Quantum, atomic and
nuclear physics
Daniel Vlk
Institute of Biophysics
Introduction
The principles and applications of modern physics, which
encompasses a lot of quantum, atomic and nuclear physics, are
extremely important for biomedical sciences.
Almost all imaging methods used in medicine are based on this
part of physics.
Basics
Before we start to describe the processes occurring on the atomic and
subatomic level of matter, it is necessary to define a “new” unit of
energy, which is very often used when speaking about individual
particles and their small assemblies – molecules. The main SI unit of
energy – the joule – is too big for that purpose. Therefore, the small
amounts of energy carried by individual elementary particles, atoms,
chemical bonds etc. are measured in units called electron volts:
1 electron volt [eV]=1.602x10-19 J.
1 electron volt is the energy obtained by one electron when accelerated
in an electrostatic field by 1 V potential difference. The potential energy
of charged particle E=W=eU changes into its kinetic energy during the
acceleration.
Basics
1 mole is such an amount of substance which contains such
number of particles as present in a sample of the nuclide carbon-
12 which mass is equal to 0.012 kg.
Avogadro‘s constant: NA=6.022x1023 mol-1
Is the number of particles in 1 mole of any substance.
Basics
Atomic mass constant is 1/12 of the mass of carbon-12
nuclide atom: mu=1.66x10-27 kg.
Molar mass (Mm) is the mass of 1 mole of a substance. It is
equal to the sum of atomic masses of the atoms forming
one molecule expressed in grams.
Molar volume (Vm) is the volume of 1 mole of a substance:
Vm=Mm/ρ
where ρ is density of the substance.
Basics
Subatomic particles (e.g. electrons, protons, neutrons) are not only
small rigid balls, but also possess some wave properties. Their
wave motion (oscillation) manifest itself as de Broglie waves, with
wavelength given by formula:
λ=h/p=h/mv,
where λ is the wavelengths of the de Broglie waves [m], h is the
Planck‘s constant and p is the momentum of the particle [kg.m.s-1].
Evidences for the wave properties of the particles are the
diffraction and interference phenomena.
Basics
Each atom consist of a nucleus and electron shells.
The internal energy of atoms is quantized. It means that only
discrete values of energy are posibble or achievable.
Basics - Electron shells
The wave function is a complex mathematical function. Is a solution of the
Schrödinger equation and determining the possible states of electrons located in
electromagnetic field.
Important is to know that the absolute squared value of wave function expresses the
probability that a particle exists in a certain part of space (in the close victinity of the
atomic nucleus) called „orbital“ (electron cloud theory…)
Electron shells
Quantum mechanics model of the hydrogen atom:
Quantum numbers:
1. Principal quantum number n = 1, 2, 3, …
It determines mainly energy of electrons.
The respective electron shells are marked by capital letters K
(n=1), L (n=2), M (n=3), …
For the simplest atom hydrogen we can write:
where E1 is the energy of the ground state of the electron (n = 1).
Note that the value of energy is negative because some work
must be done to liberate the electron from the shell.
2
1
n
E
En −=
Electron shells
Quantum mechanics model of the hydrogen atom:
Quantum numbers:
2. Orbital momentum quantum number.
It can take value 0, 1, 2, 3 ... n – 1 for each principal quantum
number n.
The respective orbitals, i.e. the defined parts of space in which
the electrons can be found with highest probability, are marked
by letters s (l = 0), p (l = 1), d (l = 2), f (l = 3)...
These numbers determine the angular momentum of the
electron, i.e. the shape of the orbital.
Electron shells
Quantum mechanics model of the hydrogen atom:
Quantum numbers:
3. Magnetic quantum number m.
This can take values 0, ±1, ±2, and ±3 ... ±l for each orbital
quantum number l.
These numbers determine the allowed spatial orientations of
electron orbitals in an external magnetic field.
Electron shells
Quantum mechanics model of the hydrogen atom:
Quantum numbers:
4. Spin quantum number s.
It can take only two values: ±½.
This number quantizes the intrinsic angular momentum of an
electron, which can be imagined as being the rotation of an
electron (say, clockwise and anti-clockwise).
Electron shells
Pauli’s exclusion principle states:
No two electrons in an atom can exist with the same set of four
quantum number values.
In atoms with a higher number of electrons, the energy of the
electron depends also partly on the quantum numbers l, m, and s.
Excitation is a process in which an atom absorbs energy.
Deexcitation is a process in which an atom emits energy in the
form of a photon.
Spectral analysis
Various physical bodies can absorb or emit ultraviolet, visible or
infrared light.
This phenomenon can be explained on the basis of excitation
and deexcitation processes.
Light is emitted mainly from hot bodies in which the atoms (or
molecules) become repeatedly excited and then returning to the
ground energetic state by deexcitation. Light absorption can be
considered an important excitation process.
The dependence of intensity of emitted or absorbed light on light
wavelength is called a spectral curve.
Spectral analysis
To obtain such a curve, it is necessary to disperse (separate)
polychromatic light into its components which differ in their
wavelength.
It can be done e.g. by prisms because the index of refraction
depends on wavelength so that light of different wavelength is
refracted through various angles. Another possibility is the use of
diffraction gratings.
The instruments used to study absorbed and emitted light are
called spectrographs (these serve only for visualization of
absorption or emission spectra on a screen) or
spectrophotometers (these serve for objective measurement of
light absorption or emission).
Spectral analysis
The emission spectrum is obtained after passage of the light
beam emitted from the substance under study through the prism
or grating.
The absorption spectrum can be seen and measured after
passage of polychromatic light through a solution of the
substance under study followed by spectral analysis of the
transmitted light.
In gases formed by atoms or very simple molecules, only light of
certain (discrete) values of wavelength is absorbed or emitted. In
such a case, we can see only bright lines on a black background
(in emission spectra) or dark lines on a background of spectral
colours (in absorption spectra) – we speak about line spectra in
both cases.
Spectral analysis
Many molecules are so sensitive to high temperatures that they
cannot emit light due to heating. Complex molecules dissolved in
solution absorb light in a relatively wide interval of wavelength
values. Thus, we can speak about band spectra. The
continuous spectra of emitted light are found in hot solid
bodies.
The existence of line spectra can be simply explained taking into
account the discrete values of electron energies in atomic
electron shells (they are given by the principal quantum numbers,
n = 1, 2, 3 ...).
Therefore, the changes of electron energy and energy values of
emitted photons can also have only discrete values.
•Chemiluminescence, the emission of light as a result
of a chemical reaction
•Mechanoluminescence, a result of a mechanical
action on a solid
•Photoluminescence, a result of absorption of photons
• Fluorescence, lifetime: nanoseconds
• Phosphorescence, lifetime: milliseconds to hours
Luminiscence
Photoelectric effect
We can observe this with electromagnetic radiation the energy of
which is high enough to liberate electrons from matter.
We already know that electromagnetic radiation propagates in
the form of small quanta of energy, which are called photons.
The energy of one photon is given by Planck’s formula:
E = hf,
where h is the Planck’s constant and f is the frequency.
Photoelectric effect
The photoelectric effect is an energy transformation which is
described by Einstein formula:
h.f = Wb + ½ mv2
where hf is the energy of the incident photon, Wb is the binding
energy of the electron (this energy must be delivered to eject an
electron from an atom), m is the electron mass, v is its velocity,
and the term ½mv2 is the kinetic energy of the ejected electron.
When the ejected electron has no or only very small kinetic
energy, i.e. hf0 = Wb, f0 is called the threshold (or limit) frequency.
When f >f0 the electrons are liberated from atoms.
Photoelectric effect
Consequence:
The energy carried by the ejected electrons does not depend on
the intensity of the incident radiation (light) but on the energy (i.e.
on the frequency and/or wavelength) of individual incident
photons.
Compton scatter (effect)
This is a phenomenon related to the photoelectric effect,
occurring at higher energies of incident photons. The photon
energy is not fully transformed into the kinetic energy of ejected
electron and the energy necessary for ejection of the electron
(binding energy Wb) so that a secondary photon of lower energy
appears. Moreover, this binding energy is (in comparison with the
energy of the secondary photon and the kinetic energy of
electron) is so small that it can be neglected. The secondary
photon does not travel in the same direction as the primary
photon – it is scattered. So we can write:
hf = (Wb) + 1/2mv2 + hf’
where hf’ is the energy of the secondary photon (f’ < f).
Compton scatter
Momentum of a photon
A photon is a discrete amount of energy connected with
oscillations of an electromagnetic field. Photons behave
sometimes as particles with zero rest mass. They have even
their own momentum which is given by the following equations:
This photon momentum manifests clearly photons obey the Law
of momentum conservation.






====
f
ch
c
hf
c
E
p λ
λ