Interference · Interference Peter Hertel X-ray di raction Laue conditions Electron and neutron di...

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


Quantumphysics



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


Quantumphysics



a1

a2

a3

a

Unit cell and primitive cell of NaCl like crystal

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



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


Quantumphysics



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


Quantumphysics



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


Quantumphysics



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


Quantumphysics



• 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


Quantumphysics



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


Quantumphysics



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


Quantumphysics



Max von der Laue, German physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



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


Quantumphysics



George P.Thomson, British physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



Louis de Broglie, French physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



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


Quantumphysics



Max Planck, German physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



Albert Einstein, German physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



David Hilbert, German mathematician

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



Erwin Schrodinger, Austrian physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



Werner Heisenberg, German physicist

Interference

Peter Hertel

X-raydiffraction

Laueconditions


Quantumphysics



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


Quantumphysics



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


Quantumphysics



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


Quantumphysics



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.

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