[Studio picture · steel beads forming lines, patches and rings]

[06 / MODEL 5]

A million sites

This model holds up to a million hydrogens, each in its computed well and tied to its neighbours. Pump them and measure whether they move in step, over what distance and whether patterns form.

[IN SHORT]

This model puts many hydrogens together in a simulated crystal. Each sits in its computed well, pulls on its neighbours with the computed couplings, loses energy and feels thermal noise. A pump makes every well stiffer and softer in turn. Loading, the arrangement of empty sites, stretch and temperature can all be set.

The two materials behave differently in kind. In nickel hydride (NiH) pumped at 0 K, the hydrogens move in step across the entire simulated box of 110,592 sites, and 99.8 percent of the pumped motion is in one uniform pattern. The in-step length of 3.85 nm equals the size of the box, so this run sets no upper limit on it. The run has not been repeated in a larger box or at 300 K.

In palladium deuteride (PdD) the hydrogens respond to the pump at every loading tried, but they stay in patches. They move in step over 0.82 to 1.75 nm and fall out of step beyond that.

At room temperature the pump adds nothing measurable in PdD.

g3_lattice g3_lattice/rerun

99.8% of the pumped motion in NiH is in one uniform pattern 3.85 nm in-step length in NiH, equal to the whole simulated box 1.38 nm in-step length in PdD at full loading 12% share in the strongest pattern for PdD at full loading

[06.1 · TWO METALS]

In step or in patches

Each site holds one hydrogen in the well computed in Model 3, and every hydrogen pulls on its neighbours. A pump stiffens and softens every well twice for each swing of the hydrogen. A hydrogen that picks up the rhythm swings at half the pump frequency. The maps show whether neighbours swing together.

[01 · THE SITES]

The hydrogen sites form a cubic crystal. These runs use 110,592 sites for NiH and 32,768 for PdD, and the first version used up to 1,000,000. Each filled site holds one hydrogen free to move in three directions. Empty sites are left out.
[02 · WELLS AND PULLS]

Every hydrogen sits in its computed well, whose walls steepen faster than a simple spring. The computed couplings tie it to five shells of neighbours. In PdD both follow the loading law, and an empty neighbour site pulls a deuterium toward itself.
[03 · THE PUMP]

The pump raises and lowers the stiffness of every well by 8 percent at twice the frequency of the collective swing. That is 20.13 THz for PdD and 42.48 THz for NiH. The response is read at half that frequency.
[04 · LOSS AND NOISE]

Each hydrogen loses energy at an assumed rate. Above 0 K it also takes random kicks from a heat bath. Quantum zero-point motion is folded into the stiffness of each well. The dynamics are classical.

[FIG 1]Pumped motion in one layer of each crystal at 0 K, drawn to the same scale

NiH110,592 sites · full loading

2 nm

In the uniform pattern: 99.8%
In-step length: 3.85 nm = box

PdD32,768 sites · loading 0.875, random

2 nm

In the uniform pattern: 0.5%
In-step length: 1.38 nm

What to notice: the NiH layer is one colour from edge to edge, because every hydrogen is at the same point of its swing. The PdD layer splits into regions that swing in opposite directions. Each map shows one (001) layer through the box, coloured by the swing at half the pump frequency over the second half of the run. Both are drawn at the same scale. PdD at full loading has the same in-step length of 1.38 nm, but no layer was saved for that case. Sources: ni_epoch part2/updated

[WHAT THE UNIFORM RESULT MEANS]

The pump acts on every site in the same way, so it drives a uniform swing. The NiH result shows that nothing else takes over. No pattern with a shorter wavelength competes, and 99.8 percent of the motion stays uniform. In PdD and in the first version of the NiH run below, patterns with a finite wavelength carry most of the motion. In PdD at full loading the uniform pattern holds 2.9 percent.

[SET BY THE BOX]

The NiH in-step length of 3.85 nm is the largest this box can give. It comes from a correlated volume of 1,437 nm³, which is the volume of the whole simulated box (110,592 sites of 0.0130 nm³ each). A larger box could show a longer length, so this run puts no upper limit on the region that moves in step. It was run once at 0 K and has not been repeated at a larger size or at 300 K.

The two metals were not run like for like. The NiH box has 110,592 sites and ran for 41 ps. The PdD boxes have 32,768 sites and ran for 4,096 steps, which is a shorter time. In the first version of the NiH study the patches were still growing with time, so part of the PdD patchiness may come from the shorter run. A PdD run matched to the NiH one in size and length has not been done.

volume moving in step: 1,437
volume of the box, nm³: 1,437

[06.2 · THE FIRST VERSION]

Stripes from a coupling that was too weak

The first version of this simulation ran NiH with a neighbour coupling about three times too small. Its pulls summed to 0.234 eV/Å², where the corrected calculation gives 0.676. That run gave a different picture, which is kept here to show what a too-weak coupling produces.

[FIG 2]One layer of the 1,000,000-site NiH box in the first version

2 nm

What to notice: the stripes. Neighbouring sheets of hydrogen swing in opposite directions (amber against crimson). They form patches a few nanometres across, and each patch is striped along a different diagonal. The map shows one (001) layer of the million-site box 41 ps into the run, in the same colours as Fig. 1. Original figure (PNG). Source: g3_lattice/coherence

[STRIPES]

Neighbouring sheets moved in opposite directions, with a period of about half a nanometre (0.53 nm in the smallest box, 0.55 nm in the larger ones). The uniform pattern held essentially none of the motion (below 10⁻⁶).
[PATCHES THAT DID NOT GROW WITH SIZE]

After 41 ps the in-step length was 1.44, 2.01 and 2.16 nm for boxes of 13,824, 110,592 and 1,000,000 sites.
[STILL COARSENING]

In one 110,592-site run, the share of the motion in the leading stripe pattern rose from 0.16 (21 to 41 ps) to 0.29 (41 to 83 ps). The runs did not show where it stops.
[WITH THE CORRECTED INPUTS]

The stripes do not appear. The same 110,592-site box runs for the same 41.45 ps and moves in step from edge to edge.

[FIG 3]In-step length against the number of sites for NiH at 0 K

first version at 41 psfirst version at 83 pscorrected inputswhole box in step

What to notice: in the first version a million sites gave about the same length as 110,592 sites, far below what the box allows. The corrected run sits on the dashed line, which is the most its box can show, so its true extent is not measured. The dashed line is (Nv/8π)^1/3 for N sites of volume v = a³/4.

What changed between the two NiH runs

Input	First version	Corrected
Summed pull of all neighbours eV/Å²	0.234	0.676
Pull of the nearest shell eV/Å²	0.370	0.486
On-site stiffness eV/Å²	1.468	1.712
Zero-point spread per axis Å²	0.0216	0.0206
Pump frequency THz	45.11	42.48 retuned
Unchanged	110,592 sites, 0 K, 8% pump, wall steepness 18.3 and 1.13 eV/Å⁴, amplitude decay time 1.59 ps

Several inputs changed together and the pump frequency was retuned to the new collective swing, so the change in outcome is not attributed to the coupling alone. The coupling changed most. It grew to almost three times its first value, while the on-site stiffness changed by 17 percent.

[06.3 · LOADING AND ARRANGEMENT]

Where the empty sites sit

Real samples have empty sites. PdD was run with 100, 87.5, 75 and 60 percent of its sites filled and the vacancies placed at random. At 87.5 percent it was also run with the vacancies ordered on a regular superlattice or bunched into clusters. Every run was at 0 K with the same pump.

[FIG 4]Three arrangements of the same number of vacancies in PdD at loading 0.875

Show

Random

2 nm

Static displacement: 0.072 Å
Pumped amplitude: 0.025 Å
In-step length: 1.38 nm

Clustered

2 nm

Static displacement: 0.067 Å
Pumped amplitude: 0.033 Å
In-step length: 1.75 nm

Ordered superlattice

2 nm

Static displacement: 0.000 Å
Pumped amplitude: 0.043 Å
In-step length: 0.82 nm

What to notice: without the pump, random and clustered vacancies push their neighbours off centre. The ordered superlattice leaves every deuterium centred because the pulls of its vacancies cancel by symmetry. With the pump the clustered case has the longest in-step length, and the ordered case swings hardest but has the shortest. Each map is one (001) layer of a 32,768-site box, and the static map shows the displacement along [100]. The ordered superlattice puts its vacancies in every other layer, and the layer shown is one without them. Source: part2/updated

[01 · VACANCY-DRIVEN DISPLACEMENT]

Each empty nearest-neighbour site pulls a deuterium toward itself with 0.0707 eV/Å. Random vacancies leave these pulls unbalanced, so their neighbours sit off centre by 0.072 Å RMS at loading 0.875, 0.093 at 0.75 and 0.105 at 0.6. This is a static displacement driven by the vacancies, not ordering.
[02 · AN ORDERED SUPERLATTICE CANCELS IT]

When the vacancies sit on a regular superlattice, every deuterium is pulled equally from opposite sides. Its static displacement is 1.8 × 10⁻⁶ Å, as small as in a full lattice (1.7 × 10⁻⁶ Å).
[03 · CLUSTERS REACH FURTHEST AND ORDER SWINGS HARDEST]

Clustered vacancies give the longest in-step length (1.75 nm) and the largest share in one pattern (0.24). Ordered vacancies give the largest amplitude of the six (0.043 Å) and the shortest length (0.82 nm).
[04 · LARGER IS NOT MORE IN STEP]

Stretching by 5 percent raises the amplitude to 0.049 Å but cuts the in-step length to 0.17 nm. Raising the pump frequency by 10 percent gives 0.063 Å and puts 0.007 of the motion in the strongest pattern.

[FIG 5]Every computed PdD case pumped at 0 K

Loading 0.875

Vacancies

2 nm

Static displacement without the pump

0.072 Å

Pumped amplitude

0.025 Å

In-step length

1.38 nm

Share in the strongest pattern

0.12

this casethe other five

How far sites stay in step: phase correlation against distance along nine lattice directions

this PdD caseNiH with corrected inputs

What to notice: every case responds to the pump, but its phase correlation falls off within a few nanometres. NiH stays near 1 across its box. Only computed cases are shown, and nothing is interpolated between them. The phase correlation is 1 for two sites at the same point of their swing and 0 for no relation. Loading 0.6 uses the loading law outside the range it was fitted on. Source: pd_summary.json

[FIG 6]Amplitude against in-step length for PdD at 0 K

loading and arrangementloading 0.875 with random vacancies and one setting changed

What to notice: no case is both large and long. The two largest amplitudes come from a 5% stretch and a 10% higher pump frequency, and they have the shortest in-step lengths.

The six loading and arrangement cases PdD at 0 K with the 8% pump

Loading	Vacancies	Static, no pump Å	Pumped Å	In-step length nm	Strongest-pattern share
1	none	0.000	0.038	1.38	0.12
0.875	random	0.072	0.025	1.38	0.12
0.75	random	0.093	0.028	1.29	0.10
0.6 law extrapolated	random	0.105	0.032	0.88	0.03
0.875	ordered	0.000	0.043	0.82	0.03
0.875	clustered	0.067	0.033	1.75	0.24
Loading 0.875 with random vacancies and one setting changed
0.875	+5% stretch	0.193	0.049	0.17	0.0008
0.875	pump frequency +10%		0.063	0.51	0.0070

Static is the RMS displacement of the deuterium from its site without the pump. Pumped is the RMS amplitude of the swing at half the pump frequency with the pump on. Both are averaged over every filled site. In-step length and strongest-pattern share are defined in eqs. (5) and (6). The fitted loading law was built from computed cells at loadings 1.0, 0.875 and 0.75, so loading 0.6 uses it outside that range.

[06.4 · TEMPERATURE]

At 300 K the pump adds nothing measurable

At room temperature the swing at half the pump frequency is set by thermal motion, and the pump does not raise it. In PdD at loading 0.875 with random vacancies, the swing at half the pump frequency is 0.00915 Å without the pump and 0.00919 Å with it. At 0 K the same pump takes it from 0.00046 to 0.025 Å.

The corrected NiH case has not been run at 300 K. The first version of the NiH run used the old inputs and showed only nearest-neighbour order at 300 K. Its in-step length was 0.27 nm, about one neighbour spacing (0.26 nm). A pump retuned 10 percent higher raised the amplitude from 0.012 to 0.032 Å but not the range (0.18 nm).

[FIG 7]Swing at half the pump frequency in PdD (Å)

0 K

No pump 0.00046

Pumped 0.0254

300 K

No pump 0.00915

Pumped 0.00919

What to notice: at 0 K the pump turns a negligible swing into a large one, but at 300 K the two bars are the same length. The runs are at loading 0.875 with random vacancies and 32,768 sites. The 300 K pair are the only ones of the 30 PdD cases run above 0 K. Source: temperatures.json

[06.5 · WHAT IS ASSUMED]

Three inputs the result leans on

The wells and couplings are computed. Three ingredients are not: how fast a vibrating hydrogen loses energy, how an empty site pulls on its second neighbours, and the quantum motion of the hydrogens, which enters only through the stiffness of the well.

The loss rate is not measured

How fast a vibration dies away sets how hard it must be pumped before it grows. No measured value exists for a single hydrogen vibration in NiH, PdH or PdD. The neutron data give either the width of a whole band of vibrations (a peak at 89 meV in NiH and at 37 meV in PdD) or the instrument resolution, and neither is a lifetime. The runs use assumed losses of 1.26 per ps for NiH and 0.48 per ps for PdD. These put the pump threshold at 1.9 and 1.6 percent stiffness modulation, and the exploratory loss of the first version gave 1.7 percent. Paulatto et al. (arXiv:1411.5628) published a calculated value for PdH at 295 K, not a measured one, and it would put the threshold near 23 percent. These are two labelled scenarios, not bounds on the real value.

1.6 to 1.9%threshold at the assumed losses 8%pump used in these runs 23.4%threshold with the calculated PdH loss

Pump thresholds are given as stiffness modulation at resonance. The 8% pump used in every run is above the threshold at the assumed losses and below it with the calculated PdH loss.

The second-shell vacancy pull is unknown

The pull of an empty site on its nearest neighbours was fitted from the computed partial-loading cells. It is 0.0707 eV/Å per empty neighbour, with −0.0001 from the third shell. The second-shell term cannot be determined from cells of this size and is set to zero. Setting it to plus or minus half the first-shell value changes which pattern leads, and the share of the motion in the uniform pattern goes from 0.005 to 0.001 or 0.155. The unknown changes the texture of the response, not only its size.

Share in the uniform pattern for PdD at loading 0.875 with random vacancies

Second shell 0 (used) 0.005

+½ of first shell 0.001

−½ of first shell 0.155

Classical motion with zero-point stiffening

Each hydrogen moves as a classical particle. Its quantum zero-point motion enters once as extra stiffness of the well. That stiffness is computed from the spread of the hydrogen at full loading (0.0206 Å² per axis for NiH). Thermal motion comes from the heat bath and is not added a second time. The NiH run used the earlier wall-steepness values (18.3 and 1.13 eV/Å⁴). With the recomputed 16.2 eV/Å⁴ along [100], the first excitation of a single hydrogen moves by −1.3 percent.

[06.6 · THE DETAIL]

Equations, method and numbers

The equation of motion

(1)m ü_i^α = −K_i u_i^α − A (u_i^α)³ − B (|u_i|² − (u_i^α)²) u_i^α + Σ_j J_ij u_j^α + F_i^α − mκ u̇_i^α + ξ_i^α(t) − δ_αx h mΩ² cos(2pΩt) u_i^x

u_i is the displacement of hydrogen i from its site (Å), with components α = x, y, z along the cube axes; m is the proton mass (NiH) or the deuteron mass (PdD). K_i is the on-site stiffness, eq. (2). A = K₄[100] and B = (3K₄[111] − A)/2 make the well steepen as in Model 3, with K₄ the quartic coefficients (eV/Å⁴). J_ij is the pull of hydrogen j on i over five neighbour shells, and it sums to 0.676 eV/Å² in NiH and 0.389 in full PdD. Missing hydrogens have no coordinate and no bonds. F_i is the vacancy force, eq. (3). κ is the loss rate, so a free swing decays in amplitude as e^−κt/2. ξ is a random force with ⟨ξ_i^α(t) ξ_j^β(t′)⟩ = 2mκk_BT δ_ijδ_αβδ(t − t′), which holds the lattice at temperature T. The pump modulates the stiffness along [100] by the fraction h = 0.08 of mΩ², where Ω is the frequency of the uniform swing and mΩ² = K − ΣJ; p = 1, or 1.1 for the raised pump frequency.

Zero-point stiffening

(2)K_i = K_ii + (3A + 2B) ⟨u²⟩₀

K_ii is the computed on-site stiffness. It is 1.712 eV/Å² for NiH. For PdD it is 0.591 at full loading, and the loading law of Model 3 changes it site by site (lattice spacing and number of empty neighbours). ⟨u²⟩₀ is the zero-point spread per axis of a hydrogen at full loading. This is the only place quantum motion enters.

The vacancy force

(3)F_i = Σ_s C_s Σ_{j ∈ shell s, empty} r̂_ij

r̂_ij is the unit vector from site i toward the empty site j. C₁ = 0.0707 eV/Å (nearest shell), C₂ = 0 (assumed), C₃ = −0.0001 eV/Å. The fit uses 80 computed force vectors from the three partial-loading cells and has an RMS error of 0.0041 eV/Å. Predicting each cell from the other two gives RMS errors of 0.0030, 0.0086 and 0.0043 eV/Å.

Reading the response

(4)u_i(t) ≈ u_i^DC + Re[a_i e^−iωt] + Re[b_i e^−2iωt]

The fit is by least squares at every site over the second half of each run, with ω half the pump frequency. The static displacement is the RMS of u^DC over filled sites in the unpumped run; the pumped amplitude is the RMS of the complex amplitude a in the pumped run. The pumped maps show a_i^x with brightness |a| and colour cos(arg a − θ). Here θ = ½ arg Σ_i (a_i^x)² is the main phase axis of the whole box. The static maps show u_i^DC along x, coloured by its sign.

Pattern shares

(5)S(q) = |Σ_i w_i e^−iq·r_i|² / (N Σ_i |w_i|²)

N is the number of filled sites and r_i their positions. The share in the uniform pattern is S(0) with w_i = a_i^x, which weights the motion by its amplitude. The strongest-pattern share is the largest S(q) with w_i = z_i = a_i^x/|a_i^x|, which uses the phase alone. A share of 1 means one pattern carries everything.

In-step length

(6)ξ = (V_q / 8π)^1/3, V_q = v Σ_d C(d) e^−iq·d

C(d) = ⟨z_i* z_i+d⟩ is the phase correlation of sites a distance d apart, averaged over filled pairs. q is the wavevector of the strongest pattern, so a striped pattern counts as in step with itself. v = a³/4 is the volume per site. V_q is the volume that moves in step. It is an integral measure, not a fitted decay length. When the whole box is in step, V_q is the volume of the box.

Method

Lattice: Octahedral hydrogen sites of the face-centred cubic metal, periodic box. PdD 32³ = 32,768 sites; NiH 48³ = 110,592; first version 24³, 48³ and 100³
Spacing: NiH a = 3.732 Å. PdD 4.084 Å at full loading, following the loading line: 4.062 at 0.875, 4.039 at 0.75, 4.013 at 0.6
Inputs: Wells, couplings and loading law from Model 3 (g3_loading); vacancy force from the rerun's part 1
Pump: h = 0.08 along [100]. PdD at 20.13 THz, response read at 10.06; NiH at 42.48 THz, read at 21.24
Loss: κ = 0.484 ps⁻¹ for PdD (amplitude decay 4.14 ps) and 1.259 ps⁻¹ for NiH (1.59 ps); assumed, not measured
Temperature: 0 K for NiH and for 28 of the 30 PdD cases; the remaining pair at 300 K
Run length: PdD 4,096 steps, second half analysed, frequency resolution 0.173 THz. NiH 17,695 steps, 41.45 ps, analysed from 20.73 ps, the same duration as the first-version 110,592-site run
Arrangements: One random draw per case (seed 13). Clustered: vacancies chosen from a smoothed random field; ordered: a periodic superlattice
Code: NumPy only. Velocity Verlet with an exact Langevin update of the velocities; couplings applied by fast Fourier transform

g3_lattice rerun/part1 rerun/part2 coherence

A · Pumped PdD cases at 0 K 32,768 sites

Loading	Vacancies	Spacing Å	Filled sites	Static Å	Pumped Å	ξ nm	Strongest share	Uniform share	Strongest pattern, wavelength nm
1	none	4.084	32,768	1.73 × 10⁻⁶	0.0380	1.377	0.118	0.0289	7.55
0.875	random	4.062	28,672	0.0716	0.0254	1.375	0.119	0.0048	7.50
0.75	random	4.039	24,576	0.0927	0.0282	1.290	0.100	0.0058	6.46
0.6	random	4.013	19,661	0.1052	0.0323	0.880	0.032	0.0128	7.41
0.875	ordered	4.062	28,672	1.76 × 10⁻⁶	0.0432	0.819	0.027	0.0002	2.98
0.875	clustered	4.062	28,672	0.0674	0.0332	1.746	0.244	0.0083	7.50

The strongest pattern is mostly a long wave close to the size of the box, but for ordered vacancies it is shorter. Uniform share uses the amplitude-weighted field and strongest share uses the phase alone, as defined in eq. (5).

B · Other PdD runs loading 0.875 with random vacancies

Case	Pumped Å	ξ nm	Strongest share	Uniform share
Reference, 0 K	0.0254	1.375	0.119	0.0048
+5% stretch	0.0487	0.172	0.0008	1.3 × 10⁻⁷
Pump frequency +10%	0.0631	0.510	0.0070	1.1 × 10⁻⁵
300 K, no pump	0.00915	0.237
300 K, pumped	0.00919	0.222
Second-shell pull +½ C₁	0.0240	1.424	0.132	0.0015
Second-shell pull −½ C₁	0.0249	1.548	0.170	0.1549

For the 300 K pair, "pumped" is the swing at half the pump frequency in each run; with no pump it is thermal motion at that frequency.

C · Pumped NiH runs full loading with the 8% pump

Run	Sites	Time ps	ξ nm	Box ceiling nm	Amplitude Å
Corrected, 0 K	110,592	41.45	3.853	3.853	0.0636
First version, 0 K	13,824	41	1.438	1.926	0.1338
First version, 0 K	110,592	41	2.005	3.853	0.1257
First version, 0 K	1,000,000	41	2.163	8.026	0.1258
First version, 0 K	110,592	83	2.443	3.853	0.1711
First version, 300 K	13,824	41	0.266	1.926	0.0116
First version, 300 K, pump +10%	13,824	41	0.184	1.926	0.0317

Box ceiling is the in-step length the box would give if all of it moved in step, (Nv/8π)^1/3. The share of the motion in the uniform pattern is 0.9984 for the corrected run (0.9828 from short 16-cycle windows and 0.99995 from the phase alone). It is below 10⁻⁶ in every first-version run at 0 K.

D · The loss rate what is known and the pump threshold it implies

Source	Loss ps⁻¹	Threshold
NiH measured: V. E. Antonov et al., Europhys. Lett. 51, 140-146 (2000), pp. 142-143, Results and Fig. 2	not measured
PdH measured: D. K. Ross et al., Phys. Rev. B 58, 2591-2595 (1998), pp. 2591-2592, Secs. II-III, Fig. 2 and Table I	not measured
PdD measured: V. E. Antonov et al., Phys. Rev. B 80, 134302 (2009), pp. 134302-1-4, Secs. II-III and Figs. 1-5	not measured
NiH, assumed in the corrected run	1.259	1.94%
PdD, assumed in these runs	0.484	1.60%
First-version exploratory loss, NiH reference mode	1.259	1.71%
PdH at 295 K, calculated (Paulatto et al.)	12.47	about 23.4%

The threshold is the stiffness modulation at which a pump at resonance just balances the loss, 2κ/Ω. The measured neutron spectra give band positions and widths, not lifetimes. NiH has its main band peak at 89 meV with a shoulder to about 130 meV. PdH has a fitted band component at 55.8 meV that is 4.8 meV wide. PdD has a band maximum at 37 meV with a shoulder to about 65 meV. A band width mixes many vibrations and the instrument response.

[WHAT THIS MEANS FOR THE NEXT MODEL]

One simulation for many inputs

This lattice is where every computed input meets: wells, couplings, loading, vacancies and the pump. The next model grows nickel on palladium and feeds the mismatch pattern into this same simulation as a map of altered wells. For NiH the first open question is how far the in-step region reaches. Answering it needs a larger box and a run at 300 K.

MODEL Model 6: nickel on palladium Next: the mismatch pattern at a layered interface and what it does to the wells beneath it MODEL Model 4: the anharmonic oscillator Previous: one hydrogen in its real well, a static route under stretch and a pumped route SOURCE The lattice simulation in the repository Code, first-version results, the loss table and the rerun on the corrected wells SOURCE The rerun on the corrected wells Part 1: oscillator and vacancy force law. Part 2: lattice, loading and NiH SOURCE Loading The computed partial-loading cells and the loading law

In short

In step or in patches

Stripes from a coupling that was too weak

What changed between the two NiH runs

Where the empty sites sit

The six loading and arrangement cases PdD at 0 K with the 8% pump

At 300 K the pump adds nothing measurable

Three inputs the result leans on

The loss rate is not measured

The second-shell vacancy pull is unknown

Classical motion with zero-point stiffening

Equations, method and numbers

The equation of motion

Zero-point stiffening

The vacancy force

Reading the response

Pattern shares

In-step length

Method

A · Pumped PdD cases at 0 K 32,768 sites

B · Other PdD runs loading 0.875 with random vacancies

C · Pumped NiH runs full loading with the 8% pump

D · The loss rate what is known and the pump threshold it implies

One simulation for many inputs