Electromagnetism Around 1800 classical physics knew: - 1/r 2 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 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 & 180 o 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 Idea superseded by quantum mechanics, - Electrons are represented as probability distributions which are solutions of Schrödinger's equation.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 s 0 with length 1/. Suppose deflected X-ray has wavevector s with legnth 1/. - The path difference is therefore p + q =  · r · (s 0 – s) p q r s0s0 s

A phase difference results from this path-difference  = - 2  (p + q) / = - 2  r · (s 0 – s) = 2  r · S Where S = s – s 0 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 s0s0 s   S -s0-s0  2 sin  /

Mathematics Interlude: Complex numbers Complex numbers derive from i = √(-1) i 2 = -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: Real axis Imaginary axis x = a + i b

Exponential functions The exponential function exp x = e x is defined by d/dx exp x = exp x Using the chain rule d/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  = i 2 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 Imaginary axis x = A exp i   Exponential representation Any complex number can be written in this form x = a + i b = A exp i  where A = √(a 2 + b 2 ) 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 2  i (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{ 2  i (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 {2  i ( x/ +  A )} + B exp {2  i ( x/ +  B )} becomes trivial – you add vectors! Real axis Imaginary axis A exp i  A B exp i  B A exp {2  i ( x/  +  A )} + B exp {2  i ( x/ +  B )}

Scattering from an atom The atomic scattering factor for an atom is described as f atom = ∫ r  (r) · exp (2  i r · S) dr  (r) is the electron density within the atom - The integration is over all space. - S = s 0 - s - | S | = 2 sin  For each point r within the atom a phase shift results  = 2  / r · S o where the 180 o comes from assuming free electrons & is usually ignored since it adds only a constant term. Assuming spherical symmetry in the electron density. f atom = 2 ∫ r  (r) · cos (2  i 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'' f O (sin  / ).

Scattering from an atom Note that the expression f atom (S) = ∫ r  (r) · exp (2  i r · S) dr is saying that f(S) is the Fourier transform of the electron density of the atom. - In this case f atom (|S|) = f atom (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.