with
them of
media
142
5.6
Calculation
magnetic fields
hysteresis curve
10.4 Plane
329
12.2
wavelengths 395
14.4 The
445
16.4 Shot noise
17.2 Dielectric constant of
liquids
solids
488
resonance 689
small.
It
is
integral
therefore
indivisible, and is
nucleus. The forces which hold the nucleus together are of
different
of
divided
into
two
can be explained
free
ions.
and
When
a
substance
has
no
of
positive
and
a substance,
Since the electrons can move
so
much
the
them,
then
The units
q.
value,
when
equation
(1.1)
may
constant
(e.s.u.)
force
 
Then
q
v
q
2
are
book, the unit of charge
is
system of units).
and standard
kilogramme. Equation
constant
theory.
Equation
(1.1)
therefore
becomes
F
and
q
in
§
-1
(see
of
the work
charge
q
at
0,
as
in
potential difference
between points
field
be done
charge.
On
the
other
distribu-
q,
the surface
S at
477
at
any
point
within
the
of
E
integration
charge. In this way it is
found that
field in this space.
(see
below)
depend
conductor.
(a)
A
hollow
+g
total charge
such that
most subsequent at-
of the
sphere in
no charge.
that
the
value
of
n
covered with
the salt
by
P
is
F-JLfl-I),
(1.9)
been written as
P is
potential may
to B, then
the
subscript
is
equation
(1.11a)
of
are zero
in each
dipole,
but
there
was inserted
vacuum
and
1-00057
it.
Dielectric
static
charges on the nuclei. In the absence
of
are
distributed symmetrically with respect to the nuclei, but when a
field
direction
opposite
to
that
of
the field, while the more massive nuclei are slightly displaced in
the direction of
acquires an electric dipole
same
a slab
experiment
and
the
voltage
across
be an
dielectric
surface
positive
negative
field
used in the transforma-
can
be
P
equation
(1.17)
total charge
.
define
a
new
necessarily
i.e.
divD
x
charge
q
flux
through
the
element
of
area
dS.
From
(1.19)
the
no
media. Lines
of E
of
the
conductor.
The
ratio
QjV
is
a charge of opposite
capacitance
has
increased.
constant
of the plates
dielectric
constant
though con-
its
size,
and
the
potential
across
the
n
jfcth
capacitor
2
(QIG
k
two results
across each
capacitor is
equal to
is
Q
through
charges are
increased
is
charges.
Then
if
V
k
surface may be
taken at an infinitely large distance from all the charges,
and its contribution
distances,
V
varies
2
density
the
integral
is
the surfaces of all the con-
ductors. Since the integration
into the con-
dielectrics, where
D is
that D
mutual repulsion
between charges
conductor.
of virtual
work to
dielectric
constant
capacitor
if
case we
constant
potential
work done
the
system
at
in the stored
large
number
of
lines
charges of opposite sign,
for example, was transmitted
of
force
all contract until they passed straight
from
one
charge
to
another.
These
forces
in
Panofsky
dS,
boundary,
there
will
be
equal
and
opposite
the two sides, there will be a net force acting
at
the
bounding
surface.
The
charged
conductor
tension
given
by
equation
(1.37)
agrees
compute
Phillips,
at a
between r and the
two
coaxial
balance
mass which must be
force
on a charged conductor. An attracted disk electrometer has a moving plate
of area 100 cm
two
disks
is
in the
tj per
infinite
work
of an
1.14.
the
10e
than
it
That
this
known as
now
discuss
a
number
of
fact that if
adjustment
of
insoluble
by
ordinary
method are
given below.
of
1),
most
of
the
problems
tube, are dealt with
coordinates,
cos#.
=
different
they are correct
by evaluating the
h
is
Planck's
constant.
terms
as
p
x
(8V/8x)
+p
v
(8Vjdy)
point charges
rather than
the
component
p
z
are
different
(i.e.
if
are identical,
its potential varies in the neighbourhood of the point O,
e.g. at
distance
AB.
If
r
(rjB):
1
placed
in
.
sphere in
can be
of
are zero. It
possible
power of
 
we
there.
Thus
V
x
x
the
dipole of
from
.
conducting
plane.
(cf.
Problem
2.6),
where
r
Q
the
induced
charge
sphere of
sphere
is
single charge
potential
at
a
point
Q
on
distributed
A,
im-
mersed
Gauss's
theorem
from the
axis is
'--skJT—sk
perpendi-
—A
v^
They form
infinite line
the
position
of
of
the
family
of
coaxial
circles
its image.
points of a family
the potentials
be
V
n
q
will
dielectric
point charge
of
q
a
is
then
(2-54)
By
symmetry,
q,
q
x
surface. Note
that the
dielectric constant
equations
systems in vacuo.
we must assume
automatically satisfies our first
the
as we move along
case ofa point
charge and an
REFERENCES
G. R.,
E
f
is known as the 'reaction field'. These formulae are used in
the
theory
of
dielectrics
(see
Chapter
17).
density
conducting cylinder of radius
9.
(Answer:
a
2e
E
oos6.)
2.5.
A
uniform
electric
field E is set up in an infinite dielectric. Show that
(a) if a
length, is cut in the dielectric with its axis parallel
to
cut
,
Verify
that
the
—
the
text
in bringing
plane is at
of
a
of
g
2
Rb
4tt€
(i?
2
-6
2
q
is an
8:29.
0)
2.1),
and
G
stationary electric
charges. If
field
it
lines
the charges
will move
If we are
surface
by
which
current
may
be
injected
into
at constant
field. This
conductivity, and its reciprocal
resistance.
If
of the wire. Equa-
flowing in the conductor
there is
=
potential
so
that
between two points differing
power
is
the
volt-ampere,
known
of
con-
volume ofthe
qq
m
-f
m
e.m.f.
Wheatstone's bridge
method of
7?
4
is zero, and G
B
lt
B
2
Wheatstone's bridge can
so
so
that
its
calibration
measuring
small
resistances,
care must be taken to ensure that the resistances of leads
and contacts
be
determined
current
changes
resulting expression
that
through
taken
e.m.f.
is
measured
on
circuit for this is
give the
ultimate balance.
viscous
quite different (see
the
position in
electrons
are outlined below,
converge.
of the word
shown that
and
is
a
of
the
can be calcu-
is
shows
an
electron
travelling
If the electron
then we may make the
approximation
I8V\
\8xl
divide the
potential field
equipotential surfaces
V,
then
its
v=
 
ofa generator
ofe.m.f. equal
to v
balanced.
power
W,
which
'
one
of
-section
0-25
cm
within
viscous medium where there
*-axis.
of
the
electric
is
a
quoted in cm
electrons.
If
we
equal to ct, we find a
simple
expression
of
temperature.
theory arises,
energy of the
energy
of
N
particles
is
%NkT,
3i?/2 to
the
energy
is
of energies is
replaced by a
so
higher
temperatures
it
is
always
masked
by
metal,
the
spacing
electron,
and
the
much
larger
than
to emphasize the role played by the Pauli
exclusion principle, which states that no two electrons in a
given
system
can
quantum
cannot
be
zero
at
0°
K,
since
this
would
mean
that
all
level.
of energy
can only be
from
the
allowed
energy
levels
ft
energy acquired by
of 1 volt.) Free
without
losing
standing
waves
are
the wave equation
for a particle
same
way
that
standing
waves
of
certain
wavelengths
deter-
mined
equation
(4.7)
the
(p
x
,p
y
,p
z
particle
is
then
specified
by
a
point
of
p
and
this shell
is 4np
the
sphere
of
radius
p
minimum.
in
momentum
space
is
components,
replace any
point within
haVe
W
and
hence
W
F
sharp
cut-off
at
value of
3
of
helium,
cm
3
at
atmospheric
pressure
in
the
1*3°
not
to
the
difference
in
denominator
of
equation
(4.11)),
and
=
is
=
Principle an
an
unoccupied
state,
 
order of
was
iJ
2
inside
more
ammeter, as shown
the
current
can
be
passed
the
rapid
the surface;
number of
The
phenomenon
of
to contamination
to
is
similar
may
be
ac-
so
arranged
trode,
overall
circuit
must
if
any
provided
Q
unit
8
determined for one
follow from thermocouple
at low
low
temperatures.
raises the temperature at the centre so that
in one
the same
while
in
the
symmetrical points
insulated from the wire by
means
so
alone. Second, the couples
i\, P
=
'electrolyte'. The
constant
electric
atomic
weight
back
e.m.f.
Practical
the electrode
electrodes.
Conductivity
Polarization
effects
do
is
then
before
such
effects
can
a
balance
tionately
electrodes so
that the
ion density
20 V/cm
current
is
pro-
portional
to
the
number
in a
ion is initially
of the
path, and
usual values
increases much
charged mole-
2
do
not.
that the electron
radius,
and
finally
it to occur,
the current through
dent of
Ionization
shown that collisions
paths, heavy ions pick
parison. The most important secondary
processes
for
are emitted
under the
action of
(a) bombardment
electric effect
processes
more
important
in certain
appear
at the
(e)
as
temperature
 
pd
follows from
the Townsend
average
the
equation
(4.41)
Inspection ofequation
a
a
of
an
electron
encounter-
formed
by
collision.
In
but the
mean free
path is
 
is
always
1953)
shows
(pd greater than about
spark,
is due
less
order of
Paschen's law breaks down.
increasingly
im-
portant
plasma
physics, has been renewed in the quest for thermonuclear power.
A
and neutral
locally. Suppose that at some instant a momentary
excess
mutual repulsion ofthe charged
the
velocity
this process sets up a
periodic disturbance known as 'plasma oscillations'
;
be
visible
as striations), and
electron or ion beams. We shall not discuss these phenomena (which
are
with a
assume
of particle,
offully ionized hydrogen
wavelength of
2 mm
tool in plasma
discharges
frequency
15
c/s.
REFERENCES
Rev.
80,
487.
in air is 6
cross
the
wires carrying
coils.
dimensions of the coils
tional to
area.
If
of
each
a
current-carrying
on
other
lines
is
that
by
making
to B
B.
wire ds carrying
a current I
AND
[5.1
circuit carrying a
vector
t
calculate
such wires a
parallel to it but passes
through
=
of
of the old
unit (the
(5.5)
showed
that
into thin strips, as in
Fig.
5.4,
of
made
up
ofa
current
is
the
current
along
the
the strip CDEF
we resolve B
Bain 9
a;-axis.
The
force
on
gradient
of
a
magnetic shells.
the elements com-
regarded
as
by integration of
using
equation
(5.3).
the
(da
a
r)
T
it is
the circuit
the
presence
of
a
mag-
netizable
being
directly
related
B-/*
M
2
are
concentric
circles
about
the
wire,
wire
one revolution,
is independent
into
be a circle
a distance z
tion of
the
first
so
P a distance z away.
If the
they
not
field
is
applied
in
the
so
and
one
or
more
short
solenoids,
multiple
of
described
in
bring
all
to Bainbridge.
these
with velocity
is
therefore
until they
independent
(to
the
'first order'
focusing of
the ions
to
H,
and
substance
away
from
that
of
internal
2=0,
induction through
the circuit
was changed.
brought about
were wound
suspended
magnet.
other experiments Fara-
(a) when the flux of magnetic induction through a circuit
is chang-
the direction
wire at
the force
uniform induction B, since (v
a
B)
moves, and
we can
direction (the
ratio
VJV
since if
a
changing
passes
through
the
mutual
inductance
between
x
of the
mechanical forces
conductors
increases
level flight at a
of
the
propeller
when
over the volume outside
there are no
the total
(thus defined)
discussed by
of the
assembly is
with the
and we
This
determines
also
the
suspension,
since
mass
of
the
coil,
the coil is t,
tion
(7.1)
the
coil
deflexion
will
be
is
just
equal
to
the
internal
resistance
instrument
should
Fig.
7.5,
the
current
R (see
attainment of
the final
the current
starts to
overshoot and
overshoot may
sources of energy loss
to the motion of the coil in the magnetic field
of the
coil equal to
induction are
in
the
the equation
coil,
it to remain
oscillate between
2
of
swing
ofthe
can
galvanometer.
secondary
circuit,
that the
the flux
current are
to
prevent
the
coil
drifting
during
the
measurement.
the dimen-
fixed,
the
current
flows
through
the
various
E
the
arm.
the torque
D, F, then the
dMjdd
apparatus.
Effects
due
to
the
for example,
in 10
a resistance of
12
cm
24
mH.
7.2.
A
ballistic
the
capacitance
throw
is
of
negligible
resistance;
the
show that the fractional
change in period is
Fig.
7.5,
of connexion
magnetization,
and
of modern atomic physics,
It
should
be
noted
that
where the
that the molecule
§
charge
q
from
sponding
potential energy is,
On
the induced moment
n atoms (or ions)
electrons in the atom,
mag-
netic
moment
in
such
a
direction
as
to
moment
is
due
field.
We
have
assumed
that
(r
2
atoms per
xm
xi
^
By
assuming
approximate values
to B is
=
sin
6
dd
o
•jr
func-
tion
of
y
a
certain
temperature
properties
change
abruptly.
hysteresis
loop
Curie-Weiss
law
(equation
(8.14))
with
a
Weiss constant 9 whose value is close to that of the Curie
point. Below
large values of magnetization are produced by quite small fields,
and
strength.
This
If
is applied,
8.3,
curve
but
=
saturation
; very large
or nearly
Initially (OA in
dis-
placements.
is
relatively
slow.
electromagnetic
induction
in
the
magnet.
m
H
m
a
from
metre
2
are
required.
up
to
about
2
weber/metre
2
the
normal
turns
is
the
/j,
and
replaced
by
investigations, and the
Fig.
8.9,
if
(1+3
to a dipole m
strip,
whose
width
increases
with
the
radius
known as
coil' and an instrument for
measuring flux.
a
former
When the coil
is placed with
its axis parallel
to a point
edge of the
coil, and the
this lower
edge
is
directed
perpendicular
the lower edge. Thus the integrated
value of
potentiometer, and
over
a
larger
volume.
vertical;
M
permanent
magnet
producing
M
at
S
2
2
the
primary
current
is
diminished
curve is attained.
for calibrating the ballistic
to its
the reversing switch to its
original position
S
and
S'
every
time, in
order to retain the cyclic state. If the direction of movement
round the
hundred years. These latter
field
components
would
currents
in
the
iono-
sphere
due
to
of maximum
sunspots, the
from the
source
Electric
that the
~
the
capacitor
dqjdt
being
assumed
that
and not that
simple additive
relation that
can
be
overcome
series
as the im-
inductance
and
given by
represented by mutually perpendicular
by
tan<£
to
operator as
voltage is
given as
of
I
number
of
by
When
a
number
of
=
three simple circuit
resistance
and
reactance
these
simple
reciprocal
its 'selectivity' is
of frequency
between the
is
voltage across the capacitance
step-up
obtained
is
also
given
approximation
to
2
(9.23)
Thus
the
resonance
frequency
is
unaltered
in
the
first
approximation,
but
the
resistance
of
the
circuit
is
increased.
The
value
of
Q
is
now
given
by
\}Q
transformer the
Thus the ratio
This, which
current
drawn
readily
appreciated
from
the
following
sum
mary
used to
=
coefficient
of
coupling
has
peak
ir
the
(9.40)
1
is
the
'leakage
inductance'
due
to
the
extreme right
in
h
Z,
up
induced
frequency
1-1
10 ohms is
tuned to parallel
series
resonance
at
1-1
Mc/s.
Find
9.3. Show
arms
at
bridge,
are
a,
b,
c,
d.
a
a frequency
resonance
that
observed to
 
regard ie
in
velocity of
where
F'
is
the
differential
of
F.
Hence,
since
v
E
y
to
a non-conducting
Z = (ju.
earlier
chapters
is
readily
(i.e. if
we have
are the
The
included
been
and
the
(10.3),
we
find
the flow
of energy
per unit
power losu
h.
The
intrinsic
of E
this reason
equation
(10.24),
we
have
V
2
J
t
density at the surface of the wire (x
=
To
we
must
calculate
the wave in
on
the
Wo
the
components
of
E
and
H
the
field
components
amplitude andphase.
rays
are
(transmitted
wave)
are equal,
Fig.
10.4):
E
z
E'
z
zcos0)/c}]
the
reflected
and
refracted
waves
are
different.
This
polarization, that
intrinsic impedance
be almost
current whose
the
direction
in
force is
in the
have?
see
ductor the
-
Hence
s
; at
S
(j
The
magnetic field
is then
(which
we
such as a
of
the pressure
introduction of a
in
Figs.
11.4
and
11.5
'characteristic' or 'iterative' impedance
be
obtained
if
a
number
some point
by
an
measured
at
the
a method of
T
into
of CD'.
portion
of
voltage by
byputting
capacitances
In fact
we may
mally small, but
so
that
case whatever
the shape
our method,
on a loss-free
(11.13)
with
v
application
of
equations
(11.9)
a
medium
equal to
finite in this
the
termination
For
simplicity
we
shall
with
the
the
case
=
by
a
reactance
has
the
same
amplitude
by
phase
change
arises
L.
On
telephone
series with
the line at regular intervals. This also reduces the attenua-
tion,
since
periodic
kept
above
the inductances
greatly
10 000 c/s.
the fractional
change in
highest frequencies,
where B
about
1-2
X
10~
4
cm.
Z
1—
exp{-
I
is
argument
(9.20)
for
a
series
space
between
preceding
sections
parallel
planes.
with
the same velocity as a wave in the unbounded medium. With
a
single
a guided
As a
as
taking
the
con-
ductors
to
at the
planes z
to
the
planes
(i.e.
g
A is
A„
in the
component in the direction
linear dimensions are
current on the end walls; for this reason it is
used in
wavemeters (see
Show
that
the
constant-fc
type
z-directions.
101
bounded by perfectly
oca
i«^y.-
 
pos-
sible
tage difference between grid
and
the
grid
alumina.
For
the
material
of
it
in
a
volatile
and
at an
elevated temperature.
takes
oxide is
maintain
a
high
vacuum
for
the
whole
life
of
to
then outgassed at red
and the
nitrogen
by
chemical
action.
12.2.
The
diode
The
simplest
the
current
there were
W
and
when
the
V,
then
vector quantity; we have
Curve
B
density
in
into
not
conducting,
being
recharged
to
the
peak
when
the
diode
conducts.
The
voltage
across
R
is
by
using
RC
is
full-wave
rectification
Supply
voltage
nSM^
Fig.
12.5.
A
the time constant of the G-R
combination is long compared with the period of the supply voltage,
so
that the voltage drop during the discharge period is only
a
With
full-wave
rectification,
shown
in
a
of
L
is
steady
voltage
output
across
R,
but
reduces
the
ratio
(l/coC
1
)/(ct)i)
For a
full-wave system
«
down
AB.
developed between
the diode consists
reaches
its
will reach the
in
an
'equivalent
when
triode.
The
constant
b
has
change of
anode
current
not being linear with the change in the applied voltages. This
is generally undesirable, since it
causes 'distortion' of
omitting
have
made
are
current
one-third power of
as
follows.
g
m
varies
from
1
to
10
mA/V,
which
is
anode
of
a
triode
voltage is
the
genera-
tor
flows
finite
problem is complicated, since the grid-cathode,
 
signal
where
A
 
impedance
even
when
the
resistance
B.
Its
value
is
then
approxi-
mately
MP =
,,,2/72
lu
B+
P
with the
grid input
capacitance. When
the anode
load is
anode
voltages,
evaporated
potential, with
a
electrons
increase
tube
This is
possible with
signals.
affects
only
the
division
of
current
usually
of
the
order
and its
in the
beams
at cathode
3/2
times
period. Show by
with
magnification
per
reduces
anode resis-
tance of
is offset by
the maximum power
the region of
l/p+l/R+2jAa>C
the
is
usually
fixed,
the follow-
ing stage then forms part of the tuning capacitance. The magnification
at
resonance
across
the
tuned
circuit
tuned circuit in
with
coupled
circuits
(see
in
as
to
v
without feed-back,
product A/3
must be positive and equal to unity. This imposes conditions on both
the
whose
a
triode
tube,
one
(13.11)
of
If
we
return
circuit
running conditions
rough guide it
order of
quartz crystal
power
amplifiers
to
to
An alternative form
Ct
Tuned
circuit
ii
E
MWVWi
c
3
nmw^
Choke
such
to the
grid, oscillations
the
amplitude
of
oscillation
is
small,
though
oscillations
can
overcome
is
provided
cycle,
capacitor
C
3
more
secondary
electrons
than
intermittent
operation
known
as
x
flows
through
r
and
C
in
parallel,
we
tube:
Since
—g
m
pr/(r-\-p)
are
positive voltage
relay, the
than the
voltage
the
grid
charac-
teristic,
which
period
fluctua-
designed to have a
be
demonstrated
by
applying
the
(low
distortion).
(2)
should
be
voltages
^
depth of
the input voltage
right
of
characteristic. The
linear detection if
anode current-grid
ofthe applied
13.9. Frequency
sometimes im-
possible, to provide sufficient amplification for this purpose at the
carrier
frequency.
A
device
known
as
frequency
changing
is
then
used,
the
same.
This
change
x
In a
n
{m
f
Although
the
side-band
used
are required
only for
input is
oscillation frequency given
an audio-frequency
circuits, due to
and
radiation.
is modified in
At
very
thermionic
small as
base is
used as
electron, t is
time is r
is virtually complete.
be
reduced
enclosed
by
be
used
for
so
rather
less
than
be
opposite.
by radiation, which is
equation
keep
the
small
electrons
which plays
glass en-
in
oscillations
will
be
O
input
to
of power in the
its
shape
is
determined
a
short
traverse
be regarded either
across
the
gap,
(ft+f)
source of signal, oscillations
parallel impedance
of the
resonator. The
if the time of
klystron
must
be
tunable,
confronted by
elec-
travel
further
tial
direction
across
tungsten
heater
and
cathode
leads
10
to
50
kV
on
the
to
the
anode
experiences
a
best
a
negative
current or voltage
which measures
power directly,
to supply
frequencies. At
whereby
the
inversely
proportional
to
the
in order that
It is usual
to apply a
in Fig.
15.3. A
1 Mc/s, but above
thermocouple
due
to
stray
capacitance.
The
sometimes joined to
separate thermo-
junction which
direct current through
of
various
semi-conducting
oxides
convenient
for
matching
Resistance variation bolometer for use at short wavelengths. The position
of the input
(Wave-
at
centimetre
wave-
lengths
is
convenient
Rectifier instruments
as a vacuum tube or valve
voltmeter.
The
the
d.c.
voltage
the
load
resistance
as in Fig. 13.19, but
without the
it
is
rectification than
also
purpose
of
triode in the
absence of an
in various
triode.
u
J?
voltage to the
full-wave square law action.
cause current to flow
current will
cut-off so that only
flow,
high
real and
determined
when,
both
Wheat-
stone's
bridge
shown
in
Fig.
of
the
arms.
For
A
and
B,
which
is
truly
zero
voltage
components
must
other,
it
does not
appear in
the other
be obtained
which can
be represented
pedances used
reactances
and quadrature,
at the
input to
of
the
harmonic.
potentiometer
before
the
grid
to
is far from
damaged by an overload.
happen that
one
purpose resistances and capacitances
an
appreciable
general rule is that the impedances of all arms should
be of
the
the real and
combination
of
C
variable
capacitance
C
2
Owing
determine
an
of
in Fig.
one arm of the
and
of
the voltage across FE is
VI{(l+jwCT)(l+QIZ)}
while the voltage across FB is
VBI(B-\-ja>L-}-r-\-S). Equating
The real and imaginary parts of this equation
give separately
(r+8)IB
nearest
approach
balance
conditions
shows
and of each other
for a
balance at
the detector
right
signs;
we
have
already
with respect to
difficulties associated
Each
arm
must be shielded. The generator and detector should also
be
shielded
and
generator
to
be included in
arms
shield. This
being
balanced
first with the unknown impedance in parallel with the variable
standard
impedance
(usually
in phase between the
reflection (the
termination ofthe
found from the
is approached.
line
point x
pressed
and measurement of
wavelengths, but
coaxial lines
and
end
intersects some of
end, and
in
maximum when
preferable
to
II
pointed
an
crystal.
These
effects
an
alternating
amplitude
of
the
resulting
A
number
of
different
sharpness of
the resonance
is no
there
Y-
axes,
120°
and
240°
about
the
Z-axis,
An
thin
slab
to the XZ-jAa,ne,
roughly
of quartz crystals
a
capaci-
tor;
tuned
circuit
at
a
frequency
just
that
the
phase
shown
ofthe
the
tive
at
such as a bridge system where
one
The
standard
suitable
The
fundamental
frequency
the standard
originated
and station
to be subject to
with harmonics rising
achieved
by
multiplying
first
This causes
it to
to run a clock
first
tuned
and the number
The variable oscillator
100
10 kc/s
unknown
by
the
ratio
the
dielectric
constant
designed
by incorporating
a specially
designed capacitor
first
with
the
A block diagram of the
apparatus is
container.
are gold-plated to maintain
sion.
A
correction
is not
If a frequency measuring
C is
evacuated so
oscillator
and
crystal.
If this frequency/' is about 1 Mc/s, then on introducing the
gas an audio-frequency beat note is produced between the
new frequency
by
comparison
with
a
ofadiabatic
capacitance
bridge
at
0-
with a
diameter of
evacuated, and then
whose
number of
workers at
different frequencies,
oscillator
found
from
the
width
the cavity.
Thus only
measurements of
length, depending
temperature
of
20°
C
diameter.
Table
15.1
certain that such impurities would
account
for
all
the
dielectric
loss.
The velocity
much effort
of the
waves.
deal of scatter,
accuracy, which
(see
Table
15.2).
to improve
A
brief
be
type
of
of
diameter
7-4
cm
and
length
8-5
number of different
3000
c
may
be
found
from
Q
was
appears as a small correction for the finite
electrical
the
radiation
'end-effects', and the
intrusion
of
the
coupling
length
of
these
probes
was
diameter
D
were
accurate
6
between
299
796
and
299
In a
move
between
successive
resonances.
but
also
includes
errors
arising
from
temperature
changes,
frequency
X
10
-6
showed a systematic
resonant conditions were
part in determin-
Gordon-Smith
resonator method
microwave analogue of the Michelson
interferometer,
as
waveguide was divided
Part of this radiation
vector sum
(a superheterodyne
L
If
R' =
0,
the
(Fig.
of
voltage
V
x
with
itself is
This equation
shows that
just
the
by finding the fractional
L
was first
motion'
never
average component
The mean
ofequipartition of energy: if
a number of terms each containing only the square of
a
variable,
applies just as much to macroscopic
objects
molecules, but the magnitude of the fluctuations in the dynamical
variable become smaller
increases, since the
motion
can
always be observed, and it sets a limit to the sensitivity of
any measuring instrument,
If this theorem is applied
to a suspension
as the sum
of two
terms, the potential energy of the suspension due to work done
against the
3
is
is the electrical tuned
and resistance
such
a
electrical
sys-
tems,
as
we
we
write
L(d
2
fluctuations
by
equation
(16.4),
distribution
frequency involves only
verified by Johnson,
galvanometer, and consider
open circuit.
to those made about
once
d(F})
of
damping
stored
that the
Brownian motion
caused
pension being
gas would be
of
thermodynamics.
The
processes
of
the
molecules
oscillation
maximum value
(6)).
The
voltage
by
the
apparent
given
by
equation
(16.21),
for
If
heard
or
viewed
on
a
than
radiation
and
limit to
one
the
amplified signal
incident on
the aerial.
be defined as that which
gives
only
2 Mc/s, before
and after the
reduction
involved, the bandwidth will
is
necessary
the
reducing the band
noise
output
is
therefore
greater
than
P
. The
thermal radiation
convenient
X
10-
21
df
(watts).
in
a uniform
electrons
of
the
integrations. As
it is
and do not
design.
ten to
of
the
as
that
of
is represented by
case R
first tuned circuit,
general
R
2
will
a voltage source
point in the receiver, and we need
only compute
the ratio
of
input
since the
when
R
2
is
the mismatch.
of sensitivity. In
2
could
compute the signal/noise
value for
have
been
2-03.
Hence
we
have
of no tube
over-coupling must not be
16.4 (a),
in order
using
a
W.
The
oscillator
is
tunable,
the
input
to
the
terminals either
some value
is
very
carefully
at
value.
The
circular tube forming a waveguide which is beyond cut-off for
the
fre-
ofany wave launched
is given by
(16.34)
may
several
the
A input from oscillator.
diameter.
L
loop
to
launch the wave
in the tube
from the end
acting
be
the
input.
lamp launches
to
by a loop connected
best
output (of
source
already roughly at
noise source is
drops out of
output
equal
to
the
principal
tance with temperature
as giving flicker effect and considerable drift in the anode
current. Ifthe anodeload is a resistance
B
with the
a half-wave
to
produce
is
such
as
a
adjusted until
by the thermo-junction is
available
signal
power).
It
is
a
1000
from the receiver is
over
the
sea
and
the
wavelength.)
classical
region
[17.1
Debye
polarizes the other ion
where the dipole moments
This,
the moments are
an over-simpli-
be
linear;
symmetrical,
dipole moment
directions until the internal
it
at
the
laboratory.
high-powered
refer
only
to
the
determines the
may be
are dealing with
of
as
the
actual
field
acting
on
each
subjected
also
to
the
electric
the
neighbouring
molecules;
approximately
average field
due to
contribution of
local field acting
moment. In
additional
random directions,
a
small
excess
in
the
number
of
dipoles
pointing
against the
Boltzmann
one
more favoured in
of the magnetic
=
as
the
same
interest
except
in
ture
(Chapter
10)
shows
that
the
refractive
index of a substance should be equal to the square root
of its
dielectric constant,
indices
the order
range
of
frequencies
to
'infinite
wave-
lengths'
the refractive index
increases as the
difficult
to
observe
because
of
but both types have
application of an alter-
000 Mcjs Optical
Appl.
which is
assumed to
displaced a distance
as
etc.,
—
w
£-|y2)i«]}.
B average to zero
correction
part
near
if the
absorption
has
resonant frequency co
unity. In practice
absorption line (from
«
quantum
mechanical
fact
that
each
electron
then
pass
through
tion
of refractive
dipole moment,
would
also
the
molecule
which
we
are
far
on
substances,
sphere,
there
are
given
in
dependent
on
temperature,
but
there
are
CONSTANT
[17.5
can
be reversed by an electric field of sufficient strength, with hysteresis
effects.
resemblances
to
ferromagnetism,
and
The
properties
a
spontaneous
undergo a
PbZrO
s
(T
c
have from
dipoles (see
that
slight
being at the
centre of a
method,
produces
a
p',
forces need be
neglected.
can be done
a
method
are used,
with those found using
factory when it can
even more marked;
field
full
a
cannot
rotate
at
any
of
the Brownian
and
P
due to induced
This
result
holds
for
the
Onsager
except that
region of
the cavity
resonant
frequency
and
Q
the
the
method
range to
large and
a
bulb
parallel
be
no
scattered
radiation
in
the
that ofa conducting
Show
that
JB-layer
at
latitude
40°
N.
particle falls as
Z>-lines
breadth
Fourier analysis of the spectrum
of an oscillator
to the
molecule due
the
maximum
and
minimum
of
the
at
the
frequencies
where
half its maximum
in
Fig.
17.5;
in
accounting for
to the electrical conductivity.
quantum
using the
de Broglie
relation and
the
wave-like
nature
is comparable with
effects.
one-dimensional case,
equation
(18.1)
de
to
they are
confined within
the form
showing that
the chance
falls off very rapidly
are a set of
the Fermi distribution
with the dimensions
wavelength
solid, where the
then varies
whose
nature
is
lowest energy
sharp
corresponding levels of
the
wave
functions
electrons and
an
almost
continuous
the
outermost
atomic
showing
how
the
as the atoms are
u
x
from
which
the
value
dynamic behaviour
if
it
were
a
particle
to allowed
correspond
with electrons.
interest
the
for
means true near
filled
*.
=
-
-*•£.
approximation).
The
energy
boundary
should
a
be measured
our treatment
assumption
that
their
motion
a correlation
that
the
overall
antisymmetric,
which
the
'correlation
energy
metal. The
a metal
the
electrons
are
firmly bound
electrons were
able
though the
to the
will
have
with the
Pauli exclusion principle,
and they will fill the energy bands from the bottom upwards. The
lowest
occur
of
k
as
have negative
values, so that the net current is zero. To establish a current flow some
electrons must be transferred fromnegative values
ofk
but
because
of the exclusion principle this is possible only if they make
transitions to unoccupied states ofhigherenergy,
the energy being
i.e.
if
the
sufficient energy from a movement
in the
substance is an
metals
such
as
lithium,
states
as
a
function
of
interatomic
distance
o
left
corresponding to the
ciably.
one
where
corresponding
filled and
there is one electron per atom in the 4s band, making
it a good
do
not
by
a
broad
Information about
the solid state
(see, for
width is practically
entirely that ofthe
00
U
temperatures. The specific
kinetic
energy,
about
20°
the
form
G
v
below
about
5°
value
of
n,
the
volume,
electron
per
two. With the transition
3d,
4d,
and
5d
(including the
electron spin)
wave
effect
a
given
elements.
electrical conductivity
velocity. Since
since
t «
10
_w
10~
6
vary
Most metals
of
an
electron
is
in
an
of
(1)
above)
should
be
independent
of
of
elastic
waves
(longitudinal
and
to those
lattice.
each
mode
the
electrons have both a wave and a particle aspect, so do the lattice
modes;
name
which
emphasizes
their
with other phonons,
electrical
resistance
at
 
zero.
showing that
with
temperature
a
at
higher
known
as
'Matthiesen's
rule').
conduction
electrons,
where there
most
metals
is
UK
of
/J
theory
(see
Problem
18.1)
273-2°
K
for
ideally
specimen the
in the thermal
of the
to the mean
the right
frequency of the phonon and v its velocity. At
any
of
carried
variation
of
residual
resistance
observed
The high frequency
the
h.f.
charged
particles
Permi surface
take part
the
anti-parallel
state,
and
of tempera-
spins
magnetic
energy
{2fiB)
is
empty
principle,
cannot
electrons in the
energy band is shown in Fig. 18.13. This differs from the
earlier
diagram
(Fig.
4.3)
two
halves,
with
their
spin
dipoles
ofelectrons
transferred
from
the
anti-parallel
to
the
parallel
orientation,
following way. We
(spin dipoles
the
value
of
(x
of
effects.
Soc. A, 211,
81,
657.
Physics
(Wiley).
volume
and
W
F
the
is
(18.23).
Verify,
by
using
equation
(4.3),
that
this
L
may
alternating electric
conductivity
at
—
in
the
in
room
temperature
the quantity (w
at
4°
path of the electrons.
of
skin depth
effect,
the
relaxation
effect.)
separation. Show
§W
S
the
conduction
band
is
easily
ionized
an extrinsic con-
rises until all
the donor impurity atoms are fully ionized, or all the acceptor levels
fully occupied. The
is known
Extrinsic and intrinsic conductivity may of course be present simul-
taneously, but the former will depend
on
the
conduc-
tivity
and
their
proper-
ties
for
any
other
semiconductor.
The
neighbours arranged in
valence
the
position
at
0°
K,
where
energy
gaps
are
bond
electron, though not neces-
germanium atom. Like
the atom it
This
uses
up
four
charged
ion
m* in an orbit
= 0-2
(an
phosphorus the observed
of
2-45
A.
Since
0-01
their
electrons
corresponding
to
fill the
four bonds which it should make on replacing a silicon or
germanium
illustrate
the
non-stoichiometric.
0°
with
energy
between
calculated
by
where
valence bands. We
and
holes
in
value
is
potential
O
of junctions.
4
to
10
5
cm
-1
at
beyond
the
of the
can still
gives,
the
energy
spread
out
coefficient
should
vary
absorption
edge
has
given
quite
connected
tum of a particle
=
conduction band.
in the
conduction band
at k
and bound electron), as
been
line spectrum
into
bands.
tion
to
detection
ofradiation.
The
the conduction
by
electrons
being
lifted
into
acceptor
of such
(19.13).
measurement of the
(w
10
5
to
10
6
cm
3
/coulomb
for
particular,
does not happen,
different
for the Hall
arises
different
averages
R
H
and
n
value of
of n
with
temperature.
(phonons) the mobility
the conductivity would
follow the law
curva-
exactly.
absorption
edge
in the
a microsecond
pulse of
B. The mean drift velocity is
Ljt, where t is
these
(through
electrons through
in time with
delay is that required by an
electromagnetic
wave
set
B and
v is
by calibrating the
velocity
v
the
relation
u
apply a negative pulse
semiconductor.
corresponding excess of
directly the
electrons in
'drift mobility',
independent
account
scattering
cross-section
determines
kinetic
energy
a
pure
material
the
(phonons);
(1960),
Electrons
and
(O.U.P.).
I
proportional
time t
complicated
band
structure
in electron momentum
in
silicon
mass
of
The scattering cross-section
proportional
to
 
but
can
be
acts as a
This
process
is
illustrated
in
Fig.
contact, before
and after
This layer
is known
between
a
current
positive
area can
an ohmic
a
single
semi-
conductor
crystal.
a
established.
19.7.
The
p-n
junction
at the junction where the doping concentration
varies from
but
provided
in the exhaustion
separate crystals.
and excess
electrons
so
negative space
of
the
forward
higher (in germanium the
the
solid-state rectifiers
a great
our discussion
may be regarded
central
section
(known
as
to the
normal
operation.
voltage v
emitter electrode to give an amplified voltage across the load
resistance R
known
is quite
at
the
base
electrode,
grounded base con-
to the
of
few volts rather
transistors
as
well
as
dVJdI
e
are
considerably
dependent
on
an 'atom' consisting of
(19.18)
and
(19.19).
Show
value at low
H
T*)
G
found by
writing down
its components
the x-
G
on
the
xy-plane
the
vector
equation
(20.2)
itself.
Since
electronic momentum
move
out
depending only
considered
above
is
that
the
An
understanding
quantum
theory
of
A
comprehensive
discussion
of
the nucleus,
this being
Planck's
The
allowed
values
of
I
are
current,
and
a
mag-
netic
dipole
That is,
a moment
directions,
owing
to
(e#/2m
the
angular
hydrogen.
that of
to
given
by
Larmor's
theorem.
shown both
momenta, both
orbit and
means
±£,
more than
states m
of
electrons
Since m
+
point
of
supplied
by
an
external
as
we
I,
when
momenta,
we have a
an atom is
angular
two
principal
types
the
'exchange
energy',
including the
vectors
s,
1,
S,
L,
etc.
(6)
Magnetic
E
reduced because of the
b
s
is
then
-m
s
interaction is to
a
resultant
S,
and
the
to equation
=
and the
more
a
the va ues of J are
therefore
integral
the
mj
determined
by
interaction
be verified
(see Problem
20.7) that
chromium ion, Cr
to
-1
magnitude smaller than
the magnetic properties
depend only on the
the
magnetic
moments
add
in
is
parallel
to
S
and
to
S and
m
L
the
factor
of
proportionality
be-
vector diagram
in Fig.
20.6. Here
.
orbit
and
spin.
which
—
L
and
the
projection
J
of
value of the
components is
m
J
trtj,
considering
is
placed
angular
momentum
vectors
spin is
other
description of
approximate
treatment
is
possible.
We
external field is
precession
round
the
external
field
is
small
compared
with
that
may
therefore
along J.
field, but
have the
and the
the degeneracy'.
the spin-orbit
splittings (separation
ofstates ofdifferent
-1
for
=
is linearly
permanent magnetic
continuous
range
of
susceptibility
and
thus,
§
dipoles do exist in atoms, and we must now examine
how the Langevin
fact that only a
an
and
g
in a
this state is exp(—M.jg$B\kT)\^ e^(-Mjg^BjkT),
where
the
all
summation
of
the
of free
by the atoms on
between
electrons
2
so that the
the
4/
shell
the
spin-orbit
coupling
interaction
changes
sign
must be
p
2
little
significance
if
this connexion
where the
appreciable
only
of different
J generally
4/
5
gives much
angular
momen-
these
10-20°
K,
for
this
paramagnetic ion in
neighbours
carry
no
ions such
as F
an electro-
energy
of
interaction
with
similar
in
of an
atom, first
neighbours
varies
lowest levels are populated,
a single
crystal may
magnetic properties. The
acquires a different
zero (except
Eu
2
absence
of
a
magnetic
state of the Er
of J; in this case the
states occur
tion
of
the
magnetic
moment.
the
same
in a
the double
degeneracy'.
4S(S+1)
between them
10000 cm
state
as the lowest level. Such a state has no magnetic moment,
and the
orbital moment
i i
1 1
effectively quenched than in the first half, so that the
values of
field. Thus,
'spin
only'
value
of
p2
make some
contribution to
the magnetic
moment, though
to do.
molecules
round
they
are
the 3d
charged
ligand
ions.
(±o,
0,
negatively charged,
higher energy when they
states) are
each zero
is
for these three states
the opposite
sign. A
role in
two
de-
sign
for
the visible
1
and
behaves
d
3
and
d
8
wave
function
xyz,
which
separate cases:
(for
spin-orbit
coupling:
(1)
spin-orbit coupling and
A the splitting
d-shell is less
g
smaller
than
the
free
spin
shown
in
giving
states
with
an
effective
>
1
(140°
tem-
perature
noticeable
[20.7
in cobalt salts.
to
the
such as K
groups in
the former
nearly regular octahedron.
parison
two
ions
localized on
so
In a complex with
along the
though
subjected
splitting is large compared with this
exchange
energy
electron,
when this electron is in the de
states.
accommodated
in
orbital momentum (corresponding to
2
cm
-1
interaction
(e.g.
nucleus about an
momentum is quantized
neutron,
spin
neutrons
boundtogether
the
number
of
protons
A,
and
the
in
a
neutrons
For nuclei
momentum
excited nuclear
the electron
gg
n
the
by
the
magnetic
(cf. Problem
This rule, known as the Lande
interval rule, no longer holds when the
electric quadrupole interaction
where
W
The
=
of
B
Q
is
\,
and B
magnetic
resonance
experiments
(see
by
neighbouring
ions.
where it follows
nucleus
nuclear
magnetic
as
and nuclear contributions to the suscepti-
bility in
York).
Kuhn,
H.
G.
1964,
 
+
...,
=
follow the trend given
these
limits.
at
900°
the
spin-orbit
coupling
so
that
the
*P
molecule is I
number
of
3
is
as the
states
2
Pj
compo-
different atoms in
equivalent to JcT with
than about
6 A,
appreciable effect
large coupling
§
two separate
vectors
g)
a
the energy for atom
expressed in terms of
substance
is
zero.
instant
be as
we
assume
and we
=
is an equation formally
of
a
to a good
Weiss
to
spontaneously
have
M
law,
moment
the further
assumption that
c
is
deriving equation
internal field, the
M\M
the
from
a
graphical
intersection, and
finite.
point
P,
thus appears
while the unmagnetized
state is unstable.
magnetization
temperature, it
the
temperature
from
the
(21.8))
for
J
(21.8)
value of J. This function
is
tion o£M
Corresponding States'
will be
field B
851110
S3
is
below
P is
in
theory
also difficulties. The fact that the unmagnetized state is
unstable
appears
in a
single crystal
magnetization within the
10
17
-10
21
atoms,
spontaneously
mag-
magnetization
of
a
single
domain,
specimen
depends
on
whether
before
the
nature
that the crystals
the
field
unit
cube
[100]
than
any
other
of
iron.
the
initial
the single
ofeasy magnetization.
with
a
hexagonal
structure
at
easy magnetization, the hexagonal crystal axis.
The excess
energy required
It
origin
is
Thus it is less for iron or nickel, which are
both cubic,
of
easy
magnetization
give
large
magnetization at
flux closure'.
on magnetization
Ifthe crystal consisted
poles' at
the ends
This is
is
reduced
still
further.
B at
directions
and shape of
fact that
small in
the substance, so
wall, and
is
found
to
thickness ofthe
whose
order
of
magnitude
is
per
unit
area
of
number, and large in size. When K
is large, particles
In large
determination of
tends to increase
is provided
when finely powdered iron
The surface
and electrolytically polished
typical Bitter
obtained
glass fibre.
normal
to
scratch parallel to the field
behaves as
a long
so that
of
beams
investigate domain
reversible
in
been
applied.
Where
 
field
the
field is removed;
could measure
specimen,
apart
the authors to
 
solenoid supplied with
of
the
is
measured
by
a
men.
This
steady
current
amplify galvanometer deflexions.
By this means
uniformity of
was 26 sec, and its rotation
was
observed
by
current at a moment
swing. The direction
was increased, and
damping,
the
and the amplitude
change for one
A
number
of
using
momentum
(21.14)
for
a
due
to
TdS
= dU--BdM;
(21.16)
P
B. The density
ferromagnet, B
T,
it
gives
a sharp drop
An
experimental
curve
showing
the
the
If
a
field
is
applied
than
in
increased
disorder
of
the
associated
below
1°
K
=
negligible
and
finite
with XM,
external
of the straight
number
of
 
(24%
Cu;
required
well below the Curie
possesses
a
to fine up the
number, and are
8
-10
10
c/s.
measuring this separation over a range
of temperature.
effect,
by a nucleus
in
a
solid
taken
whole,
varying amounts of energy
Only those y-rays which are extremely
narrow
are
be
smaller
=
§
no
hyperfine
Mc/s in metallic
the
shown that
the resonance
frequency is
quantitative
two separate
problem
clearly
we
leaving
only
that
due
to
g
moment corresponding
conduction
electrons
the
basis
of
metallic
conduction,
the
electrons
obey
the
Fermi-Dirac
statistics,
and
the
is
introduced
which we
separation of the
transferred
from
'parallel' orientation and a net
magnetiza-
tion
of
2x
fi.
§
the top ofthe
tion. Hence the total kinetic energy required to transfer x
electrons is
that the
magnetized state
the energy
and there
relation
g(W)
F
of
A
about
3d group the
of
g(W)
F
of
free
electrons
per
unit
volume,
will rise
until any
by the
of
the
Fermi
distribution
in
the
on the energy
band picture from
the
considerable
from the slope of
repulsion of the electrons,
mentioned
at
the
are
energy has little effect on the ordinary
conduction properties. However,
requires
d-
which
the
In
better starting
forward a
example, the low
moment of nickel
nickel
9
nothing
to
different
ionization
has
ionization,
typical
The
problem
conducting solid
interaction
magnetism in the iron
of
electrically
are somewhat
more detailed
magnetic, whose properties
excitations of the assembly of magnetic
carriers, known as 'spin
spins
temperature where
it
would
the
reversal
of
any
be passed on
(#'
remain localized
on any
deviation
parallelism
is
small,
waves'. In a spin
cosine of
is
proportional
giving
W
k
wavelength
energy
J.
D
0°
K
dipoles
magnetic in origin,
(4.8)
temperature
T
is
U
on
the
=
special case where they
to
the
term
(see
Problem
21.1).
existence of spin waves
In
which
proved very
difficult; more
be
repulsion
between
the
because
the
sym-
metrical
are
divided
Since
the
orbital
the
introduction
of
a
type
assumed
in
of
lower
two
one
atom
2
the
overall
metals. Since
from electronic com-
exchange interaction
resultant interaction
(see
in compounds
where the
typical plane
of atoms
exchange inter-
actions are
between ions
as A,
C. Direct
Some
degree
of
covalent
binding
is
anions.
an
oxide
MO,
where
M
interaction
(as
for
—see
§20.2),
whereasthe
vast
majority
ofinsulating
energy of
W
1
smaller
and
this. In com-
eV is required
66,
224.
Benedek,
G.
B.,
and
Armstrong,
Matthias,
B.
T.,
1960,
Bloch
wall
increases
is
of
mass
M
moving
with
velocity
is increased by
=
same
energy
the entropy in a
the lowest
dipoles
paramagnetic
in
the
where
neighbouring
here all
domly oriented, and
sub-lattices are equally
which
by
becomes steeper,
origin;
the
spontaneous
magnetization
Table
22.
1
Niel
temperature
Substance
T
N
( K)
tf>(y)
=
for
are possible.
at a temperature below the Neel
 
if
the
the
direction
magnetization
M
M
increases,
the
A), and
8M
B,
B
-\8M
B.
anti-:
stances,
as
MnF
2
in
eddy
current
Fe
3
4
technical interest
expected for
arrangements,
a
ferrimagnetic
material.
(a)
Unequal
numbers
of
sub-lattices.
(6)
Unequal
moments
on
moments,
and
a
MnFe
2
4
similar
formula
sites with
octahedral coordination
(to six
and B
is not
Thus
magnetons (the best experimental
a
number
ofinequivalent
sites
4x5
is
frequently
called,
has
a
high
value
of
(BH
he
deduced
parameters are functions
of
ions
in
each
sub-lattice.
The
general
by
Neel's
relation
as
fitted
to
in Fig.
iron garget
Curie
There are two
dependent
We
given in
Table 22.3
free tripositive
peratures
crystal field splittings
the Pr
state,
thti
is related
(23.1)
gives
Dm/Dt
coordinate
the
energy levels and the allowed
them for a nuclear spin /
=
field
of
moment due only to electron spin,
the wavelength of the radiation required for resonance in
a
field
of
systems of both
we shall
determination
magnetic moment.
of
where the phenomenon
is detected by its effect on the path of a molecule
in a molecular
magnetic
gradient, and the initial
JB
of
the
O
magnet
due
to
the
of
moment is
be
obtained
as
follows.
with angular velocity yBj,
yB
1
1
in
related
to
the dipole
moment is
\
per
thousand
(see
Table
23.1).
was
carried
absorpt:
ferromagnetic
material,
Q.
This
lines.
an
field B
r.f.
the voltages
across the
the capacitors C
tuned
circuits
nearly
to
amplifier, is
small enough
not to
varied, and
the r.f. amplifier
voltmeter after detection,
appreciable the balance
adjusted
so
(Photograph
way
right
angles
since the intensity
at the centre
centre
however,
(1949)
a neig'
10
-11
much
closely
does
their
the
neig.
shortened by
be
accomplished
resonance
unknown isotope in
§
measured at 29 Mc/s, plotted against the Debye
relaxation time
the solid
lines for
alcohol and
accordance
with
theory.
usually
achieved
by
Wimett
(1953)
below),
and
molecular
value
y,
netic moments from
the measured ratios
gauss,
determined
by
[23.5
magnetic field.
Bj is provided
coil, in which
normal
a
comparatively
tionM
p
be
parallel
laboratories.
The
last
23.10.
The
current
y
p
The two more recent
its use in determining nuclear moments, but also because nuclear
resonance
offers
a
a
magnetic
standard is
to
this is not often a serious limitation. The free pre-
cession method
such as
in
10
5
the
2-8
X
10
-5
for
hydrogen,
rising
to
10~
2
elements. This
local
addition there may
in
compounds
which
have
temperature-independent
[23.5
ofthe lattice or through interaction with neighbouring spins, and so
long
lines are obtained, shifted
average
electronic magnetization; in a paramagnetic substance we can write for
the
nuclear
precession
frequency
u>
proportional
reso-
proportional
to
the
found by
Shulman and
was
to
spread
F
they
=
 
B gives a
(see Problem
the
field
B
n
tum vector
energy
levels,
as
illustrated
constants
whose effect
value ofF into 2F-\-l levels,
provided that
x
perpendicular
to
J$
[23.6
ampared
with
the
are
showing
and the second derivative
means of
Only two examples will
atom is
c/s
(Ephemeris
time)
at
1957-0.
The
atomic
frequency
standards
in
and Retherford
are used,
(a)
(a
cavity
resonator)
which
accuracy
in
width and
effect is that
depends
'quenching'
of
the
in
T
x
which
same:
term bT
involved)
or
'Raman'
processes
between
the
than
the
first,
temperatures
predomnk;
A
resolution
work
waveguide to a loosely coupled resonant cavity
containing the
to the
cavity
resonator
poles of
reflected
from
the
a
silicon
crystal
rectifier.
quite
high,
as
few
as
10
13
parallel
diverge linearly with field)
1)},
in
have
general
the
together
wave
the
nucleus
magnetic resonance,
of
the
In more concentrated
salts the effect
the
few
direct
intensity, since all nuclear orientations
are
sensitive detector)
tiw
. A spin wave resonance curve is shown in Fig. 23.17
for
to
so that
the oscilla-
tory field
decreases accurately
these substances makes the
complicated.
ferrimagnetic substances with
angular velocity
both
ferri-
and
anti
-ferromagnetics
more
complicated
v,
the
particle
will
move
determine the angular
field
and
p
and
B,
suggeststhat
a
similar
principle
q
of
B
Then
the
x
I ion collector.
about
0-1
R
potentiometer
system
by a
nuclear magnetic
resonance experiment,
a
produced along
in a
glass tube
8 and
the
main
difference
fact
kilogauss
10
to the
effect
through
was a much
cyclotron resonance has
end of the
netic field
electron
(v
e
in Table
the measure-
being
magnetic
moment
proton
(e/Jf
also
gives
the
mass
of
resonancefrequency.
in
=^)}'
r.f.
conductivity
to
the
d.c.
con-
ductivity
against
The
x
the signal reflected by
cavity
is
immersed
carriers is short and they
are present
reflected microwave signal is there-
fore modulated at the same
frequency (usually 100 to
n-type
since the microwave
rotating the external
magnetic field, so
explored.
An
due to
cyclotron resonance
in semiconductors
 
(23.30)
which
one end of a
normal to the
I
IT
I
1
1
for
Azbel-Kaner
by the electric
jg
/
1
B(kG)
Stradling, and Kip, 1964).
the surface
(4-5
of
form shown in
surface
8,
428.
Jeffries,
J. A„ 1951, ibid.
worth).
Sandebs,
J.
H.,
1961,
The
Av
equally valid
B
x
losses.
into those
which are in
since their units oflength,
have the same
are
quite
a
such
as
these three
a
fourth unit, that of electrical charge, is defined by means
of Coulomb's
a
distance
must
equal
of
charge
per
second.
In
current
magnetic charges
in
way
m,
x
m
in the m.k.s.
system is a
(4m)
The connexion with
magnetic field produced
are already defined, this fixes the
unit
of
be
the
e.s.u.
that
waves in vacuo,
is
power
a dimension
generally
written
in
of
Table
24.1
To
(or
e.m.u.)
multiply
by
the
'absolute units', related by powers of 10 to the electro-
magnetic
units,
units
by
amounts
insignificant
except
in
e.s.u.
be
these quantities
out the
elements
is
dF
x
A
i
1
^{d8
1
A(d8,Ar)}/(4«*).
(24.19)
These
equations
ized
c.g.s.
units
(equations
(24.1),
(24.10),
and
(24.14))
not
of the
factors 47T.
equations,
which
older
to employ
a three-dimensional
U
has
(1.19)
the
force/area
3
(D/E).
Using
the
is the joule does
never
the
of
10.
As
endeavoured
to
make the text simple to follow for a person previously conversant
only
are given
for assistance.
24.4. Conversion
systems can
52, 53, 54,
Chapter 3
All numbered
equations are
48.
Chapter
5
equations:
2,
11,
37, 38, 48,
57.
II,
in
e.m.u.
replace
fi
by
1
in:
7,
1, 2, 3, 4, 5, 6, 9,
10,
17.
e.m.u.
become:
X
various
equations become
33.
2, 16,
this quantity
define
p,
Z
by
73,
74.
Replace
e.m.u. or practical units,
Chapter 18
21, 25;
units.
Chapter
17
All
numbered
expressions
in e.s.u. in:
(6),
in
=
if P and
(c). In the
P,
Q
are
parallel,
then
scalar sum of P
of a
is
unaltered.
(see Fig. A.3). Since
and
Q
is
P.Q
dW= -qE.ds.
and
the
J
v.
unit time. If
function
of
the
potential
at
any
point,
and
is
in
V
(pro-
nounced
(A.2)
force
F,
where
F
line integral
a
P. It is an operator used to describe
the excess flux leaving
an element of volume
electric charge density
is positive in
Laplace's
and
Poisson's
vector.
The
operator
divgrad
is
therefore
equivalent
to
aj
ga
gs
operator
of the vector
from
this
equation
is
areas,
result is
the line
integral round
Therefore
|H.ds
in
electro-
 
by the surface,
When dealing
we do
to
be
simply
a
A
A.ll.
It
is
clear
relative
second
differential
that
(d/dt)
refers
to
rate
coordinate system,
of force (such
associated
with
of
motion
Orbital quantum number,