On Electron Paramagnetic Resonance in DPPH Shane Duane ID: 08764522 JS Theoretical Physics 5th Dec 2010 Abstract Electron Paramagnetic Resonance (EPR) was investigated in diphenyl pecryl hydrazyl (DPPH). The Helmholtz EM coil arrangement was shown to have a linear relationship between field strength and current. A value of g = 1.85 was found for the electron g-factor, in reasonble agreement with [2]. This also validated the theory of EPR as in Weil [1]. 1 Introduction & Theory For a more detailed discussion of the theory behind this experiment see chapter 1 of [1]. The spin of the electron is one of its intrinsic properties. It has an associated spin angular momentum which can only take values M S h, where M S = ±1/2. Since the electron is charged the spin angular momentum has an associated magnetic moment and its component µ z along the direction z of a magnetic field B is µ z = γ e hm S = gµ B M S (1) where γ e is the gyromagnetic ratio, µ B = 9.2740 10 24 J T 1 [1] is the Bohr magneton and g = 2.002319 [2] is the free electron g-factor, a dimensionless quantity. The energy U of a magnetic dipole of moment µ in a magnetic field B is For the electron, (2) becomes U = µ. B. (2) U = gµ B BM S, (3) hence with the field being static there are two energy levels differing in energy by U = gµ B B. (4) The splitting of the energy levels is known as Zeeman splitting. If an electromagnetic field B 1 supplies photons of frequency ν such that U = hν, then the photons can induce a transition between the spin states of the electron where the resonance condition is hν = gµ B B. (5) 1
This flipping over of electron spin magnetic moments is known as electron spin resonance (ESR). More generally, there may be an orbital contribution to µ so that the term electron paramagnetic resonance (EPR) is also used. In this experiment the sample used was diphenyl pecryl hydrazyl (DPPH), an organic radical which gives a single line EPR spectrum. For EPR a very stable uniform static field B must be used, as variations in B will lead to variations in energy separations U. If B is not uniform over the sample, then the spectral lines will be broadened. Hence a Helmholtz coil arrangement was used as it provides the most uniform field achievable with two electromagnetic (EM) coils. The field B at distance x from the centre on the axis of a single coil of radius r with N turns is B = µ 0NIr 2. (6) 2(x 2 + r 2 ) 3/2 The field on the axis between a pair of coils at a distance x from one of them is B = 1 [ ] 2 µ 0r 2 1 I (x 2 + r 2 ) + 1, (7) 3/2 ((δ x) 2 + r 2 ) 3/2 where δ is the separation between the coils. An attempt to prove that the most uniform field occurs for δ = r was made, but was unsuccessful. 2 Experimental Method Section I Field Profile on Axis of EM Coils & B(I) Calibration A Hall probe was used to measure the steady field B between the coils. The coils used had an adjustable spacing δ so B was measured for three values of δ. The current was kept steady at 0.5 A. The field was found to be most uniform for a coil spacing of δ = r. Hence the dependence of B on I was measured in the range 0.1 0.7 A for 69mm. Section II Resistance & Impedance of EM Coils The total resistance R and impedance Z of the two coils was determined using a multimeter. To find Z we measured V cac, the AC potential difference across the coils, and I cac, the AC current running through the coils. Section III To measure B(ν) and find g for DPPH The circuit was set up as in Fig. 1 where the EPR module generated the oscillating field B 1 from its RF coil of inductance L. In the module was a variable capacitance C connected in parallel with the coil. The frequency ω 0 = 1/(LC) 1/2 of the circuit could be changed by adjusting C. When ω 0 = ν we had resonance. The current was measured for 10 values of ν and used to plot B vs ν in Graph 5. Section IV This section was not attempted since time did not allow. Section V This section was not attempted since time did not allow. 2
3 Results Errors ±e in the final significant figures of results are denoted by the error magnitude in brackets: (e). Section I Field Profile on Axis of EM Coils & B(I) Calibration The results for B vs δ are plotted in Graphs 1 to 3. In Graph 4 we see a plot of B vs I for δ = 69 mm. The slope here is B/I = 0.00352(1) T/A. The calculated value for B/I using (7) was of the order 10 5, which is not in agreement. However, if the formula (6) for a single coil is used and doubled to account for the two coils we find B/I = 0.00291 which is much closer to the experimental value. Section II Resistance & Impedance of EM Coils The resistance R of the two coils in series was found to be 13.8(1) Ω. The coil AC current and voltage were found to be I cac = 0.124(1) A and V cac = 2.53(1) V, respectively. These gave an impedance Z = V cac /I cac = 20.4(1)Ω. Section III To measure B(ν) and find g for DPPH The data for B vs ν is plotted in Graph 5. The ratio B/ν = 3.86(5) 10 11 T/Hz gives a value g = 1.85 from (5). 4 Discussion An error must have been made in the theoretical understanding of the Helmholtz arrangement due to the inability of (7) to predict the B/I ratio. the relationship between B and I was shown to be linear in Graph 4, agreeing both with (6) & (7). The value of g = 1.85 found agrees to within 7% of the accepted value from [2], which verifies the theory of spin transitions and (5). References 1. Electron Paramagnetic Resonance, J.A. Weil et al. 2. NIST Reference on Constants, Units & Uncertainty, http://physics.nist.gov/cuu/index.html 3
Appendix: Figures & Graphs Figure 1: Circuit Diagram for Section III Figure 2: Field versus position for 40mm spacing 4
Figure 3: Field versus position for 69mm spacing Figure 4: Field versus position for 100mm spacing 5
Figure 5: Field versus current for 69mm spacing Figure 6: Field versus frequency at resonance 6