2. Diffraction as a means to determine crystal structureleung.uwaterloo.ca/CHEM/750/Lectures 2007/SSNT-3-Surface-2.pdfDiffraction as a means to determine crystal ... X-ray: E (eV ...

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2. Diffraction as a means to determine crystal structure

Recall de Broglie matter waves:

He atoms: [E (eV)]1/2 = 0.14 / λ (Å) ∴ E1Å = 0.0196 eV

Neutrons: [E (eV)]1/2 = 0.28 / λ (Å) ∴ E1Å = 0.0784 eV

Electrons: [E (eV)]1/2 = 12 / λ (Å) ∴ E1Å = 144 eV

X-ray: E (eV) = 12400 / λ (Å) ∴ E1Å = 12400 eV

• X-ray diffraction techniques:

)(

1)2

(

)(2

2

photonshchE

mhE

wavematterhpwherem

pE

λυ

λ

λ

==

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Page 1 of 22SURFACE STRUCTURE

10-01-21http://192.168.1.20/CHEM/750/Lectures%202007/SSNT-3-Surface%20Structure%20II.htm

1. Laue method: fixed crystal and vary λ from 0.2 to 2 Å.

2. Rotating crystal method: use fixed λ and rotate crystal.

3. Powder method.

• Bragg law

Path-Length Difference = 2d sin θ = n λ

where θ is the angle measured from the surface plane and d is the inter-plane spacing.

Plane waves:

e i k ⋅ r e i k’ ⋅ r

d

θ



where the magnitudes of the incident and outgoing wave vectors are equal in elastic scattering because k2 = k’2, i.e. k = k’ (= 2π/λ ). Also,

k’ = k + K

where K is the momentum-transfer vector.

• Reciprocal lattice vectors

By definition, a crystal is invariant under translation of the form: T = ua + vb + wc, i.e. any physical property of the crystal is invariant under T.

If n(r) is the electron density, then n(r+T) = n(r). If we write the Fourier representation of n(r) as:

n(r) = ∑G nG e i G ⋅ r

n(r+T) = ∑G nG e i G ⋅ r e i G ⋅ T

then e i G ⋅ T = 1 = e i 2π (integer) [*]

G is called the reciprocal lattice vector and is written



in terms of the axis vectors of the reciprocal lattice: A, B, and C.

G = hA + kB + lC

where A = 2π (b × c)/ [a⋅b×c]

B = 2π (c × a)/ [a⋅b×c]

C = 2π (a × b)/ [a⋅b×c]

and h, k, l = integers.

Note: a⋅b×c is the volume of the unit cell. We can show that eqn. [*] is satisfied from the given definitions. We also get:

A⋅a = 2π B⋅a = 0 C⋅a = 0

A⋅b = 0 B⋅b = 2π C⋅b = 0

A⋅c = 0 B⋅c = 0 C⋅c = 2π

Note: The reciprocal lattices have the dimension of [length]-1 or reciprocal length.

In the case when a, b and c are orthogonal (cubic unit cell), then

|A| = 2π/|a|, |B| = 2π/|b| and |C| = 2π/|c|



and A, B, C are pointing in the same directions as the corresponding spatial vectors. But this is in general not the case.



Theorem of diffraction: The set of reciprocal lattice vectors G determines the position of x-ray reflections, i.e. K = G. Since elastic scattering requires: (k + G)2 = k2, the diffraction condition becomes:

2k ⋅ G + G2 = 0

• Brillouin zones



A Brillouin zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice. The first Brillouin zone is the smallest volume entirely enclosed by planes that are the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin. The concept of Brillouin zone is particularly important in the consideration of the electronic structure of solids. More later.









For now, the Brillouin zone gives a convenient geometrical interpretation of the diffraction condition: k ⋅ ½G = (½G)2. Any vector k from the origin to the plane which forms part of the zone will satisfy the diffraction condition. See Ewald construction later.



3. TLK model of a surface

The simplest description for a surface is the so-called “ideal” surface, which is formed by comminution of the bulk crystal along the desired crystal plane



without allowing any relaxation. The atoms in this “ideal” surface therefore have exactly the same positions before and after the termination of the “bulk” to form the surface.

• Ideal surfaces (defined above) can be described in terms of the terrace-ledge-kink or TLK model.

A terrace is a “flat” area with the same crystal orientation;

a ledge is a “narrow” step separating the terraces (this provides the “sloping ground” of the surface); and

a kink is an irregular “twist” in the ledge.





• In the TLK model, all surfaces fall into one of three categories.

Singular surfaces correspond to surface orientations lying along a low index plane of the bulk lattice (e.g. (100), (111) and (110)). These surfaces are essentially “smooth” on an atomic scale.

Vincinal surfaces correspond to orientations finitely removed from the singular orientations. These surfaces are made up of flat terraces of the closest singular orientation, separated by ledges of monatomic height, spaced appropriately to give the surface orientation relative to the singular orientation (e.g. a “stepped” surface misaligned in one direction). In the case of “stepped” surfaces



misaligned in two directions, there will also be regularly spaced kinks, one atom position deep, in order to account for the misorientation in the second direction.

Rough surfaces are disordered on the atomic scale and cannot be described by the TLK model.

Note: The TLK model does not allow for any relaxation effect (i.e. surface is ideal) nor thermally induced defects (i.e. structures basically at 0 Kelvin).



4. Surface crystallography

A surface by definition is two-dimensional and therefore should have no structure! By surface structure, we generally mean the region of the solid in the vicinity of the surface (i.e. the selvedge), or the region with only two dimensional periodicity and no periodicity in the direction normal to the surface. (The “bulk” has 3D periodicity). We will restrict our discussion only to unreconstructed “clean” surface (i.e. no adsorbate).

Note: We find in many cases that once the surface becomes free of adsorbates, the surface reconstructs (i.e. rearrange) in order to achieve a lower overall energy (e.g. all Si, Ge, GaAs, Au, Pt, etc.). This reconstruction of course changes the original structure. In the case of surface adsorption, the adsorbates may form their own surface structure (referred to here as adsorbate structure) as well as introducing changes or “reconstructions” into the structure of the “clean” surface.

• Symmetry operations (7)



Identity (or 1-fold rotation);

Rotations: 2-fold, 3-fold, 4-fold and 6-fold;

Mirror reflection in a plane perpendicular to the surface; and

Glide reflection (reflection in a line combined with translation along the direction of the line by half of the periodicity in this direction).



• Two-dimensional point groups (10)

• Bravais nets (5)



• Two-dimensional space groups (17) − This means that there are only 17 symmetrically different types of surface structure possible, although there are an infinite number of possible



surface structures.



The 2D space groups are generated by combining the 2D point groups with the Bravais nets. Note the standard international notations.

It is important to note that the surface (2D) unit net is not necessary a simple projection of a (3D) unit cell (of a lattice).



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