PROPERTIES OF ELECTRONS
9 PROPERTIES OF ELECTRONS
9.1 INTRODUCTION
Electrons possess properties that may only be understood with quantum mechanics; their intrinsic
angular momentum, or spin, and their wave nature. In the two experiments that follow you will
find compelling evidence of these properties through the observation of Electron Spin
Resonance (ESR) and Electron Diffraction.
The first day will involve the study of Electron Spin Resonance and the second day will be an
investigation into Electron Diffraction.
While these notes are designed to be self-contained students are directed to the cited references
for a more detailed treatment of the theory as some results are presented without proof. In
particular a cursory reading of the pertinent sections of “Introduction to Solid State Physics” by C.
Kittel, which is available from the Part II office, may provide some valuable insights into the
phenomenon of electron diffraction for those unfamiliar with the concept.
9.2 ELECTRON SPIN RESONANCE
9.2.1 OVERVIEW
Fine structure in the spectral lines of hydrogen alerted physicists to the need for an additional
quantum number to adequately describe the observed energy levels of hydrogen. While n, l and
ml stemmed naturally from Schrodingers’s original theory, the quantity of spin, or the electron’s
intrinsic angular momentum, was an amendment made by Dirac.
9.2.2 BASIC THEORY
a)
Orbital Magnetic Dipole Moment
An electron of mass me, charge –e, with velocity v moving in a circular Bohr radius constitutes a
current loop with an associated orbital magnetic dipole moment. It may be shown that this
orbital magnetic dipole moment, l, is related to the orbital angular momentum of the
electron, L, by the relation


l  
b L
Eq. 9.1

where b is the Bohr magneton, the magnitude of which is
b 
e
2me
Eq. 9.2
b is the basic unit of atomic magnetic dipole moments. Plank’s constant,  is the basic unit of
angular momentum.
Pre-lab Question (a)
Attach a physical significance to the negative sign in Eq. 9.1.
79
PART II LABORATORY
b)
Electron Spin
The electron possesses an intrinsic angular momentum, or spin, S, which is an analogue of the
orbital angular momentum, L. The magnitude of the electron spin is fixed and its projection onto
any fixed axis has one of only two values. This quantity is given by

S   ss  1
Eq. 9.3
where s has the single value of 1/2. The component along any fixed axis, nominally defined as z
is
S z  ms 
Eq. 9.4
with ms=1/2.
Associated with the spin is an intrinsic magnetic dipole moment s which may be calculated as


 s   g s b S / 
Eq. 9.5
It may be helpful to imagine this intrinsic magnetic dipole moment as arising from current loops
associated with the electron’s spinning charge. The factor gs appearing in Eq. 9.5 is called the gfactor of the electron and is a dimensionless number. This may be considered a measure of the
homogeneity of the electron charge distribution. However, it should be noted that the quantities
s and gs arise purely from quantal considerations and the picture given above is offered merely
as an aid to understanding.
c)
An Electron in a Uniform Magnetic Field
A magnetic dipole in a magnetic field experiences a torque. There is a potential energy of
orientation given by
 
E    .B
Eq. 9.6
A spatially uniform applied magnetic field defines a unique direction in space. This provides a
physically meaningful axis for the projection of the electron spin, which may then be either parallel
or antiparallel to B.
Pre-lab Question (b)
Show that the magnetic potential energy of an electron in a
magnetic field B is given by
Eq. 9.7
1
E   g s  b B
2
d)
Resonance Absorbtion
An electron in a stable atom will occupy one of a set of discrete energy levels. As a simplifying
assumption let us say that the electron has zero orbital angular momentum with respect to the
atomic nucleus. If the electron occupies energy level designated E0 in the absence of an applied
magnetic field, then in the presence of a field, B it may have one of two energies given by the
expression
80
PROPERTIES OF ELECTRONS
E
1

2
 E0 
1
g s b
2
Eq. 9.8
This splitting of one energy level into two through the application of a magnetic field is known as
the Zeeman effect and is represented diagramatically in Figure 9.1.
Figure 9.1 Zeeman splitting of an energy
level through the application of a
magnetic field, B. The corresponding
absorption peak is also shown.
Note that the electron can make transitions from one energy level to the other either absorbing or
emitting a photon of energy hf. Thus conservation of energy dictates the resonance condition
hf  g s  b B
Eq. 9.9
where hf is the energy of the photon as shown. As with more familiar examples, such as standing
waves, resonance here describes the maximum transfer of energy when the system is tuned to a
particular frequency, termed the resonant frequency. In general, the probability of spontaneous
absorption is negligible compared to absorption or emission being stimulated by a radiation field.
In this experiment the stimulated absorption process is further enhanced through the use of the
molecule diphenyl-picra-hydrazyl (DPPH-see Figure 9.2) for the observation of ESR. The
electrons on one of the nitrogen atoms in this sample are free, that is, not bound in closed atomic
shells or involved in interatomic covalent bonds. In this case the DPPH sample will absorb the
most energy when f is tuned to the resonant frequency for a given B.
For the sample to absorb an appreciable amount of energy in this manner other parameters must
be considered. For instance the population in the lower energy level must be larger than in the
upper level at thermal equilibrium. The lifetime of the excited molecule or atom is also important.
The processes by which the atom returns to thermal equilibrium are called relaxation processes
and may be visualised as the transferral of the absorbed energy to the crystal lattice. Generally,
it may be taken for granted that in laboratory conditions the radiation fields used do not perturb
the system far from thermal equilibrium. However, if this is not the case the magnetic field
strength, field sweep rate, modulation frequency and the time between sweeps becomes
important.
Assuming that the stated criteria for resonance absorption to be observed are met, then a
paramagnetic sample placed in a magnetic field of strength B will absorb energy from the
radiating em field peaked about a resonant frequency f given by the expression
g  B
Eq. 9.10
f  s b
h
The value of gs may then be deduced by measuring f as a function of B.
81
PART II LABORATORY
Figure 9.2 The molecule diphenyl-picrahydrazyl, or DPPH. This is a radical with
an unpaired electron on one of the
nitrogen atoms. The unpaired electrons
in the DPPH sample have no orbital
angular momentum. Thus spin
resonance may be studied in isolation.
9.2.3 EXPERIMENT
The experimental arrangement is shown in Figure 9.3. The RF oscillator unit provides the
radiation field to stimulate the absorbtion process in the DPPH. The external magnetic field is
supplied by a pair of Helmholtz coils. The Helmholtz coils are powered by an AC supply so that
the magnetic field passes slowly through the value required for resonance absorbtion compared
to the radio frequency. This ensures that the system is able to relax between successive sweeps
and the populations of the levels do not become inverted. Hence the observed drop in oscillator
current is free from instrumental variables such as sweep direction and the time between sweeps.
Figure 9.3 The apparatus used for the observation of ESR. The voltage across
the 1 resistor may be viewed on Channel 1 of the CRO. The voltage across the
RF oscillator unit may be viewed on Channel 2 of the CRO.
Pre-lab Question (c)
The particular arrangement of two coaxial
circular coils of radius R separated by distance R is known as a pair of Helmholtz coils.
Let x denote the distance from the plane of the left hand coil to any point on the axis. If
there are n turns in each coil and each turn carries a current I demonstrate that at the
midpoint between the coils the magnetic field is given by
82
PROPERTIES OF ELECTRONS
3
 4  2  nI
B  0
5 R
Eq. 9.11
directed along the x-axis. You may wish to begin with the expression for the field produced
by one such coil (B=0nIR2/2(x2+R2)3/2). Convince yourself that the field remains
essentially constant over a large volume at the centre by calculating the field at x=0.2R and
at x=0.5R. The Helmholtz arrangement in fact gives the largest possible volume of almost
constant field.
Figure 9.4 The Helmholtz
coil arrangement. Almost
constant field is obtained
over a large volume at the
center
a)
Demonstration of Resonance Absorption
Check that the DC power supply and frequency meter are connected to the radio frequency (RF)
oscillator adaptor. Note that the adaptor frequency output divides the frequency of the
control unit by 1000. Place the large plug-in coil in the RF control unit. Do not yet place the
DPPH sample in the vial inside the coil.
The plug-in coil forms part of the RF oscillation circuit. While the particulars of this circuit are not
necessary for this experiment, you should note that the coil is part of a high Q parallel resonant
circuit. The oscillation frequency is governed by both the inductance of this coil and a capacitor
whose value may be varied using the knob on top of the control unit. Using the two coils with
fewer turns it is possible to obtain frequencies for the radiating field in the coil from 30 MHz to 130
MHz. Coils with a larger number of turns have a higher inductance and hence give lower
frequencies. Connect the micro-ammeter to the RF unit via the socket marked I/(A) at its rear.
The current is the rectified current in the RF oscillator unit. Observe the current and frequency in
the RF unit.
Bring the parallel LC resonant circuit up to the RF unit until the two coils are almost touching.
Such a circuit is often known as a tank circuit and is most commonly used to select a desired
frequency out of a signal containing a variety of frequencies. The voltage gain of this circuit is
maximum when it is tuned to the resonant frequency. Observe the induced voltage across the LC
circuit on the CRO. Adjust the capacitance of the LC circuit until the voltage across it is a
maximum.
Question (a)
What happens to the current in the RF oscillator unit?
behaviour that you observe.
83
Explain the
PART II LABORATORY
b)
Electron Spin Resonance


Position the Helmholtz coils correctly using a dial caliper. Ensure that they are
connected with the same winding sense and in series to the 1 resistor and to the AC
power supply. “A” identifies the beginning of the winding and “Z” identifies the end. At
this point check with your demonstrator that the coils are connected and positioned
correctly.
Coil specifications:
 Mean diameter of coils=13.6cm
 Number of turns in each coil=320
 Maximum continuous current=1.5A(RMS)
 DC resistance of each coil=6.5(approx.)
Question (b)
What is the purpose of the 1 measuring resistance?

Connect the small plug-in coil to the RF unit and place the DPPH sample in the coil.
Note that the sample is the black powder in the vial. Position this as centrally as
possible within the coil arrangement.

When ESR occurs the sample will absorb energy from the radiating field in the inductor
coil of the RF oscillator unit and hence the impedance of the coil will increase. The
signal output (labelled “Y” on the adaptor unit) is a voltage proportional to the current in
the RF oscillator circuit. This may be observed on the CRO. Thus, at resonance, the
impedance of the circuit will be altered and a drop in the current in the RF unit will be
observed.

Familiarise yourself with Figure 9.3 and identify the features indicative of ESR in the
output of the RF unit. To begin with it may be easier to use a frequency high in the
range given by this coil to observe ESR.
Question (c)
What happens to the features when you alter the current to the
Helmholtz coils? How does this determine how you will design your experiment?

Measure the Helmholtz coil current, I, at which resonance occurs for 10 frequencies in
the accessible range. In practice this range is between 130 MHz and around 50MHz.
Take care to detail your procedure, including your efforts to minimise errors.

Plot your results and determine a value for gs.

Experiment with a small bar magnet. What happens to the features in the CRO trace
on Channel 2 when the magnet is bought towards the DPPH sample
 along the coil axis and
 perpendicular to this?
Question (d)
What is happening?
9.2.4 DISCUSSION
Compare your value with gs=2. This was the value predicted by Dirac with his relativistic electron
wave equation. Note that the value predicted by a purely classical treatment would be one.
Spectroscopic studies by Lamb have actually determined the value with extreme accuracy to be
2.00232. This discrepancy was explained through quantum electrodynamic considerations, or
the quantum mechanics of electromagnetic fields. For most applications, however, it is sufficient
to remark that the spin gyrometric ratio of the electron is twice as large as its orbital gyrometric
ratio. To see this, compare Eq. 9.1 and Eq. 9.5.
84
PROPERTIES OF ELECTRONS
Question (e)
What quantum mechanical principle ultimately governs the width of the
resonance peak? Remember that we are concerned with the energy absorbed
by the sample at resonance.
Question (f)
How is the strength of the coupling between the excited molecule and the
crystal lattice reflected in the width of the resonance peak? The energy absorbed
by the molecule at resonance is dissipated via a transferral of energy to lattice
vibrations called “phonons”.
Question (g)
What field strengths (order of magnitude estimate) would be required for
nuclear magnetic resonance (NMR)? NMR uses the magnetic dipole moment
associated with the motion of charges in the nucleus. You may wish to express
your estimate as a proportion of the fields used for ESR. For a measure of the
accessibility of these atomic resonance effects look at the nuclear and Bohr
magnetons.
9.2.5 REFERENCES

For an introduction to the quantum mechanics of angular momentum and spin see
“Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles” by R. Eisberg and
R. Resnick or any second year text on the subject.
 For a discussion of the theory of Helmholtz coils see
“Experimental methods in Magnetism” by H. Zijlstra.
 For a detailed discussion of the experimental considerations of ESR see
“Electron Paramagnetic Resonance – Techniques and Applications” by R. S. Alger.
 For supplementary information on LC resonant circuits see “Introductory Electronics for
Scientists and Engineers” by Robert E. Simpson. Section 2-12
9.3 ELECTRON DIFFRACTION
9.3.1 OVERVIEW
The postulate that matter may also possess wave-like properties was one of the most stunning
and unifying hypotheses of this century. In 1924 de Brolgie, working from arguments based on
the properties of wave packets, hazarded that the wavelength of these matter waves could be
obtained from the same relationship that held for light, namely

h
p
Eq. 9.12
where  is the wavelength, p is the momentum and h is Planck’s constant.
Davisson and Germer obtained the first experimental evidence for matter waves by reflecting
slow electrons from the face of a single nickel crystal. The wavelength that they were able to
determine from using the periodic array of nickel atoms as a diffraction grating was strikingly
similar to that calculated from the de Brolgie relation. Further experimentation by Thomson
yielded diffraction patterns from the transmission of electrons through thin polycrystalline foils.
The fact that non-relativistic electron wavelengths are comparable to interatomic distances in
solids means that electron diffraction experiments (such as Transmission Electron Microscopy, or
TEM) are powerful determinants of a solid’s structure.
85
PART II LABORATORY
9.3.2 BASIC THEORY
a)
Wavelength
The wavelength of a beam of electrons is given by

h
h

p mv
Eq. 9.13
Thus in the non-relativistic limit the wavelength is inversely proportional to the electron velocity, v.
Conservation of energy requires that for a charge accelerated through a potential the change in
kinetic energy plus the change in electrical potential energy while traversing from one point to
another must equal zero since the external forces on the charge do no net work. Thus we may
write
 mv 2 2 mv 1 2 


 2  2   eV2  eV1   0


Eq. 9.14
where V1 and V2 are the potentials at points 1 and 2 respectively. Applied to the electron
traversing an electron gun in an electron diffraction tube V2 is Va, the anode voltage and V1
equals zero. So for negative charges
mv 2
 eVa
2
Eq. 9.15
Hence the wavelength is given by

h
Eq. 9.16
2m eVa
where m and e are the mass and charge of the electron respectively. When the values of the
constants are substituted into Eq. 9.16 and the potential is in volts then the wavelength in
nanometres is given by

b)
1.2264
Eq. 9.17
Va
Electron Diffraction
Pre-lab Question (d)
Given the situation of electrons being reflected from a set of
planes, and the geometry shown below derive the condition for constructive
interference, known as the Bragg condition,
Eq. 9.18
2d sin  b  n
where d is the separation of the diffracting planes,  is the angle between the incident
beam and the crystal plane and n is the order of diffraction.
The condition for constructive interference is that the path difference between adjacent rays must
be equal to an integral number of wavelengths.
86
PROPERTIES OF ELECTRONS
Figure 9.5 The reflection of light from a set of planes.
The Bragg condition for constructive interference may
be derived purely from the geometry shown.
The experiment you will undertake replicates Thomson’s method of transmitting electrons through
a thin film of randomly oriented crystallites to investigate a ring diffraction pattern. The periodic
array of atoms in the 3-dimensional solid forms a space lattice. Diffraction from a space lattice is
somewhat more involved than the situation just depicted, however, a close analogue of the Bragg
equation still holds.
2d n sin  b  
Eq. 9.19
Here dn is the distinct interplanar distance for a particular set of planes described by the periodic
atomic arrangement. A diffraction pattern for a single crystal with a cubic structure is shown
below. R1 and R2 arise from two different sets of planes, and when analysed, will yield two
separate d values, d1 and d2.
The angle between the direct and diffracted beam is shown in Figure 9.7. You will be measuring
ring radius as a function of anode voltage.
Figure 9.6 The electron diffraction pattern from a
single crystal of a substance with a cubic
structure. The radii shown correspond in real
space to two separate sets of crystal planes and
yield two values for the interplanar spacing, d.
Pre-lab Question (e)
Using the geometry shown, derive an expression relating the ring
radius to the scattering angle, 2b. You will be measuring D=2R with a pair of dial
calipers so take care when you define this quantity.
Pre-lab Question (f)
Using the Bragg relation and the expression determined above,
derive an expression for dn, the interatomic spacing in terms of R.
Pre-lab Question (g)
The specimen you will use in this experiment is graphitised
carbon. Assume that the form of carbon is a simple cubic structure. The atomic mass
of carbon is 12 and its density is 2.3 g/cm 3. Calculate the spacing of adjacent carbon
atoms.
87
PART II LABORATORY
Figure 9.7 The geometry of the electron diffraction experiment.
9.3.3 EXPERIMENT

Ensure that the apparatus is wired up as shown in Figure 9.8.
HIGH VOLTAGES ARE USED IN THIS EXPERIMENT SO BEFORE SWITCHING THE
APPARATUS ON CHECK THAT ALL THE VOLTAGES ARE TURNED DOWN. IF THIS IS
UNCLEAR CONSULT YOUR DEMONSTRATOR.

Switch on the high voltage supply and wait for one minute for the cathode heater to
stabilise. Turn the meter to read “CT” and slowly increase the high voltage, Va, to the
approximate value of 2.7 kV. Adjust the bias voltage supply Vb to produce a clear ring
image and uniform central spot. If Vb is initially connected the wrong way reverse it,
but TURN DOWN THE HIGH VOLTAGE BEFORE DOING THIS. Current overload
causes the target to glow a dull red, so inspect the target at intervals during the
experiment. The anode current should be monitored on a meter and never allowed to
exceed 0.2 mA.

Sketch the observed ring pattern. Roam around the neck of the tube with a bar
magnet to try to increase the clarity of the ring pattern. If there is no improvement
remove the magnet. Sketch the ring pattern with the magnet in different positions.
Explain what you observe.
Question (h)
Which end of the magnet is North?

Using the dial callipers measure both ring diameters for 12 different anode voltages.
You may remark that the rings are distorted. To minimise errors four measurements
should be taken in the four quadrants and averaged. Watch for drift in the voltage and
adjust when required. Record all your measurements including the voltages.

Explain why the ring diameters increase with decreasing voltage.

Graph 1/(Va)1/2 against sinb in Sigmaplot®, and calculate the interatomic spacings from
the gradient. Compare your lattice spacings to the actual values of d11=0.123 nm and
d10=0.213 nm. From your derived values of interplanar distance suggest how the
carbon atoms are more likely to be arranged. Use a diagram to demonstrate your
conclusions.
88
PROPERTIES OF ELECTRONS
Figure 9.8 The circuit used in the electron diffraction experiment.
9.3.4 DISCUSSION
Question (i)
Why does diffraction from graphitised carbon produce a ring pattern?
What does a single crystal diffraction pattern look like? Explain the difference.
Question (j)
Could you perform a similar experiment with protons? What parameter
would you have to change?
Question (k)
Would Eq. 9.16 still be valid for a 20 MeV beam of electrons? What
expression would be more appropriate?
9.4 USEFUL DATA



h=6.62510-34 Js
b=9.27310-24 Am2
Avagadro’s number = 6.022171023 /mol
89