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PHYSICS – OSIJEK
INTRODUCTION TO SPECTROSCOPY
Dario Faj, Hrvoje Brkić
1. Structure of atoms - hystory review
2. Structure and stability of atoms and moleculas
3. Radioactivity. EM vawes
4. Interaction EM vawes and matter
5. Introduction to spectroscopy
1. Structure of atoms - hystory review
1.1.
Atomic structure
The atom consists of a central nucleus around which electrons rotate in fixed orbits. The
nucleus contains two kinds of particles, protons and neutrons, which together are called
nucleons. Both particles have nearly the same mass but the proton carries a positive electric
charge. Hence, the whole nucleus is positively charged. This charge is balanced by the
negative charges of the orbital electrons, so from outside, the atom appears electrically
neutral.
Particle Symbol Mass
(kg)
Energy Charge
(MeV)
---------------------------------------------------------Proton
p
1.672*10-27
938.2
+
Neutron
n
1.675*10 -27 939.2
0
Electron
e
0.911*10 -30 0.511
-
The different natural elements ranging from hydrogen to uranium are built of
increasing numbers of nucleons. The hydrogen nucleus has one proton and the uranium
nucleus has 92 protons and 146 neutrons. It is the number of protons and hence the number of
electrons that define the element and its chemical characteristics. The number of protons is
called the atomic number (Z) and the total number of nucleons is called the mass number of
the nucleus. All elements have different isotopes, which will have the same number of protons
but different numbers of neutrons and thus different mass numbers. For instance, for the
element carbon there exist eight different isotopes with mass numbers between 9 and 16. The
atomic number for carbon is 6 so the number of neutrons ranges from 3 to 10. It should be
stressed that an isotope of an element is not necessarily radioactive. Among the isotopes of
carbon both carbon-12 and carbon-13 are stable nuclides and the others are unstable and
hence are radioactive.
Figure. Identification of isotopes
Figure. Periodic table of elements.
1.2 Hystory review
In fifth-century BC Greeks, Leucippus of Miletus and Democritus of Abdera gave the
first atom theory we have any record of. Their theories were philosophical in origin. The basic
idea was that if you could look at matter on smaller and smaller scales (which they of course
couldn't) ultimately you would see individual atoms - objects that could not be divided further
(that was the definition of atom). They believed that atoms were in constant motion, and
always had been, at least in gases and liquids. Sometimes, however, as a result of their closelocking shapes, they joined in close-packed unions, forming materials such as rock or iron.
Basically, Democritus and his followers thought all natural phenomena could in principle be
understood in terms of interacting, usually moving, atoms.
After, a little progress in atomic theory was made over the next two thousand years,
mostly because Aristotle discredited it.
The idea rose up again in the Renaissance period. Galileo believed in atoms, and later
Newton gave a much more modern perspective on atoms and interatomic forces in the
seventeenth century (Opticks, Book 3, Part 1):
Quest. 31. Have not the small Particles of Bodies certain Powers, Virtues, or Forces,
by which they act at a distance, not only upon the Rays of Light for reflecting, refracting and
inflecting them, but also upon one another for producing a great Part of the Phenomena of
Nature? For it's well known, that Bodies act upon one another by the Attractions of Gravity,
Magnetism, and Electricity; and these Instances show the Tenor and Course of Nature, and
make it not improbable that there be more attractive Powers than these… . For we must learn
from the Phenomena of Nature what Bodies attract one another, and what are the Laws and
Properties of the Attraction, before we enquire the Cause by which the attraction is
perform'd. The Attractions of Gravity, Magnetism and Electricity, reach to very sensible
distances, and so have been observed by vulgar eyes, and there may be others which reach to
so small distances as hitherto escape Observation, and perhaps electrical Attraction may
reach to such small distances, even without being excited by friction.
In fact, although the forces binding atoms together in molecules cannot be properly
understood without quantum mechanics, many of these forces are "short range" electrical
forces - forces between bodies having overall electrical neutrality, but distorted charge
distributions. These forces could definitely be categorized as "electrical Attraction reaching to
small distances". Notice that Newton also leaves the door open for other short range forces,
which were finally discovered in the 1930's!
The first major step towards modern quantitative chemistry was taken by Lavoisier at
the end of the eighteenth century. He discovered Oxygen and set the definition of the element.
Before, there were no generally agreed on definitions of elements, principles or atoms,
although a century earlier Boyle had suggested that element be reserved for substances that
could not be further separated chemically.
In his Elements of Chemistry (1789) Lavoisier writes:
…if, by the term elements we mean to express those simple and indivisible atoms of
which matter is composed, it is extremely probable that we know nothing about them; but if
we apply the term elements, or principles of bodies, to express our idea of the last point which
analysis is capable of reaching, we must admit as elements all the substances into which we
are capable, by any means, to reduce bodies by decomposition. Not that we are entitled to
affirm that these substances we consider as simple may not be compounded of two, or even of
a greater number of principles; but since these principles cannot be separated, or rather since
we have not hitherto discovered the means of separating them, they act with regard to us as
simple substances, and we ought never to suppose them compounded until experiment and
observation have proved them to be so.
In sum, Lavoisier began the modern study of chemistry: he insisted on precise
terminology and on precise measurement, and suggested as part of the agenda the
classification of substances into elements and compounds. Once this program was truly
underway, the atomic interpretation soon appeared.
John Dalton (1766-1844) assumed that all atoms of an element were identical, and
atoms of one element could not be changed into atoms of another element "by any power we
can control". In 1811, the Italian physicist Amedeo Avogadro suggested that Dalton's picture
of atoms and molecules could be reconciled with Gay-Lussac's results on volumes if one
assumed that equal volumes of all gases, elements or compounds, contain equal numbers of
molecules. Of course, he had no idea what the number might be, but the hypothesis made
many predictions without knowing the number.
In the year 1800, 31 elements were known. By 1860, that number had almost doubled,
to 60, and the relative atomic weights, as well as many of the chemical properties, were
known. In particular, in analyzing molecule formation, a valuable emerging concept was that
of valence - the idea that each atom had a particular number of little hooks on it to attach itself
to similar hooks on other atoms.
A few years later, in 1872, the Russian physicist Mendeleev drew up his periodic
Table of the Elements.
First model of the atom was given by the J.J. Thomson in 1897. He experimentally
proved existence of electron and afterwords E. Rutherford discovered atomic nuclei and gave
solar system model of atom. In the late 19th century, it was found that the radiation from
hydrogen, as well as other atoms, was emitted at specific quantized frequencies. It was the
effort to explain this radiation that led to the first successful quantum theory of atomic
structure, developed by Niels Bohr in 1913. He developed his theory of the hydrogenic (oneelectron) atom from four postulates:
1. An electron in an atom moves in a circular orbit about the nucleus under the influence
of the Coulomb attraction between the electron and the nucleus, obeying the laws of
classical mechanics.
2. Instead of the infinity of orbits which would be possible in classical mechanics, it is
only possible for an electron to move in an orbit for which its orbital angular
momentum L is and integral multiple of .
3. Despite the fact that it is constantly accelerating, an electron moving in such an
allowed orbit does not radiate electromagnetic energy. Thus, its total energy E
remains constant.
4. Electromagnetic radiation is emitted if an electron, initially moving in an orbit of total
energy
, discontinuously changes its motion so that it moves in an orbit of total
energy
. The frequency of the emitted radiation is equal to the quantity
divided by h.
Later, A.Sommerfeld introduced aditional quantum numbers (ℓ, m) to explain some
experimental results.
Figure. Early models of atom.
The theory of atomic structure moved ahead with discovery of wave nature of
electrons (L. de Broglie). Loui de Broglie showed that atomic particles have wave
characteristics (diffraction) also and set the equation:

h
mv
Further E. Schrödinger gave the equation of vawe function of electron ψ(r):
2  
8 2 m 
ke2
E


h 
r

  0

•
Solutions are energetic stations of electrons described with quantum numbers n, ℓ, m:
•
n - main: discrete energy of electrons in electric field of nuclei; energetic shells: K, L,
M, N..; number of electrons in a shell is 2n2
•
l - orbital: orbitals: s,p,d..; can be 0, 1, ... n-1
•
m – magnetic: can be from – l to l
•
ms – spin can be +1/2 and -1/2
Figure. Model of atom today.
2. Structure and stability of atoms and moleculas
2.1.
Structure and stability of atoms
The electrons are bound to the nucleus by electrostatic forces. The binding energy of
an electron is defined as the work necessary to release the electron from its orbit. The binding
energy depends upon both the element and the position of the orbit. The different orbits or
shells are named K, L, M, N, etc. where the K-shell is the shell closest to the nucleus. The
electrostatic force is dependent on the distance between the charges which means that the
binding energy will decrease from the inner shell outwards. The electrostatic force will also
be dependent on the size of the charge, which means that the binding energy of electrons in a
specified shell is higher in an element with a high atomic number than in an element with a
low atomic number. For instance, the binding energy for an electron in the K-shell is 1.56
keV in aluminum (Z=13) and 88 keV in lead (Z=82).
In an energetic stable atom the shells are filled by electrons from the inner shell and
outwards. This structure can be changed by adding energy to the atom. The result will be
either ionization or excitation. Ionization means that the energy added is high enough to
release an orbital electron from the atom. Excitation means that an electron will be lifted to a
shell further out. This results in a vacancy in the shell originally occupied by the electron, a
vacancy which the atom tries to fill with an electron from an outer shell. When the vacancy is
filled the released energy is emitted as electromagnetic radiation (characteristic X-rays) or is
transferred to another electron which can then leave the atom (Auger electron).
The
electromagnetic radiation is called characteristic X-ray because its energy is characteristic for
the element. This means that it is possible to identify an element by its characteristic X-rays.
It is known that the nucleons, just like the electrons, can also occupy different energy
levels and that the nucleus can be present in a ground state or in an excited state. Just as in
the case of the atom, an excited state can be reached by adding energy to the nucleus. At
deexcitation the nucleus will emit the excess of energy as electromagnetic radiation. In this
case the electromagnetic radiation is called a gamma ray. The energy can also be transferred
to one of the electrons in the inner shells of the atom, which then have high enough energy to
leave the atom. This process is called internal conversion (IC). The energy of the gamma ray
will be the difference in energies between the different energy levels in the nucleus. In a
sense we can say that the energy of the gamma ray is characteristic for the nucleus and that
the nucleus can be identified by its gamma rays in the same way as the characteristic X- ray
identifies the atom.
2. 2. Quantum Numbers and electron configuration
The Bohr model was a one-dimensional model that used one quantum number to
describe the distribution of electrons in the atom. The only information that was important
was the size of the orbit, which was described by the n quantum number. Schrödinger's model
allowed the electron to occupy three-dimensional space. It therefore required three
coordinates, or three quantum numbers, to describe the orbitals in which electrons can be
found.
The three coordinates that come from Schrödinger's wave equations are the principal
(n), angular (l), and magnetic (m) quantum numbers. These quantum numbers describe the
size, shape, and orientation in space of the orbitals on an atom.
The principal quantum number (n) describes the size of the orbital. Orbitals for which
n = 2 are larger than those for which n = 1, for example. Because they have opposite electrical
charges, electrons are attracted to the nucleus of the atom. Energy must therefore be absorbed
to excite an electron from an orbital in which the electron is close to the nucleus (n = 1) into
an orbital in which it is further from the nucleus (n = 2). The principal quantum number
therefore indirectly describes the energy of an orbital.
The angular quantum number (l) describes the shape of the orbital. Orbitals have
shapes that are best described as spherical (l = 0), polar (l = 1), or cloverleaf (l = 2). They can
even take on more complex shapes as the value of the angular quantum number becomes
larger.
There is only one way in which a sphere (l = 0) can be oriented in space. Orbitals that
have polar (l = 1) or cloverleaf (l = 2) shapes, however, can point in different directions. We
therefore need a third quantum number, known as the magnetic quantum number (m), to
describe the orientation in space of a particular orbital. (It is called the magnetic quantum
number because the effect of different orientations of orbitals was first observed in the
presence of a magnetic field.)
Quantum Numbers and Electron Configurations give the distribution of electrons of an
atom or molecule (or otherphysical structure) in atomic or molecular orbitals. An energy is
associated with each electron configuration. Upon certain conditions, electrons are able to
move from one orbital to another by emission or absorption of a quantum of energy, in the
form of a electromagnetic wave (photon).
Figure. Example of electron configuration.
Rules Governing the Allowed Combinations of Quantum Numbers
•
The three quantum numbers (n, l, and m) that describe an orbital are integers:
0, 1, 2, 3, and so on.
•
The principal quantum number (n) cannot be zero. The allowed values of n are
therefore 1, 2, 3, 4, and so on.
•
The angular quantum number (l) can be any integer between 0 and n - 1. If n =
3, for example, l can be either 0, 1, or 2.
•
The magnetic quantum number (m) can be any integer between -l and +l. If l =
2, m can be either -2, -1, 0, +1, or +2.
Orbitals that have the same value of the principal quantum number form a shell.
Orbitals within a shell are divided into subshells that have the same value of the angular
quantum number. Chemists describe the shell and subshell in which an orbital belongs with a
two-character code such as 2p or 4f. The first character indicates the shell (n = 2 or n = 4).
The second character identifies the subshell. By convention, the following lowercase letters
are used to indicate different subshells.
s:l = 0;
p:l = 1;
d:l = 2;
f:l = 3
The third rule limiting allowed combinations of the n, l, and m quantum numbers has
an important consequence. It forces the number of subshells in a shell to be equal to the
principal quantum number for the shell. The n = 3 shell, for example, contains three subshells:
the 3s, 3p, and 3d orbitals.
Before we can use these orbitals we need to know the number of electrons that can
occupy an orbital and how they can be distinguished from one another. Experimental
evidence suggests that an orbital can hold no more than two electrons.
To distinguish between the two electrons in an orbital, we need a fourth quantum
number. This is called the spin quantum number (s) because electrons behave as if they were
spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital
is arbitrarily assigned an s quantum number of +1/2, the other is assigned an s quantum
number of -1/2. Thus, it takes three quantum numbers to define an orbital but four quantum
numbers to identify one of the electrons that can occupy the orbital.
2. 3. Structure and stability of molecules
Molecules are stabile assotiations of atoms with structure tending to minimal potential
energy. Molecular bonds are bonds of atoms within moleculas:
•
Energy of moleculas is always smaller than total energy of free atoms (principle of
minimum energy). Energy change can be beetwen molecular orbitals, vibration
stations and rotation stations.
Vibrations of twotwo-atomatom-moleculas
Vibration quantum number 
atom 1
E  E   1  E   
E
1 h 
1

   
   
2  2 
2
k 
2
M
h
atom 2
h 
1
1  h
   1     
2 
2
2  2
~ 0.1 eV
Energy difference between neighbour states is constant
3. Radioactivity. EM vawes.
3. 1. Radioactivity
Not all combinations of protons and neutrons in the nucleus are stable. For light elements
(mass number <20) stability is achieved if the number of protons and the number of neutrons
are about the same. Because of the positive charge of the protons there is a repelling
electrostatic force between them which is balanced by a different attractive short ranging
nuclear force between the nucleons.
In a heavy element the electrostatic force will be
considerable due to the large number of protons. Thus, to reach stability, the number of
neutrons must be relatively larger because the neutron increases the nuclear force without
increasing the electrostatic force. If the relation between protons and neutrons is altered from
the stable condition the equilibrium between the forces will be disturbed and the nucleus
becomes energetically unstable. Such an unstable nucleus is called a radioactive nucleus or a
radionuclide.
In the transformation, which we generally call radioactive decay, the nucleus loses its
excess of energy by fission or by emission of charged particles (alpha particles and beta
particles). The result of the radioactive decay is a new nucleus with a different atomic
number and in some cases a different mass number. The new nucleus will in many cases be
excited and at deexcitation the energy emitted as one or several gamma rays.
Fission is the process in which the unstable nucleus divides into two fragments of about equal
size. This process is only possible in heavy nuclides.
Radioactive decay by emission of an alpha () particle can also occur only in heavy
elements because the alpha particle itself is a comparatively heavy particle consisting of two
protons and two neutrons (a helium nucleus). In alpha decay the daughter nuclide will have
an atomic number that is two units less and a mass number that is four units less. Examples
of radionuclides with alpha decay are radium-226 and radon-222. The alpha particle energy
from different radionuclides is usually in the range of 4-8 MeV. The medical use of alphaemitting radionuclides is very limited.
Beta () decay can be one of three different kinds, -, + and electron capture (EC).
The - particle is an electron which is released in the transformation of a neutron to a proton.
In - decay the daughter nuclide will have an atomic number one unit greater than the mother
nuclide but the same mass number.
The + particle is an `electron' with a positive charge. It is called a positron and is
released in the transformation of a proton to a neutron. In + decay the daughter nuclide will
have an atomic number one unit less than the mother nuclide but the same mass number.
Electron capture is an alternative to + decay. In the process one of the electrons in
the inner shell of the atom is captured by the nucleus. No particle is emitted in the decay but
due to the vacancy in the inner shell of the atom, a characteristic X-ray will be emitted.
The transition energy released in beta-decay is divided between the beta particle and a
particle called a neutrino. This means that the kinetic energy of the beta particles from a
certain radionuclide will show a spectral distribution where the energies range from zero to a
maximum which equals the transition energy. The mean energy of the particles is roughly 1/3
of the maximum energy.
When considering a radioactive nucleus it can never be known when it is going to
decay. It can only be stated that there is a certain probability that it decays within a certain
period of time. For instance, a nucleus of iodine-131 has a probability of decaying of 0.086
(8.6 percent) per day. This is called the decay constant of the radionuclide which is different
for different radionuclides. The number of decaying nuclei per unit time in a radioactive
sample is called the activity of the radionuclide.
The unit of activity is 1 Becquerel (Bq) which is the number of decaying nuclei per
second.
One Becquerel is a very small activity. The natural body content of potassium-40 is
about 4000 Bq (4 kBq). In some nuclear medicine examinations the patients can receive 5001000 million Becquerel (500 MBq - 1 GBq) of technetium-99m.
The activity of a sample containing a certain radionuclide will continuously decrease
with a speed determined by the decay constant. Mathematically it is a monoexponential
decrease.
As an alternative to the decay constant the half-life of the radionuclide can be defined
as the time needed to reduce the activity by 50 percent.
3. 2. Electromagnetic vawes
During radioactive decay the unstable atom can release energy as a particle (α,β
radiation) or in a form of electromagnetic (EM) vawes (γ radiation).
EM spectra
Energy of EM vawe
E = h hc/
4. Interaction EM vawes and matter
The different types of radiation emitted in the radioactive decay are examples of
ionizing radiation, which means that the kinetic energy of the single particle or photon is high
enough to cause ionization, which is the process of removing an electron from the atom. The
theoretical energy limit is of the order of 100 eV. If the energy transferred is less it will not
cause ionization. Other types of ionizing radiation are charged particles from accelerators and
cosmic radiation. Also X-rays are ionizing.
Charged particles such as electrons, protons, alpha particles, etc. are called directly
ionizing radiation while neutrons and photons are called indirectly ionizing radiation. This
means that the ionization will take place in two steps, the first step being the release of a
charged particle such as an electron which is then directly ionizing.
When ionizing radiation passes through matter, it loses energy and will finally be
absorbed completely. The energy lost by the radiation is absorbed by the material, e.g., the
body. The processes involved are different for directly and indirectly ionizing radiation and
also different for heavy and light charged particles as well as for photons.
4.1. Charged particles
When a charged particle, such as an alpha particle, proton or electron, penetrates matter it will
lose energy by interaction with orbital electrons. The mode of interaction is called collision
although no true collision takes place between particles but rather collisions between the
electric fields surrounding the particles involved.
The energy transferred to the orbital
electron in the process can be high enough for the electron to leave the atom. In fact, the
kinetic energy of the ejected electron can be so high that it will act as an ionizing particle. It
is then called a delta-particle.
A heavy, charged particle such as the alpha particle, which has a mass about 7300
times the mass of the electron, will lose only a small fraction of its energy in each collision.
Due to its mass it will not change direction in the collision with a light electron. Changes in
direction can only happen in a collision with a heavy nucleus. On the one hand, such a
collision is very rare because the volume of the nucleus is so much less than the volume of the
atom, but on the other hand, due to its extensive electric field the heavy charged particle will
literally strike every atom it passes. In conclusion, this means that the path of an alpha
particle is straight and that the range in a material is quite small and well defined.
A light, charged particle (electron or positron) can lose up to half its energy in each
collision with an orbital electron. The possible high energy transfer in each collision means
that more energetic delta-rays can be produced than in the case of heavy charged particles. In
the collision, the incident particle will also change its direction which means that its path will
be irregular and curved. Due to its small size, an electron can also pass several atoms without
losing any energy or only a small fraction of its energy. Therefore, the electron has the ability
to penetrate deeper into matter than a heavy charged particle. Because of the irregular path
the range in matter will not be so well defined as that of a heavy particle.
If the light particle is a positron it will annihilate when it stops. It will recombine with
an electron and their masses will be transformed to energy and emitted as two photons in
opposite directions, each having an energy of 511 keV. This annihilation radiation is the one
used in positron emission tomography (PET), which means that the only radionuclides used in
PET-studies are those decaying by +.
When an electron comes close to the nucleus it will change direction due to the
electrostatic forces acting upon it. In each such event of deflection the energy lost by the
particle can be emitted as electromagnetic radiation called bremsstrahlung. The energy of the
emitted photon can be between zero and the whole kinetic energy of the incident electron
depending on the distance between the electron and the nucleus. The energy distribution of
bremsstrahlung emitted in the process will continuously decrease from zero energy to the
incident electron energy.
The energy loss of a charged particle passing through matter is described by the mass
stopping power which is the energy loss per unit length divided by the density of the absorber.
The unit is thus MeV·cm2/g. Its value depends on the type of particle and the particle energy .
The stopping power due to collisions is called non restricted linear energy transfer, LET.
Heavy charged particles are usually called high LET radiation and light charged particles are
called low LET radiation.
4. 2. Photons
For photons (X-rays, gamma rays) there are three main types of interaction with
matter: the photoelectric effect, the Compton process and pair production.
In all three
processes directly ionizing electrons or positrons are released or created.
The photoelectric effect is that in which the incident photon transfers all its energy to
a tightly bound orbital electron in one of the inner shells of the atom. This electron will leave
the atom carrying kinetic energy which equals the photon energy minus the binding energy of
the electron. The vacancy in the electron shell is filled and characteristic radiation is emitted
as this occurs.
In the Compton process, the incident photon collides with a loosely bound electron in
the outer shell of the atom. In the collision, the incident photon transfers some of its energy to
the electron which then leaves the atom. The photon changes its direction of movement after
the collision so the result of the interaction will be a scattered photon with reduced energy and
a recoil electron.
Pair production only occurs if the incident photon has a very high energy. When the
photon encounters the strong field around the nucleus it disappears and its energy transforms
into an electron-positron pair. Since the sum of their masses is equivalent to an energy of
1022 keV, pair production is limited to photons whose energies equal or exceed 1022 keV.
Theoretically, a photon can penetrate an absorber with no interactions at all. We can
only define a probability that a photon will interact by some of the three described processes.
This probability per unit length is called the linear attenuation coefficient. It is commonly
divided by the density of the absorber. This gives the mass attenuation coefficient which has
the unit cm2/g. The total mass attenuation coefficient is the sum of the single coefficients for
each of the three modes of interactions. The dominating process depends on the energy of the
photon and the atomic number of the absorber. Note that for human soft tissues with a mean
atomic number of 7.8, the dominating interaction process is the Compton process for all
photon energies used in medical applications (25 keV - 25 MeV).
Thickness of an absorber necessary to reduce the transmission of radiation to 50 percent
(HVL).
Radiation quality
HVL (mm)
Concrete
Lead
50 kV
4.3
0.06
100 kV
10.6
0.27
200 kV
25
0.52
500 kV
36
3.6
1 MV
44
7.9
2 MV
64
12.5
5 MV
96
16.5
10 MV
119
16.6
20 MV
137
16.3
4.3. Radiation quantities and units
Absorbed dose
The fundamental dosimetric quantity D, defined as:
D=
d
dm
where d is the mean energy imparted by ionizing radiation to matter in a volume element and
dm is the mass of matter in the volume element. The energy can be averaged over any defined
volume, the average dose being equal to the total energy imparted in the volume divided by
the mass in the volume. The SI unit of absorbed dose is the joule per kilogram (J.kg-1), termed
the gray (Gy).
Equivalent dose
The quantity H, defined as:
H = DT . WR
where DT is the absorbed dose delivered by radiation type R averaged over a tissue or organ T
and WR is the radiation weighting factor for radiation type R.
When the radiation field is composed of different radiation types with different
values of WR, the equivalent dose is:
H = WR . DT
R
The unit of equivalent dose is J.kg-1, termed the sievert (Sv).
Effective dose
The quantity E, defined as a summation of the tissue equivalent doses, each multiplied by the
appropriate tissue weighting factor:
E = WT . HT
T
where HT is the equivalent dose in tissue T and WT is the tissue weighting factor for tissue T.
From the definition of equivalent dose, it follows that:
E=
WT  WR  DT,R = WR  WT  DT,R
T
R
R
T
where WR is the radiation weighting factor for radiation R and DT,R the average absorbed dose
in the organ or tissue T. The unit of effective dose is J.kg-1, termed the sievert (Sv).
4. 3. Radiation detectors
As a detector of ionizing radiation any substance may be used that produces a measurable
signal as a result of energy deposition in the material. The signal can be electrical charge,
light, chemically changed molecules, etc. Some materials will emit the signal during the
exposure to ionizing radiation, others can retain the changes and be measured a long time
after the exposure. According to their uses detectors for ionizing radiation are divided into
counters, dosimeters and spectrometers.
A counter is a device that will only count the number of particles and photons
interacting with the detector. It will not provide information about the type and energy of the
radiation. This type of detector is generally used as a survey meter to determine if radiation is
present or not and to check for contamination of radionuclides.
A dosimeter is a device used to measure absorbed dose and absorbed dose rate, so the
signal from such a detector must be proportional to the absorbed energy in the detector over a
period of time. Dosimeters are important devices in medical applications and in radiation
protection. Their uses range from measuring the radiation output from therapy machines to
personnel monitoring.
In a spectrometer the signal is proportional to the energy of the photon or particle
interacting with the detector. This property is used in many applications in nuclear medicine.
The gamma camera has spectrometric properties used to reduce the influence of scattered
radiation on the image. The energy of the scattered photon is lower than that of the primary
photon and thus it can be sorted out by special electronics. A spectrometer can also be used to
identify radionuclides from the energy of the gamma-rays.
Gas-filled detectors
In the ionization process an ion-pair will be produced consisting of a negative electron and a
positive atom (ion). If an electric field is applied between two electrodes then the electrons
will move towards the positive electrode and the positive ions towards the negative electrode.
A current will appear in the outer circuit which is proportional to the number of ion pairs
produced per second. Depending on the strength of the electric field (high voltage) and the
design of the detector the properties of the gas detector will be different.
Luminescence detectors
Upon deexcitation some organic molecules and inorganic crystals can emit visible
light. This phenomenon is called radioluminescence. This property is used in scintillations
detectors which are commonly used in medical applications. The number of light photons
emitted upon absorption of a gamma-ray or a charged particle depends on the energy
transferred to the detector. The light photons will be converted into an electrical signal in a
device called a photomultiplier. The magnitude of this signal will depend on the number of
light photons and thus the energy transferred to the detector by the photon or the particle. The
size of the signal is electronically analyzed (pulse height analyzer).
5. Introduction to spectroscopy
Spectroscopy studies interaction between matter and radiated energy. Basic knowledge
needed to understand the spectroscopic phenomena is in atomic structure, electron states,
spins, and nucleus structure. First spectroscopic measurements were done in visible region of
electromagnetic spectra, but later on it expanded to ultraviolet and infrared regions. One of
the first instruments in spectroscopy was optical prism because it’s famous “rainbow effect”.
Important property of spectroscopy is that it has no chemical effects. The result of
spectroscopy measurements is represented by spectra of atom or molecule, under
investigation. A spectrum of atoms and molecules consists of series of lines, which represents
energy states of atom or molecule.
5. 1. Classification of spectroscopy
Usual classification of spectroscopy is by the nature of the interaction between the energy
and the material. There we have:
 Emission spectroscopy,
 Absorption spectroscopy,
 Interferention spectroscopy
Emission spectroscopy
Light is electromagnetic wave like any other electromagnetic wave, and that means it consists
of different wavelengths. When the substances under investigation are heated up, they emit
energy in form of light (E=hν). With the help of some kind of spectroscope (which will be
described later) this light can be distinguished and used in analysis. Because we study
emitted lines, this part of spectroscopy is called emission spectroscopy. Usual spectrum
observed in this kind of spectroscopy is shown in the figure.
Absorption spectroscopy
In absorption spectroscopy we measure the absorption of radiation, caused by its
interaction with matter. Absorption of electromagnetic radiation can cause several effects in
the atom, so the lines from the absorption spectroscopy are called by these effects, or
vibration lines, translation lines, and their combination. The energy that causes the quantum
mechanical change determines the frequency of the absorption line, but the frequency can be
shifted by other types of interaction, like electric and magnetic fields. Lines received from
absorption spectroscopy look like Diracks delta functions, but often there are widened by the
instrument function, and more like Gaussians or Lorenzians.
Interferention
Interferention is more commonly used in some other parts of science, like precise
measurement, optical testing, or measuring index of refraction, but it can also be used as a
very good tool in Fourier Transform Spectroscopy. Devices that produce interferention are
usually Michleson interferometer, and Faby-Perot interferometer, and both of them
superimpose several light beams with each other (two for Michelson and tens, or hundreds for
Faby Perot)
Another classification of spectroscopy is according to the electromagnetic wave energy. There
we have:
 Radiofrequent
 Microwave
 Infrared
 Visible and ultraviolet
 X-ray
Radiofrequent spectroscopy deals with magnetic spin states. There are two most common
methods in this kind of spectroscopy NMR(Nuclear Magnetic Resonance) and ESR(Electron
Spin Resonance, also known as EPR – Electron Paramagnetic Resonance). As the name
states, NMR deals with spins of atomic nuclei. Difference between spin states is shown when
material of interest is under strong external magnetic field (in order of 1 T). Relaxation times
are measured as a response to different electromagnetic pulses. EPR acts similar but it deals
with spins of electron, and weaker external fields are used.
Microwave spectroscopy is a very precise tool for the determination of molecular
structure in gas phase molecules. It can be used to determine barriers of internal rotations such
as are rotations of the CH3 group. This kind of spectroscopy has mayor role in determining
weak interactions such as are hydrogen bonds, and van der Waals interactions. It is also very
useful in astronomy when determining structure of interstellar media.
Infrared spectroscopy as the name states deals in the infrared region of electromagnetic
spectrum. It is usually connected with absorption spectroscopy. The infrared spectrum of a
sample is recorded by passing a beam of infrared light through the sample. When the
frequency of the infrared radiation is the same as the vibrational frequency of a bond,
absorption occurs, and the absorption lines are recorded.
Ultraviolet and visible spectroscopy refers to the wavelengths in ultraviolet and visible
regions of electromagnetic spectra (10 -700 nm). Measurements in this part of spectroscopy
are carried out in solutions, but they can be done with gases and solids also. Diagram of a
single beam ultraviolet-visual spectrometer is shown in a figure.
X - rays spectroscopy is divided in two main categories: Energy-Dispersive X-ray
Spectroscopy (EDS) and Wavelength Dispersive X-ray Spectroscopy (WDS). In an energydispersive X-ray spectrometer we measure the energy of incoming photons by semiconductor
detector. In a wavelength dispersive X-ray spectrometer the single crystal diffracts the
photons which are collected by a detector. Without any motion there will be just one
wavelength detected.
5. 2. Instruments in spectroscopy
The schematic layout of a prism or grating spectrometer is shown in figure. Light from
the slit is collimated by the lens L1 and dispersed by prism or grating and finally focused by
the lens L2 in the plane P, where the spectral lines can be thought of as images of slit in light
of different wavelengths. Although this schematic arrangement is normally adhered for prism
instruments, the collimating and camera lenses are usually replaced by concave mirrors, to
avoid problems of absorption and aberration problems.
Prism instruments
Conventional prism spectrograph is shown in the image below. Light from the slit S is
collimated by the lens L1, refracted by the prism, and focused by the lens L2 in image plane
P, which can be photographical plate, detector, or detector array. This setup is common for
small and medium sized instruments, for which is not important to be monocromators. Two
lenses have usually same focal length giving unit magnification, with usage of 60°prism.
Modification of this basic device is possible depending on measurement purpose.
Deviation and dispersion
In the figure below there is throughput of ray through central part of prism with angle
α.It can be seen from the geometry that deviation θ is given with θ=d1+d2 where d1=i1-r1 and
d2=i2-r2, so it follows θ=i1+i2-α. Prism instruments are usually used near the minimum
deviation position, for which can be shown that it passes prism simetrically. i1 is than equal to
i2, and ray passes through prism parallel with base, so it follows
θ=2i-α, and α=2r.
Combining of this two equations with Snell law, sin i =n sin r, for relative refraction index it
follows:
  
 
sin i  sin 
  n sin  
 2 
2
From this basic equation we can get expressions for angular dispersion:
d
2 sin(  / 2)
dn

1
/
2
dn 1  n2 sin 2  / 2 d


d
1
dn

1/ 2
2
d 1  (n / 2)
d
For standard prism (60°), this equation is simplified:


dn/dλ is a dispersion, which depends on prism material, index of refraction, and wavelength.
Rt
Resolution of prism gratings is given by expression:
dn
d ,(where t is length of prism base)
Typical value for dispersion dn/dλ in visible/ultraviolet region is 3*10-4 nm-1. If we take 100
mm as a practical upper limit for prism base, we get to 30 000 as rational limit for prism
spectrometer resolution.
Grating instruments
Diffraction grating consists from a large number of narrow, very close, equidistant
slits on plate, or concave grating. They can be setup on plane, or made by engraving, or
holographic. Collimated light from array of very close slits is transmitted or reflected by the
periodical structures formed on grooves, which act as coherent sources. For any given
wavelength secondary waves interfere constructively at certain angles and destructively under
others. Maximum of intensity is shown in the focal plane, under angles that correspond
interferention maxima. Concave gratings differ from plane gratings, in fact that grooves are
engraved in concave mirror, so grating acts as its own collimator.
For
the
light
of
wavelength
λ
condition
for
constructive
interferention
is:
m  l  d (sin   sin  ' )
d '
m

Angular dispersion for grating is observed by differentiating upper equation: d d cos  '
R
For resolution we have three equivalent expressions:
L

 Nm  D
d '
d
Maximum resolving power for grating is 500 000.
Interferometric spectrometers
Both Micleson and Faby-Perot interferometer have division of amplitude, it means that
interfering rays are produced by splitting of a input ray on partially reflecting surface. At
Faby-Perot interferometer we have multiple reflections, and Michelson uses only one partially
reflecting mirror, to get rays which are reflected back by mirrors. Scheme of Michelson
interferometer is shown in figure.
Light from extensive source behind entering aperture A is collimated by lens L1 before
entering on a beam splitter B. Lens L2 makes image of aperture A as A' in plane P (Lenses are
usually replaced by concave gratings to avoid chromatic problems). Usually two mirrors M1
and M2 setup that image M1' from M1 in partially reflecting mirror is parallel with M2 in
distance t between them.
Michelson interferometer has its extensive usage in Fourier Transform Spectroscopy
(FTS). For monocromator input FTS gives interferogram that is clear sine function. By
moving one of the mirrors, all the wavelengths are recorded, and interferogram is made from
superposition of all lines from all wavelengths. This kind of measurement imposes
characteristic modulation on all wavelengths, so we need to make inverse Fourier Transform
on interferogram to get real spectral lines. Pair of Fourier Transforms is:

I ( x) 
 B( ) cos 2xd (W )

B( ) 

 I ( x) cos 2xdx (W / cm
1
)

Where B(σ) is modificated function, and I(x) is interferogram
6. References
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2. Eisberg, R. M., and Resnick, R. Quantum physics of atoms, molecules, solids, nuclei, and
particles. John Wiley & Sons, Inc., New York, second edition, 1985.
3. Feynman, R. P., Leighton, R. B., and Sands, M. The Feynman Lectures on Physics.
Addison-Wesley, Reading, Massachusetts, 1965.
4. Griffiths, D. J. Introduction to Elementary Particles. Harper & Row Publishers, Inc., New
York, 1987.
5. Shu, F. H. The Physics of Astrophysics. Volume I: Radiation. University Science Books,
Mill Valley, California, 1991.
6. INTERNATIONAL COMMISSION ON RADIOLOGICAL PROTECTION. 1990
Recommendations of the International Commission on Radiological Protection, ICRP
Publication No. 60. Oxford, Pergamon Press, 1991 (Annals of the ICRP 21, 1-3).
7. KNOLL GF. Radiation detection and measurements 3rd edition. John Wiley and Sons,
1999
8. SORENSEN JA, PHELPS ME. Physics in Nuclear Medicine. Grune & Stratton, 1987.
9. Ervin B. Podgoršak. Radiation oncology physics: a handbook for teachers and students.
IAEA, 2005.