Sultan LeMarc 24/02/2012Theoretical Physics

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 incidence in 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 10 keV. 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 x-rays may have energies in the same range as low energy gamma rays, and the distinction 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.

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 off another 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 of the electron’s kinetic energy at once, or the loss of infinitesimal amounts of energy. As a result of such interactions, due to conservation of energy, a photon is emitted with the same energy as 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 value up to infinity, thus a continuous spectrum of x-rays are emitted as the photons have a wide distribution 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.

Figure 2: Bremsstrahlung – emission of x-rays by electron

deflection.

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 x-rays 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 Kβ characteristic 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 unique for each material. One of the most common elements 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.

Furthermore, the current of an electron beam in an x-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 on the 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 on the

energy of the incident electrons, material and thickness of the target.

Figure 3: Transitions from higher to lower

energy levels.

Figure 4: The emission spectra of a heavy metal

x-ray source.

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 -19 C) and V is the accelerating voltage. The energy E of an x-ray photon is given by:

E=hf=eV (1)

Here, h is Planck’s constant with a literature value of 6.63 × 10-34 J·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:

f= vλ(2)

The literature value for v of an x-ray photon in a vacuum is taken as c = 3.0 × 10 8 m/s. As a result, the energy of an x-ray photon and its wavelength are related by:

λ0=hceV

(3)

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 absorption which occurs at low energies, Compton (incoherent) scattering which occurs at the intermediate energy range, pair production which occurs at high energies and Thomson scattering. For x-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.

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:

dII 0

=−μdt(4)

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 coefficient µ/ρ. As a result of integrating the mathematical relationship of equation (4), intensity after the absorption is given by:

I=I 0 e−μt(5)

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.

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 6: Absorption edges of an absorption spectrum.

Figure 7: Dependence of the absorption coefficient on the incident energy, showing absorption edges of copper (Cu).

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 superimpose into 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 scattered coherently 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 scattered coherently to 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.

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:

Figure 7: X-ray diffraction pattern of a crystal structure.

n λ=2dsinθ(6)

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. 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 λ0 of 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 x-rays 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 360o circular 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. 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 Geiger-Muller 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-X-Ometer, 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.

Figure 8: Tel-X-Ometer.

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 on positioned slots fixed on the carriage arm. The first is a 3 mm slide collimator with code 562/016, which is mounted at position 13, and the second is a 1 mm 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.

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 12o and 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 λ0 deduced 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).

Results

Angle, 2θ (degrees)

Error of angle, ∆2θ(degrees)

Count per second

Error of count-rate Wavelength, λ

(nm)

Error of wavelength, ∆ λ

(nm)12.0 0.1 0.0 0.35 0.042 0.00215.0 0.0 0.35 0.053 0.00316.0 0.5 0.35 0.056 0.00317.0 1.0 0.35 0.060 0.00318.0 1.5 0.35 0.063 0.00320.0 1.5 0.56 0.070 0.00321.0 2.5 0.56 0.073 0.00422.0 2.5 0.56 0.077 0.00424.0 3.5 0.27 0.084 0.00426.0 3.5 0.27 0.091 0.00528.0 2.5 0.56 0.097 0.00530.0 1.5 0.56 0.104 0.00636.0 1.5 0.56 0.125 0.00738.0 0.5 0.35 0.131 0.00840.0 0.5 0.35 0.138 0.00841.0 1.5 0.56 0.141 0.00842.0 6.5 0.56 0.144 0.00943.0 8.5 0.56 0.148 0.00944.0 8.5 0.56 0.151 0.00945.0 4.5 0.56 0.154 0.00946.0 13.5 2.51 0.157 0.00948.0 38.5 10.0 0.164 0.01048.2 0.05 18.5 10.0 0.165 0.01048.4 13.5 2.51 0.165 0.01248.6 8.5 2.51 0.166 0.01248.8 8.5 2.51 0.166 0.01249.0 0.1 6.5 0.56 0.167 0.01250.0 1.5 0.56 0.170 0.01251.0 0.5 0.35 0.173 0.01252.0 0.5 0.35 0.177 0.012

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-X-Ometer. 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:

∆ count rate=√ (instrumental error )2+(background count error )2(6)

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:

∆ λ=|2dsinθ|√( ∆2d2d )

2

+(∆ sinθsinθ )2

(7.1)

where∆ sinθ=∆θ (cosθ )(7.2)

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 Kβ line is at 0.150 ± 0.009 nm and Kα is at 0.164 ± 0.010 nm. The value of λ0 at 30kV is deduced from this data to be 0.042 ± 0.002 nm.

Angle, 2θ± 0.1

(degrees)Counts per

secondError of count-

rate Wavelength, λ (nm)

Error of wavelength, ∆ λ (nm)

12.0 0.0 0.35 0.042 0.00218.0 0.5 0.35 0.063 0.00320.0 1.5 0.35 0.070 0.00321.0 1.5 0.35 0.073 0.00422.0 1.5 0.35 0.077 0.00424.0 2.0 0.35 0.084 0.00426.0 2.0 0.35 0.091 0.005

λ0

28.0 2.0 0.56 0.097 0.00530.0 1.5 0.35 0.104 0.00532.0 1.5 0.35 0.111 0.00634.0 2.5 0.35 0.118 0.00636.0 3.5 0.35 0.125 0.00638.0 1.5 0.35 0.131 0.00740.0 3.5 0.56 0.138 0.00741.0 4.5 0.56 0.141 0.00842.0 6.5 0.56 0.144 0.00943.0 13.5 0.56 0.148 0.00944.0 18.5 2.51 0.151 0.00945.0 8.5 0.56 0.154 0.00946.0 2.5 0.35 0.157 0.00948.0 1.5 0.35 0.164 0.01050.0 1.5 0.35 0.170 0.01051.0 0.5 0.35 0.173 0.01052.0 0.5 0.35 0.177 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.

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 Kβ line to be at 0.125 ± 0.006 nm and the Kα line at 0.151 ± 0.009 nm. The value of λ0 at 20kV is deduced from this data to also be 0.042 ± 0.002 nm, matching the same value of λ0 from the previous result.

λ0

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) × 10-34 J·s.

The error on this value for h is calculated using the following principles of error propagation:

∆ h=|λ0 eV|√(∆ λ0

λ0)

2

+(∆ eVeV )2

(8.1 )

where∆ eV=|eV|√(∆ee )2

+( ∆VV )2

(8.2)

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 0.00 0.04 0.070 0.00322.0 0.51 0.04 0.077 0.00424.0 0.00 0.04 0.084 0.00426.0 0.34 0.03 0.091 0.00528.0 0.51 0.04 0.097 0.00530.0 0.85 0.07 0.104 0.00532.0 0.34 0.03 0.111 0.00634.0 0.00 0.03 0.118 0.00636.0 0.34 0.03 0.125 0.00638.0 0.00 0.03 0.131 0.00740.0 0.34 0.03 0.138 0.00742.0 0.00 0.03 0.144 0.00944.0 0.00 0.07 0.151 0.00946.0 0.00 0.07 0.152 0.00948.0 0.00 0.07 0.152 0.00950.0 0.27 0.07 0.153 0.009

Table 3: Results for task 2 investigation of absorption using copper filter.

The error of absorbance ∆ln |ICu/I0| is calculated by:

∆ ln| ICuI 0|=∆ICuI 0

ICuI 0

(9)

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 λ0 and 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

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 time-constant 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 λ0 used in the calculation of Planck’s constant h, gave a result of (6.40 ± 0.44) × 10-34 J·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.

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 ValuePercentage

Relative Error

Percentage Difference

ErrorPlanck’s constant

h (J·s)6.63 × 10-34 (6.40 ± 0.44) × 10-34 6.8% 3.5%

CopperKαwavelength

(nm)0.154 0.157 ± 0.009 5.7% 1.9%

CopperKβwavelength

(nm)0.139 0.138 ± 0.008 5.8% 0.72%

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 Kβ wavelength 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 absorption characteristics. 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 λ0 to 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 Kα

peaks, 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 to 30 kV, as the peaks changed location, although still around a similar range of wavelengths, and the value of λ0 did not change when it should have decreased at higher voltages.

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.