solid state physics - uclucapahh/teaching/3c25/lecture05s.pdf · crystal structure thelecture...
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Crystal Structurethelecture thenextlecture
SOLID STATE PHYSICSLecture 5
A.H. HarkerPhysics and Astronomy
UCL
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Structure & Diffraction
Crystal Diffraction (continued)
2.4 Experimental Methods
Notes:
• examples showphotographic film, for x-rays.
• Can also use electronic detection for x-rays.
• Need counters (e.g.BF3) for neutrons.
• Information:
– Positions of lines (geometry)– Intensities of lines (electronics, or photogrammetry to measure
darkness of lines on films)
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2.4.1 Laue Method
1912: Max von Laue (assisted by Paul Knipping and Walter Friedrich).CuSO4 and ZnS.Broad x-ray spectrum – single crystal
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Forward scattering Laue image of hexagonal crystal.
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Shows crystal symmetry – when crystal appropriately oriented.Use for aligning crystal for other methods.Range ofλ, so cannot determinea from photographic image, but ifoutgoing wavelengths can be measured,canuse to find lattice param-eters.
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2.4.2 Rotating Crystal Method
Single x-ray wavelength – single crystal rotated in beam.
Either full 360◦ rotation (as above) or small (5 to15◦) oscillations.
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2.4.3 Powder Methods
Single x-ray wavelength – finely powdered sample.Effect similar to rotating crystal, but rotated about all possible axes.
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X-ray powder diffraction pattern of NaClO3 taken with CuKα radi-ation.
X-ray powder diffraction pattern of SiO2 taken with CuKα radia-tion.Powder diffraction patterns are often used for identifying materials.
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2.5 Mathematics of Diffraction
2.5.1 Monatomic Structure
Incoming plane wave
ψi = A exp[i(ki.r− ωt)]
Scattered by the atom in unit cellI at rI .Assume scattered amplitude isS A – all the unit cells are the same,so independent ofI.When incident wave hits atom, it is
A exp[i(ki.rI − ωt)].
It is scattered with a different wave-vector,kf , so from the atom to apoint r its phase changes bykf .(r− rI).The scattered wave is thus
S A exp[i(ki.rI − ωt)] exp[ikf .(r− rI)]
orS A exp[i(kf .r− ωt)] exp[i(ki − kf ).rI ].
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So if a plane wave with wavevectorkf is scattered from the crystal,it is the sum of the waves scattered by all the atoms, or
Total Wave = S A exp[i(kf .r− ωt)]∑I
exp[i(ki − kf ).rI ].
Write ∆k = kf − ki:
Total Wave = S A exp[i(kf .r− ωt)]∑I
exp[−i∆k.rI ],
and as the amplitude of the outgoing waveexp[i(kf .r− ωt)] is 1,
Total Amplitude ∝ S∑I
exp[−i∆k.rI ]. (1)
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2.5.2 The Reciprocal Lattice
Define a new set of vectors(A,B,C) with which to define ∆k.Require
a.A = 2π , a.B = 0 , a.C = 0b.A = 0 , b.B = 2π , b.C = 0c.A = 0 , c.B = 0 , c.C = 2π
(2)
In general,
A =2πb× c
a.b× c
B =2πc× a
a.b× c
C =2πa× b
a.b× c(3)
The vectors(A,B,C) define thereciprocal lattice.For simple cubic system, reciprocal lattice vectors are just2π/a alongthe x, y and z axes.
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Lattice Reciprocal LatticeSimple cubic Simple cubic
FCC BCCBCC FCC
Hexagonal Hexagonal
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2.5.3 The Scattered Amplitude
Let∆k = hA + kB + lC,
and remember that our structure is periodic:
rI = n1a + n2b + n3c.
Immediately we have
∆k.rI = 2π(hn1 + kn2 + ln3).
So∑I
exp[−i∆k.rI ] =∑n1
∑n2
∑n3
exp[−2πi(hn1 + kn2 + ln3)]
=
∑n1
e−2πihn1
∑
n2
e−2πikn2
∑
n3
e−2πiln3
.
Sums, in principle, go over−∞ < ni < ∞, or at least over a verylarge range1 ≤ ni ≤ Ni.Phases lead to cancellation unlessh, k and l are integers, when eachterm is 1 and total amplitude isSN1N2N3.
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So we see
• we have a strong reflection when∆k is a reciprocal lattice vector;
• remembering that ∆k is perpendicular to the reflecting plane, an(hkl) reflection has∆k = hA + kB + lC.
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2.6 The Laue Construction
This is a diagram in the reciprocal lattice.Just as the lattice is an abstract mathematical object, so is the recip-rocal lattice.Neither ki nor kf need to be reciprocal lattice vectors, butkf −ki is.
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Note that only certain special incident directions ofki will give adiffracted signal.
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2.7 Non-Monatomic Structures
2.7.1 Simple Treatment
Example: an FCC structure (thought of as simple cubic with a basisof two atoms, one at(0, 0, 0), three more at(1
2,12, 0), (1
2, 0,12), 0, 1
2,12).
For simple cubic, there is a strong reflection from(110) planes:
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but face-centred cubic has extra atoms in the orginal planes and be-tween them:
These extra planes have the same number of atoms as the original(110) planes. But if the original planes correspond to a path lengthdifference ofλ, these have path length difference ofλ/2 – their signalswill be out of phase. If the atoms are all the same, the(110) reflectionwill be missing. If the atoms are different, the amplitude of the(110)
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reflection will be reduced.
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Thesemissing orderstell us something about the structures:
• simple cubic – no missing orders;
• fcc – only see(hkl) whereh, k and l are all even OR all odd.
• bcc – only see(hkl) whereh + k + l is even.
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Summary
• Experimental methods – broad-band or single-wavelength;
• Bragg’s law explained by von Laue’s treatment;
• Scattering treatment;
• The reciprocal lattice;
• Effect of atomic basis.
Next:
• Detailed treatment of structure with a basis;
• Other information from diffraction;
• Binding of crystals.
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