Post on 20-Dec-2015
2/
...3,2,1
h
nnvrme
an integer number of wavelengths fits into the circular orbit
rn 2
where
p
h
is the de Broglie wavelength
Particle Value of
Electrons of kinetic energy
Protons of kinetic energy
Thermal neutrons (300K)
Neutrons of kinetic energy (14 MeV)
He atoms at 300K
1 eV 100 eV 10000 eV
12.2 A 1.2 A 0.12A
1 keV 1 MeV 1 GeV
0.009 A 28.6 F 0.73 F
1.5 A
9.0 F
0.75 A
A10cm10 F 1 cm;10A1 5-138
me, walking to the student union for lunch at 2 miles per hour
m1054.2 34
Electrons have a wavelength that is much shorter than visible light. The smallest detail that can be resolved is equal to one wavelength.
bu
lk
(th
ick)
foil
(th
in)
Specimen interactions involved in forming an image
Backscattered electrons can be used to identify elements in the material.
Auger electrons also can give compositional information.
Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography.
Secondary electrons are low in energy and thus can’t escape from the interior of the material. They mostly give information about the surface topography.
X-rays are produced by de-excited atoms.
Unscattered electrons are those which are transmitted through the material. Since the probability of transmission is proportional to thickness, it can give a reading of thickness variations.Elastically scattered Bragg electrons give info about atomic spacing, crystal orientations, etc.
Loss of energy by electrons is characteristic of bulk composition.
Beats
http://www.school-for-champions.com/science/soundbeat.htm
http://library.thinkquest.org/19537/java/Beats.html
How does this wave behave at a boundary?
at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave
at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change)
When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary
In this animation, the density of the thick string is four times that of the thin string …
The word “particle” in the phrase “wave-particle duality” suggests that this wave is somewhat localized.
How do we describe this mathematically?
…or this
…or this
FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite
number of harmonic waves
FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite
number of harmonic waves
dkekaxf ikx)(
2
1)(
spatially localized wave group
amplitude of wave with wavenumber k=2/
adding varying amounts of an infinite number of
wavessinusoidal expression
for harmonics
Adding several waves of different wavelengths
together will produce an interference pattern which
begins to localize the wave.
To form a pulse that is zero everywhere outside of a finite spatial range x requires adding together
an infinite number of waves with continuously varying wavelengths and
amplitudes.
hh
xpx
sin2sin
2
anglescattering
/)sin(momentum electron /)sin( hxh )sin2/( x
Uncertainty on optics:
Remember our sine wave that went on “forever”?
We knew its momentum very precisely, because the momentum is a function of the frequency, and the frequency was very well defined.
But what is the frequency of our localized wave packet? We had to add a bunch of waves of different frequencies to produce it.
Consequence: The more localized the wave packet, the less precisely defined the momentum.
Consequence: The more localized the wave packet, the less precisely defined the momentum.
kp
E