Undergraduate Physics - Investigation of the production and emission characteristics of x-rays.

1. Sultan LeMarc
Theoretical Physics
24/02/2012
Production and Emission of X-Rays
Introduction
The aim of this experiment is to investigate the production and emission
characteristics of x-rays. This will be achieved by measuring the count rate of x-rays
reflected off a crystal at varying angles of incidencein order to find the characteristic
peaks and λ0 of a copper target using the principles of Bragg’s law. In addition, the
experiment aims to examine absorption of homogeneous x-rays through the
relationship between intensity and count-rate. There will be particular consideration of
energy loss through scattering on a microscopic scale and observing the characteristics
of intensity as a function of wavelength.
X-rays are a form of ionising electromagnetic radiation, which by definition consist of
waves that have electric and magnetic fields vibrating transversely and sinusoidally to
each other. In the electromagnetic spectrum, x-rays are found in the short wavelength
and high energy end, between ultraviolet light and high energy gamma rays, with
wavelengths ranging from 0.01nm to 10nm. This range of wavelengths of x-rays is of
the order of distances between molecules and crystal lattices, making them ideal for
spectroscopic techniques for characterisation of the elemental composition of
materials. The corresponding energies of this range are 120 eV to 120 keV
respectively. As with all electromagnetic radiation, they travel at the speed of light.
X-rays can be classified into two types according to their penetrating abilities, which
relate to their wavelengths and energies although the distinction between the two is
not conclusively defined. The highest energy x-rays are called hard x-rays and
typically have energies greater than 10keV. X-rays with energies lower than this are
called soft x-rays, and are primarily used for determining the electronic structures of
materials. In contrast, as hard x-rays are more penetrative they require denser
materials to be detected and are used in medical and dental diagnostics. The hard xrays may have energies in the same range as low energy gamma rays, and
thedistinction between the two is derived from the source of radiation rather than the
wavelength.
Figure 1: Part of electromagnetic spectrum showing x-rays with corresponding wavelength
and photon energy.

2. There are two commonly used methods to produce x-rays, by an x-ray tube which is a
vacuum tube that linearly accelerates charged particles, or by a synchrotron which is a
cyclic particle accelerator that applies electric and magnetic fields.Most natural
sources of x-rays are extra-terrestrial such as the Sun and black holes but they are also
emitted by the decay of unstable nuclei on Earth.
An x-ray tube functions as a typical vacuum tube which uses the potential difference
between a cathode and an anode to accelerate charged particles. In an x-ray tube a
metal filament is the cathode and is heated by a low voltage current in order to emit
electrons in a process called thermionic emission. As a stream of electrons are
released into the vacuum, a large electric potential is applied between the cathode and
the anode; a metal target. This accelerates the electrons towards the anode due to
attraction. The accelerating voltage between the cathode and the anode affects the
speed at which the electrons travel and strike the target. The higher it is, the more the
energy the electrons have on collision, resulting to x-rays photons with higher
energies.
There are two principle mechanisms by which x-rays are produced, by bremsstrahlung
or K-shell emission. In the first, the emission of x-rays takes place when high-energy
electrons (or charged particles) are decelerated, in speed or direction, by bombarding
targets. In accordance with Maxwell’s equations, the electrons emit electromagnetic
radiation upon deceleration, a process called bremsstrahlung. This process is
fundamentally based on the deceleration of a charged particle by the deflection
offanother charged particle, typically an electron by an atomic nucleus.
These interactions are inelastic collisions with the electric fields of nuclei and involve
either the loss of all ofthe electron’s kinetic energy at once, or the loss of infinitesimal
amounts ofenergy. As a result of such interactions, due
to conservation of energy, a photon is emitted with the
same energyas the total lost by the incident electron.
The amount of kinetic energy lost by an electron in any
given interaction with the target can vary from zero up
to the total kinetic energy of the electron. Therefore it
can take several interactions with the atoms of the target
before the electrons lose all of their energy. For this
reason the wavelength of radiation from these
interactions lies in a continuous range from a minimum
Figure 2: Bremsstrahlung –
value up to infinity, thus a continuous spectrum of xemission of x-rays by electron
rays are emitted as the photons have a wide distribution
deflection.
of energy. Typically, less than 1% of the energy
supplied is converted into x-radiation during this process; the rest is converted into the
internal energy (heat) of the target.
In the second mechanism, x-rays are produced by the excitation and ionisation of the
atoms in the target. In this process transitions of electrons between atomic orbit shells
take place as the bombardment of electrons can excite and eject inner electrons from
the target atoms, provided the incident electron has sufficient energy.

3. This leaves a vacant space in the inner shell and is filled by an electron at the higher
level outer shell. In the course of this transition, the higher level electron is losing
kinetic energy as it fills the vacant inner shell.
Under the principles of conservation of energy, the transition is accompanied by the
emission of an x-ray photon with unique energy corresponding to the energy
difference between the two shells i.e. the energy lost by the shifting electron. These xrays are called characteristic x-rays with wavelengths that are distinctive for each
particular element and transition.
The innermost shell from which the incident electron has
dislodged an atomic electron is called the K shell. When the
vacancy is filled by an electron from the next higher shell, the L
shell, the photon emitted has an energy corresponding to the Kα
characteristic x-ray line on the emission spectra. When the
vacancy in the K shell is filled by an electron from the M shell,
the next higher shell after the L shell, the Kβcharacteristic x-ray
line is produced on the emission spectra.This mechanism only
works when the energy of the incident electron is above a
certain threshold value, called the excitation potential, and is Figure 3: Transitions
unique for each material. One of the most common elements from higher to lower
energy levels.
used to create x-rays in this way is copper.
The bremsstrahlung spectrum is continuous and the minimum wavelength (maximum
photon energy) is determined by the accelerating voltage.
By conservation of energy, the photon energy cannot
exceed the pre-impact kinetic energy of the electrons. The
characteristic photon energies also cannot exceed the
difference between the state energies of the K shell and the
valence shell.
Figure 4: The emission
spectra of a heavy metal
x-ray source.
Furthermore, the current of an electron beam in anx-ray
tube determines the overall intensity of the spectrum. The
accelerating voltage of the tube determines the shape and
end point of the continuous spectrum. The composition of
the target determines the wavelengths present in the line
spectrum. The combined spectrum has the characteristic
peaks superimposed onthe bremsstrahlung continuum base.
The intensity of the characteristic photons does not depend
on the energy of the incident electrons. The exact shape of
the spectrum of the emitted x-rays depends onthe energy of
the incident electrons, material and thickness of the target.

4. Figure 5: Comparison of x-ray emissions spectra produced by electrons with low energy
(red), medium energy (green), and high energy (blue). As the energy of the electron beam
increases, the critical wavelength of the x-rays decreases but the location of the characteristic
peaks does not.
The work done on each electron when it is accelerated onto the anode is equivalent to
eV, where e is the charge of the electron (literature value e = 1.602 × 10-19C) and V is
the accelerating voltage. The energy E of an x-ray photon is given by:
Here, h is Planck’s constant with a literature value of 6.63 × 10-34J·s and f is the
frequency of the photon.
As with all electromagnetic waves, the wavelength λ of x-rays is related to the
frequency f and phase velocity v by:
The literature value for v of an x-ray photon in a vacuum is taken as c = 3.0 × 108 m/s.
As a result, the energy of an x-ray photon and its wavelength are related by:
X-ray photons lose energy when they are absorbed by interactions with individual
atoms. When x-rays pass through matter they lose energy both by scattering and by
true absorption, the net effect is termed the total absorption. The four main processes
by which x-ray photons interact with matter are photoelectric absorptionwhich occurs
at low energies, Compton (incoherent) scattering which occurs at the intermediate
energy range, pair production which occurs at high energies and Thomson scattering.
Forx-ray photons the atomic photoelectric effect and Compton scattering account for
most of the absorption.
Photoelectric absorption occurs when the incident high energy x-ray photon is
absorbed by an atom and resulting in the ejection of an electron from the outer shell
i.e. ionisation of the atom. As a result, when the ionised atom returns to the neutral
state it emits a secondary x-ray which is characteristic of that particular atom. This is
also known as x-ray fluorescence.

5. Compton scattering occurs when the incident x-ray is deflected from its initial path by
interacting with an electron. As a result of this interaction, the electron gains energy
and is ejected from its orbital shell, whilst the x-ray photon loses energy and therefore
has a longer wavelength than it did prior to the interaction. This process is also called
incoherent scattering because the change in energies is not always consistent. The
change in energy is determined by the angle of scattering rather than the target
material.
When an incident homogeneous x-ray beam of intensity I0 traverses a layer of
thickness dt the intensity decreases according to the relationship:
where µ is the linear absorption coefficient of the target material. This is a
characteristic value that is unique for each material. This coefficient depends on the
density of the medium but this dependence can be removed by using the mass
absorption coefficientµ/ρ.As a result of integrating the mathematical relationship of
equation (4), intensity after the absorption is given by:
The absorption of x-rays is dependent on the target material, the thickness of the
material and energy (wavelength) of the x-rays.
However different energy x-rays are absorbed differently as materials have
characteristic absorption edges, which are points that arise when the incident photon
energy is insufficient to eject electrons and is less than the binding energy of
electrons. These points show on the graph in figure 6 as a sharp discontinuity in the
absorption spectrum. As a result of this, absorption decreases and thus the absorption
edges correspond to the electron levels in the
target atom (K edge, L edge). This means that
energies above the absorption edge are absorbed
whilst at energies below it x-rays are transmitted.
The absorption never reaches zero as there will
always be photoelectric emission taking place at
some level.
Figure 6: Absorption edges of an
absorption spectrum.
As the absorption changes near an absorption edge, it means that the absorption
coefficient depends on the wavelength which has a maximum and a minimum. As the
location of the absorption edge changes with each unique material, the absorption of
x-rays depends on atomic mass of the material due to the different energy levels
between electrons in different atoms.
Figure 7: Dependence of the absorption coefficient on the
incident energy, showing absorption edges of copper (Cu).

6. In addition to being absorbed, x-rays are also diffracted when they interact with
crystals which have lattice panes. Crystal structures are an appropriate choice for a
grating as x-rays can only be diffracted by spacing that is of same order as their
wavelength. As crystal structures have a fixed three-dimensional pattern of atom
arrangements in a periodic and symmetrical lattice, they can be used as a grating to
determine x-ray wavelengths.
There are two types of x-ray scattering, elastic and inelastic. As mentioned earlier,
during inelastic (Compton) scattering the x-rays lose energy and the resultant
wavelength is longer than the incident x-ray. In contrast, during elastic (Thomson)
scattering the x-ray photons do not lose energy and only momentum is transferred in
the process.
Diffracted waves from different atoms within the lattice can interfere with each other,
leading to a modulated resultant intensity distribution. Interference is a phenomenon
that occurs when two waves meet and superimpose to give one resultant wave. There
are two types of interference, constructive and destructive.
Constructive interference takes place when two identical phases meet and
superimposeinto a wave with combined amplitude. Destructive interference takes
place when two waves are completely out of phase with each other, resulting to the
cancelation of each other. As the atoms are in a period arrangement, the diffracted
waves interfere to produce diffraction patterns that reflect the symmetry of the
distribution of the crystal lattice.
The diffraction patterns of x-rays show that at most of the
incident angles the incident x-rays are scatteredcoherently
to interfere destructively as the combining waves are out
of phase and thus have no resultant energy. However at
some
key
incident
angles
the
x-rays
are
scatteredcoherentlyto interfere constructively, resulting in
well-defined x-ray beams leaving the crystal in various
directions. As a result, a diffracted beam can be
considered as a beam composed of a large number of
scattered rays mutually reinforcing each other.
Figure 7: X-ray diffraction
pattern of a crystal structure.
The condition to be satisfied for there to be constructive interference in x-ray
diffraction is mathematically defined by Bragg’s law, which relates the incident
wavelength λ, incident angle θ, and the spacing between the planes in the atomic
lattice d:
Bragg’s law states that intense peaks of diffracted radiation (Bragg peaks) are
produced by constructive interference of scattered waves at specific wavelengths and
incident angles.

7. Mathematically it means constructive interference of the diffracted waves occurs
when 2dsinθ is equal to an integer multiple of incident wavelengths nλ. Through the
course of this investigation, n is assumed to be 1. The scattering angle 2θ is defined as
the angle between the incident and scattered rays.
It is important to note that Bragg's Law applies to scattering centres consisting of any
periodic distribution of electron density.In this case, a crystal is used as the scattering
centre to separate the different wavelengths of incident x-rays and scattering them to
specific angles.
The literature value of 2d for the LiF crystal is 0.403 nm and the Kβ and Kα
wavelengths for copper are 0.139 nm and 0.154 nm respectively.
Method
This experiment has two tasks; the first task will investigate the characteristic peaks
and λ0of a copper (Cu) target using a lithium fluoride (LiF) crystal. In order to do this,
readings for the count-rate of x-rays over a range of angles 2θ will be taken using a
Tel-X-Ometer and a Geiger counter.
A Tel-X-Ometer is an x-ray diffraction system device which is used to detect the
absorption and reflection of x-rays i.e. a spectrometer. The device employs the
fundamental principle of an x-ray tube to produce x-rays and accelerate them towards
the target, which is mounted as the anode. The Tel-X-Ometer being used in this
investigate has two settings for the accelerating voltage, also known as extra high
voltage (EHT), at 20kV and 30kV. The filament current can also be adjusted, and
should not be allowed to exceed 80µA.
The device allows crystal to be mounted in the centre of the device to reflect the xrays for detection. It also allows numerous collimator slides to be mounted at different
numbered positions on a carriage arm assembly.A Geiger counter can be mounted at
the end of the carriage arm, which can then be moved around the device from a fixed
pivot point at the centre, allowing it to be placed at various scattering angles. In
accordance with Bragg’s law, the mounted crystal in the centre rotates to an incident
angle θ when the arm rotates at a (scattering) angle 2θ. This 1:2 ratio of the angular
displacement is maintained by gears at the central pivot point. The base of the device
has the degree scale of angles, like a large protractor,
covering the entire 360ocircular base. The carriage arm’s
minimum setting is at 12o to a maximum of 124o at each
side. Due to the limited sensitivity of this scale, with the
smallest interval being 1o, the carriage arm also has a
miniature scale at its tip that can be operated and finely
calibrated accordingly by a thumb-wheel. This has a higher
sensitivity, with the smallest interval being 0.2o.
The entire system is enclosed by a transparent plastic
scatter shield which is fitted with an aluminium and lead
back-stop directly aligned with the x-ray source.
Figure 8: Tel-X-Ometer.

8. The plastic contains a higher percentage of chlorine to absorb radiation. For health and
safety purposes, the shield must be locked and centralised into place in order to turn
on the voltage and produce x-rays. The device comes with a timer which can be
adjusted to automatically stop the emission of x-rays after a specific period of time.
A Geiger counter is a particle detector that measured ionising radiation. It is able to
detect radiation by the ionisation produced in a low-pressure inert gas in a GeigerMuller tube that contains electrodes which accelerates the electrons released by the
ionisation. Each detected particle produces a pulse of current, and an audible click,
which is translated to give a reading of the intensity as the number of pulses per
second i.e. the count rate.
Figure 9: Schematic diagram of a Geiger-Muller tube.
For the first part, the lithium fluoride crystal is mounted in the centre of the Tel-XOmeter, with the roughened large surface of the crystal directed towards the crystal
post, in the reflecting position. This side of the crystal can be identified as having a
flat matt appearance. Precaution should be taken in avoiding contact with this surface.
A primary beam collimator in the shape of a small circular disk with the code 582/001
is fitted on the glass x-ray tube. The primary collimator is positioned such that it is
targeted after the electron beam has reached a vertical orientation, so this circular disk
should be positioned with its slot vertical. In this case of x-ray photons, it is placed
after the beam has passed through the x-ray target. This experiment also employs two
slides as secondary collimators in order to align the beam, which are mounted
onpositioned slots fixed on the carriage arm. The first is a 3mm slide collimator with
code 562/016, which is mounted at position 13, and the second is a 1mm slide with
code 562/-15 at position 18. The Geiger counter is mounted at position 27 on the
carriage arm. The investigation takes readings for both x-ray tube voltages, at 20 kV
and 30 kV.
Figure 10: Diagram of Tel-X-Ometer setup.

9. Once the apparatus is set up in the specified way, the Tel-X-Ometer can be switched
on and measurements of the count-rate and incident angle can be made. Before the
readings of the x-ray are recorded, the background count over 30 seconds should be
recorded to incorporate into the data by subtracting it from the measured values. The
readings will begin with the angle 2θ at its minimum of 12oand taken over the whole
range of angles possible for each voltage.
The accuracy of the Geiger counter will be determined by the total number of counts
taken, as the instrumental error becomes smaller relative to the reading as the counts
become larger. Therefore, the average value of the count-rate over a 30 second period
is recorded. As the readings are recorded, a plot of count-rate against θ should be
made in order to establish parts of the spectrum where the count-rate increases most
and therefore take further readings within that range.
Using the literature value for the value of 2d of the LiF crystal, the value of λ
corresponding to each angle can be calculated using equation (6). In addition, plots of
count-rate against λ can be made for each of the two voltages in order to establish the
value of λ0. Furthermore, by assuming an error of the voltage at ±5%, using the value
of λ0deduced from the plots equation (3) can be used to find a value for Planck’s
constant h and its error. This is then verified by comparing it with the literature value
for h.
The final task involves the investigation of absorption by considering the intensity.
This task is based around the assumption that the intensity and count-rate are linearly
related, because the count-rate is directly proportional to number of photons entering
the detector per unit of time. Therefore this assumption can be justified as the intensity
of the radiation is directly proportional to the rate of photons emitted and thus
detected. The higher the intensity the greater the rate of photons detected and therefore
the higher the count-rate.
The setup of the system has to be adjusted for this task; first by mounting an auxiliary
slide carriage directly on the x-ray tube whilst using the primary beam collimator
582/001 to screw it in place. The 3 mm slide collimator 562/016 used in the previous
task should be moved from position 13 on the carriage arm to position 4 on the
auxiliary carriage. A slide collimator of code 562/011should be mounted at position
18 and the Geiger counter at position 26 as before.
In the first stage, with the LiF crystal in place an initial base reading of the count-rate
for angles 2θ in the range of 20o and 50o should be taken and labelled I0. For the
second stage, a copper filter is placed at position 2 in the auxiliary carriage and the
same measurements are recorded over the same angular range, labelled ICu.
This data is then used to calculate the ratio ln|ICu/I0|, which is a quantitative expression
for absorbance, and plot this against the corresponding wavelength λ at each angle
which is established in the same way as before.Applying the natural log to the
intensity ratio linearizes the exponential relationship of equation (5).

10. Results
Angle, 2θ
(degrees)
12.0
15.0
16.0
17.0
18.0
20.0
21.0
22.0
24.0
26.0
28.0
30.0
36.0
38.0
40.0
41.0
42.0
43.0
44.0
45.0
46.0
48.0
48.2
48.4
48.6
48.8
49.0
50.0
51.0
52.0
Error of
angle, ∆2θ
(degrees)
0.1
0.05
0.1
Count per
second
Error of
count-rate
Wavelength, λ
(nm)
Error of
wavelength, ∆ λ
(nm)
0.0
0.0
0.5
1.0
1.5
1.5
2.5
2.5
3.5
3.5
2.5
1.5
1.5
0.5
0.5
1.5
6.5
8.5
8.5
4.5
13.5
38.5
18.5
13.5
8.5
8.5
6.5
1.5
0.5
0.5
0.35
0.35
0.35
0.35
0.35
0.56
0.56
0.56
0.27
0.27
0.56
0.56
0.56
0.35
0.35
0.56
0.56
0.56
0.56
0.56
2.51
10.0
10.0
2.51
2.51
2.51
0.56
0.56
0.35
0.35
0.042
0.053
0.056
0.060
0.063
0.070
0.073
0.077
0.084
0.091
0.097
0.104
0.125
0.131
0.138
0.141
0.144
0.148
0.151
0.154
0.157
0.164
0.165
0.165
0.166
0.166
0.167
0.170
0.173
0.177
0.002
0.003
0.003
0.003
0.003
0.003
0.004
0.004
0.004
0.005
0.005
0.006
0.007
0.008
0.008
0.008
0.009
0.009
0.009
0.009
0.009
0.010
0.010
0.012
0.012
0.012
0.012
0.012
0.012
0.012

11. Table 1: Results for task 1 measurements of count-rate over range of angles at 30kV, with
corresponding wavelength values.Background count measured 1.50 ± 0.25 and was
subtracted from the initial count shown by the Geiger counter.
The results show that there are key angles at which the count-rate changes
significantly. A particular range where this occurs is between 48o and 49o, therefore
the count-rate for smaller intervals between this range is recorded.
The error of the angle ∆2θ is the instrumental error from the scale on the Tel-XOmeter. The error of the count-rate is a combination of the instrumental error of the
Geiger counter and the error propagated in incorporating the background count:
The results show that the wavelength increases as the corresponding angle increases,
in accordance with Bragg’s law. The error of the wavelength is calculated using the
following error propagation technique:
λ0
Figure 11: Emission spectrum of results in Table 1, highlighting the characteristic lines of Kβ
and Kα. Error bars are too small to be represented.
The graph shows that the wavelength of copper’s Kβline is at 0.150± 0.009 nm and
Kαis at 0.164± 0.010 nm. The value of λ0at 30kV is deduced from this datato be 0.042
± 0.002 nm.
Angle, 2θ
± 0.1
(degrees)
Counts per
second
Error of countrate
Wavelength, λ
(nm)
Error of wavelength,
∆ λ (nm)
12.0
18.0
20.0
21.0
0.0
0.5
1.5
1.5
0.35
0.35
0.35
0.35
0.042
0.063
0.070
0.073
0.002
0.003
0.003
0.004

12. 22.0
24.0
26.0
28.0
30.0
32.0
34.0
36.0
38.0
40.0
41.0
42.0
43.0
44.0
45.0
46.0
48.0
50.0
51.0
52.0
1.5
2.0
2.0
2.0
1.5
1.5
2.5
3.5
1.5
3.5
4.5
6.5
13.5
18.5
8.5
2.5
1.5
1.5
0.5
0.5
0.35
0.35
0.35
0.56
0.35
0.35
0.35
0.35
0.35
0.56
0.56
0.56
0.56
2.51
0.56
0.35
0.35
0.35
0.35
0.35
0.077
0.084
0.091
0.097
0.104
0.111
0.118
0.125
0.131
0.138
0.141
0.144
0.148
0.151
0.154
0.157
0.164
0.170
0.173
0.177
0.004
0.004
0.005
0.005
0.005
0.006
0.006
0.006
0.007
0.007
0.008
0.009
0.009
0.009
0.009
0.009
0.010
0.010
0.010
0.011
Table 2: Results for task 1 measurements of count-rate over range of angles at 20kV, with
corresponding wavelength values. Background count measured 1.50 ± 0.25. Errors are
calculated in same way as in Table 1.
λ0
Figure 12: Emission spectrum of results in Table 2, highlighting the characteristic lines of Kβ
and Kα. Error bars are too small to be represented.
The graph shows that the wavelength of copper’s Kβline to be at 0.125± 0.006 nm and
the Kαline at 0.151± 0.009 nm. The value of λ0at 20kV is deduced from this data to
also be 0.042 ± 0.002 nm, matching the same value of λ0from the previous result.

13. Figure 13: Comparison of both emission spectra.
Using the average of these two λ0 values, along with the literature values of e and c,
whilst assuming an error of the voltage as ±5% (30 ± 1.5 kV), the value of Planck’s
constant h is calculated to be (6.40 ± 0.44) × 10-34J·s.
The error on this value for h is calculated using the following principles of error
propagation:
Angle, 2θ
± 0.1 (degrees)
Absorbance,
ln |ICu/I0| (arb.)
Error of
absorbance,
∆ln |ICu/I0|
Wavelength, λ
(nm)
Error of
wavelength, ∆λ
(nm)
20.0
22.0
24.0
26.0
28.0
30.0
32.0
34.0
36.0
38.0
40.0
42.0
44.0
46.0
48.0
50.0
0.00
0.51
0.00
0.34
0.51
0.85
0.34
0.00
0.34
0.00
0.34
0.00
0.00
0.00
0.00
0.27
0.04
0.04
0.04
0.03
0.04
0.07
0.03
0.03
0.03
0.03
0.03
0.03
0.07
0.07
0.07
0.07
0.070
0.077
0.084
0.091
0.097
0.104
0.111
0.118
0.125
0.131
0.138
0.144
0.151
0.152
0.152
0.153
0.003
0.004
0.004
0.005
0.005
0.005
0.006
0.006
0.006
0.007
0.007
0.009
0.009
0.009
0.009
0.009
Table 3: Results for task 2 investigation of absorption using copper filter.
The error of absorbance ∆ln |ICu/I0| is calculated by:

14. Figure 14: Absorption spectrum of x-ray using copper filter, with the absorption edges
indicated by dotted red line. Error bars are too small to be represented.
There are two prominent absorption edges on the spectrum of Figure 14, one at 0.116 nm and
the other at 0.160 nm.
Analysis
The largest source of error in the investigation was the Geiger counter and its limited
sensitivity. In all tasks the measurement of the count-rate lacked the precision due to
the counter’s high relative instrumental errors. The scale of the counter had a
sensitivity that was insufficient to detect the x-rays because it was restricted by large
intervals. As a result, the count-rate could only be measured by rounding to the nearest
interval which ultimately made it difficult to probe into the ranges where there are
sharp changes. This is reflected by the high relative error of the count-rate for all three
tasks, with some percentage errors reaching as high as 13.6%. This quantity was
further affected by the error propagated through incorporating the background count.
In addition, the Geiger counter had a short time-constant which meant that the
readings were highly fluctuant over a wide range of readings, making it increasingly
hard to obtain an average reading. Furthermore, the reading is taken on an analogue
scale with a pointer, which introduces the random parallax error. Given the integral
role of this quantity throughout the investigation such as in the plots to deduce λ0and
using it as the intensity for the absorption spectrum, this error is propagated
throughout the entire data.As a result of this the integrity of the data is compromised

15. and whilst it does reflect the general principles of x-ray emission and absorption, the
results do not fully represent the true x-ray characteristics.
An improvement of this particular source of error can be made by opting to use a high
sensitivity electronic counter. The modern counters with a long and adjustable timeconstant allow readings to be taken more easily as the fluctuations are slight, although
this would mean it responds slower to change in intensity. An additional advantage of
this is that it provides digital readings on a display or read-outs on computers and
data-loggers, thus eliminating parallax error.
The results of the first task for the investigation x-ray emission successfully show the
characteristics of a typical x-ray emission spectrum. At both 20 kV and 30 kV the data
confirmed the principles of Bragg’s law in how the wavelength is related to the
incident angle and the dimensions of the crystal’s lattice structure. The results also
confirm how different wavelengths are scattered at different angles, as each particular
angle had a unique corresponding wavelength.
Both emission spectra of Figures 11 and 12 show the characteristic features of a
typical x-ray spectrum. The spectra have a broad bremsstrahlung continuum within the
shorter limits of the wavelength axis, where electrons are decelerating and emitting
radiation through the bremsstrahlung mechanism. The spectra also have the two
characteristic Kβ and Kα lines of appreciable intensity, where x-rays are produced by
the excitation and ionisation of the atoms in the target and transitions of electrons
between atomic shells.
The spectrum for 30 kV shows Kβ at 0.150 nm and Kα at 0.164 nm, whilst the
spectrum for 20 kV shows Kβ at 0.125 nm and Kα at 0.151 nm. Whilst these values are
within 6% of the literature values of Kβ and Kα wavelengths for copper, they are not
equal for the two different voltages, contrary to the theory hypothesised. These
differences can be seen in Figure 13 which compares the two spectra on one axis,
showing the characteristic peaks are at different locations.
The discrepancies between the two do not support the principle of the characteristic
wavelengths remaining unchanged at different energies. The conflict with the model
continues as both values of λ0 are identical despite that, in theory, this critical
wavelength should be smaller at higher energies.
However, although these discrepancies exist, this value of λ0used in the calculation of
Planck’s constant h, gave a result of (6.40 ± 0.44) × 10-34J·s. This has a percentage
difference of just 3.5% from the literature value and is within the range of its error,
confirming that the value of λ0 established from the data is sufficiently accurate.
The final task of investigating x-ray absorption of a copper filter produced variable
results. The absorption spectrum in Figure 14 is able to show the distinct sharp
discontinuity in absorption at two points corresponding to two different wavelengths.

16. However due to the limitations of the count-rate sensitivity, as discussed earlier, the
ratio measure of absorbance did not allow the plot to display the typical shape of
absorption spectra although the important features of the vertical absorption edges can
be recognised. The data confirms that at wavelengths longer than the absorption edge
(just above the edge), the absorption of the X-rays is considerably less than for
wavelengths shorter than the absorption edge (below the edge).The spectrum shows
the two absorption edges at 0.116 nm and 0.160 nm. As the K peaks are previously
established to be at 0.150 nm and 0.164 nm (30 kV result), the second absorption edge
this absorption spectrum is between this range, therefore the K edge can be concluded
to be at 0.160 nm.
Quantity
Literature Value
Experimental Value
Percentage
Relative Error
Percentage
Difference
Error
Planck’s constant
6.63 × 10-34
(6.40 ± 0.44) × 10-34
6.8%
3.5%
h (J·s)
Copper
0.154
5.7%
1.9%
0.157± 0.009
Kαwavelength
(nm)
Copper
0.139
5.8%
0.72%
Kβwavelength
0.138 ± 0.008
(nm)
Table 4: Comparative data analysis of experimental values and their errors.
The analysis shows that the methods for determining all three quantities were
successful in closely matching the literature values with small percentage differences,
with the experimental value of Kβwavelength being closest with a percentage difference of
just 0.72%. The literature values are within the error ranges of all three quantities.
Conclusion
In conclusion, overall the investigation has successfully supported the physics
principles of x-rays. It has met its aims to examine the production of x-rays as well as
their unique emission and absorptioncharacteristics. The experimental data supports
the validity of the methods used in the investigation, particularly the use of a crystal
structure to scatter the x-rays by separating the different wavelengths at specific
angles. The application of Bragg’s law in order to find characteristic peaks of a copper
target and using λ0to establish a value for Planck’s constant h was reliably effective as
all three experimental values closely matched the literature values for these quantities.
The plots of the emission spectra reproduced the typical features of an x-ray spectrum,
clearly showing the bremsstrahlung continuum and the two characteristic Kβ and
Kαpeaks, from which the wavelengths could be accurately deduced. The peak
wavelengths matched the literature values for copper, showing that these spectra peaks
are indeed unique to each element and are determined by the transitions within atomic
structure. However, the spectra did not follow the theorised model in the change of
voltage from 20 kV to30 kV, as the peaks changed location, although still around a
similar range of wavelengths, and the value of λ0did not change when it should have
decreased at higher voltages.

17. Finally, the results for the absorption of x-rays by the copper filter were the least
distinguished as the absorption spectrum produced by the data did not define the
absorption edges as clearly as expected. The discontinuity in absorption was not
plotted as a sharp vertical edge but rather a slightly inclined line. Whilst it was
sufficient to establish the K edge wavelength of copper, it was only made possible
through extrapolation of the knowledge of the wavelengths of copper’s peaks from the
previous task. Given the limitations of the apparatus, particularly the Geiger counter,
the values of the count-rate are severely affected by the low sensitivity which
ultimately caused the absorption ratio ln |ICu/I0|. It resulted to the ratio only having a
narrow selection of discrete values which prevented it from highlighting the gradual
continuous change in absorption beside the edge.
Direct extensions of the tasks in this investigation include using a variety of crystals
such as sodium chloride, as well as using different metal targets to compare the
spectra by analysing their atomic properties through interpreting the transition
energies from the data.Also, the scattering of x-rays could be considered in terms of
the polarization and representing them as vector waves in Fourier transform, in order
to relate the amplitude with the measured intensity. The investigation can add depth to
the way x-rays are emitted by consideration of the other product of the de-excitation,
Auger electrons. Additional extensions can include the experimentation of
wavelength’s inverse relationship with the atomic number of the target through the
principles of Moseley’s law.
Overall, the experiment can be considered a success as whole as it has achieved the
primary objective to probe into the fundamental principles of x-rays through the use of
classically verified experimentation methods, producing data that fits the theory.