A surface is not a count of broken bonds

Here is an argument that everyone meets and almost nobody checks. A solid is held together by bonds. Cutting it exposes two new faces, and every atom in a new face has lost some of its neighbours — one of six, for a simple cubic arrangement cut across a face. The energy needed to make the cut is therefore the number of bonds broken times what a bond is worth, and what a bond is worth is the cohesive energy divided by the number of bonds per atom.

Every step of that is reasonable. The conclusion is out by nearly a factor of two, and the reason is a result this field has already proved and not yet applied.

The argument is worth taking seriously rather than dismissing, because it is not a piece of folklore — it is the standard first estimate of a surface energy, it is taught as such, and it gets the order of magnitude right. What it gets wrong is the one thing it assumes without saying: that a bond has a value that does not depend on what else the atom is bonded to.

What is actually computed #

A block of a three-dimensional structure, wrapped in two directions and left open in the third, so that it has two flat faces and no edges or corners to complicate the accounting. The reference is the same block wrapped in all three directions, which has no surface at all. Both are half filled, which is the case that supplies the most binding.

The quantity to compare is not the total energy — that differs between the two by an amount that is hard to attribute — but the binding received by a site in each layer, which is a local quantity and says where the cost is paid.

The two accounts, and the size of the gap #

The computed answer agrees with the square-root rule to five hundredths of a percentage point and disagrees with the bond count by eight points. That is not a case of one estimate being rough and the other being better; it is a case of one of them being the wrong shape of argument.

Read as a cost rather than as a fraction, the numbers are starker. A bond in the bulk is worth 0.334 β. The one bond a surface atom loses costs it 0.175 β. A broken bond costs about half of what a bond is worth.

What one bond is worth, and what it is worth to whom #

It helps to put the three quantities on one line, because the argument turns on their being three and not two.

The last row is the one the count has no place for. A surface atom’s five bonds carry 1.8262 β between them, which is 0.3652 β apiece against the bulk’s 0.3336 — nine and a half per cent more. There is no single number that is “the energy of a bond” in this material, and the estimate assumed there was.

Why the surviving bonds get stronger #

The result is not mysterious and this field has already derived the fact it rests on. The bond that weakens as neighbours multiply measured binding per site of 1.272, 1.611 and 1.979 for structures with two, four and six neighbours, and binding per bond falling from 0.636 to 0.330 across the same range. An atom with more neighbours has more bonds and each one is worth less.

Run that backwards and the answer here is immediate. A surface atom has fewer neighbours than a bulk atom, so each of its remaining bonds is worth more than a bulk bond — 1.095 times as much, by the arithmetic below — and losing one bond does not cost a whole bond’s worth because the five that remain are not five bulk bonds.

The rule behind it is the one the second moment gives. The mean of the squared level energies is exactly the coordination, so a site with Z neighbours has a band whose scale goes as √Z; if the band’s shape does not change, the binding does the same. Five sixths of the bonds gives √(5/6) of the binding, and √(5/6) = 0.9129.

The bond-counting argument assumes instead that binding is proportional to Z, which would be true if a bond were an independent object with an energy of its own. It is not. A bond in this model is a shared interaction between two orbitals that are also interacting with everything else, and its share of the total depends on how much competition there is.

Where the assumption fails, and how much that matters #

The square-root rule is an approximation too and it is worth saying which one, because the agreement above is better than the rule deserves.

The derivation assumes the band’s shape is fixed and only its width scales, and that assumption is exactly the one where the states pile up shows to be false between structures of different dimension: a chain, a square net and a cubic structure have wholly different shapes, and the binding per unit width differs between them by a factor of 1.91.

The reason the rule works so well here is that nothing about the dimension has changed. The surface layer and the bulk are both parts of a three-dimensional structure; removing one neighbour from a six-connected site does not turn it into a five-connected structure of a different dimension, it perturbs a three-dimensional one. The band’s shape at the surface is close to the bulk shape, and the moment argument, whose only weakness is a shape assumption, does well.

That is worth stating as the general form: a moment argument is accurate where the things compared have the same shape and can be out by a factor of two where they do not. The same rule applied across dimensions is the one measured earlier at an exponent of 0.34 rather than 0.5.

The layer behind the surface binds more than the bulk #

The profile has a feature the accounting does not need and which is worth its own paragraph, because it is a genuine prediction and could have come out either way.

The second layer’s binding is 2.0133 against the bulk’s 2.0015. It is not merely close to bulk; it is above it. The reason is the same rule read once more: the second layer’s neighbours include surface atoms, and surface atoms have spare capacity — their remaining bonds are stronger — so a site bonded to them does better than a site bonded to fully coordinated atoms.

The effect is small, about six thousandths of a β, and it decays fast: the third layer is at 2.0031 and the fourth at 2.0058, both within three thousandths of the bulk. But the sign is the informative part. A naive bond-count picture has no mechanism for a layer to bind more than the bulk, because it has no mechanism for a bond to change its value at all.

The closed form, and the check that licenses it #

An open block of sixteen sites a side has 4,096 sites, and diagonalising a matrix that size takes minutes rather than moments. The calculation here does not diagonalise anything at that size, and the reason it is entitled not to is worth spelling out.

A wrapped or open block is a product: a level is a sum of one contribution per direction, and an eigenvector is a product of one factor per direction. A wrapped direction of length L contributes 2cos(2πk/L) with a uniform amplitude, and an open one contributes 2cos(πk/(L+1)) with a standing-wave amplitude that is what makes a layer profile possible at all.

That is a derivation, and derivations of that shape fail in a characteristic way — right about the energies, wrong about which state sits where. So the whole construction is checked against an actual diagonalisation of a 6 × 6 × 6 block: every level agreeing to 3 × 10⁻¹³ with multiplicities included, and every layer’s binding agreeing to 2 × 10⁻¹³ as well. The second check is the one that matters, because the first would pass even if the amplitudes were assigned to the wrong layers.

The measurement the whole argument rests on is an identity: the second moment of a structure’s band is its coordination exactly, its width is twice that, and its binding per site is neither. The last of the three is the quantity a surface energy is made of, and it is the one the count cannot reach.

The same rule elsewhere in this field #

The result is one member of a family the field has been building without saying so, and gathering the members makes the rule easier to trust.

The end of a chain is the one-dimensional version of the same question — how far into a structure does a boundary reach — and the answer there is a length in sites, short, a handful. The layers above are the same statement in three dimensions.

The end is the hardest place to bind is the same geometry seen from the defect side: a site at the end of a chain demands a whole β of energy difference before it traps a state, where a site in the middle traps one for any difference however small. Both essays are measuring the consequence of an atom having fewer neighbours than its fellows, and both find it larger than a proportional account would say.

Four kinds of interaction between two units of matter set the scale the whole field is measured against, each computed from a stated model. The number this essay is about is a fraction of the covalent entry, and knowing which entry matters more than knowing the fraction.

What the number is not #

Two things this does not compute, and both are the sort of thing a reader is entitled to assume unless told.

It is not a surface energy in joules per square metre. A band gap is not a bond energy makes the neighbouring point about what these units will and will not convert into. Every energy here is in units of β, the interaction between two neighbouring orbitals, and nothing has been said about what β is for any material. The result is a ratio — what a broken bond costs against what a bond is worth — and ratios are what this model can support.

And it is not a relaxed surface. The atoms in the outer layer have been left where they were; a real surface pulls them inwards, because they have unbalanced forces on them and no atoms above to hold them out. That relaxation lowers the surface energy further, so the computed figure is an upper bound in a real material, and the direction of the correction goes the same way as the correction this essay is about.

Reading the error as a measurement #

There is a more useful way to hold the discrepancy than as a mistake, and it is the way this collection tries to hold every failed rule.

The bond count and the truth differ by a factor that is not arbitrary. Write the count’s answer as Z⁻¹ of the site’s binding and the truth as 1 − √((Z−1)/Z), and their ratio is a function of the coordination alone: 1.90 at Z = 6, 1.87 at Z = 4, 1.71 at Z = 2. So the count is not merely wrong; it is wrong by a factor that can be stated in advance, and the statement is a measurement of how far a bond is from being an independent object.

That number — how much of a bond’s worth survives its neighbours being removed — is the same quantity what holds a solid together is about from the other end, where four kinds of interaction are put on one scale and the shared pair is the one whose value depends most on its surroundings.

The wider habit this is an instance of #

The bond-counting argument belongs to a family, and naming the family is more useful than fixing this one member of it.

The family is: treat an extensive quantity as a sum of independent local pieces, then count the pieces. It is behind the additivity of bond energies, behind estimating a heat of formation from a table, behind adding up bond dipoles to get a molecular one, and behind the surface estimate here. The lattice sum a finite calculation cannot do is the ionic member, where the same instinct fails for a different reason — the sum over independent pairs does not converge at all. In every case the sum is over things that are not independent, and the error is a measure of how strongly they interact.

The signature is always the same: the estimate is right about which direction, right about roughly how large, and wrong by an amount that is a fixed fraction rather than a random one. Here that fraction is very nearly a half at coordination six, and it can be computed for any coordination: the cost of one broken bond, relative to a bond’s worth, is Z(1 − √((Z−1)/Z)), which is 0.526 at Z = 6 and tends to a half as Z grows.

That last expression is the repair. It is not more complicated than the count it replaces, it takes the same input, and it is right.

Which face is cheapest, and why the repair makes crystals sharper #

The repair generalises to any number of broken bonds without any new calculation, and doing so answers the question a surface energy is usually asked in practice: not how much but which face.

An atom that keeps $Z - n$ of its $Z$ neighbours retains $\sqrt{(Z-n)/Z}$ of its binding, so exposing it costs

$$\left(1 - \sqrt{\tfrac{Z-n}{Z}}\right) \times \text{(binding per atom)}.$$

For a simple cubic structure the three low-index cuts break one, two and three bonds per exposed atom and expose them at areal densities in the ratio $1 : 1/\sqrt2 : 1/\sqrt3$. Multiplying gives the cost of each face per unit area, in units of the binding per atom over the square of the lattice spacing:

The cube face is cheapest, which is the answer bond counting also gives, so the repair does not overturn the ordering. What it changes is the spacing. Counting bonds puts the three faces at $1 : 1.414 : 1.732$; the square-root rule puts them at $1 : 1.490 : 1.941$.

The anisotropy has grown, and the reason is that a square root is concave. Losing three neighbours costs more than three times as much as losing one — 0.2929 against three times 0.0871 — because each successive bond removed leaves fewer competitors for what remains, so the ones already lost were the cheap ones. Bond counting, which prices every bond the same, cannot see that the marginal cost rises.

So the correction runs in two directions at once, and both are worth carrying. Every surface is cheaper than bond counting says — by a factor near a half at coordination six, which is this essay’s headline. And the difference between surfaces is larger than bond counting says, by about twelve per cent across these three faces.

The second has the more visible consequence. How strongly a crystal facets is decided by the ratio of its surface energies rather than by their absolute size: a material whose faces differ little grows rounded, and one whose faces differ greatly grows as flat plates and sharp polyhedra. An estimate that compresses the differences between faces therefore under-predicts exactly the thing anybody looks at a crystal and sees.

What is left #

The block here is simple cubic and cut across a face, which is the easiest surface there is: every exposed atom loses exactly one neighbour and there are no steps, edges or corners. A real crystal’s faces are not all like that, and the section above takes the arithmetic as far as the three low-index cuts of one structure. What it does not do is steps, edges and corners, where an atom loses neighbours from more than one direction and the count of what it keeps is no longer decided by the plane alone — which is where a real crystal’s growth actually happens.

The repulsion between the electrons is absent, as everywhere in this field, and at a surface its absence has a specific consequence: the electrons spill outwards into the vacuum, leaving a charge separation and an electrostatic cost that a one-electron graph model has no representation of at all. That term is a real part of a real surface energy and none of it is here.

And the whole calculation is at half filling. The relation between binding and coordination depends on where the band is filled to, and a nearly empty or nearly full band would give a different exponent — which is worth measuring, because the square-root rule is quoted as though the filling were not one of its assumptions.

The binding a band supplies against how far it is filled is the assumption everything here is read off the top of, and the arithmetic relating binding to coordination is a statement about that maximum rather than about the whole curve.