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The structure of liquid water at room temperature
C h e m i ~ d Physics 88 0 9 8 4 ) 187-197 North-Holland, Amsterdam
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T H E S T R U C T U R E O F L I Q U I D W A T E R AT R O O M T E M P E R A T U R E
A.K. SOPER Department of Physics, University of Guelph, Guelph, Ontario, Canada NIG 2WI Received 27 D e c e m b e r 1983
Full details are given o f akrecent time-of-flight neutron diffraction experiment in which partial pair correlation functions for liquid water are extracted by isotope substitution. Measurements o f differential cross sections o f mixtures o f bea_vy and light water are made, and b y performing the experiment at the high neutron energies available at a pulsed neutron source (Los A!amos) the magnitude o f dynamic corrections to the data is reduced. The problematic hydrogen incoherent scattering is removed by a subtraction technique which avoids reference to dynamic models of the liquid. The results, although they shgw good agreement with computer simulations, show serious qualitative differences with other neutron experiments on'water, and it is suggested these discrepancies are a result o f lack o f attention to the unusual properties o f hydrogen as a scatterer of neutrons.
I. Introduction
This paper presents a full account of a recent measurement of the h y d r o g e n - h y d r o g e n pair correlation function in liquid water which has already been described briefly [1]. The experiment used time-of-flight neutron diffraction at a pulsed neutron source and involved differencing diffraction d a t a sets from mixtures of light and heavy water. The experimental H - H correlation function, which will be labelled " G H H ' " in order to distinguish it from the theoretical function gHH(r) discussed below, was obtained without reference to a dynamical model of the liquid. The data have been analysed further to obtain a quantity which is a weighted sum of oxygen-hydrogen and o x y g e n - o x y g e n correlation functions, dominated by the O - H term. This second quantity is here labelled " G O H O O " . Both G H H and G O H O O are compared with r e c e n t molecular dynamic ( M D ) and Monte Carlo (MC) simulations of water. Since the first report a minor error in the data analysis was discovered: the results for the structure factors were not affected qualitatively in a n y way, but values of the H - H a n d O - H coordination n u m bers were changed slightly. The structure o f liquid water has of course been widely studied by diffraction techniques. X-ray
diffraction and neutron diffraction experiments are most often presented because X-rays d e termine approximately the centre-of-mass correlation function, while neutron diffraction is weighted in favour of the protons and so is m o r e sensitive to orientational correlations. The quantity which can be incorporated into a statistical mechanical theory is g(r12, to~, w2) [2], which is the probability of two molecules separated by a distance r12 having relative orientations wl and toz. Although this orientational correlation function is not available directly from diffraction experiments, the correlations between scattering centres, called partial pair correlation functions, g~a(r), are measured and these represent the probability of finding an a t o m o f type "'a'" a distance r from an a t o m of type " f l " . The pair correlation functions contain less information than the orientational function, but for spherically non-symmetric molecules such as water the inter-molecular p r o t o n - p r o t o n and o x y g e n - p r o t o n correlation functions yield orientational information not obtainable from the centerof-mass correlation alone. Experiments to extract these functions for liquid water h a v e not been attempted until quite recently because of a n u m b e r o f difficulties associated with scattering from protons. A single diffraction experiment obtains; after
0301-0104/84//$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Di~sion)
188
A.K. Soper / Structure of liquid water
d a t a corrections, a weighted sum o f the partial structure factors. S , # ( Q ) , which are F o u r i e r transforms into m o m e n t u m space o f the cor r es pond i ng pair correlation functions. T h e weight of each term is p ro po r tio n al to the p r o d u c t o f the scattering lengths for atoms "'¢t'" and "'fl'" In o r d e r to separate the individual terms the scattering length for o n e atomic type is varied i nde pe nde nt l y o f the others, and ideally without altering the chemistry o f the substance. This can be achieved in a numb e r o f ways, for example by using an X-ray energy n e a r an absorption edge o f one of the c o m p o n e n t s . by c o m p ar in g experiments d o n e with different types o f radiation (such as X-rays. electrons, a n d neutrons), o r by using isotopic substitution and n e u t r o n diffraction if suitable isotopes are available. Although the second m et hod was a t t e m p t e d for water [3] the weakness o f the scattering of X-rays an d electrons by protons relative to oxygen leads to uncertainty in the gHH and goH functions obtained. T h e third a p p r o a c h at present seems to be the most tractable because the hea~Lv isotope of hydrogen, deuterium, is readily available, and there is a nice contrast in neutron scattering lengths between hydrogen (b H = - - 0 . 3 7 4 × 10 -x'~ m) and deuterium ( b 0 = 0 . 6 6 7 x 10 -l~ m) [4]. N e u t r o n diffraction therefore offers a powerful technique to d e t e r min e the partial correlation functions for water and o th er hydrogeneous materials. H o w e v e r by virtue o f its small mass and large incoherent cross section, hydrogen has unique properties as a scatterer o f neutrons which r e nde r traditional methods o f data analysis inappropriate. It is the opinion o f the present aut hor that these properties have not been treated carefully enough in the literature on water diffraction studies and, although there have been two ot he r attempts at the n e u t r o n experiment [5,7] *, as well as the present one, the three results have qualitative differences. Conflicts appear at several places in the data analysis. T o d e m o n s t r a t e one example o f this disagreement, the respective results for G H H are shown in fig. 1. Th e disagreement is especially serious in the region o f the first peak (1.9-3.1 A). and the result * T h e radial d i s t r i b u t i o n functions for ref. [51 are p r e s e n t e d b y N a r t e n et al. [6].
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Fig. 1. Comparison of GHH for this experiment (solid line) with those of Thiessen and Narten [51 (dashed line) and Dore and Reed [7] {dot-dashed line).
from D ore and Reed [7.8] has a curious oscillation at larger r values, not seen in the ot her two experiments. These differences, which have led to an extensive debat e a b o u t the reliability o f any o f the results, are serious enough to warrant a t horo u g h appraisal o f the n e u t r o n experiment on water. T o clarify the issues therefore the principal aspects o f the water experiment which distinguish it fro m experiments on n o n - h y d r o g e n o u s liquids are outlined below. It is argued that if suitable precautions had been taken most of the discrepancies between the experiments would never have appeared. When analysing data from the isotope experim ent the structure of liquid water is usually assum ed to be invariant to substitution o f hydro g e n with deuterium. It has been objected however that heax~ and light water structure factors should not be equivalent on account o f the different q u a n tum-mechanical zero-point motions o f the p r o t o n a n d deuteron. This objection is substantiated b y the t h e r m o d y n a m i c properties of the two liquids. F o r example, the m ol ar volume o f heavy water is 0.35% larger than for light water at 2 5 ° C a n d Swift [9] interpreted this to imply that heavy water has a m ore o p e n structure caused by stronger h y d r o g e n bonds. Presumably such an effect would show up particularly in the H - H and O - H struc-
-A.I~ Soper / Structure of liquid water.
ture factors but how it would do so has not been quantified. I t should be emphasised however that the objections based o n - t h e non-equivalence of hydrogen a n d deuterium m a y b e m i s l e a d i n g . In the appendix it is shown on rather general grounds that even if there are measurable differences in structure between heavy and light water the f)mction measured by neutron diffraction is an average o f the structure factors for the two liquids, and the result therefore still has physie,-rl significance. For the present work it should be understood t h a t the measured quantity represents such an average structure factor. A second, technical, aspect o f the experiment is that protons have a large incoherent scattering cross section for thermal neutrons. G o o d counting statistics are required, which is not especially difficult at m o d e r n high-flux neutron sources. More serious however is the multiple scattering correction arising from neutrons which scatter more than once in the sample before being counted. The correction can be evaluated only approximately. Thiessen and Narten [5] in their diffraction experiment used a sample of light water 7 m m in diameter. Such a sample will scatter some 75% of the incident beam. Using the standard double-scattering calculation of Blech and Averbach [10] for cylindrical geometry as used by these authors the intensity of twice scattered neutrons is estimated to be = 50`% o f the once scattered intensity for this sample. Neutrons scattered three times or more will then contribute in the region of 25`% of the total scattered intensity, but the precise a m o u n t and its scattering angle dependence is not determined. In fact the only reliable way at present o f estimating the contribution from higher orders o f scattering is by an elaborate Monte Carlo simulation incorporating details of the sample scattering law a n d instrument geometry. To avoid substantial errors arising from the u n k n o w n higher order scattering it is usual to arrange that the sample scatters not more than 20`% o f the incident beam and preferably nearer 10`%. Second scattering is in roughly the same ratio to primary scattering (i.e. 20%) and so higher orders, which now contribute only = 5% to the total, can be treated quite accurately by an expansion in terms o f the ratio of secondary to primary scattering [I1]. In
189
the present study t h e level of multiple scattering h a s b e e n maintained in the region of 15-25,% b y using thin, flat plate samples. F i n a l l y , the experiment requires careful interpretation because o f dynamic effects [12], which cause a "droop:" in the differential cross section at large s c a t t e r i n g angles. The droop is slight for heavy molecules and can be corrected by an expansion in terms o f the first and second moments of the d y n a m i c scattering law, S(Q, to). For water, however, a detailed knowledge of the molecular dynamics is needed to perform this correction. A sufficiently accurate model is not available and in practice either a model involving free molecular rotations is used to calculate the incoherent cross section [13,14], or a modified expansion scheme using effective masses is derived [15] ~. Neither m e t h o d is satisfactory for fight water as there remains a serious discrepancy between experimental and predicted differential cross sections in both cases.
This difficulty is probably the most intractable and is no doubt a major cause of the experimental controversy. Both Thiessen and N a r t e n [5], and Dore and Reed [7] attempted to correct their model calculations for the self-scattering by fitting a polynomial in Q to their data. To obtain the coefficients of this polynomial either sum rules were applied to the derived structure factors or points chosen at which the distinct scattering was assumed to be zero. Unfortunately, sum rules are only a necessary condition on the data and not sufficient to define the Q dependence of the interference function uniquely, since numerous functions could be conceived which satisfy the same sum rule. The procedure of choosing positions for the zero crossing points of the interference function is equally arbitrary because for light water the interference function is at most = 7% o f the self-scattering. The important point that gets lost in the discussion is that a polynomial fit can easily misrepresent the shape of the self-scattering in the sensitive region Q - - 1 to 5 A, -1 by several percentage points. Because the interference function is such a small fraction of t h e total scattering, a An unpublished approach using effective masses due to J.G. Powles is quoted by Reed [8].
A.K. Soper/ Structure of liquid water
190
serious. Q dependent, systematic error will be introduced precisely in a region where the interference function is most sensitive to intermolecular correlations. In a previous study on liquid hydrogen chloride [16] an empirical approach for removing the hydrogen self-scattering was introduced, rather along the lines o f that used by Enderby and Neilson [17] for aqueous solutions. Although the m e t h o d cannot be rigorously justified it is at least plausible. a n d will be shown to work well for a model mixture of free molecules. It has the important virtue that no arbitrary fitting procedures are needed. Instead it is postulated that the incoherent differential cross section for a mixture containing protons and deuterons is a linear combination of the respective cross sections of the fully protohated and fully deuterated samples. The application of this method to liquid water is nov,, reported. Time-of-flight diffraction at a pulsed neutron source ( W N R at Los Alamos) has also been exploited in order to minimize the dynamic effects on the differential cross sections.
a n d n= is the n u m b e r of a atoms per molecule, b,, is the neutron scattering length, and the angular brackets < . . . ) indicate averages over the spin a n d isotope states of nucleus a_ Q is the m o m e n t u m transfer, o is the molecular n u m b e r density, a, is the energy transfer, with S(Q, a~) the d y n a m i c scattering law [lg]. In fact the quantity measured is not S(Q) but a differential cross section, Z(Qe), whose precise value is determined by the conditions of the experiment. F o r time-of-flight (TOF) diffraction using pulsed neutrons the differential cross section is measured for a set o f neutron time-of-flight channels which means that if the inelastic scattering from the sample is important there will be a range of incident and scattered flight times corresponding to each channel. This range is given by the usual kinematic rules of neutron scattering [19] plus the time-of-flight condition
1 / k i + a / k r = (1 + a ) / k ~ ,
where k,, k r and k¢ are the incident, final and elastic wave vectors respectively and a =
2. Theory for time-of-flight diffraction The quantity sought in a diffraction experiment is the static structure factor S(Q). Following van Hove [18] it is usual to split the structure factor into a " s e l f " term, Ss, corresponding to scattering from single atoms, and a "'distinct" term S ~ , corresponding to interference scattering between distinct atoms. For an N-component system the total structure factor is
(2)
sample to detector distance source to sample distance "
(3)
In terms of these quantities and the scattering angle, 20, the elastic m o m e n t u m transfer is Qe = 2ke sin 8. The measured differential cross section, which is usually presented as a function of Q~, is expressed in terms of the scattering law of the sample and a n u m b e r of instrumental parameters: z(eo)
=
CO)
x l ( k i ) E ( k , ) ( 8 k i / S k . ) ~,,
N
(4)
s(Q) = ~t N
+ E n~na(b,}(,ba)Sg#(Q),
(1)
where
S2(Q)-
÷gd~S;( fo%
[
~o)
1,
_
const.O;
sin ---~--R--dr. -
a n d we note that, unlike for the static structure factor, Q is not held constant in this integration over a~. I(kl) and E ( k f ) are the incident spectrum and detector efficiency respectively, normalized to their values at the elastic wave vector. It is straightforward to show using (2) that the jacobian is given by
1
:o-
[1 + a(ki/k,?] "
(5)
\ A.K. Soper / Structure of liquid water
T h i s term arises because the rate at ~which the incident spectrum is sampled depends on the value o f ~. _ -... Clearly ~?(Q~) only goes over to S ( Q ) in the limit o f elasticscattering, S(Q, ~ ) = S(Q)8(t~). As discussed above the procedure for going from Z ( Q , ) to S ( Q ) is non-trivial for water because of substantial inelastic scattering.-However, it is important to realiTe that inelastic "scattering will affect primarily the self terms and the distinct terms between atoms on the same molecule, the so-called intra-molecular terms. This is because these terms sample all possible molecular motions, whereas the inter-molecular terms sample only collective motions. T h e i n e l a s t i c i t y correction can be reduced by performing the diffraction experiment at small scattering angles a n d high incident energies, such as are available at a pulsed neutron source. To demonstrate this well known effect the T O F differential cross section for heavy water, as measured in this experiment, is compared in fig. 2 to the equivalent curve measured at a reactor source (ILL, Grenoble) a n d it will be seen that the large Q " d r o o p " is absent in the T O F experiment. Instead, because the data are recorded at constant
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s c a t t e r i n g a n g l e , there is a roughly c o n s t a n t s h i f t in the T O F d a t a below the self-scatter static limit. Its magnitude increases at larger scattering angles (data not shown). The choice o f 40 o scattering helps to minimize this shift. In fact calculations based on the freely rotating rigid molecule model to be described s h o r t l y showed that at 40 o the d o w n w a r d shift in the T O F d a t a would be --- 0.10 b / s r / m o l e c u l e , but at 90 ° it would be 0.46 b / s r / m o l e c u l e . Correspondingly, the droop in the reactor d a t a arises from scanning angles at fixed incident neutron energy. In this experiment measurements of Z ( Q ) were m a d e on three water samples: H 2 0 , D20 and a 5 0 : 5 0 mixture of H 2 0 and D20. If the mole fraction o f H 2 0 used to make the mixture is x it can be shown from (1), with ,Y(Q) substituted for S(Q), that
x z H ' ° ( Q ) + (1 -- x ) z D ' ° ( Q ) -- Z m-°: D,O(Q) = 4 x ( 1 - - x ) ( ( b H ) -- < b D ) ) "~ - ~ 'dH H ( Q ) + A ,
u n d e r the usual assumption that structure factors are invariant to isotope substitution. The more general scenario of structure factors being isotope dependent is discussed in the appendix. In that case the result (6) is an average of the structure factors in heavy a n d light water. Contributions to the term A arise from noncancellation of inelasticity effects in the H and D self terms, and in the O - H , O - D and O - O distinct terms, but the largest contribution, An, probably comes from the hydrogen self terms. If they do not cancel in the way postulated then the residual can be expressed in terms of the individual atomic self differential cross sections: A H -----2 X ( 1 - - x ) [ < b 2 ) ( ~ i
HzO - - Z~i HDO)
+ < b ~ ) ( , ~ 5 D:° -- ~ 5 riD°)],
0
(6)
(7)
1
0
!
5
!
10
I
15
20
Fig. 2. Time-of-flight differential scattering cross section for heavy water (upper curve), measured on general purpose diffractometer at Los Alamos. Below it is shown the same curve
measured at a constant wavelength source (ILL - Grenoble). The inelasticity shift is roughly constant with W in the TOF data.
where ~?~iH2° for example is the hydrogen self cross section in the H 2 0 molecule. In order to make a plausible estimate, ,fiH was calculated for a freely rotating a n d translating rigid molecule. This model, which used the formalism of R a h m a n [20], incorporated the correct m o m e n t s of inertia o f each of the three molecules H 2 0 , D20 a n d H D O . Translational motions of the molecules were treated in the
A.K. Soper / Structure of liquid water
192
0.2
0.1
Z dH " (O~"~"°~ecuuE : •.-
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Fig. 3. Hydrogen-hydrogen partial structure factor measured in this experiment. The thermodynamic limit for these data (not shown) occurs at -0.438. Also shown is the calculated single molecule scattering (solid line), and, on the same scale, the residue, A H- (dashed line) arising from non-cancellation of self-scattering in the model.
ideal gas approximation and the rotational and translational motions were convoluted together in the manner of Lurie [21] to form S(Q, to) for each molecule. This was then integrated numerically acc.ording to (4) to yield ~(Qe)- The result for A n is shown as the dashed line in fig. 3. Calculation of the rotational terms for a water molecule involved summation over many rotational transitions at the larger Q values, and this introduced numerical errors in the calculated ~ ( Q e ) . T o determine this erriSr-the zeroth moment of the dynamic scattering law was calculated [theoretically this quantity should be unity, see eq. (1)], and errors were found to be on the order 2,% out to a Q value of = 13 /k-~. Hence the cancellation of self terms for this model, as depicted in fig. 3 was good for this model and within numerical uncertainties. In the real liquid the molecular motions are hindered, which renders A n even smaller. Similar calculations for the distinct terms showed that their contributions to A are much smaller than A H- In the subsequent data analysis it is therefore assumed that z~ is negligible.
3. Experimental The general purpose diffractometer at the W e a p o n s N e u t r o n Research facility ( W N R ) in Los Alamos is currently being upgraded as part of the recently initiated user program for the pulsed neu-
tron scattering facility. At the time of this experiment it was a p r o t o t y p e instrument consisting of four banks of sixteen 3He proportional counters placed at scattering a n # e s of 10 o, 40 o, 90 o, and 150 o [22]. The incident flight path was. 10 m and the final flight path 1 m. A large background c o m p o n e n t from air scattering in the b e a m prevented use of the data taken at the 10 o position, and inelasticity effects appeared much larger at 90 ° and 150 °, so the present analysis will concentrate on the data taken at 40 o scattering angle. The instrument was calibrated using the Bragg reflection data from a nickel powder sample. The samples were held in a flat plate aluminium container angled at 30 o to the incident beam. The temperature was 25 ° C and atmospheric pressure at Los Alamos is = 0.78 atm. Cans of different thickness were used: 0.34 m m for H 2 0 , 0.60 mm for H 2 0 : D 2 0 mixture, and 3.3 m m for D20, chosen so that multiple scattering was 15% of the total on average and did not exceed 25% at the longest neutron wavelength (where the neutron cross sections are largest). Thicknesses were monitored to -- 3,% accuracy by transmission detectors placed before and after the sample. Data were taken for sample plus container, e m p t y container, vanadium slab (made from sintered vanadium powder, thickness 3.3 mm, and used as an instrument calibration), and air scattering background. Counting times were ---3 days per sample. After rebinning the data in constant width Q-bins (0.1 A - 1 ) and summing data from sixteen detectors, the gross counts were in excess of 106 per bin at the peak of the incident neutron spectrum. The data were corrected for self-shielding [23,24], multiple scattering, background scattering, and the vanadium data were adjusted for inelasticity effects [12]. The multiple scattering calculation used a formalism which allows all orders o f scattering to be included [11]. The sample data were then normalized point b y point to the vanadium data to obtain differential cross sections. This point-bypoint approach, although it reduces the statistical accuracy o f the result, is a more reliable w a y o f removing the incident spectrum and detector efficiency effects from the data than the conventional practice of fitting polynomials to the vanadium data, since the spectrum varies widely
A,K. Soper / Structure of liquid water over the energy range Of the experiment a n d does not have a simple d e p e n d e n c e on Q. Small errors, on the order o f 1%, in fitting polynomials c a n lead to much larger errors in the final structure factors, particularly in the case o f light water where the interference function sits on top o f a large incoherent background. In fact a similar c o m m e n t applies to analysis of reactor d a t a taken with a multi-detector array since subtle variations in detection efficiency as function o f position are certain to occur. Ignoring such variations b y smoothing with a polynomial can easily introduce spurious features in the final differential cross section. O n an absolute scale the sample differential cross sections are therefore as accurate as the known v a n a d i u m differential cross section, which, due to uncertainties in vanadium sample composition, was only known to = 5% in this ease. The relative accuracy of one sample to another however will be better than this if the relative thicknesses and cross sections o f each sample are known well. As already mentioned these latter quantities were monitored to within 3% in this experiment.
4. Results mad discussion 4.1. H - H distribution function The result of applying eq. (6) to the three sets of differential cross sections is shown in fig. 3. The data do not extend below 1.4 A, - l because of frame overlap at W N R which occurs at 8.3 ms (at a wavelength of 2.9 A on a 11 m total flight path). Also the original raw data gave a result which oscillated about a level = 0.24 above zero. This represents a cumulative error in all three data sets, but if it came only from the light water curve it would correspond to a systematic error of -- 4% in that curve, and so is within likely d a t a analysis errors. A constant shift downwards of 0.245 was applied to the d a t a before plotting. The precise shift was determined by minimiTing the amplitude o f truncation oscillations in the subsequent Fourier transform of the data. Anothei" procedure of applying a sum rule to the data [4,6] is strictly inapplicable to the water experiment because corrections for intra-molecular inelasticity effects can
193
o n l y be m a d e approximately. The Fourier transform of ,~dn(Q), eq. (8), is shown in fig. 4. Clearly the statistical accuracy of ,~dH(Q) declines beyond --10 A -1, so the d a t a have been truncated at 10.8 .~-1 (again chosen to minimize truncation effects). F o r the region Q = 0 - 1 . 4 A-~ it was necessary to extrapolate the d a t a to the t h e r m o d y n a m i c limit at Q = 0, i.e. - 0 . 4 3 8 . This was attempted by fitting various polynomials in Q but apart from a slight d o w n w a r d base line shift to the transformed data in the region r = 0 - 4 ,~ none gave significantly different results from a simple straight line fit. The linear extrapolation in fact gave the least a m o u n t of shift. The precise behaviour of the data in this region will have to be determined in another experiment. The Fourier transform calculated here is therefore
GHH(r) 1 £10.SA-~¢.),~d sin Qr~r~ = 1 + 2---~_oJ° ~- HH(e)~u~e.
(8)
it can be seen in fig. 4 that a large peak occurs at 1.5 A as well as another, unphysical peak at 0.7 /k. This second peak is now believed to be an artifact of the truncation and intramolecular inelasticity correction. To show this the single molecule structure factor as calculated from the free molecule model (solid line in fig. 3) was truncated a n d transformed in the same way as ",he data. A similar unphysical peak appears in the transform at about the same position ( d o t - d a s h e d line in fig. 4), although it is not reproduced well enough to be used as a correction for the data. The H - H intramolecular distance used for the free molecule calculation was the gas value, 1.515 ,~,. It will be seen that a peak at almost the same position occurs in the transform of the model, fig. 4: despite the substantial inelasticity a n d truncation effects the H - H distance is reproduced quite faithfully in the calculation. Therefore the peak at 1.48 A, in the experimental data is to be associated with the H - H intra-molecular distance in the liquid. The inter-molecular structure ( r > 1.9 A) bears remarkable resemblance to computer simulation predictions. Previously these d a t a were compared with the " a b initio" calculations of Lie, Clementi a n d Yoshimine [25] (LCY), a n d the "'effective
194
A.K. Soper / Structure of liquid water
GHH
2.0
!
= gHH(r)
i
!
i
h,t
t
|
EXPT (H-H
• FREE
1.5
I
DISTANCE)
MOLECULE
INELASTICITY
MODEL
PEAK
1.0
-
-
TIPS2
0.5
0.0 0
!! ! i!i i i
I
1
2
3
4
5
6
7
8
Fig. 4. Measured H H pair correlation function (solid line) compared to the TIPS2 simulation (long dashes). In the range r = 0 - 2 ,A the Fourier transform of the single molecule calculation is shown (dash-dotted), and it will be seen that the unphysical peak at = 0.7 A appears in both experimem and calculation.
structure factors was not obtained. H o w e v e r a n o t h e r useful curve was o b t a i n e d by subtracting the already d e t e r m i n e d H - H distribution f r o m the c o m p o s i t e d a t a for heavy water. This subtraction was p e r f o r m e d on b o t h the " r - s p a c e " a n d " Q space" dat a and the results were essentially identical. T h e r-space result, which is a sum o f O - H a n d O - O pair correlation functions, will be described here. As can be seen in fig. 2 the differential cross section for I:)20 oscillates a b o u t an almost horizontal line, suggesting that the inelasticity effects are constant with Q for these data. T h e relatively small shift o f this line below the static limit c o m p a r e d to d a t a taken at larger scattering angles f u r t h e r suggests that the inelasticity correction is small at this angle. As a result the heavy water cross section was Fouri er t ransform ed directly after truncating sm oot hl y at 20.4 A,-t and subtracting an a m o u n t correspondi ng to the oxygen and deuterium self-scattering.. T h e result, fig. 5, shows a sharp peak at - - 1 A correspondi ng to the O - H molecular b o n d length, and a smaller peak at 1.5 ,~ suggesting the H - H distance. T h e rest o f the
I
i
i
i
i
i
0.336goo+ 1.554gOH -F 1.780gHH -- 3.67 potential" molecular d y n a m i c study due to Stillinger an d R a h m a n [26] (ST2). H e r e com pa r i s on is m a d e with the recent simulations due to Jorgensen [27] (TIPS2). O f the available simulations the latter appears to give the best agreement with this experiment. N o t e that because this simulation uses a n "'effective" potential the effects o f m a n y - b o d y forces and q u a n t u m mechanics on the liquid structure are obscured. T h e coordination n u m b e r of the first peak, calculated for radius values from 1.9 to 3.1 ,A, gave 5.8 _+ 0.3 hydrogen atoms, which is the same as f o r the TIPS2 simulation, and p r o b a b l y n o t significantly smaller than the 6.1 h y d r o g e n a t o m s fo u n d in the L C Y simulation.
4.2. O - H plus 0 - 0
distributions
Because o f the u n k n o w n inelasticity corrections a full separation o f the da t a into three partial
2.0
x l 0 -1
/x.LCY 0.0 \ ~ P T -2.0
t ~THEOR~FICAL
LIMIT AT r = 0
!
/ 1 _,J_. -4.0
!
0
2
!
!
4
!
!
!
6
Fig. 5. Fourier transform (solid line) of heavy water data shown in fig. 2. Peaks corresponding to the O - H bond and H - H intramolecular distance are seen at = 1 . 0 A and =1.5 .-~ respectively, as well as an unphysical peak at 0.6 ,~, believed to be associated with inelasticity effects. The LCY simulation is shown as the dashed line.
A.K. Sopo"/ S t ~ c t ~ of l ~ a
cUrve is rather featurcless"but shows:good-agree-merit with earlier d a t a [28]. The un-physical peak a t 0.6 A and the smaller amplitude " ' r i p p l e s " b e y o n d 1.5 ~,, which have a characteristic period o f 0.35 ,~, are believed to b e a s s o c i a t e d - w i t h c/~mbined truncation and inelasticity effects. The data for G H H , weighted b y the appropriate neutron scattering factor (1.78 b / s r / molecule for heavy water), were subtracted from the data in fig. 5, and after renormalization the result GOHOO = 0.82go. (r) + 0.18goo(r)
(9)
is shown in fig. 6, along with a similar combination for the L C Y and TIPS2 c o m p u t e r simulations. N o t e that G O H O O is dominated b y the O - H correlation function. Truncation and inelasticity effects are apparent at small radius values as seen from the negative values for G O H O O , but a strong O - H b o n d peak at 1.0 A is present, a n d the H - H peak seen in heavy water at 1.5 A, fig. 5, has been subtracted out.
2.0
G O H O O = 0 . 1 7 8 goo + 0.822 goH
,.s
L
~ote~
~
-~-
.
195
Once : again-,the~ short range: inter'molecular structure d0es not differ veryfsignificantly from w h a t c0mpute~Lsimulations have-predicted. A n integration, between r = i . 6 : A and "r = 2.4 ",~,: of_ the first p e a k i n G O H O O yielded 2.! _+ 0.1 hydrogen-atoms-about oxygem This is close to the value for t h e : T I P S 2 simulation (1.9) but significantly. la/ger than for the L C Y - s i m u l a t i o n (1.7). It is concluded that for both G H H and G O H O O the TIPS2 potential has largely removed the earlier reported discrepancy between simulations a n d this experiment. At longer range, r > 4 ~,, a significant discrepancy between experiment and simulation appears. This can be seen best in the heavy water d a t a of fig. 5. A broad oscillation at = 7.8 ,~ appears at 6.6 ,~ in the L C Y simulation. The same discrepancy appears in the simulated differential cross section for heavy water (not shown), where the principal peak at Q = 2.0 ,~-~ is shifted to a slightly larger q value of 2.2 ,~-1. Generally, positions of peaks in diffraction work are much more. reliable than peak heights because of the simple and reliable scale calibration that is performed, so the discrepancy seen here is real. The same discrepancy is not seen in G O H O O , b u t the region r > 4 ,~ is probably masked b y truncation oscillations from the G H H function used in the subtraction. If, following Egelstaff and R o o t [29], the potential used in the L C Y simulation is regarded as a pair potential, then a long range discrepancy such as this might arise from m a n y - b o d y effects n o t included in the simulation. The same discrepancy does not occur w i t h the effective potential simulations.
5. Conclusion 0.5
t
0
~
!
2
!
I
4
I
l
!
6
Fig. 6. Measured sum of O - H and O - O pair correlation functions, compared to LCY-(short dashes) and TIPS2 (dash'-dotted) simulations.
By avoiding some of the pitfalls that have traditionally been encountered while performing the neutron diffraction experiment o n liquid water, reliable and largely model-free data have been obtained. This was achieved b y exploiting timeof-flight diffraction at a pulsed n e u r o n source and utilizing a subtraction procedure, b o t h of which help to circumvent the major obstacle of performing inelasticity corrections to the d a t a . - T h e only expense for the intrinsic reliability of the results is
196
tLK. Soper/ Structureof liquidwater
that a full separation into three partial structure factors was n o t obtained. Instead the H - H distrib u t i o n was o b t a i n e d together with a sum o f O - H and O - O distributions. T h e latter d a t a are weighted in the O - H function and so are still useful for testing theories. F o r example, the p ai r potential theories ma y need revision to account for the long-range b eh avi our in the distribution functions. This was an exploratory experiment which clearly should be followed up when the high flux pulsed sources currently in construction becom e available. In particular small Q data and better statistics at large Q are needed.
( O - O , O - H , O - D , H - H , D - D a n d H - D ) , which are in general d e p e n d e n t o n composifion¢ X. T h e distinct term in eq. (1) for this mixture is n o w
( i,o) "-Soo(Q, x) + a( bo)( bH)Son( Q, x) + 4(b.)Zsn-'Hnn(Q, x ) ,
(A.1)
where
( b . ) S o . ( Q, x) = x b . S o . ( O, x) + (I -
x)boSoD(q, x),
= x bHSuri( Q, x) + ( 1 - x)-bDSDD(O, ~ 2 ~) +2x(1
Acknowledgement
-
x)bHbDS.D(Q, x),
and
( b . > = x b . + (1 - x)bD. This work was p e r f o r m e d u n d e r the auspices o f the US D e p a r t m e n t o f Energy. I would like to t h a n k R.N. Silver for considerable s u p p o r t and e n c o u r ag emen t th r o u ghout the course o f the work. M a n y thanks also to P.A. Egelstaff for a critical reading o f the manuscript. A University Research Fellowship and grant f r om the N a t u r a l Sciences a n d Engineering Research Council o f C a n a d a is acknowledged.
Appendix I shall analyse the result of the neut r on experim e n t for the case where the structure factors o f heavy and light water are not the same. O f course H D O molecules are formed rapidly b y p r o t o n / d e u t e r o n exchange in a mixture o f light and heavy waters, b u t because neutrons are scattered by nuclei the structure factors observed in a mixture are measures o f the ensemble averaged correlation between pairs o f nuclei, so the distinction between say a p r o t o n on H z O and one on H D O is lost. T h e r e f o r e from the point o f view o f understanding the n e u t r o n diffraction experiment it is m o r e useful to consider a mixture o f heavy and light water as a t h r e e - c o m p o n e n t system consisting of H, n a n d O. T h e experiment measures a sum o f six p a r t i a l s t r u c t u r e f a c t o r s f o r t he m i x t u r e
T h e bars over the symbols therefore indicate isotope averages. T hese average structure factors are subject to asymptotic conditions: as x ---, 1,
s . . ( Q , x) ---,s ~ ° ( Q ) = s . . ( Q , 1), a n d as x ~ 0,
s . . ( Q , x) ---,s~ho(Q) = SDD(Q, 0),
(A.2)
where, for example, S~h°(Q) is the H - H structure fact or in pure light water. T h e same considerations appl y for the com posi t i on d e p e n d e n t Soo a n d SOH terms. Eq. (A.1) is a general expression for the distinct scattering. It is still not particularly useful because the individual structure factors c a n n o t be measured. Wi t hout loss o f generality, however, each o f the average structure factors can be expressed as a c o m b i n a t i o n o f its respective asym pt ot i c limits:
S,,t~(Q, x)=f(x)S~Ha'-°(Q) + (1
-f(x))S~2°(Q),
(A.3)
where
/ ( 0 ) = 0,
f ( 1 ) = 1,
but f ( x ) is otherwise arbitrary. In general it m a y be a function o f Q. If structural differences are n o t too large then a first a p p r o x i m a t i o n is f ( x ) = x .
.
:
?
-
/~.
"--" _
a . K~sop e b / -Strut t-if,e o f l i ~ d
The same analysis as Usedin tlietext toobtain the H-Hstructufe f a c t o r f r o m t h e d a t a [ e q . (6)] n o w
yields
--
4X(I--x){~'H.:o H { } -I--.~'D.o[(b H~I nzo
+bo(bH _
_
- ~
{-}]
~ " O H I J"
(A.4)
bo) z + 2bo(bH--
bo)-
N o t e t h a t t h e r e is n o w a t e r m a r i s i n g f r o m n o n cancellation of the O-H structure factors, but it involves a simple difference of the two structure factors and so will likely be small. The O-O terms still cancel identically, within the linear approximation. From (A.4) it can be seen that the quantity o b t a i n e d b y t h e a n a l y s i s is a n a v e r a g e o f t h e H - H structure factors for the two liquids. For the condit i o n s o f t h e p r e s e n t e x p e r i m e n t ( x = 0.5, a n d b n = - - 0 . 5 6 1 b D ) , t h e r e s u l t is, i g n o r i n g t h e O - H t e r m , b ~ ( 0 . 5 3 3 ~ H H ~° + 1.g04Y-HD¢), i.e. i n t h i s e x p e r i m e n t t h e r e s u l t is d o m i n a t e d the heavy water structure factor.
by
References [1] A.K. Soper and R.N. Silver, Phys. Rev. Letters 49 (1982) 471. [2] K.E. Gubbins, C.G. Gray, P.A. Egeistaff and M.S. Anath, Mol. Phys. 25 (1973) 1353. [3] O. Palinkas, E. Kalman and P. Kovacs, Mol. Phys. 34 (1977) 525.
--"
: - ' --:
:
._
: 197
-[41~L. Koestei-; H. Rai~ch, M. Herkens and K. Sehr:,der,: summary of Neutron Scattering Lengths, Report-No, JUL-1755, K F A Julieh, WestGermany (1981). i51 W.E. Tl~essen and A.H. Narten, J. Chem. Phys: 77 (i982) 2656.
where { ) = 3x(b n-
~:ater- -
-
[6] A.H. Narte n, W.E. Thiessen and L. Blum, Science 217 (1982) 1033. [7] J.C. Dote and J. Reed, to be published. [8] J. Reed, Ph.D. Thesis, University of Kent at Canterbury, U K (1981). [9] E. Swift, J. Am. Chem. So(:. 61 (1939) 198. [10] LA. Blech and B.L. Averbach, Phys. Rev. 137A (1965) 1113. [11] A.K. Soper, Nucl. Instr. Meth. 212 (1983) 337. [12] G. Placzek, Phys. Roy. 86 (1952) 377. [13] L. Blum, M. Rovere and A.H. Narten, J. Chem. Phys. (1982) 2647. [14] P.A. Egclstaff and A.K. Soper, Mol. Phys. 40 (1980) 553. [15] P.A. Egelstaff and A.K. Soper, Mol. Phys. 40 (1980) 569. [16] A.K. Soper and P.A. Egelstaff, Mol. Phys. 42 (1981) 399. [17] J.E. Enderby and G.W. Neilson, in: Water, a comprehensive treatise, Vol. 6, ed. F. Franks (Plenum Press, New York, 1979) ch. 1. [18] L. van Hove, Phys. Rev. 95 (1954) 249. [19] W. Marshall and S.W. Lovesey, Theory of thermal neutron scattering (Oxford Univ. Press, London, 1971). [20] A. Rahman, J. Nucl. Energy A 13 (1961) 128. [21] N.A. Lurie, J. Chem Phys. 46 (1967) 352. [22] A.K. Soper, in: Neutron scattering - 1981, A l P Conference Proceed/ngs No. 89, ed. J. Faber (ALP, New York, 1982) p. 23. [23] G.H. Vineyard, Phys. Rev. 96 (1954) 93. [24] V.F. Sears, Advan. Phys. 24 (1975) 1. [25] G.C. Lie, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 2314. [26] F. StiIlingerand A. Rahman, J. Chem. Phys. 60 (1974)
1545. [27] W.L. Jorgensen, J. Chem. Phys. 77 (1982) 4156. [28] G. Walford and J.C. Dote, Mol. Phys. 34 (1977) 21. [29] P.A. Egelstaff and J.H. Root, Chem. Phys. 76 (1982) 405.
.... --
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" " .. :- - "
-..
187
-
_-
:-
=
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2
T H E S T R U C T U R E O F L I Q U I D W A T E R AT R O O M T E M P E R A T U R E
A.K. SOPER Department of Physics, University of Guelph, Guelph, Ontario, Canada NIG 2WI Received 27 D e c e m b e r 1983
Full details are given o f akrecent time-of-flight neutron diffraction experiment in which partial pair correlation functions for liquid water are extracted by isotope substitution. Measurements o f differential cross sections o f mixtures o f bea_vy and light water are made, and b y performing the experiment at the high neutron energies available at a pulsed neutron source (Los A!amos) the magnitude o f dynamic corrections to the data is reduced. The problematic hydrogen incoherent scattering is removed by a subtraction technique which avoids reference to dynamic models of the liquid. The results, although they shgw good agreement with computer simulations, show serious qualitative differences with other neutron experiments on'water, and it is suggested these discrepancies are a result o f lack o f attention to the unusual properties o f hydrogen as a scatterer of neutrons.
I. Introduction
This paper presents a full account of a recent measurement of the h y d r o g e n - h y d r o g e n pair correlation function in liquid water which has already been described briefly [1]. The experiment used time-of-flight neutron diffraction at a pulsed neutron source and involved differencing diffraction d a t a sets from mixtures of light and heavy water. The experimental H - H correlation function, which will be labelled " G H H ' " in order to distinguish it from the theoretical function gHH(r) discussed below, was obtained without reference to a dynamical model of the liquid. The data have been analysed further to obtain a quantity which is a weighted sum of oxygen-hydrogen and o x y g e n - o x y g e n correlation functions, dominated by the O - H term. This second quantity is here labelled " G O H O O " . Both G H H and G O H O O are compared with r e c e n t molecular dynamic ( M D ) and Monte Carlo (MC) simulations of water. Since the first report a minor error in the data analysis was discovered: the results for the structure factors were not affected qualitatively in a n y way, but values of the H - H a n d O - H coordination n u m bers were changed slightly. The structure o f liquid water has of course been widely studied by diffraction techniques. X-ray
diffraction and neutron diffraction experiments are most often presented because X-rays d e termine approximately the centre-of-mass correlation function, while neutron diffraction is weighted in favour of the protons and so is m o r e sensitive to orientational correlations. The quantity which can be incorporated into a statistical mechanical theory is g(r12, to~, w2) [2], which is the probability of two molecules separated by a distance r12 having relative orientations wl and toz. Although this orientational correlation function is not available directly from diffraction experiments, the correlations between scattering centres, called partial pair correlation functions, g~a(r), are measured and these represent the probability of finding an a t o m o f type "'a'" a distance r from an a t o m of type " f l " . The pair correlation functions contain less information than the orientational function, but for spherically non-symmetric molecules such as water the inter-molecular p r o t o n - p r o t o n and o x y g e n - p r o t o n correlation functions yield orientational information not obtainable from the centerof-mass correlation alone. Experiments to extract these functions for liquid water h a v e not been attempted until quite recently because of a n u m b e r o f difficulties associated with scattering from protons. A single diffraction experiment obtains; after
0301-0104/84//$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Di~sion)
188
A.K. Soper / Structure of liquid water
d a t a corrections, a weighted sum o f the partial structure factors. S , # ( Q ) , which are F o u r i e r transforms into m o m e n t u m space o f the cor r es pond i ng pair correlation functions. T h e weight of each term is p ro po r tio n al to the p r o d u c t o f the scattering lengths for atoms "'¢t'" and "'fl'" In o r d e r to separate the individual terms the scattering length for o n e atomic type is varied i nde pe nde nt l y o f the others, and ideally without altering the chemistry o f the substance. This can be achieved in a numb e r o f ways, for example by using an X-ray energy n e a r an absorption edge o f one of the c o m p o n e n t s . by c o m p ar in g experiments d o n e with different types o f radiation (such as X-rays. electrons, a n d neutrons), o r by using isotopic substitution and n e u t r o n diffraction if suitable isotopes are available. Although the second m et hod was a t t e m p t e d for water [3] the weakness o f the scattering of X-rays an d electrons by protons relative to oxygen leads to uncertainty in the gHH and goH functions obtained. T h e third a p p r o a c h at present seems to be the most tractable because the hea~Lv isotope of hydrogen, deuterium, is readily available, and there is a nice contrast in neutron scattering lengths between hydrogen (b H = - - 0 . 3 7 4 × 10 -x'~ m) and deuterium ( b 0 = 0 . 6 6 7 x 10 -l~ m) [4]. N e u t r o n diffraction therefore offers a powerful technique to d e t e r min e the partial correlation functions for water and o th er hydrogeneous materials. H o w e v e r by virtue o f its small mass and large incoherent cross section, hydrogen has unique properties as a scatterer o f neutrons which r e nde r traditional methods o f data analysis inappropriate. It is the opinion o f the present aut hor that these properties have not been treated carefully enough in the literature on water diffraction studies and, although there have been two ot he r attempts at the n e u t r o n experiment [5,7] *, as well as the present one, the three results have qualitative differences. Conflicts appear at several places in the data analysis. T o d e m o n s t r a t e one example o f this disagreement, the respective results for G H H are shown in fig. 1. Th e disagreement is especially serious in the region o f the first peak (1.9-3.1 A). and the result * T h e radial d i s t r i b u t i o n functions for ref. [51 are p r e s e n t e d b y N a r t e n et al. [6].
3
I
i
I
I
GHH
i~
NARTEN et al.
i!
I! I! It
2
II !
~ THIS EXPERIMENT
1
aJ 0
0
1
,4 2
t I
ii v~-~DORE e t aL !
!
!
I
3
4
5
6
7
Fig. 1. Comparison of GHH for this experiment (solid line) with those of Thiessen and Narten [51 (dashed line) and Dore and Reed [7] {dot-dashed line).
from D ore and Reed [7.8] has a curious oscillation at larger r values, not seen in the ot her two experiments. These differences, which have led to an extensive debat e a b o u t the reliability o f any o f the results, are serious enough to warrant a t horo u g h appraisal o f the n e u t r o n experiment on water. T o clarify the issues therefore the principal aspects o f the water experiment which distinguish it fro m experiments on n o n - h y d r o g e n o u s liquids are outlined below. It is argued that if suitable precautions had been taken most of the discrepancies between the experiments would never have appeared. When analysing data from the isotope experim ent the structure of liquid water is usually assum ed to be invariant to substitution o f hydro g e n with deuterium. It has been objected however that heax~ and light water structure factors should not be equivalent on account o f the different q u a n tum-mechanical zero-point motions o f the p r o t o n a n d deuteron. This objection is substantiated b y the t h e r m o d y n a m i c properties of the two liquids. F o r example, the m ol ar volume o f heavy water is 0.35% larger than for light water at 2 5 ° C a n d Swift [9] interpreted this to imply that heavy water has a m ore o p e n structure caused by stronger h y d r o g e n bonds. Presumably such an effect would show up particularly in the H - H and O - H struc-
-A.I~ Soper / Structure of liquid water.
ture factors but how it would do so has not been quantified. I t should be emphasised however that the objections based o n - t h e non-equivalence of hydrogen a n d deuterium m a y b e m i s l e a d i n g . In the appendix it is shown on rather general grounds that even if there are measurable differences in structure between heavy and light water the f)mction measured by neutron diffraction is an average o f the structure factors for the two liquids, and the result therefore still has physie,-rl significance. For the present work it should be understood t h a t the measured quantity represents such an average structure factor. A second, technical, aspect o f the experiment is that protons have a large incoherent scattering cross section for thermal neutrons. G o o d counting statistics are required, which is not especially difficult at m o d e r n high-flux neutron sources. More serious however is the multiple scattering correction arising from neutrons which scatter more than once in the sample before being counted. The correction can be evaluated only approximately. Thiessen and Narten [5] in their diffraction experiment used a sample of light water 7 m m in diameter. Such a sample will scatter some 75% of the incident beam. Using the standard double-scattering calculation of Blech and Averbach [10] for cylindrical geometry as used by these authors the intensity of twice scattered neutrons is estimated to be = 50`% o f the once scattered intensity for this sample. Neutrons scattered three times or more will then contribute in the region of 25`% of the total scattered intensity, but the precise a m o u n t and its scattering angle dependence is not determined. In fact the only reliable way at present o f estimating the contribution from higher orders o f scattering is by an elaborate Monte Carlo simulation incorporating details of the sample scattering law a n d instrument geometry. To avoid substantial errors arising from the u n k n o w n higher order scattering it is usual to arrange that the sample scatters not more than 20`% o f the incident beam and preferably nearer 10`%. Second scattering is in roughly the same ratio to primary scattering (i.e. 20%) and so higher orders, which now contribute only = 5% to the total, can be treated quite accurately by an expansion in terms o f the ratio of secondary to primary scattering [I1]. In
189
the present study t h e level of multiple scattering h a s b e e n maintained in the region of 15-25,% b y using thin, flat plate samples. F i n a l l y , the experiment requires careful interpretation because o f dynamic effects [12], which cause a "droop:" in the differential cross section at large s c a t t e r i n g angles. The droop is slight for heavy molecules and can be corrected by an expansion in terms o f the first and second moments of the d y n a m i c scattering law, S(Q, to). For water, however, a detailed knowledge of the molecular dynamics is needed to perform this correction. A sufficiently accurate model is not available and in practice either a model involving free molecular rotations is used to calculate the incoherent cross section [13,14], or a modified expansion scheme using effective masses is derived [15] ~. Neither m e t h o d is satisfactory for fight water as there remains a serious discrepancy between experimental and predicted differential cross sections in both cases.
This difficulty is probably the most intractable and is no doubt a major cause of the experimental controversy. Both Thiessen and N a r t e n [5], and Dore and Reed [7] attempted to correct their model calculations for the self-scattering by fitting a polynomial in Q to their data. To obtain the coefficients of this polynomial either sum rules were applied to the derived structure factors or points chosen at which the distinct scattering was assumed to be zero. Unfortunately, sum rules are only a necessary condition on the data and not sufficient to define the Q dependence of the interference function uniquely, since numerous functions could be conceived which satisfy the same sum rule. The procedure of choosing positions for the zero crossing points of the interference function is equally arbitrary because for light water the interference function is at most = 7% o f the self-scattering. The important point that gets lost in the discussion is that a polynomial fit can easily misrepresent the shape of the self-scattering in the sensitive region Q - - 1 to 5 A, -1 by several percentage points. Because the interference function is such a small fraction of t h e total scattering, a An unpublished approach using effective masses due to J.G. Powles is quoted by Reed [8].
A.K. Soper/ Structure of liquid water
190
serious. Q dependent, systematic error will be introduced precisely in a region where the interference function is most sensitive to intermolecular correlations. In a previous study on liquid hydrogen chloride [16] an empirical approach for removing the hydrogen self-scattering was introduced, rather along the lines o f that used by Enderby and Neilson [17] for aqueous solutions. Although the m e t h o d cannot be rigorously justified it is at least plausible. a n d will be shown to work well for a model mixture of free molecules. It has the important virtue that no arbitrary fitting procedures are needed. Instead it is postulated that the incoherent differential cross section for a mixture containing protons and deuterons is a linear combination of the respective cross sections of the fully protohated and fully deuterated samples. The application of this method to liquid water is nov,, reported. Time-of-flight diffraction at a pulsed neutron source ( W N R at Los Alamos) has also been exploited in order to minimize the dynamic effects on the differential cross sections.
a n d n= is the n u m b e r of a atoms per molecule, b,, is the neutron scattering length, and the angular brackets < . . . ) indicate averages over the spin a n d isotope states of nucleus a_ Q is the m o m e n t u m transfer, o is the molecular n u m b e r density, a, is the energy transfer, with S(Q, a~) the d y n a m i c scattering law [lg]. In fact the quantity measured is not S(Q) but a differential cross section, Z(Qe), whose precise value is determined by the conditions of the experiment. F o r time-of-flight (TOF) diffraction using pulsed neutrons the differential cross section is measured for a set o f neutron time-of-flight channels which means that if the inelastic scattering from the sample is important there will be a range of incident and scattered flight times corresponding to each channel. This range is given by the usual kinematic rules of neutron scattering [19] plus the time-of-flight condition
1 / k i + a / k r = (1 + a ) / k ~ ,
where k,, k r and k¢ are the incident, final and elastic wave vectors respectively and a =
2. Theory for time-of-flight diffraction The quantity sought in a diffraction experiment is the static structure factor S(Q). Following van Hove [18] it is usual to split the structure factor into a " s e l f " term, Ss, corresponding to scattering from single atoms, and a "'distinct" term S ~ , corresponding to interference scattering between distinct atoms. For an N-component system the total structure factor is
(2)
sample to detector distance source to sample distance "
(3)
In terms of these quantities and the scattering angle, 20, the elastic m o m e n t u m transfer is Qe = 2ke sin 8. The measured differential cross section, which is usually presented as a function of Q~, is expressed in terms of the scattering law of the sample and a n u m b e r of instrumental parameters: z(eo)
=
CO)
x l ( k i ) E ( k , ) ( 8 k i / S k . ) ~,,
N
(4)
s(Q) = ~t N
+ E n~na(b,}(,ba)Sg#(Q),
(1)
where
S2(Q)-
÷gd~S;( fo%
[
~o)
1,
_
const.O;
sin ---~--R--dr. -
a n d we note that, unlike for the static structure factor, Q is not held constant in this integration over a~. I(kl) and E ( k f ) are the incident spectrum and detector efficiency respectively, normalized to their values at the elastic wave vector. It is straightforward to show using (2) that the jacobian is given by
1
:o-
[1 + a(ki/k,?] "
(5)
\ A.K. Soper / Structure of liquid water
T h i s term arises because the rate at ~which the incident spectrum is sampled depends on the value o f ~. _ -... Clearly ~?(Q~) only goes over to S ( Q ) in the limit o f elasticscattering, S(Q, ~ ) = S(Q)8(t~). As discussed above the procedure for going from Z ( Q , ) to S ( Q ) is non-trivial for water because of substantial inelastic scattering.-However, it is important to realiTe that inelastic "scattering will affect primarily the self terms and the distinct terms between atoms on the same molecule, the so-called intra-molecular terms. This is because these terms sample all possible molecular motions, whereas the inter-molecular terms sample only collective motions. T h e i n e l a s t i c i t y correction can be reduced by performing the diffraction experiment at small scattering angles a n d high incident energies, such as are available at a pulsed neutron source. To demonstrate this well known effect the T O F differential cross section for heavy water, as measured in this experiment, is compared in fig. 2 to the equivalent curve measured at a reactor source (ILL, Grenoble) a n d it will be seen that the large Q " d r o o p " is absent in the T O F experiment. Instead, because the data are recorded at constant
i[
J
I
t
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-
GPD
40 ° BANK
(3 DAYS)
.
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"~-- ~ A )
191
s c a t t e r i n g a n g l e , there is a roughly c o n s t a n t s h i f t in the T O F d a t a below the self-scatter static limit. Its magnitude increases at larger scattering angles (data not shown). The choice o f 40 o scattering helps to minimize this shift. In fact calculations based on the freely rotating rigid molecule model to be described s h o r t l y showed that at 40 o the d o w n w a r d shift in the T O F d a t a would be --- 0.10 b / s r / m o l e c u l e , but at 90 ° it would be 0.46 b / s r / m o l e c u l e . Correspondingly, the droop in the reactor d a t a arises from scanning angles at fixed incident neutron energy. In this experiment measurements of Z ( Q ) were m a d e on three water samples: H 2 0 , D20 and a 5 0 : 5 0 mixture of H 2 0 and D20. If the mole fraction o f H 2 0 used to make the mixture is x it can be shown from (1), with ,Y(Q) substituted for S(Q), that
x z H ' ° ( Q ) + (1 -- x ) z D ' ° ( Q ) -- Z m-°: D,O(Q) = 4 x ( 1 - - x ) ( ( b H ) -- < b D ) ) "~ - ~ 'dH H ( Q ) + A ,
u n d e r the usual assumption that structure factors are invariant to isotope substitution. The more general scenario of structure factors being isotope dependent is discussed in the appendix. In that case the result (6) is an average of the structure factors in heavy a n d light water. Contributions to the term A arise from noncancellation of inelasticity effects in the H and D self terms, and in the O - H , O - D and O - O distinct terms, but the largest contribution, An, probably comes from the hydrogen self terms. If they do not cancel in the way postulated then the residual can be expressed in terms of the individual atomic self differential cross sections: A H -----2 X ( 1 - - x ) [ < b 2 ) ( ~ i
HzO - - Z~i HDO)
+ < b ~ ) ( , ~ 5 D:° -- ~ 5 riD°)],
0
(6)
(7)
1
0
!
5
!
10
I
15
20
Fig. 2. Time-of-flight differential scattering cross section for heavy water (upper curve), measured on general purpose diffractometer at Los Alamos. Below it is shown the same curve
measured at a constant wavelength source (ILL - Grenoble). The inelasticity shift is roughly constant with W in the TOF data.
where ~?~iH2° for example is the hydrogen self cross section in the H 2 0 molecule. In order to make a plausible estimate, ,fiH was calculated for a freely rotating a n d translating rigid molecule. This model, which used the formalism of R a h m a n [20], incorporated the correct m o m e n t s of inertia o f each of the three molecules H 2 0 , D20 a n d H D O . Translational motions of the molecules were treated in the
A.K. Soper / Structure of liquid water
192
0.2
0.1
Z dH " (O~"~"°~ecuuE : •.-
'
t
0.0 -0.I
-O.2 t 5
t
10
olA -~1
1S
Fig. 3. Hydrogen-hydrogen partial structure factor measured in this experiment. The thermodynamic limit for these data (not shown) occurs at -0.438. Also shown is the calculated single molecule scattering (solid line), and, on the same scale, the residue, A H- (dashed line) arising from non-cancellation of self-scattering in the model.
ideal gas approximation and the rotational and translational motions were convoluted together in the manner of Lurie [21] to form S(Q, to) for each molecule. This was then integrated numerically acc.ording to (4) to yield ~(Qe)- The result for A n is shown as the dashed line in fig. 3. Calculation of the rotational terms for a water molecule involved summation over many rotational transitions at the larger Q values, and this introduced numerical errors in the calculated ~ ( Q e ) . T o determine this erriSr-the zeroth moment of the dynamic scattering law was calculated [theoretically this quantity should be unity, see eq. (1)], and errors were found to be on the order 2,% out to a Q value of = 13 /k-~. Hence the cancellation of self terms for this model, as depicted in fig. 3 was good for this model and within numerical uncertainties. In the real liquid the molecular motions are hindered, which renders A n even smaller. Similar calculations for the distinct terms showed that their contributions to A are much smaller than A H- In the subsequent data analysis it is therefore assumed that z~ is negligible.
3. Experimental The general purpose diffractometer at the W e a p o n s N e u t r o n Research facility ( W N R ) in Los Alamos is currently being upgraded as part of the recently initiated user program for the pulsed neu-
tron scattering facility. At the time of this experiment it was a p r o t o t y p e instrument consisting of four banks of sixteen 3He proportional counters placed at scattering a n # e s of 10 o, 40 o, 90 o, and 150 o [22]. The incident flight path was. 10 m and the final flight path 1 m. A large background c o m p o n e n t from air scattering in the b e a m prevented use of the data taken at the 10 o position, and inelasticity effects appeared much larger at 90 ° and 150 °, so the present analysis will concentrate on the data taken at 40 o scattering angle. The instrument was calibrated using the Bragg reflection data from a nickel powder sample. The samples were held in a flat plate aluminium container angled at 30 o to the incident beam. The temperature was 25 ° C and atmospheric pressure at Los Alamos is = 0.78 atm. Cans of different thickness were used: 0.34 m m for H 2 0 , 0.60 mm for H 2 0 : D 2 0 mixture, and 3.3 m m for D20, chosen so that multiple scattering was 15% of the total on average and did not exceed 25% at the longest neutron wavelength (where the neutron cross sections are largest). Thicknesses were monitored to -- 3,% accuracy by transmission detectors placed before and after the sample. Data were taken for sample plus container, e m p t y container, vanadium slab (made from sintered vanadium powder, thickness 3.3 mm, and used as an instrument calibration), and air scattering background. Counting times were ---3 days per sample. After rebinning the data in constant width Q-bins (0.1 A - 1 ) and summing data from sixteen detectors, the gross counts were in excess of 106 per bin at the peak of the incident neutron spectrum. The data were corrected for self-shielding [23,24], multiple scattering, background scattering, and the vanadium data were adjusted for inelasticity effects [12]. The multiple scattering calculation used a formalism which allows all orders o f scattering to be included [11]. The sample data were then normalized point b y point to the vanadium data to obtain differential cross sections. This point-bypoint approach, although it reduces the statistical accuracy o f the result, is a more reliable w a y o f removing the incident spectrum and detector efficiency effects from the data than the conventional practice of fitting polynomials to the vanadium data, since the spectrum varies widely
A,K. Soper / Structure of liquid water over the energy range Of the experiment a n d does not have a simple d e p e n d e n c e on Q. Small errors, on the order o f 1%, in fitting polynomials c a n lead to much larger errors in the final structure factors, particularly in the case o f light water where the interference function sits on top o f a large incoherent background. In fact a similar c o m m e n t applies to analysis of reactor d a t a taken with a multi-detector array since subtle variations in detection efficiency as function o f position are certain to occur. Ignoring such variations b y smoothing with a polynomial can easily introduce spurious features in the final differential cross section. O n an absolute scale the sample differential cross sections are therefore as accurate as the known v a n a d i u m differential cross section, which, due to uncertainties in vanadium sample composition, was only known to = 5% in this ease. The relative accuracy of one sample to another however will be better than this if the relative thicknesses and cross sections o f each sample are known well. As already mentioned these latter quantities were monitored to within 3% in this experiment.
4. Results mad discussion 4.1. H - H distribution function The result of applying eq. (6) to the three sets of differential cross sections is shown in fig. 3. The data do not extend below 1.4 A, - l because of frame overlap at W N R which occurs at 8.3 ms (at a wavelength of 2.9 A on a 11 m total flight path). Also the original raw data gave a result which oscillated about a level = 0.24 above zero. This represents a cumulative error in all three data sets, but if it came only from the light water curve it would correspond to a systematic error of -- 4% in that curve, and so is within likely d a t a analysis errors. A constant shift downwards of 0.245 was applied to the d a t a before plotting. The precise shift was determined by minimiTing the amplitude o f truncation oscillations in the subsequent Fourier transform of the data. Anothei" procedure of applying a sum rule to the data [4,6] is strictly inapplicable to the water experiment because corrections for intra-molecular inelasticity effects can
193
o n l y be m a d e approximately. The Fourier transform of ,~dn(Q), eq. (8), is shown in fig. 4. Clearly the statistical accuracy of ,~dH(Q) declines beyond --10 A -1, so the d a t a have been truncated at 10.8 .~-1 (again chosen to minimize truncation effects). F o r the region Q = 0 - 1 . 4 A-~ it was necessary to extrapolate the d a t a to the t h e r m o d y n a m i c limit at Q = 0, i.e. - 0 . 4 3 8 . This was attempted by fitting various polynomials in Q but apart from a slight d o w n w a r d base line shift to the transformed data in the region r = 0 - 4 ,~ none gave significantly different results from a simple straight line fit. The linear extrapolation in fact gave the least a m o u n t of shift. The precise behaviour of the data in this region will have to be determined in another experiment. The Fourier transform calculated here is therefore
GHH(r) 1 £10.SA-~¢.),~d sin Qr~r~ = 1 + 2---~_oJ° ~- HH(e)~u~e.
(8)
it can be seen in fig. 4 that a large peak occurs at 1.5 A as well as another, unphysical peak at 0.7 /k. This second peak is now believed to be an artifact of the truncation and intramolecular inelasticity correction. To show this the single molecule structure factor as calculated from the free molecule model (solid line in fig. 3) was truncated a n d transformed in the same way as ",he data. A similar unphysical peak appears in the transform at about the same position ( d o t - d a s h e d line in fig. 4), although it is not reproduced well enough to be used as a correction for the data. The H - H intramolecular distance used for the free molecule calculation was the gas value, 1.515 ,~,. It will be seen that a peak at almost the same position occurs in the transform of the model, fig. 4: despite the substantial inelasticity a n d truncation effects the H - H distance is reproduced quite faithfully in the calculation. Therefore the peak at 1.48 A, in the experimental data is to be associated with the H - H intra-molecular distance in the liquid. The inter-molecular structure ( r > 1.9 A) bears remarkable resemblance to computer simulation predictions. Previously these d a t a were compared with the " a b initio" calculations of Lie, Clementi a n d Yoshimine [25] (LCY), a n d the "'effective
194
A.K. Soper / Structure of liquid water
GHH
2.0
!
= gHH(r)
i
!
i
h,t
t
|
EXPT (H-H
• FREE
1.5
I
DISTANCE)
MOLECULE
INELASTICITY
MODEL
PEAK
1.0
-
-
TIPS2
0.5
0.0 0
!! ! i!i i i
I
1
2
3
4
5
6
7
8
Fig. 4. Measured H H pair correlation function (solid line) compared to the TIPS2 simulation (long dashes). In the range r = 0 - 2 ,A the Fourier transform of the single molecule calculation is shown (dash-dotted), and it will be seen that the unphysical peak at = 0.7 A appears in both experimem and calculation.
structure factors was not obtained. H o w e v e r a n o t h e r useful curve was o b t a i n e d by subtracting the already d e t e r m i n e d H - H distribution f r o m the c o m p o s i t e d a t a for heavy water. This subtraction was p e r f o r m e d on b o t h the " r - s p a c e " a n d " Q space" dat a and the results were essentially identical. T h e r-space result, which is a sum o f O - H a n d O - O pair correlation functions, will be described here. As can be seen in fig. 2 the differential cross section for I:)20 oscillates a b o u t an almost horizontal line, suggesting that the inelasticity effects are constant with Q for these data. T h e relatively small shift o f this line below the static limit c o m p a r e d to d a t a taken at larger scattering angles f u r t h e r suggests that the inelasticity correction is small at this angle. As a result the heavy water cross section was Fouri er t ransform ed directly after truncating sm oot hl y at 20.4 A,-t and subtracting an a m o u n t correspondi ng to the oxygen and deuterium self-scattering.. T h e result, fig. 5, shows a sharp peak at - - 1 A correspondi ng to the O - H molecular b o n d length, and a smaller peak at 1.5 ,~ suggesting the H - H distance. T h e rest o f the
I
i
i
i
i
i
0.336goo+ 1.554gOH -F 1.780gHH -- 3.67 potential" molecular d y n a m i c study due to Stillinger an d R a h m a n [26] (ST2). H e r e com pa r i s on is m a d e with the recent simulations due to Jorgensen [27] (TIPS2). O f the available simulations the latter appears to give the best agreement with this experiment. N o t e that because this simulation uses a n "'effective" potential the effects o f m a n y - b o d y forces and q u a n t u m mechanics on the liquid structure are obscured. T h e coordination n u m b e r of the first peak, calculated for radius values from 1.9 to 3.1 ,A, gave 5.8 _+ 0.3 hydrogen atoms, which is the same as f o r the TIPS2 simulation, and p r o b a b l y n o t significantly smaller than the 6.1 h y d r o g e n a t o m s fo u n d in the L C Y simulation.
4.2. O - H plus 0 - 0
distributions
Because o f the u n k n o w n inelasticity corrections a full separation o f the da t a into three partial
2.0
x l 0 -1
/x.LCY 0.0 \ ~ P T -2.0
t ~THEOR~FICAL
LIMIT AT r = 0
!
/ 1 _,J_. -4.0
!
0
2
!
!
4
!
!
!
6
Fig. 5. Fourier transform (solid line) of heavy water data shown in fig. 2. Peaks corresponding to the O - H bond and H - H intramolecular distance are seen at = 1 . 0 A and =1.5 .-~ respectively, as well as an unphysical peak at 0.6 ,~, believed to be associated with inelasticity effects. The LCY simulation is shown as the dashed line.
A.K. Sopo"/ S t ~ c t ~ of l ~ a
cUrve is rather featurcless"but shows:good-agree-merit with earlier d a t a [28]. The un-physical peak a t 0.6 A and the smaller amplitude " ' r i p p l e s " b e y o n d 1.5 ~,, which have a characteristic period o f 0.35 ,~, are believed to b e a s s o c i a t e d - w i t h c/~mbined truncation and inelasticity effects. The data for G H H , weighted b y the appropriate neutron scattering factor (1.78 b / s r / molecule for heavy water), were subtracted from the data in fig. 5, and after renormalization the result GOHOO = 0.82go. (r) + 0.18goo(r)
(9)
is shown in fig. 6, along with a similar combination for the L C Y and TIPS2 c o m p u t e r simulations. N o t e that G O H O O is dominated b y the O - H correlation function. Truncation and inelasticity effects are apparent at small radius values as seen from the negative values for G O H O O , but a strong O - H b o n d peak at 1.0 A is present, a n d the H - H peak seen in heavy water at 1.5 A, fig. 5, has been subtracted out.
2.0
G O H O O = 0 . 1 7 8 goo + 0.822 goH
,.s
L
~ote~
~
-~-
.
195
Once : again-,the~ short range: inter'molecular structure d0es not differ veryfsignificantly from w h a t c0mpute~Lsimulations have-predicted. A n integration, between r = i . 6 : A and "r = 2.4 ",~,: of_ the first p e a k i n G O H O O yielded 2.! _+ 0.1 hydrogen-atoms-about oxygem This is close to the value for t h e : T I P S 2 simulation (1.9) but significantly. la/ger than for the L C Y - s i m u l a t i o n (1.7). It is concluded that for both G H H and G O H O O the TIPS2 potential has largely removed the earlier reported discrepancy between simulations a n d this experiment. At longer range, r > 4 ~,, a significant discrepancy between experiment and simulation appears. This can be seen best in the heavy water d a t a of fig. 5. A broad oscillation at = 7.8 ,~ appears at 6.6 ,~ in the L C Y simulation. The same discrepancy appears in the simulated differential cross section for heavy water (not shown), where the principal peak at Q = 2.0 ,~-~ is shifted to a slightly larger q value of 2.2 ,~-1. Generally, positions of peaks in diffraction work are much more. reliable than peak heights because of the simple and reliable scale calibration that is performed, so the discrepancy seen here is real. The same discrepancy is not seen in G O H O O , b u t the region r > 4 ,~ is probably masked b y truncation oscillations from the G H H function used in the subtraction. If, following Egelstaff and R o o t [29], the potential used in the L C Y simulation is regarded as a pair potential, then a long range discrepancy such as this might arise from m a n y - b o d y effects n o t included in the simulation. The same discrepancy does not occur w i t h the effective potential simulations.
5. Conclusion 0.5
t
0
~
!
2
!
I
4
I
l
!
6
Fig. 6. Measured sum of O - H and O - O pair correlation functions, compared to LCY-(short dashes) and TIPS2 (dash'-dotted) simulations.
By avoiding some of the pitfalls that have traditionally been encountered while performing the neutron diffraction experiment o n liquid water, reliable and largely model-free data have been obtained. This was achieved b y exploiting timeof-flight diffraction at a pulsed n e u r o n source and utilizing a subtraction procedure, b o t h of which help to circumvent the major obstacle of performing inelasticity corrections to the d a t a . - T h e only expense for the intrinsic reliability of the results is
196
tLK. Soper/ Structureof liquidwater
that a full separation into three partial structure factors was n o t obtained. Instead the H - H distrib u t i o n was o b t a i n e d together with a sum o f O - H and O - O distributions. T h e latter d a t a are weighted in the O - H function and so are still useful for testing theories. F o r example, the p ai r potential theories ma y need revision to account for the long-range b eh avi our in the distribution functions. This was an exploratory experiment which clearly should be followed up when the high flux pulsed sources currently in construction becom e available. In particular small Q data and better statistics at large Q are needed.
( O - O , O - H , O - D , H - H , D - D a n d H - D ) , which are in general d e p e n d e n t o n composifion¢ X. T h e distinct term in eq. (1) for this mixture is n o w
( i,o) "-Soo(Q, x) + a( bo)( bH)Son( Q, x) + 4(b.)Zsn-'Hnn(Q, x ) ,
(A.1)
where
( b . ) S o . ( Q, x) = x b . S o . ( O, x) + (I -
x)boSoD(q, x),
= x bHSuri( Q, x) + ( 1 - x)-bDSDD(O, ~ 2 ~) +2x(1
Acknowledgement
-
x)bHbDS.D(Q, x),
and
( b . > = x b . + (1 - x)bD. This work was p e r f o r m e d u n d e r the auspices o f the US D e p a r t m e n t o f Energy. I would like to t h a n k R.N. Silver for considerable s u p p o r t and e n c o u r ag emen t th r o u ghout the course o f the work. M a n y thanks also to P.A. Egelstaff for a critical reading o f the manuscript. A University Research Fellowship and grant f r om the N a t u r a l Sciences a n d Engineering Research Council o f C a n a d a is acknowledged.
Appendix I shall analyse the result of the neut r on experim e n t for the case where the structure factors o f heavy and light water are not the same. O f course H D O molecules are formed rapidly b y p r o t o n / d e u t e r o n exchange in a mixture o f light and heavy waters, b u t because neutrons are scattered by nuclei the structure factors observed in a mixture are measures o f the ensemble averaged correlation between pairs o f nuclei, so the distinction between say a p r o t o n on H z O and one on H D O is lost. T h e r e f o r e from the point o f view o f understanding the n e u t r o n diffraction experiment it is m o r e useful to consider a mixture o f heavy and light water as a t h r e e - c o m p o n e n t system consisting of H, n a n d O. T h e experiment measures a sum o f six p a r t i a l s t r u c t u r e f a c t o r s f o r t he m i x t u r e
T h e bars over the symbols therefore indicate isotope averages. T hese average structure factors are subject to asymptotic conditions: as x ---, 1,
s . . ( Q , x) ---,s ~ ° ( Q ) = s . . ( Q , 1), a n d as x ~ 0,
s . . ( Q , x) ---,s~ho(Q) = SDD(Q, 0),
(A.2)
where, for example, S~h°(Q) is the H - H structure fact or in pure light water. T h e same considerations appl y for the com posi t i on d e p e n d e n t Soo a n d SOH terms. Eq. (A.1) is a general expression for the distinct scattering. It is still not particularly useful because the individual structure factors c a n n o t be measured. Wi t hout loss o f generality, however, each o f the average structure factors can be expressed as a c o m b i n a t i o n o f its respective asym pt ot i c limits:
S,,t~(Q, x)=f(x)S~Ha'-°(Q) + (1
-f(x))S~2°(Q),
(A.3)
where
/ ( 0 ) = 0,
f ( 1 ) = 1,
but f ( x ) is otherwise arbitrary. In general it m a y be a function o f Q. If structural differences are n o t too large then a first a p p r o x i m a t i o n is f ( x ) = x .
.
:
?
-
/~.
"--" _
a . K~sop e b / -Strut t-if,e o f l i ~ d
The same analysis as Usedin tlietext toobtain the H-Hstructufe f a c t o r f r o m t h e d a t a [ e q . (6)] n o w
yields
--
4X(I--x){~'H.:o H { } -I--.~'D.o[(b H~I nzo
+bo(bH _
_
- ~
{-}]
~ " O H I J"
(A.4)
bo) z + 2bo(bH--
bo)-
N o t e t h a t t h e r e is n o w a t e r m a r i s i n g f r o m n o n cancellation of the O-H structure factors, but it involves a simple difference of the two structure factors and so will likely be small. The O-O terms still cancel identically, within the linear approximation. From (A.4) it can be seen that the quantity o b t a i n e d b y t h e a n a l y s i s is a n a v e r a g e o f t h e H - H structure factors for the two liquids. For the condit i o n s o f t h e p r e s e n t e x p e r i m e n t ( x = 0.5, a n d b n = - - 0 . 5 6 1 b D ) , t h e r e s u l t is, i g n o r i n g t h e O - H t e r m , b ~ ( 0 . 5 3 3 ~ H H ~° + 1.g04Y-HD¢), i.e. i n t h i s e x p e r i m e n t t h e r e s u l t is d o m i n a t e d the heavy water structure factor.
by
References [1] A.K. Soper and R.N. Silver, Phys. Rev. Letters 49 (1982) 471. [2] K.E. Gubbins, C.G. Gray, P.A. Egeistaff and M.S. Anath, Mol. Phys. 25 (1973) 1353. [3] O. Palinkas, E. Kalman and P. Kovacs, Mol. Phys. 34 (1977) 525.
--"
: - ' --:
:
._
: 197
-[41~L. Koestei-; H. Rai~ch, M. Herkens and K. Sehr:,der,: summary of Neutron Scattering Lengths, Report-No, JUL-1755, K F A Julieh, WestGermany (1981). i51 W.E. Tl~essen and A.H. Narten, J. Chem. Phys: 77 (i982) 2656.
where { ) = 3x(b n-
~:ater- -
-
[6] A.H. Narte n, W.E. Thiessen and L. Blum, Science 217 (1982) 1033. [7] J.C. Dote and J. Reed, to be published. [8] J. Reed, Ph.D. Thesis, University of Kent at Canterbury, U K (1981). [9] E. Swift, J. Am. Chem. So(:. 61 (1939) 198. [10] LA. Blech and B.L. Averbach, Phys. Rev. 137A (1965) 1113. [11] A.K. Soper, Nucl. Instr. Meth. 212 (1983) 337. [12] G. Placzek, Phys. Roy. 86 (1952) 377. [13] L. Blum, M. Rovere and A.H. Narten, J. Chem. Phys. (1982) 2647. [14] P.A. Egclstaff and A.K. Soper, Mol. Phys. 40 (1980) 553. [15] P.A. Egelstaff and A.K. Soper, Mol. Phys. 40 (1980) 569. [16] A.K. Soper and P.A. Egelstaff, Mol. Phys. 42 (1981) 399. [17] J.E. Enderby and G.W. Neilson, in: Water, a comprehensive treatise, Vol. 6, ed. F. Franks (Plenum Press, New York, 1979) ch. 1. [18] L. van Hove, Phys. Rev. 95 (1954) 249. [19] W. Marshall and S.W. Lovesey, Theory of thermal neutron scattering (Oxford Univ. Press, London, 1971). [20] A. Rahman, J. Nucl. Energy A 13 (1961) 128. [21] N.A. Lurie, J. Chem Phys. 46 (1967) 352. [22] A.K. Soper, in: Neutron scattering - 1981, A l P Conference Proceed/ngs No. 89, ed. J. Faber (ALP, New York, 1982) p. 23. [23] G.H. Vineyard, Phys. Rev. 96 (1954) 93. [24] V.F. Sears, Advan. Phys. 24 (1975) 1. [25] G.C. Lie, E. Clementi and M. Yoshimine, J. Chem. Phys. 64 (1976) 2314. [26] F. StiIlingerand A. Rahman, J. Chem. Phys. 60 (1974)
1545. [27] W.L. Jorgensen, J. Chem. Phys. 77 (1982) 4156. [28] G. Walford and J.C. Dote, Mol. Phys. 34 (1977) 21. [29] P.A. Egelstaff and J.H. Root, Chem. Phys. 76 (1982) 405.