Download - 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
-
8/14/2019 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)
-
8/14/2019 Part 6. Diffraction by Crystals
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.
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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]
-
8/14/2019 Part 6. Diffraction by Crystals
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).
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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)
-
8/14/2019 Part 6. Diffraction by Crystals
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-
-
8/14/2019 Part 6. Diffraction by Crystals
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.
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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
-
8/14/2019 Part 6. Diffraction by Crystals
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