Download On Electron Paramagnetic Resonance in DPPH

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 MS h̄, where
MS = ±1/2. Since the electron is charged the spin angular momentum has an
associated magnetic moment and its component µz along the direction z of a
~ is
magnetic field B
µz = γe h̄MS = −gµB MS
(1)
where γe is the gyromagnetic ratio, µB = 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.
~ is
The energy U of a magnetic dipole of moment µ in a magnetic field B
~
U = −~
µ.B.
(2)
U = gµB BMS ,
(3)
For the electron, (2) becomes
hence with the field being static there are two energy levels differing in energy
by
∆U = gµB B.
(4)
The splitting of the energy levels is known as Zeeman splitting. If an electromagnetic field B1 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 = gµB 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=
µ0 N Ir2
.
2(x2 + r2 )3/2
(6)
The field on the axis between a pair of coils at a distance x from one of them is
1
1
1
2
B = µ0 r I
+
,
(7)
2
(x2 + r2 )3/2
((δ − x)2 + r2 )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 Vcac , the AC potential difference
across the coils, and Icac , 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 B1 from its RF coil of inductance L. In the module was a
variable capacitance C connected in parallel with the coil. The frequency ω0 =
1/(LC)1/2 of the circuit could be changed by adjusting C. When ω0 = 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 Icac = 0.124(1) A and Vcac =
2.53(1) V, respectively. These gave an impedance Z = Vcac /Icac = 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