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The Complex Permittivity
58
PART 2. MATTER IN AN ALTERNATING FIELD
VI. THE COMPLEX PERMITTIVITY VI.1. Definitions of ε* and σ* . Propagation of an electromagnetic wave Consider a plane-parallel condenser of vacuum capacitance C = £ S/d (S and d are respectively the surface area of the plates and the thickness). If an a.c. voltage V = V e is applied to this capacitor, a charge 0 = C V appears on the electrodes, in phase with the applied voltage. Q
Q
l u ) t :
o
Q
The current in the external circuit, which is the time-derivative of the charge Q: I = Q = ίω C V
(VI.1)
Q
is 90° out of phase with the applied voltage (cf. Fig.25). It is a non-dissipative displacement or induction current. We now fill up the volume between the electrodes with a non-polar, perfectly insulating material. The capacitor then displays a new capaci tance
C = Κ C
Q
with
K > 1 . The ratio
Κ
is the relative permittivity
of the sample material, with respect to vacuo, and the new current l'= Q
1
=
ί ω CV = ΚΙ
(VI.2)
is larger than above, but is still 90° out of phase with the applied voltage. Now, if the sample material is either slightly conducting or polar, or both, the capacitor is no longer perfect and the current is not exactly 90° out of phase with the voltage, since there is a small component of conduction GV in phase with the applied voltage. The origin of this in-phase component is a motion of charges. If these charges are free, the conductance G is effectively independent of the frequency of the applied voltage, but if these charges are bound to opposite charges as in oscillating dipoles, G is a function of fre quency, which will be discussed in detail in the chapter dealing with dipole relaxation.
59
? ι
> υ
3
G
V
-
/
Fig.25 - Current in a
/
leaking capacitor.
ν In either case, the resultant current is : I = (iu) C + G) \f If since
G
(VI.3)
is pure conductance due to free charges,
C = ε S/d,
the density
j*
G = o S / d , and
of the current is:
j* = ( i w E + σ ) £ The term σΕ?
iu>CI?
(VI.4)
is the displacement current density
Ό
, and
is the conduction current density. These may be considered by introducing:
either a complex conductivity σ* defined by
j* =
σ*ϊΓ:
σ* = χ ε + σ
(VI.5)
ω
or a complex permittivity
ε* defined by ε
= —
=
ε
j* = ίωε*ί?: (VI.6)
1 —
1ω
The latter quantity will usually be used in the study of dielec trics , since the dissipative term is, in general, very small with respect to the capacitive term. The loss angle
( δ in Fig.25) is such that δ a
n
dissipative term capacitive term
= ω
£
(VI.7)
Whenever the conduction (or dissipation) is not exclusively due to free charges, but is also due to bound charges, the conductivity
σ
is itself a complex quantity which depends on the frequency, so that the real part of ε* is not exactly ε , and the imaginary part is not ισί
exactly
. U)
The most general expression for the complex permittivity is: ε* =
ε
1
- ie"
(vi.8)
60 1
where ε and ε are frequency-dependent. These two components and their frequency dependence will be studied after the various types of interactions between the electromagnetic field and matter have been introduced. 11
The propagation of an electromagnetic wave in matter is based on Maxwell s equations: 1
V* χ ΕΓ = - —
,
if = μ* Η
(μ*= complex magnetic permeability)
,
D = ε* IT (ε* = complex permittivity) .
t't
V* χ Η =
— r't
—>
Elimination of
Η V
2
gives the propagation equation: Ε - ε" μ*
= ^t
0
(VI. 9)
z
In a cartesian coordinate system where the
χ
axis is chosen
along the direction of propagation, eqn. (VI.9) reduces to
4 - ε" ^ 4
= 0
(VI
-
10)
r't
CJYL
In other words, electromagnetic waves propagate through a mate* * * * * -1/2 (ε μ rial defined by ε* and μ* with a complex velocity ν* = -1/2 (ε* μ*) -1/2 whereas they propagate vacuo with a velocity c = ( Ρ ) = η ·-The ratio c/v* defines theincomplex index of refraction ; η* ik ε
0
0
1/2 = η - ik
(VI.11)
ε
\ ο^ο/ The general solution of eqn. (VI.10) is of the form: Ε = E where
Υ* = α + ί β
Q
exp(iu) t - γ* χ)
(VI. 12)
is a complex absorption coefficient. If the value of
Y* is introduced into eqn. (VI.12) this equation becomes: Ε = E e" o
a x
exp i( ω t - βχ)
(VI. 13)
In eqn. (VI.13), α appears as the absorption and ρ as the phase. Furthermore, combining eqns.(VI.lO) and (VI.12) gives:
6 1
v
*
Υ
.ω c
*
=α. —
(VI. Μ )
η
Solution for the real and imaginary parts of both members of eqn.(VI.14) gives
CM:)
In a non-magnetic material, μ* = μ , and eqn.(VI.11) reduces to ο
/
\
1 / 2
=
with
Κ
= K' - iK"
1/2
κ
(VI.15)
(relative complex permittivity).
This complex relation, which is known as Maxwell's relation, is equivalent to the system: n
2
- k
2
- K»
( 2nk which can be solved for
η
and
= K" k:
η = — (γκ' ν/2 k
2
= _ 1 _ ( γ κ»2 V2
+ Κ"
2
+ K')
1 / 2
(VI. 16) +
κ
ν2
_ ,) κ
If the dissipation is small - as is usual in dielectrics away from the critical dispersion frequencies, K"/K' = tan δ ~ δ , and
(VI.17) k = - α -
- Vk' =
Κ'
2vT From equations (VI.17), the two components of K* can be express ed in terms of the absorption coefficient a and the phase coefficient β , which are measured by microwave techniques. Finally, Κ* = ( β
2
2 - 2ίαβ) ( £ )
(VI.18)
Equation (VI.18) constitutes the basis of v.h.f. dielectric spectroscopy. Problem Using Maxwell's equations, show that the ratio between the electric and the magnetic vector of the propagating wave is a constant of the
62 1
material (its "characteristic impedance' ) , and that the value of this constant is
Ζ* =( μ* / ε * ) .
VI.2. The various types of charges and charge groups, and the corres ponding interactions In any material, there are various types of charges and charge associations, which we now consider. (a) The "inner" electrons (i.e. those of the inner electronic shells), tightly bound to the nuclei. Although little affected by the applied field, they "resonate" with high energy (~ 10^ e V ) , short wave length (~ 1 0 " ^ m) electromagnetic fields corresponding to the X-ray range. (b) The "outer" electrons (i.e. those of the outer electronic shells). These are the valence electrons, which contribute to the atomic and/or molecular polarizabilities, and also, in the case of elongated mole cular structures, to their orientation with respect to the applied field. (c) The free electrons or conduction electrons, which contribute to the "in phase" conduction. When submitted to an electric field Ε , these electrons move with a velocity ν = μ Ε . The mobility μ , characte ristic of a given material at a given temperature, accounts for all the inelastic collisions which, on average, confer to the electrons a —>
—>
velocity μ Ε in the field Ε . Note that if the electron concentr ation is not uniform (for instance whenever they accumulate near the electrodes), the "diffusion field" - DVn also contributes to the motion. (d) The bound ions, or ions bound to oppositely charged ions, forming molecular dipoles (e.g. Cl H ) or a dipole association in a lattice (e.g. lLi| " M g in LiF where [Lij is a lithium vacancy and M g a substitutional cation. These permanent dipoles experience an orientational torque in a uniform field; in a non-uniform field, a net force also acts, in addition to the torque. +
+
+
(e) The free ions, as in electrolytes and non-stoichiometric ionic crystals (e.g. excess K in Κ Cl) which move in the applied field, usually with a low mobility. Ionic dipoles such as (OH)" show both +
63 ionic and dipole characteristics. (f) Finally, the multipoles, and mainly the quadrupoles (cf. Chapter I ) , or an antiparallel association of two dipoles, which undergo only a configurational strain in a uniform field, and a slight torque in a divergent field.
Let us assume that an electromagnetic field of frequency f is abruptly applied at an instant which defines the origin of t . Depend ing on its frequency, this field puts into oscillation one or more types of charges or charge associations among those listed above. Each confi guration having its own critical frequency, above which the interaction with the field becomes vanishingly small, the lower the frequency, the more configurations are excited. Of course the critical frequency of a given configuration depends on the relevant masses and the elastic restoring and frictional forces. Whenever they exist,elastic forces such as Coulombic attraction inside molecules give the character of a resonance to the interaction, damped to a lesser or greater degree (radiation friction or Brehmstrahlung), 1
Figure 26 represents both real and imaginary components (K and K ) of a typical complex permittivity spectrum of a polar material n
containing space charges. We shall make a detailed analysis of this figure, starting from the high frequency side. As we have seen, electrons of the inner atomic shells have cri19 tical frequencies of the order of 10 Hz (X-ray range). Consequently, 19 an e.m. field of frequency higher than 10 (or of wavelength shorter than 1 A) cannot excite any vibration in the atoms; hence, it has no polarizing effect on the material, which has, for this frequency, the same permittivity ε as a vacuum. (Point 1 of the figure). ° 19 Note that for f >10 Hz, the relative permittivity (and also 2 the refractive index since Κ = η ) is smaller than unity. This means that the field propagates in the material with a velocity larger than that of light. This does not violate the law of relativity, since the velocity in question is a phase velocity. If the frequency is lower than the resonant frequency of the inner electrons, these electrons can "feel the electric component of the e.m. field, and they vibrate with the field. By so doing, they polarize the material, which raises the relative permittivity above unity. 11
64
Vibrations
ε' Space charges
Dipoles
Atoms
Valence electrons
Inner electrons Α Χ
ε
0
© 9 J. λ
ί 8
.2
10
ill
12
Relaxations
I
14
^ f\
16
,Φ
λ
1
ι
18
γ L20ι Log
10
fl
Resonances
Fig.26 - The various types of interaction between the e.m. field and matter, and the relevant relative permittivities.
This is seen at point 2 of the figure. Now, if the frequency of the e.m. field is lower than the reson ant frequency of the valence electrons which is in the range 3 χ 1 0 ^ to 3 χ 10"^ Hz (i.e. in the optical range from the u.v. (0.1 μιη) to the near i.r. (1 μπι) ) , these electrons also take part in the dielectric polarization, and their contribution again raises the permittivity. This is seen at point 3 on the figure. The same type of "resonance" process occurs at the frequencies of 12 atomic vibrations in molecules and crystals, in the range 10 to 3 χ 13 10 Hz (i.e. in the far infrared spectral range between 10 and 300 μπι), This is seen at point 4 of the figure. In all the processes mentioned so far, the charges affected by the field can be considered to be attracted towards their central posi tion by forces which are proportional to their displacements, i.e. by linear elastic forces. This mechanical approach of an electronic reson ance is only approximate, since electrons cannot be treated properly by classical mechanics. Quantitative treatments of these processes require the formalism of quantum mechanics. However, the quantum numbers of these systems are usually so large that a classical resonance model
65 including a friction term (to account for radiation damping) gives a fair description of these interactions (correspondence principle). If the frequency of the applied field is lower than that of atomic vibrations, however, a new type of interaction may appear, in which the restoring forces are not elastic as in the case of direct interactions between charged particles, but viscous in character, as a result of an irreversible thermodynamic process. The sluggish collective orientation of dipoles or the accumulation of an ionic space charge near the electrodes when a field is applied or cut off is an example of such a process which is known as "relaxation . 11
?iP2l?-??]r§5§Ei9D? which is the time-dependent polarization due to the orientation of dipoles (cf. Chapter IV) is treated in Sections VII.2 to VII.5.
§??-Ρ2ΐ§ϊί5§£ί2ί_§5^-i?i§?§£i25 in heterogeneous materials (Maxwell-Wagner effect), corresponds to the evolution from the "capacitive voltage distribution of the short time (or high frequency) heterogeneous permittivity to the "resistive" distribution of the long time (or low frequency) heterogeneous resistivity. This is discussed in Section VII.7. 11
?P52?_9harge_polarization occurs in materials containing carriers which do not recombine at the electrodes, and therefore behave, in a low frequency a.c, field, as macroscopic dipoles which reverse their direction each half period. This will be discussed in Section VII .9.
VI.3,
The response of a linear material to a variable field First consider a rectangular pulse of field of amplitude
applied to a linear material between times θ and θ + d9
Ε ,
(Fig. 2 7 )
Fig.2 7 - A pulse of applied field,
66
Fig.28- Decomposition of E(t) and D(t).
As can be seen in Figure 28, the pulse of applied field can be expressed as E(t) = where
T(u)
[r(t-9) - Γ ( ί - θ - ά β ) ] ε
(VI.19)
is the step function of unit amplitude.
From the superposition principle, the displacement D correspond ing to the pulse is the sum of the displacement D-^ due to the leading edge and D2 due to the trailing edge: D ( t ) = ε ^ Ε r(t-e) + ( c - e ) f ( t - e ) E s
x
D
( t ) 2
=
"
ε
ο ο
(VI.20)
00
(VI. 21)
r(t-e-de)-(e -e )f(t-e-de)E
Ε
s
o o
In these equations, the first terms on the righthand sides refer to instantaneous polarization, of electronic origin, which appears immediately on application of the field, whereas the second terms refer to the sluggish polarization which is due to dipole orientation, or any other frictional process. The function
f(u)
characterizes the sluggish
ness of the delayed polarization. It increases from The system is linear and the displacement E(t) at
t
due to the pulse
is: D(t) = D - W = ε ^ E(t) + ( 2
1
with
D
f(0) = 0 to f(oo)=l.
E(t) = 0
for
The derivative
ε 8
-ε
0 0
)ί
ι
(VI. 22)
Ε d9
t > θ + d9. 1
f (u)
of
f(u) , which appears in the above
equation, is a decreasing function, vanishing at infinite time, which is usually called the decay function, and denoted by Φ
.
If the field is a continuous function of time, applied at t = the displacement at instant
t
is:
67 t + (Z -ZJ
D(t) = ^E(t)
[
S
0(t-8)E(9)d6
(VI.23)
— 00
and if we use the new variable u = t - θ
(d9 = -du)
D(t) takes the form j
D(t) = e E ( t ) + QO
(VI.24)
ο r
In this form, the integral is the c on ν ο lut i on pr oduc t of Φ
and
Ε ,
usually denoted by Φ(1)
VI.4.
X
E(t)
(VI.25)
Case of an a.c. field. Kramers - Kronig relations We assume that the applied field is sinusoidal, with amplitude
E
Q
and angular frequency
ω = 2 nf: E(t)
= E
Q
cos ω t
(VI.26)
Hence, using this formula in eqn.(VI.24), we have:
D(t) =
E
q
cosut + ( ε - ε ) Ε 8
ο ο
ο
J Φ(\ι) αο (ωϋ - uu)du δ
° We deal here with times transient, so that
D(t)
t
(VI.27)
which are much longer than the
finally becomes sinusoidal with the same
angular frequency ω , but with a phase shift
φ with respect to the
applied field: D(t \
cos φ cos ω t + D sin φ sin ω t ο (VI.28) From eqn. (VI. 27) , we obtain another expression for D(t •* oo) : ->
oo)
/
= D
cos (cot N
- φ)
/
= D
o
00
D(t ->οό) = E
Q O
E
o
cosut + ( ^ " ε ^ Ε ο
r I O(u) (cosut cosuu + sinoot sinoju)du ° (VI.29)
68
D(t -> oo) = £ε + ( ε - ε ) 3
ο ο
f(u)
COSLOU d u J E
D
cosut + (VI.30)
|(ε3-ε )
j
οο
q
sincjt
By solution of eqns (VI.28) and (VI.30), we obtain:
D
D
cos(p
D
Q
εΰφ
=|ε +(ε -ε ) ο ο
8
(ε -ε ) 3
J
ο ο
j
( χ
(VI.31)
Q
(VI.32)
Q
But, by definition of the complex permittivity, D = ε* Ε = ( ε ' - ί ε " ) Ε
(νι.33)
Hence, the "in phase" component of D corresponds to
ε ' and the
"out of phase" component to ε " . It follows that (D(u) cosum du
(VI .34)
υ
ε
11
= (Sg-ε^) §
(VI.35)
The equations (VI.34) and (VI.35) show that both the real and imaginary parts of. the permittivity depend on the same decay function Φ, which can be written as a Fourier transforms: .00
2
(ΎΑχ )
Φ(") = ^ / ο
-ε
Γ
cos ux dx Zz
ε
oo
s
(VI.36)
00
ε"(χ) / ~ ε ~ ε ^ sin ux dx Ό ^s oo 0
(VI.37)
w
Consequently, the two spectra are interrelated, and the relation between them can be derived by substituting
O ( u ) , as given by eqn.
(VI.37), in eqn.(VI.34): 00-
2 ε (ω) - ε ^ =
-
no
f Γf J
ε
"
1 (VI.38) χ
s
n
( ) ^- x u dx
Changing the order of the integration gives
cos u)U du
69 '00
sin χ u cos ω u du
ε ( χ ) dx Μ
(VI.39)
The integral in the square brackets is oo
(VI.40)
/ sin xu cos ο
so that
ε' (ω) -
ω
u du =
^· ^ χ
=
-ω
I
(VI.41)
In a similar way, combination of eqns.(VI.35) and (VI.36) gives
ε "(ω) =
—
(VI,42)
Equations (VI.41) and (VI.42) constitute the Kramers-Kronig relations. They are valid for any type of dispersion, and permit one of the spectra to be obtained provided that the other has been measured throughout the complete spectral range. If we let ω = 0 in eqn.(VI.41), we have:
or (VI.43)
This shows that the total area under the curve in a plot of ε
11
vs.
log ω
is simply related to the extreme values of the permitti
vity, irrespective of the dispersion mechanism.
PART 2. MATTER IN AN ALTERNATING FIELD
VI. THE COMPLEX PERMITTIVITY VI.1. Definitions of ε* and σ* . Propagation of an electromagnetic wave Consider a plane-parallel condenser of vacuum capacitance C = £ S/d (S and d are respectively the surface area of the plates and the thickness). If an a.c. voltage V = V e is applied to this capacitor, a charge 0 = C V appears on the electrodes, in phase with the applied voltage. Q
Q
l u ) t :
o
Q
The current in the external circuit, which is the time-derivative of the charge Q: I = Q = ίω C V
(VI.1)
Q
is 90° out of phase with the applied voltage (cf. Fig.25). It is a non-dissipative displacement or induction current. We now fill up the volume between the electrodes with a non-polar, perfectly insulating material. The capacitor then displays a new capaci tance
C = Κ C
Q
with
K > 1 . The ratio
Κ
is the relative permittivity
of the sample material, with respect to vacuo, and the new current l'= Q
1
=
ί ω CV = ΚΙ
(VI.2)
is larger than above, but is still 90° out of phase with the applied voltage. Now, if the sample material is either slightly conducting or polar, or both, the capacitor is no longer perfect and the current is not exactly 90° out of phase with the voltage, since there is a small component of conduction GV in phase with the applied voltage. The origin of this in-phase component is a motion of charges. If these charges are free, the conductance G is effectively independent of the frequency of the applied voltage, but if these charges are bound to opposite charges as in oscillating dipoles, G is a function of fre quency, which will be discussed in detail in the chapter dealing with dipole relaxation.
59
? ι
> υ
3
G
V
-
/
Fig.25 - Current in a
/
leaking capacitor.
ν In either case, the resultant current is : I = (iu) C + G) \f If since
G
(VI.3)
is pure conductance due to free charges,
C = ε S/d,
the density
j*
G = o S / d , and
of the current is:
j* = ( i w E + σ ) £ The term σΕ?
iu>CI?
(VI.4)
is the displacement current density
Ό
, and
is the conduction current density. These may be considered by introducing:
either a complex conductivity σ* defined by
j* =
σ*ϊΓ:
σ* = χ ε + σ
(VI.5)
ω
or a complex permittivity
ε* defined by ε
= —
=
ε
j* = ίωε*ί?: (VI.6)
1 —
1ω
The latter quantity will usually be used in the study of dielec trics , since the dissipative term is, in general, very small with respect to the capacitive term. The loss angle
( δ in Fig.25) is such that δ a
n
dissipative term capacitive term
= ω
£
(VI.7)
Whenever the conduction (or dissipation) is not exclusively due to free charges, but is also due to bound charges, the conductivity
σ
is itself a complex quantity which depends on the frequency, so that the real part of ε* is not exactly ε , and the imaginary part is not ισί
exactly
. U)
The most general expression for the complex permittivity is: ε* =
ε
1
- ie"
(vi.8)
60 1
where ε and ε are frequency-dependent. These two components and their frequency dependence will be studied after the various types of interactions between the electromagnetic field and matter have been introduced. 11
The propagation of an electromagnetic wave in matter is based on Maxwell s equations: 1
V* χ ΕΓ = - —
,
if = μ* Η
(μ*= complex magnetic permeability)
,
D = ε* IT (ε* = complex permittivity) .
t't
V* χ Η =
— r't
—>
Elimination of
Η V
2
gives the propagation equation: Ε - ε" μ*
= ^t
0
(VI. 9)
z
In a cartesian coordinate system where the
χ
axis is chosen
along the direction of propagation, eqn. (VI.9) reduces to
4 - ε" ^ 4
= 0
(VI
-
10)
r't
CJYL
In other words, electromagnetic waves propagate through a mate* * * * * -1/2 (ε μ rial defined by ε* and μ* with a complex velocity ν* = -1/2 (ε* μ*) -1/2 whereas they propagate vacuo with a velocity c = ( Ρ ) = η ·-The ratio c/v* defines theincomplex index of refraction ; η* ik ε
0
0
1/2 = η - ik
(VI.11)
ε
\ ο^ο/ The general solution of eqn. (VI.10) is of the form: Ε = E where
Υ* = α + ί β
Q
exp(iu) t - γ* χ)
(VI. 12)
is a complex absorption coefficient. If the value of
Y* is introduced into eqn. (VI.12) this equation becomes: Ε = E e" o
a x
exp i( ω t - βχ)
(VI. 13)
In eqn. (VI.13), α appears as the absorption and ρ as the phase. Furthermore, combining eqns.(VI.lO) and (VI.12) gives:
6 1
v
*
Υ
.ω c
*
=α. —
(VI. Μ )
η
Solution for the real and imaginary parts of both members of eqn.(VI.14) gives
CM:)
In a non-magnetic material, μ* = μ , and eqn.(VI.11) reduces to ο
/
\
1 / 2
=
with
Κ
= K' - iK"
1/2
κ
(VI.15)
(relative complex permittivity).
This complex relation, which is known as Maxwell's relation, is equivalent to the system: n
2
- k
2
- K»
( 2nk which can be solved for
η
and
= K" k:
η = — (γκ' ν/2 k
2
= _ 1 _ ( γ κ»2 V2
+ Κ"
2
+ K')
1 / 2
(VI. 16) +
κ
ν2
_ ,) κ
If the dissipation is small - as is usual in dielectrics away from the critical dispersion frequencies, K"/K' = tan δ ~ δ , and
(VI.17) k = - α -
- Vk' =
Κ'
2vT From equations (VI.17), the two components of K* can be express ed in terms of the absorption coefficient a and the phase coefficient β , which are measured by microwave techniques. Finally, Κ* = ( β
2
2 - 2ίαβ) ( £ )
(VI.18)
Equation (VI.18) constitutes the basis of v.h.f. dielectric spectroscopy. Problem Using Maxwell's equations, show that the ratio between the electric and the magnetic vector of the propagating wave is a constant of the
62 1
material (its "characteristic impedance' ) , and that the value of this constant is
Ζ* =( μ* / ε * ) .
VI.2. The various types of charges and charge groups, and the corres ponding interactions In any material, there are various types of charges and charge associations, which we now consider. (a) The "inner" electrons (i.e. those of the inner electronic shells), tightly bound to the nuclei. Although little affected by the applied field, they "resonate" with high energy (~ 10^ e V ) , short wave length (~ 1 0 " ^ m) electromagnetic fields corresponding to the X-ray range. (b) The "outer" electrons (i.e. those of the outer electronic shells). These are the valence electrons, which contribute to the atomic and/or molecular polarizabilities, and also, in the case of elongated mole cular structures, to their orientation with respect to the applied field. (c) The free electrons or conduction electrons, which contribute to the "in phase" conduction. When submitted to an electric field Ε , these electrons move with a velocity ν = μ Ε . The mobility μ , characte ristic of a given material at a given temperature, accounts for all the inelastic collisions which, on average, confer to the electrons a —>
—>
velocity μ Ε in the field Ε . Note that if the electron concentr ation is not uniform (for instance whenever they accumulate near the electrodes), the "diffusion field" - DVn also contributes to the motion. (d) The bound ions, or ions bound to oppositely charged ions, forming molecular dipoles (e.g. Cl H ) or a dipole association in a lattice (e.g. lLi| " M g in LiF where [Lij is a lithium vacancy and M g a substitutional cation. These permanent dipoles experience an orientational torque in a uniform field; in a non-uniform field, a net force also acts, in addition to the torque. +
+
+
(e) The free ions, as in electrolytes and non-stoichiometric ionic crystals (e.g. excess K in Κ Cl) which move in the applied field, usually with a low mobility. Ionic dipoles such as (OH)" show both +
63 ionic and dipole characteristics. (f) Finally, the multipoles, and mainly the quadrupoles (cf. Chapter I ) , or an antiparallel association of two dipoles, which undergo only a configurational strain in a uniform field, and a slight torque in a divergent field.
Let us assume that an electromagnetic field of frequency f is abruptly applied at an instant which defines the origin of t . Depend ing on its frequency, this field puts into oscillation one or more types of charges or charge associations among those listed above. Each confi guration having its own critical frequency, above which the interaction with the field becomes vanishingly small, the lower the frequency, the more configurations are excited. Of course the critical frequency of a given configuration depends on the relevant masses and the elastic restoring and frictional forces. Whenever they exist,elastic forces such as Coulombic attraction inside molecules give the character of a resonance to the interaction, damped to a lesser or greater degree (radiation friction or Brehmstrahlung), 1
Figure 26 represents both real and imaginary components (K and K ) of a typical complex permittivity spectrum of a polar material n
containing space charges. We shall make a detailed analysis of this figure, starting from the high frequency side. As we have seen, electrons of the inner atomic shells have cri19 tical frequencies of the order of 10 Hz (X-ray range). Consequently, 19 an e.m. field of frequency higher than 10 (or of wavelength shorter than 1 A) cannot excite any vibration in the atoms; hence, it has no polarizing effect on the material, which has, for this frequency, the same permittivity ε as a vacuum. (Point 1 of the figure). ° 19 Note that for f >10 Hz, the relative permittivity (and also 2 the refractive index since Κ = η ) is smaller than unity. This means that the field propagates in the material with a velocity larger than that of light. This does not violate the law of relativity, since the velocity in question is a phase velocity. If the frequency is lower than the resonant frequency of the inner electrons, these electrons can "feel the electric component of the e.m. field, and they vibrate with the field. By so doing, they polarize the material, which raises the relative permittivity above unity. 11
64
Vibrations
ε' Space charges
Dipoles
Atoms
Valence electrons
Inner electrons Α Χ
ε
0
© 9 J. λ
ί 8
.2
10
ill
12
Relaxations
I
14
^ f\
16
,Φ
λ
1
ι
18
γ L20ι Log
10
fl
Resonances
Fig.26 - The various types of interaction between the e.m. field and matter, and the relevant relative permittivities.
This is seen at point 2 of the figure. Now, if the frequency of the e.m. field is lower than the reson ant frequency of the valence electrons which is in the range 3 χ 1 0 ^ to 3 χ 10"^ Hz (i.e. in the optical range from the u.v. (0.1 μιη) to the near i.r. (1 μπι) ) , these electrons also take part in the dielectric polarization, and their contribution again raises the permittivity. This is seen at point 3 on the figure. The same type of "resonance" process occurs at the frequencies of 12 atomic vibrations in molecules and crystals, in the range 10 to 3 χ 13 10 Hz (i.e. in the far infrared spectral range between 10 and 300 μπι), This is seen at point 4 of the figure. In all the processes mentioned so far, the charges affected by the field can be considered to be attracted towards their central posi tion by forces which are proportional to their displacements, i.e. by linear elastic forces. This mechanical approach of an electronic reson ance is only approximate, since electrons cannot be treated properly by classical mechanics. Quantitative treatments of these processes require the formalism of quantum mechanics. However, the quantum numbers of these systems are usually so large that a classical resonance model
65 including a friction term (to account for radiation damping) gives a fair description of these interactions (correspondence principle). If the frequency of the applied field is lower than that of atomic vibrations, however, a new type of interaction may appear, in which the restoring forces are not elastic as in the case of direct interactions between charged particles, but viscous in character, as a result of an irreversible thermodynamic process. The sluggish collective orientation of dipoles or the accumulation of an ionic space charge near the electrodes when a field is applied or cut off is an example of such a process which is known as "relaxation . 11
?iP2l?-??]r§5§Ei9D? which is the time-dependent polarization due to the orientation of dipoles (cf. Chapter IV) is treated in Sections VII.2 to VII.5.
§??-Ρ2ΐ§ϊί5§£ί2ί_§5^-i?i§?§£i25 in heterogeneous materials (Maxwell-Wagner effect), corresponds to the evolution from the "capacitive voltage distribution of the short time (or high frequency) heterogeneous permittivity to the "resistive" distribution of the long time (or low frequency) heterogeneous resistivity. This is discussed in Section VII.7. 11
?P52?_9harge_polarization occurs in materials containing carriers which do not recombine at the electrodes, and therefore behave, in a low frequency a.c, field, as macroscopic dipoles which reverse their direction each half period. This will be discussed in Section VII .9.
VI.3,
The response of a linear material to a variable field First consider a rectangular pulse of field of amplitude
applied to a linear material between times θ and θ + d9
Ε ,
(Fig. 2 7 )
Fig.2 7 - A pulse of applied field,
66
Fig.28- Decomposition of E(t) and D(t).
As can be seen in Figure 28, the pulse of applied field can be expressed as E(t) = where
T(u)
[r(t-9) - Γ ( ί - θ - ά β ) ] ε
(VI.19)
is the step function of unit amplitude.
From the superposition principle, the displacement D correspond ing to the pulse is the sum of the displacement D-^ due to the leading edge and D2 due to the trailing edge: D ( t ) = ε ^ Ε r(t-e) + ( c - e ) f ( t - e ) E s
x
D
( t ) 2
=
"
ε
ο ο
(VI.20)
00
(VI. 21)
r(t-e-de)-(e -e )f(t-e-de)E
Ε
s
o o
In these equations, the first terms on the righthand sides refer to instantaneous polarization, of electronic origin, which appears immediately on application of the field, whereas the second terms refer to the sluggish polarization which is due to dipole orientation, or any other frictional process. The function
f(u)
characterizes the sluggish
ness of the delayed polarization. It increases from The system is linear and the displacement E(t) at
t
due to the pulse
is: D(t) = D - W = ε ^ E(t) + ( 2
1
with
D
f(0) = 0 to f(oo)=l.
E(t) = 0
for
The derivative
ε 8
-ε
0 0
)ί
ι
(VI. 22)
Ε d9
t > θ + d9. 1
f (u)
of
f(u) , which appears in the above
equation, is a decreasing function, vanishing at infinite time, which is usually called the decay function, and denoted by Φ
.
If the field is a continuous function of time, applied at t = the displacement at instant
t
is:
67 t + (Z -ZJ
D(t) = ^E(t)
[
S
0(t-8)E(9)d6
(VI.23)
— 00
and if we use the new variable u = t - θ
(d9 = -du)
D(t) takes the form j
D(t) = e E ( t ) + QO
(VI.24)
ο r
In this form, the integral is the c on ν ο lut i on pr oduc t of Φ
and
Ε ,
usually denoted by Φ(1)
VI.4.
X
E(t)
(VI.25)
Case of an a.c. field. Kramers - Kronig relations We assume that the applied field is sinusoidal, with amplitude
E
Q
and angular frequency
ω = 2 nf: E(t)
= E
Q
cos ω t
(VI.26)
Hence, using this formula in eqn.(VI.24), we have:
D(t) =
E
q
cosut + ( ε - ε ) Ε 8
ο ο
ο
J Φ(\ι) αο (ωϋ - uu)du δ
° We deal here with times transient, so that
D(t)
t
(VI.27)
which are much longer than the
finally becomes sinusoidal with the same
angular frequency ω , but with a phase shift
φ with respect to the
applied field: D(t \
cos φ cos ω t + D sin φ sin ω t ο (VI.28) From eqn. (VI. 27) , we obtain another expression for D(t •* oo) : ->
oo)
/
= D
cos (cot N
- φ)
/
= D
o
00
D(t ->οό) = E
Q O
E
o
cosut + ( ^ " ε ^ Ε ο
r I O(u) (cosut cosuu + sinoot sinoju)du ° (VI.29)
68
D(t -> oo) = £ε + ( ε - ε ) 3
ο ο
f
COSLOU d u J E
D
cosut + (VI.30)
|(ε3-ε )
j
οο
q
sincjt
By solution of eqns (VI.28) and (VI.30), we obtain:
D
D
cos(p
D
Q
εΰφ
=|ε +(ε -ε ) ο ο
8
(ε -ε ) 3
J
ο ο
j
( χ
(VI.31)
Q
(VI.32)
Q
But, by definition of the complex permittivity, D = ε* Ε = ( ε ' - ί ε " ) Ε
(νι.33)
Hence, the "in phase" component of D corresponds to
ε ' and the
"out of phase" component to ε " . It follows that (D(u) cosum du
(VI .34)
υ
ε
11
= (Sg-ε^) §
(VI.35)
The equations (VI.34) and (VI.35) show that both the real and imaginary parts of. the permittivity depend on the same decay function Φ, which can be written as a Fourier transforms: .00
2
(ΎΑχ )
Φ(") = ^ / ο
-ε
Γ
cos ux dx Zz
ε
oo
s
(VI.36)
00
ε"(χ) / ~ ε ~ ε ^ sin ux dx Ό ^s oo 0
(VI.37)
w
Consequently, the two spectra are interrelated, and the relation between them can be derived by substituting
O ( u ) , as given by eqn.
(VI.37), in eqn.(VI.34): 00-
2 ε (ω) - ε ^ =
-
no
f Γf J
ε
"
1 (VI.38) χ
s
n
( ) ^- x u dx
Changing the order of the integration gives
cos u)U du
69 '00
sin χ u cos ω u du
ε ( χ ) dx Μ
(VI.39)
The integral in the square brackets is oo
(VI.40)
/ sin xu cos ο
so that
ε' (ω) -
ω
u du =
^· ^ χ
=
-ω
I
(VI.41)
In a similar way, combination of eqns.(VI.35) and (VI.36) gives
ε "(ω) =
—
(VI,42)
Equations (VI.41) and (VI.42) constitute the Kramers-Kronig relations. They are valid for any type of dispersion, and permit one of the spectra to be obtained provided that the other has been measured throughout the complete spectral range. If we let ω = 0 in eqn.(VI.41), we have:
or (VI.43)
This shows that the total area under the curve in a plot of ε
11
vs.
log ω
is simply related to the extreme values of the permitti
vity, irrespective of the dispersion mechanism.