Download Experiment 15. Electron Spin Resonance

Experiment 15. Electron Spin Resonance
Updated RWH April 20, 2016
1
Safety
• This experiment uses mains-powered equipment. If you suspect the equipment is not operating correctly, turn it off at the mains switch and consult a tutor.
• The main apparatus for this experiment is fully self-contained; the samples are in small sealed
glass tubes, which should be replaced in their holder when not in use. The experiment involves low magnetic fields and low-power radio frequencies.
2
Objectives
The aim of this experiment is to measure the properties of Electron Spin Resonance (ESR) in two
organic compounds, and devise a method using ESR to determine the magnitude and direction of
the Earth’s magnetic field at Sydney.
3
Introduction
The idea of electron spin and its associated magnetic moment was first introduced by George Uhlenbeck and Samuel Goudsmit1 to explain fine structure in optical spectra. These new properties
which the electron seemed to possess were put on a sound theoretical basis a few years later in
1
G.E. Uhlenbeck & S. Goudsmit, Naturwissenschaften, 47, 954 (1925)
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S ENIOR P HYSICS L ABORATORY
a landmark paper by Paul Dirac in 1928.2 A proper relativistic quantum mechanical treatment of
the hydrogen atom leads naturally to the electron possessing, in addition to its mass m and charge
e, an intrinsic or spin angular momentum with quantum number s = 1/2. The spin vector S has
magnitude
S=
q
s(s + 1)h̄ =
√
3
h̄.
2
(1)
In the presence of a magnetic field, the component of spin parallel to the field (z direction) is
quantised. It has two possible values,
1
Sz = ms h̄ = ± h̄.
2
(2)
The possible orientations of S relative to the z direction are shown in Fig. 15-1.
Fig. 15-1 : Vector model for electron spin
Since an electron has both charge and angular momentum, it also has a magnetic moment, µs,
proportional to the angular momentum, given by
µs = g
e
S,
2m
(3)
where g is the electron spin g-factor, e is the electron charge and m is the electron mass. According to Dirac’s theory, g is precisely 2. The accurate measured value, g = 2.002319304386, arises
in quantum electrodynamics (QED) from the electron’s anomalous magnetic dipole moment and
agrees precisely with QED calculations. Quantisation of spin leads to quantisation of the corresponding magnetic moment. The component values for magnetic moment are
µsz = −g
2
e
Sz = −gms µB = ∓gµB /2,
2m
P.A.M. Dirac, ‘The quantum theory of the electron’, Proc. Roy. Soc. London A, 117, 610 (1928)
(4)
E LECTRON S PIN R ESONANCE
15–3
where µB is the Bohr magneton, eh̄/2m. In the presence of a magnetic field B, there is potential
energy U associated with the orientation of the magnetic moment relative to the magnetic field,
given by
U = −µs · B = −µsz B = ∓gµB B/2.
(5)
The energy difference between the two electron states is given by
∆U = gµB B.
(6)
Transitions between these states can therefore be excited by photons of frequency νr , given by
hνr = gµB B.
4
(7)
Free radicals
In principle, we could take a group of electrons in a magnetic field B and induce them to make
transitions from one ms state to the other using a radio frequency field of frequency νr . This would
be difficult as electrons repel each other and, in any case, they would describe circular orbits at
the cyclotron resonance frequency eB/(2πm). However, a “solid” source of electrons is obtained
from an organic chemical called a free radical: the two that will be examined in this experiment
are 2,2-diphenyl-1-picrylhydrazyl (DPPH) and 7,7,8,8-tetracyanoquinodimethane (TCNQ). Their
chemical makeup is shown in Fig. 15-2.
TCNQ is an electron acceptor that is used to prepare charge transfer salts, which are of interest in
molecular electronics, while DPPH is an organic free radical that contains electrons associated with
one of the nitrogen atoms that act as though they were free. Free radicals thus exhibit g-factors very
close to the value for a free electron, 2.0023. In fact, the fractional difference is less than one part in
a thousand. Any shift in g would arise from an interaction between the electron magnetic moment
and the overall magnetic moment of the molecule. We would expect a shift equal to or less than the
fine structure shift in the H atom, and this quantity is of the order of one part in a thousand.
Fig. 15-2 : Chemical structure of the free radicals TCNQ (left) and DPPH (right)
Electron spin resonance in TCNQ and DPPH can be shown with the sample held in a small coil and
placed in a uniform magnetic field. The coil (inductance L) is connected in parallel with a capacitor
C to make an LC resonant circuit. At the resonant frequency, νr = geB/(4πm), energy must be
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S ENIOR P HYSICS L ABORATORY
supplied by the circuit in order to change the spin state of the electrons from one value of ms to the
other. As a result there will be a minute decrease in the Q or quality factor of the resonant circuit.
When the circuit is set up to oscillate weakly at a frequency ν (= 2π√1LC ) the electron spin resonance is readily seen because the tiny decrease in the Q of the resonant circuit can be easily detected
by the amplifying electronics.3 Sweeping through the resonance is best done by varying B, since
it is electronically more difficult to sweep through in frequency and maintain a constant amplitude.
In our case, the field is produced by a Helmholtz coil (see Section 5.3).
5
The apparatus
The manual for the spectrometer you will be using is on the bench and has also been placed as a PDF
file (named CWS 12 50 manual) on the computer. The pages relevant to the ESR experiment are
19, 25–41, 55, 71–72, 79, and 82. Figure 15-3 shows a schematic diagram of the ESR spectrometer.
Fig. 15-3 : Schematic diagram of the electron spin resonance experiment. The reference to NMR (nuclear
magnetic resonance) is not relevant for this experiment.
p
Note: the Q of a parallel-tuned LC circuit is given by Q = R C
. The excitation of ESR causes a slight increase
L
in loading of the tuned circuit and a corresponding decrease in the effective value of R.
3
E LECTRON S PIN R ESONANCE
15–5
The sample is placed in a small coil, inductance L, in parallel with a capacitor C. This is the
resonant circuit (also called a tank circuit), which oscillates at 50 MHz. The phase-locked loop
(PLL) is a feedback circuit that adjusts the value of the capacitance until the circuit is locked onto
the preset frequency. The variable capacitor is a reverse-biased diode, known as a varactor diode,
in which C depends on the thickness of the charge depletion layer; this is controlled by adjusting
the applied DC voltage.
5.1
Differential detection
The principle of resonance detection is simply to detect a change in the amplitude of oscillations in
this coil. This is achieved by varying the applied magnetic field in a sinusoidal manner at 38 Hz. The
50 MHz frequency is filtered on the detection side of the electronics and only the signal amplitude
difference (due to the 38 Hz modulation of B) is passed on to the detection amplifier.
A consequence of this sensitive detection method is that the derivative of the resonance (or absorption) signal is obtained. The characteristic profile of a forced resonance, either that of a damped
LC circuit or a natural transition in an atom or electron, is known as a Lorentzian. The Lorentzian
function is given by
L(ν) =
1
1
2Γ
,
π (ν − ν0 )2 + ( 21 Γ)2
(8)
with full width at half maximum (FWHM) equal to Γ. In this experiment we keep the frequency
fixed and vary B, but the form of the equation is the same, with ν replaced by B, as shown in
Fig. 15-4. By convention, Fig. 15-4 is shown in emission rather than absorption.
Fig. 15-4 : The Lorentzian line shape of an absorption (resonance) line.
The detected signal is the derivative of the Lorentzian line profile, as shown in Fig. 15-5. Differential
detection enables the precise position of the absorption peak to be determined, since it corresponds to the point where the differential signal equals zero. i.e., the zero crossings. Moreover,
one commonly-used measure of the line width of the Lorentzian can easily be obtained from the
separation between the positive and negative going peaks in the differentiated signal.
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S ENIOR P HYSICS L ABORATORY
Fig. 15-5 : The derivative of the absorption signal shown in Fig. 15-4. This is the form of the signal received
from the spectrometer.
Although the software that operates the spectrometer can do an approximate analysis for obtaining the zero crossing, line width, and integrating to obtain the absorption signal, a more accurate
measure of these quantities is obtained by fitting a Lorentzian to the integral of the differential
signal.
5.2 Significance of line-width
The natural width of an ESR spectral line is a measure of its interaction with its environment.
That is, the electrons not only interact with the applied magnetic field but any other magnetic field
arising from their environment such as those from other electron or nuclear spins. So ESR can be
an excellent probe of the electronic or nuclear magnetic environment. The line becomes broader
with increasing coupling of the electrons to their environment.
5.3 Helmholtz coil
Helmholtz coils consist of two coaxial coils of the same radius, in series and spaced a distance apart
equal to their radius. For this configuration the field is particularly uniform over a small region at
the centre of the configuration.
The magnetic field supplied by the Helmholtz coils is shown schematically in Fig. 15-6. A uniform
magnetic field of variable strength is often needed in physics. For a large field this is best done with
an electromagnet whose core is composed of a magnetic material of high relative permeability and
with flat parallel pole pieces. If this technique is used for small fields (as required in this experiment)
a difficulty arises because all magnetic materials have a remanent field: a finite field is present even
when the magnetising current is zero, making the setting of reproducible fields difficult.
E LECTRON S PIN R ESONANCE
Fig. 15-6 : Helmholtz coils, showing the
uniformity of the magnetic field in the central region. [from The Physics of Electricity
and Magnetism, W.T. Scott (Wiley, 1962)]
5.4
15–7
Fig. 15-7 : The dip circle gives the angle of the Earth’s
magnetic field.
Dip circle
The dip circle, shown in Fig. 15-7, is a compass that enables the Earth’s magnetic field direction
to be determined. Ordinary compasses allow you to determine the direction of the Earth’s field in
the horizontal plane. In reality, the Earth’s field makes an angle in the vertical plane. The dip circle
allows you to determine this angle since the compass needle can not rotate in both the horizontal
and vertical planes.
6
Experimental Procedure
1. Make sure that the spectrometer is connected as shown on page 19 of the bench manual, and
that the TCNQ sample is fully inserted in the Probehead. The Helmholtz coil and Probehead
combination are fixed to a Perspex sheet making it easier to rotate the assembly later in the
experiment. For this first part of the experiment the orientation is not important.
2. Start the CWNE software. This brings up the Startup and Acquisition page with a comprehensive menu for data acquisition.
Click on Spectrometer then on Connect in the top menu. This establishes a connection between the computer and the spectrometer. It will also automatically switch to the ESR mode
and will indicate this by a red light next to ESR on the probehead. Select Single (for a single
scan) and let Acc be initially 1 (for one scan) in the bottom right hand corner of the screen.
Set the Field Sweep to 5 gauss (Gs)4 and leave the 2nd Mod Amplit and Sweep Time at their
default values of 0.05 Gs and 0.5 min respectively.
4
The gauss is the unit of magnetic field in the old cgs system: 1 Gs = 10−4 T.
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S ENIOR P HYSICS L ABORATORY
3. Clicking on the Start button in the Acquisition section will start the 30-second scan. You
should obtain a differential plot as described in section 5.1. This will probably be a bit noisy.
Vertical scaling can be adjusted with the Gain control and the DC offset with the slider control
on the left of the screen. Experiment with both controls until you are satisfied with the result.
4. You are now ready to collect data. First you need to create your own subdirectory in the
Student files directory. Then switch back to the Acquisition screen and, under Tools → File
location, tell the software where to put your data by typing the directory path for both acquisition and data output.
5. To reduce the noise in the signal try averaging several scans by increasing Acc to around 5
and depressing the Acc button in the upper left hand part of the screen. Pressing Start will
then do the averaging. You can save your spectrum from the File menu on the top left hand
corner of the screen; it will be saved in your subdirectory with a .dcw extension.
6. The saved spectrum can now be processed by clicking on the Proc button at the top of the
screen. This will take you to another screen with upper and lower graph panels. Use the Open
File button to load your saved file into the top panel, and the 1st Derivative button on the top
left hand side of the screen to load it into the bottom panel.
You are now ready to make some basic measurements on this spectrum using the inbuilt
software. The VE button above the bottom graph panel autoscales the spectrum in the vertical
direction, and the G button alongside calculates the g-value.
7. There are two ways of measuring the linewidth: (i) the separation between the points of
inflection, which is given by the SDB button, and (ii) the full width at half maximum (FWHM)
of the absorption profile. The latter measurement is obtained by clicking the Absorption
button at the top of the screen — this integrates the 1st derivative spectrum — and then the
HDB button above the lower panel.
Record the values of g, SDB, HDB and position of the absorption peak (or zero-crossing in
the differential plot) in your logbook, along with any other relevant information about the
dataset.
8. The 1st derivative data can be saved for later analysis via the Export ASCII button at the top
of the screen; the file name now has a .txt extension.
9. Finally, go through the same procedure for the DPPH sample with the Field Sweep set to
15 Gs.
Question: Using the block diagram and description in Section 5, explain why the output of the ESR
apparatus is the derivative of the resonance curve.
C1 ⊲
E LECTRON S PIN R ESONANCE
6.1
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Lorentzian profile
In this section we will analyse the 1st derivative data using QtiPlot and test whether the absorption
profile is indeed Lorentzian, as we have assumed.
1. Import the TCNQ .txt file into QtiPlot and plot the file.
2. Before integrating the differential plot we must ensure that the baseline is set precisely to
zero. Why? Devise a method for doing this, then use the Integrate function in QtiPlot to
obtain the absorption profile (which appears as an emission profile!).
3. Fit a Lorentzian to the absorption profile and save a properly labelled hardcopy. Compare
the parameters of the fit, notably the peak position and FWHM, with those obtained earlier.
Comment on the goodness of fit, as measured by the reduced χ2 . [If you’re unfamiliar with
the concept of χ2 , click here.]
4. Repeat the above treatment for the DPPH data.
Question 1: Which of the two samples has their electrons more strongly coupled to the environment?
Question 2: Using Eqn. 8, or a simplified version5 , show that the relationship between the
√ separation of the points of inflection (SDB) and the FWHM (HDB) is given by SDB = HDB/ 3.
C2 ⊲
6.2
Measurement of the Earth’s Magnetic Field
The Geographic North Pole is actually a magnetic South Pole as shown in Fig. 15-8
• Use the dip circle to determine the direction of the horizontal component of the the Earth’s
magnetic field and the angle of dip in this plane. You will need this later when determining
the magnitude of this field. Turn off the Helmholtz coils and keep the compass away any iron
objects so as not to distort the direction of the field.
6.3
Determination of the Earth’s magnetic field and the electron g-value
If it wasn’t for the Earth’s magnetic field, it would be a simple matter to find the g-value by obtaining
the resonant frequency and using Eqn. 7. However, the Earth’s field will add to or subtract from the
applied field from the Helmholtz coil, depending on the orientation of the axis of symmetry of the
coil with respect to the horizontal component of the Earth’s field.
Note that the axis of symmetry of the Helmholtz coil is in the horizontal plane. If the horizontal
component of the Earth’s field is parallel to the axis of the Helmoltz coil, then the two fields will add
and this combined field will be applied to the electrons. However, if these two fields are antiparallel
the horizontal component of the Earth’s field will subtract from the applied field. Ideally, these two
5
eg. y = a/(x2 + a2 )
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S ENIOR P HYSICS L ABORATORY
Fig. 15-8 : Geographic north and south poles are actually magnetic south and north poles, respectively. Note
that the magnetic field lines at Sydney emerge at an angle to the Earth’s surface.
configurations will produce resonant peaks symmetric about an applied field of zero; if they don’t
there may be a small systematic error in the assumed B field from the Helmholtz coils.
• Now devise a method that will accurately determine the g-value of the electron as well as the
horizontal component of the Earth’s magnetic field.
Explain how you will use the angle of dip measured with the dip circle to determine the
magnitude of the Earth’s magnetic field.
• Outline your method in your logbook and get a tutor to check before proceeding. Don’t forget
that you are dealing with two fields that must be combined vectorially.
C3 ⊲
6.3.1
Results and Analysis
• Carry out your experiment using the TCNQ sample, save your results and analyse using
QtiPlot, including errors in your analysis.
• Use the accumulate feature in the acquisition program if you need to reduce any noise.
• Use your results to determine values for g and the total magnetic field of the Earth. Compare
your results for the Earth’s magnetic field with the accurate values from the online calculator
at the National Geophysical Data Center at
www.ngdc.noaa.gov/geomag-web/#igrfwmm,
by selecting the country and city.
• Write a conclusion.
C4 ⊲