Lasers can emit electromagnetic radiation of a single frequency, and therefore, a single wavelength, in a highly directional beam. The light from a laser is temporally coherent because all the electromagnetic waves are in phase. It is also spatially coherent because the emissions in different transverse position in the bean have fixed phase relationship. The spatial and temporal coherence of laser permits all of the energy of the beam to be focused by a lens to an extremely small spot, consequently deliver all of the power of the beam to a very small area. The goal of this experiment is to explore what characteristics of laser determine the frequency and intensity pattern emitted by a laser. In addition, I investigated a variety of the transverse patterns that can be found for a laser beam, and what changes in the characteristics of the design of the laser cause changes in those transverse pattern. Last but not the least, I also investigated how a laser can be made from discrete pieces, mirror and a gas of atoms.
As one-dimensional mode, a string fixed at both ends can vibrate in many different spatial modes. The first mode only has nodes only fixed at the ends. This mode has the longest wavelength and the lowest frequency. The second mode has three nodes, and the third mode has five nodes, etc. These one-dimensional modes are analogous to the longitudinal modes of a laser. The two-dimensional mode, one can think of it as the surface of a drum. And the circumference of the vibrating material is a node.
A laser has to have the ability to confine light and to amplify light. One way to confine the light is with two planes and mutually parallel mirrors. Light that bounds back and forth between the mirrors forms a standing wave that satisfies the boundary condition. If we set z-axis to be perpendicular to the two mirrors, and place the two mirrors at z=0 and z=L respectively, the spatially varying part of the electric field of the standing wave can be written as E(z,t)=E0sin(kz)cos(wt), where k is the wave number and w is angular frequency. An integer number of half wavelengths must fit into the distance L. Therefore, L=qλ/2, where q is an integer. Because c=fλ, the possible frequencies can be written as f=qc/2L. A discrete set of allowed frequencies is separated by Δf=c/2L. The graph below shows a schematic curve of the emission of the lasing transition. With the light being confined, it also has to be amplified. Electromagnetic waves can be created from the oscillating dipole moments in atoms when the electrons of the atoms are in a mixture of two appropriate atomic states. The energy difference between the two atomic states is determined by the frequency of the oscillation of the dipole moment: ΔE=hf. Typically, most observations of atomic dipole transition has a transition energy in the order of 107 to 109 Hz. Laser emission involves a process of stimulated emission, in which atoms are stimulated by some of the light that is present to emit their energy in phase with the initial radiation. This process amplifies an incident beam of light.
The Gauss-Hermite modes and the Gauss-Laguerre modes are the most common transverse modes. In these modes, both electric and magnetic fields are transverse to the direction of propagation. These modes are generally noted as TEMmnq, where q is the longitudinal number of half wavelengths, and m and n are the integer number of transverse nodal lines in the x- and y- direction across the beam. Generally, for each of the longitudinal modes 1, there is an infinite number of transverse modes (m,n), with the frequency increasing as the complexity of the modes increases. For the Gauss-Hermite modes, the transverse intensity can be written as:
Where x and y are the transverse coordinates and Hm and Hn are the Hermite polynomials. When z=0, the phase fronts are a plane in the resonator. Also, w(z) has its minimum value, which is called the “beam waist’. In addition, note that the intensity expression involves the product of Hermite polynomials and a Gaussian function. When x2 + y2 = w2(z), the intensity function simplifies to:
The Gauss-Laguerre modes involves both the Hermite polynomials for x and y and the Laguerre polynomials for r and θ, the polar coordinates in the transverse plane. The pattern is symmetric circles.
The apparatus of this experiment includes two lasers: one is the white Metrologic Neon laser (commercial laser). And the other is open cavity laser. There are also two Fabry-Perot interferometers. One is an old and brass one. The other is sealed inside a silver tube and mounted in a black disk. A ccd camera is also used to view the output and is connect to the computer.
I first used these apparatus to study the longitudinal modes of a laser. Firstly, I used the old-style Fabry-Perot interferometer. I first align the interferometer with the laser and the two mirrors. I adjusted the orientation of the cavity so that the laser beam is normal to the mirrors and that the dots shown up on the wall fall on top of each other. Then I inserted a diverging lens between the laser and the cavity. I saw the ring pattern that is created in transmission. When I slightly tune the adjustable mirror of the Fabry-Perot interferometer, I got circular rings shown up on the wall. When the distance between the Fabry-Perot mirrors increases or decreases by half of the laser wavelength, the rings decrease or increase in size by one ring spacing. I also noticed that when I first turned on the laser, even if I don’t turn the adjustable mirror, the rings on the wall still changes its spacing. This is because the laser got heated up inside and the cavity expands because of the rising temperature.
Nest, I used the scanning Fabry-Perot Spectrum Analyzer to display the optical spectrum of the commercial laser. This analyzer has the same function as the old Fabry-Perot interferometer that I just used, except it has a detector placed at the center of the transmitted pattern and has curved mirror so that the light is almost always limited to the fundamental transverse spatial mode. A voltage supply controls and changes the voltage on a piezoelectric crystal mounted on one of the Analyzer mirrors. The crystal changes its length in proportion to the voltage, and thereby scanning the separation between the two mirrors of the interferometers. Display the output of the commercial laser beam, and adjust the tilt of the analyzer, I saw some sharp peaks displayed on the first channel in the oscilloscope. The sharp peaks correspond to the spectral range and the smaller peaks inside the sharp peaks correspond to the longitudinal modes. Channel2 corresponds to the ramp voltage of the crystal. I measured the space between two spectral ranges and the distance between the two longitudinal modes. Knowing that the Fabry-Perot has a free spectral range (FSR) of about 7.5 GHz, I calculated the spacing between frequency, Δf, of the longitudinal modes. Therefore, I calculated the actual distance between the mirrors of the commercial lasers to be 2.1cm. I also did this for the open cavity HeNe laser, where I found the distance between the two mirrors of the open cavity laser is 0.26cm.
Then, I inserted a polarizer between the laser and the Fabry-Perot Spectrum Analyzer. By rotating the polarizer, I expected to see some peaks corresponding to the transverse peaks to vanish at certain angles because some peaks displayed on the oscilloscope correspond to the transverse waves polarizing in y direction and some corresponds to the transverse waves polarizing in the x direction. Unfortunately, I did not see the peaks vanishing with I rotated the polarizer.
To study the transverse modes of the laser, I placed a camera to capture different patterns created by the open-cavity laser. By slightly tuning the adjustable mirror, I got different patterns recorded by camera. Some of the images are shown below：
For an ideal laser, the separation between the two spots is constant. But in reality, most lasers have some small divergence. In order to calculate the divergence of the open-cavity laser, I measured the separation between the center of the two spots, as well as the distance from the output of the laser to the camera. I can see that as the distance from the output end mirror of the laser increase, the separation of the bright points in a transverse mode also increases. The two has a linear relationship. When the line hits x-axis (when the distance from the output of the laser to the camera is 0), the separation of bright points in a transverse mode is 0.1113cm. Therefore, the beam waist of my laser is 0.1113cm.
In this lab, I explored Fourier transform using an optical system.
Theoretically, an object shows up as its Fourier transform in the far field. It also shows up as its Fourier transform if it passes through a normal lens.
The formal Fourier transform could be expressed as:
The first equation tells the time dependent of function f(t) as a continuous sum of sine waves. It tells what frequency is needed to produce a signal; the second equation tells how much each sine wave is needed in the sum. It tells how to find the weighting function g(w) for a given time dependent signal. In quantum mechanics, Fourier transform is used to switch between position and momentum space.
The experimental setup, as shown below, involves a Neon-Helium laser, three lenses and a camera connected to a computer program called uEye. The laser has a “spatial filter” on front. A spatial filter is a kind of optical processor that eliminates high frequency noise on the leaser beam. Therefore, a uniform diverging beam comes out of the laser. The light then hits a collimating lens. The light that comes out of the collimating lens is then parallel. There are two lenses. The first lens takes the Fourier transform of the object. The second lens takes an inverse Fourier transform of the original object. We are essentially interested in three planes: object plane, Fourier Plane and Image plane. Object plane shows the original image. Fourier Plane shows the Fourier transformation of the original plane. Image plane transforms shows the Fourier transform of the image on Fourier plane, which transforms the image back to the original but in the up-side-down direction. If a filter is placed after the Fourier plane, the image plane will show the Fourier transform of the image after the filter. In other words, all frequencies which make up the image of the object separate in the Fourier plane. By filtering out certain frequencies, we could remove some components of the object which cannot be easily removed by blocking parts of the object itself.
Firstly, I put nothing to be my object. Theoretically, the input is a uniform brightness beam, which is a constant. The Fourier transform tells that the output should be a delta function. In fact, what I saw was a bright dot in the middle, with diminishing size dots going towards each sides. This is because I did not have a strictly uniform beam with a big enough diameter; therefore, my input is in fact a rec function. And the output of a rec function is a sinc function, which is the Fourier transform of a rec function.
Then, I used grid as my object. A grid is the superposition of vertical and horizontal lines, which form rectangular apertures. Image that I got at the Fourier plane is the superposition of vertical and horizontal sinc function. This is what I expected because according to the previous experiment, the Fourier transform of vertical or horizontal stripes should be sinc function in vertical or horizontal direction. By the Fourier transformation principal:
If h(x,y)=f(x,y) +g(x,y), where f and g are horizontal stripes and vertical strips;
The Fourier transform is just the superposition of the Fourier transform of vertical stripes and the Fourier transform of the horizontal stripes.
Knowing the Fourier transform of the grid, I could use filter to eliminate either the vertical strips or horizontal stripes. To eliminate the vertical strips and keep the horizontal strips, I had to block out the horizontal component of the Fourier transform. In order to do so, I used the vertical slit so that only vertical component could pass through after the Fourier plane. To eliminate the horizontal strips and keep the vertical strips, I had to block out the vertical components of the Fourier transform. In order to do so, I used the horizontal slit placed after the Fourier plane so that only horizontal component could pass through. Below is the vertical and component strips that I got at the image plane:
Vertical strips (only):
Horizontal strips (only):
The next thing that I did was to put some “text” on the grid as my object. My image on the object plane consists text overlapping slanted lines:
I wanted to eliminate the slanted lines, so that I adjusted the slanted lines so that they are oriented vertically. What I saw at the Fourier plane is:
The horizontal component shown up in the Fourier plane is caused by the vertical slanted lines in the object plane. In order to get rid of it, I placed a vertical slit after the Fourier plane so that the horizontal component could not pass through. This is what I got in the image plane:
The last experiment that I did was to put a fingerprint on a glass as my object. The purpose of this experiment is to get a clear image of the fingerprint. To make a periodic function with sharp edges, the series has to go to very high frequencies because the sharp edges cannot be achieved without rapidly varying intensities. The high frequencies contribute the most to the corner of an object. The frequencies which contribute more to the middle of the pulse are longer wavelength/ shorter frequencies. Without a filter, there is no apparent features showing up in the image plane because it was too bright. To get the edge of a aperture, I put a constant block after the Fourier plane. This filter only allows frequencies above certain threshold to pass through. The high frequencies which make up the edges lining passed through the filter. Below is what I got in the image plane:
In quantum mechanics, spin is an intrinsic property of electrons. An electron can either “spin up” or “spin down” depends on its energy level. When apply an external magnetic field, the spin of an electron interacts with an external magnetic field and gives up energy to or takes energy from its surroundings. This phenomenon is called Electron Spin Resonance (ESR). ESR is useful for the electrostatic and magnetic studies of complicated molecules.
There are two energy levels of an electron, differing by ΔE=2μB, where μ is the magnetic moment of the electron, and B is the external magnetic field. When one provides photons of energy E=hf, there will be a resonance when hf=2μB. When the resonance happens, a quantum mechanical spin ½ particle interacts with magnetic field and makes a transition from one state to another, corresponding to a photon of a specific frequency emission or absorption. Therefore, by setting different frequency f, one could measure the value of B that produces resonance.If we plot B on y-axis and f on x-axis, the slope of the plot will therefore be B/f=h/2μ, which shows the magnetic moment of the electron. A magnetic moment of dipole moment has an energy interaction of E=-μB=-μBzcosθ. The lower energy bound to the energy is E=-μBz for θ=0. This corresponds to when the moment is aligned with the field. The upper energy bound to the energy is E=μBz for θ=π. This corresponds to when the moment is aligned against the field. By studying the frequency of the photon as well as the energy difference between the two energy states of electron, we can investigate the technique of electron spin resonance and learn about the physics of spin.
The picture below shows the setup of the experiment. There are an oscilloscope, a 1 Ohm resistor, Helmholtz coils, the sample, a ESR adaptor box, one 120V transformer and a frequency counter. The frequency counter controls the frequency of photons produced by the coil. The 120V transformer sends current into the Helmoholz coil, which provides an external magnetic field to the sample. The ESR adapter box has +12V and -12V supplied to it by the oscillator. The unpaired electron used is a molecule called disphenylpierylhydrazil, abbreviated as DPPH. One of the electrons in this molecule behaves like an isolated electron. This sample was placed in a coil that acted as an inductor in an LRC resonance circuit. The coil was plugged into an oscillator that contained the rest of the resonance circuit. The oscilloscope displays the electric signal from both ESR and the field. When the resonance happens, paired electric signal peaks from the channel of ESR adapter could be detected.
There are two experiments in total. The first experiment is called The Resonant Circuit. The goal of this experiment is to investigate the sharpness of this resonance circuit as a function of frequency. With the spectrometer set-up, I added a passive LC circuit. Then, I display the output of the passive circuit on the oscilloscope. I found that the induced current has nothing to do with the two coils. Even if I removed the coils, I could still see the induced voltage. But if I turn the power of LRC down, there were no induced voltage anymore. The signal is due to the change of current in LRC circuit. When I adjusted the variable capacitor on the passive circuit, the voltages gets higher and then lower. This is because the maximum magnitude of voltage displayed at the point when the frequency of an LRC circuit matches with the passive LC circuit. This is how a transformer works. If the input to a transformer is V0sin(wt+theta). V0 is changed by transformer but w is not.
The second experiment is the ESR experiment. A magnetic field B is produced by a set of Helmholtz coils. The magnetic field B determines the energy difference, and in turn, determines the frequency of photons, which could provide the resonance, i.e. to flip spins from E=-μB to E=μB. There are a variety of ways to understand how the sample leads to an observed electrical signal. From one point of view, if the sample absorbs the magnetic energy created by the current in the coil, the inductance L of the coil is changed. The voltage across the coil changes when L changes. From another point of view, the alternating current in the coil is producing an ac magnetic field in the inductor. This alternating field can be thought of as a photon field. When the resonance condition is satisfied, the spins will absorb the photons and go from E=-μB to E=μB. They will also use these photons for stimulated emission and go from μB to –μB. Thus, the electron spins are constantly changing states. Since they behave like magnetic dipoles, they give rise to an oscillating magnetic field of their own and this induces a current in the coil by Faraday’s Law.
Again, the resonance condition is when 2μB=hf. To determine the magnetic dipole moment, we fixed the external frequency, changed the external magnetic field and noted where in the cycle the ESR signal occurs. Knowing that the Helmholtz pair of coils have 320 turns, the radius of the coils are 6.8cm, the distance from the center of the loop along an axis perpendicular to the plan of loop of N turns is 3.4cm, one can write the magnetic field B=(4.269*10^(-3) tesla/ampere)I, or B=(4.269*10^(-3) tesla/ampere)*(V/R). The resonance condition tells us that B=hf/2μ. Therefore, we can equate the two equations for B and calculate the magnetic moment of electron:
μ=(h*R)/[2*4.269*10^(-3)*V/f], where h is the Planck’s constant, R is the resistance of the resistor.
The slope of the plot is V/f. Plot everything into the expression for magnetic moment, we get:
μ=(h6.626*10^(-34)*1.04)/[2*4.269*10^(-3)*8/6462*10^(-9)]=(9.335±1.63)*10^(-24). The accepted value is 9.284 763 77(23)*10^(-24). Therefore, my experimental value agrees with the theoretical value within standard error.
Overall, in this experiment, by studying the frequency of the photon as well as the energy difference between the two energy states of electron, we can investigate the technique of electron spin resonance and learn about the physics of spin.
This time, the drink that I make for you is called e/k ratio and the band gap of a semiconductor.
Semiconductor is an artificial substance, which has more number of free electrons than insulator and less free electrons than conductor. When Silicon is converted to semiconductor, it is mixed with some other elements. The process of mixing impurity in Silicon is called doping. When a doped semiconductor contains excess hole, it is called “p-type”; when it contains excess free electrons, it is known as “n-type”. A single semiconductor crystal can have multiple p- and n-type regions. The p-n junctions between these regions have many useful electronic properties. In this experiment, a n-p-n type Silicon transistor TIP3055 was used to measure the current-voltage relationship of the p-n junction at different temperatures. The current-voltage relationship of a p-n junction is measured at different temperatures, permitting a determination of the ratio of e, which is the magnitude of the charge of an electron, to k, which is Boltzmann’s constant.
The theory of the experiment depends on the equation: I = Io[exp(eV/kT)-1]. In the case when eV»kT, the equation can be reduced to I = Io*exp(eV/kT). The parameters Io and e/kT can then be determined from a measurement of current versus voltage.
Below is the picture of the circuits connection:
The circuits include a negative dc voltage source using a 1kΩ and a 1.5V battery. The currents is measured with a picoammeter. The voltage is measured with a voltmeter and temperature is measure by a digital thermometer. E, C and B refer to emitter, collector and base. Below is the picture of the real setup:
In the actual experiment, I kept track of the current and voltage under 5 different temperatures, which include room temperature, water/ice mixture, boiling water, dry ice/ isopropyl alcohol and liquid nitrogen. In order to find the e/k ratio, I took the natural log of the current, and plotted it against voltage. The slope of the plot then corresponds to (e/kT), and the intercept of the plot corresponds to the natural log of Io. Below is the plot I get for dry ice:
For example, in this case, I calculated the e/k ratio by multiplying the slope(60.086) with the temperature(199K). The estimated e/k is therefore 60.086*199=11957.114. Under the other four temperature, the estimated e/k are 11323.728 for room temperature, 11397.75 for water/ice mixer, 11397.75 for boiling water, and 11853.54 for liquid nitrogen. The mean value for e/k for the five measurements are 11585.9764. This roughly agrees with the standard value, 11604.51. (e = 1.602 176 487 (40) *10^(-19) and k = 1.380 6504(24) *10^(-23). So I concluded that my experimental value agrees with the theoretical value.
To find the transistor band gap, I plotted ln(I0) versus 1/T:
The slope of the plot is is -e_gap/k and has a value of (-1.582 +- 0.0025) *10^(4). Therefore, the experimental value of energy bank gap is 1.357 +-0.02 eV. The accepted value is 1.11eV to 1.13eV. The experimental value is different from the theoretical value because the omitted temperature dependence factor.
The second piece of drink that I will make for you is called Alpha Spectroscopy. The first drink that I made in nuclear physics lab!
The purpose of this experiment is to study the detection of charged-particle radiation and the attenuation of such radiation during interaction with matters. Specifically, this experiment studies alpha particles which are helium-4 nuclei that contains 4 protons and two neutrons. Alpha particles produced from nuclear decays generally have energies between 4 and 10 MeV, and can only travel a few centimeters in air at atmosphere pressure. Therefore, this experiment needs to be done by varying degrees of vacuum. There are mainly two things that we care about the alpha particle: (1) range in which the total distance of an alpha particle travels in air which depends on the rate at which it loses energy in the medium; (2) variation in energy transfer to the medium per unit distance.
The setup of this experiment contains a surface barrier detector, an oscilloscope, a delay line amplifier, a Hastings Model 760 vacuum gauge, a multichannel analyzer(MCA), and Am-241 alpha source. The Am-241 alpha source, together with the surface barrier detector, are placed in the stainless steel vacuum chamber. In order to vary the degrees of vacuum, I need to evacuate the vacuum chamber by different extent. There are three valves along the chamber, and the chamber is open when the valve is aligned with the chamber. When evacuating the chamber, I closed all three valves at the beginning, turn on the pump, and successively opened the first and the second valve next to the pump, but always kept the third valve closed which connects to the air. When I want to raise the pressure in the chamber, I kept the the first and the second valve closed, and open the third valve until the pressure raised up to the number that I wanted. The MCA, just like what it does in the Poisson experiment, acquires the data that I need. The pulse hight is an indication of the energy deposited in the detector. The energy is displayed along the horizontal axis and the frequency of occurrence of a pulse height is displayed along the vertical axis. The frequency (number of counts in each energy channel) can be increased by acquiring data for a longer time interval. Below is the picture of the setup:
There are in total 4 experiments. The first experiment studies the Am-241 spectrum and the using of the MCA. In this experiment, I collected data for 300 seconds with 8192 bins. The data that I collected are (1)channel number with the highest peak (2) number of counts at the peak (3) the full width at the half maximum and (4) total number of particles represented in the spectrum. In this 5-minute run, I saw the three peaks in the spectrum. They are not very well resolved and each individual peak is a bit hard to be distinguished.
The second experiment studies the surface barrier detector operations. The purpose of this experiment is to determine the effect of the bias voltage on the operation of the surface barrier detector. During the experiment, I set the bias supply voltage to be 25V, 60V, 95V and 125V, and recorded the location of the largest peak, number of counts for the largest peak, full width at the half maximum and the total number of alpha particles represented. I found that the channel number of the peak height, the total number of alpha particles as well as the total number of alpha particles increase as the bias voltage increases. But they all only increase in a small extent. The full width at the half maximum decreases by a great extent as the bias voltage increases. A small FWHM gives a good resolution. The manufacturer recommends operation of the detector at 125V. This makes sense because the higher bias voltage we have, the higher channel of the peak height, the smaller full width at the half maximum and the bigger total counts of alpha particles we have. This can make the peak better resolved.
The third experiment studies data acquisition techniques. The purpose of this experiment is to become familiar with apparatus, to perform energy calibration, and to investigate the resolution of the apparatus to perform a detailed analysis of the spectrum. In the experiment I set the bias voltage to be 125V, as suggested from the result of the second experiment. I collected data for 30,000 seconds, which takes an overnight run, with 8192 channels. The spectrum that I got has three peaks. I fitted three Gaussian functions in there to describe the three peaks. The problem was that even though the dominant high energy peak can be characterized fairly accurately, each of the lower energy peaks are superimposed on there large peaks on the right. Therefore, the best way is to simultaneously fit three Gaussians at once. This takes multiple tires and adjustment. Luckily, I successfully fitted in three Gaussian functions at last. Both of the amplitude and the mean of the best fits that I got are less than what I would determine by simply looking at the data. You can see the graph as below:
The last experiment studies the energy loss of charged particles. The purpose of this experiment is to investigate the process that alpha particles are easily stopped by air, and to determine all the appropriate properties of the spectrum as a function of air pressure. The alphas from the Am-241 source are in the 5MeV range. When these massive, high-energy particle hit oxygen and nitrogen molecules in the air, they will continue on large undeflected, at least until they have lost most of their energy. In this experiment, I collected the pressure and the channel number. The channel number is related with energy of an alpha particle at a specific pressure. I plotted the energy versus pressure and found that as pressure goes up, the energy goes down. The plot of several pressures are shown below:
This is what I would expect because when there is no air in the chamber, alpha particles go straightly through the chamber, without losing any kinetic energy. When there is air in the chamber, the alpha particles collides with the air molecules. So the mean of the energy moves leftwards on the plot. In addition, since the collision of the loss of energy is random, the plot gets a larger spread. You can see the energy vs. pressure graph as shown below:
Taking the derivative of the plot above, I got the plot for the stopping power of alpha particle:
The first cocktail that I will make for you is called Poisson Statistics in Radioactive Decay.
To make this cocktail, I used the detected gamma ray emitted by one microcurie Cobalt-60 to study Poisson statistics, Gaussian statistics and Binomial statistics. Before actually making the drink, let me give you a general introduction of the mixers that we use, which are three kinds of statistical distribution and nuclear physics.
Poisson distribution is the first mixer. It is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since last event. In physics, Poisson statistics have been used to study phenomenas such as the the shot noise in electric circuits, charge hopping from one site to another in a solid conduction, etc. It has also been used to social science and all branches of natural sciences. For example, the birth defects, rare desease, traffic flow, etc.
Gaussian distribution is a continuous probability distribution that describes the distribution of real-valued random variables that are distributed around some mean value. Physical quantities that are expected to be the sum of many independent processes, for example, measurement errors, often follows Gaussian distribution.
Binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial, “success” and “failure”. In our experiment, a success corresponds to one detected gamma ray; a failure corresponds to nothing is detected.
If the three statistical distributions are the juice mixers for the drink, nuclear decay must be the spirit. It is because the randomness so encountered is an intrinsic property of the quantum process underlying nuclear decay, and the decay of each individual atom is completely independent of the fate of other atoms, we can use Cobalt-60 as our source and the detected gamma rays emitted from the source as our data.
Keeping in mind of the ingredients of the drink, I will now take you to make the drink. The setup that I use to make the drink includes a scintillation, a photo multiplier tube, a preamplifier, a voltage supply, a delay line amplifier, and a multichannel analyzer (MCA). You can see the picture as below:
When making the drink, I place the Cobalt-60 at one end of the scintillator. Starting at time t=0, the MCA counts the decays from Cobalt-60 in successive intervals of length T (dwell time) that I set. In other words, the MCA establishes bins with 0 < t < T, and T < t < 2T, and so on, and it counts the number of decays in each bins.
To study the three distributions, I measured the number of counts detected when T is set to be 0.00001s, 0.0009s, 0.005s, 0.05s, 5s, and 50s, corresponding to the mean of counts ranging from (2.00 ± 140) × 10−4 and (4.37 ± 0.0002) × 104.
When dwell time is 0.0009s, 0.005s, and 0.05s, I compared the experimental data with the theoretical values. In all of the three cases, the experimental data fits the theoretical values within the range of the error bar. For example, the graph below shows the comparison between theoretical value and experimental data of Poisson distribution when mean of counts is 4.50 ± 2.12. Therefore, I concludes that my experimental values of mean equals (7.80 ± 8.8) × 10−1, 4.50±2.12 and (4.50 ± 0.67) × 10 are well distributed according to Poisson distribution.
When the mean of Poisson distribution gets large, it approaches Gaussian distribution. For my experimental data of means equal (4.50±0.70)×103 and 4.40±0.21)×104 , I compared it with the theoretical value of Gaussian distribution. For example, the graph of the comparison between experimental data and theoretical value of both Poisson distribution and Gaussian distribution when mean counts is (4.50 ± 0.70) × 103 is shown below. In both cases, I can see that the experimental data fit the theoretical value within range of the error bar. When the mean is 4.5 × 103, there is a discrepancy between the Poisson distribution and Gaussian distribution. When the mean gets to 4.4 × 10^4 , the theoretical curve of Poisson and Gaussian distribution merge to a single line. The “Poisson noise” disappears. Therefore, we can see that Gaussian distribution is a special case of Poisson distribution when mean gets very big.
When there is only 1 count detected, or no counts detected at all in all trials, the distribution goes to Binomial distribution. For my experimental data of mean equals (2.00±140)×10−4, I plotted the experimental data and compared it with the logarithm of the theoretical value of Binomial distribution. As shown in the graph below, the experimental data first the theoretical value very well. So we can say that when dwell time is 0.00001s, the distribution follows the definition of Binomial distribution, where we only get 0 and 1 counts.
Overall, we can see that my data with different means agrees with theoretical Poisson distribution, Gaussian distribution and Binomial distribution within statistical error or my experimental data. Therefore, we can conclude that the radioactive decay of Cobalt-60 is a Poisson process. In addition, Poisson distribution is a limiting case of Binomial distribution, where the number of trials is large, and the probability of getting a success in any given one trial is small; when the mean is large, Poisson distribution approaches Gaussian distribution.
Hi friend, welcome to cocktail physics! I am Elisa, a junior physics major at Bryn Mawr College. From today on, I will be your bartender at Cocktail Physics. Every week, I will share with you some physics fun that I have inside and outside my classes. The cocktail will be made of a wide variety of ingredients, ranging from condensed matter physics to electromagnetic radiation and optics, from non-linear dynamics and chaos to chemical physics. It will be imported from the experiments that I do in the lab, physics articles that I read, and videos and cartoons that I watch. I hope you will enjoy the cocktails that I make.
Let’s have fun together with Cocktail Physics!