Difference Map
https://xraytools.com/difference?structure=cristobalite&omit=O
What a partial model is missing. The page runs the standard step — subtract, keep the phases, transform — on a structure whose absent atom you removed yourself, so the answer is known before the map is drawn.
- Before this
- This is the difference between what was measured and what a partial model calculates, so it assumes there is a model. The ordinary synthesis comes first.
- You supply
- One of the named structures and which of its sites to leave out. The amplitudes come from the complete structure and the phases from what is left of it, which is the only difference between this and a real refinement.
- Reading it
- A difference map needs most of the scattering already in the model. Over every omission this page offers, deleting a site worth less than 40 % of the cell’s scattering was found at the strongest peak every time; above that it succeeds about half as often, and the page names which case you are in. Every failure still shows the atom when given the true phases, so what fails is the model rather than the map.
Worked examples: zirconia without O1 — a small omission, found · zirconia without Zr — the heavy atom, lost · cristobalite without O — what the projection cannot show
Earlier on the path: Charge Flipping On From intensities to a structure, step 6 of 6
Notation here: F, |F| — what each one means here
See also: Fourier Synthesis and the Phase Problem · Charge Flipping · Displacement Parameters and NPD Atoms · Refinement Statistics and R Factors
What each input changes
- Omitted atom
- Which atom is left out of the calculated model. What appears in the map is what the model is missing — so this is the control that shows the map working on an answer you already know.
Teaching with this page
- Objective
- After this page a learner can read a difference map, and say what a peak in one does and does not establish.
- Start from
- this worked example
- Ask first
- A difference map shows a maximum close to a heavy atom. Is that an atom the model is missing?
- Watch for
- “Yes — the map shows what the model does not account for”
Check yourself: A difference map shows a maximum close to a heavy atom. Is that an atom the model is missing?
Not established — the model’s own errors appear there too Yes — the map shows what the model does not account for
It does show what the model does not account for, and that includes the model being slightly wrong: a position out by a little, a displacement parameter too small, or the truncation ripple that surrounds a heavy scatterer all leave residual density. Height, distance and chemistry decide together, and beside a heavy atom is where each of the three is worth checking before anything is added.
Input
What the difference map found
α-cristobalite, SiO2, P41212 — 284 reflections to 0.50 Å, with O deleted from the model.
This omission cannot be shown at this grid and resolution: even with the true phases the largest peak of the difference map is not at O. What failed is the projection, not the model — nothing computed from these amplitudes could find this atom, so the result below says nothing about difference maps.
- Si 47 %
- O 53 %
The map
The large circles are where O actually is; the small one is the strongest peak. Nothing about those positions went into the calculation. The map is drawn in the structure’s own frame with no shift and no inversion — unlike charge flipping, the phases here come from a model that already sits in a chosen origin, so there is nothing to search.
The map runs both ways. A positive peak is electron density the model does not account for — an atom you have not placed. A negative trough is the opposite: density the model claims and the measurement does not support, which is what an atom in the wrong place, or given too heavy an element, looks like. A finished refinement is judged by both being flat.
The strongest peaks
| # | x | y | Height | At the omitted site? |
|---|---|---|---|---|
| 1 | 0.219 | 0.094 | 3.6 σ | yes |
| 2 | 0.094 | 0.219 | 3.6 σ | yes |
| 3 | 0.781 | 0.906 | 3.6 σ | yes |
| 4 | 0.906 | 0.781 | 3.6 σ | yes |
| 5 | 0.281 | 0.594 | 3.6 σ | yes |
| 6 | 0.594 | 0.281 | 3.6 σ | yes |
| 7 | 0.719 | 0.406 | 3.6 σ | yes |
| 8 | 0.406 | 0.719 | 3.6 σ | yes |
Heights are multiples of the map’s own rms, which is measured from the map rather than from the observed amplitudes — these coefficients are differences, most of them near zero, so the amplitudes would be the wrong scale by a large factor.
The numbers behind the verdict
| Quantity | Value | What it says |
|---|---|---|
| Rank of the omitted site | 1 | Where the first peak that lands on O comes in the list, within one grid step. One is the only answer that counts as finding it. |
| Scattering removed | 53 % | What deleting this site took out of the cell, as electrons at s = 0. Below 40 % every omission this page offers is found at rank one. |
| Peak at the site, and elsewhere | 3.6 / 0.0 σ | The best peak that is an image of O, against the best peak that is not. This is the contrast a reader judges, and the ratio is 0.00. |
| Floor, from random phases | 0 of 20 | How often the same amplitude differences with random phases put their strongest peak on the site anyway. Anything the run achieves at this rate is worth nothing. |
| Control, from the true phases | not found | What the same amplitude differences show when given the right phases. Not found here means this projection cannot show that atom at all, and the run above is not evidence about difference maps. |
Why the floor is small here
The phases come from a model that already sits in a chosen origin, so this map arrives in the same frame as the answer and there is nothing to search: no origin shift and no choice of hand. That is the whole reason the floor here is near zero while charge flipping has to measure a floor that sometimes equals the full count — it recovers a map from amplitudes alone and fixes neither.
It is printed on every result all the same. A success criterion nobody has measured against chance is a criterion nobody should believe, whichever way it comes out — and the sibling page is the evidence for that, because there the same question has a very different answer.
Where this comes from
- Some properties of the (Fo − Fc)-synthesis
W. Cochran, Acta Cryst. 1951, 4, 408–411 · doi:10.1107/S0365110X51001355
Why the difference synthesis is the right map to look at rather than the observed one: the errors in the model largely cancel, so a peak in it is what the model is missing rather than what the model already says.