An investigation into the existence of zero-point energy in the Rock-Salt Lattice by an X-Ray Diffraction Method

In the present paper we shall attempt to collate the results of four separate lines of research which, taken together, appear to provide some interesting checks between theory and experiment. The investigations to be considered are (1) the discussion by Waller* and by Wentzel,† on the basis of the quantum (wave) mechanics, of the scattering of radiation by an atom ; (2) the calculation by Hartree of the Schrödinger distribution of charge in the atoms of chlorine and sodium ; (3) the measurements of James and Miss Firth‡ of the scattering power of the sodium and chlorine atoms in the rock-salt crystal for X-rays at a series of temperatures extending as low as the temperature of liquid air ; and (4) the theoretical discussion of the temperature factor of X-ray reflexion by Debye§ and by Waller.∥ Application of the laws of scattering to the distribution of charge calculated for the sodium and chlorine atoms, enables us to calculate the coherent atomic scattering for X-radiation, as a function of the angle of scattering and of the wave-length, for these atoms in a state of rest, assuming that the frequency of the X-radiation is higher than, and not too near the frequency of the K - absorption edge for the atom.¶ From the observed scattering power at the temperature of liquid air, and from the measured value of the temperature factor, we can, by applying the theory of the temperature effect, calculate the scattering power at the absolute zero, or rather for the atom reduced to a state of rest. The extrapolation to a state of rest will differ according to whether we assume the existence or absence of zero point energy in the crystal lattice. Hence we may hope, in the first place to test the agreement between the observed scattering power and that calculated from the atomic model, and in the second place to see whether the experimental results indicate the presence of zero-point energy or no.

Download Full-text

A quantitative study of the reflexion of X-rays from crystals of aluminium

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1929.0176 ◽

1929 ◽

Vol 125 (798) ◽

pp. 401-419 ◽

Cited By ~ 53

Keyword(s):

Single Crystals ◽

Quantitative Study ◽

Rock Salt ◽

Zero Point ◽

X Rays ◽

Point Energy ◽

Absolute Intensities ◽

Metallic Element ◽

Atomic Scattering ◽

Direct Confirmation

1. It has already been shown that the observed variations with temperature of the intensity of reflexion of X-rays from crystals of rock-salt and sylvine agree closely with those predicted by the theory of Debye as modified by Waller, from the lowest temperature at which experiments have been made, that of liquid air, up to about 500° abs. Moreover, the absolute intensities of reflexion agree closely with those calculated theoretically, if the atomic scattering factors, F, are calculated from the Schrödinger charge-distributions for the atoms, obtained by the method due to Hartree. To obtain agreement, it is necessary to assume the existence of zero-point energy of an amount half a quantum for each degree of freedom, and the experiments may perhaps do considered as furnishing direct confirmation of such energy, since the differences between the intensities calculated with and without it are considerable. The work to do described in this paper was undertaken with a view to extending the investigations to a crystal of a metallic element. Aluminium at ones suggested itself as suitable for this purpose, since it can do obtained in largo single crystals, and since its coefficient of adsorption for the X-rays employed in the experiments, Mo K a , is small.

Download Full-text

A quantitative study of the reflexion of X-rays by sylvine

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1928.0188 ◽

1928 ◽

Vol 121 (787) ◽

pp. 155-171 ◽

Cited By ~ 56

Keyword(s):

Quantum Mechanics ◽

Theoretical Result ◽

Zero Point ◽

Coherent Scattering ◽

X Rays ◽

Point Energy ◽

X Ray ◽

Quantitative Results ◽

Theory And Experiment ◽

Quantitative Treatment

In recent papers on the intensity of reflexion of X-rays from rocksalt crystals it has been shown that, from the temperature of liquid air up to about 500° abs., the dependence of the intensity of reflexion upon temperature is in accord quantitatively with a formula of the type originally deduced by Debye, if the modification suggested later by Waller is introduced, although the decrease of intensity for higher temperatures is much greater than that indicated by the law. In dealing with quantitative results of experiments on reflexion from crystals, it is convenient to consider the quantity usually denoted by F, which is a measure of the scattering power, in a given direction, of an atom for X-rays. In the course of experiments with rocksalt it has been possible to determine F both for Na and Cl, and the values so obtained, when corrected for temperature, agree very closely with the F factors calculated from Hartree’s Schrodinger density-distribution for the ions Cl - and Na + . The calculation is based upon the theoretical result, due to Wentzel and Waller that, to obtain the coherent scattering from an electron in an atom, the electron must be represented by its corresponding Schrodinger charge-density, each element of which must be supposed to scatter classically. In order to get agreement between the calculated and observed F curves, it is necessary to assume the existence of zero-point energy of amount half a quantum per degree of freedom, which is of course required by the new quantum mechanics. The agreement between theory and experiment in the case of rocksalt is extremely interesting, but, in order to place the quantitative treatment of X-ray reflexion on an entirely satisfactory basis, it appears to be of some importance to see whether a similar agreement can be obtained with other crystals. This is not quite so easy as might be supposed.

Download Full-text

Zero Point Energy

Van Nostrand's Scientific Encyclopedia ◽

10.1002/0471743984.vse7649 ◽

2005 ◽

Keyword(s):

Zero Point ◽

Zero Point Energy ◽

Point Energy

Download Full-text

Zero-point energy of linear triatomic molecules

Journal of the Chemical Society Faraday Transactions 2 Molecular and Chemical Physics ◽

10.1039/f29747000871 ◽

1974 ◽

Vol 70 ◽

pp. 871 ◽

Cited By ~ 2

Author(s):

Jan Bron

Keyword(s):

Zero Point ◽

Zero Point Energy ◽

Point Energy ◽

Triatomic Molecules

Download Full-text

Semi-automated protein crystal mounting device for the sulfur single-wavelength anomalous diffraction method

Journal of Applied Crystallography ◽

10.1107/s0021889809054272 ◽

2010 ◽

Vol 43 (2) ◽

pp. 341-346 ◽

Cited By ~ 14

Author(s):

Yu Kitago ◽

Nobuhisa Watanabe ◽

Isao Tanaka

Keyword(s):

Diffraction Method ◽

Systematic Errors ◽

Protein Crystal ◽

X Rays ◽

Anomalous Diffraction ◽

X Ray ◽

Frozen Solution ◽

Custom Made ◽

Protein Molecules ◽

X Ray Absorption

Use of longer-wavelength X-rays has advantages for the detection of small anomalous signals from light atoms, such as sulfur, in protein molecules. However, the accuracy of the measured diffraction data decreases at longer wavelengths because of the greater X-ray absorption. The capillary-top mounting method (formerly the loopless mounting method) makes it possible to eliminate frozen solution around the protein crystal and reduces systematic errors in the evaluation of small anomalous differences. However, use of this method requires custom-made tools and a large amount of skill. Here, the development of a device that can freeze the protein crystal semi-automatically using the capillary-top mounting method is described. This device can pick up the protein crystal from the crystallization drop using a micro-manipulator, and further procedures, such as withdrawal of the solution around the crystal by suction and subsequent flash freezing of the protein crystal, are carried out automatically. This device makes it easy for structural biologists to use the capillary-top mounting method for sulfur single-wavelength anomalous diffraction phasing using longer-wavelength X-rays.

Download Full-text