SPHERE OF REFLECTION
Lecture 4
092809
Consider a basketball of radius1/λ (for MoKα,1/λ = 1.407Ǻ-1, for CuKα,1/λ = 0.6486Ǻ-1).
The reciprocal lattice is rotated around the origin of this basketball, and when a reciprocal
lattice (r.l.) point crosses through the edge of the sphere of reflection, then, and ONLY
then, diffraction occurs.
Limiting sphere: take the sphere of reflection and rotate it around the origin. Therefore,
this gives the radius of the limiting sphere = 2/λ.
Bragg’s law in reciprocal space:
Imagine a crystal in a beam of x-rays, with wavelength λ.
Consider a*c* plane with the x-ray beam parallel to this plane.
C = center of circle of radius 1/ λ
P = r.l. point
O = r.l. origin
Angle OPQ is inscribed in the circle and therefore is a 90o angle.
sin OQP = sin θ = OP/OQ = OP/(2/λ)
sin θ = OP(λ/2)
But, since P is a r.l. point, length OP, by definition, = 1/dhkl
Therefore, sin θ = (1/dhkl )(λ/2)
OR
λ = 2dhkl sin θhkl
Bragg’s law is satisfied when point P contacts the edge of the sphere.
----------------------------------------------------------Reciprocal Space (revisited):
 Between the points of a crystal lattice in real space, we have Bragg planes.
 Each set of Bragg planes corresponds to one reflection.
 Each set of Bragg planes corresponds to one set of Miller indices.
 Each reflection is identified by the corresponding Miller indices (h, k, l).
 The reflections form another lattice, the reciprocal lattice.
 The vector d is perpendicular to a set of Bragg planes.
 Its length is equivalent to the distance between two Bragg planes.
 Each reflection (h, k, l) marks the endpoint of the vector d* = 1/d = s.
 The length of s in inversely related to the distance between the Bragg planes.
 We can calculate the length of s = d* from the following general equation:
d*hkl = 1/d hkl = [h2a*2+ k2b*2+ l2c*2+2hka*b*cos*+2hla*c*cos*+2klb*c*cos*]1/2
2
The reflection h, k, l is generated by diffraction of the X-ray beam at the Bragg plane
(h, k, l) which intersects the three edges of the unit cell at 1/h, 1/k, and 1/l. Sets of planes
in real space (with spacing d) correspond to points in reciprocal space (at a distance d*
from the origin). The vector d* is perpendicular to the Bragg planes and has the length
|d*| = 2sinθ/λ.
A reflection is visible when the corresponding set of Bragg planes is in reflex position,
i.e., when Bragg’s law is fulfilled. In an alternative description: a reflection is visible
when the corresponding scattering vector s = d* intersects with the Ewald sphere.
The reflections form a lattice in reciprocal space. Reciprocal unit cell dimensions are:
a*, b*, c*, etc. The dimensions and angles of the reciprocal cell are inversely
proportional to the real space cell: if a unit cell were to double in length (because we
know it has to), then the space between the X-ray reflections has to be reduced by a factor
of two.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - Number of possible reflections:
On the basis of the limiting sphere, one can calculate the total # of possible reflections:
Ntot = (4/3)π r3/V* = (4/3)π (2/λ)3/V* = 33.51/V*λ3 = 33.51 V/λ3
Ex.: for an orthorhombic cell 10 x 10 x 10 Ǻ, the Vol = 1000 Ǻ3.
Ntot = 33.51 V/λ3 = 33.51(1000 Ǻ3)/(1.5418 Ǻ)3 = 9143 reflections for CuKα
Ntot = 33.51 V/λ3 = 33.51(1000 Ǻ3)/(0.71073 Ǻ)3 = 93338 reflections for MoKα
(These HAVE to be whole numbers!!!!!)
Question: Are all of these possible reflections unique?
Answer: No. First, Friedel’s law states that Ihkl = I-h-k-l
Second, in the orthorhombic system, Laue symmetry affords three
mutually perpendicular mirror planes which make many of the data
equivalent to other data. Therefore, only 1/8 of the possible data will be
unique.
In the triclinic case, only ½ of the total possible data are unique, due to Friedels’ law .
Therefore, for CuKα, only 4571 reflections would be unique; for MoKα, 46669
reflections would be unique.
In the monoclinic case, besides Friedel’s law, the Laue symmetry (2/m) contains one
mirror. Therefore, only 1/4 of the total possible data are unique. Therefore, for CuKα,
only 2285 reflections would be unique; for MoKα, 23334 reflections would be unique.
3
In the orthorhombic case, besides Friedel’s law, the Laue symmetry (mmm) contains
three mutually perpendicular mirrors. Therefore, only 1/8 of the total possible data are
unique. Therefore, for CuKα, only 1143 reflections would be unique; for MoKα, 11667
reflections would be unique.
Intensities of the Reflections:
With the help of Bragg’s law and the Ewald construction, we can calculate the place of a
reflection on the detector (film or CCD, as examples), provided we know the unit cell
dimensions. Indeed, the position of a spot is determined alone by the metric symmetry of
the unit cell. The intensity of a spot, however, depends on the contents of the unit cell
(and, of course, on the exposure time, crystal size, etc.).
The intensity of a reflection depends, among other things, on the population of the
corresponding set of Bragg planes with atoms (if the plane is empty, or has only a few
weakly-diffracting atoms [i.e., H atoms], then the reflection is very weak or absent. If
there are many atoms on or in a plane, the reflection correspondingly will be strong).
Other factors are the thermal motion of the atoms and the atomic radius.
Atomic Form Factors:
X-rays are scattered by the electrons of an atom; therefore, approximating the atom to be
a point does not hold too well. Ergo: Bragg planes are not mathematical planes, but have
a “fuzzy” thickness, which means that there is a path difference occurring within each
individual Bragg plane.
Displacement of the place of diffraction from the ideal position of δ leads to a path
difference of δ·2π/d. This has a weakening effect on the intensity of the corresponding
reflection. This effect is a function of the distance d between the Bragg planes: it
becomes stronger with smaller distances (i.e., at higher resolution = higher diffraction
angle θ). See Massa, Fig. 5.1, p.34.
In a 1st approximation, the atomic scattering factor is dependent on the atom number
(number of electrons). With increasing resolution (i.e., decreasing Bragg plane distance,
which is equivalent to higher scattering angles, 2θ), the scattering factor becomes smaller.
See Massa, Fig. 5.2, p.34.
4
Atoms not only have a radius, they also move, and they also are held in place by covalent
bonds or other forces. With the displacement of an atom from its ideal position on the
Bragg plane, an additional path difference is added. The same principle as with the
atomic form factors applies (more weakening at high scattering angles), but this effect is
stronger when the thermal motion is higher, i.e., higher temperature. See Massa, Fig. 5.3,
p.35.
f’ = f · exp{-2π2μ2/d2}
where μ = rms amplitude of vibration
d = Bragg plane spacing
f’ = atomic scattering factor
f = atomic form factor
Since d = 1/d* = 1/(2sinθ/λ), and if we define U = μ2, then:
f’ = f · exp{-2π2Ud*2} = f · exp{-8π2U sin2θ/λ2}
If we now substitute B = 8π2U,
then f’ = f · exp{-B sin2θ/λ2} (see Massa, eq. 5.4, p.36)
where U = temperature factor => typical values are 0.05 – 0.07 Ǻ2
B = Debye-Waller temperature factor => typical values are 2 – 5 Ǻ2
These U and B values describe the atom as an isotropic sphere (basketball). It is not
surprising to learn that most real atoms do not conform to isotropic motion, but truly
should be described by anisotropic motion (football shape). This is much more
complicated and requires the use of tensors. Three mutually perpendicular vectors are
used to define the magnitude and orientation with respect to the cell. Anisotropic
vibrations can be represented by vibration ellipsoids with three principal axes U1, U2, and
U3. However, six tensors are required to give the complete anisotropic ellipsoidal shape:
U11, U22, U33, U12, U13 and U23. (See Massa, eq. 5.5, p. 36, and plot Fig. 5.4, p.37).
--------------------------------------------------------------
Wilson plot: A. J. C. Wilson found the following relationship, which allows one to
estimate the overall B and the scale factor, k, for a set of data:
Ihkl (corr.) = (Ihkl)(Ahkl)/Lp
where Ihkl = the actual intensity of a reflection & Ahkl = the absorption correction
L = Lorentz correction = 1/ sin2θ
p = polarization correction = (cos22θxtal)( cos22θmono) + 1/ 1 + ( cos22θmono)
5
The Lorentz factor (L) arises because different reflections are in positions to reflect for
different lengths of time (i.e., how fast does a r.l. point pass through the edge of the
sphere of reflection). This depends on where the reflection is located when it passes
through the edge of the sphere of reflection.
The polarization correction (p): The X-rays, as generated, are unpolarized. The rays can
be thought of as being composed of 2 components (vertical and horizontal). How does
this affect the intensities (Ihkl) of the reflections? We need to look at the interaction of
these unpolarized X-rays, first with the monochromator crystal, and then with the crystal
that we are measuring. We will take the physicists’ word that we are doing this
correction properly.
–
N
For a unit cell of N atoms, I = Σ f 2 exp(-2Bsin2θ /λ2)
abs
–
where I
i=1 i
= the average intensity of small concentric shells of data.
abs
This value depends on what is in the cell and not where it is.
Therefore, for various ranges of sinθ/λ in concentric shells:
–
–
N
I(corr.) = k I = k exp(-2Bsin2θ /λ2) Σ f 2
abs
(see Massa, eq. 8.9, p.101)
i=1 i
which is one equation with 2 unknowns = k and B.
–
–
Rearranging:
I(corr.) = k I = k exp(-2Bsin2θ /λ2)
Σ f2
–
or: ln I(corr.) = ln k - 2B (sin2θ/λ2)
Σ f2
–
Therefore, a plot of ln I(corr.) vs. sin2θ/λ2 yields the intercept = k and the slope = -2B
Σ f2
The REAL scale factor = 1/k1/2
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