Theoretical Question – T1

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Mechanics of a Deformable Lattice (Total Marks : 20)

Here we study a deformable lattice hanging in gravity which acts as a deformable physical
pendulum. It has only one degree of freedom, i.e. only one way to deform it and the
configuration is fully described by an angle . Such structures have been studied by famous
physicist James Maxwell in 19th century, and some surprising behaviors have been
discovered recently.

As shown in the figure 1, N2 identical triangular plates (red triangle) are freely hinged by
identical rods and form an N × N lattice (N > 1). The joints at the vertices are denoted by
small circles. The sides of the equilateral triangles and the rods have the same length l. The
dashed lines in the figure represent four tubes; each tube confines N vertices (grey circles) on
the edge and the N vertices can slide in the tube, i.e. the tube is like a sliding rail.

The four tubes are connected in a diamond shape with two angles fixed at 60◦ and another
two angles at 120◦ as shown in Figure 1. Each plate has a uniform density with mass M, and
the other parts of the system are massless. The configuration of the lattice is uniquely
determined by the angle α, where 0° ≤ 𝛼 ≤ 60° (please see the examples of different angle
α in Figure 1). The system is hung vertically like a “curtain” with the top tube fixed along
the horizontal direction.

The coordinate system is shown in Figure 2. The zero level of the potential energy is
defined at y = 0. A triangular plate is denoted by a pair of indices (m,n), where m, n = 0, 1, 2,
· · ·, N − 1 representing the order in the x and y directions respectively. A(m,n), B(m,n) and
C(m,n) denote the positions of the 3 vertices of the triangle (m,n). The top-left vertex, A(0,
0) (the big black circle), is fixed.

The motion of the whole system is confined in the x-y plane. The moment of inertia of a
uniform equilateral triangular plate about its center of mass is I = Ml 2 /12. The free fall
acceleration is g. Please use Ek and Ep to denote kinetic energy and potential energy
respectively.

Figure 1
 Theoretical Question – T1
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Figure 2

Section A: When N=2 (as shown in figure 3):

Figure 3

A1 What is the potential energy Ep of the system for a general angle α when N = 2? 2 points

A2 What is the equilibrium angle αE of the system under gravity when N = 2? 1 point

The system follows a simple harmonic oscillation under a small perturbation from
A3 equilibrium. Calculate the kinetic energy of this system in terms of ∆α̇ ≡d(∆α)/dt. Calculate 5 points
the oscillation frequency fE when N = 2.
 Theoretical Question – T1
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Section B: For arbitrary N :

B1 What is the equilibrium angle 𝛼E′ under gravity when N is arbitrary? 3 points

Consider the case when 𝑁 → ∞ . Under a small perturbation of angle α, the change of
B2 potential energy of the system is ∆𝐸p ∝ 𝑁 𝛾1 , the kinetic energy of the system is 𝐸k ∝ 𝑁 𝛾2 , 3 points
and the oscillation frequency is 𝑓E′ ∝ 𝑁 𝛾3 . Find the values of γ1, γ2 and γ3.

Section C: A force is exerted on one of the 3N2 triangle vertices so that the system maintains at m = 60°.

y
x

Figure 4

C1 Which vertex should we choose to minimize the magnitude of this force? 1 point

What are the direction and magnitude of this minimum force? Describe the direction in
C2 5 points
terms of the angle θF defined in Figure 4.
 Theoretical Question – T2
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The Expanding Universe (Total Marks : 20)

The most outstanding fact in cosmology is that our universe is expanding. Space is
continuously created as time lapses. The expansion of space indicates that, when the
universe expands, the distance between objects in our universe also expands. It is
convenient to use “comoving” coordinate system 𝑟⃗ = (𝑥, 𝑦, 𝑧) to label points in our
expanding universe, in which the coordinate distance
Δ𝑟 = |𝑟⃗2 − 𝑟⃗1 | = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2 between objects 1 and 2 does not
change. (Here we assume no peculiar motion, i.e. no additional motion of those objects
other than the motion following the expansion of the universe.) The situation is illustrated
in the figure below (the figure has two space dimensions, but our universe actually has three
space dimensions).

The modern theory of cosmology is built upon Einstein’s general relativity. However, under
proper assumptions, a simplified understanding under the framework of Newton’s theory of
gravity is also possible. In the following questions, we shall work in the framework of
Newton’s gravity.

To measure the physical distance, a “scale factor” 𝑎(𝑡) is introduced such that the physical
distance Δ𝑟p between the comoving points 𝑟⃗1 and 𝑟⃗2 is

Δ𝑟p = 𝑎(𝑡)Δ𝑟,

The expansion of the universe implies that 𝑎(𝑡) is an increasing function of time.

On large scales – scales much larger than galaxies and their clusters – our universe is
approximately homogeneous and isotropic. So let us consider a toy model of our universe,
which is filled with uniformly distributed particles. There are so many particles, such that
we model them as a continuous fluid. Furthermore, we assume the number of particles is
 Theoretical Question – T2
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conserved.

Currently, our universe is dominated by non-relativistic matter, whose kinetic energy is

negligible compared to its mass energy. Let 𝜌m (𝑡) be the physical energy density (i.e.
energyper unit physical volume, which is dominated by mass energy for non-relativistic
matter and the gravitational potential energy is not counted as part of the “physical energy
density”) of non-relativistic matter at time 𝑡. We use 𝑡0 to denote the present time.

A Derive the expression of 𝜌m (𝑡) at time 𝑡 in terms of 𝑎(𝑡), 𝑎(𝑡0 ) and 𝜌m (𝑡0 ). 2 points

Besides non-relativistic matter, there is also a small amount of radiation in our current
universe, which is made of massless particles, for example, photons. The physical
wavelength of massless particles increases with the universe expansion as 𝜆p ∝ 𝑎(𝑡). Let
the physical energy density of radiation be 𝜌r (𝑡).

Derive the physical energy density for radiation 𝜌r (𝑡) at time 𝑡 in terms of 𝑎(𝑡), 𝑎(𝑡0 ) and 2 points
B
𝜌r (𝑡0 ).

Consider a gas of non-interacting photons which has thermal equilibrium distribution. In

this situation, the temperature of the photon depends on time as 𝑇(𝑡) ∝ [𝑎(𝑡)]𝛾 .

C Calculate the numerical value of 𝛾. 2 points

Consider the thermodynamics of one type of non-interacting particle X. Note that the space
expansion is slow enough and thermally isolated such that the entropy of X is a constant in
time. Let the physical energy density of X be 𝜌X (𝑡), which includes mass energy and
internal energy. Let the physical pressure be 𝑝X (𝑡).

D Derive d𝜌X (𝑡)/d𝑡 in terms of 𝑎(𝑡), d𝑎(𝑡)/d𝑡,𝜌X (𝑡),and 𝑝X (𝑡). 4 points

 Theoretical Question – T2
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Consider a star S. At the present time 𝑡0 , the star is at a physical distance 𝑟p = 𝑎(𝑡0 )𝑟 away
from us, where 𝑟 is the comoving distance. Here we ignore the peculiar motion, i.e. assume
that both the star and us just follow the expansion of the universe without additional
motion.

The star is emitting energy in the form of light at power 𝑃e , which is isotropic in every
direction. We use a telescope to observe its starlight. For simplicity, assume the telescope
can observe all frequencies of light with 100% efficiency. Let the area of the telescope lens
be 𝐴.

Derive the power received by the telescope 𝑃r from the star S, as a function of 𝑟, 𝐴, 𝑃e , the
E scale factor 𝑎(𝑡e ) at the starlight emission time 𝑡e , and the present (i.e. at the observation 4 points
time) scale factor 𝑎(𝑡0 ).

If there were no gravity, the expansion speed of the universe should be a constant. In
Newton’s framework, this can be understood as that, without force, matter just moves
away from each other with constant speed and thus d𝑎(𝑡)/d𝑡 is a constant depending on
the initial condition.

Let us now consider how Newton’s gravity affects the scale factor 𝑎(𝑡), in a universe filled
with non-relativistic matter in a homogeneous and isotropic way.

As illustrated in the above figure, let us assume C is the center of our universe (this
assumption can be removed in Einstein’s general relativity, which is beyond the scope of
this question). We slice matter into thin shells around C. Let us focus on one thin shell (the
sphere in the above figure) whose comoving distance from the center is 𝑟 (recall that this
comoving distance is a constant in time).
 Theoretical Question – T2
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Use the motion of the shell to find a relation between d𝑎(𝑡)/d𝑡, 𝑎(𝑡) and the density of
F mass energy 𝜌(𝑡). (In the final relation, if you encounter a constant depending on the 5 points
initial condition, it can be kept as it is.)

Based on the model described in Part (F), is the expansion of the universe
G 1 points
(a) accelerating or (b) decelerating? Choose from (a) or (b).

For your information, in 1998, a new type of energy component of our universe is discovered. It actually
changes the conclusion in Part (G).
 Theoretical Question – T3
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Magnetic Field Effects on Superconductors (Total Marks: 20)

An electron is an elementary particle which carries electric charge and an intrinsic magnetic
moment related to its spin angular momentum. Due to Coulomb interactions, electrons in
vacuum are repulsive to each other. However, in some metals, the net force between electrons
can become attractive due to the lattice vibrations. When the temperature of the metal is low
enough, lower than some critical temperature Tc, electrons with opposite momenta and
opposite spins can form pairs called Cooper pairs. By forming Cooper pairs, each electron
reduces its energy by  compared to a freely propagating electron in the metal which has
p2
energy , where p is the momentum and me is the mass of an electron. The Cooper pairs
2me
can flow without resistance and the metal becomes a superconductor.

However, even at temperatures lower than Tc, superconductivity can be destroyed if the
superconductor is under the influence of an external magnetic field. In this problem, you are
going to work out how Cooper pairs can be destroyed by external magnetic fields through two
effects.

The first is called the paramagnetic effect, in which all the electrons can lower their energy by
aligning the electron magnetic moments parallel to the magnetic field instead of forming
Cooper pairs with opposite spins.

The second is called the diamagnetic effect, in which increasing the magnetic field will
change the orbital motion of the Cooper pairs and increase their energy. When the applied
magnetic field is stronger than a critical value Bc, this increase in energy becomes higher than
2. As a result, the electrons do not prefer forming Cooper pairs.

Recently, a type of superconductor called Ising superconductors was discovered. These

superconductors can survive even when the applied magnetic field is as strong as 60 Tesla,
comparable to the largest magnetic fields which can be created in laboratories. You will work
out why Ising superconductors can overcome both the paramagnetic and the diamagnetic
effects of the magnetic fields.
 Theoretical Question – T3
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A. An Electron in a Magnetic Field

Let us consider a ring with radius r, charge –e and mass m. The mass and the charge density
around the ring are uniform (as shown in Figure 1).

z
r
y
φ x

Figure 1

What is the angular momentum (magnitude and direction) of this ring if the ring is rotating
A1 2 points
with angular velocity ?

The magnitude of the magnetic moment is defined as M  IA , where I is the current and A is
A2 the area of the ring. What is the relationship between the magnetic moment and the 2 points
angular momentum of the ring?

Suppose the normal direction of the ring is and it makes an angle θ with the applied
magnetic field as shown in Figure 2.

For the ring described in Part (A1), what is the potential energy U of this ring if the ring is
placed in a uniform magnetic field Bz pointing to the z-direction? You should assume the
potential energy to be zero when θ = π/2.

Bz n
A3 2 points

Figure 2
 Theoretical Question – T3
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An electron carries an intrinsic angular momentum, which is called spin. We know that the
ℏ
magnitude of spin in a particular direction is 2 , where and h is the Planck’s
constant.
What are the values of the potential energy Uup and Udown for electrons with spins parallel and
A4 1 point
anti-parallel with the applied magnetic field respectively?
e
Please express your results in terms of the Bohr magneton B   5.788 105 eV  T 1
2me
and the magnetic field strength B.

According to quantum mechanics, the potential energy U up and U down are twice the values Uup
and Udown found in Part (A4). Assuming that the applied magnetic field is 1 Tesla. What is the
A5 potential energy U up and U down for an electron with spin parallel and anti-parallel to the 1 point
applied magnetic field respectively? In the rest of this question, you should use the
expressions for U up and U down for your calculations.

B. Paramagnetic effect of the magnetic field on Cooper pairs

In the question below, we consider the paramagnetic effect of an external magnetic field on
Cooper pairs (as shown in Figure 3).
Theoretical studies show that in superconductors, two electrons with opposite spins can form
Cooper pairs so that the whole system saves energy. The energy of the Cooper pair can be
p12 p2
expressed as  2  2 , where the first two terms denote the kinetic energy of the
2me 2me
Cooper pair and the last term is the energy saved for the electrons to form a Cooper pair.
Here,  is a positive constant.

B
Cooper pair

Figure 3
 Theoretical Question – T3
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Assuming that the effect of the external magnetic field is only on the spins of the electrons,
not on the orbital motions of the electrons. What is the energy ES of the Cooper pair under a
B1 1 point
uniform magnetic field ? Recall that the electrons which form a Cooper pair
must have opposite spins.

In the normal state (non-superconducting state), electrons do not form Cooper pairs. What is
the lowest energy E N for the two electrons under a uniform in-plane magnetic field
B2 pointing to the x-direction? Please use the U up and U down defined in Part (A5) 1 point
in your calculations and ignore the effects of the magnetic field on the orbital motions of the
electrons.

At zero temperature, a system will favor the state with the lowest energy. What is the critical
B3 value B in terms of Δ, such that for B  B superconductivity will disappear? 1 point
P P

C. Diamagnetic effect of the magnetic field on Cooper pairs

In the question below, we are going to ignore the effects of magnetic fields on the spins of
the electrons and consider the effects of external magnetic fields on the orbital motions of
the Cooper pairs.
At zero temperature, the energy difference between the superconducting state and the
⃗ = (0,0, 𝐵𝑧 ) can be written as
normal state for a superconductor in a magnetic field 𝐵


 2
d 2 e2 Bz2 x 2 
F       2
   dx .
  4 m e d x m e 

Here  (x) is a function of position x and independent of y. 𝜓2 (𝑥)denotes the probability

of finding a Cooper pair near x. Here,   0 is a constant and it is related to the energy saved
by forming Cooper pairs. The second and the third terms in F are related to the kinetic
energy of the Cooper pairs taking into account the effect of the magnetic field.
At zero temperature, the system prefers to minimize its energy F . In this case,  (x) takes
1
 2  4   x2
the form  x     e , with   0 .
 

Find  in terms of e , Bz , and .

The following integrals may be useful:

C1 3 points
 
 1 
e xe
 ax 2 2  ax 2
dx  , dx  .

a 
2a a
Here a is a constant.
 Theoretical Question – T3
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Work out the critical value of Bz in terms of α, at which the superconducting state is no
C2 2 points
longer energetically favorable.

D Ising Superconductors
In materials with spin-orbit coupling (spin-spin couplings can be ignored), an electron with
momentum experiences an internal magnetic field . On the other hand, an
electron with momentum experiences an opposite magnetic field . These
internal magnetic fields act on the spins of the electrons only as shown in Figure 4.
Superconductors with this kind of internal magnetic fields are called Ising superconductors.

Figure 4: Two electrons form a Cooper pair. Electron 1 with momentum experiences
internal magnetic field but electron 2 with momentum experiences an
opposite magnetic field . The internal magnetic fields are denoted in dashed
arrows.

D1 Then what is the energy EI for a Cooper pair in an Ising superconductor? 1 point
In the normal state of the material with spin orbit coupling, what is the energy 𝐸|| for the two
electrons under a uniform in-plane magnetic field 𝐵 ⃗ || = (𝐵𝑥 , 0,0) ? (Here the internal
D2 2 points
⃗ || . You should also ignore the effects of the
magnetic fields still exist and perpendicular to 𝐵
in-plane magnetic field on the orbital motions of the Cooper pairs.)
⃗ || | > 𝐵I, 𝐸|| < 𝐸I ?
D3 What is the critical value BI such that for |𝐵 1 point
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Reflected optical diffraction patterns from (Total marks: 12)

one-dimensional structures
Introduction
This experiment is based on a recent discovery of a reflected high-energy electron diffraction
phenomenon of a nanostructured Zinc-Selenide (ZnSe) surface that not only has a one-
dimensional (1D) non-uniform modulation in nanoscale, but also has periodic atomic lattice
spacing along the orientation of the 1D nanostructure (Nanotechnology, 20 (21), 215607(2009)).
Figure 1 shows the cross-sectional image of the nanostructured ZnSe surface capped with a
Au layer while the inset is an image of its surface without the Au cap.

Figure 1: A cross-sectional image of the nanostructured ZnSe surface capped with a Au layer while
the inset is an image of its surface without the Au cap.

Objective
In this experiment, you are going to explore the optical analogy of this phenomenon for
samples with 1D structure on a micrometer scale.

Figure 2: Apparatus and tools for this experiment

 Experiment Problem – E1

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List of apparatus and tools

[1] 4 Paper clips
[2] Laser diode
[3] Battery pack with on/off switch
[4] Optical platform with rotary disk, angular scale and laser-diode holder
[5] LED lamp
[6] Sample box containing Samples 1 to 5
[7] 3 screws for the observation board
[8] Pinhole
[9] Flat-head screw driver
[10] Observation board
[11] Alignment sheet
[12] Graph papers
[13] 30 cm ruler

Pictures of key apparatus and tools

Figure 3: Observation board Figure 4: Optical platform with Figure 5: Pinhole

rotary disk, angular scale and laser
diode holder

Figure 6: Laser diode with 0.5 mW

output at wavelength of 650 nm (Class
II). Figure 8: Alignment sheet
Figure 7: Sample box containing
samples labelled 1 to 5
 Experiment Problem – E1

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Figure 9: The setup used in this experiment. This shows how the observation board, the sample under
test, the optical platform with rotary disk and the laser diode should be assembled. Angle 𝝓 is the
rotation angle of the sample. Angle 𝜽 is the incident angle of the laser beam relative to the surface of
the sample.

Additional notes for items in the sample box

 All samples used in this experiment are squarely-shaped with dimensions of 1.5 cm ×
1.5 cm. They are all labelled from 1 to 5 at one side of each sample holder.
 All sample holders have dimensions of 2.0 cm × 2.0 cm with a reference point
marked as a triangle engraved near the center of one edge.
 Before placing each sample on the center of the rotary disk, you should ensure the tip
of the arrow on the rotary disk is pointing to the 0° mark of the angular scale. When
placing a new sample, try to align it with the square marking on the rotary disk of the
platform. The sample position at 𝜙 = 0° (only refers to Samples 2 to 5) is defined
when the reference point is aligned with and facing the 0° mark on the angular scale.

Sample Description of the samples

1 A plane mirror.
2 A steel plate having straight scratch grooves parallel to two edges with non-
uniform spacing.
3 A regular grating with the spacing between adjacent grooves equal to 𝑎, the grating
constant.
4 Similar to Sample 2, but with non-uniformly spaced scratch grooves oriented at an
angle of 𝝓∗ with respect to the line joining the 0° and 180° marks on the angular
scale of the rotary disk.
5 A steel plate that not only has straight scratch grooves with non-uniform spacing,
but also has pre-made grooves with equal adjacent spacing 𝑏. These pre-made
grooves lie perpendicularly to the straight scratch grooves with non-uniform
spacing.
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Safety precautions and general advice
1. Caution: Do not stare into the laser beam directly!!
2. Switch off the laser diode when it is not in use to avoid unnecessary draining of the
battery.
3. Always keep the samples in an upright position. Avoid any contact with the surface of
the samples.
4. Always hold the edges of the sample holder with your hands, when transferring the
samples.

Initial adjustments / procedures

1. Construct the experimental setup as shown in Figure 9. The observation board can be
attached to the rotary disk using the three screws, as shown in Figure 10.

Figure 10: The observation board can be attached to the rotary disk using the three screws.

2. Mount the laser diode on the laser-diode holder. [Note: Do not over-tighten the yellow
screw when securing the laser diode to the holder; otherwise the laser diode may get
damaged.]

3. Connect the battery pack to the laser diode as shown in Figure 11.

Figure 11: Laser diode connected to the battery pack.

 Experiment Problem – E1

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4. Attach a piece of paper to the observation board to observe the light emitted by the laser
diode, then adjust the aperture of the laser diode using a flat-head screw driver to obtain
a beam spot of about 1 mm in diameter.

5. Place the laser diode in the laser-diode holder such that its head is about 5 to 10 mm out
of the holder.

6. Attach the provided pinhole to the laser diode as shown in Figure 12, so that the size of
the beam spot can be further reduced.

Figure 12: Laser diode with the pinhole attached.

7. This experiment requires that the horizontal projection of the laser beam must be aligned
with the 0° and 180° marks on the angular scale and the laser beam must hit on the center
of the sample being studied. These can be achieved by adjusting the location and tilting
angle of the laser-diode holder with the screws as shown in Figure 13.

Figure 13: The location and the tilting angle of the laser-diode holder can be adjusted with the screws
labelled. Screw ‘A’ allows the height of the laser beam to vary. Screw ‘B’ allows the laser beam to be
tilted. Screws ‘C’ and ‘D’ allow the laser-diode holder to be adjusted horizontally.
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Experiment
Part A: Alignment of the setup

Construct the experimental setup based on Figure 9. Note that the distance 𝐷 between the
observation board and the center of the rotary disk has been fixed at 15 cm. Place Sample 1 at
the center of the rotary disk following the procedure as described in the additional notes for
items in the sample box (Page 3). Attach the alignment sheet to the observation board with
the 𝑥-axis matching with the top edge of the angle bar holding the observation board (This
can be done by folding the portion of the paper below the marked x-axis backward). The 𝑥-
axis is defined in this way because the experimental setup is designed in a way that the
surfaces of all the samples sitting on the rotary disk are aligned with the top edge of this angle
bar. In addition, the origin of the 𝑥-axis must be aligned with the 180° mark on the angular
scale. In this experiment, the incident angle 𝜃 of the laser beam is fixed at a certain value.
Adjust the height and tilting angle of the laser diode so that the laser beam hits the center of
Sample 1 and the spot of the reflected laser beam overlaps with the “zero-order spot” marked
on the alignment sheet.

Tasks Mark
Measure the height of the “zero-order spot” ℎ, from the origin of the 𝑥-axis.
Determine the incident angle 𝜃 of the laser beam from 𝐷 and ℎ. Write down the values of
A1 0.6
ℎ in cm and 𝜃 in degrees to three significant figures in the corresponding table in the
answer sheet.

Part B: Diffraction patterns from Sample 2

Replace Sample 1 with Sample 2 with the reference point on the sample holder aligned with
and facing the 0° mark, in which the direction of non-uniformly spaced scratch grooves of
Sample 2 is parallel to the horizontal projection of the laser beam. The angle of rotation 𝜙 of
the sample can be adjusted by turning the rotary disk and read from the angular scale.

Attach a graph paper to the observation board. Ensure that the bottom axis of the graph paper
matches with the top edge of the angle bar holding the observation board (This can be done
by folding the portion of the paper below its bottom axis backward). In addition, the origin of
the 𝑥-axis must be aligned with the 180° mark on the angular scale.

Tasks Mark
° ° ° °
Record the diffraction patterns on the graph paper for 𝜙 = 0 , 30 , 60 and 90 of the
B1 rotary disk and write down the corresponding angle of rotation 𝜙 next to each pattern. 0.8
Write down “# 2” at the top of this graph paper.
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Part C: Diffraction patterns from Sample 3

The experiments in this and the next tasks will help you to understand the physics behind the
diffraction patterns observed in Task (B1). Remove Sample 2 from the rotary disk and
replace it with Sample 3 with the reference point on the sample holder aligned with and
facing the 0° mark. For the starting position of Sample 3 (i.e. 𝜙 = 0° ), the direction of the
grating lines is parallel to the horizontal projection of the laser beam.

Attach a graph paper to the observation board. Ensure that the bottom axis of the graph paper
matches with the top edge of the angle bar holding the observation board as described in Part
(B). In addition, the origin of the 𝑥-axis must be aligned with the 180° mark on the angular
scale.

Tasks Marks
° ° ° °
Mark the centers of the diffraction spots from Sample 3 for 𝜙 = 0 , 30 , 60 and 90 on the
graph paper and write down the corresponding angle of rotation 𝜙 next to each pattern.
C1 0.8
Write down “# 3” at the top of this graph paper.

After completing Tasks (B1) and (C1), you should have realized that the reflected diffraction
patterns from Sample 2 are continuous while those from Sample 3 are discrete. The reason is
that since the spacing between the neighboring straight scratch grooves of Sample 2 is non-
uniform, so it is like a grating with its grating constant covers various values. Thus, its
diffraction patterns follow the general shapes of the diffraction patterns from a regular grating
(like those of Sample 3), however, the diffraction spots become broadened and form the
continuous patterns as observed.

Part D: Theory behind the reflected diffraction patterns from Sample 3

The origin of the reflected diffraction patterns from a regular grating like Sample 3 can be
understood with geometrical optics. In this experiment, we define the 𝑥 -𝑦 -𝑧 coordinate
system with the origin at the center of the sample as shown in Figure 14.
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Figure 14: The definition of the 𝒙-𝒚-𝒛 coordinate system used in this task, in which 𝝓 is the rotation
angle of the sample, with the observation board perpendicular to the z-axis.

Using geometrical optics, one can derive the following equations for the reflected diffraction
patterns from Sample 3,

(𝐷 cos 𝜙 + 𝑥 sin 𝜙)2

𝑦2 = − 𝑥 2 − 𝐷2
cos2 𝜃 cos2 𝜙 (1)

𝐷𝑚𝜆 cos 𝜙
𝑥= , (2)
𝑎 cos 𝜃−𝑚𝜆 sin 𝜙

where λ is the wavelength of the incident laser beam and 𝑚 is the order number of diffraction.
One can predict the 𝑥 and 𝑦 co-ordinates of the diffraction spots as a function of 𝜙 using
Equations (1) and (2), which can be demonstrated to be consistent with the observed patterns
recorded in Task (C1).

Based on Equations (1) and (2), the diffraction spots for 𝜙 = 900 should lie along the 𝑦-axis at
𝑥 = 0 with their y-coordinates expressed by the following equation:

𝑎2
𝑦 = 𝐷√ −1 (3)
(𝑎 cos 𝜃 − 𝑚𝜆)2

Tasks Marks
Equation (3) can be rearranged to obtain a quadratic equation for the grating constant 𝑎 of
Sample 3, as
𝐴𝑎2 + 𝐵𝑎 + 𝐶 = 0. (4)
D1 0.9

Derive the expressions for 𝐴, 𝐵 and 𝐶. Enter your results in the corresponding table in the
answer sheet.
 Experiment Problem – E1

Page 9 of 11
By solving this quadratic equation and using the measured 𝑦 values of the diffraction spots
for Sample 3 at 𝜙 = 900 (See Task (C1)), together with the known values of 𝐷, 𝜃 and 𝜆,
determine the grating constant a of Sample 3 in meters to three significant figures for each
D2 diffraction order from the 1st order (𝑚 = 1) up to the 6th order (𝑚 = 6) [Hints: These 1.8
orders correspond to the six spots above the zero-order spot]. Enter your results in the
corresponding table in the answer sheet.

Calculate the mean for the grating constant a in meters to three significant figures and the
D3 standard error of the mean. Enter your results in the corresponding table in the answer 0.8
sheet.

Part E: Determination of the unknown angle 𝝓∗ for Sample 4

Replace Sample 3 with Sample 4 with the reference point on the sample holder aligned with
and facing the 0° mark. In placing Sample 4 onto the rotary disk, the side of its sample holder
close to the reference point should be perpendicular to the laser beam.

Attach a graph paper on the observation board. Ensure that the bottom axis of the graph paper
matches with the top edge of the angle bar holding the observation board (This can be done
by folding the portion of the paper below its bottom axis backward). In addition, the origin of
the x-axis must be aligned with the 180° mark on the angular scale.

Tasks Marks
Along the continuous diffracted curve of Sample 4 projected on the graph paper, measure
the 𝑦-coordinates in cm for ten points starting from 𝑥 = - 1.0 cm to 3.5 cm with a step of
E1 0.6
0.5 cm. Enter your results in the corresponding table in the answer sheet.

Based on Eq. (1) given in Task (D), construct a linear equation in the form of

𝑀(𝑦, 𝑥, 𝐷, 𝜃) = 𝐼(𝐷) + 𝑆(𝜙 ∗ )𝑥 . (5)

E2 Determine the functional forms for 𝑀(𝑦, 𝑥, 𝐷, 𝜃), 𝐼(𝐷) and 𝑆(𝜙 ∗ ). Plot 𝑀 against 𝑥, using 1.6
the data recorded in E1. Determine the unknown angle 𝜙 ∗ in degrees from this graph.
Write down all the functional forms and the value of 𝜙 ∗ in the corresponding table in the
answer sheet.

Part F: Diffraction patterns from Sample 5

Replace Sample 4 with Sample 5 with the reference point on the sample holder aligned with
and facing the 0° mark. The geometry of Sample 5 is shown in Figure 15. For the starting
position of Sample 5, the direction of non-uniformly spaced scratch grooves is parallel to the
horizontal projection of the laser beam.
 Experiment Problem – E1

Page 10 of 11

Attach a graph paper to the observation board. Ensure that the bottom axis of the graph paper
matches with the top edge of the angle bar holding the observation board (This can be done
by folding the portion of the paper below its bottom axis backward). In addition, the origin of
the 𝑥-axis must align with the 180° mark on the angular scale.

Figure 15: The structure of Sample 5, where the scratch grooves are non-uniformly spaced and the pre-
made grooves are uniformly spaced. The adjacent spacing between the pre-made grooves is 𝒃.

Tasks Marks
° ° ° °
Record the diffraction patterns you observed for 𝜙 = 0 , 30 , 60 and 90 on separate
graph papers for each value of 𝜙. At the top of each graph paper, put down ‘#5’ and the
corresponding 𝜙 value. It is expected that you could observe more than 10 diffraction
F1 0.8
orders. However, you are required to record only three relatively brighter orders on each
graph paper.

The sample structure of Sample 5 can be considered as a combination of Sample 2 and

Sample 3, where the uniformly-spaced pre-made grooves are placed perpendicularly to the
scratch grooves with non-uniform spacing. This is an optical analogy of the nano-grating as
described in the introduction.

Tasks Marks
With this understanding, estimate the spacing 𝑏 in meters of the uniformly spaced pre-
made grooves of Sample 5 using the recorded diffraction pattern for 𝜙 = 0° from Task
(F1). Enter the value of 𝑏 in the answer sheet.
F2 1.6
[Note that in estimating the value of 𝑏, you are only required to take the measured data of
the first diffraction order and the estimated 𝑏 should be rounded up to three significant
figures.]
 Experiment Problem – E1

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Part G: Determination of the lattice-plane spacing for ZnSe

As mentioned in the introduction, the above phenomenon was firstly discovered on the
reflection high energy electron diffraction (RHEED) patterns of a nanostructured ZnSe
surface. Such a surface has a one-dimensional (1D) modulation in nanoscale with non-
uniform spatial separation, as well as having periodic atomic lattice planes perpendicular to
these nano-grooves. Figure 16 shows the RHEED pattern of such a nanostructured ZnSe
surface when the electron-beam is perpendicular to the nano-grooves with non-uniform
spacing (Hint: With respect to the periodic atomic lattice planes, such a condition
corresponds to 𝜙 = 0° )

Note that the spacing of the streaks in this figure, which can be measured using the given
scale below the RHEED pattern, matches exactly with the actual diffraction pattern projected
on the fluorescence screen (an analogy to the observation board used in this experiment).

cm

Figure 16: The reflection high energy electron diffraction (RHEED) pattern from a nano-structured
surface of ZnSe, when the electron-beam is perpendicular to the nano-grooves with non-uniform
spacing.

In taking the diffraction pattern shown in Figure 16, the accelerating voltage of the electron
gun was set to be 𝑉 = 13,000 volts. The corresponding wavelength 𝜆 of the high-energy
electrons incident to the center of the sample surface can be calculated by

12.247×10−10
λ= [m], (6)
√𝑉(1+10−6 𝑉)

where the relativistic effect has been taken into account.

The incident angle of the electron beam to the surface of the ZnSe sample is 𝜃 ≈ 0° and the
distance between the fluorescence screen and the incident spot of the electrons on the nano-
structured ZnSe surface is 𝐷 = 26 cm.

Tasks Marks
For the ZnSe sample, based on Figure 16 and the experimental conditions given above,
determine the lattice-plane spacing 𝑎∗ of the periodic atomic lattice planes that are
G1 1.7
perpendicular to the nano-grooves with non-uniform spacing, in meters to three significant
figures. Enter your result in the corresponding table in the answer sheet.

END
 Experiment Problem – E2

Page 1 of 10

Reflection Phase Shift of Metal (Total marks: 8.0)

Introduction
It is known that natural materials have refractive indexes (n) larger than that of vacuum (n >
no = 1). However, meta-materials (artificial materials with nano-scale structures) can exhibit
exotic phenomena, such as negative refraction which can be explained by considering
negative effective refractive index. In general, the refractive index of a material, for instance
metal, can also be a complex number when there is light absorption.

A simple method to study complex refractive index is by measuring the phase shift  when a
light beam is being reflected from the surface of a material. In optics, the phase shift due to
reflection from a surface is important for many applications especially those in holographic
measurements.

For ordinary glass, the reflection phase shift for normal incidence is 180o (or  in radian).
However,  can take different values for metals, depending on the absorptions of the metals.
Phase measurement in optics is challenging and high precision is required. Many
sophisticated techniques or methods have been developed for optical phase measurements.
However, they are too complicated to be implemented for high school students. Here, we
introduce a simple and inexpensive approach.

Objective
To study the phase shift of light reflected from Titanium (Ti) surface using Fabry-Perot laser
interferometry.

Figure 1: Apparatus for this experiment

 Experiment Problem – E2

Page 2 of 10

List of apparatus
[1] Laser diode
[2] Battery pack with batteries for the laser diode
[3] White-color-capped screws for sample holder
[4] Sample holder with two paper clips
[5] Wires with banana connectors
[6] Digital multi-meter
[7] Sample box with one Titanium (Ti)-coated Fabry-Perot etalon with specific sample number
[8] Flat-head screw driver
[9] Optical platform with rotary disk, turntable detector holder, fixed angular scale, and laser diode holder
[10] LED lamp

Description of apparatus

Figure 2: Laser diode with 0.5 mW output

at wavelength of 650 nm (Class II) Figure 3: Battery pack with 2×1.5V AA batteries and
outlet plug for the laser diode

Figure 4: Sample holder with two paper

clips
Figure 5: Digital multi-meter and wires with banana connectors
 Experiment Problem – E2

Page 3 of 10

Figure 7: Flat-head screw driver for adjusting the beam size of

the laser diode

Figure 6: Sample box and one Ti-coated

Fabry-Perot (FP) etalon (for illustration
purpose, the sample number in this figure
is #250)

Figure 8: Optical platform consisting of a rotary disk, a turntable

detector holder, a fixed angular scale, and a laser diode holder
Figure 9: LED lamp for reading and
writing
 Experiment Problem – E2

Page 4 of 10
Theory
Consider an ideal air-gap Fabry-Perot (FP) etalon as shown in Figure 10. The etalon consists
of a top thick glass plate (with refractive index ng) and a bottom sample plate (with refractive
index ns) sandwiching a thin air-gap (L ~ 5 micron) in-between.

Figure 10: Light reflections from an ideal air-gap Fabry-Perot etalon consisting of a top glass plate
and a bottom sample plate.

We use a two-beam interference approximation to model the air-gap etalon. A light beam
(beam 0) incident on the top glass plate, neglecting the reflection from the air-glass interface
of the top glass, is partially reflected (beam 1) and partially transmitted (refracted) at the
glass-air interface. The transmitted (refracted) beam then gets reflected at the air-sample
interface and then transmitted (refracted) through the top glass plate, which is labeled as
beam 2 shown in Figure 10. For beam 2, the reflection at the air-sample interface picks up an
additional phase shift s while there is no phase shift at other interfaces. The resulting
intensity of reflected light  for incident angle  is the superposition of beams 1 and 2
given by:

I ( )  I1  I 2  2 I1I 2 cos(2kL cos   s ) , (1)

2𝜋
where I1 and I2 are the intensities of beams 1 and 2, respectively, 𝑘 = is the wavenumber
𝜆

and λ is the wavelength of the incident light. Equation (1) exhibits intensity peaks and
troughs as a function of incident angle θ for fixed wavelength λ and air-gap spacing L. In this
experiment, we fix the polarization direction of the incident light and neglect polarization
effects when the beam is reflected/refracted at the interfaces. Moreover, we use a Titanium
(Ti)-coated glass plate as the bottom sample plate of the FP etalon.
 Experiment Problem – E2

Page 5 of 10

Preparatory procedures/adjustments
1. If you have chosen to do experiment E1 first, remember to remove the observation
board and also the pin hole for E1 before assembling the apparatus for this
experiment E2.
2. CAUTION: DO NOT LOOK DIRECTLY AT THE LASER LIGHT OF THE
LASER DIODE!!

3. Install the laser diode into the circular hole of the laser holder as shown in Figure 11.
Fasten the laser diode by using the yellow-color-capped screw. Make sure the body of
the laser head is horizontal with the attached yellow label (see Figure 2 the laser diode
with the yellow label) facing up (for setting the polarization of the laser light in the
vertical direction). [Note: Do not over-tighten the (yellow-color-capped) screw of the
laser holder; otherwise it will damage the laser diode!]

Figure 11: Laser diode mounted inside the laser holder

4. Connect the battery pack to the laser diode as shown in Figure 11 and switch on the laser
diode.

5. It is desirable to have the laser diode to form a beam spot about 1 mm in diameter at a
distance of 20 cm. To check the beam size, place a piece of paper at about 20 cm from
the laser diode and observe the spot size of the laser beam emitted by the laser diode.
Adjust the aperture of the laser diode using a flat-head screw driver to obtain a spot size
about 1 mm in diameter if necessary.

6. Connect the output terminals of the photo-detector pre-mounted on the detector holder to
the multi-meter CAREFULLY using the electric wires as shown in Figure 12. The
photo-detector is a photodiode connected in parallel with a resistor such that the current
produced by the photodiode when illuminated with light can be recorded by measuring
the voltage across the resistor.
 Experiment Problem – E2

Page 6 of 10

Figure 12: Photo-detector connected to the multi-meter

7. Adjust the position of the laser diode such that the laser beam shoots horizontally over
the 0o mark of the angular scale, crossing over the center of the rotary disk, and hits
directly onto the detector positioned at the 180o mark of the angular scale (as shown in
Figure 13). You can turn on the multi-meter to monitor the intensity (in terms of
voltage) of the laser beam to obtain an optimal condition. [Note: Remember to turn off
the laser diode when you are done with this step!]

Figure 13: Alignment of the laser beam

8. Mount the sample holder to the rotary disk on the optical platform using the white-color-
capped screws as shown in Figure 14.
 Experiment Problem – E2

Page 7 of 10

Figure 14: Sample holder mounted on the rotary disk

9. Then mount the Ti-coated etalon with the sample number (here #250) upright to the
sample holder using the two paper clips provided such that the bottom glass plate of the
sample is in contact with the wall surface of the holder as shown in Figure 15.

Figure 15: Sample mounted on the sample holder

10. Turn on the laser diode and adjust the vertical position of the etalon so that the laser
beam shoots at the middle area of the top or the bottom Ti strip of the etalon and gets
reflected to the photo-detector as shown in Figure 15. Then adjust the position of the
photo-detector for maximum signal (in terms of the voltage measured by the multi-
meter). Furthermore, you have to adjust the position of the sample holder using the two
white-color-capped screws such that the Ti strip rotates with the same axis as the rotary
disk, i.e. they are coaxial.
 Experiment Problem – E2

Page 8 of 10

11. Now rotate the sample holder together with the mounted Ti-coated etalon such that the
laser beam makes an incident angle  with respect to the normal of the sample surface as
indicated by the angular scale. Then move the photo-detector to an angle of about 2
and adjust the angular position of the photo-detector for obtaining a maximum signal
from the multi-meter. After completing all the above steps/procedures, the setup is now
ready for the experiment.

Note: All numerical values and answers should have three significant figures.
 Experiment Problem – E2

Page 9 of 10
The Experiment
Tasks Description Marks
Measure the reflection intensity of the Ti-coated etalon in terms of voltage as a function of
incident angle at fine enough angular steps, starting at ~ 4º such that as many intensity
1 1.0
peaks as possible are recorded. Enter your data in Table E2_1.

In order to reduce errors resulted from the mis-alignment of the laser beam with respect to
the sample holder, and also with respect to the angular scale of the optical platform, carry
out similar measurements for incident angles on the other side of the 0º mark of the angular
2 scale by first turning the sample to face the opposite side of the 0º mark of the angular scale 1.0
and then moving the photo-detector accordingly for measuring intensity of the reflected
laser beam. Enter your data in Table E2_2.

Plot the reflection intensity as a function of incident angle on a graph paper using the
recorded data in Table E2_1 and label the graph as ‘Graph E2_1’. Plot a similar graph for
the data in Table E2_2 on a different graph paper and label the graph as ‘Graph E2_2’.
3 0.9
Show the variation of the reflection intensity by drawing a smooth curve fitting the data
points in Graph E2_1 as well as in Graph E2_2.

Identify the peaks of the reflection intensity and label them with sequential peak numbers
4 in Graphs E2_1 and E2_2, starting with the peak having the largest incident angle. 0.2

Derive the condition for the reflection intensity peaks in terms of L, , ϕs ,  using Equation
5 (1). Label this condition as Equation (2). 0.3

Choose a suitable function of  X(), as a new independent variable such that the reflection
intensity peaks in Graphs E2_1 and E2_2 will be evenly spaced in a graph of reflection
6 intensity vs X(). Then calculate the corresponding values of X() and insert them, as a 0.4
new column, to Tables E2_1 and E2_2.

Construct Table E2_3 consisting of the peak number and the corresponding incident angle
for the data in Tables E2_1 and E2_2. You may label them with the notations of LHS (left-
hand side) and RHS (right-hand side), respectively, i.e. 𝜃LHS and 𝜃RHS . Then, for each
matched reflection intensity peak, calculate the average values of 𝜃LHS and 𝜃RHS , and label
7 0.6
the average as |θ|avg and X(|avg) and enter the results as new columns in Table E2_3.
[Note: The peak numbers in Tables E2_1 and E2_2 can be different. Make sure you match
the corresponding peaks obtained from Graphs E2_1 and E2_2.]

Plot a graph on a separate graph paper to show the relationship between the peak number
8 vs. X(|avg). Label this graph as ‘Graph E2_3’. 0.6
 Experiment Problem – E2

Page 10 of 10
Draw a straight line through all the data points in Graph E2_3 and then determine the slope
and y-intercept of this line. [Note: A graphical approach will also be accepted and error
9 0.4
analysis is not required.]

Rewrite Equation (2) derived in Task 5 to give an expression of the interference order m
and the function X(|avg) obtained in Task 6. Label this equation as Equation (3a). From
Equation (3a), write down the expression of m in terms of L,, and |avg as Equation (3b).
10 Hence, express the normalized reflection phase 𝜙s,n = 𝜙s /2𝜋 in terms of L,, and |avg 1.2
as Equation (3c). Write down the range of 𝜙𝑠,𝑛 . Then determine the values of m for the
peaks from the results obtained in Task 7 and insert them as a new column in Table E2_3.

Plot m vs. X(|avg) on Graph E2_3, i.e. the same graph for the plot of peak number vs.
X(|avg). Draw a line through all data points for this plot. Hence determine the air-gap
11 spacing L of the etalon and also the reflection phase 𝜙s of the Ti. [Note: Again, graphical 1.4
solution is acceptable and error analysis is not required.]

END