Electron Paramagnetic Resonance Nikki Truss February 8, 2013 Abstract In this experiment a sample of DPPH inside an RF coil, within a Helmholtz coil arrangement, was used to investigate electron paramagnetic resonance. The coil separation for the most uniform field was verified and the magnetic field calibrated against the coil current. The values for the resistance and impedance of the coils were calculated as 13.1 ± 0.1 Ω and 27.3 ± 0.9 Ω respectively, and the phase shift was calculated to be 52 ± 2. The value for the electron g factor was calculated as 2.04 ± 0.4 which is within experimental error of the accepted value of 2.002319. It was shown that a linear relationship exists between the RF field amplitude and the signal height. Parts of the experiment were unsuccessful and yielded no usable results. Introduction In this experiment, the paramagnetic resonance behaviour of an electron was investigated in order to complete the aims listed below. Five sets of experiments are carried out, each of these experiments is outlined in further detail below in the relevant sections, along with the experimental method, results and analysis for each. A fundamental property of an electron is its intrinsic spin. This spin has an associated spin angular momentum, which in any direction can only take two values, M s h where M s = ± 1 2. Due to the charge on the electron, its spin angular momentum has an associated magnetic moment, the component µ z along the z direction of a magnetic field B is given by µ z = γ e hm s = gµ B M s (1) where γ e is the gyromagnetic ratio, µ B is the Bohr magneton, a unit of magnetic moment, and g is the electron g-factor, a dimensionless quantity very close to 2. The energy U of a magnetic dipole of moment µ in a magnetic field B is For an electron, equation (2) becomes U = µ B (2) U = gµ B BM s (3) Since the field is static, there are then just 2 discrete energy levels differing in energy by U = gµ B B (4) This splitting of the energy levels is called the Zeeman splitting. An electromagnetic field B 1 of frequency ν can induce transitions between these levels if the photon energy matches the level splitting energy U, i.e the resonance condition is hν = gµ B B (5) 1
This flipping over of the magnetic moment associated with the electon spin is known as Electon Spin Resonance (ESR). In general, there may be an orbital contribution to µ so the term Electron Paramagnetic Resonance (EPR) is also used. In this experiment the sample investigated was diphenyl picryl hydrazyl (DPPH), an organic radical whose EPR spectrum is a single line. For EPR the B field used must be very stable, otherwise fluctuations in B will lead to fluctuations in the energy level separations U. To ensure a uniform B field, a Helmholtz coil arrangement was used, since it provides one of the most uniform fields achievable with two electromagnetic coils. The expression for the field B at a distance x from the centre on the axis of a single coil of radius r with N turns is derived from the Biot-Savart law, and is given by B = µ 0INr 2 2(r 2 + x 2 ) 3/2 (6) In the case of this experiment, we let x = r/2 since the field is uniform at any distance, and we multiply this expression by a fatcor of 2 since we have two coils. We are then left with B = ( ) 3/2 4 µ 0 IN 5 r (7) Experiment 1 The aims of experiment 1 are as follows; To determine how the field profile along the axis of the large coils varies with coil separation. To calibrate the centre field versus coil current for the Helmholtz pair. To show that the most uniform fild occurs when the coil spacing equals r. B versus position, d, along the axis was plotted for three values of coil spacing. The spacing for which B is most uniform was found, then for this B and I were measured and plotted against one another. The ratio B/I was then compared to the ratio calculated for this coil arrangement. Results and Analysis The plots obtained for the different coil spacings are shown below in Figure 1. (a) separation=16.5cm (> r) (b) separation=6.0cm (< r) (c) separation=7.5cm (= r) Figure 1: Magnetic field versus position along axis 2
It is clear from Figure 1c that the field is most uniform when the coil spacing equals r. Using this spacing a plot of B versus I was obtained, shown in Figure 2. The ratio of B/I was calculated as 0.00396 ± 0.00001 TA 1. This agrees very well with the value calculated from equation (7) of 0.0038 ± 0.0001 TA 1. Figure 2: Magnetic Field Calibration Experiment 2 The aims of experiment 2 are as follows; To determine the total resistance R and impedance Z of the two coils in the Helmholtz arrangement The total resistance R of both coils was measured by connecting a multimeter in series and using the resistance setting. The total impedance Z of the coils was found by using a digital multimeter to first measure V ac and then I ac. The impedance is given by Z = V ac I ac (8) 3
Results and Analysis The values obtained are summarized below in table 1. Quantity R V ac I ac Z Value 13.1 ± 0.1 Ω 5 ± 0.1 V 0.24 ± 0.01 A 21.3 ± 0.9 Ω Table 1: Experimental values for finding impedance The value for the resistance is found to be lower than the value for the impedance, which is consistent since electrical resistance is just the real component of the impedance.they are related by R = Z cos θ where θ is the phase shift between the voltage and the current. Using this relationship, the phase shift was estimated to be 52 ± 2. Experiment 3 The aims of experiment 3 are as follows; To measure B versus ν at resonance To find g for DPPH At resonance ν was recorded and the field was determined using the calibration of B versus I. The measurement was then repeated over a full range of values for ν, and the resonance field plotted against ν. From this g was determined. Results and Analysis The value for g was found to be 2.04 ± 0.4 from the plot in Figure 3 by using equation (5). 4
Figure 3: Magnetic Field versus Frequency Experiment 4 The aims of experiment 4 are as follows; To measure the full width at half height (FWHH) of the signal for various amplitudes of the 50 Hz alternating magnetic field B ac To measure the full width at half height (FWHH) of the signal for various values of the resonance field Initially ν and B dc were kept fixed and the FWHH was measured for different values of the 50 Hz alternating field. Then, B ac was kept fixed and the FWHH was measured for different values of B dc at resonance. Results and Analysis This part of the experiment proved very difficult. The results obtained are summarized below in table 2 and table 3. 5
V ac (V) FWHH (Hz) FWHH (ms) FWHH (nt) 1.22 ± 0.01 298 ± 1 3.36 ± 0.01 10.65 ± 0.04 0.95 ± 0.01 357 ± 1 2.80 ± 0.01 12.75 ± 0.04 0.53 ± 0.01 327 ± 1 3.06 ± 0.01 11.68 ± 0.04 Table 2: Measurements for FWHH for varied V ac V dc (V) FWHH (Hz) FWHH (ms) FWHH (nt) 1.24 ± 0.01 294 ± 1 3.40 ± 0.01 10.50 ± 0.04 0.88 ± 0.01 357 ± 1 2.80 ± 0.01 12.75 ± 0.04 0.36 ± 0.01 294 ± 1 3.40 ± 0.01 10.50 ± 0.04 Table 3: Measurements for FWHH for varied V dc It was not possible to obtain any data for values of V ac and V dc above 1.3 V, and so obviously some error was made with the apparatus. The data that was recorded is most likely incorrect, as no discernible trend exists. Also, equation (5) was used to convert from units of frequency to units of magnetic field, however this yielded results in nanoteslas, which seems highly improbable. Thus, some error must have been made in the theoretical understanding of this equation. The method used to measure the FWHH is shown belown in figure 4. (a) Varied V ac (b) Varied V dc Figure 4: Measurement of FWHH 6
Experiment 5 The aims of experiment 5 are as follows; To measure the signal height h To measure the dependence of h on the RF field amplitude at fixed B res To measure the dependence of h on the value of B res at fixed RF amplitude First, the resonance signal was displayed, and its height measured as a function of the RF field amplitude A. A graph of h vs A was then plotted. Then the height of the resonance signal was measured as a function of the resonance field. A graph of h vs B res was then plotted. Results and Analysis The first part of this experiment yielded good results, a linear relationship was found between h and A, shown below in Figure 5. This is consistent with the theory, as when the RF field amplitude is greater there are more photons being emitted from the RF coil that can be absorbed by the free electons in the sample, which leads to the larger signal height. Figure 5: Signal Height versus RF Field Amplitude However, the latter part of the experiment was difficult, and above a value of 0.3 mt for the the magnetic field, the signal height remained at a constant value. A linear relationship was not found here, but rather an exponential relationship, which levelled off once a value of 1 V was reached. Due to the small values of the signal height, the validity of this result is dubious. 7
Figure 6: Signal Height versus Resonance Field Conclusions Overall, this experiment was a partial success. It was shown that the most uniform magnetic field for a Helmholtz pair occurs for a spacing equal to their radius. The magnetic field was calibrated against the coil current, and the ratio of B/I was found experimentally to be 0.00396 ± 0.00001 TA 1 which agrees very well with the calculated value of 0.0038 ± 0.0001 TA 1. The total resistance of the coils was measured as 13.1 ± 0.1 Ω, and their total impedance was calculated to be 27.3 ± 0.9 Ω. Using these values, the phase shift was calculated to be 52 ± 2. From a plot of the resonance field versus frequency, the g factor was found to have a value of 2.04 ± 0.4. This value agrees very well with the accepted value of g = 2.002319. The fourth part of the experiment sadly yielded no viable results, and so no conclusions may be made about the relationship between the FWHH and the alternating magnetic field or the resonance field. The fifth part of the experiment was partially sucessful, and it was shown that there exists a linear relationship betweeen the RF field amplitude and the signal height. However, the second part of the experiment yielded possibly invalid results, showing an exponential relationship betweeen the signal height and the resonance field, up to a limiting value of 1 V at 0.3 mt. 8