The high energy-transfer region in neutron scattering vibrational spectra: What does it mean and what could it be useful for?

Abstract

We analyze the high energy transfer part of a neutron scattering spectrum from a protonated molecular crystal measured on an inverse-geometry crystal-analyzer spectrometers like TOSCA or VISION. In spite of the little information available in this range, we show that it is possible to exploit it in order to perform a new normalization procedure. Then a simple free gas model based on the impulse approximation is presented, followed by approximate methods for quickly correcting experimental data for the main unwanted effects. At the end, a practical example based on solid urea data is presented and the new normalization procedure is tested comparing the results from two different scattering laws (one being the free gas model). The two normalization constants obtained turn out to be practically identical showing the reliability of the suggested approach.

Keywords

Inelastic neutron scattering vibrational spectroscopy

1. Introduction

Inelastic neutron scattering (INS) [20] is a very well established experimental technique applied in condensed matter research, which has been invented by the Nobel laureate B. N. Brockhouse in 1955 [7], while working at Canadian Chalk River Laboratory. During the sixty years following its invention, INS developed in a substantial way and branched into several kinds of spectroscopic applications, ranging from the low energy-transfer (of the order of 1 μeV) regime, known as quasi-elastic neutron scattering, to the high energy-transfer (of the order of 10 eV) one, known as deep inelastic neutron scattering. Among this number of varieties we will focus on neutron vibrational spectroscopy (NVS) [21], that is the use of INS to study molecular vibrations in condensed matter systems. The first examples of NVS operated at a quantitative level can be found in the excellent review by Boutin and Yip [5] dating 1968. However, it was the development of the spallation neutron sources [4] in the eighties (i.e. IPNS at Argonne in USA, KENS at Tsukuba in Japan, ISIS at Didcot in UK, and LANSCE at Los Alamos in USA) which made possible to expand the spectroscopic range probed by INS towards higher energy-transfer values and better instrumental energy resolution, achieving potentialities comparable with those routinely available in standard optical vibrational spectroscopy (i.e. infrared absorption and Raman scattering).

In parallel with the neutron source evolution there was also an effort to develop original spectrometers devoted to NVS, which could combine a good energy resolution with a broad energy transfer range in one single measurement. In other words, a sort of modern time-of-flight version of the old beryllium filter spectrometers created by Woods et al. [38] for reactor sources. As the matter of fact, chemistry-oriented spectroscopists were not at all pleased to use standard time-of-flight direct-geometry chopper instruments, since the large energy transfer intervals needed for their scientific investigations had to be divided into several parts, implying changes in the incoming neutron energy through, for example, lengthy variations of the chopper parameters. This time-consuming procedure, together with the massive deterioration of the energy resolution along with the incoming neutron energy increase, prompted the development of a new concept of machine: the time-of-flight inverse-geometry crystal-analyzer spectrometer (TICS). The price to pay for this implementation of the working mode preferred by the chemistry-oriented spectroscopists was, on the other hand, expensive enough: this class of instruments is constrained to operate in a very narrow stripe of the kinematic plane (i.e. the plane described by energy and momentum transfers), strictly monotonic (see next section for details), and only very slightly changing by varying the scattering angle. Strictly speaking, differently from the aforementioned time-of-flight direct-geometry chopper spectrometers, a TICS is essentially unfit to explore large areas of the kinematic plane. However, considering the largely incoherent nature of the neutron scattering from protons, together with the abundance of these particles in the molecular systems routinely investigated, it turned out that, at least to some extent (see again the next section for details), the inability to probe a wide range of momentum transfer at a fixed energy transfer value was not regarded as a seriously limiting factor [21]. In addition, one has also to keep in mind that most of the vibrational excitations in molecular solids (not related to low energy lattice modes) are largely localized, so they come out to be either undispersed or just weakly dispersed. The first attempt to build a TICS was probably CAT (i.e. “Crystal spectrometer for high energy incoherent scattering”) operating at KENS [17], followed by TFXA [22] and its successors TOSCA-I and -II [9], both at ISIS, which improved the technique in quite a sensible way. Recently, a new and highly performing TICS, named VISION [28], has been installed at SNS (Oak Ridge, USA), while there are plans for another one, VESPA [13], to be build at the future European Spallation Source (Lund, Sweden).

The progress in the experimental side of NVS brought about by the construction of TICS’s was followed by the development of simulation codes based on the existing vibration data (either from semi-empirical or ab initio approaches), designed to produce spectra that might be compared with real neutron measurements from a TICS. Some well-known characteristics [20] of neutron spectra make INS an ideal test bed for the density functional theory (DFT) calculations of vibrational modes. In this context it is worth mentioning the program named aCLIMAX [8] that uses the normal mode output from various DFT packages to generate the calculated INS of the model molecule, making possible to establish a relatively straightforward connection between simulation and experiment. However, it is worth noting that aCLIMAX is able to compute combinations and overtones, as well as phonon wings [21], using the harmonic approximation, even though an ad hoc correction of the main anharmonicity effects is indeed possible in the spirit of the quasi-harmonic approximation [31]. Once applied to the high energy transfer region (say, larger than 250–300 meV, that is 2000–2400 cm⁻¹), the aCLIMAX spectral reconstruction confirmed in a rigorous way what had been already known since the eighties: the INS spectra of a proton-containing molecular crystal (even at low temperatures, i.e. of the order of 10–20 K), collected by a typical TICS, exhibited visible but limited information on the fundamental bands (namely, stretching modes involving protons). An interesting example is provided in Ref. [24] in the case of solid benzene. Most of the signal is coming from overtones and combinations of lower frequency modes, not only involving the internal molecular vibrations, but also molecular librations and low-energy lattice modes, the last showing a noticeable dispersion. For this reason the weak traces attributable to stretching bands come up rather broader than they should be and also affected by a peculiar blue shift, interpreted as a recoil effect of the single molecular unit [33]. However, such a recoil does not show up like a point-like particle as heavy as the molecule itself, but, on the contrary, by a much lighter fictitious particle described by the well-known formalism of the Sachs-Teller mass tensor [19].

Given this scenario and observing the aforementioned high energy-transfer trend in the large majority of TICS spectra from molecular crystals (see, e.g., the INS database of the TOSCA-I and -II spectra [1]), two questions quickly arise: (i) can the features generally observed in the high energy transfer region in a TICS spectrum be explained by a simple physical model, without making use of the complex spectral reconstruction carried out by aCLIMAX (and necessitating the detailed knowledge of vibrational eigenvalues and eigenvectors)? (ii) Is it possible to extract any valuable physical information from such a spectral range? And, in case, what kind of information? And how?

In the rest of the present paper we will try to reply to both questions and, in order to do it, we will take the following steps: in Section 2 the theoretical formalism bridging the gap between vibrational spectroscopy and single particle response will be introduced, while in Section 3 we will deal with the unwanted instrument effects that, in principle, could contaminate the high energy transfer region of a TICS spectrum. Then, in Section 4, a new spectral normalization will be introduced by exploiting the original description of this spectral zone we will have just set up. A practical example will be provided in Section 5 making use of measured and simulated data on solid urea. Finally, a short final discussion, conclusions and perspectives will be presented in Section 6.

2. The theoretical framework of high energy transfer neutron scattering

As we have already mentioned in the introduction, a TICS is characterized by an incoming “white” neutron beam (with energy $E_{0}$ function of the time-of-flight) and monochromatized out-coming scattered neutrons (with fixed energy $E_{1}$ ). In addition, we have also seen that while $E_{0}$ spans a large interval (up to more than 500 meV), $E_{1}$ is rather low, of the order of few milli-electron volts (say, $2 \div 3.8 meV$ ). In this way, while the energy transfer $ℏ ω$ is simply given by $ℏ ω = E_{0} - E_{1}$ , covering the full range normally requested for vibrational spectroscopy ( $0 < ℏ ω < 500 meV$ ), the momentum transfer $ℏ \vec{Q}$ is a monotonic function of ω and is given by: $\begin{matrix} (1) & ℏ | \vec{Q} | = \sqrt{2 m_{n} (ℏ ω + 2 E_{1} - 2 \sqrt{ℏ ω E_{1} + E_{1}^{2}} cos θ)}, \end{matrix}$ where $m_{n}$ is the neutron mass and θ is the scattering angle. It is evident in the previous equation that for $ℏ ω ≫ E_{1}$ , the square momentum transfer is dominated by a linear term in ω, while the first term containing both $E_{1}$ and θ dependences is proportional to $ω^{1 / 2}$ , and is acting as a small correction: $\begin{matrix} (2) & ℏ^{2} Q^{2} \approx 2 m_{n} ℏ ω - 4 m_{n} cos θ \sqrt{ℏ E_{1}} ω^{1 / 2} + 4 m_{n} E_{1} + \dots, \end{matrix}$ or, similarly: $\begin{matrix} (3) & Q^{2} \approx k_{1}^{2} - 2 k_{1} k_{0} cos θ, \end{matrix}$ if the incoming (and out-coming) neutron wave vectors, ${\vec{k}}_{0 (1)}$ , are introduced by: $E_{0 (1)} = ℏ^{2} k_{0 (1)}^{2} / (2 m_{n})$ . An example taken from the spectrometer TOSCA-II can be found in Fig. 3.24 of Ref. [21], where two scattering angle values have been reported: one for forward-scattering ( $θ_{F} = {42.55}^{\circ}$ ) and one for backscattering ( $θ_{B} = {137.73}^{\circ}$ ). In what follows we will use the symbols $ℏ Q_{F} (ω)$ and $ℏ Q_{B} (ω)$ to represent the quantities calculated in Eq. (1) for the two respective cases.

Moving to the theoretical framework of the NVS it is possible to verify that standard approach to it is based on the use of the incoherent approximation [20] in conjunction with the so-called Gaussian approximation (GA) [23]. This means that the corrected measured spectra (see next section for details) are interpreted according to the two following formulas for the double differential scattering cross-section: $\begin{matrix} (4) & (\frac{d^{2} σ}{d Ω d E_{0}}) = \frac{k_{1}}{k_{0}} Σ_{s} (Q, ω) = N r \frac{k_{1}}{k_{0}} \sum_{n = 1}^{r} x_{n} \frac{σ_{b, n}}{4 π} {⟨ S_{s}^{(n)} (\vec{Q}, ω) ⟩}_{\hat{Q}}, \end{matrix}$ where N is the number of crystal unit cells, $Σ_{s} (Q, ω)$ is the macroscopic self scattering law, n labels the r non-equivalent atomic species in the cell, $x_{n}$ stands for their concentrations ( $\sum_{n = 1}^{r} x_{n} = 1$ ), $σ_{b, n}$ for their bound scattering cross-sections (i.e. coherent plus incoherent), ${⟨ \dots ⟩}_{\hat{Q}}$ represents the spherical average implicit in the powder nature of the considered sample, and, finally, $S_{s}^{(n)} (\vec{Q}, ω)$ is the self scattering law [20] of the nth atomic species given by the following Fourier transform: $\begin{matrix} (5) & S_{s}^{(n)} (\vec{Q}, ω) = \int_{- \infty}^{\infty} \frac{d t}{2 π ℏ} exp (- i ω t) exp [- {\vec{Q}}^{T} ({\underline{\underline{U}}}^{(n)} (0) - {\underline{\underline{U}}}^{(n)} (t)) \vec{Q}], \end{matrix}$ with ${\underline{\underline{U}}}^{(n)} (t)$ being the time-dependent mean square displacement tensor (t-MSDT) for the nth atomic species, whose value at $t = 0$ is real, while elsewhere is complex. By appropriately choosing the reference frame it is always possible to write the aforementioned tensor in a diagonal way, but in general this is possible only for one selected atomic species, not for all of them at the same time. However, keeping in mind the incoherent approximation in Eq. (4), where each individual contribution is first calculated as a function of $\vec{Q}$ , then spherically averaged over $\hat{Q}$ , and finally summed, one can imagine to selected a diagonalizing reference frame each time in the appropriate way for any individual atomic species. So, in what follows, we will assume that all the mentioned tensors are diagonal without any loss of generality. In the spirit of the GA, a tensor element $(U_{j j}^{(n)} (0) - U_{j j}^{(n)} (t))$ , also known as $Γ_{j j}^{(n)} (t)$ , can be directly related to the corresponding element of the velocity autocorrelation tensor [23], $⟨ v_{j}^{(n)} (0) v_{j}^{(n)} (τ) ⟩$ , via the integral relationship: $\begin{matrix} (6) & U_{j j}^{(n)} (0) - U_{j j}^{(n)} (t) = Γ_{j j}^{(n)} (t) = - i \frac{ℏ t}{2 M_{n}} + \int_{0}^{t} (t - τ) ⟨ v_{j}^{(n)} (0) v_{j}^{(n)} (τ) ⟩ d τ, \end{matrix}$ where $M_{n}$ is the mass of the nth atomic species and ${\vec{v}}^{(n)} (t)$ its velocity at the time t. The term linear in t is simply connected to the recoil of the atom hit by the neutron, while the integral term is now clearly related to the vibrational dynamics of the system under investigation. This connection can be expressed in a clearer way if the harmonic approximation (HA) is assumed. It is worth noting that the HA is included in the GA, but is actually stricter than the mere GA since, for example, the diffusive behavior is not described in the former approach while can be included (although only as continuous diffusion) in latter. It is straightforward to prove that under the HA, one writes [20]: $\begin{matrix} (7) & U_{j j}^{(n)} (t) = \sum_{g, \vec{q}} \frac{ℏ | σ_{j}^{(n)} (g, \vec{q}) |^{2}}{2 N M_{n} ω_{g, \vec{q}}} [n_{g, \vec{q}} exp (- i t ω_{g, \vec{q}}) + (n_{g, \vec{q}} + 1) exp (i t ω_{g, \vec{q}})], \end{matrix}$ where g and $\vec{q}$ label the phonon branch and the phonon wavevector in the first Brillouin zone, respectively, while ${\vec{σ}}^{(n)} (g, \vec{q})$ is the phonon polarization vector for the nth atomic species, $ω_{g, \vec{q}}$ the angular phonon frequency, and $n_{g, \vec{q}}$ the corresponding Bose occupation factor. This is exactly what is calculated in the combined use of a selected DFT code and aCLIMAX: the former provides the list of ${\vec{σ}}^{(n)} (g, \vec{q})$ and $ω_{g, \vec{q}}$ , while the latter evaluates $U_{j j}^{(n)} (t)$ , $S_{s}^{(n)} (\vec{Q}, ω)$ , and finally $Σ_{s} (Q, ω)$ , according to Eqs (7), (5), and (4), respectively. Strictly speaking, the multiphonon expansion of Eq. (5) is actually implemented via the power series development of the exponential $exp (\sum_{j} U_{j j}^{(n)} (t) Q_{j}^{2})$ : $\begin{array}{rcl} S_{s}^{(n)} (\vec{Q}, ω) & = & exp (- {\vec{Q}}^{T} {\underline{\underline{U}}}^{(n)} (0) \vec{Q}) \int_{- \infty}^{\infty} \frac{d t}{2 π ℏ} exp (- i ω t) \\ (8) & \times [1 + {\vec{Q}}^{T} {\underline{\underline{U}}}^{(n)} (t) \vec{Q} + \frac{1}{2} {({\vec{Q}}^{T} {\underline{\underline{U}}}^{(n)} (t) \vec{Q})}^{2} + \dots], \end{array}$ where $exp (- {\vec{Q}}^{T} {\underline{\underline{U}}}^{(n)} (0) \vec{Q})$ is the anisotropic Debye-Waller factor [20]. The first term to be Fourier transformed (i.e. the unity) gives rise to the elastic line, the second (i.e. that proportional to $Q^{2}$ ) to the fundamental bands, while the others contain all possible overtones and combinations and in aCLIMAX are computed up to the sixth quantum event, that is up to the Fourier transform of $\frac{1}{6!} {({\vec{Q}}^{T} {\underline{\underline{U}}}^{(n)} (t) \vec{Q})}^{6}$ . However, always in the framework of the HA, the exponential would in principle allow for the exact numerical calculation of all the multiphonon terms.

The aforementioned approach based on the harmonic phonon properties is formally correct, but, it is not always the best choice for studying the high energy-transfer spectral region in NVS. In Eq. (6) we have seen that the tensorial components $Γ_{j j}^{(n)} (t)$ can be related to the corresponding elements of the velocity autocorrelation tensor, but one can easily find that, due to the fluctuation-dissipation theorem [23], it is simpler to make use of the spectral representation of $⟨ v_{j}^{(n)} (0) v_{j}^{(n)} (τ) ⟩$ , namely $F_{j j}^{(n)} (ω)$ : $\begin{array}{rcl} F_{j j}^{(n)} (ω) & = & \frac{4 M_{n}}{π ℏ ω} \int_{0}^{\infty} Im ⟨ v_{j}^{(n)} (0) v_{j}^{(n)} (t) ⟩ sin (ω t) d t \\ (9) & = & \frac{4 M_{n}}{π ℏ ω} tanh (\frac{ℏ ω}{2 k_{B} T}) \int_{0}^{\infty} Re ⟨ v_{j}^{(n)} (0) v_{j}^{(n)} (t) ⟩ cos (ω t) d t, \end{array}$ which is related to $Γ_{j j}^{(n)} (t)$ via: $\begin{matrix} (10) & Γ_{j j}^{(n)} (t) = \frac{ℏ}{2 M_{n}} \int_{0}^{\infty} \frac{F_{j j}^{(n)} (ϵ)}{ϵ} {coth (\frac{ℏ ϵ}{2 k_{B} T}) [1 - cos (ϵ t)] - i sin (ϵ t)} d ϵ . \end{matrix}$ In case of no diffusion (i.e. $F_{j j}^{(n)} (0) = 0$ ), the previous equation can be simply split and becomes fully equivalent to: $\begin{array}{rcl} (11) & \begin{matrix} U_{j j}^{(n)} (0) = \frac{ℏ}{2 M_{n}} \int_{0}^{\infty} \frac{F_{j j}^{(n)} (ϵ)}{ϵ} coth (\frac{ℏ ϵ}{2 k_{B} T}) d ϵ; \\ U_{j j}^{(n)} (t) = \frac{ℏ}{2 M_{n}} \int_{0}^{\infty} \frac{F_{j j}^{(n)} (ϵ)}{ϵ} [coth (\frac{ℏ ϵ}{2 k_{B} T}) cos (ϵ t) + i sin (ϵ t)] d ϵ . \end{matrix} \end{array}$ The advantage of the $F_{j j}^{(n)} (ϵ)$ formalism over that used in Eq. (7) lies in the possibility to perform a short-time expansion in order to describe the high energy-transfer spectral range in a physically mindful method. The rationale of this approximation is of course very simple and is based on well known properties of the Fourier transform: the high frequency behavior of any Fourier-transformed signal is dominated by the short time behavior of the original signal. The practical side of this approximation consists in performing a double limit: $t \to 0$ and $| \vec{Q} | = Q \to \infty$ , while keeping the nuclear mean travelled path of the nth species, $s_{n} = ℏ Q t M_{n}^{- 1}$ , constant. Rewriting Eq. (10) in terms of $s_{n}$ and Q, one obtains: $\begin{array}{rcl} Γ_{j j}^{(n)} (t (s_{n}, Q)) & = & \frac{ℏ}{2 M_{n}} \int_{0}^{\infty} \frac{F_{j j}^{(n)} (ϵ)}{ϵ} {coth (\frac{ℏ ϵ}{2 k_{B} T}) [1 - cos (\frac{M_{n} s_{n} ϵ}{ℏ Q})] \\ (12) & - i sin (\frac{M_{n} s_{n} ϵ}{ℏ Q})} d ϵ, \end{array}$ that now can be expanded in power series for $Q^{- 1} \to 0$ as: $\begin{array}{rcl} Γ_{j j}^{(n)} (t (s_{n}, Q)) & \approx & \frac{ℏ}{2 M_{n}} \int_{0}^{\infty} \frac{F_{j j}^{(n)} (ϵ)}{ϵ} {coth (\frac{ℏ ϵ}{2 k_{B} T}) [\frac{1}{2} {(\frac{M_{n} s_{n} ϵ}{ℏ Q})}^{2} \\ (13) & - \frac{1}{4!} {(\frac{M_{n} s_{n} ϵ}{ℏ Q})}^{4}] - \frac{i M_{n} s_{n} ϵ}{ℏ Q} + \frac{i}{3!} {(\frac{M_{n} s_{n} ϵ}{ℏ Q})}^{3}} d ϵ . \end{array}$ Making use of the following sum rules for $F_{j j}^{(n)} (ω)$ [23,29]: $\begin{array}{rcl} (14) & \begin{matrix} \int_{0}^{\infty} F_{j j}^{(n)} (ω) d ω = 1; \\ \int_{0}^{\infty} F_{j j}^{(n)} (ω) ω coth (\frac{ℏ ϵ}{2 k_{B} T}) d ω = 2 ℏ^{- 1} M_{n} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩; \\ \int_{0}^{\infty} F_{j j}^{(n)} (ω) ω^{2} d ω = \frac{1}{M_{n}} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩; \\ \int_{0}^{\infty} F_{j j}^{(n)} (ω) ω^{3} coth (\frac{ℏ ϵ}{2 k_{B} T}) d ω = \frac{2}{M_{n} ℏ} ⟨ {(\frac{\partial V}{\partial r_{j}^{(n)}})}^{2} ⟩, \end{matrix} \end{array}$ where V stands for the system potential energy, one can work out the anisotropic formulas [10] for the atomic recoil ( $\propto s_{n}$ ), the impulse approximation ( $\propto s_{n}^{2}$ ), and its first two corrections ( $\propto s_{n}^{3}$ and $\propto s_{n}^{4}$ ), also known as final state effects. Plugging these relationships into Eq. (13) one gets the correct expansion of the tensor $Γ_{j j}^{(n)}$ : $\begin{array}{rcl} Γ_{j j}^{(n)} (t (s_{n}, Q)) & \approx & \frac{ℏ}{2 M_{n}} [- \frac{i M_{n} s_{n}}{ℏ Q} + \frac{M_{n}^{3} s_{n}^{2}}{ℏ^{3} Q^{2}} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩ + \frac{i}{6} \frac{M_{n}^{2} s_{n}^{3}}{ℏ^{3} Q^{3}} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ \\ (15) & - \frac{1}{12} \frac{M_{n}^{3} s_{n}^{4}}{ℏ^{5} Q^{4}} ⟨ {(\frac{\partial V}{\partial r_{j}^{(n)}})}^{2} ⟩], \end{array}$ which can be now easily recast into a function of t (short time expansion): $\begin{array}{rcl} Γ_{j j}^{(n)} (t) & \approx & - \frac{i ℏ t}{2 M_{n}} + \frac{t^{2}}{2} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩ + \frac{i}{12} \frac{ℏ t^{3}}{M_{n}^{2}} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ \\ (16) & - \frac{1}{24} \frac{t^{4}}{M_{n}^{2}} ⟨ {(\frac{\partial V}{\partial r_{j}^{(n)}})}^{2} ⟩ . \end{array}$ However, despite its interest, the cumulant expansion in Eqs (15) and (16) cannot be directly used for the evaluation of the high energy transfer part of the vibrational spectrum, since in this case the exponential $exp (- \sum_{j} Γ_{j j}^{(n)} (t) Q_{j}^{2})$ in Eq. (5) would be impossible to be Fourier transformed because of an $exp (a^{4} t^{4})$ -type divergence. On the contrary, retaining the complete form for the recoil and the impulse approximation terms, but developing in power series the final state effect ones, one reads the so called additive approach, reported here up to $Q^{- 2}$ [10,15]: $\begin{array}{rcl} exp (- {\vec{Q}}^{T} {\underline{\underline{Γ}}}^{(n)} (t (s_{n}, Q)) \vec{Q}) & \approx & exp [i \frac{Q s_{n}}{2} - \frac{M_{n}^{2} s_{n}^{2}}{2 ℏ^{2} Q^{2}} \sum_{j = 1}^{3} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩ Q_{j}^{2}] {1 - \frac{i}{12} \frac{M_{n} s_{n}^{3}}{ℏ^{2} Q^{3}} \\ \times \sum_{k = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{k}^{(n)})}^{2}} ⟩ Q_{k}^{2} + \frac{1}{24} \frac{M_{n}^{2} s_{n}^{4}}{ℏ^{4} Q^{4}} \sum_{l = 1}^{3} ⟨ {(\frac{\partial V}{\partial r_{l}^{(n)}})}^{2} ⟩ Q_{l}^{2} \\ (17) & - \frac{1}{288} {[\frac{M_{n} s_{n}^{3}}{ℏ^{2} Q^{3}} \sum_{m = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{m}^{(n)})}^{2}} ⟩ Q_{m}^{2}]}^{2}}, \end{array}$ or, equivalently, in the time domain: $\begin{array}{rcl} exp (- {\vec{Q}}^{T} {\underline{\underline{Γ}}}^{(n)} (t) \vec{Q}) & \approx & exp [i \frac{ℏ Q^{2} t}{2 M_{n}} - \frac{t^{2}}{2} \sum_{j = 1}^{3} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩ Q_{j}^{2}] {1 - \frac{i}{12} \frac{ℏ t^{3}}{M_{n}^{2}} \\ \times \sum_{k = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{k}^{(n)})}^{2}} ⟩ Q_{k}^{2} + \frac{1}{24} \frac{t^{4}}{M_{n}^{2}} \sum_{l = 1}^{3} ⟨ {(\frac{\partial V}{\partial r_{l}^{(n)}})}^{2} ⟩ Q_{l}^{2} \\ (18) & - \frac{1}{288} \frac{ℏ^{2} t^{6}}{M_{n}^{4}} {[\sum_{m = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{m}^{(n)})}^{2}} ⟩ Q_{m}^{2}]}^{2}} . \end{array}$ This expression can be now safely Fourier transformed via Eq. (5) providing an approximate analytic formula for the self scattering law in the framework of the additive approach, $S_{s, a d}^{(n)} (\vec{Q}, ω)$ [15,25]. In mathematical terms this procedure is known as Gram-Charlier expansion and in our case reads: $\begin{array}{rcl} S_{s, a d}^{(n)} (\vec{Q}, ω) & = & \frac{1}{\sqrt{2 π} ℏ σ^{(n)} (\vec{Q})} exp [- \frac{{(ω - \frac{ℏ Q^{2}}{2 M_{n}})}^{2}}{2 {(σ^{(n)} (\vec{Q}))}^{2}}] [1 + A_{3}^{(n)} (\vec{Q}) {He}_{3} (\frac{ω - \frac{ℏ Q^{2}}{2 M_{n}}}{σ^{(n)} (\vec{Q})}) \\ (19) & + A_{4}^{(n)} (\vec{Q}) {He}_{4} (\frac{ω - \frac{ℏ Q^{2}}{2 M_{n}}}{σ^{(n)} (\vec{Q})}) + A_{6}^{(n)} (\vec{Q}) {He}_{6} (\frac{ω - \frac{ℏ Q^{2}}{2 M_{n}}}{σ^{(n)} (\vec{Q})})], \end{array}$ where ${He}_{k} (\dots)$ stands for the (probabilists’) Hermite polynomial [2] of order k (namely, ${He}_{3} (x) = x^{3} - 3 x$ etc.), while the four coefficients $σ^{(n)} (\vec{Q})$ and $A_{3, 4, 6} (\vec{Q})$ are listed below: $\begin{array}{rcl} (20) & \begin{matrix} σ^{(n)} (\vec{Q}) = {[\sum_{j = 1}^{3} ⟨ {(v_{j}^{(n)} (0))}^{2} ⟩ Q_{j}^{2}]}^{1 / 2}; \\ A_{3}^{(n)} (\vec{Q}) = \frac{ℏ}{12 M_{n}^{2} {(σ^{(n)} (\vec{Q}))}^{3}} \sum_{j = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ Q_{j}^{2}; \\ A_{4}^{(n)} (\vec{Q}) = \frac{1}{24 M_{n}^{2} {(σ^{(n)} (\vec{Q}))}^{4}} \sum_{j = 1}^{3} ⟨ {(\frac{\partial V}{\partial r_{j}^{(n)}})}^{2} ⟩ Q_{j}^{2}; \\ A_{6}^{(n)} (\vec{Q}) = \frac{ℏ^{2}}{288 M_{n}^{4} {(σ^{(n)} (\vec{Q}))}^{6}} {[\sum_{j = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ Q_{j}^{2}]}^{2} . \end{matrix} \end{array}$ If the physicists’ version [2], $H_{k} (x)$ , of the Hermite polynomials (namely, $H_{3} (x) = 8 x^{3} - 12 x$ etc.) are preferred, then the conversion is straightforward, since: ${He}_{k} (x) = 2^{- k / 2} H_{k} (x / \sqrt{2})$ . In addition, Eq. (19) can be greatly simplified if the West scaling variable y is introduced and, in parallel, a West response function, $J (\vec{Q}, y)$ (weakly depending on the momentum transfer) is also defined [36]. For the nth atomic species one writes: $\begin{array}{rcl} (21) & \begin{matrix} y_{n} = \frac{M_{n}}{ℏ^{2} Q} (ℏ ω - \frac{ℏ^{2} Q^{2}}{2 M_{n}}); \\ J^{(n)} (\vec{Q}, y_{n}) = \frac{ℏ^{2} Q}{M_{m}} S_{s}^{(n)} (\vec{Q}, ω) . \end{matrix} \end{array}$ Thus expressing the $\vec{Q}$ direction by the new symbol $\vec{κ} = Q^{- 1} \vec{Q}$ , one writes: $\begin{array}{rcl} S_{s, a d}^{(n)} (\vec{Q}, ω) & = & \frac{M_{n}}{ℏ^{2} Q} \frac{1}{\sqrt{2 π} σ_{p}^{(n)} (\vec{κ})} exp [- \frac{{(y_{n})}^{2}}{2 {(σ_{p}^{(n)} (\vec{κ}))}^{2}}] [1 + Q^{- 1} B_{3}^{(n)} (\vec{κ}) H_{3} (\frac{y_{n}}{\sqrt{2} σ_{p}^{(n)} (\vec{κ})}) \\ (22) & + Q^{- 2} B_{4}^{(n)} (\vec{κ}) H_{4} (\frac{y_{n}}{\sqrt{2} σ_{p}^{(n)} (\vec{κ})}) + Q^{- 2} B_{6}^{(n)} (\vec{κ}) H_{6} (\frac{y_{n}}{\sqrt{2} σ_{p}^{(n)} (\vec{κ})})], \end{array}$ where now the dependence on Q is explicit, while the new four coefficients $σ_{p}^{(n)} (\vec{κ})$ and $B_{3, 4, 6} (\vec{κ})$ are intrinsic properties of the system although depending on the direction of the scattering process via $\vec{κ}$ . In particular, $σ_{p}^{(n)} (\vec{κ})$ is relevant since represents the standard deviation of the momentum distribution for the nth atomic species projected along $\vec{κ}$ . The expressions of the mentioned coefficients are listed below: $\begin{array}{rcl} (23) & \begin{matrix} σ_{p}^{(n)} (\vec{κ}) = ℏ^{- 1} {[\sum_{j = 1}^{3} ⟨ {(p_{j}^{(n)} (0))}^{2} ⟩ κ_{j}^{2}]}^{1 / 2}; \\ B_{3}^{(n)} (\vec{κ}) = \frac{\sqrt{2} M_{n}}{48 ℏ^{2} {(σ_{p}^{(n)} (\vec{κ}))}^{3}} \sum_{j = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ κ_{j}^{2}; \\ B_{4}^{(n)} (\vec{κ}) = \frac{M_{n}^{2}}{96 ℏ^{4} {(σ_{p}^{(n)} (\vec{κ}))}^{4}} \sum_{j = 1}^{3} ⟨ {(\frac{\partial V}{\partial r_{j}^{(n)}})}^{2} ⟩ κ_{j}^{2}; \\ B_{6}^{(n)} (\vec{κ}) = \frac{M_{n}^{2}}{2304 ℏ^{4} {(σ_{p}^{(n)} (\vec{κ}))}^{6}} {[\sum_{j = 1}^{3} ⟨ \frac{\partial^{2} V}{\partial {(r_{j}^{(n)})}^{2}} ⟩ κ_{j}^{2}]}^{2} . \end{matrix} \end{array}$

The last point of this short theoretical section to be dealt with is the problem of the spherical average implied by the powder nature of most of the NVS samples (see Eq. (4)): $\begin{matrix} (24) & {⟨ S_{s, a d}^{(n)} (\vec{Q}, ω) ⟩}_{\hat{Q}} = \frac{1}{4 π} \int d \hat{Q} S_{s, a d}^{(n)} (\vec{Q}, ω) . \end{matrix}$ Numerically it is relatively straightforward to calculate the integral in the previous equation, but it would be also interesting to write an approximate analytical formula for it, at least for its main component $S_{s, i a}^{(n)} (\vec{Q}, ω)$ : $\begin{array}{rcl} S_{s, i a}^{(n)} (\vec{Q}, ω) & = & \frac{1}{\sqrt{2 π} ℏ σ^{(n)} (\vec{Q})} exp [- \frac{{(ω - \frac{ℏ Q^{2}}{2 M_{n}})}^{2}}{2 {(σ^{(n)} (\vec{Q}))}^{2}}] \\ (25) & = & \frac{M_{n}}{ℏ^{2} Q} \frac{1}{\sqrt{2 π} σ_{p}^{(n)} (\vec{κ})} exp [- \frac{{(y_{n})}^{2}}{2 {(σ_{p}^{(n)} (\vec{κ}))}^{2}}], \end{array}$ that is related to the impulse approximation, since the angular dependence of the final state effect terms looks extremely complicated. In the most general case one has that $σ_{p}^{(n)} (\hat{x}) \neq σ_{p}^{(n)} (\hat{y}) \neq σ_{p}^{(n)} (\hat{z})$ , so it is convenient to introduce their mean square value ${({\bar{σ}}_{p}^{(n)})}^{2}$ : $\begin{matrix} (26) & {({\bar{σ}}_{p}^{(n)})}^{2} = \frac{1}{3} [{(σ_{p}^{(n)} (\hat{x}))}^{2} + {(σ_{p}^{(n)} (\hat{y}))}^{2} + {(σ_{p}^{(n)} (\hat{z}))}^{2}] = \frac{2 M_{n}}{3 ℏ^{2}} {⟨ E_{k} ⟩}_{n}, \end{matrix}$ where ${⟨ E_{k} ⟩}_{n}$ is the single particle mean kinetic energy of the nth atomic species. At this stage it is possible to write the spherical average of the IA term, ${⟨ S_{s, i a}^{(n)} (\vec{Q}, ω) ⟩}_{\hat{Q}}$ using once again the Gram–Charlier expansion: $\begin{array}{rcl} {⟨ S_{s, i a}^{(n)} (\vec{Q}, ω) ⟩}_{\hat{Q}} & = & \frac{M_{n}}{ℏ^{2} Q} \frac{1}{\sqrt{2 π} {\bar{σ}}_{p}^{(n)}} exp [- \frac{{(y_{n})}^{2}}{2 {({\bar{σ}}_{p}^{(n)})}^{2}}] [1 + c_{4}^{(n)} H_{4} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}}) \\ (27) & + c_{6}^{(n)} H_{6} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}}) + c_{8}^{(n)} H_{8} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}}) + \dots], \end{array}$ where some important differences with respect to Eq. (22) have to be noted:

(i) no additional Q dependence is now included in the various terms containing Hermite polynomials; (ii) only even Hermite polynomials are present, since the IA self scattering law is clearly symmetric as $y_{n}$ goes into $- y_{n}$ [36]; (iii) it is straightforward to prove the general results that $c_{0}^{(n)} = 1$ and $c_{2}^{(n)} = 0$ , so the present Gram–Charlier expansion starts with $H_{0} (x) = 1$ followed by $H_{4} (x)$ . The values of the first two meaningful Gram–Charlier coefficients are given below: $\begin{array}{rcl} (28) & \begin{matrix} c_{4}^{(n)} = \frac{1}{360 {({\bar{σ}}_{p}^{(n)})}^{4}} (η_{n}^{2} - η_{n} ζ_{n} + ζ_{n}^{2}); \\ c_{6}^{(n)} = \frac{1}{45360 {({\bar{σ}}_{p}^{(n)})}^{6}} (η_{n} - 2 ζ_{n}) (2 η_{n} - ζ_{n}) (η_{n} + ζ_{n}), \end{matrix} \end{array}$ where symbols $η_{n} = {(σ_{p}^{(n)} (\hat{y}))}^{2} - {(σ_{p}^{(n)} (\hat{x}))}^{2}$ and $ζ_{n} = {(σ_{p}^{(n)} (\hat{z}))}^{2} - {(σ_{p}^{(n)} (\hat{x}))}^{2}$ have been introduced for brevity. One can easily verify that in the isotropic case, i.e. $σ_{p}^{(n)} (\hat{x}) = σ_{p}^{(n)} (\hat{y}) = σ_{p}^{(n)} (\hat{z})$ , $c_{k > 0}^{(n)} = 0$ and a simple Gaussian peak shape is retained.

Putting together Eq. (22) and Eq. (27), and dealing with the final state effects in an approximate isotropic way, it is possible to finally obtain an effective formula for ${⟨ S_{s}^{(n)} ({\vec{Q}}_{F, B}, ω) ⟩}_{\hat{Q}}$ which is meaningful, for example, in the high energy-transfer range of a TICS (roughly $300 meV < ℏ ω < 500 meV$ and $11 Å^{- 1} < Q < 15 Å^{- 1}$ ): $\begin{array}{rcl} {⟨ S_{s}^{(n)} ({\vec{Q}}_{F, B}, ω) ⟩}_{\hat{Q}} & \approx & \frac{M_{n}}{ℏ^{2} Q_{F, B}} \frac{1}{\sqrt{2 π} {\bar{σ}}_{p}^{(n)}} exp [- \frac{{(y_{n})}^{2}}{2 {({\bar{σ}}_{p}^{(n)})}^{2}}] [1 + Q_{F, B}^{- 1} {\bar{b}}_{3}^{(n)} H_{3} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}}) \\ + (c_{4}^{(n)} + Q_{F, B}^{- 2} {\bar{b}}_{4}^{(n)}) H_{4} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}}) \\ (29) & + (c_{6}^{(n)} + Q_{F, B}^{- 2} {\bar{b}}_{6}^{(n)}) H_{6} (\frac{y_{n}}{\sqrt{2} {\bar{σ}}_{p}^{(n)}})] . \end{array}$ The isotropically-approximated coefficients ${\bar{b}}_{3 - 6}^{(n)}$ (related to the final state effects) are simply given by: $\begin{array}{rcl} (30) & \begin{matrix} {\bar{b}}_{3}^{(n)} = \frac{\sqrt{2} M_{n}}{144 ℏ^{2} {({\bar{σ}}_{p}^{(n)})}^{3}} ⟨ \nabla_{n}^{2} V ⟩; \\ {\bar{b}}_{4}^{(n)} = \frac{M_{n}^{2}}{288 ℏ^{4} {({\bar{σ}}_{p}^{(n)})}^{4}} ⟨ {({\vec{\nabla}}_{n} V)}^{2} ⟩; \\ {\bar{b}}_{6}^{(n)} = \frac{M_{n}^{2}}{20736 ℏ^{4} {({\bar{σ}}_{p}^{(n)})}^{6}} {⟨ \nabla_{n}^{2} V ⟩}^{2}, \end{matrix} \end{array}$ where the mean values of the potential energy Laplacian and square gradient are linked with the spectrum, $f^{(n)} (ω)$ [23], of the imaginary part of nth atom velocity autocorrelation function, $⟨ {\vec{v}}^{(n)} (0) \cdot {\vec{v}}^{(n)} (t) ⟩$ , by the following sum rules [23,29]: $\begin{array}{rcl} (31) & \begin{matrix} \int_{0}^{\infty} f^{(n)} (ω) d ω = 1; \\ \int_{0}^{\infty} f^{(n)} (ω) ω coth (\frac{ℏ ω}{2 k_{B} T}) d ω = \frac{4}{3 ℏ} {⟨ E_{k} ⟩}_{n}; \\ \int_{0}^{\infty} f^{(n)} (ω) ω^{2} d ω = \frac{1}{3 M_{n}} ⟨ \nabla_{n}^{2} V ⟩; \\ \int_{0}^{\infty} f^{(n)} (ω) ω^{3} coth (\frac{ℏ ω}{2 k_{B} T}) d ω = \frac{2}{3 M_{n} ℏ} ⟨ {({\vec{\nabla}}_{n} V)}^{2} ⟩ . \end{matrix} \end{array}$ Needless to say, that in the framework of the HA, $f^{(n)} (ω)$ coincides with the standard nth atom-projected density of phonon states [20]: $\begin{matrix} (32) & f^{(n)} (ω) = \frac{1}{N} \sum_{g, \vec{q}} {| {\vec{σ}}^{(n)} (g, \vec{q}) |}^{2} δ (ω - ω_{g, \vec{q}}) . \end{matrix}$

In the frequent case where the scattering contributions in Eq. (4) coming from the nuclei other than protons are fully negligible (with the possible exception of D), one can directly transform $Σ_{s} (Q, ω)$ into a macroscopic West response function $Φ (Q, y_{H})$ : $\begin{matrix} (33) & Φ (Q_{F, B}, y_{H}) = N r_{H} \frac{σ_{H}}{4 π} \sum_{n = 1}^{r_{H}} x_{n} {⟨ J_{s}^{(n)} ({\vec{Q}}_{F, B}, y_{H}) ⟩}_{\hat{Q}}, \end{matrix}$ where $r_{H}$ is now the number of non equivalent protons in the crystal unit cell. This representation of the scattering law in the high energy-transfer region is particularly convenient since, as shown in Eq. (29), only its final state effect terms depend on $Q_{F, B}$ and so the total $Φ (Q_{F, B}, y_{H})$ comes out to be weakly depending on the momentum transfer, making easier to compare data coming from different instruments (exhibiting distinct scattering angles and final neutron energies).

Before concluding this subject it is worth discussing a low-temperature approximation of the final state effects which is rather coarse but is still widely used in the context of the so-called neutron Compton scattering [36] because of the modest size of these effects. The nucleus hit by the scattered neutron is dealt with as particle in an isotropic three-dimensional harmonic ground state of an effective mean field potential, exhibiting a frequency $Ω_{m}$ . Under this assumption, for $T ≪ 0.5 k_{B}^{- 1} ℏ Ω_{m}$ , one simply writes: $\begin{array}{rcl} (34) & \begin{matrix} {⟨ E_{k} ⟩}_{n} = \frac{3 ℏ^{2}}{2 M_{n}} {({\bar{σ}}_{p}^{(n)})}^{2} \approx \frac{3}{4} ℏ Ω_{m}; \\ ⟨ \nabla_{n}^{2} V ⟩ \approx 3 M_{n} Ω_{m}^{2}; \\ ⟨ {({\vec{\nabla}}_{n} V)}^{2} ⟩ \approx \frac{3}{2} M_{n} ℏ Ω_{m}^{3} . \end{matrix} \end{array}$ Thus, after plugging these approximations into Eq. (30), it is finally possible to roughly express the final state effects as a function of the single-particle momentum variance only: $\begin{array}{rcl} (35) & \begin{matrix} {\bar{b}}_{3}^{(n)} \approx \frac{\sqrt{2}}{12} {\bar{σ}}_{p}^{(n)}; \\ {\bar{b}}_{4}^{(n)} \approx \frac{1}{24} {({\bar{σ}}_{p}^{(n)})}^{2}; \\ {\bar{b}}_{6}^{(n)} \approx \frac{1}{144} {({\bar{σ}}_{p}^{(n)})}^{2} . \end{matrix} \end{array}$ This is particularly valuable in neutron Compton scattering as ${\bar{σ}}_{p}^{(n)}$ is the aim of the experimental measurement once the final state effects have been removed, and so a fitting data reduction procedure can be devised without the need of external inputs.

3. Dealing with unwanted instrumental effects

As it is well known among neutron scattering spectroscopists, there exists a data analysis procedure which aims transforming detected neutron counts into the macroscopic scattering law that we have already introduced. In this respect a TICS makes no exception. However, as we have seen in the introduction, the largely incoherent nature of the scattering process, the $ℏ ω$ range and resolution, and the single value of Q coupled to each selected $ℏ ω$ value, represent a set of unique conditions which somehow influence the data analysis procedure suitable for a standard TICS. Thus it is natural on this class of instruments to have very simple data reduction routines, transforming the time-of-flight (τ) neutron counts of an individual tube j, $C_{j} (τ, θ_{j})$ (belonging to the detector block “F” or “B”), into the corresponding experimental macroscopic scattering law, ${\tilde{Σ}}_{j} (Q_{F, B}, ω)$ . This is very standard on inverted-geometry spectrometers and so we are not going to discuss this point [37]. The only issue which is important to point out is that, due to the detector block structure, all the ${\tilde{Σ}}_{j} (Q_{F, B}, ω)$ signals exhibiting the same label “F” or “B” are actually almost identical (in terms of ω, Q, and energy resolution), except for a constant scaling factor due to small solid angle differences. So on a TICS all these signals can be safely summed together giving rise to two functions $\tilde{Σ} (Q_{F, B}, ω)$ , which are simple three-columns histograms: $ℏ ω$ , $\tilde{Σ}$ , $δ \tilde{Σ}$ (i.e. energy transfer, experimental macroscopic self scattering law, and its uncertainty), since $Q_{F, B}$ are the cited monotonic functions of ω easy to be determined. In conclusion, independently of the number of detector tubes, data size, after performing a preliminary reduction procedure, is always extremely small.

Although the large majority of neutron vibrational spectroscopists perform no further data analysis, since they usually deal with $\tilde{Σ} (Q_{F, B}, ω)$ in the same qualitative way as it is currently done with Raman and Infrared spectra, it is in principle possible to set up a series of advanced analysis routines going through the standard steps common to all the inelastic neutron scattering measurements:

sample can signal and fast neutron background subtraction;

sample self-shielding correction;

multiple scattering evaluation and removal.

Obviously the last point turns out to be particularly relevant for the high energy-transfer spectral range. Given the peculiar detector arrangement on a TICS, it is customary to host the sample in flat rectangular can (generally thin and made of an aluminum alloy) perpendicular to the incoming neutron beam and rather larger than its cross section. This situation, although not particularly convenient in terms of single-to-multiple scattering ratio, allows for an approximate analytical calculation of the elements of the data analysis procedure [3,26]. In the rest of this section we will always assume a slab-like flat sample, larger than the beam cross section and contained between two walls of a thin can having a similar shape (i.e. a plane slab). Thus we can write the following relationships among the experimental macroscopic scattering law of the sample-plus-can system (i.e. the filled can),

{\tilde{Σ}}_{fc} (Q_{F, B}, ω)

, that of the empty can,

{\tilde{Σ}}_{ec} (Q_{F, B}, ω)

, and the theoretical macroscopic self scattering law of the mere sample,

Σ_{s, s} (Q_{F, B}, ω)

\begin{array}{l} {\tilde{Σ}}_{fc} (Q_{F, B}, ω) = T_{c} (0, E_{0}) S_{s} (θ_{F, B}, E, E_{0}) T_{c} (θ_{F, B}, E_{1}) (Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω)) \\ + T_{s} (0, E_{0}) T_{c} (0, E_{0}) W (Q_{F, B}, ω) + T_{s} (θ_{F, B}, E_{1}) T_{c} (θ_{F, B}, E_{1}) W (Q_{F, B}, ω) \\ (36) & + M_{fc} (θ_{F, B}, ω); \\ {\tilde{Σ}}_{ec} (Q_{F, B}, ω) = T_{c} (0, E_{0}) W (Q_{F, B}, ω) + T_{c} (θ_{F, B}, E_{1}) W (Q_{F, B}, ω) \\ + M_{ec} (θ_{F, B}, ω), \end{array}

where:

T_{x} (θ, E)

represents the neutron transmission of the slab x (i.e. sample or can wall) of neutrons with an energy E and forming an angle θ with the line perpendicular to the slab surface,

S_{s} (θ, E_{0}, E_{1})

is the self-shielding of the sample slab acting on neutrons with initial energy

E_{0}

and scattered with final energy

E_{1}

at the angle θ,

R_{F, B} (ω)

is the instrumental energy resolution function,

W (Q_{F, B}, ω)

is the signal coming from one can wall (not corrected for its own self-shielding), and finally

M_{y} (θ_{F, B}, ω)

stands for all the multiple scattering neutron counts from the system y (i.e. filled can or empty can) also including transmission and self-shielding effects. The simple mathematical formulas for transmission and self-shielding coefficients can be found in Refs. [3,26], even though they contain a strong energy dependence of the total scattering cross-section for the protons, which cannot be disregarded. In addition, it is worth mentioning that fast neutron background counts and the so-called “dark counts” (i.e. those recorded in presence of a total neutron absorber) have been ignored, since nowadays modern TICS’s tend to be rather free from this kind of problems. As for the instrumental energy resolution of a TICS, a Gaussian peak shape is generally considered as satisfactory [9], even though its standard deviation,

σ_{F, B}^{(r)}

, has to be slowly increasing as ω grows, so, strictly speaking, one is not dealing with a pure convolution:

\begin{array}{rcl} Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω) & = & \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} \frac{d ϵ}{σ_{F, B}^{(r)} (ϵ)} Σ_{s, s} (Q_{F, B}, ϵ) exp [- \frac{{(ω - ϵ)}^{2}}{2 {(σ_{F, B}^{(r)} (ϵ))}^{2}}] \\ (37) & \approx & \frac{1}{\sqrt{2 π}} \int_{ω_{m}}^{ω_{M}} \frac{d ϵ}{σ_{F, B}^{(r)} (ϵ)} Σ_{s, s} (Q_{F, B}, ϵ) exp [- \frac{{(ω - ϵ)}^{2}}{2 {(σ_{F, B}^{(r)} (ϵ))}^{2}}], \end{array}

where

ω_{m}

and

ω_{M}

are the lower and higher limit, respectively, of the energy transfer range experimentally accessible by the TICS under consideration.

If can walls are sufficiently thin then in the formula for the empty can scattering of Eq. (36), the multiple scattering term can be neglected and the wall term $W (Q_{F, B}, ω)$ can be determined and subtracted from ${\tilde{Σ}}_{fc} (Q_{F, B}, ω)$ after taking the appropriate transmission factors into account, giving rise to the pure sample scattering, ${\tilde{Σ}}_{ps} (Q_{F, B}, ω)$ : $\begin{array}{rcl} {\tilde{Σ}}_{ps} (Q_{F, B}, ω) & = & {\tilde{Σ}}_{fc} (Q_{F, B}, ω) - T_{s} (0, E_{0}) T_{c} (0, E_{0}) W (Q_{F, B}, ω) \\ (38) & - T_{s} (θ_{F, B}, E_{1}) T_{c} (θ_{F, B}, E_{1}) W (Q_{F, B}, ω) . \end{array}$ So at this stage it is possible to write: $\begin{array}{rcl} Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω) & \approx & T_{c}^{- 1} (0, E_{0}) S_{s}^{- 1} (θ_{F, B}, E, E_{0}) T_{c}^{- 1} (θ_{F, B}, E_{1}) ({\tilde{Σ}}_{ps} (Q_{F, B}, ω) \\ (39) & - M_{fc} (θ_{F, B}, ω)), \end{array}$ since the multiple scattering term due to the filled can (properly attenuated by the sample and the can walls), is, in general, not always negligible. Given this scenario, we are left with two distinct problems before fully exploiting Eq. (39):

the total scattering cross sections of the protons included in the sample show a strong energy dependence going from 82.03 barn (bound) for ultracold neutrons to 20.51 barn (free) for epithermal ones [27]. Unfortunately the detail of this general trend are unknown and ought to be either measured via neutron transmission experiments or simulated through a certain knowledge, at least rough, of the microscopic dynamics of all the $r_{H}$ non-equivalent protons (that is, by the way, the actual goal of the experimental measurement);

the multiple scattering term in Eq. (39) generated by the sample-plus-can system is to be determined and, generally, can be simulated only if the sample microscopic dynamics is already known in detail, for example, by means of its H-projected phonon densities of states for all the $r_{H}$ non-equivalent protons.

Before suggesting possible methods to circumvent these two problems, it is worthwhile to point out that in a properly performed neutron scattering experiment they ought not to be severe as their impact is a function of the sample (single) scattering power [11],

p_{1} (E_{0})

, which for a slab-like sample is defined as:

\begin{matrix} (40) & p_{1} (E_{0}) = T_{s} (0, E_{0}) t r v_{0}^{- 1} \sum_{n = 1}^{r} x_{n} σ_{n} (E_{0}), \end{matrix}

where t the slab thickness,

v_{0}

is the crystal cell volume and

σ_{n} (E_{0})

is the energy-dependent scattering (no absorption included!) cross section of the nth atomic species, which in the H (and D) case can be substantially different from the bound one. In other words, if

p_{1} (E_{0})

is lower than

10 %

[21] for all the usable

E_{0}

range, then the effects of both self-shielding (a), and multiple-to-single scattering ratio (b) are indeed modest. In this way these two problems can be dealt with by approximate methods. In particular, the issue of

σ_{H} (E_{0})

can be tackled in various way, for example using an integrable free gas model in which the fictitious nth-species proton gas exhibits an effective temperature

T_{n}^{*}

given by

\frac{2}{3 k_{B}} {⟨ E_{k} ⟩}_{n}

. Under this approach, one writes [34]:

\begin{matrix} (41) & σ_{H, n} (E_{0}) \approx \frac{σ_{b, H}}{4 x_{n}} [(x_{n} + \frac{1}{2 x_{n}}) erf (x_{n}) + \frac{1}{\sqrt{π}} exp (- x_{n}^{2})], \end{matrix}

where

x_{n} = {[E_{0} / (k_{B} T_{n}^{*})]}^{1 / 2}

. The previous expression exhibits the correct limit for

x_{n}

growing to infinite, i.e.

σ_{H, n} (E_{0}) \to σ_{b, H} / 4

, the well-known [20] free scattering cross section of the proton. However, in the opposite limit, due to the fictitious nature of the free gas model, the value of the bound one,

σ_{b, H}

, is not exactly recovered. For this reason, it is recommended to make use of the approximation in Eq. (41) only for

x_{n} > 1

, which is generally satisfied in the high-energy transfer zone under investigation.

The multiple scattering problem on a TICS is definitely more troublesome than self-shielding, since it has not been investigated in great detail so far. The only dedicated study to the author’s knowledge is contained in an unpublished internal report [16] dealing with experiments performed in 1986 on the TFXA spectrometer [22] at ISIS. Spectra from hexamethylene tetramine at $T = 17$ K have been recorded from three slab-like samples exhibiting quite different thicknesses (namely, $d_{1} = 1.3 mm$ , $d_{2} = 1.9 mm$ , and $d_{3} = 5.4 mm$ ) and, consequently, distinct transmission values: $T_{1} = 83 %$ , $T_{2} = 76 %$ , and $T_{3} = 46 %$ (with $σ_{H} (E_{0}) = 35$ barn associated to $E_{0} = 25 meV$ ). The reported experimental results, confirmed by analytical double scattering calculations, were somehow surprising:

(A) All the three spectra (named $S_{1, 2, 3} (Q_{B}, ω)$ ) showed basically the same features.

(B) The average values of the spectral ratios came very close to the respective ratios of $(1 - T_{1, 2, 3})$ , but somehow at variance with the equivalent $d_{1, 2, 3}$ ratios, especially in the case of $d_{3} / d_{1}$ : $\begin{array}{rcl} (42) & \begin{matrix} R_{2 : 1} (ω) = \frac{S_{2} (Q_{B}, ω)}{S_{1} (Q_{B}, ω)} = 1.4 \approx \frac{1 - T_{2}}{1 - T_{1}} = 1.41 < \frac{d_{2}}{d_{1}} = 1.46; \\ R_{3 : 1} (ω) = \frac{S_{3} (Q_{B}, ω)}{S_{1} (Q_{B}, ω)} = 3.2 \approx \frac{1 - T_{3}}{1 - T_{1}} = 3.18 < \frac{d_{3}}{d_{1}} = 4.15 . \end{matrix} \end{array}$

(C) The oscillations superimposed to the mentioned spectral ratios were really small in the case of $R_{2 : 1} (ω)$ , which appeared flat and unstructured, in spite of $(1 - T_{2}) = 24 %$ . On the contrary, $R_{3 : 1} (ω)$ , exhibited two noticeable dips and one hump in the region $25 meV < ℏ ω < 100 meV$ . These findings were explained by suggesting that the multiple scattering events on a TICS are actually dominated by the elastic processes, so that in the energy-loss spectral region one can observe, together with single inelastic scattering, double, triple or further scattering in which only one inelastic event has taken place. In this way, due to the incoherent nature of the scattering and the low-temperature crystalline samples used, one is practically unable to discriminate between genuine single scattering and multiple scattering. This holds up to certain limit [say, about $(1 - T_{n}) \approx 25 %$ ], after which double (and further) inelastic scattering events start to be visible even though the general spectral features remain clearly understandable, at least up to $(1 - T_{n}) \approx 50 %$ . We think that this interpretation is essentially correct, being confirmed by the mentioned double scattering simulations, and so it implies that multiple scattering on a TICS can be roughly understood as an extra spectroscopic contribution characterized by a Q resolution lower than that pertaining to single scattering. This happens because the presence of some elastic scattering events before and/or after the only inelastic one, makes the knowledge of the real inelastic scattering angle impossible to be detected. However, since on TICS’s $\vec{Q}$ is generally dominated by ${\vec{k}}_{0}$ , the complete missing information on the inelastic scattering angle does not entail the total ignorance about the Q magnitude, but rather an uncertainty comparable to the typical distance existing between, say, $Q_{B} (ω)$ and $Q_{F} (ω)$ which is indeed very modest. Giving this scenario and focusing on Fig. 2 in Ref. [16], one can see that high energy-transfer spectral range of interested is basically flat both in $R_{2 : 1} (ω)$ and $R_{3 : 1} (ω)$ . It is just the overall intensity that is to be corrected, since it does not exactly scale as the sample thickness, but rather as $(1 - T_{n})$ . Thus a possible heuristic correction scheme for TICS data, in order to get rid of multiple scattering contributions in the high energy-transfer region, could be the following: $\begin{array}{rcl} Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω) & \approx & T_{c}^{- 1} (0, E_{0}) S_{s}^{- 1} (θ_{F, B}, E, E_{0}) T_{c}^{- 1} (θ_{F, B}, E_{1}) {\tilde{Σ}}_{ps} (Q_{F, B}, ω) \\ (43) & \times \frac{p_{1} (E_{0})}{1 - T_{s} (0, E_{0})}, \end{array}$ which holds only in the case that the neutron absorption of the sample is fully negligible. If this is not the case then $(1 - T_{s} (0, E_{0}))$ has to be replaced by the total scattering power, $p_{t} (E_{0})$ , that generalizes Eq. (40): $\begin{matrix} (44) & p_{t} (E_{0}) = T_{s} (0, E_{0}) [exp (t r v_{0}^{- 1} \sum_{n = 1}^{r} x_{n} σ_{n} (E_{0})) - 1], \end{matrix}$ so that the ratio $p_{1} (E_{0}) / (1 - T_{s} (0, E_{0}))$ becomes: $\begin{matrix} (45) & \frac{p_{1} (E_{0})}{p_{t} (E_{0})} = \frac{t r v_{0}^{- 1} \sum_{n = 1}^{r} x_{n} σ_{n} (E_{0})}{exp (t r v_{0}^{- 1} \sum_{n = 1}^{r} x_{n} σ_{n} (E_{0})) - 1} . \end{matrix}$ Plugging this result into Eq. (43) one gets the general approximate formula which completes the aim of the present section: $\begin{array}{rcl} (46) & \begin{matrix} Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω) \approx & T_{c}^{- 1} (0, E_{0}) S_{s}^{- 1} (θ_{F, B}, E, E_{0}) T_{c}^{- 1} (θ_{F, B}, E_{1}) {\tilde{Σ}}_{ps} (Q_{F, B}, ω) \\ \times \frac{μ (E_{0}) t}{exp (μ (E_{0}) t) - 1}, \end{matrix} \end{array}$ where the symbol $μ (E_{0}) = r v_{0}^{- 1} \sum_{n = 1}^{r} x_{n} σ_{n} (E_{0})$ has been introduced. This formula can be easily evaluated in our range of interest making use of Eq. (41) and keeping in mind that $Σ_{s, s} (Q_{F, B}, ω)$ still exhibits an arbitrary overall intensity and so is proportional to the macroscopic self scattering law, $Σ_{s} (Q_{F, B}, ω)$ , via an instrumental constant, C.

Fig. 1.

Values spanned by the West variable $y_{H}$ in the energy transfer interval $250 meV < ℏ ω < 500 meV$ on the crystal-analyzer inverted-geometry TOSCA-II spectrometer [9], both in backscattering (dotted line) and in forward-scattering (full line). The corresponding variation of the wave-vector transfer for the backscattering detectors, $Q_{B}$ , is also reported.

4. A new spectral normalization procedure

In the previous section we have seen how to treat experimental data obtained from a TICS measurement in order to derive the macroscopic scattering law (still broadened by the energy resolution effects) in the high energy transfer range (say, $250 meV < ℏ ω < 500 meV$ ). The only unknown quantities needed for such a data reduction procedure are the values of the mean kinetic energy of the non-equivalent protons included in measured sample. At this stage we are ready to apply the theoretical model devised in Section 2 to the experimental $Σ_{s, s} (Q_{F, B}, ω) \otimes R_{F, B} (ω)$ data. However, before taking this step it is worthwhile to investigate the kinematic range of the aforementioned energy transfer interval, i.e. the Q and $y_{H}$ values associated to ω for both backscattering and forward-scattering detectors. A plot making use of the TOSCA-II parameters [9] is reported in Fig. 1, where one can appreciate how small is the $y_{H}$ variation (both in backscattering and in forward-scattering) in the wide energy transfer interval considered. One finds the following figures for the mean and standard deviation of the West variable: ${\bar{y}}_{H, F} \pm δ y_{H, F} = (0.870 \pm 0.006) Å^{- 1}$ and ${\bar{y}}_{H, B} \pm δ y_{H, B} = (- 0.990 \pm 0.004) Å^{- 1}$ in forward-scattering and backscattering, respectively. This effect is just a consequence of the small values of $E_{1}$ (compared to $ℏ ω$ ), and so it is not limited to TOSCA-II, but, on the contrary, is present on all the TICS’s. The standard deviation $δ y_{H}$ looks particularly minute if compared to the $y_{H}$ resolution (FWHM), $Δ y_{H}$ (e.g. on TOSCA-II): $\begin{matrix} (47) & Δ y_{H} = {[{(\frac{\partial y_{H}}{\partial E_{0}} Δ E_{0})}^{2} + {(\frac{\partial y_{H}}{\partial E_{1}} Δ E_{1})}^{2} + {(\frac{\partial y_{H}}{\partial θ} Δ θ)}^{2}]}^{1 / 2} . \end{matrix}$ Making use of typical instrumental values for the uncertainties (FWHM) in the high energy-transfer range, namely: $5.53 meV < Δ E_{0} < 14.73 meV$ , $Δ E_{1} \approx 0.18 meV$ , and $Δ θ \approx {8.57}^{\circ}$ , one can work out an estimate of $Δ y_{H}$ : $0.142 Å^{- 1} < Δ y_{H, F} < 0.149 Å^{- 1}$ and $0.103 Å^{- 1} < Δ y_{H, B} < 0.108 Å^{- 1}$ in forward-scattering and backscattering, respectively. These corresponds to standard deviations ( $\approx Δ y_{H} / 2.35482$ ) of one order of magnitude larger than $δ y_{H}$ . Thus, given the present scenario and fixed the scattering angle value, we will assume in what follows that in the whole range $250 meV < ℏ ω < 500 meV$ , $y_{H}$ exhibits a constant value. On TOSCA-II this corresponds to: $y_{H, F} = (0.87 \pm 0.06) Å^{- 1}$ and $y_{H, B} = (- 0.99 \pm 0.04) Å^{- 1}$ .

Going back to Eqs (4) and (29) and making use of the constancy of $y_{H}$ in the high energy range, one can easily conclude that, if our treatment of the IA plus the final state effects holds, then the macroscopic self scattering law can be simply written, following Eq. (33), in the approximate form: $\begin{matrix} (48) & Σ_{s} (Q_{B (F)}, ω) \approx N r_{H} \frac{σ_{H}}{4 π} \sum_{n = 1}^{r_{H}} x_{n} (\frac{j_{n, B (F)}}{Q_{B (F)}} + \frac{l_{n, B (F)}}{Q_{B (F)}^{2}} + \frac{g_{n, B (F)}}{Q_{B (F)}^{3}}), \end{matrix}$ where the subscripts “ $B (F)$ ” stand for backscattering (or forward scattering), and the three types of coefficients, namely $j_{n, B (F)}$ , $l_{n, B (F)}$ , and $g_{n, B (F)}$ , assuming the validity of Eq. (35), depend only on $M_{H}$ and the nth proton momentum standard deviation, ${\bar{σ}}_{p}^{(n)}$ . This relationship is easily computable and testable (after including a proper Q resolution treatment), and, if experimentally verified at least in an approximate way, would allow for a new neutron data normalization (i.e. determining N or, to be more precise, $C N$ , since the instrumental constant is in principle unknown), fully independent of the typical additional measurements on calibration samples. It is well known that in neutron spectroscopy this is not at all a trivial problem, and especially in the case of vibrational applications, is generally solved by the study of the so-called “elastic line” at $ω = 0$ . However, the elastic line analysis exhibits two major problems: (1) it implies the knowledge of various Debye-Waller factors, at least those of the various protons present in the sample under investigation; (2) it assumes that the coherent contributions (i.e. Bragg peaks in crystalline solids) coming from the container or the sample itself are either negligible or very precisely known. On the contrary, the new normalization procedure just proposed is largely immune from these problems, since in the mentioned high energy range coherent and incoherent scattering contributions are practically identical, and so proton scattering contributions are always extremely more intense than those coming from any other nucleus. Surely, reasonable estimates of ${\bar{σ}}_{p}^{(n)}$ have to be independently determined before performing the new normalization procedure.

Fig. 2.

Experimental neutron spectra of solid urea at $T = 20.4 K$ measured on TOSCA-I. Full squares stand for raw sample-plus-cell data, ${\tilde{Σ}}_{fc} (Q_{B}, ω)$ , dotted line for raw empty-cell data, ${\tilde{Σ}}_{ec} (Q_{B}, ω)$ , while empty circles represent pure-sample data, ${\tilde{Σ}}_{ps} (Q_{B}, ω)$ (see main text for details). In addition, fully corrected single scattering data, $Σ_{s, s} (Q_{B}, ω) \otimes R_{B} (ω)$ , have been reported as a full line.

Fig. 3.

Experimental response function, $Φ_{ex} (y_{H}, Q_{B})$ , from solid urea (see main text for details) plotted as a function of $Q_{B}$ for $y_{B} = - 1.00 Å^{- 1}$ (empty circles line with error bars) together with its best fits using the approximate impulsive model (dashed line) and the complete vibrational calculation (full line).

5. A practical example: Low-temperature solid urea

We are now in the position to compare our treatment of the high energy part of a TICS spectrum with real neutron scattering data. In what follows we will deal with solid urea (i.e. CO(NH₂)₂) measured in backscattering on TOSCA-I [6] by Hudson and co-workers [18] at a temperature $T = 20.4 K$ . This instrument was characterized by a final energy $E_{0} = 3.51 meV$ and a single scattering angle $θ_{B} = {136.03}^{\circ}$ , which implies an $y_{H, B} = (- 1.00 \pm 0.04) Å^{- 1}$ as explained in the previous section. The sample exhibited a mass $m = 2.38 g$ and was about $2 mm$ thick. Its raw inelastic neutron spectra, ${\tilde{Σ}}_{fc} (Q_{B}, ω)$ , is reported in Fig. 2 together with ${\tilde{Σ}}_{p c} (Q_{B}, ω)$ and $Σ_{s, s} (Q_{B}, ω) \otimes R_{B} (ω)$ , obtained following the treatment reported in Section 3. In order to accomplish these two transformations an approximate estimate of solid urea neutron cross section was needed and has been obtained using Eq. (41) which, in turn, required the mean kinetic energy values of its composing H atoms. These physical quantities have been calculated via standard lattice dynamics simulation as explained below. The high-energy part of $Σ_{s, s} (Q_{B}, ω) \otimes R_{B} (ω)$ can be simply recast as an experimental response function, $Φ_{ex} (y_{H}, Q_{B})$ : $\begin{matrix} (49) & Φ_{ex} (y_{H}, Q_{B}) = \frac{ℏ^{2} Q_{B}}{M_{H}} Σ_{s, s} (Q_{B}, ω) \otimes R_{B} (ω), \end{matrix}$ plotted in Fig. 3 as a function of $Q_{B}$ , since $y_{H}$ is fixed to $- 1 Å^{- 1}$ for $250 meV < ℏ ω < 500 meV$ , as we have just seen. At this stage experimental data in the form of response function can be compared with equivalent physical quantities obtained through two different routes:

one exact but computationally demanding, $Φ_{GA} (y_{H}, Q_{B})$ , which is given by the full GA and includes all the non-equivalent scattering atoms [making use of Eqs (4) and (5)];

the other approximate but relatively straightforward, $Φ_{IA} (y_{H}, Q_{B})$ , i.e. the impulse approximation (plus some simplified final state effect corrections) treating only the non-equivalent protons, which exploits Eq. (48).

Both curves, reported in Fig. 3, are compared with the experimental spectrum

Φ_{ex} (y_{H}, Q_{B})

after estimating the respective best

CN

constants, namely

{(CN)}_{GA}

and

{(CN)}_{IA}

. So the purpose of this practical example is to show that, if the approximate approach is used instead of the exact one, the value of the fitted

{(CN)}_{IA}

is still reasonable when compared to

{(CN)}_{GA}

The full GA model for the response function of solid urea, $Φ_{GA} (y_{H}, Q_{B})$ , has been calculated using Eq. (49) after replacing the experimental $Σ_{s, s} (Q_{B}, ω)$ with the GA-calculated one, $Σ_{s} (Q_{B}, ω)$ , following Eqs (4) and (5). The latter physical quantity has been derived making use of the aforementioned mean displacement tensors, $U_{j j}^{(n)} (t)$ , which, in turn, needed input data containing all the relevant vibrational eigenvalues $ω_{g, \vec{q}}$ and eigenvectors ${\vec{σ}}_{g, \vec{q}}^{(n)}$ , as shown in Eq. (7). This vibrational data set has been obtained from lattice dynamics calculations performed through the general utility lattice program GULP [14], making use of a simple semi-empirical force field [12]. The initial structure, derived from low temperature crystallographic data [32], was relaxed under the constraints of the space group symmetry $P \bar{4} 2_{1} m$ giving rise to the following lattice parameters: $a = 5.549 (1) Å$ and $c = 4.696 (1) Å$ which compare well with the experimental ones: $a = 5.565 (1) Å$ and $c = 4.685 (1) Å$ [32]. Subsequently $ω_{g, \vec{q}}$ and ${\vec{σ}}_{g, \vec{q}}^{(n)}$ have been evaluated for all the 48 phonon branches g, and for $20 \times 20 \times 20$ $\vec{q}$ -points belonging to the first Brillouin zone. Final results, in the form of macroscopic self scattering law and including both the instrumental resolution broadening and the powder average, have been reported in Fig. 4 together with the single- and multi-phonon spectral components as well as the total density of phonon states of the urea crystal. Subsequently, the simulated West response function has been evaluated and scaled through the constant $CN$ so to optimize the agreement with $Φ_{ex} (y_{H}, Q_{B})$ . It turned out that the best fit was obtained with ${(CN)}_{GA} = 24.2 \pm 5.0$ . The scaled GA spectrum ${(CN)}_{GA} Φ_{GA} (y_{H}, Q_{B})$ is plotted in Fig. 3.

Fig. 4.

The simulated macroscopic self scattering law of urea crystal (including both instrumental resolution broadening and powder average) is reported in the lower panel as a full line. In addition, the single-phonon spectral component appears in the upper panel as a full line, where the multiphonon components are also plotted as a dashed line. Finally, in the inset the simulated total density of phonon states is also represented.

As for the approximate IA model of solid urea, $Φ_{IA} (y_{H}, Q_{B})$ has been evaluated as in Eq. (49) after replacing the experimental $Σ_{s, s} (Q_{B}, ω)$ with the IA-calculated one, making use of Eq. (29) in conjunction with Eq. (30). The two types of non-equivalent protons, namely $H (1)$ and $H (2)$ , have been separately considered, with the former exhibiting an H-bond (towards an O atom) of about 1.90 Å, while the latter of 2.06 Å. From the projected densities of phonon state for $H (1)$ and $H (2)$ , $f^{(H 1)} (ω)$ and $f^{(H 2)} (ω)$ , which have been derived from the mentioned lattice dynamics calculations, the following values of $⟨ E_{k} ⟩$ have been worked out: $150.708 meV$ and $168.932 meV$ , respectively. These values correspond to the following estimates of the average momentum distribution standard deviations: ${\bar{σ}}_{p}^{(H 1)} = 4.92200 Å^{- 1}$ and ${\bar{σ}}_{p}^{(H 2)} = 5.21110 Å^{- 1}$ . As for the final state effects still $f^{(H 1)} (ω)$ and $f^{(H 2)} (ω)$ have been used in connection with Eq. (31) to estimate ${\bar{b}}_{3}^{(H 1, H 2)}$ , ${\bar{b}}_{4}^{(H 1, H 2)}$ , and ${\bar{b}}_{6}^{(H 1, H 2)}$ . Concerning the anisotropic corrections, the first two coefficients, $c_{4}^{(H 1, H 2)}$ and $c_{6}^{(H 1, H 2)}$ have been calculated for both $H (1)$ and $H (2)$ . Finally, the heavy atoms (i.e. C, N and O), given their weak signals, have been completely neglected. Computational results, not including the instrumental resolution broadening which was found to be totally irrelevant, have been scaled through the constant $CN$ so to optimize the agreement with $Φ_{ex} (y_{H}, Q_{B})$ . It turned out that the best fit was obtained with ${(CN)}_{IA} = 23.1 \pm 4.8$ , which is in good agreement with ${(CN)}_{GA}$ . The scaled IA spectrum ${(CN)}_{IA} Φ_{IA} (y_{H}, Q_{B})$ is reported in Fig. 3.

Fig. 5.

Solid urea at $T = 20.4$ K: comparison among the experimental fully-corrected single scattering data (empty circles) as in Fig. 2, the properly scaled ( $CN = 24.2$ ) simulated macroscopic self scattering law (full line) as in lower panel of Fig. 4, and the equivalent quantity ( $CN = 23.1$ ) obtained via the simple impulsive approximation (dashed line).

6. Final comments and conclusions

In this ending section we will try to rationalize from a physical point of view our findings concerning the evaluation of the normalization constant $CN$ . In other words, one could wonder why both a free recoil model based on the impulse approximation and a lattice dynamics harmonic calculation (in conjunction with the Gaussian approximation) provided similar results, despite the fact that they are very different from each other, and, strictly speaking, none of them was able to exactly describe the experimental spectra in the whole energy range $2 meV < ℏ ω < 500 meV$ , as clearly shown in Fig. 5. While in the IA case the reason of this incapability is quite obvious, since the impulsive scattering law does not contain (besides the proton mean kinetic energy values) any information about the detailed distribution of the vibrational frequencies of the crystal, in the lattice dynamics case the question is more subtle and cannot be dealt with here. However, the readership interested in a better, although still not perfect, simulation of the urea experimental neutron spectra, should consult Ref. [18] where the problem has been tackled using advanced ab initio methods. Going back to our initial question, i.e. why completely distinct models of the scattering law produce very similar estimates of $CN$ , it is possible to find a qualitative explanation keeping in mind our considerations about the multiphonon components. As a matter of fact, observing Fig. 4 it is possible to appreciate (although the aforementioned coarseness of the lattice dynamics scattering model) that 99.34% of the signal in the range $250 meV < ℏ ω < 500 meV$ is represented by multiphonon components, that originate from all the various overtones and combinations of lattice phonons, molecular librations and internal vibrations. In addition, the multiphonon components, as Q grows along with ω, become progressively richer of scattering events with large P (i.e. the number of excited phonon quanta) since the strength of these terms is proportional to $Q^{2 P} / (P!)$ [20,21]. Now, considering the very general result shown first by Sjölander [30] and then proved by Warner et al. [35], i.e. that as P grows the multiphonon component on any incoherent neutron scattering spectrum becomes gradually closer to a Gaussian functional form owing to the central limit theorem, it is easy to understand that the impulsive model, being intrinsically Gaussian, is just another form of the Sjölander approximation for the multiphonon terms, once the single-phonon ones are irrelevant. So what is really important is only the fact the different scattering laws compared share the same value of single proton mean kinetic energy, which is proportional to the square of the Gaussian width.

In conclusion, we have shed some new light on the high energy transfer (say, $250 meV < ℏ ω < 500 meV$ ) part of a typical neutron scattering spectrum from a protonated molecular crystal measured at low temperature on one of the inverse-geometry crystal-analyzer spectrometers mainly devoted to vibrational studies (e.g. TOSCA or VISION). We have seen that the standard spectroscopic information available in this data range is very modest indeed, but, on the other hand, it can be used to perform a spectral normalization which is completely independent of the usual procedure based on the so called “elastic line” analysis, i.e. the study of region close to $ω = 0$ , which is neither always possible, nor convenient. Subsequently, a simple free gas model based on the impulse approximation has been presented and studied in detail, followed by approximate methods for quickly correcting experimental data for the main unwanted effects (empty container signal, self shielding, multiple scattering contaminations etc.). Finally, a practical example based on solid crystalline urea data has been presented and the new proposed normalization procedure has been tested comparing the results of a complete lattice dynamics calculation with those from the impulse approximation model. The two normalization constants obtained turned out to be practically identical (given their statistical uncertainty) showing that the new method is solid because very weakly dependent on the selected scattering law models. Thus neutron spectroscopists are strongly encouraged to apply it to their various sets of experimental data in order to perform further tests and, in case, to try including deuterated molecular crystals which have not been considered in the present study.

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