2.1 Maxwell’s Equations in the Presence of Water
55
Equations (2.16)–(2.19) connect all the main terms of the spectroscopic data, and
allow a recalculation of the corresponding frequency-dependent quantities into each
other, as according to local traditions, the different parts of the water spectrum is
represented in different terms, which complicates its comprehensive analysis. Hereinafter, we use terms of the complex dielectric function =
(ω) + i"(ω) and
dynamic conductivity, σ = " 0 ω, as the most convenient form of representation of
the broadband dielectric response of water and ice. Note that directly contributes
to (2.6), and thus represents the simplest form of the data representation, regardless
of the technique used for its determination.
2.2 The Broadband Dielectric Spectroscopy of Water
Although fully characterizes the material and is a good parameter for analysis,
it cannot be measured directly. One should first understand the basics of dielectric
spectroscopy, in order to see how the dielectric response is measured in different
frequency ranges.
Figure 2.1 shows various methods used to determine the electrodynamic parameters of water and ice, which can be recalculated as the real
and the imaginary
parts of the dielectric function. In the low-frequency range (0–10
7 Hz), the complex
impedance Z
∗ (ω) of the sample
2 is measured in a parallel-plate capacitor circuit. In
the linear response approximation, assuming that the sample is homogeneous and
isotropic, and that the field changes by the harmonic law, the dielectric function
can be found by the following formula:
ε(ω) = 1/(iωZ
∗
(ω)C 0 ),
(2.20)
where C 0 is the capacitance of the vacuum.
In the middle-frequency range (10
6 –10
11 Hz), the function is derived by
measuring the complex propagation coefficient (reflection or transmission). In this
case, a waveguide or cavity circuits are used, and the impedance of the sample is
calculated by the following formula:
Z
∗
(ω) = Z 0
1 + r
∗
(l)
1 − r ∗ (l)
,
(2.21)
where l is the length of the measuring line, r
∗ (l)=U
∗
re f /U
∗
inc is a complex reflection
factor, where U
∗
re f and U
∗
inc are the reflected and the incident signals, respectively.
2 Note that the sample is not a material, because it always has finite size and boundaries, which can
affect the dielectric measurement at low frequencies (see [2]). Thus, one should be careful with the
transfer of the sample parameters to the material properties, and use the corresponding equivalent
schemes.
55
Equations (2.16)–(2.19) connect all the main terms of the spectroscopic data, and
allow a recalculation of the corresponding frequency-dependent quantities into each
other, as according to local traditions, the different parts of the water spectrum is
represented in different terms, which complicates its comprehensive analysis. Hereinafter, we use terms of the complex dielectric function =
(ω) + i"(ω) and
dynamic conductivity, σ = " 0 ω, as the most convenient form of representation of
the broadband dielectric response of water and ice. Note that directly contributes
to (2.6), and thus represents the simplest form of the data representation, regardless
of the technique used for its determination.
2.2 The Broadband Dielectric Spectroscopy of Water
Although fully characterizes the material and is a good parameter for analysis,
it cannot be measured directly. One should first understand the basics of dielectric
spectroscopy, in order to see how the dielectric response is measured in different
frequency ranges.
Figure 2.1 shows various methods used to determine the electrodynamic parameters of water and ice, which can be recalculated as the real
and the imaginary
parts of the dielectric function. In the low-frequency range (0–10
7 Hz), the complex
impedance Z
∗ (ω) of the sample
2 is measured in a parallel-plate capacitor circuit. In
the linear response approximation, assuming that the sample is homogeneous and
isotropic, and that the field changes by the harmonic law, the dielectric function
can be found by the following formula:
ε(ω) = 1/(iωZ
∗
(ω)C 0 ),
(2.20)
where C 0 is the capacitance of the vacuum.
In the middle-frequency range (10
6 –10
11 Hz), the function is derived by
measuring the complex propagation coefficient (reflection or transmission). In this
case, a waveguide or cavity circuits are used, and the impedance of the sample is
calculated by the following formula:
Z
∗
(ω) = Z 0
1 + r
∗
(l)
1 − r ∗ (l)
,
(2.21)
where l is the length of the measuring line, r
∗ (l)=U
∗
re f /U
∗
inc is a complex reflection
factor, where U
∗
re f and U
∗
inc are the reflected and the incident signals, respectively.
2 Note that the sample is not a material, because it always has finite size and boundaries, which can
affect the dielectric measurement at low frequencies (see [2]). Thus, one should be careful with the
transfer of the sample parameters to the material properties, and use the corresponding equivalent
schemes.