xraytools.

Learning paths

https://xraytools.com/paths

The calculators are grouped by subject, and a subject is not a sequence. These routes each say who they are for, what you will be able to do at the end, which pages to work in what order, and where they stop.

How to use these

Each path is a handful of pages in a recommended order, with something to do on each. The order is the one the material builds in, not a lock: a page that needs something understood first says so in its own Before this note, and you can start anywhere. Some steps carry their example forward — the beginner path keeps coming back to rock salt — and some change subject deliberately, because the clearest demonstration of a step is not always the same material as the last one. Where a step names a worked example, that example is the state to start from; it is not a promise that the next step continues it.

There is nothing to sign up for and nothing that remembers where you got to. A result that is computed from typed input has its own address, so a step you want to come back to can simply be bookmarked. An uploaded CIF and a pasted scan are the exceptions.

From planes to a powder pattern

Somebody who has met Bragg’s law in a lecture and not used it.

At the end you will be able to

The steps

  1. Lattice Explorer Powder or single crystal

    Repeat the cell, then add a motif. Switch between the lattice points and the atoms. Explain why a point does not have to be an atom, and why eight visible corners count as one point per cell.

    Start from: the worked example

  2. Bragg Calculator Powder or single crystal

    Turn a measured angle into a spacing, then turn the spacing back into an angle. Do it in both directions before going on — the whole path is that one relation, used forwards and backwards.

    Start from: the worked example

  3. HKL Calculator Powder or single crystal

    The same crystal from the other end: a cell and a Miller index give the spacing, so the angle follows. Check the d it prints against the one you just got from Bragg’s law. They are the same number, and seeing that is the point of the step. Then compare two patterns: change the cell, the centre-site occupancy or the wavelength, one at a time.

    Start from: the worked example

  4. Reciprocal Cell Calculator Powder or single crystal

    Where the reflections actually live. Work the zone-law example: it holds with no cell constants at all, which is what tells you the relation is about indices rather than about a particular crystal.

    Start from: the worked example

  5. Space Group Reflection Conditions Powder or single crystal

    Before reading the holes, learn to read the symbol. Rock salt is Fm3m, and the symbol is two statements. F is the lattice: centred on every face, so each atom has copies half a face diagonal away. The rest is the symmetry about a point — mirrors and rotation axes. An absence always comes from an operation with a translation in it, which is why a mirror or a pure rotation never causes one, and here the centring is the only translation the symbol names. It removes every reflection whose indices mix odd and even. The page opens on 100: read which operation it says removes it, then check that each condition in the list is that one rule written for a plane or a row. Afterwards open P21/c from the examples for the other two kinds of translation: a screw axis, which rotates and then shifts along the axis, and a glide plane, which reflects and then shifts along the plane.

    Start from: the worked example

  6. Space Group from Absences Powder or single crystal

    Now the same thing backwards, which is how it happens with real data. The pattern has holes in it, and which classes of reflection are missing is what names the symmetry. The example answers the form with the conditions you have just read and nothing else, and eight space groups fit: absences fix the centring and cannot tell those eight apart. Write down what the rule says about 100 before going on.

    Start from: the worked example

  7. Structure Factor Calculator Powder or single crystal

    Why a reflection is strong, weak or missing: the atoms’ contributions add or cancel term by term. Step through the sum for the reflection you started with on /bragg — 200, where every term arrives with the same sign, so it comes out large. Then change the indices to 100 and step through that sum instead: the same atoms now cancel in pairs and the total is zero. That is the absence the previous step predicted, arrived at from the atoms rather than from the lattice type.

    Start from: the worked example

Try it before you look

In rock salt, sodium and chlorine alternate along the cell edge. Before you open it: will reflection 111 be stronger or weaker than reflection 200, and why?

Predict a number. Rock salt is cubic with a = 5.640 Å. At what 2θ does its 220 line appear, with Cu Kα1 at λ = 1.540562 Å?

°

Settle it. Open the same question on the HKL Calculator, which computes it from the cell rather than telling you about it.

Observe. Open the structure factor for NaCl and step through the sum for both reflections. Watch the sign each atom’s term carries. Open it

Explain — open this after you have looked

111 is much the weaker. For 200 every sodium and every chlorine term arrives with the same sign, so the scattering adds; for 111 the two species arrive with opposite signs and largely cancel, leaving only the difference between their scattering powers. The tempting wrong answer is that 111 has the larger d, so it comes at a lower angle where scattering factors are higher — true, and it is beaten by the cancellation. Position is set by the lattice; intensity is set by the motif, and this is the cleanest case of the two coming apart.

A plane cuts a halfway along, never meets b at all, and cuts c a quarter of the way along. What is its Miller index?

Observe. Type an index into the plane field on the reciprocal page until the “where it cuts the axes” table shows 1/2, never and 1/4. Open it

Explain — open this after you have looked

(2 0 4). Each index is the reciprocal of an intercept measured in cell fractions: 1/(1/2) = 2, 1/∞ = 0, 1/(1/4) = 4. The b index is 0 because the plane never meets that axis — not because it meets it at the origin, which is the commonest reading of a zero index and is wrong. The tempting wrong answer is (1 0 2), from reducing the triple to its lowest terms. Those are different families with different spacings, and the clearing step only ever scales up: the plane you were given cuts a at 1/2, and (1 0 2) is the family whose first member cuts it at 1. Many textbooks do reduce: there the Miller indices give only the orientation of a plane and the unreduced triple is kept for the reflection. This site keeps the factor, because the spacing is what its calculators work with.

Where this stops. That is the path. You can now say what a reflection is, where it appears, and why one the cell permits can be absent. The pattern panel on the HKL calculator is where all of it goes back together into a diffractogram.

From a measured scan to sample information

Somebody holding a diffractogram who wants numbers out of it.

At the end you will be able to

The steps

  1. Peak Finding Powder data

    Thousands of points become a handful of numbers. Start with the example that cannot be resolved: the failure is the lesson, because nothing downstream can tell you it happened.

    Start from: the worked example

  2. Powder Indexing Powder data

    Positions to a lattice. Then open the three-line example next to it — six cells account for the data equally well and the page refuses to choose, which is what an indexing result is worth on its own.

    Start from: the worked example

  3. Line Broadening Powder data

    Widths to a crystallite size. Run the size-only and strain-only examples one after the other and compare how the width changes with angle: that difference is the whole separation.

    Start from: the worked example

  4. Intensity Corrections Powder data

    What stands between a recorded count and |F|². Every factor on this page is Bragg-Brentano geometry, which is why the page is labelled powder and why none of it transfers to single-crystal data.

    Start from: the worked example

Try it before you look

Two powders give lines of the same width at 20° 2θ. One is broadened by small crystallites and the other by strain. What differs at 70°?

Predict a number. A line at 2θ = 30.00° has a full width at half maximum of 0.30° once the instrument’s own width is taken out. Cu Kα at λ = 1.541838 Å, K = 0.9. How large are the crystallites, in nm?

 nm

Observe. Run the size-only example, then the strain-only one, and compare how the width grows with angle in each. Open it

Explain — open this after you have looked

The strained sample’s lines are much wider at high angle. Size broadening goes as 1/cos θ and strain broadening as tan θ, which climbs far faster — and that difference in shape is the entire basis for separating the two. It follows that a pattern with only low-angle lines cannot separate them at all, however precisely each width is measured. The tempting wrong answer is that they stay the same because they started the same: one measurement is one equation, and there are two unknowns.

Where this stops. That is the path. You can take a scan to a cell, a size and a corrected intensity, and say in each case what the step could not have told you.

From intensities to a structure

Somebody who knows what a structure factor is and wants to see the phase problem actually solved.

At the end you will be able to

The steps

  1. Structure Factor Calculator Powder or single crystal

    The forward direction: atoms to amplitudes and phases. Note which half of each term survives a measurement and which does not.

    Start from: the worked example

  2. Fourier Synthesis and the Phase Problem Single-crystal data

    The transform back, with the phases handed to you. Then take them away: the same amplitudes with every phase set to zero put all six terms in step at the origin, so the map becomes one tall peak there and nothing anywhere else — a picture of where the atoms are not. That is the phase problem in one picture.

    Start from: the worked example

  3. The Patterson Function Single-crystal data

    A map you can compute with no phases at all, because it transforms the intensities. It contains vectors, not atoms — that distinction is the step.

    Start from: the worked example

  4. Direct Methods and the Sign Relation Single-crystal data

    The signs are not free: strong reflections whose indices add up constrain each other. Run the example where every relation holds, then the one where the weak ones fail.

    Start from: the worked example

  5. Charge Flipping Single-crystal data

    The same problem attacked by iteration instead of by statistics. Watch the score, and read what the page says a good score is worth.

    Start from: the worked example

  6. Difference Map Single-crystal data

    What a partial model is missing. This is the map that finishes a determination, and the end of the path.

    Start from: the worked example

Try it before you look

You have every measured amplitude, correct to four figures, and no phases at all — so you set every one of them to zero. What does the map show?

Predict a number. The quartz projection is summed from six terms. F(000) = 89.99 electrons and the six amplitudes sum to 61.56 electrons. If every phase is set to zero, how high is the map at x = 0, in electrons per unit x — the units of the chart’s vertical axis?

 electrons per unit x

Settle it. Open the same question on the Fourier Synthesis and the Phase Problem, which computes it from the cell rather than telling you about it.

Observe. Run the quartz synthesis and compare the two curves: the true one, and the one with every phase set to zero. Open it

Explain — open this after you have looked

One large peak at the origin, and it would be there whatever the crystal was. With every phase zero the sum is F(000) + 2∑|F| cos 2πhx, so at x = 0 every cosine reaches 1 at once and no other point can match it. The one feature the map has is therefore a property of having discarded the phases, not of the structure. Amplitudes say how much of each wave to add; phases say where to put it, and a structure is where things are — which is why the phase problem is the central problem of the subject. The tempting wrong answer is "a blurred version of the structure", by analogy with noisy amplitudes, and it is the single most useful misconception to lose. Note what the question is not: getting every phase wrong by the same 180° would simply negate the map, leaving the atoms at its minima and the structure trivially recoverable. A uniform phase error is the least damaging one there is. The difficulty is that phases are unknown individually.

Where this stops. That is the path. You can say which quantity a diffraction experiment records, what is lost, and what three different methods do about it.

How to read a published structure

Somebody handed a CIF or a paper who has to judge whether to believe it.

At the end you will be able to

The steps

  1. CIF Parser Powder or single crystal

    Start with a real file. None ships with this site, so take one from the Crystallography Open Database: search by formula or mineral name, pick a single-crystal entry recent enough to report its refinement, and download its CIF. Upload it here with the default template and find the cell, the space group and the refinement rows in the table that comes back.

  2. Interatomic Distances and Angles No diffraction data

    Distances and angles, with the uncertainties the refined parameters support. Two distances differing by less than their combined uncertainty are not resolved from each other — which is not the same as being equal, and the page says why the combined figure is approximate in the first place.

    Start from: the worked example

  3. Displacement Parameters and NPD Atoms Single-crystal data

    The atom is a probability cloud, and its site symmetry decides what shapes are allowed. Run the impossible tensor: a refinement can report one, and it means the model is wrong somewhere.

    Start from: the worked example

  4. Refinement Statistics and R Factors Single-crystal data

    The table at the end of the paper, one number at a time. Read what each quantity is a statement about — and what the whole set still cannot tell you about the chemistry.

    Start from: the worked example

Try it before you look

A paper reports two Si–O bonds as 1.605(4) Å and 1.614(4) Å and calls them significantly different. Are they?

Predict a number. A paper reports two bonds as 1.605(4) and 1.614(4) Å. What is the standard uncertainty on the difference between them, in Å to two significant figures?

 Å

Observe. Open the quartz geometry with uncertainties supplied and read the uncertainty the page puts on each distance. Open it

Explain — open this after you have looked

No. The difference is 0.009 Å and the uncertainty on a difference combines both, so it is about 0.006 Å — the gap is roughly 1.6 standard uncertainties, where three is the usual threshold for calling two values different. The tempting wrong answer is that 4 in the last place is small and the values plainly differ in the third decimal: the question is never how precise each number is, it is whether the gap between them is large compared with the uncertainty ON the gap.

A refinement reports a displacement tensor whose ellipsoid the drawing program will not render. What has gone wrong — the drawing, or the structure?

Observe. Open the impossible tensor and read what the page says about it. Open it

Explain — open this after you have looked

Neither, directly: the model has. A displacement tensor has to be positive definite — every direction must give a positive mean-square displacement — and a refinement is perfectly capable of returning one that is not, because it is fitting numbers rather than obeying physics. Such an atom is called non-positive-definite and it is a symptom: usually the wrong element at the site, an unmodelled disorder, or a bad absorption correction. The tempting wrong answer is to constrain the atom isotropic and move on, which removes the symptom and keeps the cause.

Where this stops. That is the path. You can read a determination’s own account of itself and say where it is strong, where it is silent, and which questions it does not answer at all.

Choosing the radiation

Somebody who has always used whichever tube was in the machine.

At the end you will be able to

The steps

  1. X-ray Tube Spectrum Powder or single crystal

    Where the photons come from. Two things are happening at once: a continuum from electrons being decelerated, which starts abruptly at a wavelength the tube voltage fixes, and the sharp characteristic lines, which need an electron energetic enough to knock a K electron out. Open the silver tube at 20 kV afterwards: the continuum is there and the lines are gone, which is the second mechanism switched off while the first carries on.

    Start from: the worked example

  2. Moseley Plot and Absorption Edges Powder or single crystal

    Why the filter is the element it is. An absorption edge is a step in how strongly an element absorbs, at the energy that ionises one of its shells — and a filter is the element whose edge falls between the anode’s two lines, so Kβ lands on the absorbing side and Kα does not. Read the edge column against the two wavelengths beside it before going on; the next step turns that gap into a thickness.

    Start from: the worked example

  3. Absorption Coefficient Calculator Powder or single crystal

    The same coefficient, now applied to your own sample. Work the lead sulfide example, where a crystal of that size stops almost the whole beam at both wavelengths and copper loses several orders of magnitude more than molybdenum, and read the Kβ filter panel at the bottom: it is the previous step’s edge and this step’s exponential, and nothing else.

    Start from: the worked example

  4. Friedif Calculator Single-crystal data

    The one place the choice of anode changes what can be measured rather than how long it takes. This step is single-crystal work: the anomalous signal that settles an absolute structure depends on where the photon energy sits relative to your heaviest atom’s edge, so the answer is a property of the formula and not of the radiation. Run the selenium example, then change the element and watch Friedif move at each energy.

    Start from: the worked example

Try it before you look

A nickel filter is put in front of a copper tube to remove Kβ. Before you open anything: what does it do to Kα, and roughly how much of it survives — nearly all, about half, or a few per cent?

Predict a number. A Cu tube emits Kβ and Kα in the ratio 0.20. Nickel’s mass absorption coefficient is 46.8 cm² g−1 at Cu Kα and 282.1 at Cu Kβ. What mass thickness of nickel foil brings the emitted ratio down to 1 : 100?

 mg cm−2

Observe. Open the lead sulfide example and read the Kβ filter panel at the bottom of the page. The last column is the Kα that gets through. Open it

Explain — open this after you have looked

About half of it, and that is the price of the filter. Nickel is chosen because its absorption edge lies between the two lines, not because it is transparent to Kα — it absorbs Kα too, about six times less strongly, and the thickness that kills Kβ takes roughly 45 % of Kα with it. The tempting wrong answer is “nearly all”, from reading the filter as a sieve that passes one line and stops the other. Nothing in an absorption coefficient works that way: every photon energy is attenuated, and a filter buys a ratio, always at a cost in the line you wanted.

Where this stops. That is the path. You can say what a tube emits, why one metal foil and no other cleans it up, how much of it your sample will swallow, and which of two anodes the sample itself argues for.

Teaching index

One line per page that has a teaching record: what a learner can do afterwards, what to ask them before they look, and the misconception that answer usually rests on. The same records appear on each page under “Teaching with this page”.

Each page with a teaching record: what a learner can do after it, and what to watch for. CSV
Page After this a learner can Ask first Watch for
Lattice Explorer distinguish lattice points from atoms, repeat a motif, and count the shared points in a conventional cubic cell A primitive cell shows a sphere at each of its eight corners. Does it contain eight lattice points? “Yes: every sphere belongs entirely to this cell”
Bragg Calculator convert between a measured angle and a lattice spacing in either direction, and say which quantity the instrument supplied A powder diffractometer reports a line at 31.7°. Is that the angle to put into Bragg’s law? “Yes — it is the angle the instrument measured”
HKL Calculator read a Miller index, find the reflections symmetry makes equivalent to it, and predict where the line appears A powder pattern shows one maximum at 43.3°. How many reflections produced it? “One — one maximum, one reflection”
Reciprocal Cell Calculator distinguish a plane from a direction with the same three numbers, and say when they coincide In a hexagonal cell, does the direction [100] point along the normal to the plane (100)? “Yes — they carry the same three numbers”
Space Group from Absences infer candidate space groups from which classes of reflection are missing, and say what the evidence cannot settle Every reflection with h + k odd is missing. Does that name a glide plane? “Yes — a whole class is gone, so an element removed it”
Structure Factor Calculator account for a reflection being strong, weak or exactly absent by the terms of the sum rather than by a rule A detector records 4,000 counts for one reflection. Which quantity is that, before any correction? “|F|, which is what diffraction measures”
Peak Finding turn a raw scan into a peak list and name two failures the list cannot report A peak search returns eleven peaks from a scan. Are there eleven diffraction lines in the pattern? “Yes — the search found what is in the scan”
Powder Indexing index a powder pattern and say why a cell that accounts for every line may still be wrong A cell is found that accounts for every line in the pattern. Is it the right cell? “Yes — nothing is left unexplained”
Line Broadening separate crystallite size from microstrain, and say what neither of them measures The analysis returns a crystallite size of 24 nm for a powder whose grains are visibly micrometres across. Is something wrong? “Yes — the size must match what the particles measure”
Intensity Corrections trace a recorded count through to |F|² and name the geometry every factor on the page assumes The Lorentz factor on this page is derived for a Bragg–Brentano powder scan. Does it apply to single-crystal data? “Yes — it corrects for diffraction, which is the same physics”
Fourier Synthesis and the Phase Problem state the phase problem precisely and show what a correct set of amplitudes with wrong phases produces A synthesis from six terms shows a maximum where the structure has no atom. Is the structure wrong? “Yes — density appears where the electrons are”
The Patterson Function say what a Patterson map contains, and why it needs no phases A Patterson map has a strong maximum away from the origin. Is there an atom at those coordinates? “Yes — a maximum in a map is where the density is”
Direct Methods and the Sign Relation explain why a sign relation is a property of a triplet rather than of any one reflection For three strong reflections the relation s(h) s(k) s(h+k) = +1 holds. Does that fix their three signs? “Yes — one relation for each reflection”
Charge Flipping describe the charge-flipping loop and say what a good figure of merit is and is not evidence for Charge flipping drives R to 0.000 on your data. Is the structure solved? “Yes — calculated and measured amplitudes agree exactly”
Difference Map read a difference map, and say what a peak in one does and does not establish A difference map shows a maximum close to a heavy atom. Is that an atom the model is missing? “Yes — the map shows what the model does not account for”
CIF Parser find the cell, the symmetry and the refinement table in a CIF and say what each item is A CIF gives a cell edge as 5.6400(3) Å. What is the 3? “How many decimal places can be trusted”
Interatomic Distances and Angles decide whether two reported distances are significantly different Two bonds are reported as 1.943(4) Å and 1.947(4) Å. Are they different? “No — the difference is inside the uncertainty, so they are equal”
Displacement Parameters and NPD Atoms say what a displacement ellipsoid describes and recognise a tensor that describes nothing physical One atom’s ellipsoid is far larger than its neighbours’. Is that atom moving more? “Yes — that is what a displacement parameter measures”
Refinement Statistics and R Factors read a table of crystal data one number at a time, and say what the set still cannot establish about the chemistry A structure refines to R1 = 0.028. Is the chemistry right? “Almost certainly — that is an excellent fit”
X-ray Tube Spectrum say where a tube’s characteristic lines come from, and what decides whether they appear at all A copper tube is run at 8 kV instead of 40. What does the spectrum show? “The same lines, weaker”
Moseley Plot and Absorption Edges name the filter for an anode and say what makes that element and no other the right one Nickel is the filter for a copper tube. Would nickel also filter a molybdenum tube? “Yes — nickel absorbs X-rays whatever produced them”
Absorption Coefficient Calculator decide whether a sample absorbs too much for a given radiation, and name the two things that can be changed about it Your compound gives μ = 12 mm−1 for Cu radiation, past the usual working limit. Does a smaller crystal help? “No — μ is a property of the compound and will not change”
Friedif Calculator choose between two anodes for a particular formula from its own numbers rather than from a rule of thumb The heaviest atom in your compound is bromine. Is Cu Kα still the better radiation for the absolute structure? “Yes — copper is the radiation for absolute structure”
The Ewald Construction say what a rotation, a powder and a white beam each do to the same sphere, and what each one costs A full 360° rotation about one spindle leaves the data 96 % complete. Where is the rest? “Lost at the detector edges and to shadowed frames”
Reduced Cell and Bravais Lattice explain why one lattice has many cells and what a reduction rule is choosing between A lattice point sits at the corner of every cell you draw. What is at that point? “An atom, since that is what repeats”