# T.B.A

## Check back frequently for Surprises!!!

• One would think that writers in the humanities would be delighted and energized by the efflorescence of new ideas from the sciences. But one would be wrong. Though everyone endorses science when it can cure disease, monitor the environment, or bash political opponents, the intrusion of science into the territories of the humanities has been deeply resented.

• In the Fourier Optics experiment, I got to revisit the concept of Fourier transform and see the experimental applications of Fourier transform in optical system. A Fourier transform is usually written in the following way:  (1) and (2).

Equation 1 means that sine waves times the weighting function g is being summed up, where  is the normalization factor. Equation 2 states that how the weighting function g is found for a given time dependent signal. Instead of being interested in a time dependent signal, this fourier optics experiment investigates spatially dependent signal, therefore,  is replaced by  (denotes spacial frequency) , and  is replaced by , and we get:(3) and (4).

If the Fourier transform is a constant for all x, for example let g(k) to be $\frac{1}{\sqrt{2\pi}}$, then f(x) is a dirac delta function of x. In such case, the Fourier transform is delocalized and the f(x) is localized! This reminds me of the uncertainty mentioned in the quantum course.

We used lens to realize the spatial Fourier transform of an object in this experiment. The experiment set up is demonstrated both in the diagram and photos below:

The neon-helium laser with a “spatial filter” (an optical processor that eliminates high frequency noise on the laser beam) on the front produces a nice uniform diverging beam. A collimating lens is placed in front of the laser to collimate the beam. Two lenses are used to take the Fourier transform of the object and the inverse Fourier transform of the original object, respectively. On the schematic, we label the three planes which we will mainly look at. If we place a white paper on the image plane, the original image shows up. On the Fourier plane, the Fourier transformation of the original plane is shown. On the image plane, the upside down image of the one shown on the Fourier plane can be displayed. We can place a camera to capture images which show up on those three different planes.

I started by placing nothing on the object plane, which means it is just a beam of roughly uniform brightness (a constant). Hence, the Fourier transform of the beam is a $\delta$ function:

The camera captured the following image when it was placed on the Fourier plane:

Then we can place the slit on the Fourier plane, where different frequencies can be separated, in either vertical or horizontal direction to eliminate either the horizontal strips or the vertical strips in the original image.

In addition, the width of the slit can also affect the appearance of the images:

For example: the first image was formed after placing the 0.04 nm slit on the Fourier plane, and the second image was formed after placing the 0.0 8nm slit on the plane:

We can also process texts through this experimental set up!

Even though the resolution is relatively low for the image shown on the Fourier plane, we can see that the horizontal line can be eliminated, leaving only the texts, in this case: “H”.

Other findings from this experiment also include that complementary objects can yield the same intensity pattern. Finally, let’s end this blog with the portrait of the pioneer of modern physics: Dr. Albert Einstein!

For more interesting theoretical explanation and fun activities relevant to Fourier optics, such as Fraunhofer diffraction and image transforms (with images), please refer to the following reference:

• Apparatus (Awesome apparatus from the last century, or maybe from the 19th century?!)

It has been a great pleasure using Bryn Mawr’s magnificent $\inline CO_2$ isotherm apparatus to measure both the volume and pressure of $\inline CO_2$ at different temperatures. The goal for this study is to determine both the critical point of carbon dioxide and the van der Waals parameters for $\inline CO_2$  Van de Waals equation derived by Johannes Diderik van der Waals in 19th century best describes the relationship between pressure and volume of a sample gas(which is different from ideal gas law):

Here, V is the gas volume, n denotes the number of moles of the gas, a denotes the intermolecular attraction between gas molecules and b stands for the volume per mole that are filled by the molecules. Due to the limitation of this the apparatus and our method in carrying out this experiment, the parameters we found from this experiment doesn’t agree with the literature results of $\inline CO_2$  gas, which states that the correction factor a is $\inline 3.640 L^2 bar/mol^2$ and b is 0.04267L/ mol.

In this experiment, $\inline CO_2$ gas is located in the capillary tube. The volume of the gas can be compressed when the mercury level rises in the tube as pressure is applied. Due to the limitation of the pressure gauge, we did not exceed 100kg pro cm^2 for the following measurements. As the pressure increases to certain point at a particular temperature, the gas starts to liquify. The pressure at which gas starts to liquify increases as the temperature increases. That explains why at high temperatures such as 56$^oC$ , when pressure is raised to 101 kg pro cm^2, no liquid appears in the capillary tube.

I would like to thank Bingqing for making dreams come true since we have worked together closely to take data points over the spring break and make the isothermal plots. Our data were taken at six different temperatures, including 28.7 ± 0.2 $^oC$, 29.5 ± 0.2$^oC$, 35.1 ± 0.2 $^oC$, 46.0 ± 0.2$^oC$, 51.3 ±0.2 $^oC$ and 56.6 ± 0.2 $^oC$.  The level of the mercury was read via the telescope, as shown in the setup photo. The following figure is a combination of V vs. P plots at different temperatures. Even though the value of a and b (from this experimental approach) after fitting is not at all close to the accepted value, the trend of the plot is reflected well from the plots. In general, as the gas volume decreases, the pressure that relates to the volume increases more rapidly at when volume is smaller. The temperature corresponding to the graph decreases from top plot(+) to bottom plot(*). In the future, we wish to optimize the experimental procedure to determine a and b more precisely.

.

The following P-V diagram is copied from Chapter 11-Properties of Pure Subtances from Heat and Thermodynamics by Zemansky. The P-V diagram above agrees well with the P-V diagram of isotherms of a pure substance qualitatively.

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[Update] Isotherms of Carbon Dioxide

In the previous experiment, we failed to find the constants $a$ and $b$ in the Van de Waals equation:

$(P+\frac{n^2a}{V^2})(V-nb)=nRT$ (Equation 1)

In fact, the experiment procedure can be modified to determine $a$ and $b$ experimentally! (Thanks to James for providing us the evidence of the power of this seemingly ‘outdated apparatus) We modified our experimental procedures based on a report written by Bryn Mawr physics alumna Shukyee Samantha Ho when she took the PHYS 331 Modern Physics Lab 18 years ago! :)

Back to the Van de Waals equation, in which $n$ is the number of moles of $CO_2$$R$ is the gas constant, $PVT$ denote pressure, volume and temperature of the gas respectively, and constants $a,b$ describe liquid and gas phases of the same substance.

Dividing n from both sides of equation 1, and rewrite $\frac{V}{n}=\bar V$, after rearranging the equation, we obtain:

$\bar V^3-(b+\frac{RT}{P})\bar V^2 + \frac{a}{P}\bar V-\frac{ab}{P}=0$ (Equation 2)

The new approach of the experiment will focus on measuring P versus V for a range of temperatures in the vicinity of the critical temperature since critical point (above which gas and liquid phase transition can no longer exist due to the lack of surface tension) can be associate with Equation 2. In fact, the critical point with several parameters, including its P,V,T is an inflection point. Therefore, both the first and second derivatives of P with respect to V (or $\bar V$) is 0  at a certain temperature. We can then associate the constants a and b to the parameters associated with the critical point.

Here is a plot of the our experimental results of the gas pressure versus gas height (proportional to gas volume) at different temperatures around the critical temperature of carbon dioxide—31.04 $^oC$. A quadratic fit ($y=-1.2682x^2+10.810 x +54.585$) represents the coexistence curve of both gas phase and liquid phase, which resembles the dash curve in Fig 11.1. Based on the fitting, we can find the critical pressure and critical volume ( can be calculated based on the corresponding height x = 4.2619 cm and the inner diameter of the tube d = 0.50 cm ) which are 77.62$kg/cm^2$ and 0.8364$cm^3$. We can also find the number of moles of the gas by using the ideal gas equation:$P=\frac{nRT}{V}$. We fitted a linear equation to the low pressure region on the P versus $\frac{1}{V}$ plot to find the slope, dividing the slope by R, we can get $n=5.44\times10^{-3}mol$

[Comments on our plot] This plot is in good agreement with Figure 11.1. The dots on the horizontal lines are in the range of mixture of liquid and vapor, meaning that gas and liquid can coexist in this range. As we can see, the temperatures that correspond to these lines are all below critical temperature of carbon dioxide. The black dots are on the curves close to critical isotherm, which intercept with the critical point.

So far, we have obtained the values of critical volume =$0.8364\times10^{-3}L$, critical pressure = $75.292 atm$,  critical temperature =  31.04 $^oC$ (literature) which can also be approximated by our data (31.4 $^oC$),  and the number of moles of the carbon dioxide: $n=5.44\times 10^{-3}mol$

Using the following equations derived from the first and second derivatives of Equation 2 $P=\frac{a}{27b^2},\bar V = 3b, (aka.V=3bn), T=\frac{8a}{27bR}$, we can finally calculated the two constants:

$a=4.27L^2atm/mol^2$  (or $a=4.33L^2bar/mol^2$and$b=0.513L/mol$.

Compared with the literature values for $CO_2$, with $a=3.6073L^2bar/mol^2, b=0.04282L/mol$, our experimental result this time is much better than last time. Hooray!

• $\huge \alpha$ Spectroscopy Introduction

The purpose of this experiment is to study the detection of alpha particle radiation from Am source and the attenuation of this radiation during interaction with air molecules with different densities, realized by altering pressure.

Alpha particles (helium-4 nuclei) come from the decaying process of Americium-241 source or other unstable heavy isotopes, for instance, Radium-226 and Uranium-238.

Experimental Setup:

——The vacuum system and the pump (on the ground).

Quick fact:The energies of α-particles produced from nuclear decays range between 4 MeV and 10 MeV.

Since the traveling distance of alpha particles in 1 atm air is just a few centimeters, we study alpha spectroscopy taken in the vacuum condition so that the particle can be captured by the detector. The amplifier pulses can be sent into a multichannel analyzer whose results can be displayed on the laptop interface.

For the first part of this experiment, we need to do the energy/channel calibration. First, we adjust the gain until the peaks displaying on the number of counts vs. channel number plot appear close to the edge on the right-end side. The following alpha spectrum, displayed on the multichannel analyer (MCA) was obtained after an 10 hour experiment for acquiring data under the detector bias voltage of 125 V.

Side note: Why do we use a detector bias voltage of 125 V?

Surface barrier detector operations experiment investigates the relationship between the bias voltages and the FWHM (full width at half maximum). At lower bias voltages, such as 25 V, 90 V, the FWHMs are larger, meaning a bigger energy broadening  takes place. Within the bias voltages tested, 125 V presents the most desirable spectrum with fairly concentrated energy, with more distinguished peaks and thinner spread of the energy range in the vicinity of the peaks. In addition, Ortec manual also suggested us to use 125 V as the bias voltage.

As we can see, there are three peaks. Gaussian fits to each peak will probably provide information on channel numbers of the peaks.

Since the three peaks on the first plot correspond to three energy levels of alpha particles, we did three crystal ball fits (with effect similar to gaussian fits to the data with end “tails” of the data truncated before fitting) to the plot to find the channel numbers where peaks appear. By relating the channel numbers to the alpha energy levels (5.388MeV, 5.443MeV, 5.486MeV), we can find a linear relationship between channel number and energy:

Channel Number = 1265 * Energy (MeV) +77

This is the calibration of my experiment based on the gain I have chosen, which is 100.6, as displayed on the DLA (delay line amplifier).

Why are there three major energy levels?

-In fact, 241Am can decay and there are three main alpha decays (5.388MeV—1.4%, 5.443MeV—13.6%, 5.486MeV—84.4%). The number of counts in the first figure is proportional to the percentage of decayed elements.

*The following chart(alpha-particles table) lists the percentages of 241Am decay:

http://georg.pnpi.spb.ru/english/alphas/z095.html

Since the relationship between energy and channel number can be obtained based on the knowledge above, for further measurements (displayed in terms of Channel number and counts), one can determine the energy related to each channel.

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The last part of this lab is to find energy loss of charge particles at different pressures. Air is pumped into the vacuum chamber, leading to different absorber thickness. The pressure is increased from 22 Torr to 541 Torr for acquiring data—the channel numbers of the peaks and total counts in the peak. Since channel numbers corresponds to energy, a plot of energy versus chamber pressure can be obtained, as shown below:

In some literature, for example “An experiment to measure range, range straggling, stopping power, and energy straggling of alpha particles in air” by Ouseph and Mostrovych, they call dE/dx,in which x denotes the absorber thickness, as the stopping power. The absorber thickness is proportional to the ratio between the chamber pressure and the air pressure (760 Torr), and the distance between the detector and the source.

$x=\frac{P_{chamber}}{P_{air}}D$

Therefore, dE/dP versus P  plot will shall the same trend as that of dE/dx versus x plot. The following plot shows dE/dP as a function of chamber pressure. As we can see, the slope of the plot remains positive and constant in the beginning, and then increases, at some point around P = 500 Torr, the slope decreases.

If the plot displays dE/dx (stopping power) as a function of absorber thickness, we can see that once after reaching certain absorption thickness, as the thickness keeps increasing, the energy loss starts to occur. This observation makes sense because when the absorption thickness further increases, fewer alpha particles are capable of reaching the detector. Furthermore, the higher the pressure is, the larger the absorption thickness is, indicating more collisions between air molecules and alpha particles which can lead to more energy loss before the detector detects the particle. As a result, we will see a negative slope after the spike.

The very last diagram shows the alpha spectrum taken at various pressures. As the pressure is lower (closer to vacumm state, the higher energy does the alpha peak has), the position of the alpha peak moves to a higher-energy side.

• I. Overview of the experiment

In this experiment, the  basic properties of semiconductor are investigated, and the $e/k$ ratio can be determined.

Semiconductor material is among one the three classes of solid state materials, including insulators, semiconductors and conductors. Electrical conductivity $\sigma$ for semiconductor ranges from $10^{-8} s/cm$ to $10^{-3} s/cm$, between that of insulator and conductor. When “doped” with impurity atoms, electrical insulator pure silicon transforms into a semiconductor.

Silicon semiconductor, used in this experiment, is an element semiconductor which contains atoms piling up in a three-dimensional periodic fashion. Electrons of an individual atom have discrete energy levels. When two identical atoms are far away from each other, both of them have the same degeneracy energy level.. However, as many identical atoms are closely assembled, the degeneracy energy level can split into several separate yet closely spaced energy levels as a result of atomic interactions, resulting in a continuous band of energy. As the distance between identical atoms decreases to certain degree (5.34 $\r{A}$ for silicon), the continuous band splits into two bands-conduction band and valence band, separated by a region, namely forbidden gap, with bandgap energy $E_g$. Electron energy levels cannot exist in this region. Transistors are semiconductor devices that serve as amplifiers and switches for electronic signals and powers.

The transistor studied used in this experiment is a bipolar transistor, which was first invented in 1947. In bipolar devices, both electrons and holes participate in the conduction. Hole refers to the state in which the valence band has experienced missing electrons, since the electron left the valence band and occupied the conduction band.  Conceptually, hole can be regarded as a positive charge “e” since the movement of an empty electron state (also known as hole movement) is along the electric field direction, whereas the electron moves in the opposite direction to the field. Knowing the presence of electrons and holes, scientists has defined semiconductors a materials in which electrons fill up the valence band, and the conduction band remain empty at 0 K. Different from unipolar devices, dominated by only one type of charge carrier, bipolar transistors operate with two kinds of carriers: electrons and holes. For silicon, the hole masses are much larger than that of an electron, and electrons are the major charge carriers.

Depending on whether electron is the majority carrier, n-type or p-type is used to characterize its property.  Nowadays, bipolar transistors consist of silicon substrate and two coupled p-n junctions, either in p-n-p type or in n-p-n type. A heavily doped region called emitter (E), a narrow central region named base (B) and a lightly doped part called collector (C) are the three terminals in a bipolar transistor. The current flows from collector to emitter in an n-p-n transistor as shown in the following figure,

and flows in the opposite direction in a p-n-p transistor. In this experiment, silicon transistor in the form of n-p-n was used.

Based on this experiment, the current-voltage relationship of a p-n junction in the bipolar transistor measured at five different temperatures, ranging from 84 K to 368 K, exhibits significant physical properties of these semiconductor junctions. From the relationship, the ratio of $e/k$ an be determined. Here, $e$ denotes the electron charge, and $k$ is the Boltzmann’s constant. In addition, the band gap energy $E_g$ of silicon can also be obtained based on these measurements.

II. Experimental setup

The schematic of the circuit used in this experiment is shown in the figure below:

A Keithley 6485 picoammeter with an upper range of 20 mA measured the short-circuit current$I_c$ . A Keithley 179 A multimeter served as a voltmeter, and a Tegam 869 digital thermometer monitored the temperature . An n-p-n silicon power transistor (TIP 3055) was used for the entire experiment which includes measurements taken at five distinct temperatures:

1)Liquid nitrogen—measured temperature: (80.15$\pm$0.05) K

2)Dry ice and isopropyl mixture—measured T: (198.15 $\pm$ 0.05) K

3) Water ice mixture—measured T: (274.15$\pm$ 0.05) K

4) Room temperature—measured T: (295.15$\pm$ 0.05) K

5) Boiling water —measured T: (368.15$\pm$ 0.05) K

In all circumstances except in liquid nitrogen and dry ice, the transistor was first suspended in an tube filled with oil to keep steady temperature. When at relatively low temperatures, such as in liquid nitrogen and in dry ice with isopropyl alcohol, the transistor can be directly submerged in small dewars to reach the desired temperature. At each temperature, a series of the short-circuit current $I_c$ read from the picoammeter, and the relative voltages, base-emitter (B-E) $V_{BE}$recorded on the voltmeter, were registered.

III.Results

A plot of $I_c$  vs $V_{BE}$ on semi-log graph has a slope of $e/kT$  Knowing the temperature, $e/k$ can be measured. For this experiment, $I-V$ plots at five different temperatures were obtained.

Figure:A plot of natural logarithm of current $I_C$ versus applied voltage $V$ for the  $p-n$ junction in silicon semiconductor.

Processed data based on the plot above:

So far, we have obtained the e/k ratio calculated from the experiment.

Next, we need to find the bandgap energy.

Figure: Natural log plot of the minority carrier current $I_0$ versus inverse of temperature for the  $p-n$ junction in silicon semiconductor. The slope of the lease-square fitting line is (-1.408 $\pm$  0.015) *10^4, which corresponds to $-E_g/k$

Knowing the slope of the plot above, we can calculate the bandgap energy of silicon, which yields ( 1.21$\pm$ 0.02) eV. The theoretical value  of a silicon semiconductor at 300 K is 1.12 eV. The reason why the experimental value is higher than theoretical value attributes to the relatively high impurities (higher doping level than predicted) of the transistor in this experiment, temperature variance and some simplification to factors (such as ignoring temperature factor) used in calculation.

In conclusion, the experimental results for $e/k$ is  (1.15 $\pm$ 0.04)$\times$ $10^{-4}$ C-K/J, which is in good agreement with the accepted value (1.160485 0.00006)$\times$$10^{-4}$ C-K/J. The calculated bandgap energy for the silicon semiconductor is ( 1.21$\pm$ 0.02) eV, which is reasonable for the material.

• Theoretical Background and Introduction:

When a positron (a.k.a the antimatter counterpart of the electron, or the antiparticle of the electron)  and an electron annihilate, two gamma rays are emitted. The purpose of this lab is to identify the  γ rays with opposite directions created simultaneously during the annihilation in order to determine both the angular correlation of their emission and the total activity of Na-22 source. The following diagram shows the process of electron-positron annihilation (http://tech.snmjournals.org/content/29/1/4/F1.expansion.html):

In this lab, electron-positron annihilation was studied for a radioactive source: Na-22, which decayed by positron emission to the form Ne-22:

1)${_{11}^{22}\textrm{Na}}\rightarrow _{10}^{22}\textrm{Ne*}+_{1}^{0}\textrm{e} + _{0}^{0}{\nu_e}$ (weak interaction)

2) $_{10}^{22}\textrm{Ne*}\rightarrow _{10}^{22}\textrm{Ne} + \gamma$ (strong interaction)

3) $_{1}^{0}\textrm{e}+_{-1}^{0}\textrm{e}\rightarrow \gamma + \gamma$ (weak interaction)

Here, e with a subscript “1” is a positron, and the one with a subscript “-1” is an electron.

In the third interaction above, the two gammas both have the rest mass of an electron to conserve energy. Hence, both of them have an energy of 511KeV ($E = mc^2 \,\!$). Besides, momentum is conserved since two gammas are emitted in opposite directions (180 degrees apart). Theoretically, we want to look at the simultaneous gamma rays with an

energy of 511 keV after distinguishing them from other gammas, such as those origin from second reaction, and secondary interactions in the detectors.

Experimental setup:

Figure (left hand side) 1) Na-22 source 2)Oscilloscopes 3) Gamma emission spectrum  with 0.511MeV peak eliminated by the gate (This plot was reproduced on KaleidaGraph based on data acquired by the MCA).

(right hand side) the Bicron scintillation detector.

The Bicron Scintillation Detectors close-ups:

Detector 1 is connected to the preamplifier 1, and detector 2 to the preamplifier 2. Both detectors are connected to the ORTEC counter. Each detector consists of an 1-inch diameter crystal of NaI and a photomultiplier sealed in the chrome-plated metal case to exclude water and light.

—How was a spectrum obtained in this experiment?

First of all, gamma ray incidents on the detector and produces scintillation light in the NaI crystal. As a result, photoelectrons liberations in the PMT (photomultiplier tube) is stimulated. This process very much resembles the photoelectric effect discussed in physics lectures. Therefore, we know that the number of photoelectrons are proportional to the energy of the original gamma ray. In particular, the number of photoelectrons can also be quantitatively measured by detecting their integrated photocurrent. The flow chart below explains how the signal was transferred from PMT to DLA and MCA, then was ultimately displayed by the program Genie 2000.

In this experiment, I started by placing the sodium source against the detector, so that NaI was facing the PMT (photomultiplier tube sensor). The way PMT works was based on the photoelectron effect. The detector detected the gammar rays to create electric pulses. When the voltage applied across the PMT was higher, the electron had a higher velocity and higher energy, resulting in larger pulse amplitude. PLA enhanced the electric pulse from the detector and sent it  to the DLA (delay line amplifier) which further amplified and shaped the pulse so that these pulses could be easily sorted based on pulse height. By connecting a BNC Tee between the PLA and DLA, oscilloscope could be connected to display the pulses. Multiple peaks appeared on the oscilloscope, including one that was especially bright. In fact, this peak was the signal that corresponded to the gamma ray with 0.511MeV energy. For both detectors, the same connections (PMT—>PLA—>DLA—>MCA) were set up. Oscilloscopes were used to check whether the connections worked as desired.

DLA had two output modes: unipolar output and bipolar output.  Unipolar output channel was connected to the multichannel analyzer (MCA) to acquire a spectrum of the gamma emission from Na-22 by Genie 2000 software. On the Na-22 spectrum there was a peak around channel 116, which is the large electron-positron annihilation photopeak. At around channel 307, there was a relatively small photopeak.

Once after the Na-22 spectrum was obtained for both two systems (connected to detector 1 and detector 2), we needed to select the electron-positron peak. A TSCA (Timing single channel analyzer)with time delay knobs was used to make sure that the two gamma rays arrive at the detector at the same time. The TSCA can filter pulses by establishing the upper and lower pulse height settings. By adjusting the TSCA window, we could eventually selected only the 511 KeV gamma rays from the electron-positron annihilation. Again, on the oscilloscope, the brightest peak corresponds to 511 KeV gamma rays. With the help of TSCA, we could make sure that the two gamma rays we looked at in the same time interval actually originated from the same annihilation. Then, we connected the output of DA to the BNC cable from the MCA(multichannel analyzer). MCA GATE was then used to obtain the 511KeV photopeak. On the spectrum however, the peak would not show up. The channel numbers that corresponded to the 511 KeV peak were between 99 and 135 for system I and 165 and 207 for system II. At this point, the upper and lower window for both systems should be fixed for the upcoming parts of the experiment.

The following images consist of displays from oscilloscopes and two spectra of the gamma emission from Na-22 source displayed on the Genie 2000 software during the experiment:

The following part was coincidence measurement. We needed to make sure that the pulses which originated at the same time in the beginning reached the detector at the same time as well. The distances between the source and the detectors were set to be equidistance. However, the different distance between cables of two systems could lead to signals not arriving simultaneously. To adjust, we attached the preamplifiers to a pulser whose universal coincidence unit can produce a logic output pulse when two pulses arrive simultaneously. We used an additional oscilloscope (oscilloscope 5) to adjust the TSCA delay until we could see  overlapping signals from the two systems. The following plot shows coincident pairs in 10 seconds versus delay time (coincidence-vs-delay graph):

Since$2\tau = (1.96\pm 0.02)\mu s$, the resolving time $\tau$ of the system is  (0.98 ± 0.01)μs.

Finally, after completing the set up of the apparatus, we measured the real activity of the source and the angular distribution of the coincident gamma rays.

We allowed two gamma rays that arrived at the two detectors within a time $2\tau$ to be called ‘coincident”. The actual observed count rate C includes both the true coincidence $e_1e_2fR$(where e’s are the efficiency of the detector, f is the fraction of emitted gamma ray that enters the detector) and the chance count rate $2\tau R_1R_2$. Therefore,

$C=e_1e_2fR+R_1R_2(2\tau)$Based on my measurements, R=7.80 μCi, which is almost ten times larger than the registered activity on the source label: 0.8776 μCi. So far, I haven’t found out the particular factor that has led to this error.

The angular distribution of coincidence counts:

After the Gaussian fit for this data set, we obtained:

$f(x) = (427.8\pm 13.5)\times e^{\left ( -\frac{x-(177.3\pm0.2)}{6.9\pm 0.2}}\right )^2$

The peak of the Gaussian fit appears to be at (177.3 ± 0.2) degrees, which is close to 180 degrees. The experimental result agreed with the prediction that the two gamma rays are emitted in the opposite direction. In this experiment, the source was very close to the two detectors. Considering the source diameter and the distance between the detectors and the source, we concluded that it is reasonable for the angle to be within the neighborhood angles of 180 degrees. If this experiment is repeated with two detectors a kilometer apart, the peak will appear at 180 degrees and the thinner will the width of the peak be.

• Electron Spin Resonance (ESR) is an important phenomenon which occurs as an electron responds to an external magnetic field B by switching between two distinct energy levels, absorbing or releasing energy in the form of a photon at the same time. It is the intrinsic spin of the electron that gives rise to such effect. The transition between levels can be observed by looking at absorption spectra of EM radiation from the spectrometer. In this experiment, the photon frequency can be set, the value of the magnetic field and the resonance voltage are measured. As a result, the magnetic moment of this unpaired electron can be calculated. By learning the ESR of electrons, one can study their local environment. For this reason, ESR is used extensively in physics, chemistry and biology to study materials with unpaired electrons. It is an analytical tool to study the complicated electrostatic and magnetic interactions in molecules with free electrons. NMR, which some of you might have used for analyzing proton environment in chemistry class, also based on the similar concept as ESR, except that NMR studies the behavior of proton, instead of electron.

In this experiment, we study the DPPH sample which provides an unpaired electron ( a quasi-free electron, not exactly a free electron). The ultimate purpose of this lab is to determine the g-factor of this unpaired electron. g-factor can be calculated after finding the magnetic moment of the electron.

Here is the set up of this experiment:

As you can see in the experimental setup, the ESR spectrometer consists of several parts, including the parallel Helmholtz coils which can create an external magnetic field when current follows through. The DPPH sample, inserted into a RF coil, is placed at the spatial center of the two coils. The way RF coil works is that RF coil serves as an inductor.

This diagram shows both the field voltage(sinusoidal) across the resistor VS time, and three ESR absorption spectrum obtained at different resonance frequencies. One can adjust the photon frequency by turning the knobs on top of the RF oscillator.

The red line corresponds to a frequency of 32.4 MHz, the green line relates to a frequency of a frequency of 50.1 MHz, and the blue line represents the spectrum taken at f=60 MHz. As you can see, when the frequency becomes higher, the two adjacent dips get closer, corresponding to larger external fields. As the frequency continues to increase, the two dips come even closer and no splitting can be observed. The voltage can be increased by turning the knob on the variac.

This figure below shows three ESR absorption spectrum taken at different radio frequencies and a sinusoidal wave, which corresponds to the field voltage. The red, green and blue dips correspond to energy absorption process of electrons with photon frequencies at 37Hz, 42Hz, and 50Hz, respectively. As you can see, when the frequency increases, the corresponding resonance field voltage at which resonance occurs increases, which is closer to the peak of the sinusoidal wave.

After recording the field voltage at which resonance occurs (resonance field voltage V) and the corresponding photon frequency($\nu$), we obtain a plot (V versus $\nu$) as shown bellow:

The fitted equation is $V=(8.26\pm 0.35) \times 10^{-9} \nu + 0.01$. Since the intercept 0.01 is relatively small, when calculating the ratio between frequency and voltage,  the value is approximately the value of the slope $G=(8.26\pm 0.35) \times 10^{-9}$.

This slope G is used in the calculation of Lande g-factor based on the following equation:

Knowing that $\large h= 6.626\times10^{-34} J \cdot s$$\large \inline D=\frac{B}{I}=\frac {\mu_0NR_{coil}^2} {(R_{coil}^2+z^2)^{3/2}} = (4.269\pm0.273) \times 10^{-3} T/A$, $\large R=1\pm 0.001 \Omega$, and physical constant Bohr Magnetron $\large \inline \mu_B=\frac{e\hbar}{2m_e} = 9.27400968(20)\times 10^{-24} J/T$, the Lande g-factor of the unpaired electron calculated from this set of experiment is 2.0 ± 0.2. The literature value of gDPPH is 2.00316. Therefore, our experimental result is  consistent with the literature value. Note that gDPPH is a little higher than the g-factor of a free electron which is 2.002319.

• Hi everyone, welcome to Maggie Xiao’s niche in the Experimental Physics course under the supervision of a brilliant professor (Dr. James Battat) and accompany of other seven inspiring allies (Alisha, Bingqing, Danqi, Elisa, Orsola, Wenqi, Xilei)  at Bryn Mawr College.

• (Source: wisegeek.com)