Chapter 4
Interaction of photons with matter
4.1 Very short wavelengths (< 10−12
m, γ-rays)
The method that uses γ-rays to characterize materials is called Mössbauer spectroscopy. Key to the
success of the technique is the discovery of recoilless γ-ray emission and absorption, now referred
to as the Mössbauer effect, after its discoverer Rudolph Mössbauer, who first observed the effect
in 1957 and received the Nobel Prize in Physics in 1961 for his work.
4.1.1 Mössbauer effect
Nuclei in atoms undergo a variety of energy level transitions, often associated with the emission or
absorption of a gamma ray. These energy levels are influenced by their surrounding environment,
both electronic and magnetic, which can change or split these energy levels. These changes in the
energy levels can provide information about the atom’s local environment within a system and
ought to be observed using resonance-fluorescence. There are, however, two major obstacles in
obtaining this information:
• the ’hyperfine’ interactions between the nucleus and its environment are extremely smaller,
• the recoil of the nucleus (as the gamma-ray is emitted or absorbed) prevents the resonance.
1
2 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
Just as a gun recoils when a bullet is fired, conservation of momentum requires a free nucleus
(such as in a gas) to recoil during emission or absorption of a gamma ray
• If a nucleus at rest emits a gamma ray, the energy of the gamma ray is slightly less than the
natural energy of the transition (difference is equal to the recoil energy ER)
• but in order for a nucleus at rest to absorb a gamma ray, the gamma ray’s energy must be
slightly greater than the natural energy.
Figure 4.1: Recoil of free nuclei in emission or absorption of a gamma-rays
4.1. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 3
⇒ This means that nuclear resonance (emission and absorption of the same gamma ray by
identical nuclei) is unobservable with free nuclei, because the shift in energy is too great and the
emission and absorption spectra have no significant overlap. In order to achieve resonance in
emission and absorption the loss of the recoil energy must be overcome in some way.
Figure 4.2: Resonant overlap in free atoms. The overlap shown shaded is greatly exaggerated.
4 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
Nuclei in a solid crystal are not free to recoil because they are bound in place in the crystal
lattice. When a nucleus in a solid emits or absorbs a gamma ray, some energy can still be lost
as recoil energy, but in this case it always occurs in discrete packets called phonons (quantized
vibrations of the crystal lattice). Any whole number of phonons can be emitted, including zero,
which is known as a "recoil-free" event. In this case the conservation of momentum is satisfied by
the momentum of the crystal as a whole, so practically no energy is lost.
Mössbauer realised that for the resonance effect to occur both emitting and absorbing atoms
must be bound in a lattice where the lowest vibrational excitation energy is greater than the
nuclear recoil energy ⇒ the atom is unable to recoil and γ-ray may be emitted and absorbed
without loss of energy.
Figure 4.3: Atoms fixed in lattice.
4.1. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 5
4.1.2 Mössbauer absorption spectroscopy
Mössbauer spectroscopy involves the emission and absorption of gamma-rays to determine the
chemical state of atoms in lattice. It is primarily used to study iron containing materials but not
exclusively because the effect can be observed for over 40 elements.
Mössbauer spectroscopy is only possible ("recoil-free" event) when atoms are fixed in a solid
crystal lattice at a temperature below certain critical value, the Debye characteristic temperature.
Then gamma rays emitted by one nucleus can be resonantly absorbed by a sample containing
nuclei of the same isotope, and this absorption can be measured.
In its most common form, Mössbauer absorption spectroscopy,
• a solid sample is exposed to a beam of gamma radiation,
• and a detector measures the intensity of the beam transmitted through the sample.
The atoms in the source emitting the gamma rays must be of the same isotope as the atoms in
the sample absorbing them.
If the emitting and absorbing nuclei were in identical chemical environments, the nuclear
transition energies would be exactly equal and resonant absorption would be observed with both
materials at rest. The difference in chemical environments, however, causes the nuclear energy
levels to shift in a few different ways.
Although these energy shifts are tiny (often less than a micro-electronvolt), the extremely
narrow spectral linewidths of gamma rays for some radionuclides make the small energy shifts
correspond to large changes in absorbance. To bring the two nuclei back into resonance it is
necessary to change the energy of the gamma ray slightly, and in practice this is always done using
the Doppler effect. ⇒ the source is accelerated through a range of velocities using a linear motor
to produce a Doppler effect and scan the gamma ray energy through a given range. A typical
range of velocities for 57Fe, for example, may be ±11 mm/s (1 mm/s = 48.075 neV).
In the resulting spectra, gamma ray intensity is plotted as a function of the source velocity.
At velocities corresponding to the resonant energy levels of the sample, a fraction of the gamma
rays are absorbed, resulting in a drop in the measured intensity and a corresponding dip in the
spectrum. The number, positions, and intensities of the dips (also called peaks; dips in transmitted
intensity are peaks in absorbance) provide information about the chemical environment of the
absorbing nuclei and can be used to characterize the sample.
Mössbauer spectroscopy is limited by the need for a suitable gamma-ray source. Usually, this
consists of a radioactive parent that decays to the desired isotope. For example,
• the source for 57Fe consists of 57Co, which decays by electron capture to an excited state
of 57Fe,
• then subsequently decays to a ground state emitting the desired gamma-ray.
• The radioactive cobalt is prepared on a foil, often of rhodium.
Ideally the parent isotope will have a sufficiently long half-life to remain useful, but will also have
a sufficient decay rate to supply the required intensity of radiation.
6 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
Mössbauer spectroscopy has an extremely fine energy resolution and can detect even subtle
changes in the nuclear environment of the relevant atoms. Typically, there are three types of
nuclear interactions that are observed
• isomer shift (or chemical shift) - is a relative measure describing a shift in the resonance
energy of a nucleus due to the transition of electrons within its s orbital. The whole spectrum
is shifted in either a positive or negative direction depending upon the s electron charge
density. This change arises due to alterations in the electrostatic response between the
non-zero probability s orbital electrons and the non-zero volume nucleus they orbit.
• quadrupole splitting - reflects the interaction between the nuclear energy levels and surrounding
electric field gradient (EFG). Nuclei in states with non-spherical charge distributions,
i.e. all those with angular quantum number (I) greater than 1/2, produce an asymmetrical
electric field which splits the nuclear energy levels. This produces a nuclear quadrupole
moment.
Figure 4.4: Chemical shift and quadrupole splitting of the nuclear energy levels and corresponding
Mössbauer spectra
4.1. VERY SHORT WAVELENGTHS (< 10−12
M, γ-RAYS) 7
• hyperfine splitting (or Zeeman splitting) - is a result of the interaction between the nucleus
any surrounding magnetic field. A nucleus with spin, I, splits into 2I + 1 sub-energy levels
in the presence of magnetic field. For example, a nucleus with spin state I= 3/2 will split
into 4 non-degenerate sub-states with mI values of +3/2, +1/2, -1/2 and −3/2. Each split is
hyperfine, being in the order of 10−7
eV. The restriction rule of magnetic dipoles means that
transitions between the excited state and ground state can only occur where mI changes by
0 or 1. This gives six possible transitions for a 3/2 to 1/2 transition.
Figure 4.5: Magnetic splitting of the nuclear energy levels and the corresponding Mössbauer
spectrum
8 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
4.2 Short wavelengths (10−12
to 10−9
m, including X-rays and
ultraviolet)
If photon has enough energy to excite electron to high energy levels, two effects can occur:
• internal photoeffect (e.g. photoconductivity)
• external photoeffect or the photoemission.
The kinetic energy is:
1
2
mv2
= EF ± δE − Eaf + hν − ∆E. (4.1)
At 0 K the kinetic energy is maximum
(
1
2
mv2
)max = hν − (Eaf − EF) = hν − χ = h(ν − ν0), (4.2)
where ν0 is the frequency of photoelectric effect.
In the case of photoemission induced absorption (X-ray), the photon has enough energy to
excite an electron from some inner layers of solid.
1
2
mv2
= hν − χ − EB, (4.3)
where EB is the binding energy of the inner layers. The binding energy is characteristic for each
substance, so the kinetic electrons emitted bear information related to the electronic structure
and thus, the type of material.
4.2.1 Fowler‘s theory for metal/vacuum interfaces
Fowler proposed a theory in 1931 which showed that the photoelectric current variation with the
light frequency could be accounted for by the effect of temperature in the number of electrons
available for emission, in accordance with the distribution law of Sommerfeld‘s theory for metals.
Sommerfeld‘s theory had resolved some of the problems surrounding the original models for
electron in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional
potential well (box) containing a gas of freely mobile electrons. This adequately explained their
high electrical and thermal conductivities. However, because experimentally it was found that
metallic electrons do not show a gas-like heat capacity, the Boltzman distribution law is inappropriate.
A Fermi-Dirac distribution function is required, consistent with the need that the electrons
obey the Pauli exclusion principle and this distribution function has the form
P(E) =
1
1 + exp[(E − EF )/kT]
(4.4)
At ordinary temperatures, the Fermi energy EF kT. Fowler‘s law, referred to above as the
"square law", is readily tested in practice since it predicts at T=0 K
i ∝ (ν − ν0)2
(4.5)
4.2. SHORT WAVELENGTHS (10−12
TO 10−9
M, INCLUDING X-RAYS AND ULTRAVIOLET)9
where ν0 is the photoelectric threshold frequency and ν refers to a range of energies from threshold
to a few kT above threshold. To a good approximation, the photoelectric yield per unit light
intensity near ν0 is proportional simply to the number of electrons incident per unit time. The
energy condition can be written
(1/2m)p2
+ hν > φ (4.6)
where p is the initial momentum of the electron normal to the surface. The escaping electrons
have been termed the "available electrons".
Fowler‘s derivation for the single photon does not explicitly involve the quantum mechanical form
of current; instead, a semi-classical flux of electrons arriving at the metal surface is used. The
electron gas in the metal will obey Fermi-Dirac statistics, and the number of electrons per unit
volume having velocity components in the ranges u, u + du, ν, ν + dν and w + dw is given by the
formula
n(u, ν, w)dudνdw = 2(
m
h
)2 dudνdw
1 + exp[1
2
m(u2 + ν2 + w2) − EF ]/kT
(4.7)
where u is the velocity normal to the surface and m is the electron mass. The number of electrons
per volume, n(u)du, with their velocity component normal to the surface in the range u, u + du is
given by:
n(u)du =
4πkT
m
(
m
h
)3
log[1 + exp(EF −
1
2
mu2
)/kT]du (4.8)
The solution for the photoelectric current resulted in the form
i ∝
(hν − φ)3/2
(φ0 − hν)1/2
. (4.9)
A method for determining the threshold frequency of the illuminated surface ν0 is that which
makes use of the complete photoelectric emission. When a metal surface is exposed to the total
thermal radiation from a black body with a temperature T, the total photoelectric current is given
by:
i = A‘T2
e−
hν0
kT (4.10)
were A‘ is a constant.
4.2.2 Measurement of the photoelectric work function
• Fowler isotherm: measurement of the photocurrent depending on the frequency of incident
radiation at constant temperature, the ln if/T2
is expressed as hν/kT and compared with
the theoretical curve
ln
if
T2
= ln(αA0) + ln f(x) = B + Φ[
h
kT
(ν − ν0)]. (4.11)
From the Fowler equation, giving the relation between temperature and photoelectric emission,
ν0 can be determinate by plotting the ln f(∆) as a function of ∆. A universal curve is
obtained. Then ln i
T2 is plotted against hν
kT
. A curve us obtained of the same shape as the
universal curve, which can be made to coincide with it by a parallel shift. The vertical shift
is measure of B, the horizontal is equal to hν0
kT
.
10 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
• duBridge isochromatic method: measurements at one frequency ν but different temperatures.
For this purpose ln f(∆) is plotted against ln |∆|, and again a universal curve is
obtained. From the experimental data ln if/T2
is plotted against ln 1/T. For different frequencies
the temperature dependence of the photoemission differs
• for ν = ν0 is if = αA0 = π2
/12T2
• for x 1; ν > ν0, is f(x) = π2
/6 + x2
/2 and
if =
αA0
2
h2
(ν − ν0)2
k2
+
π2
3
, (4.12)
where the parabolic dependence is applied obly when the first term in the brackets is sufficiently
small.
• for x 1, ν < ν0, is f(x) = ex
and
if = αA0T2
exp(−
h(ν0 − ν)
kT
). (4.13)
4.2.3 Interaction of excited electrons with matter
The hot-electron problem represents the study of the deviations from the linear response regime
due to heating of charge carriers above the thermal equilibrium. Studies have been carried out in
1930s, mainly by Russian physicist Landau and Davidov. This means that electron temperature
Te which , in the presence of an external high electric field, is higher that the lattice temperature
T.
Nonlinear transport deals with problems that arise when a sufficiently strong electric field is
applied to a semiconductor sample, so that the current deviates from the linear response. This
effect is typical of semiconductors and cannot be seen on metals since, owing to their large
conductivities, Joule heating would destroy the material before deviations from the linearity
could be observed.
Considering
v(i)
(t) = v
(i)
0 + a∆t(i)
(4.14)
where v(i)
(t) is the instantaneous velocity if the i-th electron at time t, ∆t(i)
is the same time elapsed
after its last scattering event, and a = eE/m is the electron acceleration, due to a constant and
uniform applied electric field E. The region of field strengths where transport starts to deviate
from linearity is called warm-electron region.
4.2.4 Photoemission from semiconductors
There are some differences when we speak about the photoemission from metals and semiconductors.
All high quantum yield photocathodes are semiconductor ones. The absorption of light is
affected by:
• internal absorption due to their band gap
4.3. INTERMEDIATE WAVELENGTHS (INCLUDING VISIBLE AND ULTRAVIOLET LIGHT)11
• external absorption from the external layer
The Fermi level is influenced by the surface layers and the type of semiconductor used. Electron
coming from different depths have different space charge and work function ⇒ as they would have
different work function for different photoelectrons excited at different wavelengths.
The density of stated around Fermi level increases strongly and we cannot assume that photon
absorption is independent from electron‘s energy state. The quantum yield increases with the
temperature faster than in metals.
4.2.5 Photocathode
• IR region: Ag-O-Cs, also called S-1. This was the first compound photocathode material,
developed in 1929. Sensitivity from 300 nm to 1200 nm. Since Ag-O-Cs has a higher dark
current than more modern materials photomultiplier tubes with this photocathode material
are nowadays used only in the infrared region with cooling.
• visible region: Sb-Cs has a spectral response from UV to visible and is mainly used in
reflection-mode photocathodes. and also bi-alkali (antimony-rubidium-caesium Sb-Rb-Cs,
antimony-potassium-caesium Sb-K-Cs). Spectral response range similar to the Sb-Cs photocathode,
but with higher sensitivity and lower dark current than Sb-Cs. They have sensitivity
well matched to the most common scintillator materials and so are frequently used
for ionizing radiation measurement in scintillation counters.
• UV region: metals and Cs-Te (Cs-I). These materials are sensitive to vacuum UV and UV
rays but not to visible light and are therefore referred to as solar blind. Cs-Te is insensitive
to wavelengths longer than 320 nm, and Cs-I to those longer than 200 nm.
4.3 Intermediate wavelengths (including visible and ultraviolet
light)
Spectrophotometry (measurements of reflectance R or transmittance T) and ellipsometry for determination
of thin film thickness and its optical properties.
4.3.1 Principles of R/T method
By reflectance (R) and/or transmittance (T) method the ratio of reflected to incident and/or
transmitted to incident light energies are measured. If the studied system is surrounded on both
sides by the same medium the reflectance and/or transmittance are given simply by the ratio of
the electric field amplitudes:
Rq =
ˆEr
ˆEi
2
q
and/or Tq =
ˆEt
ˆEi
2
q
(4.15)
where q = p, s and ˆEiq, ˆErq and/or ˆEtq denote complex amplitudes of the electrical fields of the
p-polarised and s-polarised incident, reflected and/or transmitted electrical waves.
12 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
The reflectance Rp,s and the transmittance Tp,s are obtained from their definition as
Rp,s = |ˆrp,s|2
and Tp,s = |ˆtp,s|2
, (4.16)
where ˆrp and/or ˆrs and ˆtp and/or ˆts denote the reflection Fresnel coefficient of the system for the
p-polarised and/or s-polarised wave and the transmission Fresnel coefficient of the system for the
same waves, respectively.
Usually the measurements is performed in a natural light that is represented by randomly
polarised one. The detected total reflectance and/or transmittance that does not distinguish pand
s-polarisation state can be than expressed as
R =
|ˆrp|2
+ |ˆrs|2
2
and/or T =
|ˆtp|2
+ |ˆts|2
2
. (4.17)
In the basic configuration for reflectance measurements the light intensity reflected by the
sample Ism is compared with the light intensity reflected by a reference sample Ir. Usually the
angle of incidence is less than 15 ◦
and very often it is considered to be equal to 0 ◦
because of
equation simplifying. However, this assumption is not necessary in numerical methods of optical
constant determining. Then the reflectance of sample Rsm is given by
Rsm = Rn
Ism
Ir
(4.18)
where Rn is the reflectance of the normal. A disadvantage of this method consists in the necessary
knowledge of the reference sample reflectance. The main sources of systematic errors are listed
below:
1. bad stability of the light source,
2. change of the position of incidence light on the detector area,
3. bad stability of the optical parameters of the reference sample.
The polished silicon wafer was used as the reference sample for the measurements on films deposited
on silicon. When the measurements of films on glass substrates were carried out the quartz
plate was taken as the reference.
For the transmittance measurements it is not necessary to use any reference sample because
there is no technical problem to measure transmitted and incident light intensity in the same
experimental arrangement. Therefore the third systematic error from previous paragraph is excluded.
However, when the transmittance spectra of the substrate are very complicated it can be
useful to divide the sample transmittance by the substrate transmittance to exclude the substrate
transmittance features from the measurements. Besides the possible systematic errors mentioned
in the previous section there is another one caused by different properties of the bare substrate
and the substrate with the film.
4.3. INTERMEDIATE WAVELENGTHS (INCLUDING VISIBLE AND ULTRAVIOLET LIGHT)13
4.3.2 Principles of ellipsometry
Ellipsometry is a technique enabling us to measure the polarisation state of the polarised light
wave emerging from the system studied owing to the polarisation state of the polarised light wave
incident on this system. In practice the ellipsometric measurements are mostly employed for
specularly reflected or directly transmitted light waves by the system.
If the system is optically isotropic the measured quantities are related to
ρ =
ˆrp
ˆrs
or ρ =
ˆtp
ˆts
. (4.19)
The symbols ˆrp and/or ˆrs and ˆtp and/or ˆts denote the reflection Fresnel coefficient of the system
for the p-polarised and/or s-polarised wave and the transmission Fresnel coefficient of the system
for the same waves, respectively.
Further, one can write
ρ = tan Ψei∆
, (4.20)
where Ψ and ∆ represent the ellipsometric parameters of the system for the reflected or transmitted
light waves (Ψ and/or ∆ is called the azimuth and/or the phase change).
4.3.3 Optical quantities of the thin film system
In this section an interaction of the light waves with the thin film systems corresponding to the
ideal model defined in section ?? will be described by means of the optical quantities. In case
that the system is formed by the isotropic materials it is sufficient to determine the values of
the complex Fresnel coefficients ˆrp, ˆrs, ˆtp and ˆts. Using these coefficients one can determine the
ellipsometric quantities, i. e. Ψ and ∆ (see section 4.3.2) or the reflectance R and the transmittance
T (see section 4.3.1).
Thin films on semi-infinite substrates
By solving the Maxwell equations for boundary surface of two materials (see Figure ??) under
assumption that in all the material currents caused by external electrical fields and free electric
charges will not take place we get the following expressions for the Fresnel reflection and
transmission coefficients at the j-th boundary (rjp, rjs and tjp, tjs, respectively)
ˆrjp =
ˆnj cos ˆθj−1 − ˆnj−1 cos ˆθj
ˆnj cos ˆθj−1 + ˆnj−1 cos ˆθj
, (4.21)
ˆtjp =
2 ˆnj−1 cos ˆθj−1
ˆnj cos ˆθj−1 + ˆnj−1 cos ˆθj
, (4.22)
ˆrjs =
ˆnj−1 cos ˆθj−1 − ˆnj cos ˆθj
ˆnj−1 cos ˆθj−1 + ˆnj cos ˆθj
, (4.23)
ˆtjs =
2 ˆnj−1 cos ˆθj−1
ˆnj−1 cos ˆθj−1 + ˆnj cos ˆθj
, (4.24)
14 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
where the symbols ˆnj−1, ˆnj, ˆθj−1 and ˆθj denote the complex refractive indices of both the (j −
1)-th and j-th materials and the complex angles of incidence and refraction by the boundary,
respectively. The complex angles ˆθj−1 and ˆθj fulfil the Snell’s law, i. e.
ˆnj−1 sin ˆθj−1 = ˆnj sin ˆθj. (4.25)
Let us consider the system consisting of the ambient with real refractive index n0 (the subscript
0) and a single thin film (the subscript 1) on the semi-infinite substrate (the subscript 2). The
complex symbols ˆrp,s and ˆtp,s denoting the Fresnel coefficients for reflected and transmitted waves
by the system, respectively are given as follows [?]
ˆrp,s =
ˆr1p,s + ˆr2p,se−iˆx1
1 + ˆr1p,sˆr2p,se−iˆx1
(4.26)
ˆtp,s =
ˆt1p,sˆ2p,se−iˆx1/2
1 + ˆr1p,sˆr2p,se−iˆx1
. (4.27)
where ˆx1 is a phase difference gained in the thin film, ˆr1p,s and ˆt1p,s and/or ˆr2p,s and ˆt2p,s are the
Fresnel coefficients for the first and/or the second boundaries.
If the thin film is homogeneous the Fresnel coefficients for the ambient-film boundary (ˆr1p,s)
and the film-substrate boundary (ˆr2p,s) are given by [?]
ˆr1p =
ˆn1 cos φ0 − cos ˆφ1
ˆn1 cos φ0 + cos ˆφ1
(4.28)
ˆr1s =
cos φ0 − ˆn1 cos ˆφ1
cos φ0 + ˆn1 cos ˆφ1
(4.29)
ˆr2p =
ˆn1 cos ˆφ1 − ˆn2 cos ˆφ2
ˆn1 cos ˆφ1 + ˆn2 cos ˆφ2
(4.30)
ˆr2s =
ˆn2 cos ˆφ2 − ˆn1 cos ˆφ1
ˆn2 cos ˆφ2 + ˆn1 cos ˆφ1
(4.31)
and the phase difference ˆx1 is expressed as
ˆx1 =
4π
λ
ˆn1d1 cos ˆφ1 (4.32)
where ˆn1 = nf − ikf and ˆn2 = ns − iks denote the complex refractive indices of the film and the
substrate, respectively, φ0 is the angle of incidence on the film, ˆφ1 and ˆφ2 are the complex angles of
refraction in the film and in the substrate, respectively, d1 is the film thickness and λ wavelength.
Thin films on absorbing substrates
If the analyzed thin film system is prepared on the absorbing substrate the only ellipsometry in the
reflected light or the measurements of reflectance can be performed. The light reflected from the
back side of the substrate is not needed to be considered. Then the reflection Fresnel coefficients ˆrp,
ˆrs expressed by Eq. (4.26) can be used directly for the calculation of the ellipsometric parameters
according to Eqs. (4.19) and (4.20) or the reflectance given by Eq. (4.16).
A typical example of such application is represented by thin film systems on silicon substrates
characterized in the UV-VIS spectral region.
4.3. INTERMEDIATE WAVELENGTHS (INCLUDING VISIBLE AND ULTRAVIOLET LIGHT)15
Thin films on transparent substrate
If the substrate is transparent or slightly absorbing it is necessary to take into account the light
incoherently reflected from its back side. On the other hand we can get additional information
from the transmitted light and the light reflected from the side of substrate.
If the angles of incidence differ significantly from the perpendicular direction, i. e. in the ellipsometry
measurements, and substrate is sufficiently thick the light spot reflected from the thin
film system can be distinguished from the reflection on the back side of the substrate. Then the
relations (4.26) and (4.27) for Fresnel coefficients ˆrp, ˆrs, ˆtp, ˆts can be used and we can easily
calculate the ellipsometric parameters from Eqs. (4.19) and (4.20). If the substrate is not so thick
the theory of the ellipsometric measurements becomes seemingly complicated due to the present
depolarisation.
In the R/T method carried out at the near normal incidence we can further assume that
the angle of incidence is 0 ◦
. Then the terms p- and s- polarisation lose their meaning. The
reflectance measured from the side of the thin film system on thick transparent substrate R, the
reflectance from the side of the substrate R as well as the transmittance of the same system T
can be expressed by the addition of reflected and/or transmitted intensities. In this way we get
the relation between the theoretical values of the reflectances and the transmittance of a thin film
system on the semi-infinite substrate, R, R and T, respectively and the measured values R, R
and T as follows [?]
R = R +
T2
Rs0e−2αD
1 − R Rs0e−2αD
(4.33)
R = Rs0 +
(1 − Rs0)2
e−2αD
1 − R Rs0e−2αD
(4.34)
T =
TTs0e−αD
1 − R Rs0e−2αD
(4.35)
where
α =
4πks
λ
, (4.36)
Rs0 and/or Ts0 are the reflectance and/or transmittance at the boundary between the substrate
and surrounding medium (n0 = 1)
Rs0 =
ˆns − 1
ˆns + 1
2
(4.37)
Ts0 =
4ˆns
(ˆns + 1)2
, (4.38)
and ˆns and D are the complex refractive index (ˆns = ns − iks) and the thickness of the substrate,
respectively.
In some cases, e. g. for the characterisation of films on one side polished silicon substrates in the
infrared region, it is necessary to measure transmittance of thin films on transparent substrates
with a rough back side. As it is difficult to describe the problem theoretically it is better to treat
16 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
further the ratio Trel of the transmittance of the film-substrate system T and the transmittance
of the bare substrate Tsubs. The relative transmittance Trel can be expressed as
Trel =
T
Tsubs
=
T (1 − R2
s0) e−2αD
Ts0(1 − R Rs0) e−2αD
(4.39)
where Rs0, Ts0, R, R , T, α and D have the same meaning as before.
4.4 Long wavelengths (infrared)
Infračervená spektroskopie (IR - infrared) patří mezi základní analytické metody. IR oblastí
rozumíme oblast vlnočtů od 12500 cm−1
do 250 cm−1
. Oblast od 5000 cm−1
do 250 cm−1
bývá
označována jako střední IR oblast, od 12500 cm−1
do 5000 cm−1
jako blízká IR oblast a oblast od
250 cm−1
jako vzdálená IR oblast. Elektromagnetické IR záření je využíváno v infračervené spektroskopii
k určení chemické struktury, jelikož každá látka má své unikátní IR absorpční spektrum
a většina molekul je v IR oblasti spektra aktivní. Tato metoda bývá často doplněna Ramanovskou
spektroskopií, pomocí které lze detekovat molekuly neaktivní v IR oblasti a naopak.
V molekulách se valenční elektrony na vnějších slupkách účastní tvorby vazby mezi atomy
a jsou delokalizovány na molekulových orbitalech. Naproti tomu elektrony na vnitřních slupkách
atomů zůstávají v blízkosti svých jader a jejich energie není vznikem molekuly podstatně
ovlivněna. Každému molekulovému orbitalu odpovídá určitá hodnota energie. Tyto energie tvoří
soubor nespojitých elektronových energetických hladin. Energie molekuly je také ovlivňována
pohybem jednotlivých atomů, které neustále vibrují kolem svých rovnovážných poloh. Energie vibračního
pohybu je kvantována a jednotlivým vibračním stavům (vibracím s různou amplitudou)
přísluší jednotlivé vibrační energetické hladiny. Molekuly nacházející se v jednom určitém energetickém
elektronovém stavu mohou mít různou vibrační energii, takže každá elektronová hladina
je rozštěpena na určitý počet vibračních podhladin. Molekula jako celek vykonává také rotační
pohyby, jejich energie je také kvantována. To vede k rozštěpení každé vibrační hladiny na rotační
podhladiny. Pro rozdíly energií mezi jednotlivými rotačními, vibračními a elektronovými stavy
platí:
∆Erot ∆Evibr ∆Eel (4.40)
Molekula, která je ozářena spojitým spektrem IR záření může z hlediska kvantové mechaniky
takové kvantum světla absorbovat a přejít do excitovaného vibračního stavu. Spektrum zbývajícího
záření prokazuje absorpci na určité frekvenci, která odpovídá určitému vibračnímu a rotačnímu
stavům molekuly. Molekula přejde ze stavu s nižší energií E1 do excitovaného stavu s energií vyšší
E2. Dojde k pohlcení určité energie ∆E, které odpovídá elektromagnetické vlnění o dané frekvenci
ν, pro rozdíl energií platí:
∆E = E2 − E1 = hν (4.41)
kde h je Planckova konstanta. Valenční a vnitřní elektrony atomů ani vazebné či nevazebné
elektrony molekulových orbitalů látek nejsou v IR záření excitovány. Energie IR fotonů je ale
dostatečná ke změně vibračního či rotačního stavu molekuly.
4.4. LONG WAVELENGTHS (INFRARED) 17
Existuje mnoho způsobů vnitřních vibrací celé molekuly, rozlišujeme různé vibrační módy odpovídající
různým vazebným silám a úhlům atomů v molekule. Molekula tvořená N atomy má
3N stupňů volnosti. Zajímáme-li se čistě o vibrační módy, musí se odečíst tři stupně volnosti pro
translační pohyb a tři stupně volnosti pro rotaci okolo tří os pro nelineární molekuly. Zůstává
tedy 3N − 6 stupňů volnosti z nichž každý odpovídá jedné fundamentální vibraci. Pro lineární
molekulu se pro rotaci odečítají jen dva stupně volnosti, a proto má 3N − 5 fundamentálních
vibrací. Vibrační frekvence, které odpovídají absorbovanému rozdílu energií ∆E lze určit ze
znalosti hmotností kmitajících atomů a pevnosti vazby, která je spojuje. Pro dvouatomovou
molekulu s jedním vibračním stupněm volnosti platí Hookův zákon z klasické mechaniky:
ν =
1
2π
k
µ
(4.42)
kde k je silová konstanta daná typem vazby a µ je redukovaná hmotnost dána vztahem µ = m1m2
m1+m2
.
U lineární molekuly lze rozlišit vibrace, při kterých dochází ke změně mezijaderné vzdálenosti
tzv. valenční vibrace (stretching), mohou být symetrické i nesymetrické. Vibrace, při kterých se
mění velikost valenčních úhlů jsou tzv. deformační vibrace (bending). Vibrace lineární molekuly
jsou znázorněny na obrázku 4.6. U nelineární molekuly rozeznáváme rovněž symetrickou a nesymetrickou
valenční vibraci, deformačních vibrací rozlišujeme více. Některé z těchto vibrací jsou
znázorněny na obrázcích 4.7 a 4.8.
Figure 4.6: Vibrační pohyby molekuly oxidu uhličitého CO2: A - symetrická valenční vibrace,
B - asymetrická valenční vibrace, C,D - deformační vibrace (znaménka značí vibraci nad a pod
rovinou nákresny).
Figure 4.7: Vibrační pohyby molekuly skupiny CH2: A - symetrická valenční, B - asymetrická
valenční, C - nůžková, D - kolébavá (rocking), E - kývavá (wagging), F - kroutivá (twisting).
18 CHAPTER 4. INTERACTION OF PHOTONS WITH MATTER
Figure 4.8: Vibrační pohyby molekuly skupiny CH3: A - degenerovaná valenční, B - degenerovaná
deformační, C - kolébavá, D - symetrická valenční, E - symetrická deformační.
Srovnáním s vazbami a funkčními skupinami můžeme dokonce identifikovat různé sloučeniny.
Frekvence absorpčního pásu ve spektru se zvětšuje se silovou konstantou vazby a zmenšuje se s
rostoucí hmotností atomů.
Podle mezinárodní konvence se pozice absorpčních píků vyjadřují pomocí vlnočtu (cm−1
). Toto
vyjádření je výhodné, protože je přímo úměrné absorbované energii, díky tomu je možné výsledné
absorpční spektrum lehce interpretovat.
Figure 4.9: Základní schéma FTIR spektrometru - Michelsonův interferometr s pohyblivým zr-
cadlem
Propustnost T (transmitance) je dána poměrem intenzity světla I prošlého látkou po absorpci
k intenzitě záření dopadajícího na měřený vzorek I0:
4.5. INTERACTION OF MATTER WITH VERY LONG WAVELENGTHS PHOTONS (≥ 1MM) - INCLUD
T =
I
I0
(4.43)
V infračervené spektroskopii rozlišujeme skupinové frekvence a tzv. frekvence fingerprint
(otisku prstu). Skupinové frekvence jsou charakteristické jen pro určité malé skupiny atomů,
zatímco frekvence fingerprint vznikají díky vibraci molekuly jako celku. Skupinová frekvence je
vždy nalezena ve spektru molekuly obsahující tuto skupinu a vždy se objevuje ve stejném rozmezí
frekvencí. Důležitým lomítkem ve spektru je vlnočet 1500 cm−1
, protože pro větší vlnočty se u píků
s velkou intenzitou jedná o skupinovou frekvenci. Pro vlnočty menší než 1500 cm−1
se může jednat
o skupinovou frekvenci i o frekvenci fingerprint. Se snižující se frekvencí roste pravděpodobnost,
že se jedná o frekvenci fingerprint. Skupinová frekvence pod 1500 cm−1
bývá charakterizována
velkou intenzitou, šířkou, nezvyklou ostrostí či zdvojením.
Speciálním případem infračervené spektroskopie je Fourierovská infračervená spektroskopie.
Je to měřící technika založená na časové modulaci signálu, ze kterého se následně
Fourierovou transformací získá spektrální závislost. Časová modulace signálu je prováděna pomocí
Michelsonova interferometru s pohyblivým zrcadlem viz. obrázek 4.9. Během jednoho měření se
zrcadlo posouvá konstantní rychlostí, čímž vytváří časové zpožďování paprsku. Detektor zachycuje
interferogram, ve kterém je vzdálenost interferenčních proužků d funkcí frekvence záření ν
a rychlosti pohyblivého zrcadla u. Provedením zpětné Fourierovy transformace intenzity I(d) se
získá spektrální rozložení intenzity I(ν).
4.5 Interaction of matter with very long wavelengths photons
(≥ 1mm) - including radio and microwaves
All nucleons, that is neutrons and protons, composing any atomic nucleus, have the intrinsic
quantum property of spin. The overall spin of the nucleus is determined by the spin quantum
number I. Some nuclei have integral spins (e. g. I = 1, 2, 3 . . .), some have fractional spins (e.g.
I = 1/2, 3/2, 5/2....), and a few have no spin, I = 0 (e.g. 12C, 16O, 32S, ....). Isotopes of particular
interest and use to organic chemists are 1H, 13C, 19F and 31P, all of which have I = 1/2.
A non-zero spin is always associated with a non-zero magnetic moment µ
µ = γS (4.44)
where γ is the gyromagnetic ratio.
It is this magnetic moment that allows the observation of nuclear magnetic resonance (NMR)
absorption spectra caused by transitions between nuclear spin levels in the presence of external
magnetic field (spin states splitting).
http://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/Spectrpy/nmr/nmr1.htm
Electron spin resonance (ESR) or electron paramagnetic resonance (EPR) is a related technique
in which transitions between electronic spin levels are detected rather than nuclear ones. The basic
principles are similar but the instrumentation, data analysis, and detailed theory are significantly
different. Moreover, there is a much smaller number of molecules and materials with unpaired
electron spins that exhibit ESR absorption than those that have NMR absorption spectra. ESR
has much higher sensitivity than NMR does.