Neutron Scattering Results

NEUTRON SCATTERING RESULTS K. E. Larsson Royal Institute of Technology, Stockholm, Sweden U. Dahlborg National Research Council, Stockholm, Sweden and K. Skö ld AB Atomenergi, Studsvik, Sweden 1. Neutron Method 119 1.1. Principles of the Neutron Scattering Method: Observed and Derived Quantities 119 1.2. Experimental Technique 128 2. Liquid Helium 134 2.1. Static Structure Factor 134 2.2. The Dispersion Relation and Its Related Quantities 136 2.3. Line Width of Excitation Peaks 143 3. Liquid Argon 146 3.1. Atomic Distribution 146 3.2. Atomic Motion 153 4. Hydrogen 166 4.1. Total Cross Section 166 4.2. Differential Cross Section 168 5. Methane 171 5.1. Total Cross Section 171 5.2. Differential Cross Section 172 5.3. Scattering Law 177 References 181 1* Neutron Method 1.1. PRINCIPLES OF THE NEUTRON SCATTERING METHOD: OBSERVED AND DERIVED QUANTITIES 1.1.1. Introduction The scattering of slow neutrons has proved to be a very powerful technique in obtaining information concerning the dynamical behavior 119 120 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD and the microscopic structure of the condensed state of matter. For other techniques used for this purpose, such as inelastic scattering of light, nuclear magnetic resonance (NMR), and x-ray studies, the time scale is extremely long or extremely short compared to a characteristic time interval in the liquid motion of the order of 10~13 to 10~12 sec. Thus these techniques fail to be sensitive to the detailed atomic motions in the scatterer. In contrast, a slow neutron of a wavelength comparable to interatomic distances interacts with a scattering atom or system of atoms on a time scale from 0 to about 10~ n sec. This length of observa- tion time permits the slow neutron to see both the high frequency vibratory motions and the elementary diffusive motions with relaxation times of the order of 10~12 sec. The purpose of the present chapter is to describe briefly the method of scattering of slow neutrons and to review the results gained on four liquids, namely, helium, argon, hydrogen, and methane. However, first a terminology and a framework has to be established within which the results can be discussed. For a more detailed review of the field the reader is referred to two monographs by Turchin [1] and Egelstaff [2] and to the proceedings from three symposia sponsored by IAEA on this subject [3-5]. The differential scattering cross section per atom is, according to van Hove [6], given by four-dimensional Fourier transforms of correla- tion functions G(r, t) and Gs(ry t) - g £ = «c 2 oh^S c o h (K ) £ o) (1.1) jnk j — = ßincoh T - ^Incohi*, ω ) (1.2) did α ω R0 where 1 f Scoh(K, ω ) = r - exp[i(Kr — wt)] G(r, t) dt dt (l-3a) 1 r Sincoh(*, ω ) = ^ J exp[i(xr - ω ή ] Gs(r, t) dv dt (1.3b) The functions S, sometimes called ' 'scattering laws," depend only on the properties of the scatterer; Ϋ Ι Υ . and ϋ ω stands for the momentum and energy transfers in the scattering process and are given by κ = k - k0 ω = (ä /2m)(£2 - V ) (L4) where k0 = 2π /λ 0 and k = 2π /λ are the initial and final neutron wave vectors, λ 0 and λ are neutron wavelengths, m is the neutron mass, and  NEUTRON SCATTERING RESULTS 121 2 π ί is Planck's constant. The coherent and incoherent scattering lengths are aloh and a^coii · The classical interpretation of the correlation functions G(r, t) and G s (r, t) is as follows: given a particle at the origin at time zero, G s (r, t) gives the proba- bility that the same particle is at position r at time t and G(r, t) gives the probability that any particle is at position r at time t. Other quantities often used in connection with discussions of the scattering law are the intermediate scattering functions corresponding to the spatial part of the transforms just given /(κ , t) = -!- j dt exp(*xr) G(r, *) (1.5a) and / s (*, t) = i - J dv exp(mr) Gs(r, t) (1.5b) Particularly the function /(κ , 0) has a clear and simple physical meaning as discussed later. The cross section is separated into two parts, the so-called coherent and incoherent cross sections. The properties of incoherence or coherence depend on the details of the interaction between the neutron and the nucleus in the scattering process and will not be discussed here. It is seen from the equations that the self part of the correlation function enters into the incoherent cross section while the coherent cross section is determined by the complete correlation function. Most nuclei scatter both incoherently and coherently. Important exceptions in connection to the present treatment of some simple liquids are the cross sections for hydrogen, which is almost completely incoherent, and for helium, which shows a 100% coherent cross section. 1.1.2. Incoherent Case In case the scattering nucleus has an incoherent cross section all the motions of the scatterer are revealed in the scattered spectrum. The fact that various atomic motions occur on different time scales is thus most easily and without discrimination seen for this case. The low frequency— or equivalently the long time motions—corresponding to frequencies smaller than 1012 cps or to times of about 10~12 sec or longer, give rise to a more or less narrow peak centered round the ingoing neutron energy and is often called the quasi-elastic peak. The high frequency or short time motions, corresponding to frequencies larger than 1012 cps 122 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD or times shorter than 10~12 sec, give rise to a broad inelastic neutron spectrum. Considerable effort has gone into the interpretation and understanding of the quasi-elastic neutron scattering. Various attempts were made to separate it from the inelastic part, and more or less complex models were created to throw some light on the nature of the diffusive atomic motions. Particular attention was given to the width of this quasi-elastic peak, which is found to be a function of κ (or the scattering angle). The width is an observable quantity sometimes easy to obtain but mostly rather difficult to define accurately due to the problems involved in the separa- tion of the quasi-elastic peak from the rest of the observed spectrum. For several simple models the full width (Δ Ε ) at half-maximum of the quasi-elastic peak was given as follows: (a) The simple diffusion model [7] Δ Ε = MDK2 (1.6) where D is the self-diffusion coefficient. (b) The simplest jump diffusion models [8, 9] 2W 2ft/ 2ft / e~2W \ "-^("-■ nnsd TQ\ 1 + Z)iA 0 , <■ · *> or 2ft Λ sin κ ΐ Δ Ε ~(>-==r· ) <»"· > where τ 0 is the residential time between jumps, 2W' is> the Debye-Waller factor, and / is the jump length. (c) The modified jump diffusion model [10] where D0 is a smaller diffusion coefficient describing a slow continuous motion of the vibrating particle during the residential time r 0 . (d) A modified gas model [11] Δ Ε = 2^(2 In 2)1/2(Z)/r)1/2* (1.9) where r is a delay time before diffusion sets in and which transforms to the simple gas model, if τ = tß = MD/kBT, where M is the atomic or molecular mass. This formula should be valid for somewhat larger K-values and not in the limit κ —> 0.  NEUTRON SCATTERING RESULTS 123 (e) The complex model [12] involving motions of the center of gravity of the molecule as well as internal atomic motions relative to the center of gravity (such as free or hindered rotations of a simple molecule like CH 4 ). Two extreme cases are: (i) High viscosity (τ ' 0 > r 0 ) AE = — [1 - F(K, I) e x p ( - 2 ^ · - 4We)] (1.10) T oo where l/r 00 = l/r 0 -f l/^o a n d To^s t n e residential time for a proton before jumping, r' 0 is the time for which free diffusive motions of the molecule are hindered, and 2Wi and 2We are the Debye- Waller factors for the proton and center of gravity vibrations, respectively; F(K> I) is an integral which mainly depends upon the K-value and the internal protonic jump length (for instance partial rotation), the value of which varies between 1 and 0. It may also be of oscillatory character. (ii) Low viscosity τ [ ^> r' 0 and r 0 Δ Ε = 2h[DK* + (l/r 0 ) - F(Ki I) e x p ( - 2 ^ · ) ] (1.11) where τ [ is the time for which the molecule is free to diffuse (a fraction of time of ^/(TQ + τ [) &t I for case (ii)). A variety of models were thus created to assist in the understanding of the incoherently scattered quasi-elastic neutron intensity from a liquid, and attempts were made to relate its width to macroscopic properties such as diffusion as well as to microscopic phenomena such as relaxation time for elementary atomic or molecular steps of motion. In connection with experimental and theoretical neutron scattering studies it was shown [13] that in the absence of interference scattering a generalized frequency distribution p(ß) in a liquid, corresponding to the distribution of normal modes f(ß) in a solid, can be derived from the inelastic neutron spectrum through p(ß) = ß* lim (1/α ) SincohK ß) (1.12) where α = #2/c2/2 MkBT and β = fi<jù \kBT. For a solid the relation between p(ß) and f(ß) is (L13) l*n = -£m Thus, p(ß) may be obtained directly from a series of scattering measure- ments for very small momentum transfers. The value of p(0) is shown to 124 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD yield the self-diffusion constant through p(0) = MD/nkBT. The observed similarity between liquid and solid state frequency distributions has stimulated the use of solid state formalism in the derivation of f(ß) from the inelasticly scattered neutron spectrum. This method, which is identical to a phonon description of the atomic motion, is of course to be considered merely as an aid in the interpretation and may not, without a careful and critical comparison to other evidences, be used as a proof for the existence of phonons or quasi-phonons in a liquid. In general, the atomic motion revealed in the inelastic part corresponds to energy transfer also observable by use of other radiation scattering techniques. Thus in the more general case the energy transfer observed in the inelastic neutron spectrum is also seen in light scattering (infrared or Raman spectra). The main difference is that in neutron scattering not only energy transfer but also momentum transfer is involved. 1.1.3. Coherent Case In the case when coherent scattering occurs or in the more general case of a mixed coherent and incoherent scattering (such as for liquid argon) one has to resort to the general definitions of the cross sections as Fourier transforms of the correlation functions G(r, t) and G s (r, t). According to the definitions given above a complete experimental mapping of the scattering functions £(κ , ω ) and Ss(x, ω ) allows a deter- mination of the correlation functions by way of a Fourier inversion of the data. From the definitions of the correlation functions it is found that G s (r,0) = 6(r) (1.14a) G(r,0) = 8(r)+g(r) (1.14b) where g(r) is the static pair distribution function, which gives the average particle density around a given particle at the origin. This function is well known from x-ray studies of liquids. Experimentally, G(r, 0) is obtained from an angular distribution study. Evidently one obtains for a liquid -^JFT = acoh J S(yty ω ) ά ω = a2e0h j exp(ntr) G(r, 0) dv = <&Λ /(Χ , 0) = <z2coh(l + J exp(mr)[£(r)</r]) = 4oh[l + y(x)] (1.15a) From a complete mapping of S(x, ω ) the pair distribution function ^(r) is obtainable by a Fourier inversion of the integral over all energy transfers tiw. In general, a direct angular distribution study by use of neutrons—a determination of dacoh/dü —gives the desired liquid  NEUTRON SCATTERING RESULTS 125 structure factor 1 + γ (χ ) only if the ingoing neutron energy E0 (or fiœ0) is much larger than all energy transfers occurring in the energy exchange between the neutron and the liquid system. This follows from the fact that in general the differential cross section cPojdQ dœ contains the factor k/k0 = [(ω + ω ο )/ ω ο ] 1 / 2 a s a multiplier in front of £(κ , ω ). Thus, only if ή ω 0 ^> ϋ ω one finds that d*a , 2 J ^ aCOh j S(yt}œ)dco (1.15b) dQdi This is called the "static approximation,'' valid only if fiœ0 ^> fiœ. A few attempts were made to create models for the liquid atomic motion such that theory could predict S(x, ω ). In general, 5(κ , ω ) is determined by the correlation function G(r, t) = G s (r, t) + Gd(r, i) where Gd(r, t) is the time-dependent pair correlation function. The main problem was to find a reasonable and physically plausible construc- tion for Gd(r, t). The oldest attempt—the so-called convolution ap- proximation—describes Gd(r, t) as a convolution of G8(r> t) with the static pair correlation function g{r). This results in a cross section [7] i2 2 J2 d crCOh tfcoh & tfincoh π ι / vi /i i^:\ [1 κ )] L16 ΊΩ Ϊ^ = <^ -Ί Ω Ί ^ +* ( > which, however, has failed to describe the observed £(κ , ω ) when exposed to a critical test, the main reason being that due consideration is not given to the existence of an atom at the origin. In fact, the motions of the atom at the origin and the neighboring atoms might be—and most probably are—coupled, such that correlated motions occur. Assuming that correlated motions of the phonon type occur within a correlation range R a round each atom in the liquid, a correction to the convolution approximation was created to yield a cross section formula [14, 15] J2 2 J2 u o"coh #coh « ° "incoh Γ 1 ι / \ ■ i ar/r> νι /ι ι τ \ where q is the absolute value of wave vector of a quasi-phonon of energy ϋ ω \ L(R, K, q) is a complicated function given by 3R r° ° 1 r+q (r / R2 \ m K q) = dk exp ( k + x) ' ^ v z /„***> 2ç L I t ( - T « - *) - e x p ( - f (K + k + χ ή ] - [exP ( - f (* - K)2) - ex P ( - ? (* + *)2)] j ä x (1.17b) 126 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD Still a third attempt was made to calculate the cross section on the basis of an extreme polycrystalline model for a liquid with the assumption that not only is there a correlation range R (within which ordered and coupled atomic motion occurs for a time τ ) long enough to permit the development of quasi-phonons, but there is also a geometrical order with a—perhaps partly destroyed—regular lattice structure. This rather extreme quasi-crystalline model also predicts a dependence of cross section on the polarization of the phonons. The cross section is given by Egelstaff [16]. (For a review of this and other models the reader is referred to Dahlborg and Larsson [17].) J2 2 aJ 2 u aCoh #coh O'lncoh ~/ /i ι ο Z(q, /c, Θ^\) (1.18a)\ dQ doj ^incoh ά Ω d( where Z(q, /c, Θ ) is a dynamic liquid structure factor in a simplified way dependent also on the angle Θ between the phonon polarization vector eq and the direction eK of the momentum transfer vector κ . Here, Z(qy Ky Θ ) is given by with cos Θ = eq · eK for the longitudinal vibrations and sin Θ = eQ · eK for the transverse vibrations; r m i n and r m a x are obtained from K— q K+ q n 1Q v Tmin = —^ r m a x = -7J (l.lö C) The limits within which scattered intensity is allowed are then identical to the corresponding limits in the case of a polycrystalline solid for which intensity corresponding to the Bragg reflection rhkl and a certain q value occurs between the limits {27rrhkl -f- q) and (2π τ Μ Ι — q). The two later models have so far had some success in picturing the observed S(K> ω ). For a quantum liquid such as helium below the λ -transition a dispersion relation for the single excitations is theoretically well established, and such dispersion relations were experimentally determined. Above the λ -transition in helium and for other normal liquids such as condensed argon, the definition of a dispersion relation is not clear because the meaning of single excitations is unclear in a medium so highly excited that the interactions between the excitations make their lifetimes and mean free paths small, perhaps smaller than or comparable to their oscillation period and wavelength, respectively. Nevertheless, attempts were made to define dispersion relations ω = w(q) for simple liquids  NEUTRON SCATTERING RESULTS 127 such as argon. Such attempts are logically motivated from the possible success of polycrystalline models for liquids. A more detailed discussion for the dispersion relations is given in connection to the presentation of the results on liquid helium and liquid argon. 1.1.4. Some General Rules From the discussion given it is clear that the scattering functions 5(κ , ω ) may not so far be calculated from first principles. Only by use of phenomenological models may some qualitative and simple quantitative deductions be made. It was, however, shown that the scattering functions have to obey exactly some very general and simple rules. These rules are as follows: (a) The detailed balance condition relating the energy loss and gain parts of the scattering functions and the cross sections: S(H, -ω ) = exp ( - - f ^ ) S(x, ω ) (1.19) and σ (£0-+£,Ω 0->Ω ) σ (Ε -*£0,Ω ->Ω 0) E exp[-ElkBT] E0 txp[-E0lkBT] ^' ^ (b) The sum rules and the moment relations. Defining <ω η > = f ω η £(κ , ω ) dœ (1.21) the most important relations are: <w° >incoh = 1 <>° >coh = 1 + y(*) UK2 <w>incoh = <>>coh = ^ f n γ £\ Δ <<^>incoh = -jÇ j- κ 2 kM BT 1 +K γ (κ ) where M is the mass of the scattering atom. In the derivation of the second moments the quantum effects and the recoil of the atoms have been neglected. The higher moments depend implicitly on the internal potential energy of pairs and are very involved. In the present status of 128 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD neutron spectroscopy they are of little interest because so far enough accurate data have not been obtained. The relations given above give little information about £(κ , ω ) itself but merely serve as a check of the consistency of different theoretical models and of experimental results. 1.2. EXPERIMENTAL TECHNIQUE The main experimental problem in the neutron scattering techniques is to obtain a beam of incident neutrons with a well-defined energy and to determine the energies of those neutrons scattered in a selected FIG. 1. Vertical section of the slow chopper time-of-flight spectrometer in Stockholm (from Larsson et al. [18]).  NEUTRON SCATTERING RESULTS 129 direction from the sample. These two determinations give the energy and momentum transfers from the conservation relations (Eq. 1.4).The problems of monochromation and of energy and angular analysis can be met in several ways and some of them will be briefly discussed here in order to elucidate the reliability of different types of measurements. In Table I some pertinent data for five different types of neutron spectrometers are collected. The reason for this particular choice of instruments is that with their aid some of the most extensive measure- ments were made on the four liquids to be discussed below. Most of the data in Table I have been taken from a compilation of Brugger and Harker [23] of time-of-flight neutron spectrometers. The operational properties of different instruments are now well understood, while some measurements performed at an early stage can be impaired by systematic errors. Table I needs some explanatory remarks. The monochromatizing and analyzing actions can be performed by the time-of-flight technique, by the diffraction technique, or by a combination of the two. The equipment shown in Fig. 1 utilizes the fact that the total cross section of a polycrystal possesses a cut-off energy above which the cross section is very high. Thus if a piece of polycrystalline beryllium is placed in a neutron beam with a Maxwellian energy distribution, only neutrons with energies less than about 5 meV will be transmitted. The energy spread of the neutrons hitting the sample is rather large, about 5 0 % . After the scattering in the sample, the energy of the neutrons in a selected direction is recorded by the slow chopper time-of-flight technique. In spite of the large energy spread of the incident neutrons a relatively good resolution for elastic scattering is achieved by making use of the sharp filter cutoff. However, the analysis has to be performed with the greatest care. An improvement of this equipment is the apparatus shown in Fig. 2, where the monochromating properties of a chopper is utilized. This spectrometer was used for extensive measurements on liquid argon. The chopper is placed before the sample, thus performing the combined action of reducing the width of spectrum of incident neutrons and triggering of a time-measuring device. An advantage of this equipment compared to the previous spectrometer is that, as the pulsing device is placed before the sample, simultaneous measurements in many scattering angles can be made. The uncertainty in energy of the impinging neutrons is about 1 5 % at 5 meV. In their measurements on methane, Harker and Brugger [24] used the phased rotor velocity selector shown in Fig. 3. The principle is the following: The first chopper (A, Fig. 3) produces a short burst of neutrons, while the second (Z), Fig. 3), placed a certain distance from A,  TABLE I o CHARACTERISTICS OF SOME NEUTRON SPECTROMETERS Type of Monochromating Analyzing Width of the spectrum Resolution of Resolution of analyzer spectrometer device device of incident neutrons analyzer for elastic for inelastic scattering and location with energy E0 meV scattering to energy Ef meV Slow chopper, > Stockholm, Time of 13% f o r £ 0 ~ 4 m e V CO Sweden [18] Be filter flight 50% a t 5 m e V 4% for E0 = 5 meV and Ef = 25 meV O 2 Semimonochromating chopper, Studsvik, Be filter plus Time of 4.5%forE0 = 5meV > Sweden [19] chopper flight 15% a t 5 m e V 2.4 % for EQ = 5 meV and Ef = 25 meV xr w o Phased chopper velocity selector, 0 Idaho Falls, Time of Variable. Typical 10% for £Ό = 55meV > USA [20] Phased rotors flight value: 2 % at 55 meV 4.8 % for E0 = 55 meV and Ef = 25 meV ö Triple axis crystal spectrometer, Single Chalk River, Single crystal crystal O: Canada [21] Al (111) Pb(lll) Variable 3.3 % for £ 0 = 5 meV ö Rotating crystal spectrometer, Chalk River, Single crystal Time of 1 % for E0 = 5 meV Canada [22] Al (111) flight -1 % for 5 < E0 < 50 meV 3.4 % for E0 = 5 meV and Ef = 25 meV  NEUTRON SCATTERING RESULTS 131 Detectors / H Lead ^ Iron ES3 Borate paraffin Ξ Water Sample Chopper Be-filter Be-filter fcmWWKfl FIG. 2. Horizontal section of the time-of-flight spectrometer in Studsvik, Sweden (from Holmryd et al. [19]). ROTATING COLLIMATORS FITTED SHIELDING SHIELDED BEAM. <* \ SCATTERING HOLE X M <Λ ROOM v PLUG " COUNTERS VACUUM PUMP BEAM MONITOR FIG. 3. Cutaway drawing of the MTR velocity selector (from Brugger and Evans [20]). 132 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD opens a preset time after the burst is produced at A. Thus by using a given distance and time lag between the choppers a burst of neutrons with a certain energy is obtained. The aim of the two rotating collimators is to reduce the background. The resolution of this instrument is rather poor as seen in Table I. Its main advantage lies in the possibility of  NEUTRON SCATTERING RESULTS 133 easily changing the wavelength of the incident neutrons, thus allowing measurements of 5(κ , ω ) over a wide region of (AC, o>)-space. Another method of obtaining a monochromatic neutron beam is by use of single crystals. An instrument of this type is the double crystal spectrometer used by Henshaw and Woods [21] for measurements of the dispersion relation in liquid helium (Fig. 4). From the white neutron beam only neutrons with a specific energy are reflected in a certain direction as given by the Bragg formula. The energy analysis of the neutrons scattered in the sample is made by the second spectrometer. Neutrons from higher order reflections are eliminated by inserting a beryllium block in the channel leaving only cold neutrons for first- order reflection. The rotating crystal spectrometer shown in Fig. 5 is a combination of the time-of-flight technique and the crystal technique. The mono- chromatization and the pulsing of the neutrons are performed by a rotating crystal. Each time a set of crystal planes satisfies the Bragg condition a burst of monoenergetic neutrons is produced. This technique FIG. 5. Schematic diagram of the Chalk River rotating crystal spectrometer (from Woods [25]). 134 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD was used for measurements on helium by Woods [25] and on methane by Dasannacharya and Venkataraman [26]. When comparing the various instruments of Table I it should be remembered that the higher the resolution, the lower the useful neutron flux. The cold neutron technique making use of the full beryllium-filtered neutron spectrum as the incident beam has a relatively limited usefulness but gives a high intensity. The double rotor system or a rotating crystal spectrometer tends to give one or two powers of ten lower intensity, which thus is the prize paid for the higher resolution. 2* Liquid H e l i u m 2.1. STATIC STRUCTURE FACTOR As discussed briefly in the preceding section, information about the atomic distribution in liquids can be obtained from a neutron diffraction pattern. Measurements performed on liquid helium cover a wide range of temperatures as well as pressures. The three neutron diffraction studies published were all made at Chalk River [27-29], where also most measurements on inelastic scattering were performed. Figure 6 shows the liquid structure factor 1 + y(K) as a function of I.61 1.4 1.2 i.o| ^ 0.8| + 0.6 0.4 0.2 * = ^ S I N (φ /2) FIG. 6. The liquid structure factor for liquid helium under its normal vapor pressure at ( · ) 2.29° K and ( o ) 1.06° K. The effect of the λ -transition causes a lowering and broadening of the main maximum (from Henshaw [28]).  NEUTRON SCATTERING RESULTS 135 the momentum transfer for liquid helium under its normal vapor pressure at 2.29° and 1.06° K (that is, on both sides of the A-transition which occurs at 2.19° K). The wavelength of the incident neutrons was 1.064 A. The momentum transfer κ is given by κ = 4π /λ sin <f>/2y where λ is the neutron wavelength and φ is the scattering angle. Unless the scatter- ing is elastic κ is not given as above, but this condition is nearly fulfilled if the ingoing neutron energy E0 is much larger than the possible energy transfers Δ Ε in the scattering system (E0 ± Δ Ε ^ E0). The circles do not correspond to measured intensities but are taken from smooth curves obtained from the experimental data after correction for experimental effects and for multiple scattering. The broken curves for small /c-values are extrapolations from the first experimental point to the known value of the zero angle scattering L0 given by L0 — nkBTifjT, where n is the particle number density and ψ τ is the isothermal compressibility. From Fig. 6 it is seen that between the two temperatures no drastic change occurs when passing the λ -point but rather small differences in detail occur in the region of small /c-values. The main peak is at κ = 2.03 A - 1 beyond which there is a small second maximum at κ — 4.3 A - 1 . The ratio of the main peak height of the curve at 2.29° K to that at 1.06° K is 1.047. This is close to 1.05 which has been deduced from x-ray measure- ments. It is interesting to note that there is an indication of a small bump at κ ~ 0.8 A - 1 in the 1.06° K measurement. The experimental error in this /c-range is, however, comparatively large so nothing definite can be concluded about its reality. In Table II the main results are collected from neutron diffraction work on liquid helium. It is worth noting that the data of Hurst and Henshaw [27] are not corrected for multiple scattering effects while the others are. This is probably the reason why these earlier results system- atically differ from the more recent ones. The distribution function 4nr[p(r) — p 0 ], where r is the distance from the atom chosen as origin, p(r) the atomic density at distance r, and p0 the mean atomic density in the liquid, is obtained from the liquid structure factor 1 + y(K) through 4nr[p(r) - Po] = - f κ γ (κ ) sin(n<) ά κ (2.1) π J o As it is only possible to cover a finite /c-region experimentally, errors might be introduced by taking the Fourier transform integral from zero to infinity. Although the uncertainties of some quantities derived from the transform may be quite large, some qualitative conclusions can be drawn: 136 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD TABLE II RESULTS FROM LIQUID HELIUM ATOMIC DISTRIBUTION FUNCTIONS BY NEUTRON SCATTERING Number of Position Liquid Position of nearest where temp. Pressure Density maximum neighbors from 4irrp(r) rises Ref. (° K) (atm) (gm/cm 8 ) in 4π τ ρ (τ ) symmetric peak from zero (A) in 4nrp(r) (A) (atoms) 1.06 NVP 0.145 3.80 9.8 2.35 [28] 1.65-2.25 NVP 0.146 3.70 8.6 2.25 [27] 2.29 NVP 0.146 3.80 9.7 2.40 [28] 4.24 NVP 0.125 3.72 8.1 2.25 [27] 5.04 NVP 0.095 3.94 7.0 2.20 [27] 2.05 15.0 0.166 3.60 9.2 2.35 [29] 4.2 51.3 0.184 3.55 10.2 2.26 [29] (a) The nearest distance of approach is nearly independent of density, approximately 2.30 A. (b) The mean radius of the shell of nearest neighbors is decreasing with increasing density from about 3.9 A at p = 0.095 gm/cm 3 to about 3.5 A a t p = 0.184 gm/cm 3 . (c) The number of nearest neighbors is increasing with increasing density. (d) From Fig. 7 it is obvious that a change in density induced by pressure variation has a larger effect on the radial distribution function than the corresponding change caused by temperature variation. Not only is the number of nearest neighbors increased with increasing pressure but also a more marked structure further out in the liquid seems to be introduced. 2.2. T H E DISPERSION RELATION AND ITS RELATED QUANTITIES In 1957, Cohen and Feynman [30] suggested that it should be possible to determine the energy-momentum relation for the elementary excitations in liquid He II by inelastic scattering of slow neutrons. The experiment should in principle be of the same type as those which at that time already had been performed in order to measure dispersion relations in crystals. Before 1957 some unsuccessful experiments [31, 32]  NEUTRON SCATTERING RESULTS 137 0 1 2 3 4 5 6 7 β 9 10 RADIAL SPACING ( A N G S T R O M S ) FIG. 7. The radial distribution functions 4nrp(r) for five different densities. The straight lines are 4irrp0 (from Henshaw [29]). were performed to establish an effect of the λ -transition in the total cross section. As these now are of less interest they will not be discussed here. In 1960, Palevsky [33] made a summary of these early results and also of the inelastic scattering experiments which had been performed at that time. 138 Κ . Ε . LARSSON, U. DAHLBORG, AND K. SKÖ LD A typical scattering pattern taken from Henshaw [34] is given in Fig. 8 where the spectrum of 4.14 A neutrons scattered from liquid helium at different temperatures using a rotating crystal spectrometer 44 40 36 32 28 24 6 20 * 16| 12 8 4 0| to&S+H****** J_ _L 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 5.3 NEUTRON WAVELENGTH(ANGSTROMS) FIG. 8. The spectrum of neutrons scattered from liquid helium at several temper- atures using a rotating crystal spectrometer. Angle of scattering, 80° . The vanadium curve gives the wavelength distribution of incident neutrons uncorrected for the resolution of the instrument. The liquid helium curves have been corrected for the wavelength sensitivity of the instrument and normalized on the basis of liquid density. The curves at 2.08° and 4.2Γ Κ have been corrected for instrument resolution (from Henshaw [34]). is plotted. The operation of a rotating crystal spectrometer has been discussed above. The vanadium curve gives the wavelength distribution of the incident neutrons. The change in energy E and momentum p of the neutron in the scattering process which equals the energy and momentum of the created excitation is calculated through the con- servation formulas F-Jt(J LA (2.2a) 2m \ V A,1 / 87r2COS<ft-|1/a (2.2b)  NEUTRON SCATTERING RESULTS 139 where λ ^ and Xf are the incident and scattered neutron wavelengths, φ is the scattering angle, m is the neutron mass, and h is Planck's constant. The first result was published by Palevsky et al. in 1957 [35] and demonstrated the existence of excitations with long mean free paths in He II. Since then the dispersion relation in He II has been determined with high accuracy at different temperatures and pressures [21, 25, 34, 36—41]. In Fig. 9 some measured points obtained from the liquid at its 20r- 1 1 1 71 15 Δ Δ Δ ,4 Δ Δ »Q · · D Δ Jf Δ P Δ D V'* m 10 L D· z • 1 ·· Υ "Δ ^Δ 1 1 1 0 1 2 3 4 -1 MOMENTUM ( A ) FIG. 9. The dispersion relation at different temperatures and pressures. The data, which are taken from different publications, do not pretend to be complete, ( θ ) Palevsky et al. [37] NVP, ( □ ) Yarnell et al [39] NVP, ( · ) Henshaw and Woods [21] NVP, ( + ) Woods [25] NVP, and ( Δ ) Henshaw and Woods [40] 25.3 atm. normal vapor pressure and at 25.3 atm are plotted as a function of the momentum pjfi. The energy is given in degrees Kelvin. It is clear that the results are fitted very well by a curve of the shape predicted first by Landau [42, 43] and later by Feynman [44]. On the whole the agreement between different measurements is extremely good, which is very satisfactory because a variety of different experimental methods were used (cold neutron time-of-flight spectrometer, cold neutron crystal spectrometer, rotating crystal spectrometer, and triple axes crystal spectrometer). All results at normal volume and pressure (NVP) are not performed at one temperature but fall in the temperature interval 1.1 ° K < T < 1.6° K. We have preferred not to try to adjust the data to one temperature as the complete temperature dependence of the excita- 140 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD tion curve is unknown. From fitted curves through the measured points at the two pressures the following facts can be extracted: (a) The slope of the curves for small momenta, the phonon region, corresponds very closely to the measured velocity of ordinary sound in He II. (b) The maximum of both curves fall at E ~ 13.8° K and Pin~ l.io A- 1 . (c) The minimum of the curves, the "roton" part, may, according to Landau, be fitted by a theoretically predicted parabola {Ρ Ρ ο )2 Ε = Δ + ~μ (2.3) within a very narrow interval. In Table III values of the parameters TABLE III PARAMETERS OF THE ROTON MINIMUM Experimentalist Temp. Pressure Δ Po/à (° K) (atm) (° K) (A"1) (%e) Palevsky et al., 1958 [37] 1.44 NVP 8.1 ± 0.4 1.90 ± 0.03 0.16 ± 0.02 Larsson and Otnes, 1959 [38] 2.03 NVP 6.7 ± 0.3 1.94 ± 0.02 0.13 ± 0.02 Yarnell et al, 1959 [39] 1.1 NVP 8.65 ± 0.04 1.92 ± 0.01 0.16 ± 0.01 1.6 NVP 8.43 1.92 ± 0.01 0.16 ± 0.01 1.8 NVP 8.15 1.92 ± 0.01 0.16 ± 0.01 Henshaw and Woods, 1960 [40] 1.1 25.3 7.0 2.05 — Henshaw and Woods, 1961 [21] 1.12 NVP 8.65 ± 0.04 1.91 ± 0.01 0.16 obtained by different experimenters are collected. Unfortunately, all the experiments have been performed at different temperatures except for those of Yarnell et al. [39] and Henshaw and Woods [21] which, however, show remarkable agreement. The data of Palevsky et al. [37] seem to fall somewhat below the others. This is probably a pure experimental effect because enough consideration has not been given to the influence of the (although small) transmission of a 20-cm-thick Be filter between 3.58 and 3.95 A. (d) The slope of the NVP curve after the minimum is equal to or slightly less than the slope at very small momenta. For the liquid at  NEUTRON SCATTERING RESULTS 141 25.3 atm, however, the slope is well below the one corresponding to the velocity of ordinary sound at this pressure. (e) The NVP curve has a plateau at 17.9° K for 2.7 < p\fi < 3.5 A" 1 and shows a tendency to rise for larger values of pjh. The energy of the plateau is approximately twice the energy at the roton minimum. However, Woods claims a definite suspicion that the energy of the plateau (17.9 ± 0.6° K) is significantly greater than 2Δ (17.3° K). The 25.3-atm curve shows a tendency to flatten off in the same way, but measurements for very large momenta are missing. The limiting value must in this case be larger than 2Δ (14.0° K) as the last measured point is at 15.0° K. A measurement of significance for the understanding of excitations in normal liquids was made by Woods [45] who studied the temperature dependence of a long wavelength phonon by use of a rotating crystal spectrometer. As mentioned above the relation E = cp> where c is the velocity of ordinary sound, was found to hold at 1.1 ° K for small momenta. Below the λ -point in the superfluid state the existence of a phonon is accepted, but above the λ -point in He-I, which is a normal liquid, the existence of phonons is still partly an open question. The results given in Fig. 10 clearly show that for temperatures up to 2.57° K long wave- length excitations with momentum of 0.38 A - 1 exist in the liquid and that the velocity of the excitation is the same as the velocity of ordinary sound, represented by the broken curve, or slightly higher. No change _ 300 It X. 200 o o ö l 100 > 1.0 1.5 2.0 2.5 TEMPERATURE ( "K ) FIG. 10. Phonon velocity calculated from the observed neutron distributions com- pared with the measured velocity of sound. The vertical bars on the points correspond to the full width at half-maximum of the peaks. The instrument resolution is 2° K and is the width observed at temperatures below 1.9° K. The broken curve represents the measured velocity of ordinary sound. The point at 1.1 ° K is taken from Henshaw and Woods [21], while the other data ( o ) are taken from Woods [45]. 142 Κ . Ε . LARSSON, U. DAHLBORG, AND K. SKÖ LD is seen at the λ -point. Also, the widths of the neutron distributions are fairly small indicating a relatively long mean free path. Even above 2.57° K peaks were found in the scattered neutron distributions, but the uncertainty in assigning a definite energy was too large due partly to the increasing importance of multiphonon interactions. Measurements of the excitation energy in the momentum range corresponding to the roton minimum on the other hand show a very strong temperature dependence. A systematic study of this effect has been made by Larsson and Otnes [38], Yarnell et al. [39], and Henshaw and Woods [21]. The distributions of scattered neutrons corrected for instrumental effects were found to be symmetrical in energy around the mean energy change for liquid temperatures in the range of 1.78° to 4.21 ° K. (Note that in Fig. 8 the scattered intensity is plotted in a wave- length scale.) When the temperature is approaching the λ -point, the "gap" energy is decreasing to about 5° K as can be seen in Fig. 11. At Vf - 6 é > o cr LU 4 h\ t 2 3 TEMPERATURE ( ° K ) FIG. 11. The temperature variation of the energy of the excitation with momentum corresponding closely to the roton minimum. The data are taken from different publications: (O) Larsson and Otnes [38], (*) Yarnell et al. [39], and ( · ) Henshaw and Woods [21]. the λ -point a marked change in the rate of variation occurs, and above the transition (that is, for He I) only a slow decrease is seen. In He I the gap energy should not be interpreted as a roton energy but rather as a mean energy change of the neutrons in the scattering process. This mean energy change may be compared to the mean energy change expected if the neutrons were scattered from free helium atoms. It turns out that the mass of such an atom should be 4.2 helium masses at the λ -temperature and then increasing to about 4.6 helium masses at 4.2° K. This is equivalent to saying that at 4.2° K an apparent collective of  NEUTRON SCATTERING RESULTS 143 4.6 helium atoms is the neutron scattering unit. When comparing the data in Fig. 11 it is again seen that the results of Larsson and Otnes seem to fall below those of Henshaw and Woods, but as discussed earlier a possible systematic error in their interpretation may be the reason for this discrepancy. The two sets do, however, show the same tendency. The measurements of the gap energy are all made at constant scattering angle, and any variation of ρ 0/ϋ with temperature has been neglected. This is not strictly correct as the energy and the momentum of a one- quantum peak is coupled via the conservation laws, but as the minimum of the dispersion curve is rather flat the introduced errors are not serious. As a matter of fact, Yarnell et al. found no variation of p^jfi in the tem- perature range 1.1° to 1.8° K, while Larsson and Otnes found pjfi = 1.94 ± 0 . 0 2 A- 1 at 2.03° K compared to pjfi = 1.90 ± 0 . 0 3 A"1 at 1.44° K. Also of interest is the scattering cross section for the production of a single excitation in the liquid. Figure 12 shows the relative differential scat- tering cross section obtained by integration of the intensity under a peak at two different wavelengths of incident neutrons. The data are taken from Henshaw and Woods [21] and Woods [25]. The cross section has a maxi- mum at a momentum of about 2.0 A~l. This number may be compared to the position of the main maximum of the diffraction pattern 2.03 A - 1 . It is clearly demonstrated from the figure that measurements at large momenta are very difficult to perform as the intensity (for instance, at pjfi = 3.36 A"1) is only 1 % of the intensity at pjfi = 2.0 A"1. This was the reason why Woods could not continue his dispersion curve measure- ment and establish the rise for pjfi > 3.5 A - 1 with certainty. At small momenta there is a tendency for a flat maximum. In 1961, Woods [46] made a measurement to see if macroscopic rota- tion of liquid helium at 1.54° K caused a change of dispersion curve around the roton minimum. No effect was observed. 2.3. LINE WIDTH OF EXCITATION PEAKS The energy resolution in the experiments on liquid helium has in all cases been about 2° K. This means that a natural line width of an excitation of about 1° K and larger has given an observable broadening of the scattered neutron distribution. If the uncertainty relation Δ Ε - At = 2fi is used with Δ Ε = kBT it is found that the observable broadenings correspond to lifetimes of an excitation in a range smaller than about 2 X 10 - 1 1 sec. Unfortunately, line width studies hitherto performed are incomplete and are mostly concentrated to investigations near the roton minimum.  MOMENTUM CHANGE (A" 1 ) 2.0 2.5 3.0 40 50 60 70 80 90 100 SCATTERING ANGLE, φ (DEGREES) Ί — I — I — I — I — I — I — I — I — I — I — I — Γ 1.4 MOMENTUM CHANGE (A*1) FIG. 12. The relative partial differential cross section for the production of a single excitation in the liquid at two different wavelengths of incident neutrons (from Henshaw and Woods [21] and Woods [25]). (a) Helium temperature 1.6° K, λ 0 = 2.77 A and (b) liquid helium temperature 1.1 ° K, λ 0 = 4.04 A.  NEUTRON SCATTERING RESULTS 145 1.0 1.5 20 2.5 MOMENTUM ( p / f i IN A -1 ) FIG. 13. Energy spread of the excitation spectrum of liquid helium at various momenta and temperatures as inferred from the observed width of the cutoff in the scattered neutron distribution (from Yarnell et al. [39]). The measurements of Yarnell et al. [39] (Fig. 13) show that the mean lifetime of the excitations are also dependent on the momentum transfer ϋ κ . Although the errors are relatively large, it must be concluded that the energy spread of the distributions is less at the roton minimum and is larger both for smaller and larger momentum values. This is valid in the temperature range of 1.1° to 1.8° K. Figure 14 shows the line width of an excitation with momentum t^15 x I— Q LU t »It ,v 2 3 TEMPERATURE (*K) FIG. 14. Line width of an excitation with momentum corresponding to the roton minimum as a function of temperature. The data are taken from different publications: (O) Larsson and Otnes [38]; (*) Yarnell et al. [39], and ( · ) Henshaw and Woods [21]. 146 Κ . Ε . LARSSON, U. DAHLBORG, AND K. SKÖ LD corresponding to the roton minimum as a function of temperature. The data are taken from Larsson and Otnes [38], Yarnell et al. [39], and Henshaw and Woods [21]. The same objections as before can be raised to the Larsson and Otnes results. Also, Henshaw and Woods have not taken into account the variation of the resolution as a function of the energy change of the scattered neutrons. The main features of the two sets of data are, however, the same. The width is continuously increasing up to the λ -temperature where a drastic change of the slope occurs. In the He I region there is only a slight increase in the slope up to 4.2° K. Above the λ -temperature the measured widths are in good agreement with the ones calculated for a gas of free atoms. The masses used in this calculation are the ones derived from the measurements of the mean energy change of the neutrons (see discussion in connection with Fig. 11). When comparing Figs. 11 and 14 it is seen that close to the λ -temperature the energy of the roton excitation is about 6° K while the energy width is of the order of 10° K. This observation, taken together with the result that the mean energy transfer as well as the width of the scattered neutron distribution correspond to an apparent recoiling mass of more than four helium masses, indicates that the measured distribution is a multiphonon distribution in which the one quantum component is hidden, its intensity being relatively low and its width very large. This means that the interaction between excitations is very strong in this momentum region and that the concept of elementary excitations is questionable. On the other hand, in other and denser simple liquids, the concept of quasi-phonons might still be a valuable hypothesis to use in attempts to understand experimental facts. Also, it is to be noted that the measurements of Woods show that the width of an excitation peak with momentum 0.38 A - 1 is small even above the λ -temperature. 3+ Liquid Argon 3.1. ATOMIC DISTRIBUTION Measurements of the atomic distribution in liquid argon have been reported by Henshaw et al. [47], Henshaw [48], and also by Dasannacharya and Rao [49] (henceforth referred to as DR). Henshaw used the conventional diffraction technique in which the total scattered intensity is recorded as a function of scattering angle. The intensity pattern obtained by Henshaw is shown versus κ in Fig. 15 where the error flags include counting statistics only. The data are corrected for background, resolution, and double scattering. A correction is also  NEUTRON SCATTERING RESULTS 147 FIG. 15. The intensity pattern from liquid argon obtained by Henshaw [48] (dots with error flags) shown together with the curve obtained by Dasannacharya and Rao [49] (full curve): T = 84° K, λ 0 = 1.04 A [48]; T = 85.5° K, λ 0 = 4.06 A [49]. applied for the change in the number of scattering atoms with angle. The wavelength of the incident neutrons was 1.04 A, and it is assumed that the change in wavelength on scattering is small so that the spread of /c-values within the distribution scattered at a certain angle may be neglected (see above discussion). The static approximation is reasonable for wavelengths not much larger than 1 A but becomes rapidly worse when the wavelength increases and cannot be used at all for the wave- length of 4.05 A employed by DR. In this case a mapping of the cross section over the (ω -κ ) plane must be made and the integral over ω must then be evaluated at each separate value of κ . The procedure adopted by DR will be considered in detail in connection with the discussion of the dynamical studies below. The result for the intensity pattern obtained by them using 4.06 A neutrons is shown by the solid line in Fig. 15 where it is seen that the two curves are in fair agreement but that minor differences, especially for κ < 1 A - 1 , are also observed. The region of K covered in this way by DR is too narrow for a derivation of the atomic distribution function, and the curve that was used for this analysis was obtained by combining time-of-flight data taken with 4.06 A incident neutrons and crystal spectrometer data taken with shorter wavelength incident neutrons. The complete function obtained by DR is shown by the t = 0 curve in Fig. 20 and the function tabulated in Table IV. 148 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD TABLE IV INTERMEDIATE SCATTERING FUNCTIONS /(*, t) FOR LIQUID ARGON AT 84.5° K DERIVED FROM DATA SHOWN IN FIG. 20 t in units of 10"13 sec K (A-> ) o 1 2 3 4 5 7 10 20 25 30 40 0.1 0.2 0.345 0.345 0.345 0.345 0.34 0.34 0.335 0.32 0.285 0.265 0.24 0.19 0.3 0.355 0.355 0.35 0.35 0.35 0.345 0.335 0.32 0.275 0.25 0.225 0.175 0.4 0.360 0.360 0.355 0.36 0.355 0.35 0.34 0.325 0.295 0.28 0.27 0.24 0.5 0.380 0.380 0.375 0.37 0.365 0.36 0.35 0.335 0.29 0.275 0.255 0.235 0.6 0.385 0.385 0.38 0.38 0.37 0.36 0.345 0.31 0.25 0.22 0.19 0.135 0.7 0.39 0.39 0.385 0.38 0.37 0.365 0.35 0.325 0.26 0.24 0.22 0.19 0.8 0.39 0.395 0.39 0.38 0.37 0.36 0.34 0.31 0.25 0.22 0.20 0.17 0.9 0.390 0.390 0.39 0.375 0.37 0.355 0.335 0.31 0.22 0.21 0.19 0.15 1.0 0.385 0.38 0.38 0.37 0.355 0.34 0.32 0.299 0.19 0.19 0.17 0.14 1.1 0.385 0.39 0.38 0.365 0.350 0.335 0.30 0.275 0.20 0.16 0.125 0.085 1.2 0.40 0.395 0.385 0.37 0.355 0.355 0.30 0.27 0.18 0.17 0.145 0.11 1.3 0.41 0.405 0.39 0.37 0.35 0.33 0.28 0.245 0.19 0.15 0.125 0.095 1.4 0.44 0.435 0.420 0.395 0.365 0.335 0.28 0.235 0.145 0.12 0.09 0.05 1.5 0.485 0.480 0.46 0.425 0.39 0.34 0.29 0.23 0.145 0.115 0.09 0.05 1.6 0.59 0.580 0.56 0.51 0.465 0.415 0.325 0.25 0.13 0.09 0.065 0.025 1.7 0.905 0.895 0.855 0.80 0.73 0.66 0.533 0.405 0.20 0.15 0.105 0.06 1.8 1.21 1.19 1.15 1.01 1.00 0.93 0.78 0.625 0.32 0.23 0.17 0.10 1.20° 1.18« 1.11« 1.00 « 0.94« 0.865« 0.775« 0.675« 1.306 1.9 1.765 1.75 1.59 1.605 1.5 1.4 1.215 1.00 0.58 0.44 0.34 0.22 2.0 2.16 2.14 2.075 1.975 1.86 1.75 1.525 1.30 0.785 0.615 0.485 0.31 2.055« 2.03« 1.95« 1.845« 1.735« 1.625« 1.47« 1.30« 2.1 1.84 1.815 1.74 1.625 1.5 1.37 1.15 0.925 0.49 0.365 0.275 0.135 2.2 1.35 1.315 1.235 1.11 0.985 0.855 0.66 0.49 0.215 0.14 0.09 0.03 1.235« 1.19« 1.087« 0.925° ' ■ 0.785°1 0.67« 0.475° 1 0.205« 1.18» 2.4 0.825« 0.775« 0.665« 0.52« 0.41« 0.34« 0.25« 0.180« 0.80° 2.6 0.70« 0.65« 0.525« 0.395« 0.29« 0.23« 0.18« 0.115« 0.806 2.8 0.725° ■ 0.64« 0.465« 0.34« 0.28« 0.225« 0.14« 0.08« 0.50° 3.0 0.81 0.75 0.57 0.45 0.35 0.30 0.865° 1 0.775° 1 0.58« 0.42« 0.30« 0.245° 1 0.15« 0.10« 0.92° 3.2 0.97« 0.885° ' 0.685° 1 0.49« 0.36« 0.29« 0.20« 0.115« 0.90* 3.4 1.135° ' 1.01« 0.775° 1 0.56« 0.44« 0.35« 0.22« 0.115« 0.85°  NEUTRON SCATTERING RESULTS 149 TABLE IV {continued) 3.6 1.215a 0.135° 0.90" 3.8 1.165° 1.06° 0.825° 0.58° 0.41° 0.305° 0.18° 1.325" 4.0 0.99 0.875 0.635 0.44 0.34 0.30 1.105a 0.98a 0.71° 0.455° 0.305° 0.22° 0.14° 1.07" 4.2 0.925 0.775 0.49 0.31 0.20 0.085 0.835" 4.6 0.96 0.695 0.395 0.23 0.15 0.105 0.80" 4.8 0.875 0.69 0.385 0.225 0.175 0.140 0.83" 5.0 0.975 0.75 0.395 0.205 0.13 0.11 0.795" 5.4 1.135 0.88 0.49 0.32 0.24 0.20 0.950" 5.6 1.035 0.78 0.41 0.195 0.085 0.075 0.925" 5.8 1.07 0.76 0.395 0.155 0.18 0.16 0.80" 6.0 0.97 0.705 0.275 0.05 0.0 --0.03 0.90" ° Correspond to open circles in Fig. 20. 6 Correspond to crosses in Fig. 20. The intensity pattern obtained by Henshaw et al. and Henshaw was used to derive the atomic density function p(r) from the relation: 2r r° ° 4rrr2[p(r) — p0] = — KI(K) sin(r*) ά κ (3.1) 7Γ J 0 where i(#c) = [Ι (κ ) — /(oo)]/[/(oo) — J ] , Ι (κ ) is the coherent intensity at the value κ of the wave vector transfer, Δ is the ratio of the incoherent to the coherent cross section, and p0 is the mean atomic density. The value of Δ was obtained by calculating p(r) for values of r smaller than the distance of closest approach (3.4 A) and adjusting Δ such that p(r) closely approximated the value zero for those values of r. The value for Δ that was obtained in this way was 0.325, which is in good agreement with the result obtained from a consideration of the limiting scattered intensity for small and large values of κ . Using Δ = 0.325, the radial distribution function was calculated from Eq. (3.1) for 0 ^ r ^ 20 A. The result is shown in Fig. 16 where the function 4nr2p0 with 150 Κ . Ε . LARSSON, U. DAHLBORG, AND K. SKOLD β 10 12 RADIUS (ANGSTROMS) FIG. 16. The transform 4τ τ τ 2 [p(r) - Po\ for liquid argon (T = 84° K). The smooth curve is — 4nr2p0, where p0 is the average density equal to 2.13 X 10~2 atoms A~ s . (from Henshaw [48]). pQ = 2.13 X 10~2 atoms/A 3 is also shown. The oscillations in the radial distribution function for r < 3 A arise because the integration of Eq. (3.1) is terminated at κ = 7 A - 1 before the intensity pattern has attained its limiting value. The atomic density distribution function 47rr2p(r) is shown by the solid line in Fig. 17, where the average density function 4π τ 2ρ 0 is also shown (dashed line). The function 4nr2p(r), obtained by DR by per- forming the Fourier transforms over ω and κ of the experimental scattering law data, is shown by the open circles in Fig. 17; p(r)> which is identical to Gd(r, 0), is tabulated in Table V together with the distinct correlation function at finite times. We refer to the discussion below for details of the analysis. The method adopted by DR is experimentally rather difficult as two transforms must be performed and termination errors will enter from both, but on the other hand the method is free from the systematic error inherent in the static approximation which is used by Henshaw. Also, the transform over ω is in this case taken at zero time and is thus simply the area of the scattering law. It seems that the two sets of data shown in Fig. 17 should be assigned about the same degree of reliability and the differences, as far as they are not due to the small difference of 0.5° K in temperature, are not understood. The number of nearest neighbors is calculated by Henshaw from the area under the first peak in 4π τ 2ρ (τ ). Depending on the shape assumed for the peak when extrapolating it to the right (compare Fig. 17), the value obtained for the number of nearest neighbors is 8.0 to 8.5. This  NEUTRON SCATTERING RESULTS 151 ou Ί Ί Τ — ι — ι 1 ι 1 1 r 28 \\ 26 in V 24 F I 22 - A /1 t cl 20 A ^ // #/ / A < •fr ιβ O i l —\ 2 O I -o, l/ l Ci Γ Ρ O £ 16 — A C \ 1 X'4 — 1 I / / A J/ // // ~~\ o ^r o JT 10 // —\ o/\ / o/ 8 /l / / / \ / / 6 1 \ /J A 4 A L A 2 J>^ *Ό /7 ο ^ο O 30Ö *T^^L ο -2 1 J J_ l 1 1 L_J L 10 RADIUS (A) FIG. 17. ( ) The radial distribution function 4nr2p(r) measured by Henshaw [48] (T = 84° K), (O) the result obtained by DR [49] (T = 84.5° K), and ( ) the average atomic density 4nr2p0 with p0 = 2.13 X 10~2 atoms A - 3 . should be compared to the value 10.2 to 10.9 obtained at T = 84.25° K by x-rays [50]. The data by DR seem to indicate a higher value for this number than the one obtained by Henshaw, but those data were not used to evaluate the number of atoms in the first shell. The first peak in Henshaw's curve is at 3.86 A and the distance of closest approach, which is given by the point where the curve goes to zero, is 3.05 A. The corresponding numbers from the curve by DR are 3.80 and 3.16 A, respectively.  ^- ^- © © © © © ^wbbob\^is)booo\^Nbbob\^wbbobN^k)bboo\^N)bboo\^|gboo^^ s) b bo o\ 4^ to b x > III III II p p p o o p p p p o o o o o o o ο ο ο ρ ο ο ο ο ο ρ ρ ο ρ ρ ο ο ο ο ο ο ρ ο ο ο ο ö ö ö o ö b o b ö o o ö ö b O Q O O o o ö b Ö O Ö Ö Ö Ö Ö Ö Ö O Ö O Ö O Ö Ö Ö Ö Ö κ ω ^ Ν ) σ \ ο ο υ ι θ \ ο ο υ ΐ ο ο Ν ) \ ο ο ι ο · 1 Λ Α Ι Λ * ^ .« . . Λ H 2 5 c > I I I H o o o o o o o o o o p o o o o o p p p p p p o o δ CO o o o o o o o o o s: CO © © © © © © © © © © © © © © © © © © © © © © © © O © © © © © © © © © O O O O V O O S J W V J O O N J W V J N ) 0 \ 0 0 W I ^ O O W » 0 N W I U I > 1 0 \ 0 0 O W \ > 4 0 N O O S ) O \ W O ^ - V 1 M K ) O « J W > ? 2 o ρ ρ ρ ρ ρ ρ ρ ρ ρ ο ο ο ο ο ο ο <ζ >ο ο ρ ο <ο ο ρ ρ ρ ρ ρ ρ ρ ρ α >ρ H δ S3 > © © O © © © © © © © © © © © © © © © © © _ . _ _ _ . _ . © © © © © © 228888 M O O 00 ON O «G Cd r VO -4 · >! ON oo M μ w N) g V 0 K ) 0 0 W 0 0 M M O K ) 0 0 V Û 0 N V C O W S ) ^ 0 0 O M Cd o © I p p p p p o o p p o o o o o o o o o o p - © © © © © © © © I © I © I © © © © > © © © © © © © © © © © © © © © O © © © © © © © Ift < 00 00 VO © ^ k K>l IΟ *>ι «^ï »— - J .£> N© - ^ H - © ^ OO «^1 f » ( .» i L Λ Λ 2222222SSgSgS88888|88 tv k. ■ » /"S i L I .■ » f · . Λ Λ „ , U W M 0 0 ^ N O ^ W U 0 0 J i U i 0 -^ © \ ^ 0 0 0 0 ^ 0 0 .Ä .. ^ .. . - > CO I I » o O: © p p © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © r o © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © © 2 Ö · — · — >— Ν > Ν > Ν > Ν > Κ > Κ > Κ > Ν > Ν > ^ - Η - μ - > — i— Η - μ - ι — N > O J ^ 4 ^ - ^ S ) H - © © © © © © ^ 0 0 y 0 O S > - ^ Û > N Î v J U i W O > J y i N ) ^ N ) ^ 0 > N 0 y i ^ y i \ 0 W > j M | o N ) W O O i i i . V O ^ ^ V O > J 0 0 0 0 0 0 O U > N ) 0 0 V O N ) 0 \ - ^ O N ^ - ^ - ^ ^ ( ^ U ) y o O - ^ - ^ ^ > J 0 N - ^ O J \ C w  NEUTRON SCATTERING RESULTS 153 TABLE V {continued) 8.6 0.0183 0.0185 0.0177 0.0181 0.0179 0.0188 8.8 0.0187 0.0187 0.0185 0.0185 0.0187 0.0190 9.0 0.0196 0.0196 0.0194 0.0194 0.0198 0.0196 9.2 0.0208 0.0208 0.0206 0.0206 0.0206 0.0204 9.4 0.0217 0.0219 0.0217 0.0217 0.0217 0.0211 9.6 0.0221 0.0227 0.0227 0.0227 0.0225 0.0219 9.8 0.0225 0.0232 0.0234 0.0230 0.0230 0.0227 10.0 0.0228 0.0234 0.0242 0.0232 0.0232 0.0230 a According to Dasannacharya and Rao [49]. 3.2. ATOMIC M O T I O N Information about the dynamical properties may be obtained from neutron scattering results either by Fourier transformation of the experimentally observed scattering law, in which case the van Hove correlation functions are obtained, or by comparing the experimental results with model calculations of the scattering law. The first method, which is experimentally rather difficult as data must be collected over a large region of the (ω -κ ) plane, was applied to liquid argon at 84.5° K by Dasannacharya and Rao. The other method, which is difficult because there is no adequate theory of the liquid state on which to base the models, was applied to liquid argon at 88° K by Kroo et al. [51] and later at 94.4° K by Skold and Larsson [52]. Other, not quite as comprehensive, measurements at 85° K are reported by Chen et al. [53]. 3.2.1. Correlation Functions In the experiment by DR a large region of the (o>-/c) plane was covered by combining results from time-of-flight measurements in which 4.06 A incident neutrons were used with results obtained from crystal spectrometer measurements. The crystal spectrometer measurements were made using the constant κ method with the wavelength of the scattered neutron kept constant and equal to 1.808 A for distributions in the range 1.8 A - 1 < κ < 4.0 A - 1 and equal to 1.425 A for distributions in the range 4.0 A - 1 ^ κ ^ 6.0 A - 1 . Data were also taken at κ = 1.8, 2.0 and 2.2 A" 1 with λ = 1.808 and at κ = 3.0 A" 1 with λ = 1.425 A. The consistency of these overlapping data is discussed in connection with the intermediate scattering function below. The time-of-flight results, which are originally energy distributions at constant angles, were converted to constant /c-distributions for the scattering law by multiplying with the k0/k factor and then forming 154 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD a grid of κ versus ϋ ω . Examples of the curves obtained in this way are shown for certain values of κ in Fig. 18 where in fact not fiœ but λ , the wavelength of the scattered neutron, is used as variable. The dashed curve at K = 2.0 A - 1 shows the experimentally observed resolution function, while the dashed curve at κ = 0 A - 1 shows the resolution function K--2.0 κ -\Α κ -\.0 K =0.6 K-0.2 Κ --0Α S(/c,X) 3.0 e3.5 4.0 4.5 \'(Angstroms) FIG. 18. The scattering surface for liquid argon at 84.5° K, 550 mm, shown as function of wave vector transfer κ and wavelength of scattered neutrons (from Dasannacharya and Rao [49]). which is obtained by extrapolation of the full width at half-maximum of the resolution function observed at higher value of κ and assuming the resolution function to be Gaussian. The curve at κ = 0.2 A - 1 is obtained by interpolation between the curve at κ = 0 and curves at higher values of κ and represents the energy distribution that should be observed at this value of κ . Typical constant /c-distributions taken with the triple axis crystal spectrometer are shown in Fig. 19 where the resolution functions are also shown. The curves in the left and right halves of Fig. 19 were taken with λ = 1.808 A and λ = 1.425 A, respectively. Energy distributions were in both cases measured at intervals of κ equal to 0.2 A - 1 .  NEUTRON SCATTERING RESULTS 155 2 0 - 2 Δ Ε IN mev FIG. 19. Typical energy distributions for constant κ observed from liquid argon at 84.5° K with triple axis spectrometer Ingoing neutron energy 25.01 meV: (a) κ = 2.0 A - 1 , (b) K = 3.0 A - 1 , (c) K = 4.0 A - 1 ; Ingoing neutron energy 40.2 4 meV: (d) κ = 4.0 A - 1 , (e) K — 5.0 A - 1 , and (f ) κ = 6.0 A - 1 . Resolution function for the two outgoing energies are shown in (c) and (f ) (from Dasannacharya and Rao [49]). The intermediate scattering function derived by Eqs. (1.5a) and (1.15) from the data described above is shown in Fig. 20 for various times in the range 0 < t < 50 X 10~13 sec. The function is also given in Table IV. The resolution is removed by dividing the cosine transform of the observed distributions with the cosine transform of the resolution function. Absolute normalization of /(/c, t) for all t was obtained by normalizing Ι (κ , 0) to the diffraction pattern by Henshaw [48]. It is seen that overlapping data are in agreement within experimental errors, and this gives confidence in the experimental procedure as well as in the method of analysis. Correlation functions derived from the intermediate scattering functions of Fig. 20 are shown in Figs. 21 and 22. For times larger than 156 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD MA) K-(A) FIG. 20. The intermediate scattering function for different times (in units of 10~13 sec). The Gaussians at t = 1 and t = 2 are calculated using a perfect gas model (from Dasannacharya and Rao [49]). 10 X 10~13 sec, the curves in Fig. 20 were arbitrarily extrapolated from K = 2.2 A"1 to K = 4.0 A- 1 . For small times the /(/c, t) curves are still oscillating for the maximum observed values of /c. To decrease the termination errors transforms were taken not of Ι (κ > t) but of /(/c, t) — Ι σ (κ , t) where Ig(f<y t) is the gas distribution. This method was used for times smaller than 5 X 10~13 sec only. When the total scattering law is transformed one obtains a weighted combination of Gs(r, t) and Gd(r, t)y but for small times the two functions do not overlap and a separation is then possible. Using the value 0.66 for the coherent to the total scattering cross section the separation was made for times smaller than 20 X 10~13 sec. The separated Gd(r, t) and Gs(r, t) are shown in Figs. 21a and Fig. 22, respectively, and these functions are also given in Tables V and VI. The weighted combination that is directly obtained in the transform is shown for t > 20 X 10~13 sec in Fig. 21b. The oscillations that are observed for small values of r in Fig. 21a are due to termination errors. It was found by fitting that G8(r, t) for t < 20 X 10~13 sec was Gaussian. By assuming that Gs(r, t) is Gaussian for all times it was possible to derive the full width of half-maximum of Gs(r, t) even for  NEUTRON SCATTERING RESULTS 157 FIG. 21. (a) Pair distribution function of liquid argon for different times (in units of 10~ 13 sec). (b) A weighted combination of the self-correlation and pair correlation functions of liquid argon (84.5° K, 550 mm) for large times (from Dasannacharya and Rao [49]). w 1 i l l 1.6 Ί \ Gs(r,t) 1.2 -\ (b) A \t=3 0.8 -\ 0.4 A \ t =5 n 1 N^^l 0 0.2 0.4 0.6 0.8 1.0 0 0.4 0.8 1.2 1.6 2.0 1.0 2.0 3.0 4.0 Γ (Α ) Γ (Α ) Γ (Α ) FIG. 22. Self-correlation function of liquid argon at 84.5° K for different times (in units of 10~13 sec). The ordinate on the right-hand axis of (a) applies to t = 2 (from Dasannacharya and Rao [49]). 158 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD TABLE VI T H E SELF-CORRELATION FUNCTION G8(r, t) FOR LIQUID ARGON AT 84.5° K a t in units of 10~13 sec t in units of 10~13 sec t in units of 10~ 13 sec r(A) 1 2 r(A) 3 5 r(A) 10 20 0.0 27.56 17.43 0 1.66 0.68 0.0 0.2492 0.0961 0.05 25.64 17.18 0.1 1.58 0.67 0.2 0.2397 0.0923 0.10 20.83 16.28 0.2 1.40 0.62 0.4 0.2075 0.0841 0.15 14.74 14.74 0.3 1.10 0.55 0.6 0.1612 0.0740 0.20 11.54 13.14 0.4 0.79 0.47 0.8 0.1126 0.0607 0.25 4.6 11.22 0.5 0.52 0.37 1.0 0.0708 0.0474 0.30 2.05 9.29 0.6 0.30 0.28 1.2 0.0392 0.0354 0.35 0.83 7.31 0.7 0.20 0.16 1.4 0.0202 0.0259 0.40 0.32 5.51 0.8 0.13 0.08 1.6 0.0138 0.0183 0.45 0.13 3.97 0.9 0.08 0.04 1.8 0.0069 0.0126 0.50 — 2.82 1.0 0.05 0.01 2.0 0.0051 0.0089 0.55 — 1.98 1.1 0.03 0.003 2.2 0.0032 0.0063 0.60 — 1.35 1.2 0.025 — 2.4 0.0019 0.0063 0.65 — 0.77 1.3 0.015 — 2.6 0.0013 0.0069 0.70 — 0.45 1.4 0.0125 — 2.8 — 0.0095 0.75 — 0.26 1.5 0.01 — 3.0 — 0.0126 0.80 — 0.19 1.6 0.006 — 3.2 — 0.0146 0.85 — 0.13 1.7 — — 3.4 — 0.0177 0.90 — 0.06 1.8 — — 3.6 — 0.0196 0.95 — 0.04 1.9 — — 3.8 — 0.0208 1.00 — — 2.0 — — 4.0 — 0.0202 ° According to Dasannacharya and Rao [49]. times for which Gs(r, t) and Gd(r, t) overlap. This was achieved by using the observed peak height and the normalization condition of G s (r, t). The values obtained in this way are shown in Fig. 23 where the three curves show the prediction by Fick's law for diffusion for three values of the diffusion constant. It was concluded by DR that the observations were, within the experimental errors, consistent with the simple diffusion results. The variation with temperature of the full width at half-maximum of the energy distributions of the scattering law also indicates that the diffusion occurs in a simple fashion. The logarithm of the width plotted versus T~x yields a straight line from the slope of which the activation energy E = 700 ± 200 cal/mole is derived. 3.2.2. Model Comparisons In the experiment by Kroo et al. [51] the analysis was made by com- parison of the experimental data to cross sections calculated from certain  NEUTRON SCATTERING RESULTS 159 τ 1 1 r O 10 20 30 40 T I M E (10~13SEC) FIG. 23. The width at half-maximum of the self-correlation function for liquid argon at 84.4° K as a function of time: ( o ) obtained from the peak height of Gs by assuming that the function is Gaussian and using the condition that the area of Ga is unity (from Dasannacharya and Rao [49]). simple models discussed in Section 1. It was concluded that the poly- crystalline model by Egelstaff [16] was in qualitative agreement with the observations, and a dispersion relation for the thermal vibration was derived. An experiment by Chen et al. [53] gave further evidence of phonons although this measurement was less extensive. The experiment recently reported by Skold and Larsson [52] is of the same type as the one by Kroo et al. but covers a larger range of /c-values and is of higher statistical quality. The experiment by Skö ld and Larsson was made with a time-of-flight spectrometer using 4.1 A incident neutrons (see Fig. 2). Intensity distributions observed at 17 scattering angles from liquid argon at 94.4° K are shown in Fig. 24, where the shape of the incident spectrum is also shown. The observed intensities are given as double differential cross sections although no correction has been made for the width of the incident spectrum. Absolute values were obtained by normalizing the diffraction pattern that was evaluated from the curves in Fig. 24 to the diffraction pattern obtained by Gingrich and Tompson [50]. The cross section is represented as function of κ at various constant 160 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD rj ■ · ' — -- . 7-—i— . . . , . . . . _ r i Î i i i i i i i I "1 Γ IT I I 1 I i ! 1 1 1 1 A 56° 60° / \ Lη ν ^ 1Λ *"\, f Λ / 66 e 70 e 72° 1 ·' ' f· / \% J K 6r *. I 5\ 76 e 80e ,* 83° 1 ? 4 Τ * 3 I ]/ y ■ j v 5 6 I 5 87.5 e 90° 96° % η A ■ / 102e 110e 115° A •v 0.5 1.0 1.5 B r a g g - p e a k from -Bragg-peak Bragg-peak from A I - ccontainer o n tail Γ . Λ Ι - container Al-containc 130° ,1k ■ "' A 0.5 1.0 1.5 0.5 1.0 1.5 .[lo'Sec 1 ] FIG. 24. Intensity distributions scattered from liquid argon at 94° K shown as double differential scattering cross section versus final neutron frequency for 17 angles of scattering. The incident spectrum is shown by the solid curves in the upper part of the figure (from Skö ld and Larsson [52]).  NEUTRON SCATTERING RESULTS 161 - I j T 1 r-q fc 'T 1 ' 1 T I .. 079-10 11 JF 1.57« 10* I • ** 1 : J ίE · · *· ' · · ... : :..· · "· 1 Γ * ^-" Γ i 1 i 1 i H F i 1 i 1 i - b 1 ' 1 ' - 1 1 ' : : .· · ... 2.36-10" : 3.93-10* r .· β • ^ ^ , • β β ······ c - ·' \ - - —i 1 ι 1 ι _ 1 1 1 ι <*>\Έ r —i 1 1 1 1— 1 1 1 . 1 ■ 1 -ι — _ 1 1 A.71.10* o ' I - 5.50-1012 - 6.2Θ -1012 o ο · o° ^ , •β 0 ^"^ . .· · · ' ^ ^ r = ! = s^ - \ - - - i 1 i 1 i ι 1 ι 1 _i 1 _1 ' 1 n— 1—1 ! r 9.42-1011 = ■ 11.00-1012 .^ r -. i 1 1 i 1 J_ i - FIG. 25. Double differential scattering cross section for liquid argon at 94° K shown as function of κ for various frequency transfers. The solid line shows the calculated multiphonon cross section (see discussion in the text) (from Skö ld and Larsson [52].) values of the frequency transfer in Fig. 25. This mode of representation clearly displays how the structure in the ^-distributions is smeared out as ω increases and is convenient when discussing coherent scattering. Theoretical cross sections to which data are compared often include only one-phonon terms, and the multiphonon contribution must therefore be subtracted from the experimental data before the comparison is made. The normalization of the multiphonon term is a difficult procedure and  τ _ 60 Λ ο ) Os 50 • 40 T 3 0 1 20 • "u 10 *£^ '· L· 3 -(e) 2 > S en O 2 A > ^> 8 r L5 2.0 2.5 3.0 3.5 w o 50 O > Ö 03 O: r Ö FIG. 26. One-phonon scattering cross section for liquid argon at 94° K shown as function of κ for various frequency transfers, (a) ω = 0.00 x 1012 sec"1; (b) ω = 1.57 X 1012, ( ) R = 10 A and q = 0.1 A"1, ( ) R = 20 and q = 0.1; (c) ω = 2.36 X 1012, ( ) R = 10 and q = 0.2, ( ) R = 20 and q = 0.3; (d) ω = 3.14 X 1012, ( ) R = 10 and q = 0.4, ( ) R = 20 and q = 0.4; (e) ω = 3.93 x 1012, ( )R = 10 and q = 0.4, ( ) R = 20 and q = 0.5; (f) ω = 4.71 X 1012, ( ) R = 10 and q = 0.5, ( ) R - 20 and q = 0.6; (g) ω = 5.50 X 1012, ( ) R = 10 and ? = 0.60, ( ) R = 20 and q = 0.65; and (h) ω = 6.28 x 1012, ( ) R = 10 and g = 0.7, ( ) R = 20 and q = 0.8. The solid lines and the dashed lines show the cross section calculated from Singwi's formula (1.17a, b) for various values of R and q for liquid argon at 94° K (from Skö ld and Larsson [52]).  NEUTRON SCATTERING RESULTS 163 may easily introduce errors in the resulting quasi-elastic and one- phonon term. Its subtraction from the total neutron scattering distribu- tion or scattering function is of the same nature as the subtraction of all the inelastic spectrum from the total spectrum in hydrogenous liquids to give the separated quasi-elastic peak. In both cases subjective plausibility conditions and few objective conditions are used to justify the subtrac- tion. The experimental basis for subtraction of the multiphonon term in the incoherent form is given by a series of observations of scattering of cold neutrons from a single crystal of aluminum [54], a polycrystal of aluminum at high temperature, and, finally, liquid aluminum [17]. From these series of observations it was found that (a) The shape of the large energy transfer region stayed the same in single crystal, polycrystal, and liquid. (b) This high energy transfer region was for the single crystal case well described by the multiphonon term calculated in the incoherent approximation even for the 100% coherent scatterer. Consequently, the multiphonon expansion valid for a solid [55] and in the incoherent approximation was used and normalized to the cross- section data at the highest energy transfer. The computed multiphonon term is given by the solid line in the curves of Fig. 25. Examples of the one-phonon cross sections are shown in Figs. 26 and 27 where also the corresponding cross section calculated from the model by Singwi [14, 15] and the model by Egelstaff [16] are shown. Singwi's model which includes the coherence parameter /?, the frequency spectrum/(ω ), and the wave vector q as parameters is drawn in Fig. 26 for the values of the parameters that gave the best fit at the corresponding ω . By fitting these parameters, points on the frequency spectrum and on the dispersion curve were obtained. The ω -q relation is shown in Fig. 28, and the frequency spectrum is shown in Fig. 29. There is some uncertainty in the value of R and the wave vector q, the value of which depends on the value choosen for R. The ω -q relation is therefore derived for R = 10 A and R = 20 A, respectively. The true value of R is believed to be within these limits. EgelstafFs model, in which the longitudinal and transverse phonons are treated separately, also includes the wave vector and the frequency spectrum as parameters. It was found by fitting that the best description of the data is obtained if it was assumed that only longitudinal phonons contribute. The curves obtained for the cross section being described as the sum of a longitudinal and a transverse component and for the longitudinal part only are shown in Fig. 27 by dashed and solid lines, respectively. The values of q that were obtained from this analysis are  4t 1 1 ι ι ι -(e) > 5ö CO Γ CO O S5 0.5l· 1 1 1 1 1— > L5 2J0 2£ 3.0 3.5 w o 1.01 > 0.5 ^Ku^^t Ö CO O: 1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5 f Ö FIG. 27. One-phonon scattering cross section shown as function of κ for various frequency transfer, (a) ω = 0.00 X 10" sec - 1 ; (b) ω = 1.57 X 10", q = 0.1 A"1; (c) ω = 2.36 X 10", q = 0.2; (d) ω = 3.14 X 10", q = 0.3; (e) ω = 3.93 X 10", q = 0.4; (f) ω = 4.71 x 10", q = 0.45; (g) ω = 5.50 x 10", q = 0.6; and (h) ω = 6.28 x 10", q = 0.7. The lines show the cross section calculated from Egelstafï 's formula (1.18a, b,c) including only longitudinal phonons (solid lines) and including both transverse and longitudinal phonons (dashed lines) (from Skö ld and Larsson [52]).  NEUTRON SCATTERING RESULTS 165 I I "Ί —i 1 ι 1 1 1 "Ί — 8 - c = 8.2*104 y 7 -J 6 ·■ / o a / i—i a J ?o 5 _ / ■ a· / o X D ' 7 A · ■ /O ~1 D Φ x«0 /- Ü ^ 3 _ a 3 ·■ ^ 2 8" 1 I I _j __l 1 1 1 1 I _J | L· 0.5 1 [A"' ] 1.0 FIG. 28. The ω -q relations obtained by Skö ld and Larsson [52] for liquid argon at 94° K (circles and filled squares) and the ω -q relations obtained by Kroo et al. [51] for liquid argon at 88° K (open squares). The dashed line shows the linear relation corre- sponding to the sound velocity. ( · ) Singwi model, R = 10 A, ( o ) Singwi model, R = 20 A, (■ ) Egelstaff model, this experiment, and (D) Egelstaff model, earlier experiment. shown in the ω -q plot in Fig. 28 where the dashed line is the linear dispersion relation corresponding to the value 8.2 X 104 cm/sec for the velocity of sound. The results obtained in the rather similar study by Kroo et ai. are also shown in Fig. 28. Rather than give the dispersion relations derived from the Egelstaff model—and also the Singwi model— starting out from q = 0, it would be closer to the truth and entirely in the spirit of this model to start plotting them from κ = 2 A - 1 corre- sponding to the main peak of the liquid structure factor in argon. In these models this peak is a manifestation of a certain periodicity of the liquid lattice. The periodicity is of course of a limited extension ranging over a distance of 2R\ 2R is to be compared to a dimension of polycrystal, the lifetime of which is of the order of 10~12 sec. In the spirit of this model a frequency distribution is meaningful. Points on the frequency spectrum curve were consequently derived from the cross section curves given at constant energy transfer. The method of deriving these data was by determining the normalization factor in the cross section, which mainly is determined by the frequency 166 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD Μ ω ) 0 1 2 3 A 5 6 7 ω [sec'^IO" 1 2 ] FIG. 29. The frequency spectrum observed by Skö ld and Larsson [52] for liquid argon at 94° K compared to the computer calculations by Rahman [56] (—). (O) This experiment, Singwi's model and ( ■ ) this experiment, EgelstafFs model. distribution /(ω ). These results are shown in Fig. 29 where the solid line shows the result of a computer calculation by Rahman [56]. It was concluded by DR that the variation with time of the width of Gs(r> t) was consistent with the predictions of Fick's law for diffusion. This conclusion may seem to be in contradiction of the results described in this section which include a dispersion relation of phonons and a frequency spectrum rather different from the Lorentzian curve that should be observed if the motion is of the simple diffusion type. It was, however, shown by Rahman [56] that the mean square displacement of the particles may be very similar to the simple diffusion result even if the frequency spectrum is very different from the Lorentzian curve. The results may therefore not be conflicting if experimental errors are considered. 4* Hydrogen 4.1. TOTAL CROSS SECTION Total cross section measurements on liquid parahydrogen (99, 7 5 % ) and on a mixture of 7 5 % ortho- and 2 5 % parahydrogen were performed  NEUTRON SCATTERING RESULTS 167 by McReynolds and Whittemore [57] using neutrons from an electron linear accelerator. The hydrogen sample was held at its boiling point of 20.4° K at atmospheric pressure. The cross section studies were performed in the range 0.0005 eV to about 0.25 eV. The aim of these total cross section studies was to find out in what way the energy transitions between the low-lying rotational levels / = 0 —► / = 1 corresponding to a para-ortho transition is observed in the total cross section and on the whole how well the Schwinger-Teller theory [58] for the cross section of the free (gaseous) hydrogen molecule describes the facts. The observed total cross sections for pure para- hydrogen and the mixture are given in Fig. 30. It is seen that for the parahydrogen below 0.010 eV the cross section varies as expected from the theory for a gas molecule. The step in the cross section immediately above 0.010 eV should correspond to the inset of the para-ortho transition / = 0 - > / = l . As shown at the inset in the figure which gives the step in the cross section on an enlarged scale Ί 1 I I I lll| 1 1 I I III | Γ ORTHO H2 GAS sl0!l 0 0.01 0.02 0.05 o e| oo 0 o° c iC ENERGY(eV) tPO f 9 el '<*»,'<**%> r z «^θ ο ο ο ο ο ο ο ,α ^ ο < 00 z o O in UJ , v </> f PARA H 2 GAS ·· % V) O 5 < < I I 1 I Nil J I I I I I 8I I I J L I I I I III J L 10" 6 β 10-2 4 e é 10-i 10- NEUTRON ENERGY ( e V ) FIG. 30. Total neutron cross section of ( · ) liquid parahydrogen and (O) mixed para-orthohydrogen at 21 ° K as measured by transmission of neutrons from electron linear accelerator. Energy determination was by time-of-flight, points in the inset having been measured at higher resolution to investigate sharpness of minimum (from McReynolds and Whittemore [57]). 168 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD the step seems to be fairly sharp and occur at approximately 0.015 eV. Quantum mechanically the rotational states of the free hydrogen molecule are given by E =M~iJU+l)=0M5J(J+l) eV (4.1) where M is the proton mass, a is the distance between the protons, which is 0.75 A, and / is the rotational quantum number. A freely recoiling gas molecule would give the step J = 0 —> J = l a t 0.023 eV because the law of conservation of momentum has to be satisfied. This measurement of the total cross section of parahydrogen has thus given the apparently conflicting results that the free gas cross section explains the behavior below 0.010 eV in contrast to the observation that the para-ortho transition occurs at about 0.015 eV as it would do if the molecule were completely hindered from performing a free recoil. Also, the sharpness of the step at 0.015 eV is, on the other hand, indicating— qualitatively—a free rotation. A free rotation of the molecule in a quasi- stable cave of neighboring molecules would probably create a cross- section result of the type observed. 4.2. DIFFERENTIAL CROSS SECTION A differential cross section measurement at 20.4° K on almost pure parahydrogen and on an ortho-parahydrogen mixture of ratio 2 : 1 was performed by Whittemore and Danner [59], who used an electron linear accelerator time-of-flight spectrometer. The resolution was relatively poor, 0.1 < Δ Ε \Ε < 0.2, and the ingoing energies were 0.065 and 0.040 eV, which are high. These factors prevented the structure (if there is any) of the transitions being observed. The scattering angle was 90° , and the incident energies were selected in such a way that one or more of the scattering components (caused by differential rotational states) were suppressed. In fact, the average outgoing energy E for scattering at 90° is related to the incident energy E0 as E = ^(E0 — 2J), where Δ is the energy difference between two rotational levels. The sample thicknesses were chosen to have a transmission greater than or equal to 0.85, and the effects of multiple scattering were believed to be small. Examples of the results exhibiting the main features for parahydrogen are given in Fig. 31. The elastic peak that occurs at 0.065 eV is a back- ground peak and is not caused by liquid hydrogen. The main features of the spectrum are:  NEUTRON SCATTERING RESULTS 169 ENERGY (eV) 0.100 α ο 2β 0.014 0.010 90 110 130 150 170 190 NUMBER OF Ι β μ SEC CHANNELS FIG. 31. (—) Experimental data for 0.065 eV neutrons scattered by a 1 mm layer of parahydrogen. ( ) Theoretical predictions based on a perfect gas model from Sarma [59a] (from Whittemore and Danner [59]). (a) No structure is observed because of the high ingoing neutron energy used in the experiment combined with a relatively poor resolution. (b) The observed energy distribution is broader than predicted by a gas model taking into account the possible rotational transitions. For the case of parahydrogen the expected contribution is simple enough—the transition J = 0 -+ J = 1 being dominant—so that a comparison of the theoretical prediction to the experimental curve may have some meaning. With this in mind a further conclusion may probably be drawn from these measurements: (c) The position of the peak of the observed distribution agrees with the prediction of a gas model. On the other hand, the width of the distribution may be interpreted as an indication of hindrance of molecular motions. The status of our knowledge of the molecular dynamics of liquid hydrogen on the basis of this measurement as well as on the measurement of the total cross section consequently is that hindrance for the free translations may exist but probably the rotations occur fairly freely.  170 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD As a contrast to the studies of the differential scattering cross section with high ingoing neutron energies, Egelstaff et al. [60] used the cold neutron scattering technique to measure the scattered neutron spectra from liquid hydrogen at 15° , 18° , and 21 ° K. The specimen of hydrogen was contained in a series of parallel stainless steel tubes of 1 mm diam with a wall thickness of 0.05 mm, in a cryostat. The ratio of ortho- to parahydrogen was a few percent less than the room temperature value of 3 : 1, since the conversion rate at 20° K is around 1 % per hour. The cross section for the scattering of neutrons from parahydrogen is about 4 barns for neutrons unable to excite the para-ortho transition. As this is ~ l / 2 0 of the orthohydrogen scattering cross section, the contribution of parahydrogen scattering to this experiment was 2%. Typical experimental results are shown in Fig. 32, which shows distributions in reciprocal velocity of 4 A neutrons scattered from liquid hydrogen at 15° K at various angles. The ortho-para conversion line at 510 /xsec/meter and the quasi-elastic peak at 1030 /xsec/meter are resolved. ENERGY GAINED BY THE NEUTRON ( m e V ) 30 20 10 5 3 2 1 0 - 1 -2 30 20 15 1 Π 1 1 1 1 1 1 1 1 ! 500 - (a) UJ z ° »0° % 90 * υ <250 I J*· V,. H V) ^ Ζ α Λ η ^ ^ . * · ^ ^ ^ 0 o CO cr UJ • g 250 * · 45e J 4& f• o · Ί l 0 ^ « J ^ S t a tmH^/ ° UJ "S,*J Û (/>750 o - Z — 2 3 500 J £ o o u. 2 250 o ° 20- "J u 0 ° GO <^^M^^Pi^ 3 4^ < V^ -1 2 400 600 800 1000 1200 1400 400 500 600 700 800 RECIPROCAL VELOCITY OF THE SCATTERED NEUTRON (/^S/m) FIG. 32. The scattering of 4 A neutrons at various angles from liquid hydrogen at 15° K. (a) The distribution of detected neutrons showing the quasi-elastic and ortho-para peaks, (b) The ortho-para conversion peak shown on an enlarged scale. The resolution function of the apparatus shown as full width at zero height is independent of the angle of scatter (from Egelstaff et al. [60]).  NEUTRON SCATTERING RESULTS 171 Figure 32b shows the ortho-para conversion line on an enlarged scale. It is clear from this figure that the flight time corresponding to maximum intensity varies with angle, but this effect disappears when the data are corrected for the detailed balance factor and the time-of-flight scale is converted to a constant energy interval scale. The data at all tem- peratures were consistent with an ortho-para conversion energy of 0.0152 ± 0.0005 eV. The widths of the quasi-elastic and ortho-para conversion lines were observed to be a function of angle of scattering. The instrumental resolution is indicated. Knowing the diffusion coefficient from measure- ments using spin-echo techniques of magnetic resonance by Hass et al. [61] the expected diffusive broadening WDK2 was calculated. The observed line widths are greater than IHDK2, by about a factor of 2. The contribution due to acoustic modes was estimated empirically from the variation with angle and, after allowing for this, the residual width due to the splitting of the triplet state was about ± 0.0005 eV. This splitting of the ortho state will produce a line made up of two components, one elastic peak and distributions on either side shifted from it by the average splitting of the ortho levels. At low angles the central peak should be dominant and with increasing angles the side peaks should increase. Some evidence for the total peak having this behavior can be seen in Fig. 32 for 90° angle of observation. 5* Methane 5.1. TOTAL CROSS SECTION Total cross section measurements on methane have been performed by Rogalska [62] and Whittemore [63]. The results of Rogalska obtained at 117° K and in an energy range from 0.0057 to 0.101 eV are given in Fig. 33. These experimental results are compared to the cross section curve calculated for a gas at the same temperature, and the conclusion may be drawn that the simple gas model in the Krieger-Nelkin version [64] fits and that therefore the molecular rotations are free in liquid methane. Although the precision of total cross section measurements is normally quite good, the total cross section is an integral magnitude. In general, very few conclusions regarding the details of molecular dynamics may be drawn from such information. This is verified in the case of methane, if the present conclusion that the molecular rotations are free in the liquid state is compared to the conclusions from high resolution differential neutron cross section studies. 172 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD 30 I I I I I I I I I I 0.4 0.6 0.8 1.0 2 4 6 8 100 200 En(meV) FIG. 33. Total scattering cross section per proton in the CH 4 molecule; the curve is calculated according to the Krieger-Nelkin theory for the gas. Experimental results are for the liquid (from Rogalska [62]). 5.2. DIFFERENTIAL CROSS SECTION Using the beryllium-filtered spectrum and time-of-flight method, Janik et al. [65] measured the differential cross section for methane at 100° K and a scattering angle of 90° . It is observed (compare Fig. 34) that the intensity of the quasi-elastic peak varies strongly with angle and that its contribution to the total energy distribution at the 90° angle is small—as a matter of fact quite insignificant compared, for instance, to the predictions for methane in the gaseous state by Griffing [66, 67]. Neither does the gas model in the Krieger-Nelkin version fit the observed distribution nor does a modified version of the Rahman [68] and Griffing theory fit the data. In this modified theory the translational motions of the molecule is described by simple diffusion. The inelastic part of the observed neutron spectrum is also broader than expected on the basis of any known theory. This fact, together with the angular variation of the intensity of the quasi-elastic peak, indicates  NEUTRON SCATTERING RESULTS 173 NEUTRON WAVE LENGTH ( A ) 2 3 4 10 5 NEUTRON ENERGY ( m e V ) FIG. 34. Scattered neutron spectrum from liquid methane. Scattering angle, 90° ; temperature, — 173° C. (1) Krieger-Nelkin theory. Mett = 4 proton masses (2) Krieger- Nelkin theory, Mett = 2.1 proton masses; (3) Boltzmann-like distribution (from Janik et al. [65]). rather that some hindrance exists for the molecular rotations in liquid methane. The complexity of the molecular motions already for a "simple" symmetric molecule like methane is thus at once demonstrated in a differential cross section measurement. The measurements reported by Hautecler and Stiller [69] on methane at 102° K were carried out by use of the inverted beryllium filter method. A great uncertainly was caused by the use of a beryllium filter of only 7 cm thickness, which caused a considerable transmission of neutrons of energies greater than 5.2 meV. After appropriate corrections a scattered spectrum rich in detail was obtained. The various energy transfers observed were compared to transitions between rotational levels assuming that the molecule is free to rotate, and most of the levels observed were identified as being due to transitions between free rotator levels. However, later measurements by other research workers using a narrow ingoing neutron spectrum and high resolution (see below) did not show any isolated energy transfers. Dasannacharya and Venkataraman [26] studied methane at 98° K using a rotating crystal spectrometer giving 4.1 A neutrons with a 174 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD resolution of 0.0004 eV. The range in scattering angle was 15° to 90° . The measurements were performed with a 1-mm-thick sample in transmission geometry. Some typical distributions corrected for con- tainer contribution (very small), air scattering, and variation of detector efficiency with energy are shown in Fig. 35. These distributions are given on an arbitrary scale and are not normalized to each other in any way. Comparing the results in the region of high energy transfer with that of Janik et al., it was concluded that multiple scattering is insigni- ficant, at least in a qualitative sense. It is, however, to be noticed that a sample thickness of 1 mm is quite considerable and probably results in TIME OF FLIGHT (USEC/W) 200 400 600 800 1000 1200 i—i—i—r Ί 1 1 Γ Γ Ί Γ (a) h ■ TIME OF FLIGHT (tf SEC/m) j _ I I I I I I I I I I I I 200 400 600 800 1000 1200 1 1 1 1 Γ τ τ π —r 1 I I I I I I I I I Γ h (b) -\ h (O y ^ V ^ ^ \ ^<4 -U I I II I I 1 I I I I I L I I I I I I I I I I I I I Ί —i—i—i—i—i—i—i—rr~\—r 1 1 1 T7"T 1—i 1—I 1 r . (d) · ·· J |_ (e) %V· · -· . J_L J I 1 I I I I I I I I I I l I I I L 80 40 20 10 8 6 80 40 20 10 8 6 ENERGY ( m e V ) ENERGY ( me V ) FIG. 35. Corrected time-of-fiight distributions observed on scattering 4.1 A neutrons by liquid methane at 98° K (E0 = 4.87 meV). (a) 30° , (b) 45° , (c) 75° , (d) 60° , and (e) 90° . The dashed lines denote the inelastic background subtracted in order to isolate the quasi-elastic scattering (from Dasannacharya and Venkataraman [26]).  NEUTRON SCATTERING RESULTS 175 some multiple scattering if a critical comparison to theories is to be attempted on the basis of these results. In order to get reliable values for the widths of the quasi-elastic peaks the variation of resolution with angle was carefully studied. By introduc- ing a Soller collimater between crystal and sample, the resolution for 15° and 22.5° angle was cut down to 0.0002 eV. By interpolating the inelastic background, the dashed curve in Fig. 35, the quasi-elastic peak was separated and true widths were extracted by fitting with a computed 477SIN0 (A"') FIG 36. Line widths for (a) normal and (b) deuterated methane, (c) Liquid structure factor for normal methane (from Venkataraman et al. [70]). 176 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD peak. The quasi-elastic peak was computed on the basis of a convolution of the Gaussian resolution with an assumed Lorentzian width function. The widths obtained in this manner are plotted against κ in Fig. 36a, from which the conclusion was drawn that the translational motions of the methane molecule are described by a simple diffusion formula. The variation of the integrated intensity of the quasi-elastic peak as a function of K was not studied. In extending the results obtained for CH 4 the same authors [70] have measured time-of-flight spectra for CD 4 at 95° K. In contrast to ordinary hydrogen, deuterium has a relatively big coherent neutron cross- section, which is why one could expect interference effects. As in the case of CH 4 the data at all angles showed a quasi-elastic peak and a broad inelastic bump even if the former was generally more pronounced. Using the same procedure as for CH 4 the true full width at half-maximum TIME OF FLIGHT (^SEC/m) 200 300 400 500 600 700 800 900 1000 1100 1200 — Γ " *5 Vol <ν ι \ 1 (\j \Jl V To fO (VJ 1 1 1 1 1 1 1 1 1 1 "~ 1 •j^80 40 30 2 0 1 8 16 14 12 10 9 I I I 8 7 I 6 ENERGY (meV) FIG. 37. Spectrum observed at 15° scattering angle with high resolution from liquid methane at T = 98° K. (λ 0 = 4.1 A, E0 = 4.87 meV.) The insert shows the spectrum expected if the rotations are free (from Dasannacharya and Venkataraman [26]).  NEUTRON SCATTERING RESULTS 177 was derived, the result of which is shown in Fig. 36b together with the liquid structure factor in Fig. 36c measured by Petz [71] using x rays. The oscillatory character of the line width observed earlier in other coherently scattering liquids was observed also for CD 4 . An explanation of this fact in terms of a variable apparent mass of the scatterer was attempted. The primary interest in the inelastic spectrum was to find whether discrete rotational levels exist in liquid methane in the same sense as they do in the gas. A computed spectrum for a scattering angle of 15° , corresponding to a resolution of 0.0002 eV for full width at half-height, is shown as an inset in Fig. 37. A comparison with the measured spectrum seems to indicate that no free rotations exist in liquid methane, at least no low energy transfers. 5.3. SCATTERING LAW Harker and Brugger [24] used a phased chopper facility to investigate the scattering law for methane at 99° K and a pressure of 0.705 atm. They covered 15 different angles and used 3 different ingoing energies in their experiment. Differential cross-section values at specific energy transfers were obtained by a four-channel second degree interpolation. The displayed form of the data is the reduced partial differential cross- section S(K, ϋ ω ) which is defined: S(K, hœ) = (EJE)1' * exp(-ha>l2kBT) · σ (Ε 0 , £, Θ ) (5.1) where hw = E -E0, κ = k - k0 (5.2) σ (Ε 0 , Ε , Θ ) is the partial differential cross section with respect to the scattered neutron energy and solid angle. This reduced form of the cross section allows a combination of the data from runs at different incident energies. In performing this there are regions in the variable κ where the data overlap. The /c-scale was therefore divided into equal increments of log κ and arithmetic averaging of the data in each increment was made. Since the number of target molecules was not accurately known, the data were normalized to the measured total cross section as obtained by Rogalska [62] and by Whittemore [63]. The data obtained are shown in Fig. 38, which displays S(K, fiœ) for each reduced energy transfer e = —iï œjkBT with hand-drawn fits. For small momentum transfers the measured neutron distributions should for an infinite resolution appear considerably more peaked than what is observed. Due to insufficient resolution, the intensity values derived from the measurements are consequently too low for the smallest energy and  104 Ί 1—Γ "Γ β *=0 3^ 4 5 β 7 β 9 10 I^KANGSTROMS"*1) FIG. 38. Composite of the reduced partial cross sections for liquid methane (T = 99° K, c = ( - M / 0 . 0 0 8 5 3 eV). The solid curves are hand-drawn fits to the data (from Harker and Brugger [24]).  NEUTRON SCATTERING RESULTS 179 momentum transfers. For κ = 1.0 A - 1 5(/c, fiœ) should be 40% higher. At K near 4.0 A - 1 the correction would be negligible. For e above 1.5, multiple scattering may add to S(K, fiœ) such that the value of the 1.2 \0) n o 0.8 h 0.4 \- · - P~a · 1.2 r 1 T r QbT— o o 0.8 Lo o° """"l 0.4 I I __L . 1 l 1.2 1 1 1 [(c) 1 o o o Γ ° o • 0.4 • H 1 1 1 .£. " l ~ 1 I 1 1 (d) 0.8 - o o o 0 o • • 0.4 • 0 1 1 1 1 I _^ 3 4 -1 Gvi ( A N G S T R O M S ) FIG. 39. Ratios of the observed values to the expected values for a monatomic system of the zero and first moments of energy transfer with respect to the reduced partial cross section versus κ . (a) methane gas, T = 294° K, (b) methane gas, T = 125° K, (c) liquid methane, T = 99° K, and (d) solid methane, T = 83° K; (O) <e0>exp/<e0>mOnatomic and ( · ) <e1>exp/<(€ l)monatomic · The solid and dashed curves are the ratios of the expected values of the molecular gas in the classical limit to the expected values for a monatomic system of the zero and first moments, respectively (from Harker and Brugger [24]). 180 Κ . Ε . LARSSON, U. DAHLBORG, AND Κ . SKÖ LD scattering function is lower than shown in the figure (however, the transmission was 9 0 % ) . The zero and first moments of energy transfers with respects to the reduced partial cross section were also computed. The defining equations are Zero moment = <€ ° > = A f cosh(€ /2) S(K, hœ) de (5.3a) J o First moment = <€ *> = A f € sinh(c/2) S(/c, hœ) de (5.3b) *o where A = 8π Α Α Γ /σ & , and ah is the total bound scattering cross section. The results of these computations are as given in Table VII, both for the expected moments based upon a monatomic system with hydrogen as the principal scatterer and for the expected moments based upon the classical treatment of a freely rotating and translating molecule. TABLE VII EXPERIMENTAL AND THEORETICAL MOMENTS FROM METHANE DATA*1 K (A-1) <€ ° >0bS \ e /monatomic \€ /molecular <c1>obs \ e /monatomic \€ /molecular 1.1 1.0 1.0 0.10 0.3 0.093 1.4 0.80 1.0 0.99 0.17 0.5 0.15 1.7 — 1.0 0.99 0.24 0.7 0.22 2.0 0.80 1.0 0.99 0.47 1.0 0.30 2.9 0.86 1.0 0.97 0.76 2.0 0.58 4.1 0.83 1.0 0.92 1.2 4.0 1.1 5.0 0.77 1.0 0.88 ° According to Harker and Brugger [24]. The observed and expected molecular moments are compared with the expected monatomic moments in Fig. 39. 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