[08 / SPECULATION]
Engineered strain
[SPECULATION]This page looks ahead. It builds on computed results, and every statement carries its label: computed, established, expected or unknown.
Stretch a metal lattice and the pocket each hydrogen sits in goes soft. Our calculations show how steeply it softens and that the direction of the stretch matters. Layered metals stretch a lattice by design. That makes strain something to engineer. It can decide where hydrogens are free to move together and in which direction.
- [COMPUTED]a result in the repository
- [ESTABLISHED]published surface science
- [EXPECTED]our forward look
- [UNKNOWN]open and not settled by any model here
[IN SHORT]
What we computed, what we expect and what is open
What we computed
- A 1 percent uniform stretch takes the bare PdD well from 0.181 to −0.013 eV/Ų. NiH loses about a fifth of its stiffness.
- Direction matters. Stretching along the hydrogen's motion softens it most. In a thin film, in-plane strain moves the softness between motion within the layer and motion across it.
- A nickel layer on palladium makes a regular strain pattern with a wavelength of 2.75 nm on bare Pd and 1.73 nm at the PdD spacing. At the PdD spacing it carries lines of soft deuterium sites.
- When pumped at 0 K, NiH moves in step across the whole simulated box. PdD stays in patches that move in step over about 1 to 2 nm.
What we expect
- Before the Pd is fully loaded, a substrate can already hold it at the spacing where the deuterium well goes soft.
- Strain or the way empty sites are arranged can lengthen the distance over which PdD moves in step.
- A mismatch pattern fixes the soft sites at a period set by the choice of metals, and pumped motion gathers there.
- The d band is the electronic route from stretch to well.
What is open
- Whether hydrogen stays in a strongly strained layer.
- Whether the strain survives loading.
- Where each hydrogen settles once it is no longer held at the centre of its site.
- How fast the hydrogen motion loses energy.
[08.1 · WHY STRAIN IS THE LEVER] [COMPUTED]
Stretch the lattice and the well goes soft
Each hydrogen sits in a pocket between metal atoms. Its stiffness says how hard the pocket pushes back when the hydrogen moves. We computed it for all the hydrogens moving together, with the lattice squeezed and stretched by 1 percent in three ways: uniformly, along the line the hydrogen moves on and across it.
[FIG 1] Well stiffness against strain in NiH and PdD
Uniform (hydrostatic)
NiH −22.0 PdD −113.3 per unit strain
Uniaxial along the motion
NiH −23.9 PdD −80.9 per unit strain
Uniaxial across the motion
NiH +1.2 PdD −15.9 per unit strain
The chart needs JavaScript. Every value is in the tables under The detail.
In absolute terms both metals lose stiffness at a similar rate per unit of uniform strain: −20.5 eV/Ų for PdD and −23.1 eV/Ų for NiH. The PdD well starts almost six times softer (0.181 against 1.050 eV/Ų), so the same stretch takes it to zero.
Direction matters. A stretch along the hydrogen's line of motion softens it most. A stretch across that line stiffens NiH slightly (+1.2) and softens PdD five times less than a stretch along it (−15.9 against −80.9).
[FIG 2] Where the softness goes in a thin PdD film and under shear
Thin film under in-plane strain with its height relaxed
Shear along [100]
The chart needs JavaScript. The film and shear tables are under The detail.
A negative bare stiffness marks a direction in which the bottom of the well is flat or has two hollows. Zero-point motion and the anharmonic walls still hold the deuterium. With those included, the static model on the corrected wells puts the ordering of PdD at 5.3 percent expansion at 0 K (7.1 at 300 K), PdH at 7.5 (8.8) and NiH at 8.1 (8.2). An earlier run on coarser wells gave 3.0 (3.5), 3.8 (4.0) and 8.3 (8.5). These strains are extrapolated from calculations at plus and minus 1 percent strain. Part of the change comes from a different extrapolation of that strain response and not only from the corrected well.
[EXPECTED] So strain is a design variable. A substrate, a clamp or a neighbouring layer sets the spacing. The spacing in turn decides whether hydrogens can move collectively, where they can and in which direction.
[08.2 · THE D BAND] [ESTABLISHED]
Why a stretched metal binds differently
In a transition metal the outer d electrons fill a band of energies up to the Fermi level, which is the top of the filled states. Stretch a layer and its atoms overlap less, so the band narrows. To keep the same number of electrons below the Fermi level, a band that is more than half full has to move up. Its centre rises toward the Fermi level.
[FIG 3] A d band under tensile strain
Schematic
The interactive schematic needs JavaScript. The text beside it describes the same change.
Hammer and Nørskov turned this into the d-band model: the closer the d-band centre sits to the Fermi level, the more strongly a metal surface binds what lands on it [1]. Mavrikakis, Hammer and Nørskov then computed that stretching a metal surface shifts its d states up and strengthens adsorption [2]. Kibler and co-workers laid single Pd layers on a series of metal crystals, from gold, which stretches the layer, to metals that compress it. How strongly each layer held hydrogen and how fast it ran surface reactions both followed the strain and the expected d-band shift [3].
The same change in the electrons that alters hydrogen binding at a surface should act inside the metal too. It plausibly changes the stiffness of the interstitial well and the way the electrons screen one proton's charge from its neighbours. Our calculations see the net effect, which is the softening in Fig 1. They have not yet separated the d-band part from the rest.
The calculation that separates them can be done now. It is the projected density of states against strain in the same PdD cells. Rerun the uniform, film and shear cases with the states projected onto the Pd d orbitals and the D s orbital, read off the d-band centre and width at each strain, and plot the well stiffness against the centre. If the stiffness follows the centre across all three strain families, the d band is the lever. If it does not, the softening comes from elsewhere, such as the geometry of the cage of metal atoms around each site.
[08.3 · A MENU OF METAL PAIRS] [ESTABLISHED] [EXPECTED]
The lattice mismatch of each metal pair
Grow one metal on another and the layers have to reconcile two spacings. The mismatch decides whether you get a flat film, a strained film or a regular pattern. Seven pairs of common metals with the same face-centred cubic structure run from almost matched to almost 10 percent apart. Select a pair to see what it would do to a deuterium well.
[FIG 4] Seven pairs on a mismatch axis
[EXPECTED] What each pair would do to a deuterium well
This pair has no palladium, so there is no computed well. It is on the menu as the near-perfect match. A Pd layer grown on either metal would be held within 0.2 percent of the spacing of fully loaded PdD (4.084 Å, a published extrapolation from measurements at 77 K).
A Pd layer on Pt is stretched by 0.9 percent before any hydrogen enters. As it loads, its own spacing passes that of Pt and the layer ends up compressed. Held at the Pt spacing, a fully loaded layer sits 3.9 percent below the spacing of fully loaded PdD (4.084 Å, a published extrapolation from measurements at 77 K). That is the compressed end of the film law, where motion within the layer is stiff and motion across it is soft. It makes a good control for the Pd-on-Au idea.
Fully loaded Pd is larger than Pt, so a coherent layer is compressed by 3.9 percent. That is the compressed end of the film law, where motion within the layer stiffens (0.868 eV/Ų at −4 percent) and the bare stiffness across it is negative.
This pair has no palladium. It is on the menu as a reference for the few-percent range, where a coherent film forms first and dislocations come later.
This pair has no palladium. It shows the size of strain a few-percent pair imposes. A coherent Pt film on Au is stretched by 3.9 percent.
This is the case for the first design idea. Au holds the layer at 4.078 Å, which is within 0.2 percent of the spacing of fully loaded PdD (4.084 Å, a published extrapolation from measurements at 77 K). Unloaded Pd is stretched by 4.8 percent to get there. A fully loaded layer on Au sits at the zero row of the film law, where motion within the layer is already soft (0.084 eV/Ų, under half the cubic 0.181). At modest loading the layer would still have the full-loading spacing.
At full deuterium content Pd's own spacing nearly equals that of Au. The mismatch falls to 0.1 percent and the layer is almost unstrained. The substrate does its work at lower loading, where it holds the Pd open at the full-loading spacing.
Here Pd is the substrate. A Cu film would print a periodic strain pattern on the top Pd layers as Ni does, but with a longer period of about 5.1 nm. This pair is not modelled yet.
On loaded Pd the mismatch grows to 11.5 percent and the period shortens to about 3.1 nm, which is close to the Ni-on-Pd pattern. This pair is not modelled yet.
This pair is modelled on the previous page. The pattern repeats every 38.9 Šwith 11 Ni atoms on 10 Pd, and its strongest modulation runs along the diagonals with a wavelength of 2.75 nm. On bare Pd nearly every deuterium well comes out stiffer than in the loaded crystal. A few interface sites soften by at most 0.075 eV/Ų.
At the PdD spacing 7 Ni atoms sit on 6 Pd and the pattern wavelength drops to 1.73 nm. With the full strain law the Pd layer at the interface splits into one soft line in each direction per repeat, with stiffer islands between. The local strains run from −18 to +8 percent. Every site lies outside the computed strain range and the geometry has no hydrogen, so the map is an extrapolation.
[ESTABLISHED] These are general tendencies of growing one crystal on another. Very small mismatch grows flat and almost strain free. With a few percent, the first several layers form a coherently strained film matched atom for atom to the substrate. Above a critical thickness misfit dislocations appear and relieve the strain. Larger mismatch gives a periodic dislocation network or moiré pattern with a period of roughly the lattice spacing divided by the mismatch. Near 10 percent, layer-by-layer growth is hard to keep and islands form. Temperature, growth rate, surface orientation and chemistry move all of these boundaries.
Only a thin layer held coherently by its substrate feels the full mismatch strain. A thicker film relaxes, and the strain survives near the interface. That is why the pairs that matter here are a few atomic layers of one metal on a crystal of another. Pd on Au is a well-known case of a stretched Pd layer whose hydrogen behaviour changes [3]. The pair modelled on the interface page is Ni on Pd, whose spacings differ by about 9.4 percent.
[COMPUTED] Hydrogen moves the target. Loading Pd to full deuterium content expands it by 5.0 percent, from 3.89 to 4.084 Å (the published extrapolation for fully loaded PdD). That changes every pair with Pd in it. The interface model runs both cases. Ni on bare Pd puts 11 Ni atoms on 10 Pd with a repeat of 38.9 Å. Ni on Pd at the PdD spacing puts 7 on 6 with a repeat of 24.5 Å. Measured against nickel's own spacing, that second mismatch is about 16 percent, which is the strain a coherent Ni film would carry.
Lattice constants are standard room-temperature values for the pure metals: Au 4.078, Ag 4.086, Pt 3.924, Pd 3.890, Cu 3.615, Ni 3.524 Å. They are reference data, not computed in this project. The mismatch f = (afilm − asubstrate) / asubstrate and the period ≈ afilmasubstrate / |asubstrate − afilm| are computed here from those constants. The PdD spacing of 4.084 Å is the repository's reference. It is an extrapolation to full loading of measurements made at 77 K by Schirber and Morosin [4]. The hydrogen notes are expected behaviour read off the computed film law.
[08.4 · THREE DESIGN IDEAS] [EXPECTED]
Three ways to put strain to work
Each idea starts from a computed result and ends with a test that the existing models can run cheaply, before anything is grown.
-
[IDEA 1]
Stretched Pd with a soft well at modest loading
[COMPUTED]The PdD well loses all its bare stiffness under a 1 percent uniform stretch. In the film law at the full-loading spacing of 4.084 Å, motion within the layer is already soft (0.084 eV/Ų, under half the cubic 0.181) and crosses zero before +2 percent.
[EXPECTED]Gold's spacing of 4.078 Å is within 0.2 percent of that reference. A coherent Pd layer on gold holds the full-loading spacing whatever its loading. Its deuterium well at modest loading would sit near the softening point that bulk Pd reaches only when full. Pd on Pt does the opposite once loaded and makes the control.
[TEST IN THE MODEL]Compute PdD wells at partial loading with the in-plane spacing fixed at 4.078 Å and the height relaxed, using the settings of the PdD strain law (its 35 runs took 200 core-minutes). Feed them to the static branch of the lattice model and see whether order appears below full loading.
-
[IDEA 2]
From patches to one region in PdD
[COMPUTED]When pumped at 0 K on the corrected inputs, NiH locks in step. In a box of 110,592 sites, 99.8 percent of the motion is in the uniform pattern. Its in-step length of 3.85 nm is set by the size of the box, so the run puts no upper limit on it. The run has not been repeated in a larger box or at 300 K. PdD under the same pump stays in patches that move in step over 0.82 to 1.75 nm in six cases (Table 6). Clustered vacancies gave the longest in-step length and ordered vacancies the shortest. A 5 percent uniform stretch raised the amplitude (0.025 to 0.049 Å) but cut the in-step length from 1.38 to 0.17 nm.
[COMPUTED]The two materials differ in balance. The neighbour couplings add up to 0.39 of the on-site stiffness in NiH and 0.66 in PdD. The PdD well is far softer (0.591 against 1.712 eV/Ų) and more anharmonic: its walls lift one deuterium's first vibration level 40 percent above the harmonic value, against 19 percent for hydrogen in nickel.
[EXPECTED]We expect the higher coupling ratio, the soft anharmonic well or both to be what keeps PdD in patches where NiH locks. The runs have not separated them. Strain and the arrangement of empty sites change the well and the couplings, so both are levers on the in-step length. The open question is which strain or which vacancy arrangement lengthens it.
[TEST IN THE MODEL]Rerun the pumped PdD lattice with a one-directional (uniaxial) strain and with couplings that change with strain. The current runs apply only uniform stretch and hold the couplings fixed. Watch the in-step length and the share in the strongest pattern. The 30 PdD cases of the rerun took about an hour on one thread.
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[IDEA 3]
A built-in array of soft sites
[COMPUTED]Ni on Pd makes a square pattern whose strongest modulation runs along the diagonals, with a wavelength of 2.75 nm on bare Pd and 1.73 nm at the PdD spacing. At the PdD spacing and with the full strain law, the Pd layer at the interface splits into one soft line in each direction per repeat, with stiffer islands between. Under the pump the motion there rises about tenfold (0.0013 to 0.0146 Å) and gathers in the soft regions. The softest quarter of the interface sites holds 2.86 times its even share of the pumped energy. The motion does not copy the pattern's period. Site by site its correlation with the softness is 0.25, and most repeats have no strongly moving site. Every site lies outside the computed strain range, and the geometry has no hydrogen.
[COMPUTED]Empty deuterium sites make soft sites too. In a 64-atom PdD cell at loading 0.875, a D with two empty neighbours is softer than one with none (0.529 against 0.703 eV/Ų) and is pushed toward the vacancy. With the same four empty sites bunched instead of spread, that D is softer still (0.431), pulls on its neighbours about three times more weakly (0.128 against 0.399) and is pushed harder (0.109 against 0.070 eV/Å). So the arrangement of the vacancies matters as well as their number.
[EXPECTED]The pair sets the spacing. Across the menu the expected period runs from about 2.6 nm (Ni on loaded Pd) through 3.7 nm (Ni on Pd) and 5.1 nm (Cu on Pd) to tens of nanometres for the closely matched pairs. Any such pattern fixes where the soft regions are, and pumped motion should gather in them. It is open whether some pair makes the motion follow the pattern, so that the pattern acts as a template and not only as a set of hot regions. A gentler mismatch keeps the strains closer to the computed range.
[TEST IN THE MODEL]Relax other pairs with the same script. Its EMT potential has parameters for all six metals on the menu, and the whole Ni-on-Pd run took about two CPU-minutes. Map each pair with the full strain law and the loading law as the rerun did for Ni on Pd. Then run the pumped lattice and compare the enrichment and the correlation.
Electrochemical atomic layer deposition is one route that can lay down such layers one atomic layer at a time.
[08.5 · WHAT IS UNKNOWN] [UNKNOWN]
Four questions no model here has settled
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Does hydrogen stay in a strongly strained layer?
A stretched layer and a compressed one should take up hydrogen differently, and a thin layer can give it up to its neighbours. Nothing in these models lets hydrogen move between layers.
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Does the strain survive loading?
Loading expands Pd by 5.0 percent. A layer pinned to its substrate stores that as stress and may relieve it with dislocations, by buckling or by coming loose. The interface relaxations contain no hydrogen.
-
Where does each hydrogen settle?
Every calculation here held the hydrogens on ideal site centres. Several soft points on this page are places a hydrogen would move away from, such as the hump across an unstrained PdD film and the deuterium pushed toward an empty neighbour. The first priority of the next pass is to find where it settles and how stiff its well is there.
-
How fast does the motion lose energy?
The loss rate of the in-step vibration is unmeasured. No measured decay rate for a single hydrogen vibration was found for NiH, PdH or PdD. The loss rate sets the pump threshold, which two labelled scenarios put at 1.7 percent or about 23 percent stiffness modulation. Strain and interfaces could move it either way.
[08.6 · THE DETAIL]
Definitions, equations and the numbers
The well and its strain response [COMPUTED]
u: displacement of the hydrogens from the site centre (Å) as they all move together. K: collective stiffness (eV/Ų), which is the curvature at the centre with the metal atoms held fixed. K0 is its unstrained value. K4: quartic coefficient (eV/Å⁴). A negative K means the centre is flat or double-bottomed and the hydrogen is held by the quartic walls and zero-point motion.
ε: strain, the fractional change in length. Uniform: all three axes scaled by 1 + ε. Uniaxial: εxx = ±0.01 and the other components zero, with the hydrogen moving along x (along) or y (across). For uniform strain d ln K/d ln V = s/3, which is −7.32 for NiH and −37.8 for PdD.
Shear: F is the deformation of the cell and γ the engineering shear along [100]. Film: in-plane strain ε is imposed on x and y, and the height cz relaxes until the stress across the layer vanishes. All PdD cases use a two-atom cell at a0 = 4.084 Å.
Mismatch and pattern period [ESTABLISHED]
af, as: cubic lattice constants of film and substrate. f: mismatch relative to the substrate, negative when the film is smaller. ε∥: in-plane strain of a film matched atom for atom to the substrate (coherent).
L: length over which the two lattices come back into register along a cube axis. On a (001) face the strongest modulation runs along the diagonals with period L/√2. In the interface model a 38.9 Å repeat gives 27.5 Å and a 24.5 Å repeat gives 17.3 Å.
In-plane strain of a fully loaded PdD layer held at the substrate spacing, measured from the reference of the computed film law. This is how Fig 4 places a pair on that law [EXPECTED].
Settings
GPAW 26.7, PBE, plane waves at 500 eV, Fermi smearing 0.1 eV, metal atoms fixed, hydrogen displaced by ±0.01 Å. NiH at k = 8, PdD at k = 12. With the cell free, the same setup prefers 4.134 Å for PdD, which is 1.2 percent larger than 4.084 Å. At that spacing the collective stiffness is −0.055 eV/Ų. Strain thresholds quoted from 4.084 Å would be about 1.2 percentage points smaller counted from 4.134 Å. The film with no imposed strain grows 3.2 percent taller for the same reason.
Table 1 · Collective stiffness under ±1 percent strain [COMPUTED]
| Material, strain | K(−1%) | K(0) | K(+1%) | s |
|---|---|---|---|---|
| NiH, uniform (hydrostatic) | 1.295 | 1.050 | 0.834 | −22.0 |
| NiH, uniaxial along the motion | 1.314 | 1.050 | 0.812 | −23.9 |
| NiH, uniaxial across the motion | 1.033 | 1.050 | 1.057 | +1.2 |
| PdD, uniform (hydrostatic) | 0.397 | 0.181 | −0.013 | −113.3 |
| PdD, uniaxial along the motion | 0.339 | 0.181 | 0.046 | −80.9 |
| PdD, uniaxial across the motion | 0.209 | 0.181 | 0.152 | −15.9 |
K in eV/Ų. s: fractional change per unit strain. Source: g2_input/adopted-strains.json.
Table 2 · PdD film with the height relaxed [COMPUTED]
| In-plane strain | Height / a0 | K in-plane | K across layer |
|---|---|---|---|
| −4% | 1.103 | 0.868 | −0.341 |
| −2% | 1.066 | 0.395 | −0.290 |
| 0% | 1.032 | 0.084 | −0.181 |
| +2% | 1.000 | −0.127 | 0.057 |
| +4% | 0.971 | −0.281 | 0.421 |
K in eV/Ų. Source: g3_skew/partA/strain-table.json. The +2% row kept its starting height because that already met the stress tolerance (residual stress 1.6 × 10⁻⁴ eV/ų against a tolerance of 5 × 10⁻⁴). It lies on the trend of the other four rows, which interpolate to about 1.001.
Table 3 · PdD under shear [COMPUTED]
| Shear | 0 | 2% | 4% | 6% | 8% | [110], 8% |
|---|---|---|---|---|---|---|
| Lowest K | 0.181 | 0.155 | 0.119 | 0.078 | 0.035 | 0.030 |
Shear along [100] unless marked. K in eV/Ų. Source: g3_skew/partA/strain-table.json.
Table 4 · Ordering strain in the static model [COMPUTED]
| Material | 0 K | 300 K |
|---|---|---|
| NiH | 8.1% | 8.2% |
| PdH | 7.5% | 8.8% |
| PdD | 5.3% | 7.1% |
Linear expansion at which the collective mode goes soft, computed in mean field with zero-point and thermal motion on the corrected wells. The values are extrapolated from calculations at ±1 percent strain. Source: g3_lattice/rerun (part1/pd_x1_quantum/result.json; part2/updated/ni_epoch/quantum.json). An earlier run on coarser wells gave 8.3 / 8.5 (NiH), 3.8 / 4.0 (PdH), 3.0 / 3.5 (PdD) (g2_refit/refit.json). Part of the change comes from a different extrapolation of the strain response.
Table 5 · The metal pairs [ESTABLISHED]
| Film on substrate | af / as (Å) | f | ε∥ | L | εref |
|---|---|---|---|---|---|
| Ag on Au | 4.086 / 4.078 | +0.2% | −0.2% | > 100 nm | · |
| Pd on Pt | 3.890 / 3.924 | −0.9% | +0.9% | 45 nm | −3.9% |
| PdD on Pt | 4.084 / 3.924 | +4.1% | −3.9% | 10 nm | −3.9% |
| Ni on Cu | 3.524 / 3.615 | −2.5% | +2.6% | 14 nm | · |
| Pt on Au | 3.924 / 4.078 | −3.8% | +3.9% | 10 nm | · |
| Pd on Au | 3.890 / 4.078 | −4.6% | +4.8% | 8.4 nm | −0.1% |
| PdD on Au | 4.084 / 4.078 | +0.1% | −0.1% | > 100 nm | −0.1% |
| Cu on Pd | 3.615 / 3.890 | −7.1% | +7.6% | 5.1 nm | · |
| Cu on PdD | 3.615 / 4.084 | −11.5% | +13.0% | 3.1 nm | · |
| Ni on Pd | 3.524 / 3.890 | −9.4% | +10.4% | 3.7 nm | · |
| Ni on PdD | 3.524 / 4.084 | −13.7% | +15.9% | 2.6 nm | · |
PdD rows use the repository's 4.084 Å; the other lattice constants are standard room-temperature values (reference data, not computed here). εref is given for Pd films only.
Table 6 · Pumped PdD lattice at 0 K [COMPUTED]
| Loading, vacancies | Static (Å) | Pumped (Å) | In-step length (nm) | Strongest- pattern share |
|---|---|---|---|---|
| 1.0 | 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 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 |
| 0.875 random, +5% uniform stretch | · | 0.049 | 0.17 | · |
The pump is at 8 percent stiffness modulation. Static is the displacement with no pump, and pumped is the amplitude of the oscillation. *Below loading 0.75 the loading law is used outside the range it was fitted on. For comparison, NiH at full loading (110,592 sites) has 99.8 percent of the motion in the uniform pattern and an in-step length of 3.85 nm, which is the size of the box. Source: g3_lattice/rerun/part2/updated/part2.md, ni_epoch/phase_summary.json.
- [1] B. Hammer and J. K. Nørskov, Electronic factors determining the reactivity of metal surfaces, Surf. Sci. 343, 211 (1995).
- [2] M. Mavrikakis, B. Hammer and J. K. Nørskov, Effect of strain on the reactivity of metal surfaces, Phys. Rev. Lett. 81, 2819 (1998).
- [3] L. A. Kibler, A. M. El-Aziz, R. Hoyer and D. M. Kolb, Tuning reaction rates by lateral strain in a palladium monolayer, Angew. Chem. Int. Ed. 44, 2080 (2005).
- [4] J. E. Schirber and B. Morosin, Phys. Rev. B 12, 117 (1975), cited in the repository for the PdD lattice constant.
[WHAT THIS MEANS FOR THE NEXT MODEL]
Three calculations turn this page into results
The projected density of states against strain in the cells of the PdD strain law tells whether the d band is the lever. PdD wells at partial loading in a film held at the gold spacing test the first idea. A pumped PdD lattice under uniaxial strain with couplings that follow the strain tests the second. Each reuses cells and scripts that already exist. Before any of them, the next pass lets the hydrogens leave their ideal site centres.