5.
History of Brownian motion
1785: Jan Ingenhousz observed irregular motion of coal dust particles in
alcohol.
1827: Robert Brown watched pollen particles performing irregular motion in
water using a microscope. He repeated his experiments with dust to rule out
that the particles were alive.
1905: Einstein provided the first physical theory to explain Brownian motion.
1908: Jean Perrin did experiments to verify Einstein’s predictions. The
measurements allowed Perrin to give the first estimate of the dimensions of
water molecules. Jean Perrin won the Nobel Prize of Physics in 1926 for this
work.
6.
y Random walk
R = qe1 + qe2 + qe3 + ... + qeN where ei are unit vectors
For random walk we require that R = 0
Example (assume only two steps)
qe2
qe1
x
2
(
2
) (
) (
R 2 = ( qe1 + qe 2 ) = q 2 e 2 + e12 + 2e 2 ⋅ e1 = q 2 1 + 1 + 2e 2 ⋅ e1 = q 2 2 + 2e 2 ⋅ e1 )
Average over M experiments
2 1 m 2 1 M 2 2 q2
R = ∑ R k = M ∑ q ( e1 + e2 + e3 + ... + e N ) = M (MN + ∑ ei ⋅e j )
M k =1 k =1 i≠ j
t
If we assume that each step is random and takes a time τ and the total time is t, then N =
τ
2 t 2 q 2 2q 2 q 2
We may write R = Nq = q = 4Dt, where D =
2
= x = x where q 2 = q 2 + q 2 = 2q 2
x y x
τ 4τ 4τ 2τ
Each step in the x and y directions are random,
but otherwise equal, such that qx2=qy2
7.
Random walk
MSD
y
x
t
2 q2
Mean Square Deviation = MSD = R = 4Dt, where D = x
2τ
1D: MSD=2Dt
2D: MSD=4Dt try to show this yourself!
3D: MSD=6Dt
8.
Fick’s law of diffusion
Adolf Fick (1855):
J
dC
J = −D
A
dx
J= flux of particles (number of particles per area and time
incident on a cross-section) [m-2s-1]
D= diffusion coefficient [m2s-1]
C=concentration of particles [m-3]
(sometimes n is used instead of C to represent concentration )
9.
Random walk is due to thermal fluctuations!
v
ma = 0 = −fv + R(t) f = 6πrη for a spherical particle where r = radius of particles
R(t) is a random force due to collision with water molecules
fv
R(t)
k BT
D= (Einstein relationship, 1905)
f
10.
Diffusion coefficients in different materials
k BT
D= (Einstein relationship, 1905)
f
State of matter D [m2/s]
Solid 10-13
Liquid 10-9
Gas 10-5
12.
Photons and Electromagnetic Waves
• Light has a dual nature. It exhibits both wave and
particle characteristics
– Applies to all electromagnetic radiation
13.
Particle nature of light
• Light consists of tiny packets of energy, called photons
• The photon’s energy is:
E = h f = h c /λ
h = 6.626 x 10-34 J s (Planck’s constant)
14.
Wave Properties of Particles
• In 1924, Louis de Broglie postulated that because
photons have wave and particle characteristics,
perhaps all forms of matter have both properties
15.
de Broglie Wavelength and Frequency
• The de Broglie wavelength of a particle is
h h
λ = =
p mv
• The frequency of matter waves is
E
ƒ=
h
16.
Dual Nature of Matter
• The de Broglie equations show the dual nature of matter
• Matter concepts
– Energy and momentum
• Wave concepts
– Wavelength and frequency
17.
X-Rays
• Electromagnetic radiation with short wavelengths
– Wavelengths less than for ultraviolet
– Wavelengths are typically about 0.1 nm
– X-rays have the ability to penetrate most materials
with relative ease
• Discovered and named by Röntgen in 1895
18.
Production of X-rays
• X-rays are produced when high-speed electrons are
suddenly slowed down
29.
Scattering by an object of finite volume
Scattered
beam
Incident
beam
Janin & Delepierre
30.
Schematic for X-ray Diffraction
• The diffracted radiation is very
intense in certain directions
– These directions correspond
to constructive interference
from waves reflected from the
layers of the crystal
31.
Diffraction Grating
• The condition for maxima is
d sin θbright = m λ
• m = 0, 1, 2, …
32.
X-ray Diffraction of DNA
Photo 51
http://en.wikipedia.org/wiki/Image:Photo_51.jpg
34.
Bragg’s Law
• The beam reflected from the lower
surface travels farther than the one
reflected from the upper surface
• Bragg’s Law gives the conditions for
constructive interference
2 d sinθ = mλ; m = 1, 2, 3…
38.
Scattering and diffraction of neutrons
Institut Laue-Langevin,
Grenoble, France
39.
Why use neutrons?
Electrically Neutral
Microscopically Magnetic
Ångstrom wavelengths
Energies of millielectronvolts
40.
The Electron Microscope
• The electron microscope depends on
the wave characteristics of electrons
• Microscopes can only resolve details
that are slightly smaller than the
wavelength of the radiation used to
illuminate the object
• The electrons can be accelerated to
high energies and have small
wavelengths
41.
Nuclear Magnetic Resonance (NMR) spectroscopy
Superconducting magnets 21.5 T
Earth’s magnetic field 5 x 10-5 T
http://en.wikipedia.org/wiki/Nuclear_magnetic_resonance
42.
Spin and magnetic moment
• Nuclei can have integral spins (e.g. I = 1, 2, 3 ....): 2H, 6Li, 14N
fractional spins (e.g. I = 1/2, 3/2, 5/2 ....): 1H, 15N
or no spin (I = 0): 12C, 16O
• Isotopes of particular interest for biomolecular research are
1
H, 13C, 15N and 31P, which have I = 1/2.
• Spins are associated with magnetic moments by:
m = γħ I
43.
Larmor frequency
A Spinning Gyroscope A Spinning Charge
in a Gravity Field in a Magnetic Field
ω = γ B0
http://www.cem.msu.edu/~reusch/VirtualText/Spectrpy/nmr/nmr2.htm#pulse
54.
A 2D NMR spectrum
http://www.bruker-nmr.de/guide/
55.
Nuclear Overhauser Effect Spectroscopy (NOESY)
provides information on proton-proton distances
NOE ~ 1/r6
http://www.cryst.bbk.ac.uk/PPS2/projects/schirra/images/2dnosy_1.gif
56.
Information obtained by NMR
• Distances between nuclei
• Angles between bonds
• Motions in solution
57.
Today’s lesson:
• Molecules in solution; Brownian motion
• X-ray
• Scattering and diffraction
• Neutron scattering
• Electron Microscopy (EM)
• Nuclear Magnetic Resonance (NMR) spectroscopy
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