+ vermiculite samples at fixed p/p0.The QENS data are fitted to a model consisting of an elastic and a single Lorentzian quasi-elastic component plus a flat background. The quasi-elastic linewidth and the relative intensities of the three components are adjustable parameters. The resultant quasi-elastic broadenings were approximately five times smaller than the values derived previously from measurements at lower energy resolution. Also, the quasi-elastic scattering intensity increased with p/p0 and decreased with increasing scattering angle, the latter being the reverse of the behaviour observed at the broader energy resolution.The low- and high-resolution measurements taken together indicate the occurrence of two different phases of translational motion of water molecules not directly coordinated to the exchangeable (Ca2+) cations. These are attributed (i) to rapid localized motions in ‘cages’ bounded by the silicate sheets and the hydrated cations and (ii) to slower, longer-range inter-cage diffusion. In addition, the relative intensities of the quasi-elastic scattering at high and low wavevector transfer suggest that the high-resolution measurements also observe rotations of the complete six-fold coordinated hydration shells of the cations about one of the C3 axes of the octahedron. To explain the observed width, this rotation must have a correlation time of ca. 10–10 at ambient temperature.The present QENS data yielded effective diffusion coefficients for water in Ca2+–montmorillonite which increased rapidly from < 1.0 × 10–10 m2 s–1 at p/p0= 0.15 to ca. 4.5 × 10–10 m2 s–1 at p/p0= 0.33. At higher values of p/p0 the effective diffusion coefficients remained approximately constant within the limits of experimental error. For water adsorbed by vermiculite having Na+, Ca2+ or Mg2+ as the exchangeable cation the diffusion coefficient measured at p/p0= 0.76 was found to be significantly smaller than the corresponding value for Ca2+–montmorillonite. This may be attributed to the hindering effect of the more densely packed network of hydrated cations in the more highly charged vermiculite.