A cubic cell has fourteen chances to fail

Two conditions are not one condition twice asked how a measured plane cell should be tested for the extra conditions a higher symmetry needs. A square cell needs a right angle and two equal edges. There were three rules. The direct one computes the cell’s distance from the square cells as a $\chi^2$ on two degrees of freedom. The walk tests each added condition on its own, climbing the diagram one step at a time and accepting the square only if every step on the way passes. The third divides the level between the walk’s two steps. Twenty thousand cells drawn around a true square cell said that the direct test and the divided walk both reject five per cent of them, as they claim to, and that the undivided walk rejects nearly ten.

It ended on the question the plane cannot ask. In three dimensions the strata a cell can sit on are not a chain but a lattice of inclusions, several paths lead up to one stratum, and some steps add several conditions at once. A walk that divides its level among the steps of every path divides it among more than two, and whether each step is then left enough power to reject anything is a question of arithmetic. The calibration that answers it is the same one, run on cells drawn around a true cubic cell with the six parameters of a real refinement and their covariances.

This essay runs it. The walk has fourteen steps on ten paths, the plane’s remedy undercorrects and the obvious generalisation overcorrects, and the level that is actually right, measured on the draws, gives a walk that is more sensitive than the direct test to the distortion it most needs to see.

Ten strata above a cubic cell #

A cell in space is six numbers, the edges $a$, $b$, $c$ and the angles $\alpha$, $\beta$, $\gamma$. Every condition that the lattice strata above a primitive cubic cell impose is linear in them: an angle equal to $90^\circ$, two edges equal, two angles equal. A stratum is a set of such conditions, and going up costs the cell a parameter for each of them. The strata a primitive cubic cell lies on are ten. The triclinic imposes nothing. There are three monoclinic ones, each needing two right angles, and the orthorhombic, needing all three. The three tetragonal strata add two equal edges to the orthorhombic. The rhombohedral needs three equal edges and three equal angles, at any value. The cubic needs three right angles and three equal edges.

In the plane the corresponding list had five members, the strata of the five plane lattices, and the square was two steps up from the oblique cell by either of two routes. The rhombohedral stratum is the one that makes the diagram in space a lattice. A primitive cubic cell is a rhombohedral cell whose angle happens to be $90^\circ$, which is the observation a lattice described on somebody else’s axes makes about the fourteen Bravais lattices. So the cubic stratum can be reached from the rhombohedral by adding one right angle, as well as from any of the three tetragonal ones by adding one pair of equal edges.

The picture at the head of this essay is that diagram, computed rather than drawn from memory. Two strata are joined when the second’s conditions imply the first’s, as linear algebra rather than as lists, since equal $a$ and $b$ and equal $b$ and $c$ imply equal $c$ and $a$ without anyone saying so. They are joined only when no stratum lies between. There are fourteen steps. There are ten paths from the triclinic to the cubic. Nine have four steps, through one of the monoclinic strata, the orthorhombic and one of the tetragonal strata. One has two steps, through the rhombohedral. The first step of that short path adds four conditions at once.

For each stratum the cell’s distance is its least Mahalanobis distance to the flat the conditions cut out, a $\chi^2$ with as many degrees of freedom as conditions. Because the conditions are linear and nested, the increase in $\chi^2$ from one stratum to the next along a step is itself a $\chi^2$ on the conditions the step adds, and the increases along one path are independent. That is what lets a step be tested on its own.

The measurement #

The cells are drawn around a true cubic cell of edge 10 Å, with standard uncertainties of 0.0005 Å on each edge and 0.005° on each angle. The edges are correlated at 0.5, because they come from one refinement against one wavelength and one set of detector distances, and an error in any of those moves all three together. These are round numbers of the size a good single-crystal diffractometer reports, stated rather than taken from a particular compound. The angles are left uncorrelated with each other and with the edges.

The correlation matters in one place. A test of equal edges looks at $a - b$, whose uncertainty is $\sigma\sqrt{2(1 - \rho)}$. At a correlation of 0.5 that is exactly the uncertainty of a single edge, 0.0005 Å. The plane essay found that this covariance, which structure reports rarely give, decides equal-edges verdicts, and here it sets the scale on which a tetragonal distortion is measured. Two standard uncertainties of $c - a$ is a difference of 0.001 Å on 10, one part in ten thousand.

Four rules and what they reject #

The direct test computes the cell’s $\chi^2$ from the cubic stratum on its five degrees of freedom, the two-condition test of the plane with three more conditions added, one for each further parameter a cubic cell has to fix. It rejects 4.96 per cent of the twenty thousand true cubic cells, which is five per cent within the sampling error, as a calibrated test must.

The walk, every step at five per cent, rejects 28.8 per cent. A true cubic cell has fourteen chances to fail and needs to pass them all, and even though the chances are not independent, over a quarter of true cubic cells fail one of them. In the plane the same rule rejected 9.8 per cent, which was bad enough.

Dividing the level between the steps of a path, the plane’s remedy, puts each step at $1 - 0.95^{1/4} = 1.27$ per cent, since a path has four steps. It rejects 9.0 per cent. The division is right for one path and wrong for the walk, because the walk does not test one path. It accepts the cubic stratum only when every path passes, and the ten paths between them have fourteen distinct steps.

Dividing among all fourteen, $1 - 0.95^{1/14} = 0.37$ per cent a step, rejects 2.9 per cent. That overcorrects, because the formula assumes the fourteen steps are independent and they are not. Nine of the ten paths pass through the orthorhombic stratum, so the three steps into it share its three right angles between them. The three steps out of it test pairs of equal edges, any two of which imply the third. A cell that fails one of those steps has an unusually large difference between two edges, which makes it more likely to fail another, and the fourteen chances overlap.

The level, measured #

The rules are all the same rule with different per-step levels, and the right level can be read straight off the draws. A true cubic cell fails the walk at a per-step level $\ell$ exactly when its smallest step probability is below $\ell$. So the false-rejection rate as a function of the level is the distribution of each cell’s smallest step probability, and the level that gives five per cent is that distribution’s fifth percentile.

It is 0.66 per cent a step, between the two formulas’ values of 0.37 and 1.27. In the formula $1 - 0.95^{1/m}$ it corresponds to $m = 7.7$, so the fourteen steps behave, for this purpose, like fewer than eight independent tests. The number has no reason to be a whole number, since the steps are partly dependent rather than divided into independent groups.

What it does not depend on is more surprising than what it does. The same calibration run with the edges uncorrelated, and again with a correlation of 0.9 and angles four times less precise, gives 0.67, 0.66 and 0.66 per cent. The reason is that under a true cubic cell every step’s statistic is a $\chi^2$ measured in the cell’s own uncertainties, so the uncertainties divide out. What is left is the geometry of the conditions: which steps share which conditions, and at what angles the flats they cut out meet once the covariance has been used to measure them. With the three edges measured equally well, and the three angles equally well, a correlation between the edges changes the scale of every edge difference by the same factor and leaves that geometry alone. So the level of 0.66 per cent is a property of the diagram, not of the diffractometer. It would change if the three edges were measured unequally well, as they are when one axis of a crystal is much longer than the others and a cubic cell is not in question, or if the angles and edges were correlated with each other. Near-symmetry, and the tolerance that is not here argued that no single tolerance can serve every pattern. The measured level is the opposite kind of number. It is a tolerance on probabilities rather than on parameters, and within this diagram one serves every cell.

The plane never needed this, because its two steps were independent and the formula was exact. In space the formula is exact only for a chain, and the strata above a cubic cell are not one.

What each search calls a cubic cell #

A structure report does not just accept or reject the cubic stratum. It names a lattice, the most symmetric stratum the rule accepts, and a cubic cell that fails the cubic test is reported as something. The lead question was how often a cubic structure measured this well would be called tetragonal.

The direct rule, accepting each stratum whose own $\chi^2$ passes, calls 95.0 per cent of the cubic cells cubic, 2.9 per cent tetragonal, 1.2 monoclinic and 0.7 rhombohedral. The undivided walk calls only 71.2 per cent cubic, 13.6 per cent tetragonal and 12.1 per cent rhombohedral. The rhombohedral calls are the ones the plane had no analogue for: a cubic cell whose angles are fine but one of whose right-angle steps has failed can still reach the rhombohedral stratum by its own path, which tests the angles for equality rather than for a particular value. The walk at the measured level of 0.66 per cent calls 95.0 per cent cubic, 2.7 per cent tetragonal and 2.0 rhombohedral.

The second half of the picture shows a failure the calibration alone does not show. On a cell that is plainly tetragonal, six standard uncertainties long in $c$, the undivided walk calls 81.3 per cent tetragonal and 14.9 per cent monoclinic. A tetragonal stratum is accepted only when every path up to it passes, and three paths lead to each tetragonal stratum through three different monoclinic ones. Every test on the way that has nothing to do with the distortion is still a chance to fail. The walk’s false rejections are not confined to the stratum being asked about. They happen at every level below it too.

Which rule notices a distortion #

A rule that never rejects anything has a perfect false-rejection rate. The question that matters is how often each rule notices a real distortion. The natural one to try is the smallest one a cubic cell can have: $c$ slightly longer than $a$ and $b$, a tetragonal cell.

With $c - a$ one standard uncertainty long, the direct test declines to call the cell cubic 11.7 per cent of the time; with two, 39.1; with three, 76.4; with four, 96.5. The undivided walk declines far more often, 42.1 per cent at one, but most of that is its false-rejection rate, which is 29.9 per cent in this sample with no distortion at all. The fully divided walk is the least sensitive, at 34.8 per cent at two.

The walk at the measured level is more sensitive than the direct test at every separation: 42.6 against 39.1 per cent at two standard uncertainties, and 81.4 against 76.4 at three. At the same false-rejection rate, it is the better detector of the distortion. The reason is that a tetragonal distortion violates one condition, and the walk has a step that tests exactly that condition on one degree of freedom. The direct test spreads its attention over all five conditions, four of which the distortion leaves alone. A test that concentrates on the right degree of freedom has more power for the same level. The plane could not show this, because its alternative was always one step from the square.

The advantage is specific to that kind of distortion. A distortion that moved all six parameters a little would favour the direct test, which pools them, and a rhombohedral distortion, the angles all moving together away from $90^\circ$, would be caught by the step from the rhombohedral stratum. Where the distortions that occur in real cubic materials sit on that range is a fact about materials, not about the diagram.

What the calibration does not decide #

The measurement model is stated, not measured. Real refinements report correlations between the angles as well as the edges, and between edges and angles in low-symmetry settings. Every number above depends on the covariance, except the facts that the direct test is calibrated and that the measured level calibrates the walk.

The steps are tested at one level each, not at levels suited to what they test. A step that adds four conditions, like the one from the triclinic to the rhombohedral stratum, could be given a different share of the level from a step that adds one, and a walk that weighted its steps by how often a real distortion violates them would be more sensitive still. The measured level treats all fourteen alike, which is the simplest rule that is calibrated, not the most powerful one.

Only strata a primitive cubic cell lies on are used. A body-centred or face-centred cubic cell has other strata above it, and centring is not innocent here: centring one face of a cube removes its three-fold axes altogether. The strata above a centred cubic cell include the body-centred tetragonal and the face-centred orthorhombic, whose conditions are stated on a different cell. Twenty-five cells, and fourteen lattices counted the cells those strata come from. The diagram would change and so would the measured level, though not the method.

A step is tested with $\chi^2$ on its own added conditions. That is exact for nested linear conditions and Gaussian errors. It is approximate for a real refinement’s standard uncertainties, which are themselves estimates, and at the tails that decide a one-per-cent test the approximation is not guaranteed.

And a verdict is not a structure. A cell that passes the cubic test is a lattice that could carry a cubic group. Whether the atoms do is a second question, answered by the intensities and not by the cell, and it can go either way: where a symmetric motif lifts the group found the plane’s cases in which the contents raise a pattern’s symmetry beyond what their site asks for, and a lattice with more symmetry than its contents is the everyday case.

Who worked out the pieces #

Testing a measured cell for higher metric symmetry is what searches descended from Le Page’s program of 1982 do, and Andrews and Bernstein’s six-dimensional description of cells is the space they search in; the International Tables catalogue the strata as the Bravais types’ metric conditions. The multiple-testing problem the walk runs into is the one statistics met in the 1950s, and Šidák’s correction of 1967 is the $1 - (1-\alpha)^{1/m}$ used above. What this essay adds is the diagram’s own count, fourteen steps on ten paths, and a level measured for it.

What the calibration must refuse #

The refused claim is the natural extension of the plane’s remedy, and it is wrong in the direction of too many rejections. The test that an exactly cubic cell lies on every stratum is the check on the conditions themselves. A sign error in one of them, or a missing implication, would put an exact cubic cell at a non-zero distance from some stratum it lies on, and the walk would then reject it for a reason with nothing to do with measurement.

Still open: the cells a cubic lattice can be written on #

The diagram here starts from a primitive cubic cell written on its own edges. A refinement reports whatever cell the indexing chose, often a reduced one, and a cubic lattice written on a reduced cell of a centred lattice, or on a skewed primitive cell, sits on the same strata through conditions that are no longer on single parameters but on combinations of them. How far one lattice is from another found in the plane that a distance between lattices that no change of basis can move has to be measured in a hyperbolic geometry rather than on the cell parameters directly. The diagram of strata for every cell a cubic lattice can be reported on, with the level measured for each, would say whether the calibration depends on the cell as well as on the lattice. It has not been built here.