Theory

 

1. Nuclids
2. Nuclear decay and kinds of radiation
3. Law of distance
4. Law of absorption
5. Geiger-Müller detector
6. Statistics and error of measurements
7. Rules of protection against radiation
8. Lorentz force

 

1. Nuclids

The atom consists out of a core (nucleus) and an outer electron cloud. Each nucleus contains positively charged protons and electrically neutral neutrons. In the following, index A describes the amount of all nucleons (protons and neutrons) in one nucleus, that is the atomic mass number. Different elements are different with respect to the number of their positively charged nucleons (protons) and respective number of their shell electrons. This number is named atomic number Z in the periodic system of elements.

Atoms (nuclei) with different atomic mass number A but the same atomic number Z are called isotopes (isotopic nuclei or nuclides). Each element, which exists in nature, is in general a mixture of different isotopes. The agreed abbreviation for an element in the periodic system is

with A atomic mass number A, atomic number Z, element symbol X and number N = A - Z of neutrons.

 

 

2. Nuclear decay and kinds of radiation

The quality of certain radio-nuclei, to transform spontaneously in a more stable nucleus via radiation, is called radioactive decay. This radioactive radiation is of different kinds, which are called historically α-, β- and γ-radiation. In addition, emission of neutrons and electron annihilation radiation must be stressed, but which are not treated here in this remote experiment. Each radioactive substance has its specific lifetime, described in that model by its half life, that is the time interval in which the activity of a radioactive material (i.e. number of decays per second, called Bequerel (Bq)) is reduced to half of the initial value.

 

a) α-radiation

α-radiation consists out of Helium nuclei, which a decaying parent nucleus X is emitting. The parent nucleus is transformed into a daughter nucleus Y. This process will be described by the following decay equation:

For example the decay equation for the Americium-241 used in the experiment is

.

Each α-particle of one kind of α-radiation possesses the same spatial range due to their discrete kinetic energy. For example, typical energies of α-particles are in the range of ca. 4 - 9 MeV, their range in air then is accordingly about 3 - 8 cm. α-particles as positively charged Helium nuclei can be deflected by either electric or magnetic fields.

According to the energy level diagram Americium-241 (parent nucleus) decays into Neptunium-237 (daughter nucleus) while emitting α- and γ-radiation, whereas the α-radiation is classified by 5 different kinetic energies:

Fig. 1: Energy-level diagram of Am-241.

 

This nucleus can decay under different decay channels, i.e. different possibilities for an instable nucleus to transform. For example, via a channel which is called α3). In that process an α-particle with kinetic energy of 5.486 MeV is emitted. Since the daughter nucleus is an excited state the process is followed by an emission of γ-quanta (γ1 or γ4). The probability for this decay channel is 85.2 %. At the end of these processes the stable nucleus Neptunium-237 is formed. Americium has a half-life of 433 years.

 

b) β-radiation

One differentiates between β-- and β+-radiation. In the first cases the decaying nucleus X is emitting an electron e- and an antineutrino νe-quer, in the second case a positron e+ and a neutrino νe. Here we will discuss further on only the relevant β--decay. The standard description for the β--decay is

.

For the Sr-90 (radioactive source used in this RCL) here the equation of decay reads:

Here the symbol Y stays for Yttrium element. The emitted electrons have a continuous distribution of kinetic energies, with a maximum kinetic energy, specific for the decaying nucleus and ranging somewhat between 0.5 - 2 MeV. Most of the electrons do not reach the maximum kinetic energy. On the other hand also in microphysics conservation of energy and momentum must be fulfilled, therefore, Wolfgang Pauli (~1930) presumed, that a third particle besides nucleus X, Y and electron must be involved in this process. This third particle was found experimentally later. This particle antineutrino νe-quer carries the missing kinetic energy i.e. the difference between maximum kinetic energy and kinetic energy of electrons. The experimental verification was very difficult, because the probability of interaction with other particles is very small, since neutrinos carry no charge and possess a very small mass, if at all. In air the range of β--radiation is about 1.5 to 8.5 m depending on their kinetic energy. Because of the electric charge of the electrons these particles can be deflected by electric and magnetic fields.

Fig. 2: Energy-level diagram of Sr-90.

 

The parent nucleus is decaying into Yttrium, with a half-life of 64.1 hours, emitting electrons with a maximum kinetic energy of 0.546 MeV. This intermediate nucleus can further on decay by two different decay channels into Zirconium which is a stable nuclide. The probability that the Yttrium nucleus is emitting electrons via the channel with maximum kinetic energy of 2.284 MeV is 99.98 %. The probability to emit electrons via the other channel (electrons with maximum kinetic energy of 0.523 MeV) is about 0.02 %. In a second stage the excited nucleus will emit a γ-quantum. Sr-90 has a half-life of 28.8 years.

 

c) γ-radiation

The γ-radiation is an electromagnetic radiation consisting of so called γ-quanta, which is emitted by an excited nucleus. In general γ-radiation is present with α- or β-radiation (see Fig. 2). γ-radiation is produced if one nucleus Y* in an excited state relaxes to a new state with less energy. The equation of decay for Co-60 (γ-source used here in this RCL) reads

.

A star (*) means here that the nucleus is in an energetically excited state. γ-quanta possess only discrete energies, lying typically between 10 keV and 10 MeV. Since it is an electromagnetic wave γ-radiation has an infinitely large range in principal. Applying electric or magnetic fields will not directly influence this γ-radiation.

Fig. 3: Energy-level diagram of Co-60.

 

Co-60 is decaying into Ni-60 emitting first electrons and then γ-quanta. The decay may proceed through two channels: in the first channel by emitting electrons with maximum kinetic energy of 0.318 MeV with a probability of about 99.9 %. The daughter nucleus Ni* is in an excited state which relaxes by emitting γ-quanta with discrete energies of 1.173 MeV and subsequently of 1.332 MeV. The other decay channel with probability of 0.1 % is the emission of electrons with maximum kinetic energies of 1.491 MeV and a subsequent emission of γ-quanta with 1.332 MeV. The half life of Co-60 is 5.27 years.

 

 

3. Law of distance

The intensity I of radiation detected at distance r from a point-like radioactive source emitting isotropically with activity A is described by the following law (where S is the area of the sphere of radius r centered on the source):

.

Therefore the counting rate Z decreases when doubling the distance r to the forth part of the initial value (Fig. 4).

Fig. 4: Increase of the surface of a sphere with increasing radius.

 

Taking into account detection efficiency and other parameters which we will summarize by a factor b then we can write

.

Apart from the detector area which can be neglected at larger distances the equation describes a pure geometrical consideration: in particular, the inverse quadratic dependence on distance r is relevant. This geometrical law is the basis of one of the rules, which must be considered when protecting against radioactive radiation: "Stay away from a radioactive source as far as possible".

 

 

4. Law of absorption

Penetrating through matter of thickness d the counting rate Z(d) will be lowered from an intitial value Z(0) without matter. For γ-radiation and approximately for β-radiation the law of absorption is:

The coefficient µ (in cm-1) is a measure for the scattering and absorption of radioactive radiation by a specific material. If half of the radiation is blocked by the material this respective thickness is called dh:

Between the decay meter dh and the coefficient µ of absorption we get the relation

.

The law of absorption does not hold for α-radiation, because α-particles are losing their kinetic energy via coulomb interaction with electrons of atomic shells. The range of α-particles in matter is proportional to the kinetic energy of this α-particles. This range in solid matter is so small that α-particles are already blocked by a sheet of paper. Therefore, we did not plan to study details of absorption of α-particles in this remote lab. On the other hand, although of small range α-particles can be absorbed by important parts of living cells (penetration depth ~ a few ten micrometers) and, therefore, can damage epithelial tissues: Ingestion of sources of α-radiation must be avoided.

 

 

5. Geiger-Müller detector

This detector (Fig. 5) consists out of a metallic cylinder filled with Ar-gas, and an isolated, metal wire along the cylinder axis. High voltage U of several hundret V is applied to the metallic cylinder as cathode and wire as anode:

Fig. 5: Schematic setup of a Geiger-Müller detector
with connected counting system.

 

If a radioactive radiation enters the tube, part of the gas molecules will be ionized by interaction (collision with particles or quanta). These ions will be accelerated in the electric field (anode - cathode) and will produce further charges in a secondary process; as a consequence one will register a short electric current signal. This signal will be amplified by means of an electronic device and then registrated by a counting system.

 

 

6. Statistics and error of measurements

If one is measuring the number of decays (for example 10 times a number x of decays) of a radioactive sample with constant activity within a fixed time interval, one gets for example 6 decay processes 3 times, 4 decay processes 2 times and so on. As a consequence one cannot predict when a radioactive process will happen or which atom/nucleus will decay. What can be stated after these measurements is that 6 decay processes had happened with a relative probability of 3/10 = 0.3 (30 %), 4 decay processes with a relative probability of 2/10 = 0.2 (20 %) and so on. One gets an experimentally determined probability distribution which can be modelled approximately by a so called Poisson distribution P(x); P is the probability that x decay processes will appear:

.

For small averaged values (e.g. reference measurements without a radioactive source) the distribution P(x) is not symmetric around the average value, for larger average values (e.g. measurements with a radioactive source) P(x) is getting more symmetrically:

Fig. 6: Transition between Poisson´s distribution and the Gauß normal distribution
with increasing mean value.

 

For large average values (> 10) we get

and the Poisson distribution will be replaced by a Gaussian distribution:

.

Here σ is the well-known standard deviation:

Fig. 7: Gauß´s normal distribution for different mean values.

 

The uncertainty Δx of the measurement is √x. The relative error Δx/x = 1/√x. To increase the precision of the measurement one has the increase the number of measurements or, what is in principle the same, to increase the time interval of the measurement.

 

 

7. Rules of protection against radiation

It is agreed that one has to obey some rules how to encompass radioactivity:

 

 

8. Lorentz force

If electrically charged particles (with charge q ) are freely moving through space with a magnetic field B then these particles are deflected by a force FL (called Lorentzforce, after H. A. Lorentz 1853 - 1928).

.

In case the velocity of particles v is perpendicular to the magnetic field B, also the force is perpendicular to both of them.

 

Fig. 8: Direction of Lorentzforce for a positive (left) and a negative charged particle
(right) and the case that the magnetic field strength is directed into the drawing plane.

 

In general one gets the unknown velocity v of a particle from the kinetic energy Ekin = 0.5mv2, which is the electric energy Eel = qU of particles accelerated by an voltage U.