Electromagnetism• Around 1800 classical physics knew:

- 1/r2 Force law of attraction between positive & negative charges. - v ×B Force law for a moving charge in a magnetic field.

• About 1850 Maxwell ``unified'' electricity & magnetism:- Seen as two aspects of the same phenomenon.- A ``deeper'' physical basis for this unification was found by Einstein's theory of relativity in 1905.

• One prediction of Maxwell's theory was that light was electromagnetic waves.

- All of optics incorporated into the same theory.

Scattering of Light• Charged particles are accelerated by electric fields.• Charge particles are the source of electric fields:

- Acceleration of a charged particle perturbs the electric field.- Accelerating electrons radiate photons!

• When an electron interacts with a electromagnetic wave it oscillates at the same frequency of the wave.

- Generates electromagnetic radiation with the same frequency & 180o out of phase.- Called ``scattering''.

Structure of atoms • Bohr proposed electrons orbit the nucleus like planets at specific radii.

- ie. specific angular momentum.• Idea superseded by quantum mechanics,

- Electrons are represented as probability distributions which are solutions of Schrödinger's equation. - Each with specific angular momentum.

Scattering from an atom• Due to different path-differences the X-rays scattered from electrons within an atom do not-necessarily add in phase.

- As the scattering angle gets wider you lose scattering power.

Adding and subtracting waves • If we add two cosines:

cos (2 x / + A) + cos (2 x / + B)

= cos {2 x / + (A + B)/2 } × 2 cos {(A - B)/2 }

where is the wavelength & is the phase.• If A = B then the term 2 cos {(A - B)/2} = 2 cos(0) = 2

- The resulting wave has twice the amplitude.- Add ``in phase''.

• If A = B + then the term 2 cos ((A - B)/2) = 2 cos(/2) = 0

- The resulting wave has zero amplitude.- Add ``out of phase'' & therefore cancel.

Two electron system.• Consider two electrons separated by a vector r.• Suppose incoming X-ray has wave-vector s0 with length 1/.• Suppose deflected X-ray has wavevector s with legnth 1/.

- The path difference is therefore p + q =· r · (s0 – s)

pq

rs0

s

• A phase difference results from this path-difference= - 2 (p + q) / = - 2r · (s0 – s) = 2 r · S

Where S = s – s0

• The wave can be regarded as being reflected against a plane with incidence & reflection angle and

|S| = 2 sin / - Note that S is perpendicular to the plane of reflection.

r

s0

s

S

-s0

2 sin /

Mathematics Interlude: Complex numbers• Complex numbers derive from i = √(-1)

i2 = -1• Any complex number can be written as a sum of a real part and an imaginary part:

x = a + i bwhich can be drawn on an Argand diagram:

Real axis

Imag

inar

y ax

is

x = a + i b

Exponential functions• The exponential function

exp x = ex

is defined by d/dx exp x = exp x

• Using the chain ruled/dx exp ax = a exp ax

• Hence, if i = √-1 then d/dx exp ix = i exp ix

• Note cosine & sine functions have similar rules:d/dx sin x = cos x

d/dx cos x = - sin x

Exponential representation of complex numbers• Assume

exp i = cos + i sin • Check by going back to previous definition of exp ix.

d/d exp i = d/d {cos + i sin }= - sin + i cos = i2 sin + i cos = i {cos + i sin }

= i exp i• Since exp i = cos + i sin

Real{ exp i } = cos Imaginary{ exp i } = sin

Real axis

Imag

inar

y ax

is

x = A exp i

Exponential representation• Any complex number can be written in this form x = a + i b = A exp iwhere A = √(a2 + b2)

• On an Argand diagram.

Description of waves• Common to write a wave as

y = A cos 2 (x/ - t + ) • This can equally well be written

y = Real { A exp 2i (x/t+ ) }• In physics, if you are careful that your measurables are always real then you can drop the requirement to write `Real´ all the time. • An electromagnetic wave frequently written as

= A exp{ 2i (x/ + t + ) }• The intensity (probability of detecting a photon)

I = * = ||• Always a real number even though the wave function is a complex exponential.

Adding and subtracting waves again • If we add two cosines:

cos (2 x / + A) + cos (2 x / + B)

= cos {2 x / + (A + B)/2 } × 2 cos {(A - B)/2 }

It rapidly gets complicated, especially if they have different amplitude. • Using the complex representation

A exp {2i ( x/ + A )} + B exp {2i ( x/ + B )}

becomes trivial – you add vectors!

Real axis

Imag

inar

y ax

is

A exp iAB exp iB

A exp {2i ( x/+ A )} + B exp {2i ( x/ + B )}

Scattering from an atom• The atomic scattering factor for an atom is described as

fatom = ∫r (r) · exp (2i r · S) dr (r) is the electron density within the atom - The integration is over all space.- S = s0 - s - | S | = 2 sin

• For each point r within the atom a phase shift results = 2/ r · S + 180o

where the 180o comes from assuming free electrons & is usually ignored since it adds only a constant term. • Assuming spherical symmetry in the electron density.

fatom = 2 ∫r (r) · cos (2i r · S) dr- now integrate over half the atomic volume.- Guaranteed real.

Again scattering from an atom• As the scattering angle gets wider you lose scattering power.

-eg. an oxygen atom will scatter with the power of 8 electrons in the forward direction ( = 0) but with less power the further from the forward direction ( > 0).

• Mathematically described by the ``atomic scattering factor'' fO (sin / ).

Scattering from an atom• Note that the expression

fatom(S) = ∫r (r) · exp (2i r · S) dris saying that f(S) is the Fourier transform of the electron density of the atom.

- In this case fatom(|S|) = fatom(2 sin ) since the electron density distribution is symmeteric.

- ie. Adding up all the scattering contributions of a function of electron density as a complex exponential leads naturally to a Fourier Transform.