[Springs, steel balls, bar magnets, tuning fork · parts before assembly]

[03 / MODEL 2]

What could make
them order

Coupling the hydrogen nuclei to the light field leaves each one vibrating on its own. This model puts five candidate ingredients into one small model of hydrogen in a metal, switches each on and off, and records which combinations give a stable state in which the hydrogens move together.

g0_model g1_rank

[COMBINATIONS] 32 on and off settings of five ingredients, run for NiH, TiH_1.84 and PdD
[ORDER BY ITSELF] 1 pair attraction between neighbours plus a well that stiffens
[ROUTES] 2 static: the hydrogens shift together. Pumped: a drive sustains a collective oscillation
[WHAT DECIDES IT] 5.0 decades moved by the screening length, which ranks first

[IN SHORT]

Two ingredients and two routes

One pair of the five ingredients produces a stable ordered state on its own. The pair is an attraction between neighbouring hydrogens and a pocket that stiffens as a hydrogen moves further out. With that pair the hydrogens all shift the same way and stay there. A ferroelectric crystal orders in the same way.

Whether a real hydride reaches that point depends on how strongly the metal's electrons screen the attraction. With the simplest estimate of that screening, none of the three hydrides does.

A second route does not need weak screening. A pump that changes the stiffness of each pocket at twice the vibration frequency can keep a collective oscillation going.

Both routes turn on two numbers the metal sets, and both can be computed.

1.55 Å screening NiH needs to order. The estimate is 0.56 Å

1.09 Å screening PdD needs. The estimate is 0.59 Å

1.6–1.8% stiffness modulation that starts the pumped oscillation

5.0 decades moved by screening, which ranks first

[03.1 · THE INGREDIENTS]

Five ingredients in every combination

The model follows one hydrogen nucleus that vibrates in its pocket in the metal and couples to the light field as in the previous model. Five extra ingredients can each be switched on or off. Five switches make 32 combinations, and each was run for three hydrides at two strengths of electron screening.

D[01]

Dipole attraction

A nucleus that moves off-centre pulls its neighbours the same way through its electric charge. The metal's electrons crowd round each nucleus and weaken this pull beyond a distance called the screening length.

Alone: with weak screening the shift runs away. Nothing stops it.
Q[02]

Anharmonic well

The pocket gets steeper the further the hydrogen moves. This is a quartic term in its energy. It cannot cause order, but it stops a shift from growing without limit.

Alone: no order.
E[03]

Electron response

The conduction electrons follow a moving nucleus and push back on it.

Alone: no order.
L[04]

Drive and loss

An outside force shakes the hydrogens at their vibration frequency, and friction drains the motion.

Alone: a forced response that stops when the drive stops.
G[05]

Gradient

Neighbours that move differently cost energy, so any pattern is smoothed over a correlation length.

Alone: no order, and no domain size is picked.

[FIG 03.1]32 combinations × 3 hydrides × 2 screening lengths

Hydride

Electron screening

Ingredients

NiH · weak screening, 5 Å · D Q

Ordered

The hydrogens all shift the same way and stay shifted. The state lasts without any drive.

Threshold loading: 0.55 H or D per metal atom
Shift of each hydrogen: 0.22 Å
Energy gained: 22 meV per H or D
Vibration frequency: 29.2 THz
Ring-down after the drive stops: – ps
Response to a local drive: – nm across
Correlation length: – nm

NiHmetal's estimate, 0.56 Å

NiHweak, 5 Å

TiH_1.84metal's estimate, 0.61 Å

TiH_1.84weak, 5 Å

PdDmetal's estimate, 0.59 Å

PdDweak, 5 Å

Ordered
Ordered with a forced response
Forced response only
Runaway (rejected)
No order
Rows: D dipole attraction, Q anharmonic well. Columns: E electron response, L drive and loss, G gradient. · means none.

Each square is one combination of switches. Notice that order (amber or orange) appears only with weak screening and only in the bottom row, where D and Q are both on. With the metal's own screening estimate, nothing orders in any hydride. Select a square or flip the switches to read its numbers. Source: g0_model/domain.json

Drive and loss on their own give a forced response. The hydrogens follow the drive at 21.5 THz in NiH, 36.1 THz in TiH_1.84 and 8.9 THz in PdD. The response to a local drive is about 2 to 2.6 nm across. It rings down within 0.9 to 4 ps once the drive stops, so it lasts only as long as the drive.

The gradient term gives a correlation length but does not pick a domain size. An ordered region in this model has no preferred size.

[03.2 · ROUTE ONE: STATIC]

The lattice softens and then the hydrogens shift together

The attraction lowers the stiffness of every hydrogen's pocket in proportion to how much hydrogen is loaded. At a threshold loading the stiffness reaches zero. Above it the hydrogens all settle at a position shifted in the same direction. Ferroelectric crystals order in the same way. The quartic term sets how far they shift.

[FIG 03.2]Threshold against screening

Screening length 0.56 Å

Threshold loading (H or D per metal atom) against screening length λ (Å). A hydride orders where its curve falls below its full loading (dashed). Notice that the metal's own estimate (open circles) sits well to the left of where NiH and PdD cross, and that TiH₂ never crosses. The curves come from the closed form in g1_rank

[FIG 03.3]The vibration softens and then stiffens

Hover or tap the chart to read the frequency and the shift at each loading.

Lowest vibration frequency (THz) against loading (H or D per metal atom), with weak 5 Å screening, all five switches on and the drive off. Notice the dip to nearly zero at each threshold (dotted). That softening is the sign that the hydrogens are about to shift. Above it the frequency rises as the square root of the excess loading. The curves are evaluated with g0_model/domain.py

[WITH WEAK 5 Å SCREENING]

NiH orders above 0.55 H per Ni atom and PdD above 0.29 D per Pd atom. At full loading each hydrogen shifts by 0.22 Å in NiH and 0.43 Å in PdD. The ordered state is lower in energy by 22 meV per H and 104 meV per D. TiH_1.84 would need 2.4 H per Ti and does not order.

[WITH THE METAL'S OWN ESTIMATE]

The simplest (Thomas-Fermi) estimate of screening is 0.56 to 0.61 Å. The thresholds are then 9.5 H per Ni, 18.3 H per Ti and 6.0 D per Pd, which are all far above full loading. To order at full loading NiH needs 1.55 Å and PdD 1.09 Å. TiH₂ does not order even with no screening at all.

[THE OPEN NUMBER]

The 5 Å value is a placeholder chosen to show what order looks like. The first input the next model computes is the real screening between neighbouring hydrogens at the distance that matters.

[03.3 · ROUTE TWO: PUMPED]

A pump keeps a collective oscillation going

If energy flows into the hydrogen vibration faster than friction drains it, the vibration grows until its own nonlinearity caps it. It then keeps going at a steady amplitude, much as a laser does above threshold. In the model this happens once the gain G exceeds the loss κ. It does not need weak screening.

[AT G = 2κ AND 300 K]

NiH

23.5 THz

Linewidth
8.4 GHz

Gain length
4.8 nm

Decay after the pump stops
1.6 ps
TiH₂

39.5 THz

Linewidth
21.4 GHz

Gain length
4.1 nm

Decay after the pump stops
0.9 ps
PdD

9.7 THz

Linewidth
3.4 GHz

Gain length
5.3 nm

Decay after the pump stops
4.1 ps

Which pump could do it

Four ways to feed the vibration were tested against what a laboratory can deliver.

NO

Steady or pulsed current

The electrons have to drift faster than the vibration wave travels, which takes more than 10¹⁴ A/m². A 10¹³ A/m² pulse reaches 3 to 10 percent of that.
NO

Hot hydrogen arriving during loading

In PdH one atomic layer needs about 6 × 10³⁰ arrivals per m² per second. That takes about 270 bar of H₂ with every atom sticking hot, and a 5 nm layer needs about 5000 bar. Heating a harmonic vibration does not invert it in any case.
AS A PROBE

Light at the vibration frequency

It gives a forced response at any strength and has no threshold. At 21.5, 36.0 and 8.9 THz (NiH, TiH₂, PdD) an internal field of 2 to 11 MV/m gives a 0.02 Å amplitude. It is useful for measuring coupling and loss.
YES

Parametric drive at twice the frequency

Modulating the pocket stiffness by 1.6 to 1.8 percent starts the oscillation. That corresponds to a strain near 0.3 percent, and existing mid-infrared pulsed lasers can reach it.

[FIG 03.4]Schematic only

A child on a swing pumps it by standing and squatting twice per swing. Here the pump stiffens and relaxes each hydrogen's pocket at twice the vibration frequency, and the vibration grows at exactly half the pump frequency.

The parametric pump is the one carried forward. The vibration locks to half the pump frequency, so the pump runs at 46.9 THz for NiH, 78.7 THz for TiH₂ and 19.4 THz for PdD. It has a threshold, a saturated amplitude and two possible phases.

The strain figure uses a placeholder that assumes the stiffness changes five times faster than the strain. The computed values on the wells page are 22 for NiH and 113 for PdD. The strain needed is smaller in proportion, by a factor of about 4 for NiH and 23 for PdD. What is still missing is how strongly a pump field changes the pocket stiffness in a loaded hydride.

[03.4 · WHAT MATTERS MOST]

What the answer depends on

Several inputs here are placeholders. Each was swept across a plausible range while the others stayed fixed, and the ranking counts how many factors of ten (decades) the results moved. Screening leads by a wide margin. Three inputs can switch order on or off; the rest change only how large the effect is.

[FIG 03.5]Sensitivity of the static route

Computable by electronic structure
Look up or measure
Neither
Can switch order on or off

Screening length λ ◆, can switch order on or off

5.0

0.2 to 5 Å · classification: computable by electronic structure
Quartic coefficient β

2.0

0.1 to 10 · classification: computable by electronic structure
Loss γ

2.0

0.001 to 0.1 · classification: look up or measure
Choice of vibration peak E₀ ◆, can switch order on or off

1.6

peak or shoulder: NiH 89 to 130 meV, PdD 37 to 65 meV · classification: look up or measure
Gradient stiffness c

1.0

0.01 to 1 · classification: computable by electronic structure
Loading x ◆, can switch order on or off

0.7

NiH 0.01 to 1, TiH₂ 1.74 to 2, PdD 0.6 to 1 · classification: look up or measure
Field mode frequency ω_c

0.28

5 to 20 E₀ · classification: neither
Conduction electrons per metal atom n_e

0.10

0.5 to 2 · classification: computable by electronic structure
Lattice constant a

0.02

NiH 3.726 to 3.738, TiH₂ 4.453 to 4.455, PdD 4.080 to 4.088 Å · classification: look up or measure
Electron response strength χ

0.001

0 to 4 · classification: computable by electronic structure
Temperature T

0

5 to 600 K · classification: look up or measure

Each bar is the largest number of decades that any of seven results (threshold loading and density, frequency, shift, energy, correlation length, ring-down) moved across the input's range in any of the three hydrides. Notice that the top input is screening, which an electronic-structure calculation can supply. Source: g1_rank/rank.json

[03.5 · WHAT AN EXPERIMENT COULD SEE]

Three signatures

[STATIC ROUTE]

A hydrogen vibration slows as loading approaches the threshold. Then a static distortion appears, and the vibration stiffens as the square root of the excess loading.
[PUMPED ROUTE]

Under a pump at twice the vibration frequency there is a threshold in pump strength. Above it, an oscillation at exactly half the pump frequency saturates, and it decays over picoseconds once the pump stops.
[ISOTOPE]

In the same metal, the D frequency is close to the H frequency divided by √2, while the static threshold and the ordering energy do not change.

[THE DETAIL]

Equations, inputs and tables

This is a mean-field model of one representative proton or deuteron in reduced units, with five switchable terms. Scripts: g0_model/domain.py g1_rank/rank.py g1_rank/pumps.py

1. The model

Energies are in units of E₀, the measured optic vibration energy of the host (89 meV for NiH, 149.4 meV for TiH_1.84, 37 meV for PdD). Lengths are in ℓ₀ = ħ/√(mE₀) and times in ħ/E₀. The Hamiltonian per site is

H = ½(p + gA)² + ½(Π² + ω_c²A²) + ½r q² + ¼β q⁴

+ ½[p_y² + ω_e²(y − √χ q/ω_e)²] + ½c(∇q)²

r = k₀ − x J₁(λ), J₁(λ) = ⅓(ε_p/E₀)² (1 + d/λ) e^−d/λ

q, p: displacement and momentum of the proton or deuteron
A, Π: transverse field mode and its momentum; frequency ω_c = 10E₀
g: field coupling, √x ε_p/E₀, with ε_p the proton plasma energy at one H per metal atom
y, p_y: conduction-electron response (switch E) at the electron plasma energy ω_e with strength χ =1; the counterterm keeps it from shifting the static stiffness
r: effective stiffness of the pocket; k₀ is fixed once so that the disordered model reproduces E₀ at the reference loading with Thomas-Fermi screening
x: loading, H or D per metal atom
J₁, λ, d: screened dipole attraction per unit loading (switch D), screening length, and exclusion radius d = a/√2 for NiH and PdD, a/2 for TiH (2.64, 2.89 and 2.23 Å)
β: quartic coefficient (switch Q), 1
c: gradient coefficient (switch G), 0.1d²

Switch L adds a viscous force of −γ times the velocity on each coordinate, with γ = 0.01. It also adds a drive of amplitude f = 0.001 at the lowest mode. The forced amplitude s = |q|² follows the first-harmonic Duffing relation s[(Re D + ¾βs)² + (Im D)²] = f², with D(ν) the linear response of the coupled proton, field and electron system. The Thomas-Fermi screening length is λ_TF = ½√(πa_B/k_F), with k_F = (3π²n_e)^1/3 for one conduction electron per metal atom.

2. Closed forms of the static route

x_c = k₀ / J₁(λ)

q₀² = −r/β (for r < 0)

ΔE = −E₀ r² / (4β)

ξ = √(c/k), k = r (disordered) or −2r (ordered)

Order needs r < 0, which happens when the loading x exceeds x_c. Without the quartic term (β = 0) a negative r has no minimum and the case is rejected as a runaway. Above threshold the curvature −2r = 2J₁(x − x_c) grows linearly with the excess loading, so the vibration frequency grows as its square root. ξ is a correlation length; a positive gradient term does not select a finite domain.

Screening needed to order at full loading
Material	Critical screening λ (Å)	Thomas-Fermi estimate (Å)
NiH	1.55	0.56
PdD	1.09	0.59
TiH₂	none: does not order even unscreened	0.61

Full loading: 1 for NiH and PdD, 2 for TiH₂. From g1_rank/rank.json.

3. Forced response (switch L)

Forced response with all five switches on and Thomas-Fermi screening
Material	Frequency (THz)	Response to a local drive (nm across)	Ring-down (ps)
NiH	21.5	2.4	1.6
TiH_1.84	36.1	2.0	0.9
PdD	8.9	2.6	4.1

Computed with all five switches on and Thomas-Fermi screening. Size is the e-fold diameter of the response to a localised drive. From g0_model/domain.json.

4. The ordered state

[RESULTS WITH D AND Q ON]

Static route results with dipole attraction and quartic well on
Material	Screening λ (Å)	Threshold x_c	Ordered at reference loading	Shift (Å)	Energy (meV per H or D)
NiH	0.56 (Thomas-Fermi)	9.5	no
TiH_1.84	0.61 (Thomas-Fermi)	18.3	no
PdD	0.59 (Thomas-Fermi)	6.0	no
NiH	5	0.55	yes	0.22	−22
TiH_1.84	5	2.4	no
PdD	5	0.29	yes	0.43	−104

Switches D and Q are on. The reference loading is 1 for NiH and PdD and 1.84 for TiH_1.84. From g0_model/domain.json.

[ISOTOPE SWAP]

The physical spring and quartic coefficients are held fixed and only the mass changes. The frequency of the ordered state's vibration moves; the threshold and the ordering energy do not.

Matched isotope comparison at 5 angstrom screening
Host	Vibration as loaded (THz)	Isotope swapped (THz)	Threshold x_c, both	Energy (meV), both
NiH → NiD	29.2	21.0	0.55	−22
PdD → PdH	21.4	28.1	0.29	−104

Computed with 5 Å screening, all switches on and the drive off. From g1_rank/rank.json.

5. The pumped route

dn/dt = (G − κ) n − s_n n²

ℓ_g = 2v_g / (κμ), μ = G/κ − 1

(δk/k)_th = 2κ/Ω

n is the occupation of a proton optic mode at wavevector qd = π/2, G the gain, κ the energy loss rate and s_n the saturation. Above G = κ the mode settles into a self-sustained oscillation. ℓ_g is the gain length, with v_g the group velocity. For a parametric pump at 2Ω the threshold is a fractional stiffness modulation of 2κ/Ω; the strain threshold divides this by d ln k/dε, taken as 5 for illustration.

Pump candidates
Pump	Threshold	Reachable
Steady or pulsed current	electron drift faster than the mode's phase velocity: above 10¹⁴ A/m² (1.05 to 3.05 × 10¹⁴); a 10¹³ A/m² pulse reaches 3.3 to 9.5%	no
Hot hydrogen during loading (PdH)	arrival flux above 5.8 × 10³⁰ m⁻² s⁻¹ for one atomic layer (270 bar of H₂, every atom sticking hot); 4986 bar for a 5.3 nm layer	no
Resonant light	none: a forced response at any strength; 2.0 to 11.2 MV/m internal field gives 0.017 to 0.024 Å	as a probe
Parametric, at twice the mode frequency	stiffness modulated by 1.6 to 1.8%	yes, mid-infrared pulsed lasers

Self-sustained oscillation at G = 2 kappa and 300 K, and the parametric pump
Material	Frequency (THz)	Linewidth (GHz)	Gain length (nm)	Decay (ps)	Parametric pump (THz)	Stiffness threshold (%)	Strain threshold (%)
NiH	23.5	8.4	4.8	1.6	46.9	1.7	0.34
TiH₂	39.5	21.4	4.1	0.9	78.7	1.8	0.36
PdD	9.7	3.4	5.3	4.1	19.4	1.6	0.32

At G = 2κ and 300 K. From g1_rank/rank.json and g1_rank/pumps.json.

6. Ranking of the inputs

Each input is swept over 41 values across its range while the others are held at the weak-screening baseline (5 Å, all switches on, drive off). For each of seven results y, the spread is log₁₀(max|y| / min|y|); the score is the largest spread over results and hydrides. A result that passes through zero because order switches on or off is flagged rather than scored.

Sensitivity ranking
Input	Range swept	Decades	Switches order	Can be supplied by
Screening length λ	0.2 to 5 Å	5.03	yes	computable by electronic structure
Quartic coefficient β	0.1 to 10	2.00	no	computable by electronic structure
Loss γ	0.001 to 0.1	2.00	no	look up or measure
Choice of vibration peak E₀	peak or shoulder: NiH 89 to 130 meV, PdD 37 to 65 meV	1.58	yes	look up or measure
Gradient stiffness c	0.01 to 1	1.00	no	computable by electronic structure
Loading x	NiH 0.01 to 1, TiH₂ 1.74 to 2, PdD 0.6 to 1	0.74	yes	look up or measure
Field mode frequency ω_c	5 to 20 E₀	0.28	no	neither
Conduction electrons per metal atom n_e	0.5 to 2	0.10	no	computable by electronic structure
Lattice constant a	NiH 3.726 to 3.738, TiH₂ 4.453 to 4.455, PdD 4.080 to 4.088 Å	0.02	no	look up or measure
Electron response strength χ	0 to 4	0.0015	no	computable by electronic structure
Temperature T	5 to 600 K	0	no	look up or measure

Ranges are exploratory, not measured intervals. From g1_rank/rank.json.

7. Inputs and placeholders

Measured references and placeholder values
Input	Value	Status
Lattice constant a	NiH 3.732, TiH_1.84 4.454, PdD 4.084 Å	measured
Optic vibration energy E₀	NiH 89, TiH_1.84 149.4, PdD 37 meV	measured (neutron scattering)
Quartic coefficient β	1	placeholder
Loss γ	0.01	placeholder
Drive amplitude f	0.001	placeholder
Gradient coefficient	0.1 d²	placeholder
Electron response strength χ	1	placeholder
Field mode frequency ω_c	10 E₀	placeholder
Conduction electrons per metal atom	1	placeholder
Weak screening length	5 Å	placeholder
Temperature	300 K	scenario

[WHAT THIS MEANS FOR THE NEXT MODEL]

Compute the two numbers

Both routes come down to quantities the metal sets. The static route needs the screened force between neighbouring hydrogens at the distance between them; the pumped route needs how much a pump field or a strain changes the stiffness of each hydrogen's pocket. The next model computes the pockets and the force constants from electronic-structure calculations.

g0_model g1_rank

MODEL Model 3: the computed proton wells Next: the shape of each hydrogen pocket and the force constants from electronic structure MODEL Model 1: protons and the field Back: why the field alone does not order the hydrogens SOURCE g0_model and g1_rank in the repository Scripts, JSON outputs and READMEs

Two ingredients and two routes

Five ingredients in every combination

Dipole attraction

Anharmonic well

Electron response

Drive and loss

Gradient