07 5 / 2013

## PVT in Carbon Dioxide with A Different Approach

Thank to the paper of a graduated physics major, Shukyee Samantha Ho, digged out by James, Maggie and I are able to return to the carbon dioxide experiment and to find the constants a and b in the state equation of the Van der Waals gas with a new approach.

The equation of state can be rewritten as a cubic equation in molar volume by some mathematical arrangement of the original state equation.

The equation above can be rewritten as the follow:

One of the remarkable feature of the equation of state expressed as a cubic equation of molar volume is that it can describe both the gaseous and the liquid phases for a substance. Last time, we measured the isothermal curves for six different temperatures ranging from 29.0 celsius to 56.6 celsius. This time, we focused on the measurement under the isothermal condition with the temperature in the neighborhood of the critical temperature of carbon dioxide, 31.04 celsius. The critical temperature is the temperature of the critical point of a certain substance, above which a distinct gas-liquid phase transition does not exist any more due to the disappearance of the surface tension. Below the critical temperature, a clear gas-liquid coexistence can be observed from an isothermal P-V curve as shown in the figure below.

The contour of the blue shadow area in the figure is called the coexistence curve, because any point under the curve corresponds to an equilibrium of gas-liquid coexisting. Noticeably, as the temperature gradually approaches the critical temperature, the flat coexistence region on the isothermal curve becomes narrower and eventually disappears at the critical temperature. As shown in the figure, the critical point with T_{c}, V_{c}, and P_{c} is an inflection point, which indicates that at constant temperature T_{c}, the critical pressure P_{c} and the critical volume V_{c} must fulfill the following conditions:

By taking the first derivative and the second derivative of the equation of the state of Van der Waals carbon dioxide, the critical volume, the critical temperature and the critical pressure can be expressed in terms of the constants a and b:

It means that the constants a and b can be determined once we find the critical point for the carbon dioxide.

The experimental setup is exactly the same as the last time. The only difference is that temperatures were controlled near the critical temperature while doing the measurements. Five measurements of P-V were carried out at 24.1 celsius, 26.9 celsius, 29.9 celsius, 30.2 celsius and 31.4 celsius. The plot pressure versus the height of gas for each isothermal condition is shown in the figure below.

The different colors represent different temperatures. The coexistence curve is determined based on the flat region of each isothermal curve and fitted with a quadratic function. The maximum y value of this quadratic function and the corresponding x value are the critical pressure and the critical volume (in terms of the height of the gas) that we want to find—77.62 kg/cm^2 and 4.2619 cm. The pressure is then converted to “atm” and the volume of the gas is (4.2619 cm)*(0.25^2)*pi=0.8364 cm^3. This time, we also recalculated the value of n, the moles of carbon dioxide in the capillary tube. The volume-pressure data of the measurement at the highest temperature from the last time was used. To approximate the ideal gas condition, only the data points of P and 1/V at low pressure region were plotted and fitted with a linear function. The number of moles is determined with the state equation for the ideal gas: P=(nRT)*(1/V). With this method, the moles of carbon dioxide is found to be 5.44*10^-3 mol.

By now, we have all the values we need to know in order to calculate the Van der Waals constants a and b: critical temperature=31.04 celsius, critical volume=0.8364*10^-3 L, moles of carbon dioxide=5.44*10^-3 mol, and critical pressure=75.292 atm. The result of a is 4.273 L^2*atm/mol^2, and b is 0.05125 L/mol. The accepted values for a and b are 3.6073 L^2*atm/mol^2 and 0.04282 L/mol respectively.

06 5 / 2013

## Relativistic Beta Spectroscopy

Beta particles are electrons or positrons emitted during a beta radioactive decay. When a nucleus of certain unstable atom has too many neutrons or protons, it can be stabilized by emitting a relativistic electrons and an antielectron neutrino or a relativistic antielectron and a neutrino. As a result, the original nucleus decays into a different nucleus with the atomic number +1 or -1. Based on the conservation of energy, in the weak interaction decay described above, the electron and the antielectron neutrino or the antielectron and the neutrino share a certain amount of kinetic energy resulted from the difference in rest energy between the original particles before decaying and the new particles. In this experiment, a radioactive source is placed into a magnetic field so that the electrons moving in the perpendicular direction with respect to the magnetic field with the charge of -e generated from the beta decay experience a magnetic force, the Lorentz force. The Lorentz force applied on the electrons gives them a centripetal acceleration to have them move in a circular orbit in a brass chamber. The electrons with right kinetic energy can finish the circular trajectories (the red dashed line) and arrive at the detector as shown in the following figure.

According to Newton’ second law, the magnetic force applied on the moving charged electrons equals the time derivative of momentum; and relativistically, the momentum is related to the mass and velocity of the electrons. Therefore, the magnitude of the magnetic field and the corresponding moment have the relationship of evB=dp/dt. With the energy-momentum four-vector, the relationship between the maximum kinetic energy of the electron detected by the beta detector and the magnetic field can be expressed as:

where e is the electrical charge of the electrons, R is the radius of the circular trajectory of the electrons, c is the speed of light and E_{0} is the rest energy of the electrons. By measuring the kinetic energy K corresponding to the maximum counts of electrons emitted during the beta decay from a certain radioactive source for the varying magnitude of the magnetic field, the B^{2}/K versus K curve can give the information to find the elementary electrical charge and rest mass of the electrons.

The experimental setup includes a brass vacuum chamber which hosts the radioactive sources and the beta particle detector, a vacuum pump and a gauge to create and monitor the vacuum environment during the measurement, an electromagnet to generate the external magnetic field applied on the electrons and a current supply for the electromagnet. The Hall probe is inserted into the probe holder in the brass chamber and the magnitude of the magnetic field can be read by the Gaussmeter connected to the probe. Same as the data acquisition part of the alpha spectroscopy, an oscilloscope, a bin combining both of the detector bias supply and the amplifier (NIM), a multiple channel analyzer and a voltmeter to check the voltage supply to the detector.

The measurement in this experiment started with the energy calibration. Six radioactive isotopes were used to take the beta spectra in non-vacuum environment without the external magnetic field. The radioactive source was directly put in front of the detector. The beta spectra of these six isotopes for the calibration are shown as below.

The best estimated channel number corresponding to the maximum kinetic energy for each isotope and the uncertainty are plotted. The conversion between the channel number read from the spectra and the energy can be obtained from the linear function in the figure below. (The data points from Cs-137 and Sr-90 were omitted when plotting the channel-energy curve because they considerably deviated from the linearity.)

Then, the brass chamber was connected to the vacuum pump and placed into the electromagnet. The magnetic field changes linearly with the current flowing through the coil inside the electromagnet. During the measurement, the seven different magnitude of current was set from 0.289 A to 2.813 A to generate the applied magnetic field with different magnitudes. The radioactive isotope used in this measurement was Cl-36. The beta spectrum of Cl-36 under different magnetic fields was taken and the energy (channel) at peak intensity for each spectrum was recorded. The different colors of data in the following figure represent different magnitude of the magnetic field.

To find the elementary charge and the rest mass of the electrons from beta decaying, the square of the magnetic field over the kinetic energy at peak instenisty of electrons (B^2/K) in Tesla^2/Joules versus the maximum kinetic energy (K) in Joules is plotted. Based on the equation described earlier, the slope of the linear function yields the inversed (eRc)^2. The radius R is 0.0508 m and the c is the speed of light which is 2.998*10^8 m/s. The calculation result for the e is 1.35*10^-19 C, which is closed to the accepted value of e, 1.60*10^-19 C. The y-intercept of the plot give the information to calculate the rest energy of electron. The calculated rest energy of electron is 4.14*10^-14 Joules. The accepted value of E_{0} is 8.19*10^-14 Joules.

The errors which led to the discrepancy from the accepted value might include: 1) the magnetic probe was not fixed perfectly in the center of the brass chamber; 2) the estimations on the maximum kinetic energy were not very accurate; 3) the beta detector was supplied with 90 V voltage (suggested voltage should be 125 V), which resulted in yielding low resolution spectra.

05 5 / 2013

## Alpha Particle Spectroscopy

Alpha particles, first discovered by the English scientist Ernest Rutherford, are a type of charged particles from radioactive sources. An Alpha particle has the same structure as the nucleus of helium, which is composed of two protons and two neutrons, resulting in a positive charges of +2 for the overall particle. Certain large unstable atoms with high atomic number tend to obtain stabilization by emitting alpha particles in a form of ionized radiation, if they have a relatively small ratio of neutrons to protons in their nuclei. In this experiment, the alpha particles emitted from the radioactive source Americium-241 are detected with a silicon surface barrier detector. The attenuation of this ionized radiation during interaction with air are studied based on the alpha spectra obtained from measurements under different pressures.

The experimental setup is shown in the photos where most apparatus are labeled as follows. The radioactive source and the silicon surface barrier detector was placed in a stainless steel vacuum chamber, which was connected to a vacuum pump and a vacuum gauge to generate and monitor the different levels of vacuum environment during the experiment. A nuclear instrumentation module (NIM) bins functioned as a delay line amplifier, as well as supplied power to the surface barrier detector. A multiple channel analyzer was connected to a laptop installed with the data acquisition software MAESTRO. A voltmeter was connected to the detector bias supply to monitor the voltage of the detector. During the experiment, in order to get high resolution of the spectrum, voltage was set to be about +125V. The data pulses could be observed via the oscilloscope.

The measurements in this experiment includes two parts: 1) the energy calibration via an ten-hour data acquiring; 2) the energy loss of charged loss. The figure below shows the Am-241 alpha spectrum obtained from 36,000-seconds acquisition. The expected spectrum is dominated by three characteristic alpha-emission peaks: 5.486 MeV, 5.443 MeV and 5.388 MeV.

The spectrum can be simultaneously fitted with three Gaussian distribution functions. The mean of each functions yields the channel number of a corresponding peaks 5643+/-543, 5606+/-2 and 5585+/-6. Known the energies of the three peaks, the conversion between the channel number and the energy can be expressed as a linear function as shown in the figure below.

The alpha spectra of Am-241 taken at different levels of vacuum are shown in the figure below. The pressure was increased with a step size of about 30 Torr, starting from 0 Torr until the peak disappearing at 567 Torr. Each peak can be fitted with a Gaussian distribution function. The mean value, the integral of the under-curve area and the full width at half maximum (FWHM) give the information to study the attenuation of the charged alpha particles in air.

Here shows the alpha counts as a function of air pressure in the chamber. The unit of the pressure is converted to Psi. As shown in this figure, the alpha particles detected by the detector decreases dramatically as the pressure of the vacuum chamber reaches around 10 Psi. Since I did not take enough data points in the range of 10 Psi to 11 Psi, the range straggling parameter could not be determined precisely.

This figure shows the energy of alpha radiation at different thickness of air. The absorber thickness is defined as the distance between the detector and the source multiplied by the ratio of pressure in chamber to the pressure of air: (5.9 cm)*(Pressure of the chamber/760 Torr).

By taking the derivative of the plot above, the stopping power, dE/dx, as a function of the absorber thickness can be obtained as shown below. The value of dE/dx agree with previous reported values (around the 2 MeV region). Also, the plot shows the trend that the velocity dependence of the stopping increases with the decreasing velocity.

The energy straggling variations for different vacuum environments are plotted as a function of absorber thickness in the following figure. The increased FWHM results from the increment in energy straggling. Therefore, as shown in this plot, the straggling increases rapidly with the increased absorber thickness after the thickness reaches about 2.8 cm.

01 4 / 2013

## PVT in Carbon Dioxide

An ideal gas is distinguished from real gases by two special properties of ideal gas particles. First, an ideal gas particle has the negligible volume and mass; second, ideal gas particles collide with each other elastically, during which the kinetic energy of gas molecules remains constant. An ideal gas is a theoretical gas which does not exist in reality; however, at high temperature and low density, most real gases behave qualitatively like an ideal gas. It is because the kinetic energy is high due to the high temperature, which makes the energy loss during the inelastic collision less significant, and the size of the gas molecules becomes negligibly small compared to the giant empty space between them. In those cases, the state of these gas molecules can be described with the ideal gas law: PV=nRT, where P is the inner pressure, V is the volume of the gas, n is the number of moles, T is temperature in Kelvin and R is the gas constant (8.314J/K*mol). However, in some cases, the gas particles have a non-zero volume and considerable intermolecular interactions with each other. The Van der Waals equation is an adjustment to the ideal gas law to more accurately describe the state of gas molecules in the situation above. Two parameters are introduced in the Van der Waals equation: [P + a/(V^2)](V - b) = RT, where a is the correction term for the intermolecular force and b is the compensation term for the finite molecular size of gas particles. The V here is the molecular volume.

In this experiment, Maggie and I used the “truly magnificent CO_{2} isotherm apparatus” to measure the relationship between pressure P and volume V for a fixed amount of carbon dioxide at an isothermal environment. For the ideal gas, P=nRT/V; for the Van der Waal gas, P=[RT/(V/n - b)] - [a/(V^2)]. Here is the photo of the magnificent apparatus which was built about 100 years ago. The experimental setup includes a 120V transformer as the voltage supply, a water sink as a thermal reservoir to keep the isothermal environment of the glass jacket and the carbon dioxide gas in the capillary tube. A oil-mercury pump is used to apply the external pressure to the gas molecules by suppressing the gas with mercury. Assuming that this is a reversible process, the external pressure applied equals the inner pressure of the carbon dioxide gas. The telescope should be carefully arranged in a fixed angle and position with respect to the capillary tube (except for the height) after calibration with a meter. After the water from the thermal reservoir flowed to the glass jacket and created the stable isothermal environment for the carbon dioxide gas, we started to pump the mercury into the capillary tube and recorded the pressure and the corresponding height of the mercury. The P-H curve under six different isothermal conditions were measured: 29.0 celsius, 35.1 celsius, 40.7 celsius, 46.0 celsius, 51.3 celsius and 56.6 celsius. The uncertainty of the temperature read is 0.2 celsius. One of the noticeable trend is that as the temperature increases, the pressure needed to liquify the carbon dioxide increases. The volume of the carbon dioxide can be calculated from the read of the height of the mercury by: (radius of the capillary tube)^2*pi*(height of the capillary tube - height of the mercury). The pressure here is measure in kg/cm^2, which can be converted to the atm or bar by multiplying by 0.968 or 0.98. Then we are able to plot the P-V curve for each measurement from lowest temperature to highest temperature (from the bottom to the top). The isothermal P-V curve plotted from our data are qualitative well agree with the theoretical prediction, from which the the change of pressure increases for the small values of the volume. Here is the theoretical plot .

Then I tried to find out the n, the number of moles of the carbon dioxide inside the capillary tube. First, I tried to record the temperature, pressure and the volume of the liquified carbon dioxide, so that I can later search the density of liquid carbon dioxide at that P and T and obtain the mass of the carbon dioxide. However, the idea is nice but naive. There is no comprehensive record that includes all density for carbon dioxide for all PV conditions.

Next, I tried to plot the P versus 1/T curve for each temperature to see which one behaves most like an ideal gas whose P-1/V plot should be linear. It turned out the lowest temperature is the most linear one. I fitted the P and 1/V data with linear function. The slope of the line should equal nRT. The n value I got is 0.0028155 mol. Everything looked so promising so far. I used this value to calculate the molar volume of the carbon dioxide in the capillary tube, and fitted the P-V curve from the previous figure with the Van der Waals equation. The fitting was horrible. The parameter a and b obtained from fitting is unacceptably off. (For your information, the accepted values for a and b are 3.6073 L^2*atm/mol^2 and 0.042816 L/mol.

It was disappointed that we were not able to gain the reasonable result for a and b. However, it was somewhat comforting that we obtained the qualitatively desired P-V curve for the carbon dioxide in isothermal environment.

(Source: chem.queensu.ca)

27 3 / 2013

## Poisson Statistics

Nuclear decay is a stochastic random process. In such a process, even with the precise detection of previous events of the emission of gamma ray photon by a radioactive nucleus, there is no way to predict the happening of the next event. There are three main distribution functions of interest describing the random events: the Binomial distribution, the Poisson distribution, and the Gaussian distribution. Each of these three distribution function has its own optimal fitting situation.

The Binomial distribution describes the discrete probability distribution of obtaining n events in N trials. The outcome of the trials described by the Binomial distribution can only be 1 or 0 (“yes” or “no”). This probability has two parameters, n, the total number of events in N trials, and p, the probability of getting one “yes” in one trial. The Poisson distribution expresses the discrete probability distribution of getting n events in a certain interval such as time, distance and area. In order to build the Poisson distribution, the mean value of events occurring during the selected fixed interval must be determined. Also, the events should be independent of the time since the previous event. Different from the Binomial and the Poisson distribution, the Gaussian distribution, also called normal, distribution is a continuous probability distribution, which is featured by two important parameters—the variance and the mean of the distribution. Both the Poisson and the Gaussian distribution can be derived from the Binomial distribution. When the two parameters of the Binomial distribution n is large and p is small, the Binomial distribution becomes the Poisson distribution; when n is large and p is approximately 0.5, the Binomial distribution becomes the Gaussian distribution. Also, for large mean value, the Poisson distribution can be approximated by the Gaussian distribution.

In this experiment, one microcurie Cobalt-60 was used to detect the radioactive decay event and study the nature of the random events, as well as find the empirical relationships between three main distribution function in limiting cases. The experimental setup includes a high voltage supply, a multichannel analyzer (MCA, introduced in detail in Gamma Spectroscopy) and an amplifier as shown in the photo below. The multichannel scaling mode (MCSR) was applied to read all the counts for a dwell time t at each channel. The dwell time could be set to be a number ranging from few microseconds to many seconds depending.

Here is the raw data output from MCA for the dwell time of 0.5 seconds. The red dash line shows the approximate mean value of counts (about 430 counts). With this mean value and the corresponding dwell time, I can choose other dwell times, so that I have other sets of data, which have the different mean values of counts ranging from about 5 to 20,000. For other four measurements, the dwell times were set as 0.01 seconds, 0.3 seconds, 2 seconds and 20 seconds.

The figure below is the histogram plot of frequency versus bins for the data of dwell time = 0.5 seconds. The data set was divided into 50 bins and the frequency of getting events occurred at each bin was plotted.

Next, the plot of frequency versus the number of Co-60 decay can be plotted, and fitted with function of both the Gaussian and Poisson distribution. The red curve represents the Gaussian function fitting the data while the blue curve is the Poisson distribution fitting. As shown in this plot, at dwell time = 0.5 seconds, both the Gaussian and the Poisson distribution can fit the gamma decay of the Co-60 well.

With the same process, I plotted the frequency of obtaining events as a function of number of decay for each other time dwell setting. For the two extreme cases of my measurement, plots of the dwell time of 0.01 second and 20 seconds are shown in the figures below. For the 0.01 seconds dwell time, the Poisson distribution fits better than the Gaussian distribution and obvious Poisson noise appears as fitting with the Gaussian function. However, for the 20 seconds dwell time, the fitting curves from the Poisson and the Gaussian can almost overlap with each other and Poisson noise does not exist any more. Therefore, it proves the theories discussed earlier that the Gaussian distribution can be treated as the approximation of the Poisson probability distribution with large mean value.

The experimental setup and data analysis method got updated and improved (right after I finished the experiment). Enjoy.

23 3 / 2013

## High Resolution Gamma Spectroscopy

Gamma radiation is very high-frequency electromagnetic radiation, usually above 10^18 Hz. Gamma radiation happens when unstable and radioactive nuclei decay to other nuclei, accompanying with the emission of positrons, electrons and alpha particles. When the daughter nuclei at excited states relax back to ground state, gamma radiation comes in well-defined quanta of high energy, ranging from several keV to several MeV. In this experiment, the gamma radiation from the radionuclides is detected with a high-resolution liquid nitrogen cooled Germanium crystal detector. The high-energy gamma radiation emitted is absorbed by the detector and liberate the localized electron from the valence band of the semiconductor Germanium crystal to the conduction band. The number of free electron is proportional to the energy carried by the gamma radiation, accumulating in a capacitor. As the capacitor is discharged, voltage pulse is produced with the height proportional to the energy of emitted gamma radiation. The pulse is amplified, collected and then digitalized by a Multichannel Analyzer (MCA). The MCA outputs the histogram plotting the integer counts corresponding to the number of the pulse versus the channel number representing the height of the pulse. Therefore, the energy of the gamma radiation for certain radionuclides can be deduced from the gamma radiation spectrum after calibrating the relationship between the energy and the channel number. The picture below shows the experimental setup, which include a detector, a high voltage supply and a MCA.

In this experiment, I measured the gamma radiation from several radioactive sources. First, I used the spectrum, collected for 5 minutes, measured from well-studied radionuclides, including Cobolt-60, Cesium-137 and Sodium-22 to make a calibration graph. Here shows the example gamma radiation spectru of Co-60. Each photo peak represents the situation that the incoming gamma radiation fully converts its kinetic energy to a photoelectron and is destroyed afterward.

The Co-60 spectrum has two high-energy photo peaks, which are at 1173 keV and 1332 keV referring to the literature. For each radioactive isotope, I recorded the channel number and the number of counts at the peak position(s) shown in each spectrum. With the profile of the photo peak(s) for the normal radioactive decay for each element, the the calibration graph of the energy versus the channel number can be plotted, as shown in the figure below. With the calibration graph obtained from the measurement of Cobolt-60, Cesium-137 and Sodium-22, the relationship between the integer channel number and the energy can be found with the linear function of y=1.1787+0.16723x. Then with this calibration, the energy of photo peak(s) for other radioactive decay from other sources can be determined.

The Ba-133 spectrum has 5 photo peaks at the channels of 484, 1649, 1807 2124 and 2290 at relative low channel range as shown in the figure above. From the calculation with the calibration determined earlier, the energy of each photo peak is found to be 82.12 keV, 276.94 keV, 303.36 keV, 356.37 keV and 384.13 keV respectively. These values are very close to the literature value of these peaks (80.998 keV, 273.398 keV, 302.853 keV, 356.017 keV and 383.851 keV), which indicates that the calibration measurement is reliable.

With the gamma radiation spectrum, the Compton scattering and Compton. The Compton scattering happens when the incoming gamma radiation undergoes scattering and converts partial energy to the electron. The maximum energy that an electron can received from the Compton scattering happens as the scattering is 180 degree. The electron undergoes back scatters through 180 degree appears at the Compton edge and the photoelectron absorbing the back-scattered gamma radiation results in “back scatter peak”. Theoretically, the sum of the energy of the Compton edge and the energy of back scatter peak equals the energy of photo peak. For example, Na-22 has the two photo peaks at 511 keV and 1277 keV. The Compton edge and the back scattter peak are at the channel of 1981 and 1073 corresponding to the 511 keV peak, and 6324 and 1296 corresponding to the 1277 keV peak. The energy of summation of the Compton edge and the back scattter peak is 513 keV for the first set and 1277 keV for the second set.

In the end, I ran an overnight experiment to measure the gamma radiation of Brazil nuts. Here is the gamma spectrum taken for 24 hours. The result shows that the Brazil nuts has unexpected high radioactivity. Its spectrum shows peaks at 76.09, 85.96, 93.48, 186.63, 239.31, 295.67, 295.33, 338.48, 352.02, 511.23, 609.72 and 911.74 keV.

12 2 / 2013

## Electron Spin Resonance

Under the influence of an external magnetic field, an electron with the intrinsic spin can either absorb energy from or release energy to the environment by changing the spin state—spin-up or spin-down. The energy level of each spin state is given by the equationE=-μB=-μBcos(θ). Accompanying with the change of the spin state of the electron, a photon with certain amount of energy is either absorbed or emitted. The phenomenon called electron spin resonance (ESR) occurs, as energy of the photon, hf, equals the energy difference between two quantized energy levels of electron’s spin states, 2μB, schematically described in the figure below. ESR is useful for the electrostatic and magnetic studies of complicated molecules.

In this experiment, I studied the unpaired electron in diphenylpierylhydrazil (DPPH) molecules as the quantum mechanical spin ½ particles interacting with an external magnetic field, as well as derived the value of magnetic moment of electrons ewith the basic equation hf=2μB.

The experimental setup was shown in photo where most apparatus were labeled. The frequency counter was applied to record the frequency of the photon produced by the coil inserted in oscillator; and the oscillator was connected to a ESR adapter supplied by -12V and +12V potential source. The 120V transformer supplied the AC current for Helmholtz coils to create the external magnetic field which was constantly changing with time. A 1 Ohm resistor was connected to the circuit in series with Helmholtz coils. The oscilloscope displayed the electric signals from both the ESR and the field simultaneously. In order for ESR to happen, the photons must provide right amount of energy hf = 2µB. This could be observed by the electric signal. The inductance L of the coil inserted to oscillator would change if the unpaired electron in DPPH absorbed the photon created by the current in the coil. The voltage across the coil changed in consequence. Therefore, as resonance occurred, paired electric signal peaks from the channel of ESR adapter could be detected. In this experiment, data collected were the frequency read from the counter and the voltage from the channel of the external field, corresponding to the peak position of ESR signal.

Given the basic property of the Helmholtz coil such as the turns of coils, distance between two parallelled coils and radius of the coils, the magnetic field B created could be express as a linear function of current I, B = K*I, where K is a constant. (With the Helmholtz coil I used, K = 4.269E3 T/A. The equation could be then converted to B = K*(V/R). Since 2μB = hf at resonance, B = hf/2μ; therefore, the magnetic moment of electrons could be calculated with μ = (h*R)/[(2K)*(V/h)], where h is the Planck’s constant, R is the resistance (1.046 Ohm by measurement). The term of (V/f) could be simply obtained from the slope of the voltage vs. frequency plot, where data points with uncertainty were fitted with a linear function.

The value of μe from my experimental result was (9.25+/-0.25)E-24 J/T, which well agreed with the accepted value 9.2848E-24 J/T.

12 2 / 2013

## e/k and Band Gap

As pure silicon is doped with atoms of certain other elements, it transforms from an insulator into a semiconductor. Based on the types of dopants and current carriers in silicon, the semiconductor can be categorized into two classes—p type whose current carriers are negative electrons and n type whose current carriers are positive holes. In this e/k ratio and transistor’s band gap energy experiment, a n-p-n type of silicon transistor TIP3055 was applied to measure the current-voltage relationship of p-n junction at different temperature. This I-V relationship enabled the determination of the e/k ratio with the equation, I=(Io)*[exp(eV/kT)-1], where e is the elementary electric charge, k is the Boltzmann’s constant and T is the temperature in Kelvin.

The circuit was connected as shown in the figure above. A 1.5 V battery and a 1 K resistor together provided the variable potential source. A digital thermometer, a voltmeter and a picoammeter were used to track the temperature, the emitter-base voltage and the collector current respectively. Here is the photo of the real experimental setup.

The measurements were carried out at five different temperature environments—room temperature, ice/water mixture, boiling water, dry ice/isopropanol mixture and liquid nitrogen. The first three experiments were conducted in the oil bath while the last two were in dewar bottle.

As the voltage was applied, eV/kT was much larger than 1; then, the relationship between the current and voltage became I=(Io)*exp(eV/kT). The semi-logarithmic plot of current vs. voltage was expected to show linear relationship with the slope of the e/kT value and the y-intercept of the ln(Io).

The data points were fitted with the least-square straight line. The slope of each line were used to calculate the e/k value at each temperature. The mean value of e/k of the five measurements was (1.16+/-0.02)E4 C*K/J, where the quoted error was from both the linear regression and the uncertainty in the temperature measurement. The mean value of e/k from measurements was in good agreement with the accepted value of e/k, (1.160485+/-0.00006)E4 C*K/J. Then, an universal function I0=AT^(3/2)exp(-Eg/kT) was applied to calculate the energy of the silicon transistor’s band gap. My calculation of Eg was based on the assumption that temperature dependence of band gap energy was negligible and therefore minority carrier current Io was proportional to exp(-Eg/kT) only. The natural log of Io was in the linear relationship with inverse temperature for the p-n junction of the silicon transistor. The plot of ln(Io) vs. 1/T had the slope of -Eg/k, which was (1.4218+/-0.0022)E4 C*K/J. The uncertainty here came from the linear fitting. The final result of Eg obtained from calculation was 1.22+/-0.02 eV, which was slightly higher than the reference literature, 1.11 eV to 1.13 eV. Since the calculation of band gap energy omitted the factor of temperature dependence, the slope of plot of ln(Io) vs. 1/T actually was not exact the value of -Eg/k. In consequence, the band gap energy of p-n junction was overestimated.

Here are some links to the similar experiment done by others and some information about semiconductor. You might find them interesting.

http://www.ajur.uni.edu/v7n1/Low%20et%20al%20pp%2027-32.pdf

http://www.physics.ohio-state.edu/~gan/teaching/summer04/Lec5.pdf

27 1 / 2013

## Aloha~

Hi, everyone. My name is Bingqing. After two years’ struggling in Bryn Mawr College, I finally found my “true love”—Physics and our beloved physics department. Physics is the coolest subject in the globe. For me, physicists are like detectives and solve cases in physics world. Besides doing physics experiments, watching animes while lying on the beach is another most enjoyable things in the world for me. I believe you will agree with me soon.

To see what we get here, let’s open the Schrodinger’s box…

(Source: redbubble.com)