Figure 1

(Color online) Energy bands for bcc Fe at its experimental lattice constant, $2.87\text{Å}$. The coloring is such that $s$ character is green and $d$ character is blue. The Fermi energy is indicated by a horizontal line. The upper panels are majority and the lower panels minority spin bands. Far left is the tight-binding $d$-band model and in the center our nonorthogonal $sd$ model. To the right are bands calculated in the LSDA-GGA.Reuse & Permissions

Figure 2

Densities of states of pure bcc Fe in the orthogonal $d$ (a) and nonorthogonal $sd$ (b) tight-binding models compared to the LSDA-GGA (c). Majority- and minority-spin states are shown in the upper and lower panels of each. The zero of energy is shifted to the Fermi energy.Reuse & Permissions

Figure 3

(Color online) Contributions to the energy-volume relation in the orthogonal $d$ (left) and nonorthogonal $sd$ (right) tight-binding models. The lower panel shows the volume dependence of the magnetic moment. The dotted lines refer to the hcp crystal structure and show the rapid collapse of the moment under pressure. The solid line is the bcc moment and may be compared with the circles which are LSDA-GGA calculations. The upper panel shows the band-structure energy (blue) and the magnetic energy (green) and their sum in red. Solid lines refer to bcc and dotted lines to hcp. The pair potential energy favors bcc in both models. A vertical line indicates the equilibrium volume in bcc Fe.Reuse & Permissions

Figure 4

(Color online) Pair potentials in the orthogonal $d$ (dotted line) and nonorthogonal $sd$ (solid line) tight-binding models. Note how these are negative at some of the critical bond lengths. While this helps to stabilize bcc against hcp, the real benefit is in obtaining a correct elastic constant $C^{\prime }$. For reference below we also show here the Fe-H pair potential in blue (see Table ). Vertical lines are placed at first- and second-neighbor distances in bcc Fe and at first-neighbor distances in hcp Fe.Reuse & Permissions

Figure 5

(Color online) To illustrate the tetrahedral (T) and octahedral (O) interstices in the bcc (upper figure) and fcc (lower figure) crystals. Note that in the bcc lattice the octahedral site is at the center of a distorted octahedron, unlike the fcc where it is regular. The distance to the two apical atoms, shown here as a horizontal bond, is shorter by a factor $1/\sqrt{2}$ than the distances to the equatorial atoms, two of which are shown here in the upper face. This leads in general to the well-known tetragonal distortion of the bcc lattice near octahedral interstitial atoms, for example, in martensite. For details see Refs. 46, 69. Neither is the tetrahedral interstitial site in the bcc lattice regular—indeed both octahedral and tetrahedral bcc interstices have tetragonal symmetry. The fcc crystal structure with all the octahedral sites occupied becomes that of cubic rocksalt adopted by many transition metal carbides and nitrides. In fcc, the tetrahedral site is regular; when half these sites are occupied the resulting crystal structure is that of zinc blende.Reuse & Permissions

Figure 6

(Color online) Cohesive energy and magnetic moment as a function of volume per Fe atom in the four FeH phases calculated within the LSDA-GGA (left). Dotted lines denote nonmagnetic phases. The cohesive energy is with respect to solid $\alpha \text{-Fe}$ and molecular ${\text{H}}_2$ also calculated using the same energy functional and hence is an approximation to the heat of formation. Note that on this basis none of the phases is expected to exist. On the right we show the same quantities calculated in the nonorthogonal $sd$ tight-binding model. We expect that the almost exact degeneracy of bcc TET and fcc TET is accidental.Reuse & Permissions

Figure 7

(Color online) Energy bands for bcc tetrahedral FeH, calculated at the lattice constant of pure bcc Fe. The upper panels show majority-spin and the lower minority-spin states. The coloring is such that $\text{H}s$ character is red and $\text{Fe}d$ character is blue. $\text{Fe}s$ bands are green. The Fermi energy is indicated by a horizontal line. Note that the $\text{Fe}4s$ band has been pushed above the $d$ bands. Bands on the left are from our tight-binding model and on the right are bands calculated in the LSDA-GGA.Reuse & Permissions

Figure 8

(Color online) Hopping and overlap integrals as functions of bond length, $r$, in the $sd$ nonorthogonal model. Except in the case of $dd\delta$ the dotted lines are the overlap integrals corresponding to the hopping integrals of the same color. Vertical dotted lines indicate the Fe-H bond length in bcc tetrahedral FeH at equilibrium volume, and the Fe-Fe bond lengths of the first six neighbors in pure bcc Fe.Reuse & Permissions

Figure 9

(Color online) Illustrates the translations of the bcc intersitials in constructing our adiabatic potential surfaces, after the three-dimensional drawing of Fig. 1 in Krimmel et al. (Ref. 73). The figure represents an (010) face of the bcc lattice with Fe atoms as black circles at each corner. The octahedral sites are shown as squares, the central, filled one being the one occupied in octahedral FeH. Of the four tetrahedral sites, shown as triangles, one is occupied in tetrahedral FeH and this is shown filled in here. The point, S, is midway between two tetrahedral sites—the expected diffusion path of H in Fe (Ref. 74) which is highlighted in red here and in Fig. 10. Those displacements which are in the (010) plane are indicated.Reuse & Permissions

Figure 10

(Color online) Adiabatic potential surface sections of bcc FeH: left LDA (Ref. 68), right TB. These curves show the energy as a function of the displacement of the H sublattice relative to the Fe sublattice. The curves which start at the point “O” refer to displacements from the octahedral site phase; a H atom initially at position $\left[\frac{1}{2}0\frac{1}{2}\right]$ translates in the directions indicated. Along the [001] direction it eventually falls into a vacant tetrahedral site (see Fig. 9). This curve hence represents the transition to the tetrahedral-site phase. Translation along [101] takes the H atom to a position midway between two, vacant, tetrahedral sites—this point is marked S. For a H atom initially occupying a tetrahedral site, translation along [101] moves it to an adjacent, unoccupied, tetrahedral site, the half-way point being the same point S. The translation labels are vectors referred to Fig. 9. For each case, LDA and TB, the calculations are at fixed atomic volume, namely, the equilibrium volume of the bcc tetrahedral phase of FeH, see Table ; $a_0$ is the corresponding equilibrium lattice constant.Reuse & Permissions

Figure 11

Adiabatic potential surface sections of fcc FeH: left LDA (Ref. 68), right TB. At the point O we have the rocksalt phase, from which translation of the H sublattice along a $⟨111⟩$ direction transforms the structure to the zincblende phase in which tetrahedral sites are occupied. The energy maximum between O and T is located close to where the H atom squeezes between an equilateral triangle of Fe atoms in the (111) plane. At the maximum along ${⟨110⟩}_o$, and along ${⟨001⟩}_t$, the H is positioned midway between two nearest-neighbor Fe atoms (see Fig. 1 of Ref. 75). Note that both these two energy barriers are predicted by the TB model with quantitative accuracy. The calculations are at the calculated equilibrium volume of the fcc octahedral (rocksalt) phase of FeH, see Table ; $a_0$ is the corresponding equilibrium lattice constant.Reuse & Permissions

Figure 12

(Color online) Three possible binding sites of H on the (001) surface of Fe, after Ramasubramaniam et al. (Ref. 1). Four large circles represent the Fe atoms at the corners of a unit cell of the (001) face of the bcc lattice. At the center is the “hollow” site, a smaller circle; this may be displaced along [100] by an amount $\delta$ to become the QT site indicated by a triangle. The “bridge” site is shown as a square. It is important to recognize that the bridge and hollow sites in the plane of the truncated bulk surface are octahedral interstices whereas the QT site at $\delta =0.25a_0$ is a tetrahedral site. If a H atom at the bridge site is displaced up or down by $0.25a_0$ then it comes to occupy a tetrahedral site. Here, $a_0=2.87\text{Å}$ is the equilibrium pure $\alpha \text{-Fe}$ lattice constant.Reuse & Permissions