Neutron scattering studies for analysing solid-state hydrogen storage

6 Neutron scattering studies for analysing solid-state hydrogen storage D. K. R O S S, University of Salford, UK

6.1

Introduction

In the search for useful hydrogen storage materials, whether solid state or H2 adsorbed on surfaces, the most immediately relevant data are, of course, the macroscopic thermodynamics and kinetics of the ab(ad)sorption/desorption process. However, if we seek to use scientific reasoning to guide our search, it is clearly necessary to understand the behaviour of hydrogen on the atomic scale both within the material and when entering or leaving it. It is in the latter context that thermal neutron scattering can make a crucial contribution, particularly when linked to ab initio modelling. Neutron scattering is a widely used technique for studying the structure and dynamics of solids and liquids. This is because a thermalised neutron emerging from a moderator at a nuclear reactor or pulsed neutron source has a wavelength that is comparable to interatomic distances and an energy that is comparable to the energies of atomic excitations in the solid. Moreover, we have good techniques for measuring diffraction patterns and energy transfers very accurately. There are many cases where neutron diffraction (ND) has significant advantages over X-ray or electron diffraction while inelastic neutron scattering (INS) often has advantages over IR and Raman measurements, for instance, in the measurement of phonon dispersion curves in single crystal samples. Other areas of application include small angle neutron scattering (SANS) for measuring longer range spatial correlations, where neutrons have many advantages over X-rays and quasi-elastic neutron scattering (QENS), which probes the diffusive motions, particularly of hydrogen, in a unique way. One notable property of the neutron, which we will not be particularly concerned with here, is its magnetic moment which allows a whole array of methods for investigating the magnetic structure and dynamics of solids. General textbooks are available on the subject, such as Squires [1]. A full description of quasi-elastic neutron scattering may be found in the book by Bee [2]. 135

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6.2

The neutron scattering method

6.2.1

Advantages of the neutron scattering technique for investigating solid-state hydrogen storage systems

The particularly significant contribution that neutron scattering can make to the study of hydrogen in solids arises from a number of factors: •

•

•

• •

• •

The neutron scattering cross-section of the proton (80 × 10–28 m–2 or, colloquially, 80 barns) is between one and two orders of magnitude greater than that of any other isotope so that it is easy to detect hydrogen in quite low concentrations in solids. The hydrogen cross-section is predominantly incoherent (σinc = 79.7 b, σcoh = 1.8 b) for reasons described below and so the measured scattering is dominated by scattering from individual atoms summed over all H atoms present. Incoherent inelastic scattering from polycrystalline solids reproduces the frequency distribution of the hydrogen vibrations in a solid, weighted with the square of the hydrogen vibration amplitudes and inversely with the mass of the hydrogen atom and so normally other elements can be ignored in a solid containing hydrogen. It should be noted that this amplitude-weighted density of vibrational states can also be directly derived from ab initio simulations of the solid’s dynamics and hence provides a direct check on the accuracy of these calculations. Moreover, the enthalpy change on hydriding/dehydriding can be estimated using this density of vibrational states. This technique is particularly useful for electrical conductors because IR and Raman techniques are not available. However, the technique is still very valuable for insulators because the electromagnetic techniques only provide frequencies at the zone centre and the intensities of these features are strongly influenced by the selection rules involved in the electronic response. Incoherent quasi-elastic neutron scattering from diffusing hydrogen atoms allows us to investigate the elementary jumps involved in the tracer diffusion process. On the other hand, the deuteron (2D), has a relatively large cross section which is predominantly coherent (σcoh = 5.6 b and σinc = 1.8 b) and so the diffraction pattern from a crystalline solid contains terms that enable us to locate D in crystal structures through the interference between scattered waves from all pairs of nuclei. H and D have scattering lengths of opposite sign so that there are a whole range of phase contrast techniques that can identify the role of hydrogen, for instance in SANS. A general advantage of neutrons is that they are relatively penetrating so

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that samples can be contained in cans that are able to withstand considerable gas pressures and temperatures. This feature enables us to measure the actual process of (de)hydrogenation in situ under working hydrogen pressures and so we can directly investigate processes that determine the kinetics of hydrogen cycling into and out of a hydrogen store. Being the lightest isotope, hydrogen has very pronounced quantum mechanical properties. As the neutron interacts with the nucleus, it directly observes these properties, formally observing the transition state probability between two quantum states of the solid. Of particular importance to us here is the nature of the scattering from para- and ortho-molecular hydrogen because selection rules require that these molecules have antisymmetric total wave functions. Hence we find that para-hydrogen, with anti-parallel spins has only even rotational quantum numbers (J) while ortho-hydrogen, with parallel spins can only have odd values of J. Moreover, the scattering lengths combine in such a way that para-hydrogen only scatters significantly via a spin-flip process involving a J = 0 to 1 transition while ortho-hydrogen has large cross-sections for both spinflip and non-spin-flip scattering. Also, the energy difference between the lowest energy para-state (J = 0) and the lowest ortho state (J = 1) for free hydrogen molecules is 14.7 meV, which is easily measured with high accuracy. These features make it instantly possible to identify whether hydrogen is present in its molecular form and to probe the trapping states with progressively less trapping energy as hydrogen is added. The basic theory of neutron scattering from molecular hydrogen has been described by Young and Koppel [3].

The application of neutron scattering to understanding the role of hydrogen in solids has been described in various general reviews [4,5]. Specific applications to intermetallic compounds are described by Richter et al. [6]. Its use in vibrational spectroscopic studies in chemistry, biology, materials science and catalysis are described by Mitchell et al. [7]. For a review of the basic properties of metal–hydrogen systems, we would refer the reader to the book by Fukai [8].

6.3

Studies of light metal hydrides

Because of the need to store hydrogen in a system with a low mass so that it can approach the US Department of Energy (DoE) gravimetric target of 6– 9% of the storage system mass, we will be interested in applying the technique to complex metal hydrides such as the alanates, borohydrides and amides and to intermetallic hydrides based on magnesium. In these novel systems, various interesting features can be investigated such as the existence of continuous phase transitions as have been observed in in situ diffraction

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studies of the lithium amide system [9,10] and in similar investigations into the hydrogen absorption/desorption process in alanates. Borohydrides are also of interest but from our point of view suffer from the disadvantage that samples would have to be made from the 11B isotope because of the strong neutron capture cross-section of 10B.

6.4

Studies of molecular hydrogen trapping in porous materials

The alternative proposed method of storing hydrogen in a solid matrix involves the physisorption of H2 on a high surface area material. The original interest in this approach was based on some very promising reports of high mass uptakes in single-walled carbon nanotubes [11] and carbon nanofibres [12]. Although these results seem to have been affected by residual water in the vacuum system, there have since been a wide range of experiments on a variety of carbon frameworks, zeolites, ice clathrates, metal oxide frameworks (MOFs), etc. While the normal van der Waals adsorption energy is rather weak (~ 40 meV/atom) and would require that the adsorber was cooled to 80 K to get a reasonable uptake, recent work suggests that it may be possible to enhance this adsorption energy with suitable catalysts, possibly involving spillover effects. The work to be described here (Section 6.7) reported below shows how neutron inelastic scattering can be used to characterise the potential energy surface at different trapping sites in a solid.

6.5

The basic theory of neutron scattering

The neutron is an uncharged nucleon with a mass essentially the same as a proton, a spin of 1/2 and a magnetic moment of –1.913 nuclear bohr magnetons. As we can neglect consideration of the magnetic interaction, being only interested here in the behaviour of hydrogen, we can confine our attention to the neutron–nucleus interaction. Full discussion of the theory can be found elsewhere [1,2] so the present treatment is considerably simplified with emphasis on physical principles rather than mathematical rigour. The probability of a neutron interacting with a nucleus is given in terms of the microscopic cross-section, σ, defined as the rate of interaction/nucleon in unit incident neutron flux (one neutron/unit area/second). Of the possible types of interaction, we are only interested in scattering processes, for which the cross-section is σs. In general, scattering events can result in changes to both the direction and the energy of the neutron so we define a double differential cross-section, d2σ(E0 – E′, θ)/dE′dΩ to be the rate of scattering from an initial energy E0 to a unit range about the final energy E′ and through a scattering angle, θ, into unit solid angle, Ω. Assuming that the nucleus does not change its internal energy level, the scattering process would be a simple

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two-body collision until the neutron energy is lowered to near thermal energies where the binding of the target atom in the solid becomes significant and hence influences the scattering process. It is, of course, this near thermal energy range (0–1 eV) that is of interest to us here. From the definitions above, we can write: ∞

∫ ∫ ∞

4π

d 2 σ( E0 – E ′ , θ ) d Ω dE ′ = dE ′ d Ω

∫

4π

d σ( E0 , θ ) d Ω = σ s ( E0 ) dΩ [6.1]

For a fixed nucleus at the origin, this quantity is easily derived in quantum mechanical terms. If we represent the incident flux as a plane wave, we can write ψ(r) = exp (ik0 · r), where k0 is the incident wave vector and hk = mv is the momentum of neutron (with m = neutron mass and v = neutron velocity). On this basis, the neutron probability density in the incident beam, ψψ*, is unity so that the incident flux is v neutrons/unit area/s. We can now represent the scattered wave as an expanding spherical wave centred on the origin which can be written ψ′(r) = (–a/r) exp(ik′ · r). Here, k′ is the scattered wave vector and a is known as the neutron scattering length. The – sign is conventionally inserted here because, for most isotopes, the ‘compound nucleus’ formed from the neutron and the target nucleus is far from a stable nuclear state so we get ‘potential scattering’ with a 180° phase change in the scattering process. The scattering length a obviously depends on the nuclear properties (and is different for every isotope). In particular, if the nucleus has a finite spin I, the neutron–nucleus interaction is spin-dependent and so we need to define a+ and a– for parallel and anti-parallel orientations of the neutron spin relative to the nuclear spin. Here the total spin of the compound nucleus, J+/– = (I +/– 1/2), needs to be defined because it determines the statistical weight of the a+ and a– states. It should be noted that if the compound nucleus has an energy anywhere near the energy of a stable state, the values of a can vary quite widely and indeed can be either positive or negative as here there is strong coupling between the incident neutron and the target nucleus. The proton is a good example of this because in the anti-parallel state, the compound nucleus formed is very close to the ground state of the deuteron (spin zero!). This is why a– is negative and so much larger than for any other nucleus. It also follows that the scattering cross-section for an isolated fixed nucleus will be dσ/dΩ = a2 = σ/4π. We can now consider the scattering from a set of N nuclei rigidly fixed at positions ri (so that the neutron does not exchange energy with the target nucleus). If we observe the scattered waves a long way from the scattering nuclei, we can sum them and then define the scattered flux by multiplying the summed wave function by its complex conjugate and hence we obtain [1] an expression for the angular cross-section defined/nucleon:

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dσ/dΩ = (1/N)| ∑i ai exp(iQ · ri) |2 = (1/N) ∑ij aiaj exp [iQ · (ri – rj)]

[6.2]

Here, Q (= k′ – k0) can be thought of as the wave vector transport as hQ is the momentum transferred in the collision. The matrix on the right-hand side of the equation can be simplified by first averaging over the diagonal terms, yielding and then over the non-diagonal terms, where (remembering that the values of ai and aj are completely uncorrelated with the position vectors, rij) we get 2 ∑ij(i≠j) exp[iQ · (ri – rj)]. Now by reinserting diagonal terms 2 into the summation and subtracting the same quantity from the leading term, we can write: dσ/dΩ = [ – 2] + (1/N) 2 ∑ij exp[iQ · (ri – rj)]

[6.3]

Here the first term, [ – 2], is independent of the angle of scattering. It is just the sum of the scattering from each nucleus taken independently and is called the incoherent scattering. Note that this term is zero if a has the same value for all nuclei. If we integrate over a solid angle, we get an expression for the incoherent scattering cross-section, σinc = 4π [ – 2]. Also, the second term, (1/N) 2 ∑ij exp[iQ · (ri – rj)] is called the coherent scattering and we can similarly write σcoh = 4π 2. This term depends on the positions of all the atoms present taken in pairs and defines the diffraction pattern for the solid. It is analogous to X-ray diffraction except that, because the scattering is from the nucleus, there is no atomic form factor. This last expression, (6.3), has been derived for fixed nuclei but the separation process into incoherent and coherent terms carries over into all kinds of neutron scattering theory. Moreover, it is particularly important for hydrogen. Remembering that the number of quantum states associated with a compound nuclear spin of J will be 2J + 1, we find that the average scattering length taken over the two spin states of hydrogen can be written:

=

(2 J + + 1) a + + (2 J – + 1) a – (2 J + + 1) + (2 J – + 1)

[6.4]

Substituting J+ = (I + 1/2) and J– = (I – 1/2), for the case of the proton with I = 1/2, we find that = 3/4 a+ + 1/4 a–. As mentioned above, a– is large and negative while a+ is positive and not quite a third of the value of a– so that is small and negative. Numerically, this means that the incoherent crosssection is 79.7 b while the coherent cross-section σcoh (= (4π 2)) is only 1.8 b and so the proton is a strongly incoherent scatterer. On the other hand, the deuteron turns out to have σcoh = 5.6 b and σinc = 2.0 b. Hence, to use neutron diffraction to identify the position of hydrogen in a crystal structure, it is more or less essential to prepare the deuterated version of the material, because, for H, the incoherent scattering gives an intense flat background to

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the diffraction pattern. Also, as is negative for hydrogen and positive for deuterium, it is possible to make an isotopic mixture that has exactly zero. Scattering from tritium is also of interest in some cases. It has a total scattering cross-section of 3.1 b (σcoh = 2.89 b, σinc = 0.14 b). Tabulations of the values of cross-sections and scattering lengths can be found in many places [13,14]. To complete our discussion of these general features of neutron scattering, we will formally extend the quantum mechanical treatment introduced above to include energy transfers, i.e. by assuming that the atoms in our set are held together by a set of mutual interactions. The initial state is therefore described by a particular set of quantum numbers which define the Hamiltonian or total energy of the system. As a result of an inelastic collision, any of these numbers can be changed by one or more, each integer change being associated with a quantum of energy and an equal amount of energy will be transferred to or from the neutron. This process is described by Fermi’s Golden Rule [1], which may be written: d 2 σ (E 0 – E ′, 0) = Σ i Σ j Σ n pn Σ n ′ ( k ′ / k 0 ) d E ′ dΩ

∫ ψ ( r ) a a δ [ r – ( r – r )] n

i

j

× ψ *n ′ ( r ) exp[ i ( Q ⋅ r )] d r δ ( E 0 – E ′ – E n ′ + E n )

i

j

[6.5]

In this expression, the scattering system is assumed to be in thermal equilibrium and so the cross-section has to be averaged over each possible initial state of the system n, weighted by the appropriate Bose–Einstein thermal occupation number pn and then summed over all final states n′. The delta function in r describes the short range neutron–nucleus interaction and the energy ensures energy conservation. k0 and k′ are the neutron wave vectors for the incoming and outgoing neutron respectively, and the wavefunction for the nth quantum state is ψn(r). As before Q = k′ – k0. The full usefulness of this equation may be found in the standard textbooks [1]. It will be seen that, if we separate out the parts of the equation that involve the neutron wave function, the rest of the equation is only a function of Q and ω. Van Hove [15] pointed out that this was a general property of neutron scattering cross-sections. Thus the expression for the cross-section can be reduced to the form of the scattering function which is a function of only two variables (in contrast to the three variables required to define the inelastic scattering cross-section). This relationship can be written:

d 2 σ ( E0 – E ′, θ )  σ   k ′  = S ( Q, ω )  4π   k0  d E ′d Ω

[6.6]

where S(Q, ω) is known as the scattering function, again with coherent and incoherent versions.

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6.6

Theory of inelastic neutron scattering

6.6.1

Theory of inelastic neutron scattering from solids

As described above, inelastic neutron scattering measures the energy that the system can exchange with the neutron and the probability of each exchange (see equation 6.5). Given that we are interested only in hydrogen, we can initially restrict ourselves to incoherent scattering, i.e. those terms in equation (6.5) for which i = j, where the initial and final quantum states both refer to the same nucleus. Actually, we can set this problem up in two different ways – either as the wavefunctions of individual protons at occupied sites in the lattice or in terms of phonons in a periodic lattice. The first approach works best for isolated protons which do not interact strongly with neighbouring atoms, but the method is easily adapted to the case where the proton is situated in an anharmonic potential. The second approach works well if the material has a periodic stoichiometric structure but is limited to harmonic interactions. Alternatively, we can use the deuteride and measure the coherent inelastic scattering. Here, we would normally adopt the second approach using a single crystal sample. Being a coherent process, we obtain information on the relative motion of pairs of atoms which can be described in terms of phonon dispersion curves. This is, of course, the most direct way of investigating the interactions between pairs of atoms in the crystal structure. The most direct experimental approach is to use a triple axis spectrometer on a steady state (reactor) source, the method pioneered by Brockhouse and Stewart [16]. Another approach that works for polycrystals, and has become possible through the extensive modelling software now available, is to calculate the spherically averaged (polycrystalline) cross-section for direct comparison with the experimental data and to adjust the model to improve the fit [17]. Incoherent inelastic scattering from a proton in a harmonic potential well: the Einstein oscillator model The simplest analytic model for an isolated proton in a lattice assumes that it is situated in a potential well centred on an interstitial site. This model is particularly appropriate to protons in a transition metal lattice, where the electron from the hydrogen atom can be accommodated in the d-band of the metal, but is also applicable to many other cases as well – e.g. to molecular hydrogen trapped in ion-exchanged zeolites (see Section 6.8.2 below). The model assumes that there are no interactions between neighbouring hydrogen atoms and that there is little coupling with the lattice modes. This implies that M/mp >> 1 where mp is the mass of the proton and M is the mass of the lattice atom. In transition metals, with face-centred cubic (FCC), body-centred cubic (BCC) or hexagonal close packed (HCP) lattices, the proton normally sits on either octahedral or tetrahedral sites. In more complex intermetallic

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lattices, such as Laves phases or AB5 alloys, the site geometries can be much more complex. However, the shape of the potential well can be expanded using Taylor’s theorem. Now the first term will be a constant, the second will be zero if the origin is taken at the centre of the well and the third term is the harmonic (parabolic) term which can be separated into independent terms in the x, y and z directions ( V ( r ) = 1 / 2 m p ( ω 2x x 2 + ω y y 2 + ω 2z z 2 )) . If the higher terms in the expansion can be neglected, the wave function of the proton can be represented as the product of three independent one-dimensional Hermite polynomials where the corresponding energy levels are equally spaced with energies, En = (n + 1/2)hω. Fermi’s Golden Rule (Eq. 6.5) can now be used to calculate the cross-sections explicitly (see Elsaesser et al. [23]). The result is: S(Q, ω) = exp[–2W(Q)] exp (hω/2kT) × Σ ∞k , l , m – ∞ I k ( s k ) I l ( s l ) I m ( s m ) δ ( hω – k hω k – lhω l – mhω m ) [6.7] where k, l, m represent the change in the existing quantum numbers in each of the Cartesian directions, subject to the condition that the initial and final quantum numbers must lie between 0 and ∞. In the equation, In(s) are the modified Bessel functions where

 hQi2   hω i  si =   csch  2 kT   2mp ω i 

[6.8]

and Qi is the component of Q in the ith Cartesian direction. In equation (6.7), 2W(Q) is the Debye Waller factor in the direction, Q, which can be written: 2 W ( Q ) = Σ i (< ni > + 1 / 2 )( h / m p ω i ) Qi2 = Σ i < ui2 > Qi2

[6.9]

where is the thermal average phonon occupation number in the ith direction and < ui2 > is the mean square displacement of the proton from its mean position in the ith direction. This equation implies that if the wave vector, Q, has a component in a particular Cartesian direction i, then there is a finite probability that the proton can be transferred to any energy level in the ith direction that is consistent with energy conservation. Consideration of the modified Bessel functions show that a term involving n quanta being transferred will vary as Q2n times a form factor that is the Fourier transform of the product of the initial and the complex conjugate of the final wavefunctions of the proton. It will be noted that if the parabolic potential is known, then the cross-sections can be calculated explicitly with no selection rules that depend on the electronic structure – as is required when dealing with IR and Raman measurements.

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Perturbation analysis of anharmonic and anisotropic effects in inelastic neutron scattering from H in solids Because the proton wavefunction is not localised, it samples the potential energy surface in a volume close to the centre of the interstitial site. Hydrogen, being the lightest nucleus, samples the potential further from the centre of the site than any other nucleon (with the exception of the µ+ meson). Its wavefunction is therefore more subject to the effects of the higher terms of the Taylor expansion than any other scatterer. The allowed set of such terms depends on the symmetry of the interstitial site. For octahedral sites in a cubic system, the potential can be written – for terms up to the quartic – as: V(x, y, z) = c2(x2+ y2 + z2) + c4(x4+ y4+ z4) + c22(x2y2 + y2z2 + z2x2)

[6.10]

From this equation, we can obtain a reasonable estimate of the wavefunctions and energy levels to be expected for this potential using perturbation theory [19]. The energy levels are given by the expression: Eklm = (hω0/2) + β(j2 + j + 1/2) + γ[(2k + 1)(2l + 1) + (2l + 1)(2m + 1) + (2m + 1)(2k + 1)]

[6.11]

with j = k + l + m. From this, we can derive a set of energy transfers: ε100 = ε010 = ε001 = hω0 + 2β + 4γ (fundamental terms) ε200 = ε020 = ε002 = hω0 + 6β + 8γ (first harmonic) ε110 = ε011 = ε101 = 2hω0 + 4β + 12γ (combination vibration) where these parameters are related to the parameters defining the potential by ω0 = √2c2/m, β = 3hc4/4mc2, γ = h2 c22/c2m Here, the notation ε110 implies that the Cartesian operators have been raised by unity in the x and y directions and this implies that Q has to have finite components in the x and y directions. This result implies that the energy levels are no longer evenly spaced and that the intensity of the corresponding inelastic peak will vary strongly with the direction of Q relative to the crystal axes. This model works well for α-Pd/H [20]. Here the respective energy levels were observed at 69, 138 and 156 meV, yielding the following values for the parameters: hω0 = 50 meV, β = 9.5 meV and γ = 0. It should be noted that these parameters imply rather too large a perturbation for perturbation theory to work reliably. Comparison with ab initio calculations Over the past decade or so, ab initio calculations of lattice dynamics using density functional theory (DFT) have become increasingly accurate. In these

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calculations, the total energy of an assembly of atoms is calculated using DFT to represent the effects of electron–electron correlations. Then the nuclear coordinates are relaxed so as to minimise the total energy. As long as the initial configuration of the atomic positions is sufficiently accurate, the relaxed structure should coincide with the real system. The process of relaxing to the minimum energy effectively provides force constant data for dynamical calculations for direct comparison with neutron scattering data. The situation is somewhat more complicated when we consider proton dynamics because the quantum properties of the proton have also to be calculated. This is done in practice using the Born–Oppenheimer approximation, namely that motions on different timescales can be separated. Specifically, this means that we can solve the Schrödinger equation for the electrons over a whole range of different proton positions so that the variation of the total energy of the system with proton position yields a potential energy surface within which to calculate the wavefunctions and energy states of the proton. This method, sometimes referred to as the ‘frozen phonon’ method, was pioneered by Ho et al. for the NbH1.0 system [21,22]. The calculation was done for a periodic lattice and the protons were all moved in phase with each other but in anti-phase with the metal atoms (the zone centre phonons). The various excited states of the proton were calculated and agreed well with experiment. The approach was extended to the Pd-H system by Elsaesser et al. [23] and to a single crystal Pd-H0.85, by Kemali et al. [24] where full ab initio calculations of the inelastic scattering cross-section were compared with experimental neutron scattering measurements. Having determined the full three-dimensional wavefunctions of the proton, the full inelastic cross-section was calculated using equation (6.5). For a harmonic spherically symmetric wavefunction, the energy levels would be fully degenerate and the scattering would be spherically symmetric. However, for an anharmonic system such as Pd-H, the (1,1,0) level is split from the (2,0,0) level which is itself split into a singly degenerate state, |C> and two degenerate states, |A> and |B>. These energy levels can be easily identified using their directional dependence: the (1,1,0) level gives maximum scattering in the (110) and (111) directions and is zero in the (100) direction, while the (2,0,0) level gives a maximum in the (100) direction (because Q lies entirely in one Cartesian direction, only one Cartesian wavefunction can change its energy level). Similarly, the (111) level has its maximum in the (111) direction. The experimental energy levels agree pretty well with the calculated values (Fig. 6.1). Incoherent scattering from systems with interacting protons The discussion above has been concerned with systems in which the protons do not interact significantly with each other so that we can analyse the scattering on the basis of scattering from individual protons in a fixed potential

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S(Q, ω)

0.06 0.04

[110]

0.02 [001] 0.00 0

40

80

120

160 200 240 280 Energy transfer (meV)

320

360 400

6.1 The experimental and calculated incoherent scattering functions S(Q,ω), against the energy transfer (points represent the data, lines represent the theory). For clarity, the spectra for the [110] and the [111] directions have been shifted vertically. The calculated spectra have been convolute with the experimental resolution function [24].

energy well. However, for crystalline systems in which there is significant interaction, we have to analyse the scattering in terms of the phonon dispersion curves. In this approach, which is limited to harmonic potentials, we describe the motion of the atoms in terms of their periodic displacements in the crystal, i.e. plane waves with wave vectors q. The full theory can be found in the standard textbooks [1,18]. By using the periodicity of the lattice, the equations of motion for a given phonon yield the ‘dynamical matrix’. This matrix is of dimension 3n, where n is the number of atoms in the unit cell and the 3 represents the number of Cartesian axes. Each element, i, j, in this matrix is determined by the force constant relating the relative displacements of two atoms, each in a particular Cartesian direction. This matrix equation can be solved, yielding 3n eigenvalues, ωs(q), which are the frequencies of the 3n solutions for the particular q used. For each ωs(q), there is an eigenvector, representing the corresponding displacements of the n atoms in the unit cell in each Cartesian direction. These displacements are in general complex numbers, implying that the phase of the displacements of a given atom may differ from the phase of the phonon solution assumed. Now we can plot the 3n ‘dispersion curves’, ωs(q), as a function of q. We can also derive the corresponding neutron scattering cross-section for the creation or annihilation of a phonon which now depends on satisfying both energy conservation and momentum conservation, where the phonon has an energy hω and a momentum hq: hω = E0 – E′ and Q = k0 – k′ = ␶ + q

[6.12]

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Here ␶ is a reciprocal lattice vector. The intensity of the scattering at a particular value of q is determined from the relative vibration amplitudes of the atoms taken in pairs over the unit cell and weighted by the scattering lengths of each atom. The classical method of using neutrons to measure these dispersion curves involves the use of single crystal samples and a triple axis spectrometer [1,16]. However, we are mainly interested in samples where the scattering will be dominated by hydrogen and, as we have seen, the coherent scattering of hydrogen is small compared with the incoherent cross-section. Hence the inelastic scattering will be dominated by the self (diagonal) term in the full cross-section. All the information about the conservation of momentum disappears and we are left with a hydrogen vibrational amplitude-weighted contribution from each phonon mode in the unit cell. Of course, it would be possible to use deuterium but if we did, we would still have the problem of making single crystals of hydrides (deuterides). As most hydrides are formed in the solid state by a first-order transition involving a significant lattice parameter change (with the notable exception of palladium [24] and a couple of other systems), it is therefore generally impossible to get adequate single crystals. We are therefore forced to use hydrogen and analyse the incoherent scattering. Here we can show [1] that the incoherent inelastic scattering cross-section can be written: 2 d 2 σ inc = k ′ [< n ( ω ) + 1>] Q Θ ( ω ) 2ω dω dΩ 4 π k0

[6.13]

where we have replaced (E′ – E0) by hω, is the thermal population of modes of frequency, ω, and Θ(ω) is the amplitude-weighted density of states given by: Θ ( ω ) = f ( ω ) Σ ρ ( σ ρinc / M p ) exp (– Q 2 < uρ2 >) Σ j | eρs ( ω ) | 2 [6.14] Here f(ω) is the normalised number of phonon modes of frequency ω summed over all phonon modes, s, and wave vectors, q, in the Brillouin zone of the crystal, eρs ( ω ) is the displacement of the ρth hydrogen atom, of mass Mp, in the sth mode summed over all the phonons of frequency ω, and < uρ2 > is the mean square amplitude of displacement of the ρth hydrogen averaged over all modes. For the case of a binary hydride with two atoms/unit cell, there are six modes. The first three of these are ‘acoustic’, that is to say that the metal atom and the hydrogen atom are roughly in phase with each other and that both have about the same amplitude of vibration. The other three modes, however, are ‘optical’, i.e they are roughly in antiphase but with an immobile centre of gravity. This means that the square of the H vibration amplitude increases in proportion to the mass of the metal atom. Thus, the scattering from the optical modes will normally outweigh that from the acoustic modes

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by more than an order of magnitude. Further, if there is no interaction between adjacent H atoms, the frequency range within a given dispersion curve will vary as Mp/Mm where Mm is the mass of the metal atom. For heavy metals, this range is small and so the scattering appears as a rather narrow intense peak. However, for lighter metals, particularly where H–H interactions can be significant, the spectral shape can become much more complex. However, with the recent development of ab initio calculations, it has become possible to calculate the inelastic scattering cross-section directly for comparison with neutron scattering data. Examples of such measurements will be given below.

6.6.2

Inelastic neutron scattering from molecular hydrogen

An exception to the above considerations of the incoherent scattering from hydrogen comes in the case of molecular hydrogen where, owing to the light mass of the atoms and the weak interaction with the surroundings, the overall wavefunction of the molecule has to be anti-symmetric. Thus, if the nuclear spins are anti-parallel (para-hydrogen), the angular momentum, J, must be even – whereas, for parallel spins (ortho-hydrogen), it must be odd. The energies of the rotational states for a free hydrogen molecule are given by: EJm = EJ = BJ(J + 1)

[6.15]

where B = h 2 /2 µd e2 is the rotational constant, = 7.35 meV for H2. Here, µ is the reduced mass for the molecule, de is the equilibrium separation of the H nuclei (0.0741 nm) and mh is the component of the angular momentum parallel to the quantisation axis where m = –1,0,1 for the J = 1 level. It is obvious that this association of the relative nuclear spin states with the angular momentum state is contrary to our assumption above that the spin direction (and hence the scattering length) of a given proton is independent of its surroundings, and this has a profound effect on the neutron scattering from molecular hydrogen. The full theory of this process has been given by Young and Koppel [3]. These authors showed that the scattering can be divided up into terms corresponding to transitions between different J states – if J′–J is even, then the neutron spin stays unchanged but if J′–J is odd, then the neutron has exchanged spin with one of the nuclei and the nuclear spin of the molecule is changed from odd to even (ortho to para) or vice versa. Also, of course, if J changes, the neutron has to lose or gain the corresponding quanta of rotational energy. Thus, the cross-sections can be written:

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d 2 σ J ,J ′ = ( k / k 0 ) FJ – J ′ ( Q ) exp [–2 W ( Q )] δ ( E – E ′ + E j – E j ′ ) d Ω dE′ ∗ Smol(Q, ω)

[6.16]

where ∗ represents a convolution, the FJ–J′ (Q) are the form factors for the rotational transitions and Smol(Q, ω) is the scattering function for the molecule centre of mass. Explicitly, the first few rotational transition form factors can be written: F0–0 = 4(σcoh/4π) j 02 F0–1 = 12(σinc/4π) j12 F1–0 = (4/3)(σinc/4π) j12 F1–1 = 4[(σcoh/4π) + (2/3)(σinc/4π)] (2 j 22 + j 02 )

[6.17]

Here σcoh and σinc are as defined above for the hydrogen nucleus and j0, j1 and j2 are the zero, first and second spherical Bessel functions with argument Qde/2. Remembering that σinc = 79.7 b and σcoh = 1.8 b, it is clear that for para-hydrogen in its ground state, the scattering will be dominated by the J = 0 to J′ = 1 (or J = 0 to J′ = 3, 5…) neutron energy loss transition. As the J = 0 state has zero rotational energy and the J = 1 has an energy J(J + 1)B = 14.7 meV, the neutron energy loss cross-section will be dominated by a delta function at 14.7 meV. The above expressions for the rotational form factors are based on the assumption that the molecule is situated in a spherical potential so that the different m states are degenerate. However, if the molecule is trapped in a perturbed potential well, the degeneracy of the m levels will be lifted. Thus, for ellipsoidal perturbation of the J = 1 state, we expect to find the m = ±1 levels split from the m = 0 state. The energy level diagram for this situation can be found in, for instance, Mitchell et al. [7]. Simple perturbation theory would suggest that the mean energy should remain fixed at 14.7 meV. If the mean energy decreases, we can interpret this to mean that the H–H distance (2de) has increased owing to interaction with the trapping medium. As can be seen in equation (6.16), this rotational transition part of the cross-section has to be convoluted with the scattering function for the molecule centre of mass. If the molecule is rigidly bound, Smol (Q, ω) = δ(ω). If it is free to recoil, its cross-section will be as for scattering from a perfect gas. The assumption here is that the initial state of the molecule has a defined momentum taken from a Maxwell–Boltzmann distribution and that all final states are available for the recoiling molecule. In this case, the molecular scattering function can be written:

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Solid-state hydrogen storage

 ( hω – h 2 Q 2 / 4 M ) 2  SPG ( Q, ω ) = ( M / π h 2 Q 2 k B T )1/2 exp  – ( h 2 Q 2 k B T / M )   [6.18] where M is the mass of the molecule, kB is Boltzmann’s constant and T is an effective gas temperature. It will be noted that this has a Gaussian shape, with its mean value displaced to the average recoil energy, h2Q2/4M. Note that here Q is a scalar because the scattering is isotropic. If, on the other hand, the molecule is trapped in a harmonic potential well, the simple harmonic oscillator scattering function SSHO(Q, ω) will take the usual form [18] and the neutron will scatter the molecule into one of a series of equally spaced energy levels, the spacing being proportional to the steepness of the parabolic well. At low temperatures there will be a probability that it starts and finishes in its ground state. Here, the cross-section for elastic scattering will vary as exp(–Q2)δ(ω) where is the mean square displacement of the molecule in the potential. However, if the molecule is trapped in a sufficiently shallow potential well it ends up in one of a number of closely spaced energy levels. Summing the scattering over these closely spaced states will give a continuous distribution that will approximate to the perfect gas distribution. The total scattering function for this system can be well represented by: SSHO (Q, ω) = exp(–Q2)δ(ω) + [1 – exp(–Q2)] SPG(Q, ω) [6.19] Now, this expression can be substituted into (6.16) to yield the full expression including both a rotational transition combined with the molecular centre of gravity response. In particular, if the molecule starts on the para-ground state (J = 0), the scattering will be dominated by the transition to the J′ = 1 (ortho) state. Now, performing the convolution integral, the scattering will consist of a peak at J(J + 1)B (= 14.7 meV) due to elastic scattering from the molecule and a broad Gaussian peak centred on J(J + 1)B + h2Q2/4M. Similar terms will appear at larger energy transfers for J′ = 3,5,… etc. Because the transition rate between the ortho- and para- states is low except in the presence of magnetic catalysts, we can equilibrate the gas at a given temperature in the presence of a magnetic catalyst. For low temperatures, in the presence of a catalyst, the system will become virtually entirely para while for high temperatures it will tend to 25% para and 75% ortho. We can therefore do neutron scattering experiments with different predetermined ratios of the two forms so that both cross-sections can be extracted.

6.6.3

The theory of quasi-elastic neutron scattering

The treatment above refers to the probability of the neutron causing a transition of the scattering system from one fixed quantum state to another. In contrast

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to this, quasi-elastic scattering describes a situation where the scattering nucleus is free to diffuse around in space and this causes a kind of Doppler broadening, the final neutron velocity (energy) acquiring a continuous range of values symmetrically around the original velocity (energy). Quasi-elastic scattering therefore provides us with a very direct technique for studying diffusion of an atomic species in either a liquid or a solid phase. Given that, again, hydrogen is easily studied, it is a potentially useful way of investigating dynamic mechanisms in hydrogen storage systems. The basic formulation of this problem was given by Van Hove [25] in the form of his space–time correlation functions, Gs(r, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick’s law diffusion. Formally the equations can be written: S inc ( Q, ω ) = (1/2 π )

∫∫ G ( r , t ) exp [i ( Q ⋅ r + ωt ) dr dt

S coh ( Q, ω ) = (1/2 π )

∫∫ G ( r , t ) exp [i ( Q ⋅ r + ωt ) d r dt

s

[6.20]

Here, Gs(r, t), the self-correlation function, is defined to be the probability of finding a nucleus at position r at time t if that nucleus was at the origin at t = 0. Similarly, G(r, t) is the probability of finding an atom at position r at time t, if any atom was at the origin at t = 0. Strictly speaking, the scattering functions are not symmetric in ω because the probabilities of neutron energy gain is related to energy loss by the Boltzmann factor (detailed balance), exp(–hω/kT), but so long as hω << kT, the above equations will hold and the correlation functions are real and do not need to be treated as operators. It should be noted that in practice, we have to construct models for the diffusion process, calculate the corresponding scattering function and compare the resulting scattering function with experiment. The simplest such model is that due to Vinyard [32] who noted that in Einstein diffusion, for times significantly longer than the mean times between individual diffusive jumps (assumed to be uncorrelated to each other) the self-correlation function will be a Gaussian of mean square deviation = 6Dtt, where Dt is the tracer diffusion coefficient. On substituting this into (6.15) above, we obtain: S inc ( Q, ω ) =

(1/ π )( Dt Q 2 ) ( Dt Q 2 ) 2 + ω 2

[6.21]

This expression is a Lorentzian of half width at half maximum, DtQ2. Thus

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Solid-state hydrogen storage

by measuring the quasi-elastic width and plotting as a function of Q2, we can extract Dt. Now in practice, when we plot data of this kind, we get a good straight line at low Q2 but the curve flattens out at higher Q. This is due to the fact that at higher Q, the observation time becomes shorter and the details of the individual jumps become important. The Chudley–Elliott model There are a variety of models in the literature to describe this kind of situation in a liquid or other disordered solid. Full details of these models can be found in the book by Bee [2]. As we are mainly interested in diffusion in a periodic lattice, we shall move directly to consideration of the Chudley– Elliott model [26]. Here the basic model is that the diffusing species is able to hop between equivalent sites on a regular Bravais lattice where the probability of a given atom jumping from a given site occupied at t = 0 is given by a Poisson distribution. A simple derivation of the result can be found elsewhere [5, 27]. The resulting equation is: S( Q, ω ) =

(1/ πτ )[1 – F ( Q)] ω 2 + (1/ τ ) 2 [1 – F ( Q )] 2

[6.22]

Here, if we assume that the jumps are to the m nearest neighbour sites at distance l so that the spatial distribution after one jump is f (r) = (1/m)∑l δ(r – l), then F(Q) = (1/m)∑l exp(iQ l). From this, it is clear that the quasielastic broadening is a Lorentzian in ω as before but that now the broadening in Q is an oscillatory function which returns to zero at the reciprocal lattice points of the defect lattice. Thus, although the quasi-elastic scattering is Lorentzian in shape for a single crystal sample, in a poly-crystal, its form involves integration over a range of widths. In practice, authors have often assumed that this situation can be approximated by the assumption that the first jump is to the surface of a sphere of radius equal to the nearest neighbour jump length [28]. This assumption yields a Lorentzian peak shape with a broadening function, F(Q) = sin(Ql)/Ql. It is clearly better to use the full formula and average over orientation. The classic example of the Chudley–Elliott model in practice is the α-Pd– H system where the protons occupy octahedral sites on an FCC lattice and so diffuse on a Bravais lattice [29,30]. The experimental measurements on a single crystal of α-Pd–H demonstrated that the jumps were indeed between nearest neighbour octahedral sites and that jumps to further sites could be ruled out. It is interesting to note that the first attempts to predict this process theoretically suggested that once a proton escaped from its site, it moved through several sites before retrapping [31]. However, when the interaction with conduction electrons was included, the jump was limited to the nearest neighbour site so the agreement with experiment was excellent [33]. The

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original Chudley–Elliott model is valid only for Bravais lattices. However, it has been extended to non-Bravais lattices. This results in a model consisting of a set of different Lorentzians having Q-dependent widths and intensities [34]. For polycrystalline samples, after averaging over direction, the resulting peak shapes can be well fitted with a sum of two Lorentzians for comparison with the experimental data [35]. The model can also be extended to describe diffusion over different types of site [36]. An important assumption in this theory is that there is no correlation between successive jumps and this is generally a good assumption at low H concentrations. However, at larger concentrations, this is not strictly true because correlation effects become significant, i.e. if an atom jumps from a filled site to an empty one, the site vacated is, at this instant, empty whereas the other sites are occupied with a probability c where c is the overall fractional occupation, so that the chance of jumping back is enhanced. This effect was first considered by Ross and Wilson [37] who showed by Monte Carlo simulation that, at finite concentrations, the quasi-elastic peak deviates from the Lorentzian shape. This was the first example of the need to resort to Monte Carlo simulation of the diffusion process to obtain Gs(r, t) in situations where the diffusion process becomes significantly complicated. This is likely to be important in efforts to understand the diffusive process in complex hydride stores. Localised diffusion: the elastic structure factor If the diffusing atom is limited to a small region of space, it is clear that Gs(r, t) tends to a constant shape, Gs(r, ∞) at sufficiently long time. This shape is actually the convolution of the probability distribution of the atom in the confined space convoluted with itself (because the initial point is distributed over the space with the same probability distribution). Clearly, on Fourier transforming in time, this will yield a delta function in ω, i.e. elastic scattering. Thus, on substituting into equation (6.15), we get S inc ( Q, ω = 0) = (1/2 π )

∫ G ( r , ∞ ) exp (i Q ⋅ r ) dr s

[6.23]

This elastic scattering term is known as the elastic incoherent structure factor. It decreases from unity at Q = 0 to 0 at large Q. As the area of Sinc(Q, ω) in the ω direction is unity, there is an additional quasi-elastic component that increases from 0 at Q = 0 to unity at large Q. The form of the quasi-elastic component depends on the nature of the localised diffusion. In the simplest case, where the jumping is between two trapping sites, the quasi-elastic term is a Lorentzian with a Q-independent width which is just 1/τ where τ is the mean residence time on either site. Two specific models will be noted here (a) random jumping round a ring of sites, the Barnes model [38] and (b)

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Solid-state hydrogen storage

diffusion between two rigid boundaries, the Hall–Ross model [39]. Further examples can be found in Bee [2]. The Encounter model The ‘Encounter model’ [40] is appropriate for vacancy diffusion models, where the diffusing species can jump only when it exchanges with a vacancy. In the limit of low vacancy concentrations, the vacancy diffuses just like an interstitial at low concentrations. It can approach from infinity and make several exchanges with a given tracer atom before it disappears again to infinity again (an encounter). Using Monte Carlo, we can determine the average displacement of the tracer atom in an encounter. As the displacements in successive encounters are uncorrelated, the process is a Markov chain and the total displacement can be determined by successive convolutions of the displacement probability function in a single encounter with itself. Thus, the quasi-elastic scattering can be determined using the Chudley–Elliott model, but for the jump displacements in an encounter rather than in a single jump, with the mean time between jumps now being the mean time between encounters. This model appears to be relevant to diffusion in complex hydrides.

6.7

Inelastic scattering measurements on solid-state hydrides

6.7.1

Inelastic neutron scattering (INS) measurements on Laves phase hydrides

Laves phases, mainly of C15 (cubic) and C14 (hexagonal) geometry, have a variety of site geometries, the relative sizes of which depend on the relative radii of the A and B atoms. They form hydrides of composition up to the composition of about AB2H4. As the metals involved are transition metals or rare earths, the resulting gravimetric ratios are in the range 1–2% but various examples have been used in practice, e.g. ZrMn2H3.6. In the C15 structure, there are three different tetrahedral interstitial sites available for hydrogen occupation, namely g (A2B2), e (AB3) and b (B4). Each unit cell contains 96 g sites, 32 e sites and 8 b sites. The distribution of hydrogen over the sites can be analysed in terms of the Westlake criterion [41], namely that the radius of the site should be greater than 0.4 Å and that the separation between H sites should be greater than 2.1Å. Normally, the g sites are occupied first, followed by the e sites, as can be confirmed by neutron diffraction from the deuteride (see for example Fruchart et al. [42]). INS can be used to investigate the potential wells occupied by H in these systems. Because interactions between H on adjacent sites can be neglected, the spectra consist of a series of Gaussian peaks, one for each distinct mode

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of vibration, up to three/site, depending on the site symmetry. By measuring the spectra for different H contents and comparing with the diffraction data, it is possible to relate specific peaks to specific sites. These vibrations have been successfully modelled for the case of hydrides of ZrTi2, ZrCr2 and TiCr1.85 [43]. The measured cross-sections are given in Fig. 6.2. The Westlake criterion implies that, for Zr2Ti, the e sites will be filled first, giving a doublet, because the e site has tetragonal symmetry, whereas Ex,y Ez

(a) ZT (TFXA)

Counts/meV

3 2

200 K

1 20 K 0 80

100

120

5×104

140

Ex

160

Ez

200

(b) ZC (IN1B)

Ey

4×104

Counts

180

3×104 2×104 1×10

190 K

4

20 K

0 100

125

150

Ey,z

Counts

3×10

Ex

4

175

200

(c) TC (IN1B)

2×104

200 K 100 K

1×104

20 K 0

100

125 150 175 Energy transfer (meV)

200

6.2 INS spectra for three C15 Laves phase hydrides: (a) ZrTi2H3.6 measured on TFXA at ISIS, (b) ZrCr2H0.9 measured on IN1B at the ILL, Grenoble, and (c) TiCr1.85H0.4 [43].

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Solid-state hydrogen storage

the g site is occupied first in the other two cases, giving three peaks (orthorhombic symmetry) because of the different rA/rB ratios for the different compounds. By assuming that the potential well occupied by the H atom is determined by the summation of different Born Mayer potentials for each H–metal atomic pair – where each metal atom is assumed to have the same potential in any Laves phase in which it appears (as A or B atom), the authors obtained potential energy surfaces with the correct symmetry – and hence could calculate the corresponding Einstein frequencies. Hence, they were able to fit the parameters of the B–M potentials for each metal using the observed energy levels (Fig. 6.2). The overall fit was good and enables them to predict the threshold energies for H jumping between sites.

6.7.2

Inelastic neutron scattering (INS) on alanates

If we now turn to the neutron scattering spectra of light hydrides, first considering the sodium alanates (NaAlH4 and Na3 AlH6), which, when doped with TiCL3, show reasonably fast hydrogen absorption/desorption [44]. Here the H appears tetrahedrally coordinated to the Al atom through largely covalent bonding while the more distant Na atoms are more ionically bonded. Inelastic neutron scattering measurements by Íñiguez et al. [45] (see Fig. 6.3) show excellent agreement with the corresponding ab initio calculations once NaAlH4 (exp.) 1 + 2 phonons (cal.)

Neutron intensity

1 phonon (cal.)

0

50

100 150 200 250 300 Neutron energy loss (meV)

350

400

6.3 INS measurements on NaAlH4 (top line), measured using the filter analyser spectrometer at NIST, compared with ab initio calculations (bottom dotted line) and these calculations with multiphonon contributions added (bottom full line). The inset shows the structure of NaAlH4 where the tetrahedrons show the positions of the hydrogen atoms surrounding the Al site [45].

Neutron scattering studies

157

multiphonon contributions are added. In particular, the sharp features between about 50 meV and 110 meV are attributed to a mixture of bending and stretching modes in the AlH4 ion while the peak at 210 meV is due to pure stretch modes. The sharpness of the features is a result of the relatively low coupling between adjacent AlH4 groups. The measurement was repeated on a sample containing 2% of Ti but no change could be observed. Similar agreement between theory and experiment was observed with Na3AlH6. Very similar results were observed for this system by Fu et al. [46], using the somewhat higher resolution of the TOSCA instrument at ISIS and similar ab initio calculations. However, these authors also measured scattering from these materials after the removal of most of the hydrogen and found three sharp peaks around 50 meV which they attributed to AlH3 or some of its polymorphs, particularly the Al4H12 molecule. This observation is a strong indication that AlH 3 plays an important role in the hydrogenation/ dehydrogenation mechanism.

6.7.3

Inelastic neutron scattering (INS) measurements on hydrides of magnesium and its compounds

The normal hydride of magnesium is MgH2, which has a slightly distorted rutile structure in which H atoms form an HCP lattice with Mg on alternate octahedral sites. Clearly the H atoms interact with each other producing significant dispersion. Also the Mg–H bonding is partly ionic and hence long range. The first measurements were by Santisteban et al. [47], using IN1b at the ILL. Their measured spectrum shows a band between 20 and 40 meV attributed to largely in-phase motions of H and Mg, a broad band between 50 and 90 meV, attributed to strongly dispersed optical phonon branches, with a second optical range from 115 meV to about 160 meV. These authors also report Raman measurements of the zone centre optical phonons which come at either the top or bottom of the density of states bands measured by neutrons, thus identifying the symmetry of the vibrations in each case. Later measurements by Schimmel et al. [48] on TOSCA, have been compared with ab initio calculations which yielded a density of states curve that is an excellent match to the measured spectrum. In particular, the calculations show that the strong and remarkably narrow peak that appears in both theory and experiment at 75 meV is due to a local flat region of one of the optical dispersion curves in the vicinity of the A symmetry point in the reciprocal lattice. This point is clearly going to be sensitive to disorder in the MgH2 lattice. Schimmel et al. then followed up these measurements by comparing this spectrum from well-ordered MgH2 with MgH2 produced from Mg that had been ball-milled to speed up the process of hydrogenation [49]. The results for this experiment illustrate the usefulness of INS in the characterisation of

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Solid-state hydrogen storage

nanoscale material. It is known that the ball-milled material contains some metastable γ-phase, MgH2, mixed with the normal rutile phase. This γ- phase is normally stable only at high temperatures. It must therefore be in a metastable form when produced by the ball milling. The measured inelastic spectrum shows a much reduced and broadened peak at 75 meV and a series of lowenergy peaks that are not seen in the HCP phase but that are predicted in the ab initio simulation of the γ-phase, suggesting that the sample contains about 50% of the γ-phase, presumably in a highly stressed condition. Now, following several hydrogenation/dehydrogenation cycles, it is found that the scattered spectrum has reverted to the rutile phase spectrum but with some broadening, due to the nanoscale structure. It is notable that the absorption/desorption times are still about 10 times faster than for the bulk hydride. This important result suggests that the rapid uptake is associated with particle size rather than with damage and stress in the lattice. It was noted that the relief of the strain and reversion to the rutile structure did not slow down the H2 absorption process so we can conclude it is speeded up by the small particle size and hence the limiting process can be attributed to slow diffusion in the bulk hydride phase. It is also notable that the cycled material should have (nearly) recovered the heat of absorption of the bulk materials, except for any contribution due to the surface energy. Reducing this quantity would, of course, be important as this would yield a lower desorption temperature.

6.7.4

Inelastic neutron scattering (INS) measurements on borohydrides

There is currently considerable interest in the use of borohydrides as hydrogen stores, given the high fractional hydrogen mass of the BH 4– ion. For instance, the recently characterised magnesium tetrahydridoborate, Mg(BH4)2, contains 14.9% H2 by mass [50]. Early experiments by Tomkinson and Waddington [51] established the nature of the vibrations in these systems. Basically, the BH 4– ion acts as an independent entity, with internal modes that do not vary much for different alkali metal cations. At higher temperatures, the BH 4– ion is free to rotate and the structure is FCC. At lower temperatures, the rotation is frozen out with generally a lower symmetry space group. More recent work by Allis and Hudson [52] compares measurements at higher resolution using FANS (NIST) and TOSCA (ISIS) with ab initio calculations on NaBH4 and KBH4. For both these compounds, the spectra break into three regions, 6–25 meV, acoustic vibrations with ions basically in phase, 25–37 meV, optic translation and torsion and a librational peak at around 47 meV. Two sets of the internal modes of the BH 4– ion, ν2 and ν4, are visible in the neutron data as single peaks. Raman and IR spectroscopies are able to resolve individual modes within these bands where selection rules allow whereas the neutron technique cannot resolve the individual peaks but can provide a proper

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(amplitude squared) weighted average and range. As expected, the two internal modes are very similar in energy for Na and K salts. In applying neutron scattering to borates, we have to face the difficulty that the isotope 10B has a very large neutron absorption cross-section. This problem is even worse for the lithium salt because of the additional absorption of 6Li. Hence, it is desirable to use 11B and 7Li and recently INS data have been published for the 7Li11BH4 compound, which, with natural isotopic ratios, contains 18.5% H by mass [53]. The structure of this compound again consists of BH 4– tetrahedra with Li atoms distributed between them. The low-temperature phase has Pnma space group (orthorhombic) while above 400 K, it is P63mc (hexagonal). The inelastic spectrum measured at 5 K is shown in Fig. 6.4. Here the features between 20 and 35 meV are attributed to translational lattice vibrations. The librational mode of the BH 4– group is at ~ 52 meV. Its width is significantly larger than the resolution width (~1.3 meV). This is attributed to dispersion caused by interaction between the tetrahedra and splitting due to different modes of vibration. Two internal modes of the BH 4– group are at about 130 and 160 meV, which is in agreement with the Raman data.

6.8

Inelastic neutron scattering from molecular hydrogen trapped on surfaces

6.8.1

Hydrogen trapped in carbon nanotubes

The interest in hydrogen adsorption onto carbon nanotubes was initiated by reports that remarkably high gravimetric adsorption could be obtained on Translatory vibrations

νL

2000 1800

Intensity (arb, units)

1600 1400

ν2

ν4

1200 1000 800 600 400 200 0 10

30

50

70 90 110 Energy (meV)

130

150

170

6.4 INS data for the 7Li11BH4 sample taken at 5 K [53].

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Solid-state hydrogen storage

bundles of single walled nanotubes (SWNT). Dillon et al. [54] measured the adsorption on a soot containing 0.1–0.2% SWNTs at 133 K from which they estimated a gravimetric adsorption on nanotubes of 5–10%. Ye et al. [55] reported a gravimetric uptake of about 8% on crystalline ropes of SWNTs at 80 K and at pressures > 12 MPa. Also Chambers et al. [56] claimed that tubular, platelet and herringbone forms of carbon nanofibre were capable of adsorbing in excess of 11%, 45% and 67% respectively at room temperature and 12 MPa. While these estimates proved to be rather high, probably because of difficulties in removing all the water vapour from the apparatus and of estimating the purity of the samples, nanotubes have proved to be a very useful model system for studying the energy of adsorption of hydrogen molecules on sites of well-determined geometry. The first INS measurements on H2 adsorbed on SWNT were reported by Brown et al. [57] who observed a broad peak at an energy transfer of 14.5 meV having a width of about 2.4 meV which was about twice the instrumental resolution (1.1 meV) and was independent of temperature. The desorption of H2 with increasing temperature suggested a trapping energy of around 60 meV. The peak was attributed to H2 trapped on the convex external surface. Subsequently Ren and Price [58] observed a similar broad peak but with a width that increased linearly with temperature from 4.2 and 35 K. From their data, they concluded that H2 was physisorbed in the interstitial tunnels in the SWNT bundles. Subsequently, a similar sample was examined in some detail by Georgiev et al. [59,60] using the TOSCA inverse geometry spectrometer at ISIS, Rutherford Appleton Laboratory, UK (Fig. 6.5). This instrument provides probably the best available resolution in the 15 meV energy transfer range and, having good statistics, was capable of observing the splitting of the rotational peak. The forward and backward detector banks on this instrument give two distinct Q values. The sample was held at 20 K and was loaded with incremental amounts of hydrogen so that if different sites are present giving different spectra, the relative occupation of the two sites can be determined as a function of the total H2 adsorbed. For low coverage, these data clearly show the peak to be split into two components, the higher energy peak at about 15.1 meV having about half the area of the lower energy peak at about 13.5 meV. This suggests that the higher energy peak corresponds to the J = 1, m = 0 state while the lower energy peak corresponds to the J = 1, m = ±1 state. This is as would be expected for a molecule trapped in a potential well with a fairly narrow minimum in the direction normal to a surface but quite wide parallel to the surface. As the coverage is increased, a new broad peak appears first as a single peak at 14.5 meV and then, above 100% coverage, as a broader peak that requires two Gaussian peaks centred on 14.2 and 14.6 meV to fit it properly. The temperature dependence of the desorption suggests that the split peak has a trapping energy of around 80 meV while the single peaks

Neutron scattering studies

161

S(Q, ω)

(002)

0.5

0

100

1.0

1.5

200 300 400 Neutron energy loss (cm–1) (a)

3.5

500

600

Empty carbolex 30%, 17 K 50%, 17 K 80%, 30 K 100%, 17 K 144%, 17 K

3.0 2.5

S(Q, ω)

2.0

2.0 1.5 1.0 0.5 0.0 0

10

11

12

13 14 15 16 Energy loss (meV) (b)

17

18

19

20

6.5 Inelastic neutron scattering measurements on H2 adsorbed on single walled carbon nanotubes measured at 20 K on the TOSCA spectrometer at ISIS, Rutherford Appleton Laboratory, UK. (a) The full energy spectrum with the diffraction pattern measured from the nanotube sample as an inset. (b) The energy spectra on an expanded scale showing the rotational peak for a series of different hydrogen coverages. (c) The variation of the areas of fitted Gaussian peak with H2 surface coverage [59, 60] (see page 162).

162

Solid-state hydrogen storage 2.2

E = 13.5meV E = 14.2 meV E = 14.6 meV E = 15.1 meV

2.0

Peak integral intensity

1.8

Rotationally free H2

1.6 1.4 1.2 1.0

J = 0,m = 0 to J = 1,m = ±1 transition of the perturbed adsorbate

Perpendicular vibration?

0.8 0.6 0.4

J = 0,m = 0 to J = 1,m = 0 transition

0.2 0.0 –0.2 –20

0

20

40 60 80 100 Surface coverage (%) (c)

120

140

160

6.5 (Continued)

have a much lower trapping energy of around 40 meV. The authors concluded that the more strongly trapping site was along the groove sites on the surfaces of the bundles while the single peak at 14.5 meV was due to trapping on the convex surfaces. They also suggested that above 100% coverage, a second layer forms over the convex surface, which is responsible for the new 14.6 meV peak, while the first layer, now being more strongly trapped, shows its peak at 14.1 meV. It will be noted that the weighted mean of the split peak is at 14.0 meV, significantly lower than the free molecule value (14.7 meV). This suggests that the H–H distance has been increased by the interaction with the surface. We may note that Raman scattering studies of H2 on SWNTs [61] showed slight shifts in the stretch frequencies (Q band), showing one component with an upshift relative to the gas molecule of around 0.23 meV associated with a trapping site and a downshift of –0.13 meV associated with a convex surface site. The neutron scattering data yields other valuable information. By fitting the Young and Koppel model [3] to the recoil humps, the data give accurate values of the effective Maxwell–Boltzmann temperatures of the trapped molecules while a comparison of the forward and backward spectra yield values of the Debye–Waller factor and hence of the mean square displacement of the molecule in its site. As expected, when the mean square displacement decreases, the effective temperature increases. Thus, as the H2 coverage increases, there is firstly a small increase in the effective temperature from 130 K as molecules in the groove site are pushed closer together. Then, as molecules start to trap on the convex surface, the average temperature begins

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to decrease, reaching 80 K at 140% coverage. Presumably at even higher coverage, the average temperature would asymptotically approach the physical temperature of the surface. Very similar results were reported by Schimmel et al. [62], except that these authors found that the higher energy peak observed at 14.72 meV has almost twice the area of the low energy component observed at 13.63 meV. They also concluded that the split peak is due to H2 adsorbed on groove sites. Also, Liu et al. have reported measurements on H2 adsorbed in boron-substituted SWNTs [63]. These studies, with the lower-resolution FANS spectrometer at NIST, showed the same constituent peaks at 13.5 and 15.1 meV with little difference due to the boron loading, possibly because only a few per cent of B atoms were incorporated in the SWNT structure. The experiment also compared the spectra from arc and laser-produced samples, the former having somewhat broader peaks, attributed to a greater range of tube diameters and other defects in this sample. Other high surface area carbon systems have since been investigated. Georgiev et al. [64] observed H2 on activated carbons derived from pine wood. Here, at low coverages, a pure recoil spectrum is observed, suggesting completely free recoil on a graphene surface. At higher coverage, a broadened peak appears and this is attributed to H2 in nanoscale slit pores where the adsorption energies on both surfaces augment each other. Schimmel et al. [65] reached similar conclusions from measurements on activated charcoal, carbon nanofibres and SWNT samples using a lower-resolution spectrometer at the Delft reactor.

6.8.2

Hydrogen trapped in zeolites

Zeolites are compounds of Si, Al and O which consist of a framework containing a regular array of cavities. The framework has a net negative charge which is balanced by the ‘exchange’ cations which are held electrostatically within the cavities and can therefore be exchanged with other cations using an appropriate aqueous solution. Zeolites have also been investigated as possible hydrogen storage media [66,67]. For these systems, adsorption isotherms measured at 80 K absorb up to 2.2% by weight in the case of Ca-X. Actually, the amount adsorbed in different zeolites depends on the size of the cavities and the exchange ion. Experimental data have been reviewed and compared with molecular dynamics simulations by Vitillo et al. [68]. Although the amounts of hydrogen adsorbed are not very promising for practical applications, they have proved to be useful test bed systems for understanding the nature of the H2 adsorption process. Thus, for instance, data due to Jhung et al. [69] for H2 adsorption in Y zeolite exchanged with H, Na, K and Cs at 80 K, imply that the heat of adsorption is in the range 5.7– 6.6 kJ/mole for these systems, showing a tendency to increase with the

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S(Q, ω)/arb. units

electrostatic field within the zeolite and also to decrease as the amount of hydrogen adsorbed increases. Inelastic neutron measurements have been performed on TOSCA for H2 on zeolite-X having Na+, Ca2+ and Zn2+ exchange ions (Fig. 6.6) [70,71]. Measurements were performed on para-H2 in zeolite X exchanged with Na+, Ca+ and Zn+. These spectra show a new feature of the inelastic scattering, namely the series of uniformly spaced peaks above the complex peak derived from the perturbed rotational states around 14.7 meV. It is clear that these peaks represent the vibrational states of the H2 molecules which are convoluted with the rotational scattering as shown in Eq. 6.16. Also, clearly, a set of equally spaced levels suggests that the molecules are located in rather well-defined parabolic potentials. It will also be noted that the spacing and therefore the steepness of the parabola increases in the sequence Na+, Ca2+, Zn2+. Indeed, the spacing is proportional to the polarising potential of the

0

20 40 60 Neutron energy loss (meV)

80

6.6 The INS spectra of para-H2 in (from top to bottom) zeolites Na–X, Ca–X, Zn–X. Also shown are the spectra decomposed into Gaussians. The Zn–X and the Na–X spectra had their backgrounds removed using a smooth function. These spectra are all measured at low H2 uptakes of around 0.15 wt% [71].

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cations, given by Z/r, where Z is the charge on the cation and r is the ionic radius. Recently, remarkable results have been reported on Cu exchanged ZSM5 [72–74]. Firstly, Ramirez-Cuesta and Mitchel [72] reported a split rotational level at approximately 12 and 14 meV, the 12 meV peak decreasing in energy as the H2 loading was increased. The reduction in the mean energy of the rotational level was attributed to a small increase in the H–H distance, giving rise to a reduced rotational constant, an indication that there is a significant electronic interaction with the Cu ion. Subsequently, Georgiev et al. [73] reported INS measurements on the IN5 spectrometer at the ILL, showing complex tunnelling peaks in the energy range up to 2 meV energy transfer. Here, the sample had been carefully outgassed in a vacuum in order to convert the Cu ions to their monovalent state. It is not clear whether the observed scattering is from ortho- or para-hydrogen. Unfortunately, the TOSCA and IN5 energy transfer ranges do not coincide so it is difficult to establish whether both sets of peaks at ~2 and 12–14 meV coexist. However, it is clear that the trapping energy for the hydrogen is remarkably high for Cu+. This strong interaction also shows up in the IR measurements where the H2 stretch frequencies in this system are at 3070 and 3125cm–1 compared with 4161cm–1 in the unperturbed molecule. In contrast, samples exchanged with alkali metals show downward shifts of this frequency of only 50–90 cm–1 [73], supporting the idea that the interaction with Cu+ is dramatically higher. To confirm this, in subsequent observations on hydrogen adsorption measured volumetrically, Georgiev et al. [74] found heats of adsorption of 75 kJ/mole at low coverages decreasing to 40 kJ/mole at higher coverages (0.4 H2/Cu atomic). These values are remarkably an order of magnitude higher than values reported for other zeolite systems [75], suggesting that this is an example of non-dissociative chemisorption. However, as mentioned before, although the heat of adsorption is close to practical values, the gravimetric adsorption is too low to be useful.

6.8.3

Hydrogen trapped in ice clathrates

Recently there has been considerable interest in the use of ice clathrates as hydrogen stores. Under kilobar pressures, it was possible to get (32 + x) H2 molecules into the clathrate cubic structure II containing 136 H2O molecules based on there being one H2 in each of the 16 small cages and four (x = 16) in each of the 8 larger cages up to a decomposition temperature of 180 K [76]. At higher temperatures, it is necessary to use a larger guest molecule such as tetrahydrofuran (THF) in the larger cage to stabilise the structure up to its melting point and at 50 bar [77]. In this case, only the smaller cage is occupied by H2 molecules with one molecule/cage. This latter system has been studied with INS with interesting results [78]. To minimise the scattering

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Solid-state hydrogen storage

from the host, both the ice and the THF were deuterated. Again, the results could be interpreted in terms of the convolution of the perturbed J = 0 to J = 1 rotational level with the energy levels of the H2 molecule in the small cage. However, in contrast to the zeolite case above [72], the potential well is very anharmonic, which is not surprising given the rigid cage structure. The authors modelled this potential by adding model potentials between the molecule of H2 and the host structure over the 512 nearest D2O molecules and then solving Schröedinger’s equation for H2 in the resulting flat-bottomed potential well. The agreement of the experimental peaks with this model is remarkably good.

6.8.4

Molecular hydrogen storage in metal oxide frameworks

Metal–organic framework compounds (MOFs) constitute a relatively new class of porous compounds, which employ molecular building units with predefined functions and geometries which are used to assemble periodic solids containing a regular array of pores of controllable volume with interconnecting tubes with controllable diameters. Thus, for instance, frameworks build by linking octahedral Zn4O(O2C–)6 groups yields some of the highest porosities known [79]. The metal oxides that link the organic clusters also create trapping sites for hydrogen that could in principle provide the molecular chemisorption that is necessary to yield the right kind of trapping energy. Of the huge number of MOFs now known, only a small fraction have been checked for hydrogen uptake [80 and references therein] and of these only a few have been investigated using INS. The majority of these measurements have been made with the QENS spectrometer at IPNS at the Argonne National Laboratory. In these experiments the sample was kept at low temperatures but the hydrogen was not converted to the para state before admission to the sample capsule so it is presumed that it contained about 75% of ortho-hydrogen. The first MOF to be measured (MOF-5 = IRMOF1) [81] shows peaks at 10.3 and 12.1 meV at low loadings which are attributed to the J = 0–1 transitions on two different sites, the former being attributed to a site at the Zn atom and the latter to sites on the benzenedicarboxylate (BCD) linking molecules. The 10.3 meV band is presumed to correspond to the more strongly adsorbing site because at higher loadings, the 12.1 meV peak increases in intensity. At higher loadings still, the 12.1 meV line seems to break into four components, suggesting a number of different sites on the linkers. The model for the J = 1 energy levels involved assumes that the energy is lower for an H2 orientation parallel to the plane (m = ±1) as compared to one normal to the plane (m = 0) [82]. This suggests that the peaks observed correspond to transitions from the J = 0 level to the J = 1, m = ±1 level and that therefore there should be

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a further peak for the J = 1, m = 0 level beyond the range of the measurement (>18 meV). An alternative explanation would be that there is little orientational dependence but that the rotational level is considerably lowered by a significant increase in the H–H distance. However, given the presence of ortho-H2, we might also expect to see the transition between the J = 1, m = ±1 level to the J = 1, m = 0 level so the interpretation must remain tentative. In a subsequent paper [80], the measurements were extended to IRMOF-8, IRMOF-11 and IRMOF-177. The measured spectra for these compounds behave in a similar way to IRMOF-1, in that, at low loading, there are two single peaks corresponding to two sites, which the authors attribute to (a) sites adjacent to the Zn atom (stronger binding – peak at lower energy) and (b) sites on the organic linker (weaker binding, higher energy and with relatively higher intensity at higher loading). For higher loadings, the spectra become even more complex than in IRMOF-1, as is to be expected for these somewhat less symmetric systems.

6.9

Quasi-elastic scattering measurements on hydrogen diffusing in hydrides

6.9.1

Quasi-elastic scattering studies of hydrogen in Laves phases

The most notable series of measurements of quasi-elastic neutron scattering in intermetallic hydrides in recent years has been that by Skripov and coworkers on hydrogen diffusion in cubic Laves phases (C15-type). In these systems, quasi-elastic neutron scattering measurements show that the hydrogen has two types of jump motion with different characteristic frequencies [83]. The faster jump motion corresponds to localised jump motions while the slower motion gives rise to long range diffusion. This corresponds to the case discussed on page 153. If only the localised motion were present, there would be an elastic structure factor with a Q-dependence determined by the Fourier transform of the H–H correlation function at long time. The remainder of the peak intensity would be quasi-elastic with an energy width determined by the inverse of the mean time between jumps on the hexagon. If the jumps were between two sites, the width of the quasi-elastic peak would be independent of Q. For larger numbers of sites, it will in general be made up of Q-dependent proportions of several Lorentzian broadenings of fixed widths, as determined by the localised site geometry. Now, if we introduce a slower long-range diffusion, then the localised S(Q, ω) is convoluted with the appropriate version of the Chudley–Elliott model. In practice, this means that the elastic peak becomes broadened by an amount that increases with Q2 at low Q, thus defining the tracer diffusion coefficient for hydrogen in the system. In most of the compounds examined, of composition AB2Hx, at low

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Solid-state hydrogen storage

and intermediate hydrogen concentrations (up to x~ 2.5) H occupies only the tetrahedral g sites, with A2B2 coordination, while the tetrahedral e sites (with AB3 coordination) only begin to fill at higher H concentrations. The g sublattice consists of hexagons lying perpendicular to the <111> direction. Each g site has three nearest neighbours, two g-sites on the same hexagon at distance r1 and one on an adjacent hexagon at distance r2. The ratio r2/r1 depends on the actual H site, which, in turn depends on the relative ratio of the metallic radii, rA and rB. For most cases investigated, [84–89], rA/rB ≤ 1.25 and here r1 ≤ r2 and, in consequence, there is a fast localised diffusion around the g sites on the hexagon and long-range diffusion that depends on the slower jumps between hexagons . However, for the case YMn2 [90], rA/rB = 1.425 and hence r2 < r1 and so the localised diffusion is between g sites on adjoining hexagons and long-range diffusion results from jumps around a given hexagon.

6.9.2

Quasi-elastic scattering from hydrogen in alanates

Now, turning to quasi-elastic scattering measurements on complex hydrides which offer the possibility of higher gravimetric H contents, we find that the measurements are rather difficult because the jump rate of the hydrogen atoms is either too slow or, only a small fraction of the atoms take part in the motion (vacancy or defect diffusion). Moreover, the material is extremely sensitive to atmospheric moisture, which creates practical difficulties. To date, the most interesting measurement is that due to Vegge and co-workers [91–93] on sodium alanates, NaAlH4 and Na3AlH6. For these materials, it has been found that activation using TiCl3 considerably accelerates the hydrogen absorption/desorption rate [94]. The samples were measured using the backscattering spectrometer at KFA Jülich, Germany. For both compounds, with or without TiCl3, it was possible to detect any quasi-elastic scattering only at quite high temperatures. For NaAlH4, doped or undoped, there was no detectable quasi-elastic scattering below 315 °C. In fact, only around 0.5% of the hydrogen was mobile at 390 °C in the undoped sample. For Na3AlH6, at 350 °C, only the doped sample showed any broadening (2% of atoms) while at 390 °C, the undoped sample showed significant quasi-elastic scattering (13%) with clear indication of the Chudley–Elliott form with jump lengths of around 2.8 Å. Taken in conjunction with the ab initio calculations, it is inferred that for NaAlH4, there is a localised motion involving the transfer of an H atom from an (AlH4)– ion to a faulted AlH3 cluster. Activation energies for further jumps are significantly higher than this localised motion. As expected, the QENS shows a broadening that is more or less independent of Q, confirming a localised motion. For the Na3AlH6 case, as mentioned, the doped sample broadening displayed substantial Chudley–Elliott model behaviour, which is consistent with the ab

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initio calculations which showed that, assuming that the Ti atom was on an Al site, there is a lower barrier to long-range diffusion (~ 0.36 eV) but for only a few per cent of the hydrogens. These results are fully consistent with the rather low rates of H2 evolution even for finely divided material at high temperatures.

6.9.3

Quasi-elastic broadening due to H2 diffusion on surfaces

Finally, in this section, we should note that QENS can be used to study the diffusion of H2 on surfaces. This has been demonstrated by Fernandez-Alonso et al. [95] who adsorbed hydrogen direct from room temperature (i.e. 75% ortho-H2) onto a sample of nanohorns and observed the quasi-elastic scattering using IRIS at ISIS. This spectrometer provides 9 µeV resolution in the quasielastic region but also allows one to make inelastic measurements around the rotational level at 14.7 meV with a resolution of about 70 µeV. The quasielastic measurements were possible because the sample contained ortho-H2. Measurements were made at 25 and 15 K. The quasi-elastic intensity decreased with increasing Q as expected owing to the size of the H2 molecule and the mean square displacement of the molecule on the surface. The broadening was fitted with a jump diffusion model, yielding values of the tracer diffusion coefficient of 0.96 Å2 ps–1 at 15 K and 6.5 Å2 ps–1at 25 K with corresponding jump lengths of 5.4 Å at 15 K and 6.81 Å at 25 K. This fit was obtained by allowing for an elastic contribution to the peak with intensity about four times the quasi-elastic peak. This was attributed to an immobile fraction although some extra elastic component might be expected as a result of averaging over the orientation of Q relative to the surface normal. In principle, because of the convolution in equation (6.16), the inelastic peak around 14.7 meV should also be quasi-elastically broadened due to the motion of the molecule centre of mass but the data recorded for this peak region were measured at 1.5 K. It showed similar substructure due to splitting of the J = 1 rotational line seen for H2 on nanotubes [59].

6.10

Conclusions

In this chapter, we have attempted to show the valuable role that neutrons can play in developing our understanding of how hydrogen behaves in the new kinds of hydrogen absorbers that are now being investigated. For the moment, fairly simple experiments are being undertaken. Increasingly, however, experimentalists will need to do more sophisticated experiments such as varying the ortho–para ratio in the H2 being adsorbed. Experiments will also be used to monitor changes in the structure of absorbers as hydrogen is cycled in and out for the many hundreds of cycles that would be needed in

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practice. The better our understanding of the behaviour of these materials subjected to these variables, the easier it will be to adjust the composition or other physical properties to enhance performance. It would be difficult to exaggerate the importance of finding materials that will operate successfully as a fuel system in a hydrogen-fuelled car and neutron scattering will inevitably play an important role in this effort.

6.11

References

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