Electromagnetism

1

• 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.

2

• 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''.

3

• 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• Idea superseded by quantum mechanics,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.

4

• 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.

5

• Adding and subtracting waves
• If we add two cosines
• cos (2p x /l fA) cos (2p x /l fB)
• cos 2p x /l (fA fB)/2 2 cos (fA -
  fB)/2
• where l is the wavelength f is the phase.
• If fA fB then the term 2 cos (fA - fB)/2
  2 cos(0) 2
• - The resulting wave has twice the amplitude.
• Add in phase''.
• If fA fB p then the term 2 cos ((fA - fB)/2)
  2 cos(p/2) 0
• - The resulting wave has zero amplitude.
• - Add out of phase'' therefore cancel.

6

• Two electron system.
• Consider two electrons separated by a vector r.
• Suppose incoming X-ray has wave-vector s0 with
  length 1/l.
• Suppose deflected X-ray has wavevector s with
  legnth 1/l.
• - The path difference is therefore
• p q l r (s0 s)

7

• A phase difference results from this
  path-difference
• Df - 2p (p q) / l - 2p r (s0 s) 2p r
  S
• Where
• S s s0
• The wave can be regarded as being reflected
  against a plane with incidence reflection angle
  q and
• S 2 sin q / l
• - Note that S is perpendicular to the plane of
  reflection.

r
q
s0
q
q
2 sin q / l
S
s
-s0
8

• Mathematics Interlude Complex numbers
• Complex numbers derive from i v(-1)
• i2 -1
• Any complex number can be written as a sum of a
  real part and an imaginary part
• x a i b
• which can be drawn on an Argand diagram

9

• Exponential functions
• The exponential function
• exp x ex
• is defined by
• d/dx exp x exp x
• Using the chain rule
• d/dx exp ax a exp ax
• Hence, if i v-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

10

• Exponential representation of complex numbers
• Assume
• exp iq cos q i sin q
• Check by going back to previous definition of
  exp ix.
• d/dq exp iq d/dq cos q i sin q
• - sin q i cos q
• i2 sin q i cos q
• i cos q i sin q
• i exp iq
• Since exp iq cos q i sin q
• Real exp iq cos q
• Imaginary exp iq sin q

11

• Exponential representation
• Any complex number can be written in this form
• x a i b A exp iq
• where A v(a2 b2)
• On an Argand diagram.

q
12

• Description of waves
• Common to write a wave as
• y A cos 2p (x/l - nt f)
• This can equally well be written
• y Real A exp 2pi (x/l - nt f)
• 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
• Y A exp 2pi (x/l nt f)
• The intensity (probability of detecting a
  photon)
• I Y Y Y
• Always a real number even though the wave
  function is a complex exponential.

13

• Adding and subtracting waves again
• If we add two cosines
• cos (2p x /l fA) cos (2p x /l fB)
• cos 2p x /l (fA fB)/2 2 cos (fA -
  fB)/2
• It rapidly gets complicated, especially if they
  have different amplitude.
• Using the complex representation
• A exp 2pi ( x/l fA ) B exp 2pi ( x/l fB
  )
• becomes trivial you add vectors!

A exp 2pi ( x/l fA ) B exp 2pi ( x/l fB
)
14

• Scattering from an atom
• The atomic scattering factor for an atom is
  described as
• fatom ?r r (r) exp (2pi r S) dr
• - r (r) is the electron density within the atom
• The integration is over all space.
• S s0 - s
• S 2 sin q/l
• For each point r within the atom a phase shift
  results
• Df 2p/l 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 (r) cos (2pi r S) dr
• now integrate over half the atomic volume.
• Guaranteed real.

15

• 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 (q 0) but
  with less power the further from the forward
  direction (q gt 0).
• Mathematically described by the atomic
  scattering factor'' fO (sin q / l).

16

• Scattering from an atom
• Note that the expression
• fatom(S) ?r r (r) exp (2pi r S) dr
• is saying that f(S) is the Fourier transform of
  the electron density of the atom.
• - In this case fatom(S) fatom(2 sin q/l)
  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.