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A26537
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School of Physics and Astronomy
DEGREE OF B.Sc. & M.Sci. WITH HONOURS
FIRST-YEAR EXAMINATION
03 22132
LC INTRODUCTION TO NANOSCALE PHYSICS
SUMMER EXAMINATIONS 2015
Time Allowed: 1 Hour
Students should answer two questions.
If more than two questions are attempted only the best two attempted questions will
be marked
Calculators may be used in this examination but must not be used to store text.
Calculators with the ability to store text should have their memories deleted prior to
the start of the examination.
A table of physical constants and units that may be required
will be found at the end of this question paper.
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Students must answer two questions out of three.
If you answer more than two questions, only the best two will be marked.
1. (a) Calculate the number of atoms in a 5 nm diameter platinum spherical nanoparticle.
Consider a FCC unit cell with lattice constant of a = 3.91Å.
[4]
(b) The bulk density of platinum is 21.5 g cm 3 . How does it compare with the
value obtained in (a)? (Atomic mass of platinum AWP t = 195.1 g mol 1 )
[4]
(c) Calculate the number of surface atoms in the particle. You may find it useful
to assume that the surface is a shell of one lattice constant thick.
(d) Explain the interest in small platinum particles for catalysis.
[4]
[4]
(e) Suppose that there are three available shapes for particles to use in catalysis:
a sphere, a cube and a tetrahedron. Which shape is preferable for catalysis?
Justify your answer.
[4]
2. (a) The diagram below shows a schematic representation (not to scale) of the
geometry of an Atomic-Force Microscope (AFM). Using the parameters shown
on the diagram, derive the magnification factor, M = A/a, where a is the
height of an object which bends the cantilever tip upwards, and A is the
resulting displacement of the laser on the detector.
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(b) Use the magnification factor to calculate the displacement A for the following
typical values of an AFM experiment: a = 1 nm, ✓ = 600 , l = 1 µm, and
L = 1 cm.
[3]
(c) A protein of a 10 nm diameter is imaged by an AFM probe of a similar radius.
Estimate the full width of the image at its half-height. Include a sketch to
support your estimation.
[3]
3. (a) The figure below shows spectra of absorption and luminescence of CdSe
colloidal quantum dots of various diameters. Derive a quantitative estimate
for the spectral shift between quantum dots of 1.2 nm and 4.5 nm diameter
and compare this with the values from the figure. What is a conventional
name used in scientific literature for this phenomenon?
[6]
Figure 1: Question 3(a). Absorption (solid line) and luminescence (dashed line) spectra
of colloidal CdSe quantum dots of various diameters.
(b) Estimate the dimensions of a metallic nanoparticle at which the conductivity
will deviate from its bulk counterpart. Recall that the conductivity of bulk
2
materials is given by = neme⌧ . Use in your estimate typical values for the
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conductivity, = 5.9 ⇥ 107 S/m, electron effective mass, m = 1.47me , Fermi
velocity, vf = 1.57 ⇥ 106 m/s, and electron density, n = 8.5 ⇥ 1028 m 3 .
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Physical Constants and Units
Acceleration due to gravity
g
Gravitational constant
G
Ice point
Tice
Avogadro constant
NA
Gas constant
R
Boltzmann constant
k, kB
Stefan constant
Rydberg constant
h/2⇡
~
Speed of light in vacuo
c
~c
Charge of proton
me
Rest energy of electron
mp
Rest energy of proton
1.381 ⇥ 10
23
JK
1
eV K
1
Wm
4
1.097 ⇥ 107 m
1
2
13.606 eV
6.626 ⇥ 10
1.055 ⇥ 10
34
34
J s ⌘ 4.136 ⇥ 10
J s ⌘ 6.582 ⇥ 10
15
eV s
16
eV s
2.998 ⇥ 108 m s
1
19
C
197.3 MeV fm
1.602 ⇥ 10
9.109 ⇥ 10
31
1.673 ⇥ 10
27
1.66 ⇥ 10
27
kg
kg
kg
931.5 MeV
✏0
Magnetic constant
µ0
Bohr magneton
µB
Nuclear magneton
µN
↵ = e2 /4⇡✏0 ~c
c
= h/mc
Bohr radius
a0
angstrom
Å
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1
K
8
5.670 ⇥ 10
u
Electric constant
barn
mol
1
4
⌘ 0.862 ⇥ 10
Atomic mass unit energy equivalent
torr (mm Hg at 0 C)
1
938.3 MeV
One atomic mass unit
Compton wavelength of electron
273.15 K
0.511 MeV
Mass of proton
Fine-structure constant
2
8.314 J K
e
Mass of electron
N m2 kg
[N.B. 1 mole ⌘ 1 gram-molecule]
R1 hc
h
2
6.022 ⇥ 1023 mol
R1
Planck constant
11
6.673 ⇥ 10
9.81 m s
torr
8.854 ⇥ 10
4⇡ ⇥ 10
9.274 ⇥ 10
5.051 ⇥ 10
Fm
1
7
Hm
1
24
A m2 (J T 1 )
27
A m2 (J T 1 )
7.297 ⇥ 10 3 = 1/137.0
2.426 ⇥ 10
12
m
11
m
10
10
m
5.2918 ⇥ 10
133.32 Pa (N m 2 )
b
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m2