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The calculation of the specific charge of an electron has been done by various methods and by numerous scientists over time. This report describes one method using the splitting of yellow and green lines by the Zeeman Effect. This was done by measuring the distance between satellite fringes and plotting the results against magnetic field strength B. The results showed a linear relationship with the line equation for yellow and green light being and respectively. Uncertainties in the data and possible explanations for them are given as well as suggestions for an improved experiment.
In an external magnetic field B, a magnetic dipole has an amount of potential energy Um that depends upon both the magnitude Î¼ of its magnetic moment and the orientation of this moment with respect to the field (figure 1).
where Î¸ is the angle between Î¼ and B. The torque is a maximum when the dipole is perpendicular to the field and zero when it is parallel or antiparallel to it. To calculate the potential energy Um, a reference configuration in which Um is zero must be established. It is convenient to set Um = 0 when Î¸ = Ï€/2 to the angle Î¸ that corresponds to that orientation.
When Î¼ points in the same direction as B, then Um = -Î¼B, its minimum value.
The magnetic moment of the orbital electron in a hydrogen atom depends on its angular momentum L. Hence both the magnitude of L and its orientation with respect to the field determine the extent of the magnetic moment of a current loop, given by
where I is the current and A is the area it encloses. An electron that makes f rev/s in a circular orbit of radius r is equivalent to a current of -ef and its magnetic moment is therefore
Because the linear speed Ï… of the electron is 2Ï€fr its angular momentum is
Comparing the equations for the magnetic moment Î¼ and the angular momentum L shows that
for an orbital electron (figure 2). The quantity (-e/2m) is called the gyromagnetic ratio. The minus sign means that Î¼ is in the opposite direction to L and is a consequence of the negative charge of an electron. While the above expression for the magnet moment of an orbital electron has been obtained classically, quantum mechanics yields the same result.
The magnetic potential energy of an atom in a magnetic field is therefore
which depends on both B and Î¸.
Figure : (a) shows a magnetic moment of a current loop enclosing an area A. (b) shows a magnetic moment of an orbiting electron of angular momentum L.
Area = A
Figure 2 shows (a) the magnetic moment of a current loop enclosing area A and (b) the magnetic moment of an orbiting electron of angular momentum L.
From figure 3 it can be seen that the angle Î¸ between L and the z-direction can have only the values specified by
with the permitted values of L specified by
l = 2
ml = 2
ml = 1
ml = 0
ml = -1
ml = -2
Figure 3 shows space quantisation of orbital momentum where the orbital quantum number is l = 2 and there are accordingly 2l + 1 = 5 possible values of the magnetic quantum number m1, with each value corresponding to a different orientation relative to the z axis.
The magnetic energy that an atom of magnetic quantum number ml has when it is in a magnetic field B can be found by substituting in equations X and X into equation X to give
The quantity is called the Bohr Magneton:
Thus, in a magnetic field, the energy of a particular atomic state depends on the value of ml as well as n. A state of total quantum number n breaks up into several substates when the atom is in a magnetic field, and their energies are slightly more or less than the energy of the state in the absence of the field. This phenomenon leads to a "splitting" of individual spectral lines into separate lines when atoms radiate in a magnetic field. The spacing of the lines depends on the magnitude of the field. The splitting of spectral lines by a magnetic field is called the Zeeman Effect after the Dutch physicist Pieter Zeeman who first observed it in 1896. The Zeeman Effect is a vivid confirmation of space quantisation (Beiser, 1995).
Each line splits into a number of components distributed symmetrically about the position of the zero field or "parent" line. The spacing of the components observed increases linearly with the field, and those at right-angles to the field direction are plane polarised whilst those observed along the field direction exhibit circular polarisation.
The normal Zeeman Effect
The normal Zeeman Effect consists of the splitting of a spectral line of frequency f0 into three components whose frequencies are
Figure 4 below shows the splitting of two spectral lines with the presence of a magnetic field.
Figure shows the normal Zeeman Effect.
l = 2
l = 1
ml = 2
ml = 1
ml = -1
ml = 0
ml = -2
ml = 0
ml = -1
ml = 1
No magnetic field
Magnetic field present
Figure 4 shows the normal Zeeman Effect.
In ferromagnetic materials, the magnetic field due to the magnetic moments is often greater than the magnetising field by a factor of several thousand. Figure 5 shows a plot of B against the magnetising field Bapp (Tipler and Mosca, 2008). As the current is gradually increased from zero, B increases from zero to along part of the curve from the origin O to point P1. The flattening of this curve near point P1 indicates that the magnetisation is approaching its saturation value, at which magnetic moments are aligned. When Bapp is gradually decreased from point P1, there is not a corresponding decrease in magnetisation. The shift of the domains in a ferromagnetic material is not completely reversible, and some magnetisation remains even when Bapp is reduced to zero, as indicated in the figure. This effect is called hysteresis, and the curve in figure 5 is called a hysteresis curve. The value of the magnetic field at point P4 when Bapp is zero is called the remnant field Brem. At that point the material is a permanent magnet. If the current is now reversed so that Bapp is in the opposite direction, the magnetic field B is gradually brought to zero at point c. The remaining part of the hysteresis curve is obtained by further increasing the current in the opposite direction until point P2 is reached, which corresponds to the saturation in the opposite direction, and then decreasing the current to zero at point P3 and increasing again in its original direction. It can be seen therefore that the magnetisation depends on the previous history of the material.
Figure 5 shows a plot of B against applied magnetising field Bapp. The outer curve is called the hysteresis curve.
Calibration of the magnet
The relationship between the B field and the current I depends on the history of the material in the magnet, therefore the magnet needed to be given a well defined history by cycling the current up and down several times. Measurements of B were made at intervals of 1A from 0-12 A and then back down from 12-0A. The current was not allowed to exceed 12A otherwise the coil overheats. These measurements of B were made using a Hall Effect flux meter. The probe remained in the same position between the poles of the magnet at all times. The same cycle was used throughout all measurements so that the history remained well defined and the results obtained more accurate as a result. As the magnet exhibits hysteresis a calibration graph of the magnetic flux density B verses the coil current I was plotted (figure X).
The set up of the apparatus was as shown in figure 6 below.
Mercury lamp between the poles of the magnet
filter (green or red)
Figure 6 shows the experimental setup.
In order to produce sharp interference fringes the Fabry-Perot interferometer had to be carefully adjusted: all the optical bench elements had to be aligned along a common axis; the condenser frame had to be placed as close to the mercury lamp as possible without touching it; the collimator had to be shifted to give a reasonably parallel beam where the etalon is; the telescope lens had to be shifted to give a sharp image of the lamp in the image plane; the plates of the etalon had to be parallel and perpendicular to the axis; the eyepiece had to be focused onto the image plane and the polariser had to be rotated to select just the parent or satellite fringes.
Measurement of line splitting
Measurements of the yellow line splitting were made by taking readings of the values of r1 and r2 as shown in figures 7 and 8 below. Using the following equation
d values were calculated and plotted as a function of B.
Readings were impossible to make when satellite fringes merged, however, by increasing B further the satellites reappeared on the other side of the crossover and further measurements could be made.
By using linear regression on the positions of the two satellite fringes the crossover field could be found fairly accurately. This experimental method was then repeated using a green filter.
Figure 7 shows the fringes as seen through the telescope.
Figure 8 shows the parent and satellite fringes and illustrates the measurement made in the experiment.
Figure 9 shows a graph of magnetic flux density B against current I.
Figure 10 shows a graph of the distance between adjacent satellites and magnetic flux density for a yellow filter. The corresponding errors for both sets of values are not distinguishable on the graph.
Figure 11 shows a graph of the distance between adjacent satellites and magnetic flux density for a green filter. The corresponding errors for both sets of values are not distinguishable on the graph.
Calculation of specific charge
The theoretical value of specific charge can be found using the following equation
The experimental value of Bc is found be rearranging the line equations to find x where y = 0.
For the yellow filter,
and for the green filter
The specific charge is therefore
for the yellow filter and
for the green filter.
The error calculations for the above values are shown in Appendix A.
Figure 9 of the calibration graph shows an "S" shape curve as is expected from a magnetic exhibiting hysteresis. The curve is similar to the one shown in figure 5 in the introduction. This shows that the current I and magnetic field strength B readings are fairly accurate as they are able to show a similar trend. Unfortunately, due to time restrictions, re-calibration at the end of the experiment was not undertaken. This would have shown how much hysteresis took place during the experiment and thus given a good reflection on the reliability of the results obtained.
Figure 10 and 11 appear to demonstrate a non-linear relationship between magnetic flux density and the distance between adjacent satellites, this does not agree with the relevant theory. Assuming that magnetic flux density and the distance between adjacent satellites does have a linear relationship, as shown with the lines of best fit on both graphs; the error for the green filter is large compared to that of the yellow filter. This can be seen on figure 10, which shows the line of best fit running through two points for the yellow filter graph, whilst no points hit the line of best fit on figure 11 for the green filter. This is reinforced by the errors in the line equations, giving errors in the gradient of Â±0.00018 and Â±0.00075 and errors in the intercept of Â±0.1 and Â±0.4030 for the yellow and green filters respectively. This clearly indicates that there is more error associated with the results for the green filter. The consequence of this is higher errors in the green filter result than the yellow filter for the specific charge, 1.250Ã-1011 Ckg-1 and 3.548Ã-1010 Ckg-1 respectively.
Although the values found for both specific charge values are within error, the errors are fairly large and therefore do not ensure accuracy.
This report has described an experiment to calculate the specific charge using the Zeeman Effect. As aforementioned in the discussion, figures 10 and 11 appear to demonstrate a non-linear relationship. If this is valid, then this method for calculating the specific charge of an electron is flawed due to the use of a line of best fit to find BC. Consequently a different technique other than a line of best fit would need to be used to find the cross over point.
The error for the green filter is much larger than the error in the yellow filter. Possible reasons for this include inaccuracies in the positioning of the magnetic sensor, difficulty in reading the r1 and r2 values in the duller green light and a less well defined history in the current cycle.
The position of the magnetic probe could have been altered by accident either during the experiment or between lab sessions. This would have resulted in different measurements of the B and hence the specific charge. The duller green light made it more difficult to measure r1 and r2. This could have resulted in less accurate readings being taken and therefore larger errors in the distance between adjacent satellites, giving a different specific charge value. A less well defined history could have occurred from losing some of the original calibration, meaning that the B readings would be slightly different and ergo the specific charge value.
The reliability of all the results could have been improved by further repeats. Unfortunately, due to time restrictions, this wasn't possible, and further experimentation will be required to validate the results obtained. Re-calibration of the magnetic to show its hysteresis (as seen in figure 5) would also be useful to undertake in further experiments, as mentioned in the discussion.
The values for the specific charge of 2.470Ã-1011 Â± 3.548Ã-1010 Ckg-1 for the yellow filter and 2.560Ã-1011 Â± 1.250Ã-1011 Ckg-1 for the green filter are of the same order of magnitude as the theoretical specific charge of an electron 1.758Ã-1011 Ckg-1 (Avison, 1989). The value for the yellow filter is closer to the theoretical value than the green filter value however the value for the green filter is within error whilst the value for the yellow filter is not. This shows that although the higher error in the green filter arose from inaccuracies in the apparatus, this error is justified given the results obtained.
Possible improvements that could be made in further experiments include using more precise and accurate versions of the existing equipment; such as better a quality micrometer, telescope, magnetic probe, filters, and etalon. A better quality telescope and filters would make readings of r1 and r2 less difficult to make, by giving a larger and clearer image to make readings from. A better quality micrometer and magnetic probe would give more accurate measurements of the adjacent satellite spacing and magnetic field strength B respectively. And a more precise etalon would reduce the errors in the results obtained. Another improvement would be making more precise optical adjustments. Optical elements could be aligned microscopically using for example a micrometer. This would result in a much better optical alignments giving clearer fringes to make readings from and hence reducing the error in these readings.
There will always be a reading error unless human error is eliminated from the readings taken. This could be achieved by using an electronic device to calculate the relative distances between two light intensities, i.e. the distance between adjacent satellites. This would vastly improve the accuracy of the results, giving a very accurate specific charge.