Electron-Positron Annihilation

The goal of this experiment is to investigate the physics of positron-electron annihilation. Much of the work in this experiment is to get very familiar with using fast electronics associated with nuclear instrumentation wired in a complicated way and understanding the role of each piece of apparatus.

The physical process involved in this experiment is the decay of sodium-22. The sodium-22 source decays by positron emission to neon-22. Each positron annihilates with an electron in the surrounding or in the source itself. The two steps are

The asterisk means the nucleus is in an excited state.

The two gamma rays that are emitted when a positron and an electron annihilate can be detected using temporal coincidence techniques. By identifying the gammas from the electron-positron annihilation, the angular correlation of their emission and the total activity of the source can be determined.

There will be two identical detecting systems. The signal from the PMT goes to a Preamplifier and from there to a Delay Line Amplifier (DLA). The DLA amplifies and shapes the pulse. The output of the DLA is then input to the “Multi Channel Analyzer” (MCA) which measures the pulse height, counts the number of pulses with given pulse heights, and plots a histogram of the number of events versus pulse height, known as the channel number.

Each detector consists of a one-inch diameter crystal of NaI and a photomultiplier hermetically sealed in the chrome-plated metal case to exclude moisture and light. When a gamma is incident on the crystal, it ejects a number of photoelectrons.

Now we set up for system 1 and put a sodium-22 source up against the detector. A pattern of pulses is shown on the oscillascope:

Now do the same process for system 2 and a similar plot can be observed. Then we would like to acquire a spectrum of the gamma emission from the sodium-22.

Identify that the relatively small photopeak on the right (higher energy) is the gamma line, and the large electron-positron annihilation photopeak is on the left (lower energy).

The next thing is to selecting the electron-positron peak. Now the Timing Single Channel Analyzer (TSCA) should be connected in the systems. The TSCAs filter pulses having a particular range of voltage heights using a window that is established with the upper and lower pulse height settings. The other important feature of a TSCA is that the time delay of the logic pulse it outputs relative to its input pulse is variable. The time delay is varied by adjusting the delay setting on the TSCA.

After several trail and error practices, a fixed group of values for upper and lower limit of the window settings of the TSCAs as well as the time delay can be found. And the 0.511 MeV gamma-rays from electron-positron annihilation can be selected. At this point, two new spectra were obtained, an example of the two annihilation-peak-specific spectra for the two systems is shown:

Now proceed to setting up for the Coincidence Measurement.
Now the two detectors on the angular correlation apparatus should be 180 degrees apart.

We need to make sure that pulses originating simultaneously arrive simultaneously in order to measure coincidence rates. We will use a pulser to send identical signals simultaneously to both preamplifiers. Use the Universal Coincidence unit to measure the coincidence rate. We expect to see two systems to have very similar count in a same period of time. The following figure is what we would expect to see as the dependence of coincidence rate on delay time. The width of the peak on the figure is 2τ which is twice the resolving time. And from this we have calculated the resolving time for our system to be 0.97 microseconds.

Now set the delay for a maximum coincidence rate for pairs of pulses. And put the sodium-22 source half way between the detectors. Observe the counts for coincidences.

Now proceed to the angular correlation of the detection of annihilation gamma radiation. At 180 degrees apart, the average coincidence rate is 53. A range of coincidence rate for separation angles from 170 to 179 degrees are also measured and a plot is obtained:

Chaos in LRD Circuits

The goal of this experiment is to use a diode circuit to investigate non-linear behavior.
We will be first investigating the LRC circuit which involves three main components resistors R, capacitors C, and inductors L. They are “linear devices”, as the voltage across them depends on the charge q and its time derivatives, or the current, but not on any non-linear terms like q2. If the three components are connected in series and a single sine wave voltage at a frequency is applied, the output voltage across any one of continuous two of the three components will also be a single sine wave at this frequency.
When the capacitor C is replaced by a diode D, then the output gets very complicated, as many different frequencies will be present at the same time, because of the fact that the voltage across a diode involves non-linear terms in the charge q.
The experiment setup contain a digital oscilloscope that can accurately view relatively high frequencies,

a digital function generator whose output amplitude, dc offset, and amplitude can be adjusted,

and a circuit board.

The electronic response of an LRC circuit to a sinusoidal input can be viewed below. As a range of frequencies from 10 kHz to 100 kHz is swept across the circuit, a plot is acquired on the oscilloscope. The output frequency is always a single sine wave. And the resonance was found to be approximately 53 kHz.

As input frequency changes, phase and amplitude of output frequency change as well:

Now proceed to the non-linear response of an LRD circuit.
This time, a x-y mode plot on the oscilloscope are useful to see the response of the circuit. In this mode the input frequency signal is plotted on the x-axis and the output frequency signal is plotted on the y-axis. As x increases, the x-y plot shows a series of complicated change.

In the figure above, from left to right, top to bottom, frequency increases, and as a result, the crossing points on the plot increases, which shows an increasing number of output frequencies.

The input amplitude also becomes a variable since the degree of non-linear response depends on it. There was hysteresis observed in the frequency dependence of the output amplitude, depending on the dc offset and the input amplitude, meaning that the frequency response is different depending on whether the frequency is increased or decreased and it depends on the starting frequency.
The results of this experiment can be very complicated and a detailed quantitative analysis may take many years of research. What would be investigated are a multi-dimensional space with frequency, input amplitude, and dc offset as the independent variables and output amplitude as the dependent variable. Enjoy!

Optical Spectroscopy of Hydrogen and Deuterium

Atomic optical spectroscopy can study the electromagnetic spectrum of elements and determine the elemental composition. In atoms, electrons reside in different levels of well defined energies. In order to move from lower to higher energy level, an electron has to absorb energy, and to move from higher to lower energy level, an electron will emit energy. This energy is in the form of light. Every element has its unique electronic structure, and the emitted or absorbed electromagnetic radiation has wavelength determined by the electronic transition. Therefore, the study of the wavelength of this spectral line leads to the study of the unique element.

In this experiment, ionized gas of hydrogen and deuterium in gas filled tube was provided with electric discharge, and the free electrons accelerated under the effect of electric field collide with the electrons in the atomic orbitals, which move to higher energy levels. Radiations of characteristic energies were emitted as excited electrons decay to lower energy levels. The emission spectra of hydrogen and deuterium may consist of various possible emission lines, and in this experiment the Balmer α, β, γ, and δ transition lines were measured.

The spectra of deuterium and hydrogen are the subject of study in this experiment. Hydrogen has atomic number of one, and is the lightest element. Deuterium is one of the two stable isotopes of hydrogen. While hydrogen has one proton in its nucleus, deuterium contains one proton and one neutron. Deuterium has an abundance of 1.5 x 10^−4 compared to 0.99985 for ordinary hydrogen.

The goal of this experiment is to calculate the mass ratio between hydrogen and deuterium by measuring the wavelength splitting between the hydrogen and deuterium emission lines. The energy levels of a hydrogen atom are given by
E_n = −(Z^2)(α^2)μ(c^2)/2(n^2)
where Z is the nuclear charge; α is the fine structure constant; μ is the reduced mass defined by μ=(m_e)(m_N)/(m_e+m_N), in which m_e is the mass of electron and m_N is the mass of the relevant nucleus; c is the speed of light in vacuum, and n is the principle quantum number in the Bohr theory. The energies of corresponding levels in hydrogen and deuterium differ by approximately 0.25% due to the different reduced masses of these two atomic systems.

A schematic diagram of energy levels associated with the Balmer α, β, γ, and δ transitions for atomic hydrogen is shown below.

As seen in the diagram, emission line from energy level 6 to 2 is the Balmer δ emission line, from energy level 5 to 2 is Balmer γ emission line, from energy level 4 to 2 is Balmer β emission line and from energy level 3 to 2 is Balmer α emission line. The vacuum wavelengths λ_(Hvac) of the Balmer α, β, γ, and δ lines for hydrogen as calculated from the equation above are given in the following table. Since the equation calculates transition wavelengths in vacuum, a correction must be applied to obtain wavelengths as measured by a diffraction spectrometer in air at standard pressure and temperature. The formula for this correction is:
λ_air = λ_vac/n_air
where n_air is the index of refraction of air. We will be using it as 1.000277 at STP.

All experimental parameters used were recorded including slit widths, integration times, incremental steps, and ranges of wavelength scan. The emission peaks look Gaussian in shape. In the lamp discharges, far fewer atoms are produced in the high atomic states so the Balmer δ lines were much less intense than Balmer α lines. And as mentioned before, slit widths needed slight adjustments when scanning for different Balmer lines in order to obtain well resolved spectra for all four transition lines.

The measured values of spectra wavelengths will be used to calculate the wavelength separation, (λ_Hair – λ_Dair), between the hydrogen and deuterium peaks of the four Balmer spectra. The deuteron-to-proton mass ratio can be determined from single Balmer transition
using:
E = hc/λ
λ_D/λ_H = (1/μ_D)/(1/μ_H) = (1+m_e/m_d)/(1 + m_e/m_p)
The equation can be re-written as:
m_d/m_p = (λ_H m_e/m_p)/(λ_D m_e/m_p + λ_D − λ_H)
where the electron-proton mass ratio m_e/m_d is known to be 5.446 x 10^−4. By method of error propagation, the uncertainty in m_d/m_p is:
δ(m_d/m_p) = ((∂(m_d/m_p)/∂_λH)2δ_λH+(∂(m_d/m_p)/∂_λD)2δ_λD)^(1/2)

This experiment studies the emission spectra of light output of several gas discharge lamps. The light of the lamp is generated via electrical discharge through an ionized gas. To do this, one need to separate the light by its wavelength in order to measure the spectra. A grating can be used to achieve this purpose. It can diffract light into separate beams traveling in different directions which depend on the spacing of the rulings on the grating and incident angles of the light. In this experiment, a spectrometer with such a grating was used to obtain dispersed emission spectra of the light output of several gas discharge lamps.

The spectrometer consists of two mirrors and a grating, which direct and collimate incident light from an entrance slit. Connected to the spectrometer at its exit slit are a photomultiplier (PMT) tube and a motor. The PMT is a vacuum tube and a very sensitive detector of electromagnetic spectrum, which amplifies the light signal. The motor scans the angle of the grating to send light of various wavelengths to the PMT. A schematic diagram of the instrument is shown.

Spectra of three sources were measured here: mercury, hydrogen, and deuterium. The mercury lamp was used to verify the calibration of the spectrometer by looking for a line at wavelength 5460.7 angstroms and a doublet at 5769.6 and 5789.7 angstroms. A spectrum of mercury is shown in, in which a high peak at 5472.0 angstroms and a doublet at 5781.0 and 5802.0 angstroms were detected. This measurements show that the measurements by the spectrometer are approximately 11 to 13 angstroms higher than the actual values. Therefore, when scanning for spectral peaks in deuterium and hydrogen transition lines, wavelength values of slightly higher than theoretically calculated values are expected. Adjustments were also performed to test the optimization of the slit widths and integration time. Generally, if slit widths of the entrance and exit were too large, spectra will too noisy. But different elements and different transition lines in the same elements may need to be individually adjusted to achieve the best looking spectra. A reasonably long integration time would be sufficient to obtain a good spectrum, such as 50 or 100 ms.

Spectra of both hydrogen and deuterium can be measured by using the deuterium gas filled tube, because the deuterium tube contains not purely deuterium but also hydrogen. Spectra were taken several times with different slit widths, integration times, scan ranges, and increments, and the best resolved spectra are presented here. The calculation of the exact energy of the Balmer α, β, γ, and δ transitions in vacuum of hydrogen and deuterium can be done, in which Z=1 for both deuterium and hydrogen, α = 7.72973525689 x 10^−3, m_e=9.10938291 x 10^−31 kg, m_p=3.34358348 x 10^−27 kg, m_d=1.672621777 x 10^−27 kg, c=299792458 m/s, n=2, 3, 4, 5, 6. Therefore for hydrogen and deuterium.

Using these calculations, wavelengths of transition lines can be obtained through:
λ = hc/(E_n1 − E_n2)
where h = 6.62606957 x 10^−34 J s. So the wavelengths of hydrogen and deuterium in the four Balmer transitions in vacuum and air (at STP) can be calculated.

Spectra and the Gaussian fit of the Balmer α, β, γ, and δ transitions are shown. In each of the graphs, the larger peak is the deuterium peak and the smaller peak is the hydrogen peak. Values of the mean peak positions and their associated uncertainties were obtained from the gaussian fit of the spectra and presented in Table 2. The Balmer α, β, γ, and δ lines were obtained with same entrance slit of 5 mm, exit slits of 35, 23, 23, and 26 mm, respectively, wavelength range of 10, 10, 6 and 7 angstroms, increment of 0.2, 0.1, 0.1, and 0.05 angstrom, and same integration time of 100 ms.

The deuteron-proton mass ratio and associated uncertainties were calculated from each of the spectra for the Balmer α, β, γ, and δ lines and were shown in the table below:

The average value of the deuteron-proton mass ratio based on the four calculated means and four uncertainties is 1.98 ± 1.11.

Poisson Statistics and Radioactive Decay

The goal of this experiment is to study the nature of random events and the model of poisson statistics. To do that, we will make use of fast nuclear electronics instrumentation and study radioactive decay of radioisotope cobalt-60.

First, before the experiment, we will have a review of the several distribution functions. There are three main types of distributions to be introduced here: the binomial distribution, the gaussian distribution, and the poisson distribution.

Before looking at the binomial distribution, we will introduce the binomial probability functions. In situations with repeated trials, if there are two possible independent results A and B of each trial, and their probabilites are p and q, respectively, then this repeated trials with constants p and q are called Bernoulli trials and the formula to calculate probability of exactly x times of A out of n trials is:
f(x)=C(n,x)(p^x)(q^(n-x))

To figure out the probability of no more than x times of A in n trials, then the formula is:
F(x)=f(0)+f(1)+…+f(x)
=C(n,0)(p^0)(q^(n))+C(n,1)(p^1)(q^(n-1))+…+ C(n,x)(p^x)(q^(n-x))

In this case, f(x) is the binomial probability function and F(x) is the binomial distrubition function.
Second, with an approximation to the binomial distribution for large n, it is possible to achieve the following formula:
f(x)=C(n,x)(p^x)(q^(n-x))~exp[(-(x-np)^)/(2npq)]/(2πnpq)

When n goes to infinity, the left hand side and right hand side of this formula will be equivalent. The right hand side is the Gaussian approximation to the binomial distribution.

Finally, we will look at the Poisson distribution. The Poisson distribution is useful when probability of some event is small and constant. Let us looking at an example of counting the occurrence of radioactive decay. Assuming observation period is much shorter than the half life of the isotope, then the probability of one decay during Δt is μΔt where μ is constant, if Δt is so short that there could be either zero or one decay during. Let P_n(t+Δt) be denoted as probability of n decays in time interval t+Δt. For n>0, this can be written out as:
P_n(t+Δt)=P_n(t)P_0(Δt)+P_n-1(t)P_1(Δt)= P_n(t)(1-μΔt)+P_n-1(t) μΔt

Rearranging it and let Δt approach zero, we have:
dP_(t)/dt=μP_n-1(t)- μP_n(t)

Solving for the equation, the final formula can be obtained:
P_n=(μ^n)exp[-μ]/n!

This is the probability of exactly n counts per unit time, μ is the mean counts per unit time, and this probability function is the Poisson distribution.

Now proceeding to the next part of the experiment: collecting occurrences of the radioactive decay. The radioactive isotope used here is a cobalt-60 source. This source will be placed near a detector which can detect the gamma radiation emitted by the source. Here are some photos of the gamma spectroscopy experimental set-up. The small red piece in figure 1 is the cobalt-60 source, placing next to the detector. The detector is connected to a pre-amplifier which then goes into a multichannel analyzer (MCA). The MCA reads the counts for a “dwell time” t and plots a spectrum of the number of counts to each channel.

The dwell time is set to a range of different values from 10^3 μs to 2×10^7 μs. By varying the dwell time, we can investigate how the distribution changes. By looking at the spectrum from the MCA screen, we can have a feel of an approximate mean count. For small means μ=np, the data can be fitted to Poisson distribution, and for large μ, the Poisson distribution can be well approximated by the Gaussian distribution.

For means well below 1, the occurrence of zero count will be much higher than one. So there there is no need to plot the data. The dwell times used here are: t = 1 x 10 ^2 μs, 2 x 10^2 μs, and 5 x 10^2 μs.

For means of about 0.3 to 5, the data can be compared to theoretical Poisson distribution. The dwell time used here are: t = 1 x 10 ^3 μs, 2 x 10^3 μs, and 5 x 10^3 μs. Following is an example of a set of data of small mean (or short dwell time), in particular, 10^3 μs, and its fit of Poisson distribution:

For large means, the Gaussian approximation of the Poisson distribution can be useful in fitting the data:
P(n)=exp[-((n-μ)^2)/2σ^2]/σ(2π)^(1/2)
where σ is standard deviation and σ=(μ)^(1/2)

The dwell time used here are ranging from 1 x 10 ^4 μs to 2 x 10 ^7 μs. Following is an example of a set of data of large mean (or long dwell time), in particular, 0.5 sec, and its fit of Gaussian distribution:

It turned out that the data of is indeed well approximated by Gaussian distribution. The calculated parameters in the fit for the mean of this data is 411.9, and the standard deviation is 28.44.

High Resolution Gamma Spectroscopy

Gamma ray spectroscopy delivers important information about energy spectra which are characteristic to the gamma emitters. It can be used to identify nuclear sources in the laboratory, in our surrounding environment, and in astrophysical background. Gamma radiation has very high frequencies, ranging from approximately 30 to 300 exahertz, which corresponds to energy range of approximately 1 to 10 MeV. The high energy of gamma ray can ionize atoms and molecules and is biologically hazardous.

In this lab, we study and analyze the gamma radiation spectroscopy of three radioisotopes cobalt-60, cesium-137, and sodium-22 to obtain calculation of electron mass. The spectroscopy was acquired by using a liquid-nitrogen-cooled high-purity germanium detector.

When the incident gamma photons are absorbed in the detector, electrons whose amount is proportional to the gamma ray energy are liberated to flow as current. The electrons are then accumulated on a capacitor in a preamplifier connected to the detector, and result in a voltage proportional to the amount of charge. This voltage is therefore proportional to the energy of the gamma ray. The Multichannel Analyzer (MCA) amplifies the pulse and measures the pulse height. It then records the number of pulses with each value of the pulse height, and plots a spectrum of the number of pulses with that height versus pulse height, called the channel number of the detector, which is proportional to the energy of gamma ray. A schematic diagram of the system of apparatus is as the following.

The spectroscopies of the three sources can be seen below. On the spectrum of cobalt-60, two peaks at channel numbers 7015 and 7969 are observed, which correspond two photo peaks at 1.173 MeV and 1.332 MeV.

For cesium-137, one peak at channel number 3952 was recorded, corresponding to the 0.662 MeV photo peak, as shown below.

For sodium-22, two peaks at channel numbers 3052 and 7623 were observed, which correspond to the two peaks at 0.511 MeV and 1.277 MeV. The higher energy peak here is the gamma decay photo peak. The lower energy peak here is the peak of electron-positron annihilation, since the gamma decay in sodium-22 follows a beta decay that involves the emission of positrons.

Based on data taken at the five peaks of these elements, energy scale calibration was performed and a graph of channel number versus energy was plotted.

Now before proceeding to the calculation of electron mass, we will introduce some information about Compton scattering of the photons and electrons. Inside the semiconductor crystal of the detector, electrons are localized in valence band. During the gamma ray detection, as photons are absorbed into the detector crystal, electrons may be scattered to delocalized conduction band through Compton scattering and are thus set free to move in an applied voltage, a photoelectron may completely absorb the energy of a photon and the photon is destroyed in the process. When the energy of a photon is absorbed fully and the photon is destroyed, a photo peak with energy equivalent to the gamma photon energy will appear on the spectroscopy. When Compton scattering occurs, gamma photons collide with electrons inside the valence band of the detector crystal and lose only partial energies.

The electrons obtain kinetic energy from the collision and are taken to the conduction band. In this process, gamma photons may escape the crystal after collision or continue to lose energy to electrons in further collisions. During this process, gentle collisions resulted in low energy transfer and were therefore observable at low channel numbers. When the amount of energy transferred to electrons through Compton scattering reaches the highest possible value, the electron energy is observed at the Compton edge. This occurs when a gamma photon collides and backscatters at 180 degrees with an electron. When the backscattered gamma photons later convert all their energies to photoelectrons, the corresponding energy is at the backscatter peak. A schematic diagram of the Compton scattering is shown as the following. The Incident photon with indicated energy and momentum collide with an electron at rest, resulting in the photon and electron leaving to travel at some angles with respect to the incident direction.

The theoretical formula for Compton scattering is as the following:
Δλ=hc(1/(E_1)-1/(E_2))=h(1-cos(θ))/(m_0)c
When θ=180 degrees, we obtain the rest energy of electron E_0 as a function of the energy of photo peak E_1 and the energy of backscatter peak E_2:
E_0=(m_0)c^2=2(E_1)(E_2)/((E_1)-(E_2))

So finally, by using the energy scale calibration of the detector as well as the data on the spectrum of cesium-137, the electron mass can be calculated. The channel numbers and uncertainties of photo peak and backscatter peak of cesium-137 are 3952±7 and 1151±40.
E = (C)/k
where C is channel number, and k=5984.4±8.2. So the electron mass in the form of rest energy becomes:
E_0 = 2E_1E_2/(E_1- E_2) = 2(C_1)(C_2)=k(C_1 – C_2)
To calculate the error of the electron mass calculation, the Propagation of Uncertainties method was used.
And according to propagation of uncertainties, the error of the electron rest energy is:
δE_0 = ((δC_1 (δE_0/δC_1))^2 + (δC_2 (δE_0/δC_2))^2 + (δ_k(δE_0/δk))^2)^1/2
So the result of electron rest energy is
0:54 ± 0:03 MeV
The accepted value of electron rest energy is 0.511 MeV. So the experimental result is consistent with expected value within experimental uncertainty.

Lexie’s Log of Physics

Hello, welcome to my blog! This is Lexie. I’m a physics student and a fan of experiments! The purpose of this blog is to post my discoveries to be encountered during various kinds of interesting physics experiments that I will be doing. These would include different experiments in the condensed matter physics, nuclear & particle physics, optics & electromagnetic radiation, chemical physics, and non-linear dynamics & chaos. As we all know the fascination and profound impact physics can bring to our understandings and our lives, I hope you would enjoy and feel inspired by the experiments findings posted throughout the updating of this blog, and I welcome your comments and discussion!