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Structure of deuterium gas near the critical point
Physica B 156 & 157 (1989) 125-127 North-Holland, Amsterdam
STRUCTURE OF DEUTERIUM GAS NEAR THE CRITICAL POINT M. ZOPPI’, R. MAGLI’,
W.S. HOWELLS
and
A.K. SOPER
‘Consiglio Narionale delle Ricerche, Istituto di Elettronica Quantistica, Via Panciatichi 56130, I-50127 Firenze, zlJniverstci degli Studi di Firenze, Dipartimento di Energetica, Via di S. Marta, I-50139 Firenze, Italy 3Rutherford Appleton Laboratory, Neutron Division, Chilton, Didcot, Oxon, OX11 OQX, England
Italy
The structure factor and pair correlation function for gaseous deuterium at densities and temperatures near the critical point have been measured with neutron diffraction using the Liquids and Amorphous Diffractometer, LAD, at ISIS. No evidence is found for the enhanced ordering which had previously been predicted from Raman scattering data.
In a recent article, Clouter, Deacon and Kiefte (CDK) [l] found that the vibrational Raman Q-branch spectrum exhibited a pronounced fine structure, of periodicity 6 GHz, near the liquidgas critical point (T, = 38.3 K). The experiment was performed, using high resolution light scattering, by analysing the vibrational transitions (u = O+ 1, J = 0) and (u = O+ 1, J = 1) in both hydrogen and deuterium near the critical point. The spectral shape for deuterium, however, was different from hydrogen because the expected broadening of the Raman line was accompanied by the appearance of a regular fine structure which was absent in the case of hydrogen. The effect occurred in the temperaure range 39-60 K along the critical isochore. After a thorough analysis of the experiment the authors made several specific predictions about the structure in the gas which in principle would be straightforward to confirm by neutron diffraction. The principal conclusion was that the effect should be accompanied by an “unusually narrow first peak in the radial distribution function”, although until the present measurements no experimental data have been available on deuterium gas in this region. The object of the experiment was therefore to obtain, by neutron diffraction, the structure of deuterium gas at densities and temperatures close to critical conditions: any narrowing effect on the main peak in g(r) as predicted by CDK would be readily discernible by this technique. The following is a preliminary account of these
measurements: a longer more detailed account is in preparation. The sample was contained in cylindrical pressure cell (diameter 20.5 mm, height 60 mm and wall thickness 0.5 mm) manufactured from a vanadium sheet by electron beam welding. The top and bottom stainless steel flanges were fitted to the wall by the same welding technique. Two calibrated Rh-Fe resistance thermometers were located in the top and bottom flanges. When the cell was tested empty, a temperature difference was measured, between the two sensors, of 0.6 K. Deuterium entered the container through a stainless steel capillary that connected the scattering cell with the gas handling system. Three different thermodynamic states were measured along the T = 41.8 K isotherm and a fourth one at T = 46.6 K. The pressure was set at 20.9, 23.2 and 27.0 bar, for the lower temperature and to 33.2 bar for the higher temperature. Because of a lack of published PVT data for gaseous deuterium we have estimated the densities of the samples by scaling PVT data for hydrogen [2] with the critical parameters for deuterium. The temperatures above refer to the upper sensor because we found that, for unknown reasons, during the experiment the lower sensor gave a temperature reading higher by 1.5-2.0 degrees. We assumed the upper sensor reading to be correct because estimates of the density based on this sensor were in good agreement with the densities obtained by the analysis of the
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
126
M. Zoppi et al. 1 Structure of deuterium ,qus
neutron transmission data for each sample. Therefore, in spite of the good stability of pressure (0.3%) and temperature (0.1%) the accuracy on the absolute value of the density is quite poor and cannot be better than 10%. HOWever, the range of densities used is sufficiently large to ensure that we have spanned the critical density region. The time-of-flight data, measured on the Liquids and Amorphous Diffractometer, LAD, at ISIS, were normalized to the scattering from the empty vanadium pressure cell, to remove the effects of the incident neutron spectrum and detector efficiency, and then corrected for attenuation, multiple scattering and container scattering in the usual way. Fig. 1 shows, as an example, the measured differential scattering cross sections at 20 degrees scattering angle for all four densities as a function of Q, the momentum transfer. Each curve represents the average of several scans. Generally the scans overlapped well, although some variations occurred between detectors, usually at larger scattering angles. which could not be accounted for. Therefore final normalization to remove any residual discrepancies between different scans was achieved by comparison with the expected single molecule scattering at large Q values. The data at 5 and 10 degrees scattering angles (not plotted here) are
4 .i
Fig. 1. Differential cross section for deuterium gas near the critical region. Curve (a): T= 46.6 K, P = 33.2 bar, p = 9.6 molecules/nm3. Curves (b), (c) and (d): T= 41.gK; p = 20.9, 23.2, 27.Obar; p =7.9, 10.5. 13.6 molecules/rim’.
found to agree with the 20 degrees data. on the overlapping Q region, and showed the expected enhance small (2 scattering as determined from the high compressibility of the gas. It is virtually impossible to perform an accurate inelasticity correction on these data because the available models for S( Q, w) for the hydrogen and deuterium molecule [3,4] do not apply to the range of Q values encountered in this experiment. (The value of Q,,, is about SO A--’ but to make a reliable inelasticity correction to TOF data it is necessary to specify S( Q, w) for (2 greater than 100 & ‘). However, the subsequent analysis to the pair correlation function only involves the data measured at 5, 10 and 20 degrees, at which scattering angles the effects of inelasticity are believed to be small. The scattering cross section in fig. 1 shows an intermolecular structure peak near Q = 2 A- ‘. and a long range oscillation at larger Q values, corresponding to the intramolecular bond distance. The main peak increases monotonically in height with increasing density, but apart from this expected variation there is no sign of any unexpected features, nor is there any evidence for the pron&nced structuring at the critical density as predicted by CDK. From the experimental cross section we derived the pair correlation function, g(r), by means of an inversion procedure which is described elsewhere [5], and which makes use of a one-dimensional Monte Carlo simulation of the atom-atom distribution functions. Starting from an assumed pair distribution function which satisfies the required compressibility constraint [S(O) = p,ykBT, where p is the number density and x is the isothermal compressibility] the atoms are moved by random amounts from one bin to another and the move is accepted with a probability determined from a Boltzmann-like distribution. Once a fit to the data is achieved to the desired level of accuracy, the distribution is sampled at regular intervals (typically 5000 moves). The resulting pair correlation functions are shown in fig. 2: the error bars are the standard deviations about the mean for 100 samples and the intramolecular peak at 0.7 A has been removed for clarity.
M. Zoppi et al. I Structure of deuterium gas
,,ll’l’
,r
“‘,‘WP”
0 0
I
2
3
4
5
6
7
(A) 8
, 9
10
Fig. 2. Pair correlation function for deuterium gas near the critical region. The labels are the same as in fig. 1. The critical point of deuterium is: T, = 38.34 K, P, = 16.65 bar, p, = 10.43 molecules/nm3.
Curve (a), of fig. 2, refers to the higher temperature (T = 46.6 K), while (b), (c) and (d) correspond to the isotherm T = 41.8 K. The density for curve (a) is similar to that for curve (c). These pair correlation functions do not converge to unity until r = 25 8, but it is believed this is an artifact associated with the size of the sphere used in the simulation (R,,, = 25 A) together with the high compressibility values. However, the qualitative behaviour of g(r) is clear and can be used to draw some conclusions. On the T = 41.8 K isotherm the height of the first peak, as well as its width decreases with increasing density and a second maximum develops in the region around 7 A. Therefore the general behaviour of the radial distribution func-
127
tion with density appears normal and no unexpected additional structure is evident at the critical density. In order to evaluate the temperature variation of g(r) we have performed a classical MonteCarlo simulation of a monatomic Lennard-Jones system near the critical point (T: = 1.35, pr = 0.35) and we have found that a change in temperature similar to the experimental one gave a variation of the height of the peak of the order of 5% to be compared with a value of 2.5% obtained by comparison of curves (a) and (c). At the same time the main peak in both data and simulation narrows with increasing density. In conclusion we can state that, as far as the static structure of deuterium is concerned, the present experiment does not show any evidence for enhanced ordering near the main peak of the pair distribution function and that its width is monotonically narrowing, as density increases, as is expected to happen for a classical monatomic system under the same thermodynamic conditions. References [l] M.J. Clouter, C.G. Deacon and H. Kiefte, Phys. Rev. Len. 58 (1987) 1116. [2] H.M. Roder, G.E. Childs, R.D. McCarty and P.E. Angerhofer, N.B.S. Technical Note No. 641 (1973). [3] J.A. Young and J.U. Koppel, Phys. Rev. A 33 (1964) 603. [4] V.F. Sears, Proc. Phys. Int. Sot. 86 (1965) 965. [5] A.K. Soper, in: Proc. Workshop on Static and Dynamic Properties of Liquids, Dubrovnik, Yugoslavia, 1988, in preparation.
STRUCTURE OF DEUTERIUM GAS NEAR THE CRITICAL POINT M. ZOPPI’, R. MAGLI’,
W.S. HOWELLS
and
A.K. SOPER
‘Consiglio Narionale delle Ricerche, Istituto di Elettronica Quantistica, Via Panciatichi 56130, I-50127 Firenze, zlJniverstci degli Studi di Firenze, Dipartimento di Energetica, Via di S. Marta, I-50139 Firenze, Italy 3Rutherford Appleton Laboratory, Neutron Division, Chilton, Didcot, Oxon, OX11 OQX, England
Italy
The structure factor and pair correlation function for gaseous deuterium at densities and temperatures near the critical point have been measured with neutron diffraction using the Liquids and Amorphous Diffractometer, LAD, at ISIS. No evidence is found for the enhanced ordering which had previously been predicted from Raman scattering data.
In a recent article, Clouter, Deacon and Kiefte (CDK) [l] found that the vibrational Raman Q-branch spectrum exhibited a pronounced fine structure, of periodicity 6 GHz, near the liquidgas critical point (T, = 38.3 K). The experiment was performed, using high resolution light scattering, by analysing the vibrational transitions (u = O+ 1, J = 0) and (u = O+ 1, J = 1) in both hydrogen and deuterium near the critical point. The spectral shape for deuterium, however, was different from hydrogen because the expected broadening of the Raman line was accompanied by the appearance of a regular fine structure which was absent in the case of hydrogen. The effect occurred in the temperaure range 39-60 K along the critical isochore. After a thorough analysis of the experiment the authors made several specific predictions about the structure in the gas which in principle would be straightforward to confirm by neutron diffraction. The principal conclusion was that the effect should be accompanied by an “unusually narrow first peak in the radial distribution function”, although until the present measurements no experimental data have been available on deuterium gas in this region. The object of the experiment was therefore to obtain, by neutron diffraction, the structure of deuterium gas at densities and temperatures close to critical conditions: any narrowing effect on the main peak in g(r) as predicted by CDK would be readily discernible by this technique. The following is a preliminary account of these
measurements: a longer more detailed account is in preparation. The sample was contained in cylindrical pressure cell (diameter 20.5 mm, height 60 mm and wall thickness 0.5 mm) manufactured from a vanadium sheet by electron beam welding. The top and bottom stainless steel flanges were fitted to the wall by the same welding technique. Two calibrated Rh-Fe resistance thermometers were located in the top and bottom flanges. When the cell was tested empty, a temperature difference was measured, between the two sensors, of 0.6 K. Deuterium entered the container through a stainless steel capillary that connected the scattering cell with the gas handling system. Three different thermodynamic states were measured along the T = 41.8 K isotherm and a fourth one at T = 46.6 K. The pressure was set at 20.9, 23.2 and 27.0 bar, for the lower temperature and to 33.2 bar for the higher temperature. Because of a lack of published PVT data for gaseous deuterium we have estimated the densities of the samples by scaling PVT data for hydrogen [2] with the critical parameters for deuterium. The temperatures above refer to the upper sensor because we found that, for unknown reasons, during the experiment the lower sensor gave a temperature reading higher by 1.5-2.0 degrees. We assumed the upper sensor reading to be correct because estimates of the density based on this sensor were in good agreement with the densities obtained by the analysis of the
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
126
M. Zoppi et al. 1 Structure of deuterium ,qus
neutron transmission data for each sample. Therefore, in spite of the good stability of pressure (0.3%) and temperature (0.1%) the accuracy on the absolute value of the density is quite poor and cannot be better than 10%. HOWever, the range of densities used is sufficiently large to ensure that we have spanned the critical density region. The time-of-flight data, measured on the Liquids and Amorphous Diffractometer, LAD, at ISIS, were normalized to the scattering from the empty vanadium pressure cell, to remove the effects of the incident neutron spectrum and detector efficiency, and then corrected for attenuation, multiple scattering and container scattering in the usual way. Fig. 1 shows, as an example, the measured differential scattering cross sections at 20 degrees scattering angle for all four densities as a function of Q, the momentum transfer. Each curve represents the average of several scans. Generally the scans overlapped well, although some variations occurred between detectors, usually at larger scattering angles. which could not be accounted for. Therefore final normalization to remove any residual discrepancies between different scans was achieved by comparison with the expected single molecule scattering at large Q values. The data at 5 and 10 degrees scattering angles (not plotted here) are
4 .i
Fig. 1. Differential cross section for deuterium gas near the critical region. Curve (a): T= 46.6 K, P = 33.2 bar, p = 9.6 molecules/nm3. Curves (b), (c) and (d): T= 41.gK; p = 20.9, 23.2, 27.Obar; p =7.9, 10.5. 13.6 molecules/rim’.
found to agree with the 20 degrees data. on the overlapping Q region, and showed the expected enhance small (2 scattering as determined from the high compressibility of the gas. It is virtually impossible to perform an accurate inelasticity correction on these data because the available models for S( Q, w) for the hydrogen and deuterium molecule [3,4] do not apply to the range of Q values encountered in this experiment. (The value of Q,,, is about SO A--’ but to make a reliable inelasticity correction to TOF data it is necessary to specify S( Q, w) for (2 greater than 100 & ‘). However, the subsequent analysis to the pair correlation function only involves the data measured at 5, 10 and 20 degrees, at which scattering angles the effects of inelasticity are believed to be small. The scattering cross section in fig. 1 shows an intermolecular structure peak near Q = 2 A- ‘. and a long range oscillation at larger Q values, corresponding to the intramolecular bond distance. The main peak increases monotonically in height with increasing density, but apart from this expected variation there is no sign of any unexpected features, nor is there any evidence for the pron&nced structuring at the critical density as predicted by CDK. From the experimental cross section we derived the pair correlation function, g(r), by means of an inversion procedure which is described elsewhere [5], and which makes use of a one-dimensional Monte Carlo simulation of the atom-atom distribution functions. Starting from an assumed pair distribution function which satisfies the required compressibility constraint [S(O) = p,ykBT, where p is the number density and x is the isothermal compressibility] the atoms are moved by random amounts from one bin to another and the move is accepted with a probability determined from a Boltzmann-like distribution. Once a fit to the data is achieved to the desired level of accuracy, the distribution is sampled at regular intervals (typically 5000 moves). The resulting pair correlation functions are shown in fig. 2: the error bars are the standard deviations about the mean for 100 samples and the intramolecular peak at 0.7 A has been removed for clarity.
M. Zoppi et al. I Structure of deuterium gas
,,ll’l’
,r
“‘,‘WP”
0 0
I
2
3
4
5
6
7
(A) 8
, 9
10
Fig. 2. Pair correlation function for deuterium gas near the critical region. The labels are the same as in fig. 1. The critical point of deuterium is: T, = 38.34 K, P, = 16.65 bar, p, = 10.43 molecules/nm3.
Curve (a), of fig. 2, refers to the higher temperature (T = 46.6 K), while (b), (c) and (d) correspond to the isotherm T = 41.8 K. The density for curve (a) is similar to that for curve (c). These pair correlation functions do not converge to unity until r = 25 8, but it is believed this is an artifact associated with the size of the sphere used in the simulation (R,,, = 25 A) together with the high compressibility values. However, the qualitative behaviour of g(r) is clear and can be used to draw some conclusions. On the T = 41.8 K isotherm the height of the first peak, as well as its width decreases with increasing density and a second maximum develops in the region around 7 A. Therefore the general behaviour of the radial distribution func-
127
tion with density appears normal and no unexpected additional structure is evident at the critical density. In order to evaluate the temperature variation of g(r) we have performed a classical MonteCarlo simulation of a monatomic Lennard-Jones system near the critical point (T: = 1.35, pr = 0.35) and we have found that a change in temperature similar to the experimental one gave a variation of the height of the peak of the order of 5% to be compared with a value of 2.5% obtained by comparison of curves (a) and (c). At the same time the main peak in both data and simulation narrows with increasing density. In conclusion we can state that, as far as the static structure of deuterium is concerned, the present experiment does not show any evidence for enhanced ordering near the main peak of the pair distribution function and that its width is monotonically narrowing, as density increases, as is expected to happen for a classical monatomic system under the same thermodynamic conditions. References [l] M.J. Clouter, C.G. Deacon and H. Kiefte, Phys. Rev. Len. 58 (1987) 1116. [2] H.M. Roder, G.E. Childs, R.D. McCarty and P.E. Angerhofer, N.B.S. Technical Note No. 641 (1973). [3] J.A. Young and J.U. Koppel, Phys. Rev. A 33 (1964) 603. [4] V.F. Sears, Proc. Phys. Int. Sot. 86 (1965) 965. [5] A.K. Soper, in: Proc. Workshop on Static and Dynamic Properties of Liquids, Dubrovnik, Yugoslavia, 1988, in preparation.