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Models and calculations The single-molecule scattering can be exactly calculated by modelling the hydrogen molecules as freely rotating harmonic oscillators [3], taking into account the density and temperature dependence of the centre-of-mass kinetic energy of a quantum system [4]. Wavelength-dependent detector efficiency, attenuation, and the influence of the finite size of sample, container, and detectors, can all be taken into proper account. Experimental data The diffraction pattern of saturated liquid parahydrogen was measured with the D4C diffractometer of ILL, Grenoble Sample data: temperature T = 17.1 K pressure p = 29.9 bar molecular number density n = 22.95 nm -3 concentration of para species > 0.9995 (by using a catalyst in the sample container) Results The determination of the static structure of liquid hydrogen is demonstrated to be a feasible, though difficult, task. Reliable data can be obtained by the joint use of neutron diffraction and quantum-mechanical simulation [5]. U. Bafile (a), M. Celli (a), D. Colognesi (a), F. Formisano (b), E. Guarini (c), R. Magli (c,d), M. Zoppi (a) www.ifac.cnr.it/idrogeno bafile@ifac.cnr.it (a) Istituto di Fisica Applicata Nello Carrara, Consiglio Nazionale delle Ricerche, Italy (b) Istituto Nazionale per la Fisica della Materia, Operative Group in Grenoble, France (c) Istituto Nazionale per la Fisica della Materia, Unità di Firenze, Italy (d) Dipartimento di Chimica, Biochimica e Biotecnologie per la Medicina, Università di Milano, Italy Neutron scattering studies of quantum fluids: hydrogen www.ifac.cnr.it www.infm.it The microscopic structure of liquid hydrogen is a still open problem [1], because x-ray and neutron spectroscopy are of difficult application to such a system, a much harder case than deuterium, already solved a decade ago [2]. The microscopic dynamics of condensed hydrogen The physical problem is the study of the single particle dynamics in condensed quantum systems. From the high-energy, high-wavevector region of hydrogen spectrum (E > 100 meV, Q > 80 nm -1 ) one can obtain information on the momentum distribution and the density-dependent mean kinetic energy of the particle. From the low-energy region one can extract information on the Fourier transform of the velocity autocorrelation function (liquid) and on the phonon density of states (solid). Experimental data The incoherent scattering function of liquid and solid para-hydrogen was measured using inelastic neutron scattering by the TOSCA spectrometer at the pulsed neutron source ISIS (UK). Sample data:- pH 2 at low pressure and at seven temperatures (12 < T/K < 21) [6] and - along the isotherm T = 19.3 K, crossing the melting transition, with pressure 17 bar to 636 bar [7]. Models and calculations The scattering cross section is obtained in the hypothesis of the translational motion of the molecular center of mass uncoupled from the molecular internal motion. The internal dynamics is described by means of a quantum free rotating harmonic oscillator model [3]. The self dynamics of the molecular center of mass requires different models according to the energy and wavevector range investigated. References [1] F.J. Bermejo, K. Kinugawa, C. Cabrillo, S.M. Bennington, B. Fåk, M.T. Fernández-Díaz, P. Verkerk, J. Dawidowski, and R. Fernández-Perea, Phys. Rev. Lett. 84, 5359 (1998); A. Cunsolo, G. Pratesi, D. Colognesi. R. Verbeni, M. Sampoli, F. Sette, G. Ruocco, R. Senesi, M.H. Krisch, and M. Nardone, J. Low Temp. Phys. 129, 117 (2002). [2] M. Zoppi, U. Bafile, R. Magli, and A.K. Soper, Phys. Rev E 48, 1000 (1993); E. Guarini, F. Barocchi, R. Magli, U. Bafile, and M.-C. Bellissent-Funel, J. Phys.: Condens. Matter 7, 5777 (1995); M. Zoppi, U. Bafile, E. Guarini, F. Barocchi, R. Magli, and M. Neumann, Phys. Rev. Lett. 75, 1779 (1995). [3] J.A. Young and J.U. Koppel, Phys. Rev A 135, 603 (1964); M. Zoppi, Physica B 183, 235 (1993); E. Guarini, J. Phys.: Condens. Matter 15, R775 (2003). [4] M. Celli, D. Colognesi. and M. Zoppi, Eur. Phys. J. B 14, 239 (2000). [5] M. Zoppi, U. Bafile, M. Celli, G.J. Cuello, F. Formisano, E. Guarini, R. Magli, and M. Neumann, J. Phys.: Condens. Matter 15, S107 (2003); M. Zoppi, M. Neumann, and M. Celli, Phys. Rev B 65, 092204 (2002). [6] M. Celli, D. Colognesi, M. Zoppi, Eur. Phys. J. B. 14, 239 (2000). [7] M. Zoppi, D. Colognesi, M. Celli, Eur. Phys. J. B. 23, 171 (2001). [8] M. Zoppi, D. Colognesi, M. Celli, Europhys. Lett. 53, 34 (2001); M. Celli, D. Colognesi, M. Zoppi, Phys. Rev. E 66, 021202 (2002) Low-Q region results The agreement between the experimental self dynamic structure factor of the molecular center of mass with a quantum simulation is impressive [8]. High-Q region results As expected, a strong density dependence of the center of mass mean kinetic energy, characteristic of a quantum systems, is evident and the comparison with the simulation is excellent [6,7]. Experimental spectrum

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©D.D. Johnson and D. Ceperley 2010-2011 MSE485/PHY466/CSE485 1 Scalar Properties, Static Correlations and Order Parameters What do we get out of a simulation?

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