Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Interference
Peter Hertel
University of Osnabruck, Germany
Lecture presented at APS, Nankai University, China
http://www.home.uni-osnabrueck.de/phertel
Spring 2012
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Crystal
• a crystal is a regular array of identical unit cells
• location xl = l1a1 + l2a2 + l3a3
• l1 = −M1,−M1 + 1, . . . ,M1 − 1,M1
• l2 and l3 likewise
• each unit cell serves as an antenna
• it is excited by a primary electromagnetic wave
• and emits a secondary wave
• response is described by a complex number f
• which describes the responsiveness and the retardation
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
a1
a2
a3
a
Unit cell and primitive cell of NaCl like crystal
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Source, crystal and detector
• the X-ray source is at xs = −Rsnin
• the detector is at xd = +Rdnout
• what is the distance between source and a unit cell?
• rsl = |xs − xl| = |Rsnin + xl|• = Rs|nin +R−1
s xl|• = Rs
√1 + 2R−1
s xl · nin + . . .
• = Rs + xl · nin + . . .
• distance between detector and unit cell likewise
• rdl = Rd − xl · nout + . . .
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Spherical wave
• for simplicity, we work with a scalar electric field E
• wave equation ∆E + k2E = 0
• vacuum wave number k = ω/c = 2π/λ
• spherical solution for primary field
Epr = Aeikr
r• field strength at distance r from source
• secondary field emitted by unit cell
Esd = fEpr(rsl)eikr
r• field strength at distance r from unit cell
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
S
D
l
nin
nout
X-ray scattering.Primary field emitted at source S. A secondaryfield is emitted by the unit cell labeled l. It is detected at D.
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Detected intensity
• field strength of secondary wave at detector
El = AfeikRs
Rs
eikRd
Rdeiknin · xl e
−iknout · xl
• this is the contribution from unit cell l
• the entire field strength is the superposition
E =∑l
El = AfeikRs
Rs
eikRd
Rd
∑l
e−i∆ · xl
• wave vector transfer
∆ = k(nout − nin)
• detected intensity of secondary waves
|E|2 = |A|2|f |2
R2sR
2d
∣∣∣∣∣∑l
e−i∆ · xl
∣∣∣∣∣2
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
• the sum has three factors, Σ = F1F2F3
• each of the form
F =
M∑l=−M
e−i l∆ · a
• with
z = e−i∆ · a
• the factor is
F =
M∑l=−M
zl =sin(N∆ · a/2)sin(∆ · a/2)
• where N = 2M + 1 (number of unit cells in thisdimension)
• intensity at detector is proportional to
|Σ|2 = |F1|2|F2|2|F3|2
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
0 2 4 6 8 10 12 14 16 18 200
5
10
15
20
25
|F |2 versus ∆ · a for N = 5
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Laue conditions
• Each factor is tiny unless ∆ ·ai is an integer multiple of 2π
• this must be fulfilled for a1 and a2 and a3
• Laue conditions
∆ · ai = νi 2π
• sharp refraction peak if Laue conditions are fulfilled
• recall that ∆ = k(nout − nin) depends on angles andwave number
• different kinds of X-ray diffraction experiments
• Max von Laue 1912, Nobel prize 1914
• William Henry and William Lawrence Bragg, father andson, 1914, Nobel prize 1915
• proof that X-rays were not particles, but electromagneticwaves
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Max von der Laue, German physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Massive particle diffraction
• In 1937, George P. Thomson discovered that electronsproduced the same refraction pattern as X-rays
• provided the momentum p was identified with ~k• predicted before by Louis de Broglie
• in 1946 the same was discovered for neutrons
• today preferred because only nucleons are visible
• dedicated nuclear reactors and spallation sources
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
George P.Thomson, British physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Louis de Broglie, French physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Quantum physics
• normally: equal equations have equal solutions (Feynman)
• now: same solutions must come from same equations
• guess the mathematical framework of quantum physic
• probability
• complex probability amplitudes
• interference
• Hilbert space, linear operators, observables, states,expectation values
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Max Planck, German physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Albert Einstein, German physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
David Hilbert, German mathematician
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Erwin Schrodinger, Austrian physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Werner Heisenberg, German physicist
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Differential cross section
• the target has n scatterer per unit volume
• beam intensity I = I(x)
• the beam is weakened by scattering
dI(x) = −I(x)σ ndx• therefore
I(x) = I(0) e−σnx
• cross section is solid angle integral
σ =
∫dΩ σd(ϑ)
• differential cross section depends on scattering angle ϑ
• but normally not on azimuth φ
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Neutron-O2 scattering
• oxygen nuclei at ±an/2• O2 molecule held together by common electron cloud
• ignored by neutrons
• differential cross section is
σd =
∣∣∣∣f e−ia∆ · n/2 + f e+ia∆ · n/2
∣∣∣∣2• i.e.
σd = 4|f |2 cos2 a∆ · n2
• average over direction n (randomly oriented molecules)
σd = 2|f |21 +
sin(∆a)
∆a
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
Constructive, destructive and no interference
• M refers to the randomly oriented molecule, A to the atom
• differential cross section
σMd = 2
1 +
sin(∆a)
∆a
σAd
• scattering on an atom is isotropic
• cross section for scattering on a randomly orientedmolecule depends on scattering angle
• via
∆ = k|nout − nin| = 2k sinϑ
2• for ∆a→ 0 (slow or forward): amplitudes add, cross
section for scattering on molecule is four times as large asfor atom, full interference
• fast neutrons: cross section for molecule is twice that foratom - no interference
• intermediate: constructive or destructive interference
Interference
Peter Hertel
X-raydiffraction
Laueconditions
Electron andneutrondiffraction
Quantumphysics
Neutron-moleculescattering
From full tono interference
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
4
The ratio of molecular to atomic differential cross section isplotted vs. scattering angle ϑ. The curves are for ka = 1, 2, 4,and 8, according to decrease at forward direction. Valuesbelow 2 mean destructive interference.