xray crysto n fourier transform

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When a monochomatic X‐ray di ff racts o ff a crystal it performs part of a mathematical operation: the Fourier transform...

Tony PhillipsStony Brook University

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Introduction

X‐ray crystallography has been essential, since the beginning of the 20th century, to our understanding of mater; recently, asknowledge of the chemical composition of proteins has progressed, the determination of their 3‐dimensional structure has becomeindispensable for the correct interpretation of their functions. Our main access to this information is X‐ray crystallography. (X‐rays areused because their wavelengths are on the order of inter‐atomic distances in molecules, in the range 1‐100 Å ; one Å is m).Mathematics enters into the process at two stages. The study of space groups tells what periodic configurations are possible in three‐space; in fact these are often called ʺcrystallographic groupsʺ by mathematicians. The other connection, the subject of this column, is thesurprising and pleasing fact that when a monochomatic X‐ray diff racts off a crystal it performs part of a mathematical operation: theFourier transform (developed in the 19th century in completely diff erent contexts); when the incidence angle is varied, the completetransform is produced. The flaw in this lovely picture is that we cannot measure all the details of the diff racted wave; otherwise theentire molecular structure could be calculated by inverting the Fourier transform. There are many online resources devoted to X‐ray crystallography; I have especially profited from Randy Readʹs Protein

Crystallography Course and from Kevin Cowtanʹs Interactive Structure Factor Tutorial. Thanks also to my Stony Brook colleaguesPeter Stephens and Miguel Garcia‐Diaz for helpful correspondence and remarks.

Fourier series: temperature distribution in a wire

Fourier series and the Fourier transform were invented as a method of data analysis. For example, let us follow Jean‐Baptiste JosephFourier (1768‐1830) in studying the time evolution of the temperature distribution in a circular loop of circumference , given an initialdistribution of temperature ; (we require ). We start by calculating what are now known as the Fouriercoe fficients of :

It can be proved, if is sufficiently well‐ behaved, that the linear combination

(the Fourier series of ; when these formulas simplify pleasantly) converges everywhere to .

re Column from the AMS http://www.ams.org/samplings/feature-column/fc

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(Taken as an initial distribution separately, each of the and functions determines a simple solution, as does aconstant function; the linearity of the heat equation allows these separate solutions to be combined, with coefficients and , to givethe complete solution to the problem.)

It is useful to simplify the formulas by using Eulerʹs identities , and grouping terms

to yield

and

in general, the are complex numbers.

Electron density distribution in a crystal

The dual operations of integration: and summation: can be realized in other contexts. In this columnwe will consider the function that gives the electron density distribution in the crystalline state of some compound. Suppose,for simplicity, that the unit building block, corresponding to a molecule of the compound, is a rectangular solid; say with edge‐lengths

and ; these by by solids are stacked in three‐space so as to give a structure repeating every units in the ‐direction, every units in the and every units in the . Then the function will be triply periodic, with periods and , and consequentlycan be represented as a triple Fourier series

where

directly generalizing our formulas for the circle. If the coefficients are known, the electron density distribution can be calculated,and then the structure of the molecule can be determined. It is therefore remarkable that the diff raction paterns formed when the

crystal is bombarded with X‐rays contain precious information about the . Roughly speaking, we can imagine the complexnumbers placed at the vertices of a 3‐dimensional latice; each X‐ray diff raction patern projects this latice onto the planeof the image plate. If we label by the projection produced by a beam meeting the crystal at a generalized angle , a vertex which is good position with respect to (this condition also depends on the wavelength of the radiation) will appear on the plate asa spot at location and of intensity proportional to the square of the absolute value . Varying will bring a new set ofvertices into good position; eventually the latice can be reconstructed, along with the absolute value of the coefficients at the vertices.

The phase problem. The absolute values alone, although they contain a great deal of information about the molecule inquestion, do not allow to be completely reconstructed. A simple example comes from temperature distributions likethose studied by Fourier. Consider (with circumference ) the temperature distributions

and

. So the coefficents for are all , while those for are

. The absolute values are the same, but the distributions are diff erent, as shown below. X‐ray

crystallographers have devised many ways to get around this limitation, called ʺthe phase problem;ʺ they are beyond the scope ofthis column.


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The temperature distributions (red) and (blue), on a circular loop ofcircumference (ploted radially) are quite diff erent but have Fourier coefficients with the same absolute values.

The reciprocal latice

X‐rays interact with a crystal through interaction with parallel families of planes. Suppose as before that the unit cell in the crystal is an rectangular parallelipiped (when and are all diff erent, this structure is called orthorhombic). Every triple of

integers defines a family of planes through the crystal, defined by the equation

Let us change coordinates to , , . (We are now in the reciprocal latice; for a non‐rectangular crystal the change of

coordinates is only slightly more complicated). Then the equation defining the planes becomes

for an integer. For each this family of parallel planes fills up the crystal, in the sense that each unit cell vertex lies in (exactly)one of them, as is easy to check. We call these the latice planes. Graphically, it is easier to represent the analogous concept in two

dimensions, so we suppress and for the moment. Our crystal is then an array of rectangles; in the coordinates thesemeasure . The pair gives the lines , for an integer. These lines are parallel to the ‐axis, and slice through the base of each unit cell. On the other hand, gives the lines , or , gives , and

gives . (See image).

Latice lines: the lines corresponding to are shown in blue, those for in orange, and those for in purple. Not alllines are shown.

X‐ray diffraction: how a monochromatic plane wave performs Fourier analysis on the electron density


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distribution.

This diagram labels the vectors involved in the calculation of diff raction. An incoming plane wave diff racts off an electron at as awave with the opposite incidence angle , and also off another electron at relative position . The wave vectors and have thesame magnitude, which we set for convenience to be , the reciprocal of the wavelength. The diff racted rays travel diff erent lengths:

the path through is longer by than the path through , as shown. is the projection of along and has length. is minus the projection of along , and so has length . This gives, for the diff erence in path

lengths, or . Since is the wavelength, the ray traveling through will have its phase retarded by ;corresponding to a factor . The length of the vector is ; so , which is equal to the component of in the direction of times thelength of , is equal to ; constructive interference, which corresponds to , an integer, thus also corresponds to

.

This

is

Braggʹs law;

its

discovery

in

1912

started

the fi

eld

of

X‐

ray

crystallography.

The

diff

raction

corresponding

to

adiff raction vector and a single electron at position multiplies the amplitude of the scatered wave by a phase factor . If is the electron density function in the crystal, the eff ect on will sum to

So the structure factor appears as the Fourier transform of the electron density function . (Straightforward diff ractionexperiments only measure the (square of) the absolute value of , which shows up as the intensity of the spot corresponding to .If the phases were also known, the Fourier transform could be inverted to give an exact picture of the electron density ). It is possible to rewrite this integral in terms of the vectors in reciprocal space. For each such vector we write, using thereciprocal coordinates

In this case the inverse Fourier transform

where is volume in reciprocal space, can be approximated by a Fourier series

which can be compared with the Fourier series for given at the start of this column.

Here

is

a

nice

example,

from

Kevin

Cowtanʹ

s

Interactive

Structure

Factor

Tutorial.

The

example

is

2‐

dimensional,

and

shows

howrapidly the structure factors, with their phases , converge to the target structure. The target electron density function looks like this:


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Cowtanʹs simulation leads to the approximate Fourier synthesis of the target from just the seven largest structure factors: thosecorresponding to . Here is how the synthesis proceeds, step by step,each time adding in the next structure factor. These images are from his tutorial , and are used with permission. The unit cell (notorthorhombic!) is outlined in dots.


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Comments (2)

Dear Tony:

Thank you for a nice article. I have meant to sit down

and understand the relationship between wavefronts

and Fourier transforms. It is important in radar, thefield I work in, and in particular, in antenna theory.

All the best. -- Greg Coxson

#1 - Greg Coxson - 10/10/2011 - 13:17

Response to Greg

Dear Greg, Thanks for the kind words. It's great to

hear from a reader. Best wishes, Tony

#2 - Tony Phillips - 10/18/2011 - 15:58

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Tony PhillipsStony Brook University

Email Tony Phillips


xray crysto n fourier transform

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