[Studio picture · two wire meshes laid one over the other]

[07 / MODEL 6]

Nickel on palladium

Nickel atoms sit closer together than palladium atoms. Lay one crystal on the other and the rows drift in and out of step in a fixed, repeating pattern. This model asks what that built-in pattern does to the deuterium wells in the palladium beneath it and to their pumped motion.

[IN SHORT]

Nickel's crystal repeats every 3.52 Å and palladium's every 3.89 Å, so the two differ by about 9.5 percent. Deposited nickel takes its orientation from the palladium, so the mismatch cannot be tuned away. It prints a fixed pattern of good and bad fit into the crystal.

The pattern repeats every 2.75 nm on bare palladium and every 1.73 nm once the palladium is swollen to the spacing of fully loaded palladium deuteride (PdD). Its strain is strongest in the palladium layer that touches the nickel and fades within a few layers.

A strain law computed from the electrons up turns that strain into softer and stiffer deuterium wells. When every well's stiffness is shaken at a fixed frequency, the deuterium at the PdD-spacing interface moves about ten times more, mostly where the wells are soft. But the motion does not repeat with the pattern.

The interface makes hot regions but not a template.

g3_skew g3_skew/partA g3_lattice

1.73 nm repeat of the pattern at the PdD spacing and 2.75 nm on bare Pd 10× more pumped motion at the PdD-spacing interface 2.9× pumped energy in the softest quarter of sites compared with an even spread 0.25 correlation of the pumped map with the soft-site map

[07.1 · THE PATTERN]

Eleven on ten

Lay a ruler with slightly shorter marks along a longer one. The marks line up, drift apart and line up again further on. That is a vernier scale, and nickel on palladium makes one in two dimensions.

[FIG 1]How two spacings make one repeat

Case

Palladium spacing 3.890 Å

[REPEAT]

2.75 nm

Fit: 11 Ni on 10 Pd
Pd / Ni row spacing: 2.751 / 2.501 Å
Nickel: stretched 0.46% to fit exactly
Gap as a share of the Pd / Ni lattice constant: 9.5 / 10.5%

Move the slider and the nickel returns to its own spacing of 3.52 Å. The two presets are the cells the model relaxes. In each one the nickel is squeezed or stretched slightly so that a whole number of nickel atoms fits.

What to notice: where a nickel atom sits in the hollow between palladium atoms it continues the palladium crystal (in step, amber). Half a repeat further on it sits on top of a palladium atom (out of step, purple). The closer the two spacings, the longer the repeat. Distances run along a close-packed row of the (001) face, where neighbouring atoms are a/√2 apart for a lattice constant a, and the repeat is L = d_Pdd_Ni / (d_Pd − d_Ni). The positions are schematic and shown before relaxation. Source: g3_skew

[01 · IN STEP AND OUT OF STEP]

Good fit and bad fit

A nickel atom over a hollow continues the palladium crystal. One sitting on top of a palladium atom does not. Between those spots the layers have to bend and shear, and that strain reaches down into the palladium.
[02 · WHOLE NUMBERS]

A cell that repeats

The model needs a cell that repeats exactly, so it takes the nearest whole-number fit. On bare palladium 11 nickel spacings fit on 10, with the nickel stretched by 0.46 percent. At the PdD spacing 7 fit on 6, with the nickel squeezed by 0.55 percent.
[03 · LOADING MOVES IT]

Swollen palladium gives a shorter repeat

Deuterium swells palladium. At the 4.084 Å spacing of fully loaded PdD, the mismatch grows to about 16 percent of nickel's spacing and the repeat shrinks from 2.75 to 1.73 nm.

The atom positions come from a cheap classical model. A slab of six palladium layers sits under four nickel layers on the (001) face. The bottom two palladium layers are held fixed, and every other atom moves until the forces fall below 0.02 eV/Å. The forces come from EMT (effective-medium theory), a simple interatomic potential for metals built into the ASE toolkit. The bare-palladium cell has 2,168 atoms (1,200 Pd, 968 Ni) and repeats every 38.9 Å along a cube axis. The PdD-spacing cell has 824 atoms (432 Pd, 392 Ni) and repeats every 24.5 Å. Both relaxations together take about two CPU-minutes. EMT places the metal atoms only.

[07.2 · THE STRAIN BENEATH]

A pattern pressed into the top layers

The relaxed positions give the local strain at every deuterium site, which is how far the crystal around it is stretched, squeezed or sheared. The pattern is strongest in the palladium layer that touches the nickel and fades with depth.

[FIG 2]Local strain at the deuterium sites in each layer

Strain

Layer

Point at a site to read its values, or focus a map and use the arrow keys.

What to notice: at the PdD spacing the interface layer carries strong in-plane stretch and squeeze in a square pattern (one repeat is outlined with a dashed line). Three layers down the pattern is almost gone. Bare palladium shows the same shape, but weaker and with a longer repeat. Both maps are drawn to the same scale with the close-packed rows along the axes. Strains are measured from the 4.084 Å spacing at which the strain law was computed, so bare palladium starts out squeezed. The bracket on the colour scale marks the range of that strain component the law was computed over. A site lies inside the law's range only if every component does, and no site does. Each tick above the scale is one site of the selected layer. Source: g3_skew g3_lattice/rerun

[FIG 3]How deep the pattern reaches

Show

bare Pd at 3.89 Å PdD spacing of 4.084 Å layer held fixed

What to notice: the strain pattern is largest in the interface layer. By the fourth layer it falls to 0.30 of that value at the PdD spacing and 0.40 on bare Pd. The bottom two layers are held fixed, so the slab is too thin to give a decay length. The pumped view comes from the lattice rerun in section 07.4 and uses a logarithmic scale. The strain pattern is the root-mean-square size of the strain tensor's variation across each layer. Source: g3_skew

[A SMOOTH CORRUGATION]

The relaxed pattern is smooth and square and strongest along the diagonals of the cube cell. Its wavelength is 27.5 Å on bare palladium and 17.3 Å at the PdD spacing. No network of separate misfit dislocations forms in this relaxation (g3_skew README).

[STRAINS OF SEVERAL PERCENT]

At the PdD spacing the stretch and squeeze at single sites run from −17.7 to +8.3 percent, and the largest shear component reaches 10.0 percent. On bare palladium the same reference gives −8.4 to +0.5 percent. One percent of uniform stretch is enough to change the bulk PdD well sharply (Model 3).

[THE LAYERS PULL TOGETHER]

When the palladium is held at the PdD spacing in the plane, its layers in this metal-only model sit 11 to 13 percent closer together than in the cubic crystal. The vertical strain in Fig 2 is negative everywhere. On bare palladium the layer spacing changes by 0.3 percent at most.

[07.3 · WELLS ON THE PATTERN]

Soft lines and stiff islands

Each deuterium sits in a well between metal atoms. A strain law computed from the electrons up gives the stiffness of that well for any local strain. Applying it site by site turns the strain map into a map of wells that are softer or stiffer than in the loaded bulk crystal.

[FIG 4]Change in well stiffness from the strain in each layer

Layer

Point at a site to read its values, or focus a map and use the arrow keys.

What to notice: at the PdD spacing the interface layer splits into lines of softened wells and islands of stiffened ones. Each repeat has one soft line in each direction. The second layer holds weaker soft patches, and the two deeper layers come out stiffer everywhere. On bare palladium almost every well is stiffer than in the loaded crystal. The map plots the lowest eigenvalue of ΔK in eV/Å², where ΔK is the change in the well's stiffness tensor relative to bulk PdD at loading 0.875. Amber is softer and purple stiffer. Source: g3_skew/partA g3_lattice/rerun

[AT THE PDD SPACING]

In the interface layer the change runs from −1.51 to +1.33 eV/Å². Its size is larger than the stiffness of the bulk well it is added to, which is about 0.70 eV/Å² for one deuterium with no empty neighbours at this loading (Model 3). At the softest site the soft direction lies in the plane of the interface along a close-packed row.

[ON BARE PALLADIUM]

This case keeps the palladium at its unloaded spacing while its wells are those of loading 0.875, so the crystal sits squeezed against the loaded bulk it is compared with. Nearly every well comes out stiffer, and a few interface sites soften by at most 0.075 eV/Å².

[THE FIRST PASS]

A cheaper first pass used only the response to ±1 percent strain and left shear out. It put 6 percent of the deuterium sites at the PdD spacing below zero stiffness. They were all in the two layers nearest the interface and formed isolated patches, with no lines and no grid. On bare palladium every site stayed stiff, and the lowest was at 0.45 eV/Å². The full strain law, which includes shear, replaces that map.

[THE MAIN UNCERTAINTY]

Every site lies outside the computed range

The strain law was computed for uniform and one-directional strain of ±1 percent, shear up to 8 percent, and thin films strained in the plane by −4 to +4 percent with their height free. At the PdD spacing the interface strains run from about −18 to +8 percent, and no deuterium site in either case falls inside the computed range. Much of the excess is the vertical squeeze of section 07.2. EMT has no hydrogen in it, so nothing holds the palladium layers apart the way the deuterium in a loaded crystal would. The well map is therefore an extrapolation built on a geometry without hydrogen.

The check is a first-principles calculation of the wells in a small cut-out around one soft region. It has not been done and belongs to the next pass. One row of the thin-film series behind the law was not height-searched. At +2 percent the starting height already met the stress tolerance (residual stress 1.6 × 10⁻⁴ eV/Å³ against a tolerance of 5 × 10⁻⁴), so its height is the starting value of 1.000. That value lies on the trend of the other four rows, which interpolate to about 1.001.

Computed in-plane strain, %: −4 to +4
Interface at the PdD spacing, %: −18 to +8

[07.4 · PUMPED]

Hot regions but no template

The lattice simulation of Model 5 takes the well map as input. Every deuterium well is modulated by 8 percent in stiffness, and the motion at half the modulation frequency is read off. The question is how much of that motion gathers in the soft wells and whether it repeats with the pattern.

[FIG 5]Soft wells and pumped motion in the interface layer at the PdD spacing

Overlay

Soft wells · change in stiffness, eV/Å²

Pumped motion · amplitude, Å

Point at a site to compare the two maps there, or focus a map and use the arrow keys.

What to notice: the bright sites in the pumped map sit inside the outlined soft quarter far more often than chance, but most repeats have none and no two repeats look alike. Each map covers 17.3 nm, which is 10 repeats in each direction. Black cells in the pumped map are empty sites (loading 0.875). The pumped map shows the amplitude of the motion along the pump direction at half the pump frequency on a square-root colour scale. The rerun is at 0 K with the pump at 8 percent stiffness modulation. Source: g3_lattice/rerun

0.0013 → 0.0146 Å

Pumped amplitude in the interface model at the PdD spacing without and with the pump. The pump raises it about tenfold.

2.86×

Share of the pumped energy in the softest quarter of interface sites compared with an even spread. The motion gathers where the wells are soft.

0.25

Site-by-site correlation between softness and pumped amplitude in the interface layer. A value of 1 would mean the motion copies the soft-site map.

1.35

Difference between the pumped map and a copy of itself shifted by one repeat, measured in units of its own spread. A map that repeats with the pattern gives 0, and two unrelated maps give about 1.4.

At the PdD spacing the pump lifts the motion at half its frequency from 0.0013 to 0.0146 Å (root mean square over the four palladium layers). Almost all of it sits in the two layers nearest the nickel (0.023 and 0.018 Å, compared with 0.0013 and 0.0006 in the two deeper layers). Sites moving more than 0.05 Å form 124 separate clusters, and the largest has 7 sites.

On bare palladium the effect is negligible, with the motion going from 6.45 to 9.68 × 10⁻⁵ Å.

A first pass gave a cleaner picture. On the first-estimate well map with the earlier wells, the pumped response sat almost entirely in the interface layer (over 96 percent). It appeared as one hot site per repeat and so inherited the pattern's period, with 25 sites on bare palladium and 81 at the PdD spacing (g3_lattice README). The rerun on the corrected wells uses the loading law and the full strain law, and it replaces that first pass. The hot sites remain but the regular spacing does not.

[WHY IT MATTERS]

A deposited layer builds strain into a crystal on purpose, and the choice of metals sets its repeat. Here that strain is large enough to change the deuterium wells by more than their own bulk stiffness, and the changed wells gather pumped motion. The engineered-strain page takes up whether some other pair could make the pattern act as a template, with its repeat matched to the size of the pumped patches.

[07.5 · THE DETAIL]

Equations, method and numbers

Local strain from atom positions

(1)F_i = arg min_F Σ_j |y_ij − F x_ij|²

x_ij is the vector from palladium atom i to neighbour j in the unrelaxed slab for every neighbour within 0.8 a. y_ij is the same vector after relaxation. F_i is the 3 × 3 deformation that best maps one set onto the other (least squares).

(2)F = R U, E = U − I

The polar decomposition splits F into a rotation R and a symmetric stretch U. E is the strain tensor (Biot strain) and I is the identity. A deuterium site takes the mean F of the four palladium atoms around it in its own layer.

(3)E_abs = (a_Pd / 4.084 Å)(I + E) − I

This is the strain measured from the spacing at which the strain law was computed. a_Pd is the lattice constant the slab was built at (3.89 or 4.084 Å). The maps in Fig 2 plot the in-plane mean (E_xx + E_yy)/2, the vertical E_zz and the largest shear component. A shear component of 4 percent corresponds to 8 percent tilt of the cell.

The strain law

(4)K(E) = K₀ I + Σ_m=1¹² c_m B_m(E)

K is the 3 × 3 stiffness tensor of the deuterium well for all deuterium moving together (eV/Å²), with the metal held fixed. In the unstrained cubic crystal K₀ = 0.181 eV/Å². The B_m are the 12 terms of first and second order in E allowed by cubic symmetry: three linear (stretch along an axis, stretch across it, shear) and nine quadratic. The coefficients c_m are fitted to the computed strain cases of Table C, whose shear and film cases took 35 GPAW runs and 200 core-minutes. The fit has a root-mean-square error of 0.0081 eV/Å² and a largest error of 0.044. Leaving one case out and predicting it misses by up to 0.16. A linear law misses by up to 0.39, so the quadratic one is used.

(5)K_lab = R K(E_abs) R^T

The law is written in the crystal's own axes, and the local rotation R from eq. (2) turns it into the fixed axes of the slab.

Counting strain and loading once

(6)ΔK = R [K(E_abs) − K(E_bulk)] R^T, E_bulk = (a(x) / 4.084 Å − 1) I

a(x) = 4.084 Å · exp[0.044 (x − 1)] is the bulk lattice constant at loading x. It is 4.0616 Å at x = 0.875, where the law gives K(E_bulk) = 0.296 eV/Å². ΔK is added to the on-site stiffness that the loading law gives each site, and the couplings between neighbours are left unchanged. The loading law already contains the stiffening that comes with the smaller lattice of a less loaded crystal. Subtracting K(0) instead of K(E_bulk) would count it twice and make an error of 0.115 eV/Å² on every site.

Reading the pumped maps

(7)η = (Σ_soft |u|² / Σ |u|²) / f_soft, D = √(⟨(|u| − |u|_→L)²⟩ / var |u|)

|u| is the amplitude of the motion at half the pump frequency at each site of a layer. "Soft" is the set of sites at or above the 75th percentile of softness −min eig ΔK, and they make up a fraction f_soft of the layer. η is the enrichment, which is 1 for an even spread. |u|_→L is the amplitude map shifted by one repeat. D = 0 for a map that repeats exactly and √2 for an unrelated one. ⟨ ⟩ is the mean and var the variance over the sites of the layer. The correlation is Pearson's correlation between softness and |u| over the same sites.

Method

Geometry: ASE with its EMT potential; FIRE optimiser, cell held fixed, forces below 0.02 eV/Å on every moving atom. (001) slab: six Pd layers, four Ni layers, bottom two Pd layers fixed, no lateral offset and no twist between the crystals
Cells: 11 Ni on 10 Pd, 38.9 Å square, 2,168 atoms; 7 Ni on 6 Pd, 24.5 Å square, 824 atoms
Strain law: GPAW, PBE, 500 eV plane waves, 12 × 12 × 12 k-points, two-atom PdD cell at 4.084 Å, metal fixed, deuterium displaced by ±0.01 Å
Lattice: One repeat tiled 10 × 10: 60 × 60 × 4 = 14,400 deuterium sites, 12,600 occupied at random (loading 0.875); 0 K; stiffness modulated by 8 percent at 20.13 THz, response read at 10.06 THz; open top and bottom, periodic in the plane
Compute: About two CPU-minutes for both relaxations; 200 core-minutes for the shear and film cases of the strain law

g3_skew g3_skew/partA g3_lattice

A · The two slabs

Case	Bare Pd	PdD spacing
Pd lattice constant	3.89 Å	4.084 Å
Coincidence	11 Ni on 10 Pd	7 Ni on 6 Pd
Ni spacing imposed	3.536 Å (+0.46%)	3.501 Å (−0.55%)
Atoms (Pd, Ni)	2,168 (1,200, 968)	824 (432, 392)
Repeat along a cube axis	38.9 Å	24.5 Å
Wavelength along the diagonals	27.5 Å	17.3 Å
Optimiser steps; largest force left	188; 0.016 eV/Å	286; 0.020 eV/Å
Normal strain range, from 4.084 Å	−8.4 to +0.5%	−17.7 to +8.3%
Largest shear component	4.6%	10.0%

The imposed Ni spacing is the nickel lattice constant that makes the coincidence exact, shown with its strain from nickel's own 3.52 Å. Strain ranges and shear are over the palladium atoms of the four moving layers. The wavelength is the repeat divided by √2.

B · Strain pattern by layer

Layer	Bare Pd: depth Å	RMS strain pattern	PdD spacing: depth Å	RMS strain pattern	Mean vertical strain
1 · interface	0.00	0.0664	0.00	0.1244	−11.2%
2	1.95	0.0333	2.04	0.0610	−12.3%
3	3.89	0.0218	4.08	0.0255	−12.7%
4	5.83	0.0266	6.13	0.0369	−13.1%
5 fixed	7.78	0.0137	8.17	0.0180	−6.8%
6 fixed	9.72	0.0000	10.21	0.0000	0%

The RMS strain pattern is the root mean square over the layer of the Frobenius norm of E minus its layer average. The mean vertical strain is the layer average of E_zz at the PdD spacing, taken from the slab as built. The fixed layers carry strain only through their moving neighbours.

C · The cases behind the strain law two-atom PdD cell, eV/Å²

Case	E_xx, E_yy, E_zz %	Largest shear component %	Lowest stiffness	Fit error	Left out and predicted
Cubic, unstrained	0, 0, 0	0	0.181	0.000	·
Uniform	−1.0, −1.0, −1.0	0	0.397	0.002	0.014
One direction	−1.0, 0, 0	0	0.209	0.025	0.058
Uniform	+1.0, +1.0, +1.0	0	−0.013	0.002	0.002
One direction	+1.0, 0, 0	0	0.046	0.009	0.011
Shear along [100]	0, 0, +0.01	1.0	0.155	0.001	0.001
Shear along [100]	−0.02, 0, +0.06	2.0	0.119	0.004	0.004
Shear along [100]	−0.04, 0, +0.13	3.0	0.078	0.003	0.003
Shear along [100]	−0.08, 0, +0.24	4.0	0.035	0.007	0.011
Shear along [110]	−0.04, −0.04, +0.24	2.8	0.030	0.006	not identified
Film, height free	−4.0, −4.0, +10.3	0	−0.341	0.020	0.161
Film, height free	−2.0, −2.0, +6.7	0	−0.290	0.044	0.063
Film, height free	0, 0, +3.3	0	−0.181	0.024	0.026
Film, height free	+2.0, +2.0, 0	0	−0.127	0.004	0.005
Film, height free	+4.0, +4.0, −2.9	0	−0.281	0.013	0.152

Fit error is the largest error of any tensor entry for that case. "Left out and predicted" is the same error with that case removed from the fit. The single [110] shear case cannot be left out because it alone fixes one cross-shear coefficient. The unstrained case is the reference K₀, not a fitted point. Shear components are half the tilt of the cell, so 4 percent here is 8 percent shear. The +2 percent film row kept its starting height (see the main uncertainty above).

D · The pumped interface in the rerun loading 0.875 at 0 K

Layer, PdD spacing	Amplitude Å	Enrichment η	Correlation	Shift difference D
1 · interface	0.0230	2.86	0.25	1.35, 1.35
2	0.0180	3.59	0.34	1.34, 1.31
3	0.0013	2.42	0.19	1.38, 1.36
4	0.0006	0.53	0.18	1.35, 1.33
All layers, PdD spacing	0.00134 → 0.01457
All layers, bare Pd	0.0000645 → 0.0000968	3.60 interface	0.25	1.33, 1.33

The amplitude per layer is the root mean square of the motion at half the pump frequency with the pump on. The "All layers" rows give it without → with the pump. The shift difference is eq. (7) for a shift of one repeat along each of the two row directions. The lattice uses the corrected wells of Model 3, the loading law and the vacancy force law. Strain does not change the neighbour couplings.

E · First-pass well map ±1% derivatives without shear

	Bare Pd	PdD spacing
Lowest stiffness, deuterium sites eV/Å²	0.448 to 1.380	−0.754 to 1.256
Lowest stiffness, palladium sites eV/Å²	0.368 to 1.391	−0.812 to 1.254
Deuterium sites below zero	0	6.3%
Deuterium sites below half of 0.181, layer 1	0	13.9%, isolated spots or finite patches
The same, layer 2	0	11.1%, isolated spots or finite patches
The same, layers 3 and 4	0	0

The first pass used the stiffness response to ±1 percent strain along and across the motion (−14.6 and −2.9 eV/Å² per unit strain) and no response to shear. Its maps are kept in g3_skew next to the geometry, and the arrays let the law be swapped without redoing the relaxation.

[WHAT THIS MEANS FOR THE NEXT MODEL]

Check one soft region and then try other pairs

The interface gathers pumped motion into its soft regions without ordering it. The next pass computes the wells from first principles in a small cut-out around one soft region to see whether the soft wells survive. Then the same map can be built for other metal pairs, whose patterns repeat at other spacings.

MODEL Engineered strain Next: other metal pairs and three design ideas MODEL What an experiment could test The predictions, the open inputs and the next pass MODEL Model 5: many sites together Previous: the lattice simulation this page feeds SOURCE The mismatch geometry and first-pass maps EMT slabs, per-site strain arrays and the first well map SOURCE The PdD strain and shear law Every computed case and the fitted law SOURCE The pumped interface in the rerun Prepared maps, lattice cases and the interface figure

In short