[Studio picture · a row of pendulums with the middle ones swinging together]

[05 / MODEL 4]

The anharmonic oscillator

One hydrogen is treated as a quantum particle in its computed well and tied to its neighbours. It gives the vibration energies that neutron experiments measure. It also gives two routes by which the hydrogens could fall into step: stretch the metal or pump the well.

[IN SHORT]

Each hydrogen in the metal sits in a pocket and rattles. This model treats one hydrogen as a quantum particle in its computed pocket with the steep walls included. It ties the hydrogen to its neighbours with the computed pulls.

It gives the energy of the vibration in which all the hydrogens move in step: 85.6 meV for nickel hydride, 62.3 for palladium hydride and 39.8 for palladium deuteride. Neutron experiments find the vibration peaks at 89, 55.8 and 37.

It also gives two routes to collective behaviour. Stretch the metal and that in-step vibration slows to a stop, so the hydrogens shift together. Or make the well stiffer and softer in a rhythm at twice the vibration frequency, and above a threshold the vibration grows by itself.

The model was run twice. In the first run the palladium energies were much further from the measured peaks. Recomputed wells brought them to within 8 to 12 percent.

g3_lattice/rerun/part1 g3_lattice/rerun/part2 g2_refit

39.8 meV for palladium deuteride with every D in step. The measured peak is 37 85.6 meV for nickel hydride with every H in step. The measured peak is 89 5.3% stretch at which the D in palladium shift together at 0 K 1.51 H to D ratio of the single-atom vibration in palladium. The measured ratio is 1.51

[05.1 · THE MODEL]

Three ingredients for one hydrogen

The model keeps what the vibration needs and nothing else: the shape of the pocket, the quantum nature of a particle as light as hydrogen, and the pull of the neighbours. Every input comes from the calculations in Model 3; nothing is fitted to the measured peaks.

[01 · THE WELL]

A bowl with steep walls

The well has a stiffness at the bottom and walls that rise fast toward the metal atoms and gently between them. The pattern repeats along each of the three cube axes, following the symmetry of the crystal. The bottom has a stiffness of 0.59 eV/Å² in palladium and 1.71 in nickel.
[02 · QUANTUM]

A particle that never sits still

Hydrogen is so light that it cannot rest at the bottom. Even at absolute zero it spreads over about 0.14 to 0.17 Å along each axis. Its vibration energy comes in fixed steps, which are the levels of the well.
[03 · NEIGHBOURS]

A pull that favours moving together

When one hydrogen moves, its neighbours are tugged along. Summed over all of them, the tug lowers the energy of the one vibration in which every hydrogen moves in step. That vibration is the collective mode.

[WHAT COMES OUT]

Vibration energiesOne atom alone and the collective mode compared with the peaks measured by neutron scattering.
The static routeHow far the metal must be stretched before the collective mode stops and the hydrogens shift together.
The pumped routeThe rhythm and strength at which modulating the well makes the collective mode grow by itself.

[05.2 · LEVELS]

A ladder with uneven rungs

A quantum particle in a well can only hold certain energies. In a perfect parabolic bowl the steps between them are all the same. These wells steepen, so the steps are larger and the second one splits three ways depending on how the motion is shared between directions.

[FIG 1]Energy levels of one hydrogen in its well

Material

The well and its levels

Steps above the lowest level

PdD: lowest level 65.1 meV above the bottom of the well. First step 48.9 meV; second step 47.3, 57.4 or 60.5 meV. A perfect bowl of the same stiffness: 35.0 meV every step.

along [100] toward a metal atomalong [111] between threeperfect bowl of the same stiffness

What to notice: the steps are larger than in a perfect bowl, and the second step is not equal to the first. Toggle between PdH and PdD. The well is the same, but the heavier D sits lower and its steps are closer together. The levels are those of the full three-dimensional well and were found by direct diagonalisation in the source runs (nothing is solved in your browser). The curves are cuts through the well along two directions. Sources: g3_lattice/rerun/part1 g3_lattice/rerun/part2

[05.3 · AGAINST MEASURED]

Calculated against measured

Neutrons fired into the metal lose energy to the hydrogen vibrations, and the energy they lose shows up as a peak. The collective mode of the model lands 3.8 percent below that peak in nickel hydride and 7.6 and 11.6 percent above it in palladium.

[FIG 2]Vibration energy, meV

NiH

89 measured 85.6 100.6

PdH

55.8 measured 62.3 74.0

PdD

37 measured 39.8 48.9

all in step (collective mode)one atom alone with neighbours heldmeasured peak

What to notice: letting the neighbours follow lowers the energy, and the collective value sits close to each measured peak. Model values are the first excitation of the three-dimensional well and the collective mode built from it. Measured peaks: NiH, Antonov et al., Europhys. Lett. 51, 140 (2000); PdH, Ross et al., Phys. Rev. B 58, 2591 (1998); PdD, Antonov et al., Phys. Rev. B 80, 134302 (2009). Sources: g3_lattice/rerun/part1 g3_lattice/rerun/part2

What the peaks are. Each measured value is the peak of a whole band of vibrations recorded by inelastic neutron scattering on a powder or plate. It is an average over many modes of the crystal and not exactly the single all-in-step mode the model computes.

One atom against all. One hydrogen moving alone with its neighbours held still needs more energy: 100.6, 74.0 and 48.9 meV. Let the neighbours follow and the pull between them brings the energy down to the collective value.

One input in NiH is older. The NiH run used the corrected stiffness and coupling but an earlier pair of quartic coefficients (18.3 and 1.13 eV/Å⁴). With the recomputed [100] coefficient of 16.2, the single-atom excitation moves from 100.6 to 99.3 meV, a change of −1.3 percent.

[H AGAINST D: A CHECK ON THE STEEP WALLS]

The isotope ratio

A deuteron is twice as heavy as a proton. In a perfect bowl that would lower its vibration energy by the square root of two, which gives a ratio of 1.414. Steep walls push the lighter hydrogen up further because it spreads further into them, so the ratio rises. The measured ratio in palladium is 1.508. The model gives 1.515 for one atom in its well and 1.563 for the collective mode.

1.414perfect bowl 1.508measured 1.515model, one atom 1.563model, all in step

ENERGY OF H ÷ ENERGY OF D, PALLADIUM

[05.4 · BEFORE AND AFTER]

Better wells bring the model closer to measurement

This model was run twice. The first run used single-atom wells computed with coarse numerical settings (a 400 eV cutoff, a single k-point). The wells were then recomputed at tighter settings and the oscillator was run again.

For palladium the on-site stiffness fell from 1.09 to 0.59 eV/Å² and the pull of the neighbours rose from 0.064 to 0.389. For nickel the pull rose from 0.234 to 0.676 and the on-site stiffness rose from 1.468 to 1.712.

[FIG 3A] Collective mode above the measured peak, %

PdH measured peak +44.5 +11.6

PdD +47.4 +7.6

NiH +2.0 −3.8

[FIG 3B] Stretch at which the hydrogens shift together at 0 K, %

PdH 3.8 7.5

PdD 3.0 5.3

NiH 8.3 8.1

first run on coarse wellscurrent run on recomputed wells

In palladium the collective mode moved from 44.5 and 47.4 percent above the measured peaks (PdH and PdD) to 11.6 and 7.6 percent above. One atom alone moved from 47.4 and 50.6 to 32.7 and 32.1. At 0 K the predicted ordering strain rose by 3.7 percentage points for PdH and 2.2 for PdD. Part of that rise comes from the strain law: the rerun uses a curved law fitted to every computed strain where the first run used a straight line through ±1 percent. With the new well and the old straight line, PdD would order at 4.1 percent and PdH at 5.1. Source of the first run: g2_refit

[05.5 · THE STATIC ROUTE]

Stretch until the vibration stops

Stretching the metal softens the collective mode. When its stiffness reaches zero, nothing holds the hydrogens at the centres of their sites any more. They shift together to a new position, which is an ordered state that needs no pump.

[FIG 4]Collective stiffness against stretch

Temperature

NiHPdHPdDother temperatureextrapolated

What to notice: each line falls to zero where the hydrogens shift together. Deuterium gets there first, and warming to 300 K pushes every crossing to larger stretch. The stiffness r sets the square of the collective frequency, ω² = r/m, so the vibration stops where a line meets zero. It includes the push back from zero-point and thermal motion (eq. 3 in The detail). Stretch is the uniform change in length of the metal. Sources: g3_lattice/rerun/part1 g3_lattice/rerun/part2

[STRETCH AT WHICH THE HYDROGENS SHIFT TOGETHER]

Material	0 K	300 K
PdD	5.3%	7.1%
PdH	7.5%	8.8%
NiH	8.1%	8.2%

Zero-point motion pushes back. Because the hydrogen is spread over its well, it feels the steep walls even at rest. That adds stiffness: 0.63 eV/Å² for D and 0.83 for H in palladium. The lighter H spreads further, gains more, and needs more stretch. Heat spreads both a little more, so 300 K needs more stretch than 0 K.

These are extrapolations. The response of the well to strain was computed at −1, 0 and +1 percent. The crossings at 5 to 9 percent extend those calculations well beyond that range.

All palladium numbers use a lattice spacing of 4.084 Å. How a different spacing would shift every strain threshold is set out on the wells page.

[05.6 · THE PUMPED ROUTE]

Pump it like a swing

A child on a swing can go higher without being pushed by standing up as the swing passes the bottom and crouching at each end. That shortens and lengthens the swing twice in every full swing. If the child keeps that rhythm, each pass goes higher than the last.

[FIG 5]Schematic of pumping a swing

The hydrogen version. Make the well stiffer and softer over and over at twice the frequency of the collective mode. Each cycle of that rhythm feeds a little energy into the vibration. If the gain beats the loss, the vibration grows by itself. It locks to the rhythm of the pump in one of two phases half a period apart. This is parametric resonance.

How hard to pump. The threshold is the fractional change in stiffness at which the gain just equals the loss: 1.9 percent for NiH, 1.0 for PdH and 1.6 for PdD. It is set by how fast the vibration loses energy, and that loss rate has not been measured for any of the three metals. The values here assume a loss rate of 0.48 per picosecond for palladium and 1.26 for nickel. A faster loss raises the threshold in proportion.

Collective mode, pump rhythm and threshold
Material	Collective mode THz	Pump rhythm THz	Threshold % of stiffness
NiH	20.7	41.4	1.94
PdH	15.1	30.1	1.02
PdD	9.6	19.3	1.60

[FIG 6]An illustration of how parametric resonance works

Material

Stiffness modulation

2.80%

What to notice: below the marked threshold the swing dies away, and above it the swing grows. This is one oscillator with the computed collective frequency of the chosen material and the loss rate given above. Its stiffness is raised and lowered by ± the slider value at twice that frequency, and it is solved step by step in your browser. It only illustrates the mechanism. The lattice simulation is Model 5. The threshold marks are the model's values.

[05.7 · THE DETAIL]

Equations, method and numbers

The single-site well

(1)H = p²/2m + ½K Σ_a u_a² + ¼A Σ_a u_a⁴ + ½B Σ_a<b u_a²u_b²

u = (u_x, u_y, u_z) is the displacement of the hydrogen from the centre of its octahedral site along the cube axes (Å), p its momentum and m the mass of H or D. K = K_ii is the on-site stiffness for one hydrogen moved with every other nucleus held (eV/Å²). A = K₄[100] and B = (3K₄[111] − A)/2 (eV/Å⁴), chosen so that the well is ½Ku² + ¼K₄[100]u⁴ along a cube axis and ½Ku² + ¼K₄[111]u⁴ along a cube diagonal. The levels come from direct diagonalisation in a harmonic-oscillator basis of 10 states per axis in each parity block (blocks of up to 1000 states). Going from 8 to 10 states per axis moves no Pd level by more than 0.0007 meV.

The collective mode

(2)1 = J₀ χ(E), χ(E) = Σ_n 2ε_n |⟨n|u_x|0⟩|² / (ε_n² − E²)

ε_n = E_n − E₀ are the excitation energies of the well of eq. (1) and ⟨n|u_x|0⟩ the matrix elements of the displacement; χ is the response of one site (Å²/eV), with every computed transition kept. J₀ = Σ_j J_ij is the summed pull of all other hydrogens (eV/Å², from Model 3). The root E is the energy of the collective mode with all hydrogens in step (the random-phase approximation at zero wavevector). For a harmonic well it reduces to E = ħ√((K − J₀)/m). The single-atom value is ε₁.

The static route

(3)r(T, ε) = K − J₀ + ΔK(ε) + (3A + 2B) ⟨u_x²⟩_T

r is the collective stiffness (eV/Å²) and ε the uniform strain. ΔK(ε) is the computed change of the collective stiffness with strain. For Pd it is −20.50ε + 88.57ε² eV/Å² from the strain law fitted to all computed PdD strain cases. For NiH it is −23.06ε eV/Å² from the two-atom cells at ±1%. ⟨u_x²⟩_T is the mean-square displacement along one axis from the levels of eq. (1), in the ground state at 0 K or thermally averaged at 300 K. The last term is the Hartree (mean-field) treatment of the quartic terms. Averaged over the spread of the hydrogen, they add (3A + 2B)⟨u_x²⟩ to the curvature. The spread and the quartic coefficients are held at their unstrained values. The ordering strain is the smallest ε > 0 with r = 0; beyond it the displacement grows along the cube diagonal, where the walls rise least.

The pump threshold

(4)q̈ + κ q̇ + Ω²[1 + h cos 2Ωt] q = 0, h_th = 2κ/Ω

q is the amplitude of the collective mode, Ω = 2πf its angular frequency, h the fractional modulation of its stiffness and κ the loss rate (an undriven amplitude decays as e^−κt/2). Above h_th the amplitude grows; FIG 6 integrates this equation directly. The values used are κ = 0.484 ps⁻¹ for Pd (amplitude decay time 4.14 ps) and 1.259 ps⁻¹ for NiH (1.59 ps). A light field could change the stiffness only through its square, so the light itself would sit near the mode frequency rather than at twice it. How strongly it couples was not computed.

Method

Inputs: The single-site wells and summed pulls of Model 3: 64-atom cells, one hydrogen moved, 500 eV, 3×3×3 k-points, at full loading
Lattice: NiH a = 3.732 Å; PdH and PdD a = 4.084 Å, the same electronic well with the H or D mass
NiH quartics: 18.32 and 1.13 eV/Å⁴ from the earlier, coarser 64-atom calculation; the recomputed [100] value 16.17 enters only the single-atom check in Table C
Levels: Exact diagonalisation of eq. (1), parity blocks, NumPy
Collective: Eq. (2), all computed dipole transitions kept
Static: Eq. (3) at 0 K and 300 K
Pumped: Eq. (4), on resonance, weak drive
Compute: 4.1 s on one thread for both Pd isotopes; 1.7 s for the NiH check with the recomputed quartic

g3_lattice/rerun/part1 g3_lattice/rerun/part2 g2_refit

A · Inputs eV/Å², eV/Å⁴

Material	K_ii	J₀	K_ii − J₀	K₄[100]	K₄[111]	B
NiH	1.712	0.676	1.036	18.32	1.128	−7.467
PdH	0.591	0.389	0.202	11.62	2.228	−2.466
PdD	0.591	0.389	0.202	11.62	2.228	−2.466

PdH and PdD share one electronic well; only the mass differs.

B · Levels meV above the bottom of the well

Material	Lowest	First	Second, two axes	Second, one axis (a)	Second, one axis (b)	Perfect-bowl step
NiH	140.6	241.2	336.8	352.3	361.8	84.3
PdH	97.1	171.1	242.5	259.3	264.7	49.5
PdD	65.1	114.0	161.3	171.4	174.5	35.0

The first level holds one quantum along any of the three axes. The second level splits into one quantum along each of two axes or two along the same axis, and the cubic term mixes the latter into two levels. Steps from the first level: NiH 95.6, 111.1, 120.6; PdH 71.4, 88.2, 93.6; PdD 47.3, 57.4, 60.5. The perfect-bowl step is ħ√(K_ii/m).

C · Vibration energy against measured meV

Material	One atom, ε₁	All in step, eq. (2)	Measured peak	All in step against peak
NiH	100.6	85.6	89	−3.8%
PdH	74.0	62.3	55.8	+11.6%
PdD	48.9	39.8	37	+7.6%
NiH, recomputed K₄[100] = 16.17	99.3
H/D ratio, palladium	1.515	1.563	1.508	perfect bowl 1.414

With the recomputed NiH quartic the two-step lines move to 94.9, 109.3 and 117.6 meV above the first level. The NiH collective mode and the static and pumped NiH results were not recomputed with it.

D · Static route eq. (3)

Material, T	Spread, rms Å per axis	Stiffening eV/Å²	r unstrained eV/Å²	Ordering strain
NiH, 0 K	0.143	0.823	1.860	8.07%
NiH, 300 K	0.146	0.854	1.890	8.20%
PdH, 0 K	0.167	0.834	1.036	7.46%
PdH, 300 K	0.175	0.915	1.117	8.77%
PdD, 0 K	0.145	0.633	0.834	5.27%
PdD, 300 K	0.164	0.803	1.005	7.05%

Stiffening is (3A + 2B)⟨u_x²⟩, with 3A + 2B = 40.02 eV/Å⁴ for NiH and 29.91 for Pd.

E · Pumped route eq. (4), on resonance

Material	Mode f THz	Pump 2f THz	Loss κ ps⁻¹	h_th collective stiffness	h_th lattice stiffness
NiH	20.70	41.39	1.259	1.94%	1.85%
PdH	15.06	30.11	0.484	1.02%	0.93%
PdD	9.63	19.26	0.484	1.60%	1.47%

The last column is the same threshold expressed as a modulation of the zero-point-stiffened lattice stiffness that the lattice model of Model 5 uses.

[WHAT THIS MEANS FOR THE NEXT MODEL]

From one site to a million

The oscillator fixes what each site brings: its levels, the stiffening from its own spread, its collective frequency and the rhythm that pumps it. It cannot say whether the hydrogens stay in step over a large region or what empty sites do. The next model puts these sites on a lattice of up to a million, with loading, vacancies, stretch, temperature and pump as settings.

MODEL Model 5: many sites together Next: do the hydrogens move in step on a large lattice? MODEL Model 3: the computed proton wells Previous: the wells and pulls this model starts from SOURCE Rerun on the corrected wells, part 1 The Pd oscillator: levels, collective mode, static and pumped routes SOURCE Rerun on the corrected wells, part 2 The NiH oscillator and the recomputed quartic check SOURCE The first run The same model on the earlier, coarser wells

In short