mathematics

Patterson function

Patterson function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Patterson function rather than just read about it. In short: The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT.

Key takeaways

Patterson function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Patterson function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Patterson function from memory before moving on to harder problems.

Reference excerpt

The Patterson function is used to solve the phase problem in X-ray crystallography. It was introduced in 1935 by Arthur Lindo Patterson while he was a visiting researcher in the laboratory of Bertram Eugene Warren at MIT. The Patterson function is defined as

P ( u , v , w ) = ∑ h , k , ℓ ∈ Z | F h , k , ℓ | 2 e − 2 π i ( h u + k v + ℓ w ) . {\displaystyle P(u,v,w)=\sum _{h,k,\ell \in \mathbb {Z} }\left|F_{h,k,\ell }\right|^{2}\;e^{-2\pi i(hu+kv+\ell w)}.}

It is essentially the Fourier transform of the intensities rather than the structure factors. The Patterson function is also equivalent to the electron density convolved with its inverse:

P ( u → ) = ρ ( r → ) ∗ ρ ( − r → ) . {\displaystyle P\left({\vec {u}}\right)=\rho \left({\vec {r}}\right)*\rho \left(-{\vec {r}}\right).}

Furthermore, a Patterson map of N points will have N(N − 1) peaks, excluding the central (origin) peak and any overlap. The peaks' positions in the Patterson function are the interatomic distance vectors and the peak heights are proportional to the product of the number of electrons in the atoms concerned. Because for each vector between atoms i and j there is an oppositely oriented vector of the same length (between atoms j and i), the Patterson function always has centrosymmetry.

One-dimensional example Consider the series of delta functions given by

f ( x ) = δ ( x ) + 3 δ ( x − 2 ) + δ ( x − 5 ) + 3 δ ( x − 8 ) + 5 δ ( x − 10 ) . {\displaystyle f(x)=\delta (x)+3\delta (x-2)+\delta (x-5)+3\delta (x-8)+5\delta (x-10).\,}

The Patterson function is given by the following series of delta functions and unit step functions

P ( u ) =

5 δ ( u + 10 ) + 18 δ ( u + 8 ) + 9 δ ( u + 6 ) + 6 δ ( u + 5 ) + 6 δ ( u + 3 ) + 18 δ ( u + 2 ) + 45 δ ( u ) + 18 δ ( u − 2 ) + 6 δ ( u − 3 ) + 6 δ ( u − 5 ) + 9 δ ( u − 6 ) + 18 δ ( u − 8 ) + 5 δ ( u − 10 ) . {\displaystyle {\begin{aligned}P(u)={}&5\delta (u+10)+18\delta (u+8)+9\delta (u+6)+6\delta (u+5)+6\delta (u+3)\\&+18\delta (u+2)+45\delta (u)+18\delta (u-2)+6\delta (u-3)+6\delta (u-5)\\&+9\delta (u-6)+18\delta (u-8)+5\delta (u-10).\end{aligned}}}

References

External links "Structural resolution. The Patterson function and the Patterson method".

Worked examples

Example 1 — a first encounter with Patterson function

Start with the simplest possible case. Write down what Patterson function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Patterson function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Patterson function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Patterson function

In research: Patterson function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Patterson function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Patterson function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Crystallography, so understanding it makes those chapters shorter.
In everyday life: Look for Patterson function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Patterson function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Patterson function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Patterson function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Patterson function in simple terms?

Why does Patterson function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Patterson function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Patterson function.