[02 / MODEL 1]

Protons and the field

[Facing mirrors · a light beam · beads that stay still]

Each hydrogen nucleus in a metal is a charge that vibrates in its own small pocket. The starting idea for a coherence domain is that the electromagnetic field the nuclei share could lock those vibrations together. Model 1 writes that idea down in its simplest exact form. The field on its own does not make the nuclei act together.

[IN SHORT]

The field alone does not order the nuclei

The model treats every hydrogen nucleus as a charged mass on a spring and couples it to the electromagnetic field, which also carries light.
It solves the motion exactly and asks whether the slowest pattern of motion becomes unstable. That is how an ordered state would first show itself.
When the field is treated the same way in both places it enters the energy, the answer is no at every coupling strength.
An instability appears only when a direction factor of 8π/3 is applied to one of those places and not to the other. Model 2 therefore looks for a different ingredient.

0 unstable modes at any coupling when the field is treated the same in both terms

0.3455 coupling above which the factor-on-one-term version turns unstable

8π/3 the direction factor that makes the difference

g0_stability g0_scales

[02.1 · THE IDEA]

Charges on springs in one shared field

The picture behind a coherence domain is simple to state. This model keeps it that simple so that it can be solved without approximation.

[01]

A charge on a spring

A hydrogen nucleus sits in a pocket between metal atoms and vibrates about its centre like a mass on a spring. It carries a charge, so its motion stirs the electromagnetic field and the field pushes back on it.
[02]

One field links them all

Every nucleus feels the same field, so in principle the field could draw them into a single shared motion. That is the starting idea for a coherence domain: a region where many nuclei move in step.
[03]

Order shows as a soft mode

The model finds the normal modes: the patterns in which everything moves at one frequency. If the slowest pattern's frequency falls to zero, the evenly spread state is unstable and the nuclei settle into a shared displacement. If it stays above zero, nothing orders.

[02.2 · TWO TERMS]

The field enters the energy twice

The energy of a charge moving in a field is a square. Its momentum is shifted by the field and then squared. Opening the square gives three kinds of piece. One is the motion of the nucleus. Two are coupling pieces, in which the nucleus and the field trade energy (physicists call this the p·A term). The last is a corner piece called the A² term, which stiffens the field wherever charges sit.

One common way of reducing the problem comes from Gamberale and Modanese (arXiv:2205.14015). It adds up the coupling over every direction in which the field can travel. That multiplies the coupling pieces by a fixed factor. If the same factor also multiplies the corner, the square stays whole. If it multiplies the coupling only, part of the corner is left out.

[SAME FIELD IN BOTH TERMS]

A complete square. It is never negative, so no pattern of motion can pull the energy below its resting value. Every mode is stable.

[FACTOR ON THE COUPLING ONLY]

A square with a piece missing. The energy is the full square minus the left-out part of the corner. Once the coupling is strong enough, that difference can fall below zero.

[HOW TO READ IT]

Motion of the nucleus. The energy of its own vibration.
Coupling (the p·A term). Nucleus and field trade energy. The direction sum multiplies each piece by √s.
The A² term. It stiffens the field where charges sit. When it is kept whole, it carries the full factor s.
Left out. When only the coupling carries the factor, s − 1 parts of the corner out of s are missing.

s = 8π/3 is the factor from summing over every direction of travel. The cut corner is drawn to scale for that value.

[02.3 · THE RESULT]

The slowest mode stays stable at every coupling

The chart follows the slowest pattern of motion as the coupling grows. Above the zero line it is a stable vibration. Below it, the evenly spread state gives way and the nuclei would order. Move the slider or hover over the chart.

[FIG 02.3] Lowest normal mode against coupling

Direction factor

Coupling ε 0.52

The chart needs JavaScript. Every value it shows is in the table of normal modes below.

At ε = 0.52

Same field in both terms 0.192 stable

Factor on the coupling only −0.505 unstable

Values are ν²/ω′², the slowest mode's frequency squared in units of the oscillator frequency squared.

What to notice: the dashed curve crosses zero at ε = 0.3455 and keeps falling. The solid curve never crosses; it only reaches zero at ε = 1, where the bare field frequency itself is zero. Dots are the values in stability.json; the lines are the closed forms from the same script.

[WHERE THE MATERIALS SIT]

The coupling ε is the plasma frequency of the nuclei divided by their oscillator frequency. The plasma frequency follows from how densely the nuclei are packed. The published scales are an oscillator at 0.41 eV and a proton plasma frequency of 0.2 to 0.3 eV, which puts ε in the shaded band at about 0.5 to 0.7.

NiH

0.240 eV

Bare proton plasma energy at one hydrogen per nickel atom. It lies inside the published range. Divided by the published 0.41 eV oscillator energy, it gives ε ≈ 0.59. The chart marks that point.
PdD

0.149 eV

Bare deuteron plasma energy at one deuterium per palladium atom. The published work treats PdD as a different mechanism and gives it no oscillator energy, so it has no point on the chart.
The solid curve is above zero for every ε below 1. Where a material sits on the axis does not change the answer.

[02.4 · WHY]

A square is never negative

The energy of a charge in a field is its shifted momentum squared plus the energy of its pocket, which is never negative. However the field is arranged, that total cannot fall below what the nuclei have with no field at all.

An ordered state needs the energy to fall. Coupling of this kind cannot provide that in this model or in any reduction of it that treats the field consistently. The ingredient that orders the nuclei has to come from somewhere else.

[02.5 · THE DETAIL]

The model in equations

The model is a quadratic Hamiltonian of charged oscillators and field modes in the dipole approximation, solved exactly by normal modes. It makes no rotating-wave approximation and keeps every counter-rotating term. It has no fitted parameters.

1. The starting energy

H = (p + eA)² / 2m + V, V ≥ 0(1)

This is for one nucleus of mass m, charge e and momentum p in a pocket potential V and a field with vector potential A. Opening the square gives p²/2m, the coupling term (e/m) p·A and the diamagnetic term e²A²/2m. Every piece of (1) is non-negative, so no consistent reduction of it can lower the energy below the uncoupled ground state.

2. Bright and dark modes

N identical oscillators in a dipole-sized region couple to the field only through their uniform, collective motion. That gives one bright block per Cartesian direction: one matter mode and one field mode. The other 3(N − 1) matter modes are dark and keep ν² = ω′². Everything below is about the bright block, in units where ω′ = 1 unless written out.

3. The factor on the coupling only

In the published reduction the A² term is absorbed mode by mode into the field frequency, ω′ = √(ω² + ω_p²). The p·A coupling is then summed over all propagation directions, which multiplies it by √s. With every counter-rotating term kept:

H = ω′ (x² + p² + q² + r²) / 2 + √s ω_p p q(2)

ν² = ω′² (1 − √s ε), ε = ω_p / ω′(3)

The lowest mode reaches zero at ε = 1/√s = √(3/(8π)) = 0.3455 and is unstable above it. The result is exact; the rotating-wave approximation plays no part in it. For the paper's fixed-axis comparison with s = 4, the crossing is at ε = 0.5.

4. The same field in both terms

If the field that enters p·A (uniform over the sites, summed over directions) is also the field that enters A², the direction sum multiplies both:

H = (P + gQ)²/2 + ω′²X²/2 + Π²/2 + ω²Q²/2, g = √s ω_p(4)

This is a sum of squares and cannot have an unstable mode. The squared frequencies z = ν²/ω′² are the roots of

z² − Bz + C = 0, B = 2 + (s − 1)ε², C = 1 − ε²(5)

Both roots are positive for 0 ≤ ε < 1, since their product C and their sum B are positive. The lower root reaches zero only at ε = 1, where the bare field frequency ω = ω′√(1 − ε²) is zero.

5. Where the two differ

Written in the variables of (4), block (2) is

H₍₂₎ = H₍₄₎ − (s − 1) ω_p² Q²/2(6)

The two differ by exactly the part of the A² term that the direction factor did not multiply, which is the left-out corner of the square in figure 02.2. That missing piece is the source of the instability in (3).

Symbols

x, p · X, P: collective (bright) matter coordinate and momentum; lower case scaled by ω′
q, r · Q, Π: field-mode coordinate and momentum, scaled and unscaled
ω′: oscillator frequency; also the field frequency once the A² term is absorbed
ω: bare field-mode frequency
ω_p: plasma frequency of the nuclei, √(n e²/ε₀m), for number density n and bare charge e
ε: coupling, ω_p/ω′ (not to be confused with the vacuum permittivity ε₀)
s: direction factor: 8π/3 for the sum over all propagation directions, 4 for the fixed-axis comparison
ν: normal-mode frequency; ν² < 0 means the mode is unstable

Lowest normal mode (ν²/ω′²)

The table gives every row of g0_stability/stability.json. Both matrices were also diagonalised directly as a cross-check and agree with the closed forms.

ε	s = 8π/3 · ALL DIRECTIONS		s = 4 · FIXED AXIS
ε	FACTOR ON COUPLING ONLY	SAME FIELD IN BOTH	FACTOR ON COUPLING ONLY	SAME FIELD IN BOTH
0	1.000	1.000	1.000	1.000
0.1	0.711	0.745	0.800	0.814
0.25	0.276	0.471	0.500	0.585
0.3455	0	0.348	0.309	0.465
0.4	−0.158	0.291	0.200	0.405
0.5	−0.447	0.206	0	0.307
0.52	−0.505	0.192	−0.040	0.289
0.56	−0.621	0.165	−0.120	0.256
0.69	−0.997	0.097	−0.380	0.160
0.75	−1.171	0.072	−0.500	0.123
0.9	−1.605	0.024	−0.800	0.043
0.99	−1.865	0.002	−0.980	0.004
1	−1.894	0	−1.000	0

Measured scales

Values are from g0_scales/scales.json, with four metal atoms per cubic cell. The plasma energy ħ√(n e²/ε₀m) uses the bare nuclear charge and is not a screened mode. Ti/H is the other host the published work treats.

MATERIAL	LATTICE a (Å)	SITE	H OR D PER METAL	DENSITY OF NUCLEI (m⁻³)	BARE PLASMA ENERGY (eV)	OPTIC VIBRATION (meV)	MEASURED VALUES FROM
Ni/H	3.732	octahedral	1	7.70 × 10²⁸	0.240	89	Antonov et al., EPL 51, 140 (2000)
Ti/Hδ-TiH₂	4.454	tetrahedral	2	9.05 × 10²⁸	0.261	149	Khidirov and Kayumova (2012), INIS 44129203; Hattori et al., PRB 106, 134309 (2022)
Pd/D	4.084	octahedral	1	5.87 × 10²⁸	0.149	37	Antonov et al., PRB 80, 134302 (2009)

This is the quadratic model in the dipole approximation as the paper sets it up. It does not cover a finite domain treated beyond the dipole approximation, anharmonic wells or any mechanism outside minimal coupling. The paper's gauge argument about what counts as radiation was not evaluated.

g0_stability: script, output, README g0_scales: measured inputs Gamberale and Modanese, arXiv:2205.14015

[WHAT THIS MEANS FOR THE NEXT MODEL]

Keep the field. Look for what can lower the energy.

Coupling to the field leaves the nuclei disordered at every coupling strength when it is treated consistently. It stays in the next model as a stable baseline. Model 2 adds five candidate ingredients on top of it and tests them alone and in every combination. They are dipole attraction between the hydrogen nuclei, anharmonic wells, coupling to the metal's electrons, drive and loss, and a gradient term between neighbouring sites.

NEXT · MODEL 2 What could make them order Five candidate ingredients tested in every combination PREVIOUS · 01 The method Start with the cheapest whole model and add resolution where it matters SOURCE g0_stability in the repository stability.py, stability.json and the README with the closed forms