Physics307L:People/Benedict/Formal

Steve Koch 02:39, 15 December 2010 (EST):Excellent report. Very cool analysis. I love how you created a superset of data from the other students' open data. I was a bit disappointed that you didn't engage me on the issues of mean free path and filament voltage. But it's a great study nonetheless.

An Accounting of the Sources of Error in the Determination of the Charge to Mass Ratio for the Electron Using the Fine Beam Tube Method
Author: Alex Benedict University of New Mexico, Department of Physics and Astronomy MSC07 4220, 800 Yale Blvd NE, Albuquerque, New Mexico 87131-0001 USA Contact info: abenedic@unm.edu

Abstract
The determination of the e/m ratio for the electron using the fine beam method is well known to be prone large systematic errors. In this paper we will present an account of a typical lab using this method, and the common method of determining e/m from experimental observations. We will then explore the effect of changing any of the 3 parameters in the common model, to see how large of an effect small changes in observed voltage, current, and radii of curvature would have to account for error in the experimental observation of parameters used in the calculation of the e/m ratio. We then explore corrections to the model for relativistic effects, and energy loss from collisions of the electrons with the helium gas.

It was found that the most probable source of error was the loss of energy due to collisions with the helium gas, and the accelerating voltage not accurately reflecting the actual voltage by which the electrons are accelerated due to the electric field not being uniform. 

Introduction
The first investigation into the determination of the charge to mass ratio of the electron was performed by J.J. Thomson[2] around 1897, his results were communicated on Friday, April 30[4]. One of the first uses of the ratio was in approximating the mass of the electron, which was found to be almost on the order of an atom[4]. This was taken as proof that atoms are composed of smaller particles that compose cathode rays. More precise measurement of the charge of an electron was made by Millikan in his famous experiment[3], which lead thanks to the e/m ratio already known to a more accurate measurement of the mass of an electron. Accurate measurement of these fundamental constants was useful in proving the predictions of the then new field of quantum mechanics, which predicted the energy levels of hydrogen and those predictions required accurate values for the charge and mass of the electron. Overall the charge to mass ratio is a useful quantity, and accurate determination of it is valuable even today.

The most common method used in basic physics laboratory classes for the measurement of this ratio is the fine beam tube method, wherein an electron gun is used to transmit electrons through a helium gas whose electrons are excited by the collisions and begins to glow. The electrons are accelerated by an electric field of known voltage, which provided an estimate of the kinetic energy of the electrons. There is also a pair of Helmholtz coils which provide a mostly uniform magnetic field oriented so as to be perpendicular to the plane of the beam, which causes the beam to curve. By measuring the radius of the curvature, and the strength of the magnetic field, and the accelerating voltage a crude approximation to the e/m ratio can be found. The most common method which is referenced in many physics manuals uses an assumption of constant kinetic energy, constant radius of curvature, and a uniform magnetic field to derive a formula used in the calculation of the e/m ratio. A derivation of the simple model can be found below, however all of those assumptions are flawed to some degree as the beam must lose kinetic energy to collisions with the helium gas, and correspondingly the radius of curvature will change as the beam moves through the loop. The assumption of a uniform magnetic field is actually the strongest assumption due to the properties of the pair of Helmholtz coils, and deviations from the expected values are small.

Below we will analyze the various sources of error inherent in the determination of the e/m ratio using the fine beam tube method, and attempt to analyze their relative magnitudes so as to propose a more accurate model for use in future physics labs.

Materials and Assembly
The materials used in this lab consists of three power supplies, a Gelman Deluxe Regulated Power Supply(6-9V 2A), Soar DC Power Supply model PS 3630(6.0V 1.5A), Hewlett Packard 6236B(150-300V 40mA). We also used three Digital Multimeters, two for measuring voltage, one for current. Most importantly we used a Uchida Yoko e/m Experimental Apparatus model TG-13.

The e/m experimental apparatus is just a tube filled with helium gas a low pressure, and containing a device for heating a filament to release electrons which are accelerated by a generated electric field. Then the accelerated electrons are exposed to a magnetic field, through the pair of Helmholtz coils on the device. The applied magnetic field produces a Lorentz force on the electrons which causes them to move in a circle. Information on the assembly of the materials can be found in [1].





Procedure and Methods
For this lab we first turned on the power supply connected to the filament heater, as it takes about 3 minutes to heat up. We then set the accelerating voltage to a reasonable starting voltage, of ~175V, and the current to the Helmholtz coils to ~1A. We then took measurements of the radius of curvature of the beam, for a constant current and varying potential, and for a constant potential and a varying current. We measured the radius of curvature by measuring it on both sides of the ruler and averaging the values to correct for the loss of radius as the electrons lose energy to collisions with the helium gas.

Results and Discussion
The most common method for computing the e/m ratio for electrons starts with considering the kinetic energy of the electrons, given as usual by $$T=\frac{1}{2}mv^2$$, and we know the kinetic energy given to the electrons by the accelerating voltage such that $$T=eV_{acc}=\frac{1}{2}mv^2$$, which gives $$v^2= \frac{2eV_{acc}}{m}$$. The Lorentz force from the applied magnetic field has a magnitude of, $$F=evB\sin \boldsymbol{\theta}$$, We also know that the magnetic field from the Helmholtz coils is perpendicular to the plane of the beam, such that $$\sin \boldsymbol{\theta}=1$$ where the B field is given by $$B=7.80\times 10^{-4} \mbox {tesla/ampere}\, \times I$$, where I is the current. . We also know that any centripetal force is of the form $$F=\frac{mv^2}{r}$$ where r is taken as the radius of curvature in this case. Putting this all together gives, $$\frac{e}{m}=\frac{2V_{acc}}{B^2r^2}$$ we will always assume from here on that r is measured in cm, B in tesla, and Vacc in volts.

Using Data which can be found here e/m Lab Notebook, the results of using the data and directly calculating e/m gives an average value of $$\frac{e}{m}= (2.17\pm .02) \times 10^{11} \frac{C}{Kg}$$, which is a relative error of 23.6% and a fractional error of 0.9%. Forming a least squares fit of the data for the constant current, varying voltage or for varying current and constant voltage gives similar results, which leads to the conclusion that the majority of the error in the measurement of e/m must be systematic error. This means that either the method of measuring the values used is flawed, or that the model used to attain the value is flawed. We will now explore these two scenarios.

We begin by estimating the amount of error introduced by taking poor quality measurements, and checking their magnitudes. To start we form a function which will give us the observed value for e/m for given values of the error of the radius, B field, and voltage, such that $$F(r_e,B_e,V_e)= \frac{2*(V_{acc}+V_e)}{(B+B_e)^2(r+r_e)^2}$$, where $$(r_e,B_e,V_e)$$ are taken to be the error in the measurement of the radius, the B field, and the accelerating voltage respectively. We then take the taylor series of the function and keep only term of the first order, since we will assume all measurement errors to be small getting, $$F(r_e,B_e,V_e)\approx \frac{2*V_{acc}}{B^2r^2} + \frac{2*V_{e}}{B^2r^2} - \frac{4*V_{acc} * B_e}{B^3r^2} - \frac{4*V_{acc} * r_e}{B^2r^3}$$. This then gives us an error function $$E(r_e,B_e,V_e)= \frac{2*V_{e}}{B^2r^2} - \frac{4*V_{acc} * B_e}{B^3r^2} - \frac{4*V_{acc} * r_e}{B^2r^3}$$. now if we assume the we are only off by small amounts, then typical values of the error of the radius of curvature would be about one tenth of a centimeter, the typical error of the B field would be about 0.00001 T, and typical error in the voltage would be about 5 volts. Then the magnitude for error given average values of r of 5cm, B of 0.00117 T, and Vacc of 200V, would be 1.66% of the magnitude of the actual value of e/m for voltage, 1.13% for the B field, and 2.56% for the radius. In the worst case measurement error could account for about 5% of the total error. Which makes it an important contribution to the error, but it still is short of the around 20% error typically attained while performing this experiment.

We can also use the above error function to investigate the magnitude of the effect that the earth's magnetic field would have on the experiment. We know that the strength of the Earth's magnetic field is about 5*10^-5 T, and using that value in the error function would give an error of about 5% in the calculation of e/m.

We must then investigate contributions to the measurement error from sources not included in the model typically used. We first begin with the obvious problem of the energy loss due to collisions of the electrons with the helium particles. It can be easily seen from measurement of the radius that the the radius as measured on the left side of the tube is usually about 0.1 cm smaller than the radius as measured near the opening of the electron gun, therefore it is easy to assume that the beam must lose some energy as it travels through the helium gas. We know from the manual for the device, which is obtainable from the Pasco Scientific website(Pasco just resells the Uchida Yoko TG-13 as the SE-9638) that the pressure of the helium gas in the Fine Beam Tube is about 10^-2 mmHg, given the large imprecision in the value given, we will assume that it is about 2 Pascals to make the following calculations simpler. From this we can find the mean free path of the gas given that the average diameter of the helium molecule is 0.98 Angstrom, and that the temperature is about 300 K using $$l=\frac{kT}{\sqrt{2} \pi d^2 p}$$, where k is boltzmann's constant, T is temperature, d is the diameter of the particles, and p is the pressure. This gives $$l \approx 4.7 cm$$.

The mean free path is the average distance a particle of the gas can travel in the gas without colliding, so if we assume the same idea applies to electrons in the gas on average the electrons with travel about 4.7 cm before colliding and losing energy. Since the electrons are traveling in a circle of radius r, we know the total distance they travel, and so we can estimate the average number of collisions taking r as about 5 cm as $$N \approx \frac{2 \pi r}{l} = 6.7$$, so on average each particle experiences about 7 collision as it moves through the loop. We also know that the wavelength of light emitted from the collision is about 500 nm, using the relation $$ E= \frac{hc}{\lambda}$$ gives a typical energy of $$ E \approx 2.5 ev$$, which means on average, the electrons should lose about 18 ev of energy. We could then be justified in correcting the model to account for the energy loss, by taking the accelerating voltage as being 3.3*r volts less than it is measured, this is of course an over estimate but it is about the same order of magnitude as the actual effect. Given more precise measurement of the pressure of the helium gas, and knowing the temperature at which it was inserted would motivate a more careful analysis. However that information is not available at this time. This implies a new model for the calculation of the e/m ratio of $$\frac{e}{m}=\frac{2(V_{acc}-3.3r)}{B^2r^2}$$.

Using this new model on the same original set of data gives a value for the e/m which improves the relative error by about 7%, and gives $$ \frac{e}{m} =(2.05 \pm 0.02)*10^{11} \frac{C}{Kg} $$.

Next is an investigation into whether relativistic effects would contribute in a meaningful way to the error in determining the e/m ratio. To see if relativistic corrections are going to be helpful we first determine the ratio of the velocity of the electrons to the speed of light, using $$ \frac{v}{c}=\frac{eBr}{mc}$$ which was used in the initial derivation, this gives $$\frac{v}{c} \approx 0.027$$ using B=0.00117 T, and r=4 cm. And since that is less than 10% of the speed of light, there is no need for taking relativistic effects into account.

The last reasonable source of error to investigate is the uneven acceleration of the electrons, given that there is a hole in the anode which would lead to a nonuniform electric field. This effect is much harder to take into account without accurate measurement of the electron gun assembly, which is unfortunately not available at this time. However this can be assumed to be large effect given that the dimensions of the chamber in which the cathode and anode are contained is around 2 cm long and about 1 cm in diameter(based on a rough attempt of measurement through the glass, and some estimation), even a tiny 1 mm hole in the anode would lead to a large change in the electric field in the gun. We will assume a 10% loss of voltage from this effect for the purposes of investigation, and motivated by the hole being about 10% of the size of the diameter of the assembly. Combining this with the above corrected model gives a new sample model of $$\frac{e}{m}=\frac{2(0.9V_{acc}-3.3r)}{B^2r^2}$$. Using this model with the same data set as before gives $$ \frac{e}{m} =(1.83 \pm 0.02)*10^{11} \frac{C}{Kg} $$, and a relative error of only 4%. This is a huge improvement over the simple model used in most experiments of this type.

The next reasonable step to take would be to perform a numerical search on a larger set of data to find the coefficients which minimize the error in the model. To do this I created a super set of data, from the lab notebooks of Nathan Giannini, Kirstin Harriger, Alexandra Andrego, Alex Benedict. I would like to thank them for making their data easily available online, I assembled the assorted data here:. I Performed a simple search through a restricted set of the possible coefficients (Code Here :User:Alex_G._Benedict/Notebook/Physics_307L:_Junior_Lab/SimpleSearchScript) instead of using least squares fitting since the inclusion of Andrea's data led to ridiculous values and I wanted to search only through a restricted set of information anyway. The result of this simple search was $$\frac{e}{m}=\frac{2(0.8V_{acc}-10r)}{B^2r^2}$$ when the radius is measured in meters. This just further reinforces the ideas presented above, since there is a model which performs better than the simple model for all data, as can be seen in the last column of the data superset.



Conclusion
Through a careful accounting of the sources of error in the experiment we were able to account for a significant percentage of the sources of error in the experiment. We were able to lower the relative error from 23% to just 4%, through corrections to the simple model used. Using a numerical search we were able to reduce the error for a superset of data from ~30% error to ~5% for the reasonable sets, and even for the flawed set we were able to reduce the relative error tremendously. There is still a need for further investigation of the sources of error, and more precise knowledge of the pressure of the helium gas and the temperature it was bottled at would help significantly. It would also be helpful to know how many electrons are emitted by the electron gun at a given voltage. Along with a knowledge of the precise dimensions of the electron gun assembly, to aid in the estimation of the actual accelerating voltage. Unfortunately most of the records of such information would be held by the corporation that manufactured the device(Uchida Yoko), and their headquarters in Tokyo were unable to help when I tried calling. However, if anyone has a broken Uchida Yoko TG-13 tube lying around, and would like to measure those things that could be very helpful.

One of the more useful results of the above analysis is that the we were able to determine a reasonable estimation of the error in measuring the radius of the beam, and the relative effect of having an external magnetic field. There is also still a need for a more through investigation into the behavior of the beam in the low voltage, high current regime, which produces some odd effects.

Acknowledgments
I would like to thank Joseph Frye for the excellent pictures used in this paper. I would also like to thank Katie Richardson for the suggestion to investigate the mean free path in the gas.