part 6. diffraction by crystals

8/14/2019 Part 6. Diffraction by Crystals

1/16

Scattering from Two Electrons

so

s

r

Vectors representing the incoming andoutgoing x-ray waves are called so and s,by convention they have length = 1/ .Their difference, s-so, is called S.

e1

e2

The two electrons areseparated by thevector r.

The path difference forx-rays scattered by e2 vse1 = - .r.S =- (hx+ky+lz)

This corresponds to aphase difference of

2r

.S

= 2 (hx+ky+lz)Key Result!

Note that this result is inradians, in degrees the phasedifference is360r.S =360(hx+ky+lz)

Part 6. Diffraction by Crystals


2/16

Phase of a scattered x-ray, wrt origin, = 2 r.SThe result on the previous slide can be generalized. The phase of ascattered wave, with respect to the origin, is 2 r.S. Where r is theposition vector of the scattering electron and S is difference betweenoutgoing and incoming waves.

Thus, the structure shown below on the left, gives the scatteringvectors (and resultant) shown on the right. Note that the individualscattering vectors all have length = 1e.

Unit Cell

e1

e2

e3r1 r2

r3

Scattering Vectors

2 r3.S

2 r1.S

2 r2.S

Resultant

F(hkl)


3/16

Scattering from an Atom (Form Factor)It is generally more useful to think in terms of atomsrather than individual electrons. The atomic scatteringvector, defined with respect to the atomic nucleus, isknown as the form factor. It is the sum of scattering from

every bit of electron density associated with the atom.

This is a number that varies with the magnitude

of the scattering angle, but not with itsdirection. In other words, it depends upon themagnitude of S but not its orientation

Form factor for Carbon

(S) = 2 (r)cos(2 r.S)dr

S = 1/d =

Pathdifference

When |S|=0, =0, f= #e in atom.

Form factors for a varietyof atoms and ions.

Units of electrons

Reduction in scatteringwith increasing S (=higherres. and ), due to thesize of the electroncloud. X-rays scatteredfrom opposite sides ofthe atoms interferedestructively to some

extent.


4/16

Temperature (B) factorThe rate at which the form factordiminishes with increasing scattering angledepends upon the size of the atom The

effect is magnified if the atom is vibratingand thereby occupies a larger volume. Thevibration of atoms is usually modeled by anexponential term known as the B factor.

B factors are usually refined for eachindividual atom in a structure. Usually,the B factor is modeled as an isotropicterm that causes an exponential fall off inscattering with increasing angle.

B = 8 2 x u2 u = mean displacement

B = 80 2; u ~1 B = 20 2; u ~ 0.5


5/16

Scattering by a Unit Cell (Structure Factor)

The resultant scattering from a unit cell isequal to the sum of scattering vectors fromall of the atoms in the unit cell.

The scattering vector from the entire contents of a unit cell isknown as the Structure Factor, and is denoted F(S).

The Structure Factor Equation: F(S) = fj(S)exp[2 irj.S]

Consider a unit cell that is comprised ofthree atoms at positions that are givenby the vectors r1, r2, and r3.

0

1 2

3r1 r

2r3

f1

f2

f3F(S)

The length of the scattering vectorfrom each atom is equal to the formfactor for an atom of that element, Bfactor, and scattering angle (S). The

phase is given by the position of theatom in the unit cell and S.

Atomic scattering factor, wrt 0= fj = fjexp[2 irj.S]


6/16

Scattering from a Crystal: Laue Conditions

Scattering from a single unit cell is a continuous function (like the 6holes in the optical grid earlier). The reason that crystals only give

diffraction in discrete directions is because scattering from each ofthe individual unit cells must be in phase with scattering from all ofthe other unit cells. This can only occur in directions that correspondto the reciprocal lattice. We will not prove it here, but thisrequirement for diffraction is embodied within the Laue conditions:

a.S = h, b.S = k, and c.S = l; where h, k, l are integers.

The Laue conditions are equivalent to Braggs law.

The integers h, k, and l are the Miller indicies.This is why we can write the structure factor as a function of S, as inF(S), or as a function of hkl, as in F(hkl).


7/16

Braggs law is equivalent to areciprocal lattice point lying onthe surface of Ewalds sphere.sin = (|S|/2) / (1/ ) = |S|/2

Remember that |S| = 1/d sin = /2d; = 2d sin

Ewald constructions are used to show when reciprocal lattice points will bein diffraction condition for a given orientation of the crystal. A sphere (Ewaldsphere aka sphere of reflection) is drawn with radius 1/ . The crystal isimagined to be at C (the center of the Ewald sphere). The origin of thereciprocal lattice is at O (the point at which the undeflected x-ray beam wouldleave the sphere). Reciprocal lattice points diffract when they cross Ewaldssphere. When this happens, a diffracted x-ray propagates in the direction s untilit hits the detector.

X-ray beam O

Reciprocal lattice origin

Diffracted x-ray

Reciprocal lattice

point in diffraction

condition

Crystal at center ofsphere of radius = 1/

Miller planes

1/d


8/16

Diffraction when rl points touch surface of Ewald sphere

O Beam trap stopsdirect beam andh=0, k=0, l=0reflection

Reciprocal latticepoints that touchthe surface of

Ewalds spheregive rise todiffracted x-raybeams.

Diffracted raysthat interceptthe detector willbe recorded

Max resolution

possible = /2


9/16

Data Collection is Limited by Resolution of Diffraction

Data can only be collected to the resolution towhich the crystal diffracts. This can berepresented as a sphere, centered about thereciprocal lattice origin, that has radius =1/dmin . Remember, the reciprocal lattice, andthe Ewald construction, have dimensions of -1 .

S = h.a* + k.b* + l.c* = 1/d (-1 )

Rotating the crystal causes the reciprocal lattice to rotate. This is equivalent torotating Ewalds sphere in the opposite direction -- which is much easier to draw!

Rotation axis verticalView down/alonthe rotation axis

Reflections come into diffraction condition by rotating the crystal


10/16

The Laue Method

Ewald sphere for theshortest wavelengthx-rays.

Ewald sphere for thelongest wavelength x-rays.

Reciprocal lattice points intercepting an Ewaldsphere from the continuous spectrum.

Rather than rotating the crystal, the Laue method movesEwalds sphere by changing the wavelengths by using abroad spectrum of white synchrotron x-rays ( = 0.5 to2.5 ). Favorable cases allow an essentially completedataset to be collected from one or a few images. TheLaue method is not usually used because the data obtainedare of lower quality than normally achieved usingmonochromatic radiation. However, the Laue method isvaluable for time-resolved crystallography, since, for

favorable cases, essentially complete data can be collectedin milliseconds.

Factors that restrict the useof Laue crystallographyinclude:

(1) The crystal must be of

very high quality.(2) Intermediateconformational states mustbecome highly populatedduring the reaction without destroying thecrystal quality.

(3) The reaction must becoordinated throughout the

crystal every moleculemust be doing the samething at the same time.This can sometime beachieved by starting areaction with a pulse oflaser light.

h l l b h


11/16

Friedels Law This explains our earlier observation thatthe diffraction pattern is centrosymmetric.

(S) = fj(S)exp[2 irj.S] = F(hkl) = fj(hkl) exp[2 i(hx + ky + lzThe equivalence of these

expressions follows because:

r = a.x + b.y + c.z

r.S = a.S.x + b.S.y + c.S.z

remember Laue conditions:

a.S=h; b.S=k; c.S=l

r.S = hx + ky + lz

Start from the structure factor equation:

+45

-45

F(hkl)

F(hkl)

|F(hkl) | = |F(hkl) |

(hkl) = - (hkl)

Consider f(hkl) and f(hkl) for one atom:

The lengths of these vectors isidentical (form factor varies with

resolution but not with direction).The phase of these vectors is: (hkl) = 2 (hx + ky + lz) (hkl) = 2 (-hx + -ky + -lz) = - (hkl)


12/16

Anomalous ScatteringFriedels law breaks down in the case ofanomalous scattering. Anomalousscattering occurs when the form factor(atomic scattering vector) has animaginary component (i.e. a phasechange). This occurs when the energy ofthe x-ray photon is similar to that of anabsorption edge (electronic transitionwithin the atom).

The form factor of ananomalously scattering atomis a function of wavelength:

f( ) = fo + f( ) +i f( )

f( )

fo

f( )

f()

F(hkl)

F(hkl)

Friedels law isequivalent to sayingthat the complexconjugate of F(hkl)= F(hkl). This nolonger holds in thecase of anomalousscattering.

F(hkl)

F(hkl)

F(hkl) = F+ F(hkl) = F-


13/16

Anomalous Scattering

In many cases where the protein is expressed recombinantly, it is possible tointroduce specific anomalously scattering atoms by incorporatingselenomethionine in place of methionine. Data are then collected at severaldifferent wavelengths at a synchrotron and the structure determined by theMAD (multi-wavelength anomalous diffraction) method. This is now oftenthe preferred method for determining new protein crystal structures.

Anomalous scattering is not significant for most protein crystals, becauseelements like H, C, O, N, and S do not have absorption edges that are close to

the energy of x-rays used for data collection. However, many proteins containmetal ions that do have significant anomalous scattering (e.g. Fe). Moregenerally, many of the heavy atoms (e.g. Hg, Pt, U) that are used to determineprotein crystal phases by the MIR method (see later) have quite substantialanomalous scattering effects. For these crystals the F(hkl) and F(-h-k-l)reflections are treated separately, and the differences between them can be

used to help determine phases for the protein crystal structure.


14/16

Electron Density EquationThe electron density equation relates the diffracted x-ray waves (vectors)to the structure. It can be derived from the Structure Factor equation:

F(hkl) = fjexp[2 i(hx+ky+lz)]

F(hkl) = (xyz) exp[2 i(hx+ky+lz)]This can be recast explicitly in terms of electron density:

This is a Fourier transform. An important property of FTs is that if A is thetransform of B, then B is the transform of A. Therefore...

(xyz) = (1/V) F(hkl) exp[-2 i(hx+ky+lz)]

(xyz) = (1/V) |F(hkl) |exp(i )exp[-2 i(hx+ky+lz)]

This can be simplified to an expression in cosine terms:

(xyz) = (1/V) |F(hkl) |cos[2 (hx+ky+lz)- (hkl) ]

V = volume of unit cell. F = |F|exp(i )

l h El E


15/16

Visualizing the Electron Density Equation

(x) = (1/a) |F(h) |cos[2 (hx)- (h) ]Consider the electron density equation in one dimension:

Each of the terms in this summation corresponds to an electrondensity wave that has amplitude = |F|, makes h oscillations perunit cell, and is shifted from the origin by .

h=1, =0

h=2, =180

h=3, =90, small |F|

h=4, =120, big |F|

El D W d h


16/16

Electron Density Waves are Summed to give the Structure

Diffracted x-ray waves correspond to electron density waves. Any

periodic function can be reconstructed exactly by an infinite seriesof sinusoidal electron density waves.

(x) = (1/a) |F(h) |cos[2 (hx)- (h) ]

X-ray wavesIndividualed waves

Summeded waves

h=1

h=2

h=3

h=4

Structure: 2atoms perunit cell

part 6. diffraction by crystals

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