Radiation Dosimetry

Alpha decay (or α-decay and also alpha radioactivity) represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:

As can be seen from the figure, alpha particle is emitted in alpha decay. Alpha particles are energetic nuclei of helium. Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Alpha particles are relatively large and carry a double positive charge. They are not very penetrating and a piece of paper can stop them. They travel only a few centimeters but deposit all their energies along their short paths.

In practice, this mode of decay has only been observed in nuclides considerably heavier than nickel, with the lightest known alpha emitters being the lightest isotopes (mass numbers 106–110) of tellurium (element 52). In nuclear reactors alpha decay occurs for example in the fuel (alpha decay of heavy nuclei). Alpha particles are commonly emitted by all of the heavy radioactive nuclei occuring in the nature (uranium, thorium or radium), as well as the transuranic elements (neptunium, plutonium or americium).

Theory of Alpha Decay – Quantum Tunneling

Among the variety of channels in which a nucleus decays, alpha decay has been one of the most studied. The alpha decay channel in heavy and super heavy nuclei has provided information on the fundamental properties of nuclei far from stability, such as their ground state energies and the structure of their nuclear levels.

Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. ¹²C).

The height of the Coulomb barrier for nuclei of A « 200 is about 20-25 MeV. The alpha particles emitted in nuclear decay have typical energies of about 5 MeV. On the one hand an incoming 5 MeV alpha particle is scattered from a heavy nucleus and it cannot penetrate the Coulomb barrier and get sufficiently close to the nucleus to interact via the strong force. On the other hand, a 5 MeV alpha particle bound in a nuclear potential well is able to tunnel that same Coulomb barrier.

By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. They assumed that the alpha particle and the daughter nucleus exist within the parent nucleus prior to its dissociation, namely the decay of quasistationary states (QS).  A quasistationary state is defined as a long-lived state that eventually decays. Initially, the alpha cluster oscillates in the potential of the daughter nucleus, with the Coulomb potential preventing their separation. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei. The main success of this model was the reproduction of the semi-empirical Geiger-Nuttall law that expresses the lifetimes of the α emitters in terms of the energies of the released α particles. It must be noted, that other common forms of decay (e.g. beta decay) are governed by the interplay between both the nuclear force and the electromagnetic force.

Special Reference: W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467.

 

Gamma decay or γ decay represents the disintegration of a parent nucleus to a daughter through the emission of gamma rays (high energy photons). This transition (γ decay) can be characterized as:

As can be seen, if a nucleus emits a gamma ray, atomic and mass numbers of daughter nucleus remain the same, but daughter nucleus will form different energy state of the same element. Note that, nuclides with equal proton number and equal mass number (thus making them by definition the same isotope), but in a different energy state are known as nuclear isomers. We usually indicate isomers with a superscript m, thus: ^241mAm or ^110mAg.

In most practical laboratory sources, the excited nuclear states are created in the decay of a parent radionuclide, therefore a gamma decay typically accompanies other forms of decay, such as alpha or beta decay. Typically after a beta decay (isobaric transition), nuclei usually contain too much energy to be in its final stable or daughter state.

Gamma rays are high-energy photons with very short wavelengths and thus very high frequency. Gamma rays from radioactive decay are in the energy range from a few keV to ~8 MeV, corresponding to the typical energy levels in nuclei with reasonably long lifetimes. As was written, they are produced by the decay of nuclei as they transition from a high energy state to a lower state. Since the gamma rays are in substance only a very high-energy photons, they are very penetrating matter and are thus biologically hazardous. Gamma rays can travel thousands of feet in air and can easily pass through the human body.

In contrast to alpha and beta radioactivity, gamma radioactivity is governed by an electromagnetic interaction rather than a weak or strong interaction. As in atomic transitions, the photon carries away at least one unit of angular momentum (the photon, being described by the vector electromagnetic field, has spin angular momentum of ħ), and the process conserves parity.

Special Reference: W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467.

See also:

Radioactive Decay

In general, nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts (lighter nuclei). The fission process often produces free neutrons and photons (in the form of gamma rays), and releases a large amount of energy. In nuclear physics, nuclear fission is either a nuclear reaction or a radioactive decay process. The case of decay process is called spontaneous fission and it is very rare process.

Spontaneous fission is also possible if we will study the nuclear binding curve. This type of decay is energetically possible for a nucleus having A > 100.  Although spontaneous fission is expected to become more probable as the mass number increases, it is still a very rare process even in uranium.

Spontaneous fission is feasible over practical observation times only for mass numbers greater than about 232. For example, ²³²Th, ²³⁵U, and ²³⁸U are primordial nuclides and have left evidence of undergoing spontaneous fission in their minerals.

For heavy transuranic elements the spontaneous fission transition rate increase with the mass number and it may become the dominant decay mode at mass numbers greater than about 260.

Similarly as for alpha decay, also spontaneous fission occurs due to quantum tunneling. Spontaneous fissions release neutrons as all fissions do, so it contributes to neutron flux in a subcritical reactor. Radioisotopes for which spontaneous fission is not negligible can be used as neutron sources. For example, californium-252 (half-life 2.645 years, SF branch ratio about 3.1 percent) can be used for this purpose.

The spontaneous fission of naturally occurring isotopes of uranium (uranium-238 and uranium-235) does leave trails of damage in the crystal structure of uranium-containing minerals when the fission fragments recoil through them. A radiometric dating technique based on analyses of these damage trails, or tracks, left by fission fragments in certain uranium-bearing minerals and glasses is known as fission track dating.

Spontaneous Fission of Certain Nuclei

The main isotopes, which have to be considered in the fuel cycle of all commercial light water reactors, are:

Isotopes of uranium

• ²³⁸U. ²³⁸U belongs to the group of fertile isotopes. ²³⁸U decays via alpha decay to ²³⁴Th with half-life of ~4.5×10⁹ years. ²³⁸U occasionally decays by spontaneous fission with probability of 0.000055%. Its specific activity is very low ~3.4×10^-7 Ci/g.
• ²³⁵U. ²³⁵U belongs to the group of fissile isotopes. In fact, ²³⁵U is the only existing fissile nucleus from naturally-occurring isotopes and therefore it is a highly strategic material. ²³⁵U decays via alpha decay (by way of thorium-231) into ²³¹Pa with half-life of ~7×10⁸ years. ²³⁵U occasionally decays by spontaneous fission with very low probability of 0.0000000072%. Its specific activity is very low ~2.2×10-6 Ci/g.
• ²³⁴U. ²³⁴U belongs to the group of fertile isotopes. ²³⁴U decays via alpha decay to ²³⁰Th with half-life of 246 000 years. ²³⁴U occasionally decays by spontaneous fission with very low probability of 0.0000000017%. Its specific activity is much higher ~0.0063 Ci/g.

Isotopes of plutonium

• ²³⁸Pu. ²³⁸Pu belongs to the group of fertile isotopes. ²³⁸Pu decays via alpha decay to ²³⁴U with half-life of 87.7 years. ²³⁸Pu generates very high decay heat and has very high rate of spontaneous fission.
• ²³⁹Pu. ²³⁹Pu belongs to the group of fissile isotopes. ²³⁹Pu decays via alpha decay to ²³⁵U with half-life of 24100 years. This isotope is the principal fissile isotope in use.
• ²⁴⁰Pu. ²⁴⁰Pu belongs to the group of fertile isotopes. ²⁴⁰Pu decays via alpha decay to ²³⁶U with half-life of 6560 years. ²⁴⁰Pu has very high rate of spontaneous fission and has high radiative capture cross-section for thermal and also for resonance neutrons.
• ²⁴¹Pu. ²⁴¹Pu belongs to the group of fissile isotopes. ²⁴¹Pu decays via negative beta decay to ²⁴¹Am with half-life of 14.3 years. This fissile isotope decays to non-fissile isotope with high radiative capture cross-section for thermal neutrons. An impact on reactivity of the nuclear fuel  is obvious.

Other isotopes

• ²⁴²Cm. ²⁴²Cm decays via alpha decay with half-life of 162 days. ²⁴²Cm occasionally decays by spontaneous fission with probability of 0.0000061%.
• ²⁵²Cf. ²⁵²Cf  (half-life 2.645 years, SF branch ratio about 3.1 percent) can be used in the primary neutron source for better monitoring of reactor startup and shutdown operations when the reactor is subcritical.

Spontaneous Fission and Source Neutrons

The term “source neutrons” refers to neutrons other than prompt or delayed fission neutrons. In general, they come from sources other than neutron-induced fission. These neutrons are very important during reactor startup and shutdown operations when the reactor is subcritical, because they allow to monitor the subcriticality of a reactor usually via source range excore neutron detectors.

In nuclear reactors, also spontaneous fission is very important from this point of view. In an irradiated nuclear fuel assemblies, there are nuclides, that undergo a process of spontaneous fission. As was written, spontaneous fission is, in fact, a form of radioactive decay that is found only in very heavy chemical elements (especially transuranic elements). For example, ²⁴⁰Pu has very high rate of spontaneous fission. For high burnup fuel, the source neutrons are provided predominantly by spontaneous fission of curium nuclei (Cm-242 and Cm-244).

Primary Source of Neutrons

Sometimes source neutrons must be artificially added to the system. External source of neutrons contains a material, that emits neutrons and cladding to provide a barrier between reactor coolant and the material. External sources are usually loaded directly into the reactor core (e.g. into guide thimble tubes). Neutron sources , that are based on spontaneous fission are known as primary sources. Primary source of neutrons does not need to be irradiated to produce neutrons. These sources can be used especially in case of a first core (i.e. a core that consist of fresh fuel only). Primary source of neutrons are based on the spontaneous fission reaction. The most commonly used spontaneous fission source is the radioactive isotope californium-252. Cf-252 and all other spontaneous fission neutron sources are produced by irradiating uranium or another transuranic element in a nuclear reactor, where neutrons are absorbed in the starting material and its subsequent reaction products, transmuting the starting material into the SF isotope.

See also:

Radioactive Decay

X-rays, also known as X-radiation, refers to electromagnetic radiation (no rest mass, no charge) of high energies. X-rays are high-energy photons with short wavelengths and thus very high frequency. The radiation frequency is key parameter of all photons, because it determines the energy of a photon. Photons are categorized according to the energies from low-energy radio waves and infrared radiation, through visible light, to high-energy X-rays and gamma rays.

Most X-rays have a wavelength ranging from 0.01 to 10 nanometers (3×10¹⁶ Hz to 3×10¹⁹ Hz), corresponding to energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays. The distinction between X-rays and gamma rays is not so simple and has changed in recent decades.  According to the currently valid definition, X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus.

Since the X-rays (especially hard X-rays) are in substance high-energy photons, they are very penetrating matter and are thus biologically hazardous. X-rays can travel thousands of feet in air and can easily pass through the human body.

Discovery of X-Rays – Wilhelm Conrad Röntgen

X-rays were discovered on November 8, 1895 by German physics professor Wilhelm Conrad Röntgen at the University of Würtzburg in Germany. He was studying electric discharges in glass tubes filled with various gases at very low pressures. In these experiments, Röntgen had covered the tube with some black paper and had darkened the room. He then discovered that a piece of paper painted with a fluorescent dye, at some distance from the tube, would glow when he turned on the high voltage between the electrodes in the tube. He realized that he had produced a previously unknown “invisible light,” or ray, that was being emitted from the tube and a ray that was capable of passing through the heavy paper covering the tube. Röntgen referred to the radiation as “X”, to indicate that it was an unknown type of radiation.

Realizing the importance of his discovery, Röntgen focused all his attention on the study of this new radiation that had the unusual property of passing through black paper. Through additional experiments, he also found that the new ray would pass through most substances casting shadows of solid objects such as blocks of wood, books and even his hand. He found that X-rays propagate in straight lines from which they are deflected neither by electric nor magnetic fields. The first x-ray image was a picture of his wife’s hand on a photographic plate formed due to X-rays. His discovery spread quickly throughout the world and Wilhelm Conrad Röntgen received the first Nobel Prize in Physics for his discovery.

Characteristics of X-rays

Key features of X-rays are summarized in following few points:

• X-rays are high-energy photons (about 100 – 1 000 times as much energy as the visible photons), the same photons as the photons forming the visible range of the electromagnetic spectrum – light.
• X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. It may range from about 20 kV up to 300 kV. Radiation with low voltage is called “soft” – and radiation with high voltage is called “hard”.
• Photons (gamma rays and X-rays) can ionize atoms directly (despite they are electrically neutral) through the Photoelectric effect and the Compton effect, but secondary (indirect) ionization is much more significant.
• X-rays ionize matter via indirect ionization.
• Although a large number of possible interactions are known, there are three key interaction mechanisms  with matter.
  • Photoelectric effect
  • Compton scattering
  • Rayleigh scattering
• X-rays travel at the speed of light and they can travel hundreds of meters in air before spending their energy.
• Since the hard X-rays are very penetrating matter, it must be shielded by very dense materials, such as lead or uranium.
• The distinction between X-rays and gamma rays is not so simple and has changed in recent decades.  According to the currently valid definition, X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus.
• For X-rays generated by X-ray tube, there are two different types of X-rays spectra:
  • Bremsstrahlung
  • Characteristic X-rays
• Characteristic X-rays frequently accompany some types of nuclear decays, such as internal conversion and electron capture.

X-ray – Production

Since X-rays are high-energy photons, which have electromagnetic nature, they can be produced whenever charged particles (electrons or ions) of sufficient energy hit a material. It is similar to the photoelectric effect, where photons can be annihilated when they strike the metal plate, each one surrendering its kinetic energy to an electron.

X-rays can be generated by an X-ray tube, a vacuum tube that uses a high voltage to accelerate the electrons released by a hot cathode to a high velocity. The cathode must be heated in order to emit electrons. Electrons, accelerated by potential differences of tens of thousands of volts, are aimed at a metal target (usually made of tungsten or another heavy metal) in a vacuum tube. The larger the voltage between the electrodes the higher energy will the electrons attain. Upon striking the target, the accelerated electrons are abruptly stopped and X-rays and heat are generated. Most of the energy is transformed into heat in the anode (which must be cooled). Just 1% of the kinetic energy of the electrons is converted into X-rays. X-rays are usually generated perpendicular to the path of the electron beam.

A specialized source of X-rays which is becoming widely used in research is particle accelerator, which generates radiation known as synchrotron radiation. When ultra-relativistic charged particles move through magnetic fields they are forced to move along a curved path. Since their direction of motion is continually changing, they are also accelerating and so emit bremsstrahlung, in this case it is referred to as synchrotron radiation.

X-rays can also be produced by fast protons or other positive ions. The proton-induced X-ray emission or particle-induced X-ray emission is widely used as an analytical procedure.

Soft and Hard X-rays

X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. X-rays with high photon energies (above 5–10 keV) are called hard X-rays, while those with lower energy (and longer wavelength) are called soft X-rays. Due to their penetrating ability, hard X-rays are widely used to image the inside of visually opaque objects. The most often seen applications are in medical radiography. Since the wavelengths of hard X-rays are similar to the size of atoms, they are also useful for determining crystal structures by X-ray crystallography. By contrast, soft X-rays are easily absorbed in air. The attenuation length of 600 eV X-rays in water is less than 1 micrometer.

X-Ray Spectrum – Characteristic and Continuous

For X-rays generated by X-ray tube, the part of energy that is transformed into radiation varies from zero up to the maximum energy of the electron when it hits the anode. The maximum energy of the produced X-ray photon is limited by the energy of the incident electron, which is equal to the voltage on the tube times the electron charge, so an 100 kV tube cannot create X-rays with an energy greater than 100 keV. When the electrons hit the target, X-rays are created by two different atomic processes:

• Bremsstrahlung. The bremsstrahlung is electromagnetic radiation produced by the acceleration or deceleration of an electron when deflected by  strong electromagnetic fields of target high-Z (proton number) nuclei. The name bremsstrahlung comes from the German. The literal translation is ‘braking radiation’. From classical theory, when a charged particle is accelerated or decelerated, it must radiate energy. The bremsstrahlung is one of possible interactions of light charged particles with matter (especially with high atomic numbers). These X-rays have a continuous spectrum. The intensity of the X-rays increases linearly with decreasing frequency, from zero at the energy of the incident electrons, the voltage on the X-ray tube. Changing the material from which the target in the tube is made has no effect on the spectrum of this continuous radiation. If we were to switch from a molybdenum target to a copper target, for example, all features of the x-ray spectrum would change except the cutoff wavelength.
• Characteristic X-ray emission. If the electron has enough energy it can knock an orbital electron out of the inner electron shell of a metal atom. Since the process leaves a vacancy in the electron energy level from which the electron came, the outer electrons of the atom cascade down to fill the lower atomic levels, and one or more characteristic X-rays are usually emitted. As a result, sharp intensity peaks appear in the spectrum at wavelengths that are a characteristic of the material from which the anode target is made. The frequencies of the characteristic X-rays can be predicted from the Bohr model.

Characteristic X-rays and Internal Conversion

Internal conversion is an electromagnetic process, by which a nuclear excited state decays by the direct emission of one of its atomic electrons.

Since the process leaves a vacancy in the electron energy level from which the electron came, the outer electrons of the atom cascade down to fill the lower atomic levels, and one or more characteristic X-rays are usually emitted. Sometimes X-ray may interact with another orbital electron, which may be ejected from the atom. This second ejected electron is called an Auger electron. This is very similar to electron capture, but in case of electron capture, a nucleus changes its atomic number. As a result, the atom thus emits primary high energy electron, characteristic X-rays or secondary Auger electron, none of which originate in that nucleus.

Interaction of X-rays with Matter

Although a large number of possible interactions are known, there are three key interaction mechanisms with matter. The strength of these interactions depends on the energy of the X-rays and the elemental composition of the material, but not much on chemical properties, since the X-ray photon energy is much higher than chemical binding energies. The photoelectric absorbtion dominates at low-energies of X-rays, while Compton scattering dominates at higher energies.

• Photoelectric absorption
• Compton scattering
• Rayleigh scattering

Photoelectric Absorption of X-rays

In the photoelectric effect, a photon undergoes an interaction with an electron which is bound in an atom. In this interaction the incident photon completely disappears and an energetic photoelectron is ejected by the atom from one of its bound shells. The kinetic energy of the ejected photoelectron (E_e) is equal to the incident photon energy (hν) minus the binding energy of the photoelectron in its original shell (E_b).

E_e=hν-E_b

Therefore photoelectrons are only emitted by the photoelectric effect if photon reaches or exceeds a threshold energy – the binding energy of the electron – the work function of the material. For very high X-rays with energies of more than hundreds keV, the photoelectron carries off the majority of the incident photon energy – hν.

At small values of gamma ray energy the photoelectric effect dominates. The mechanism is also enhaced for materials of high atomic number Z. It is not simple to derive analytic expression for the probability of photoelectric absorption of gamma ray per atom over all ranges of gamma ray energies. The probability of photoelectric absorption per unit mass is approximately proportional to:

τ_{(photoelectric)} = constant x Z^N/E^3.5

where Z is the atomic number, the exponent n varies between 4 and 5. E is the energy of the incident photon. The proportionality to higher powers of the atomic number Z is the main reason for using of high Z materials, such as lead or depleted uranium in gamma ray shields.

Although the probability of the photoelectric absorption of photon decreases, in general, with increasing photon energy, there are sharp discontinuities in the cross-section curve. These are called “absoption edges” and they correspond to the binding energies of electrons from atom’s bound shells. For photons with the energy just above the edge, the photon energy is just sufficient to undergo the photoelectric interaction with electron from  bound shell, let say K-shell. The probability of such interaction is just above this edge much greater than that of photons of energy slightly below this edge. For photons below this edge the interaction with electron from K-shell in energetically impossible and therefore the probability drops abruptly. These edges occur also at binding energies of electrons from other shells (L, M, N …..).

Compton Scattering of X-rays

The Compton formula was published in 1923 in the Physical Review. Compton explained that the X-ray shift is caused by particle-like momentum of photons. Compton scattering formula is the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays. In the case of Compton scattering the photon of frequency f collides with an electron at rest. Upon collision, the photon bounces off electron, giving up some of its initial energy (given by Planck’s formula E=hf), While the electron gains momentum (mass x velocity), the photon cannot lower its velocity. As a result of momentum conservation law, the photon must lower its momentum given by:

So the decrease in photon’s momentum must be translated into decrease in frequency (increase in wavelength Δλ = λ’ – λ). The shift of the wavelength increased with scattering angle according to the Compton formula:

 

where λ is the initial wavelength of photon λ’ is the wavelength after scattering, h is the Planck constant = 6.626 x 10^-34 J.s, m_e is the electron rest mass (0.511 MeV)c is the speed of light Θ is the scattering angle. The minimum change in wavelength (λ′ − λ) for the photon occurs when Θ = 0° (cos(Θ)=1) and is at least zero. The maximum change in wavelength (λ′ − λ) for the photon occurs when Θ = 180° (cos(Θ)=-1). In this case the photon transfers to the electron as much momentum as possible. The maximum change in wavelength can be derived from Compton formula:

The quantity h/m_ec is known as the Compton wavelength of the electron and is equal to 2.43×10⁻¹² m.

Rayleigh Scattering – Thomson Scattering

Rayleigh scattering, also known as Thomson scattering is the low-energy limit of Compton scattering. The particle kinetic energy and photon frequency do not change as a result of the scattering. Rayleigh scattering occurs as a result of an interaction between an incoming photon and an electron, the binding energy of which is significantly greater than that of the incoming photon. The incident radiation is assumed to set the electron into forced resonant oscillation such that the electron re-emits radiation of the same frequency but in all directions. In this case, the electric field of the incident wave (photon) accelerates the charged particle, causing it, in turn, to emit radiation at the same frequency as the incident wave, and thus the wave is scattered. Rayleigh scattering is significant up to ≈ 20keV and like Thomson scattering, is elastic. The total scattering cross section becomes a combination of the Rayleigh and Compton bound scattering cross sections. Thomson scattering is an important phenomenon in plasma physics and was first explained by the physicist J. J. Thomson. This interaction has great significance in the area of X-ray crystallography.

X-ray Attenuation

As the high-energy photons pass through material, their energy is decreasing. This is known as attenuation. The attenuation theory is valid for X-rays and gamma rays as well. It turns out that higher energy photons (hard X-rays) travel through tissue more easily than low-energy photons (i.e. the higher energy photons are less likely to interact with matter). Much of this effect is related to the photoelectric effect. The probability of photoelectric absorption is approximately proportional to (Z/E)³, where Z is the atomic number of the tissue atom and E is the photon energy. As E gets larger, the likelihood of interaction drops rapidly. For higher energies, Compton scattering becomes dominant. Compton scattering is about constant for different energies although it slowly decreases at higher energies.

See also: X-ray Attenuation

Shielding of X-rays

In short, effective shielding of X-rays is in most cases based on use of materials with two following material properties:

• high-density of material.
• high atomic number of material  (high Z materials)

However, low-density materials and low Z materials can be compensated with increased thickness, which is as significant as density and atomic number in shielding applications.

A lead is widely used as a X-rays shield.  Major advantage of lead shield is in its compactness due to its higher density. A lead is widely used as a gamma shield. On the other hand depleted uranium is much more effective due to its higher Z.  Depleted uranium is used for shielding in portable gamma ray sources.

In nuclear power plants shielding of a reactor core can be provided by materials of reactor pressure vessel, reactor internals (neutron reflector). Also heavy concrete is usually used to shield both neutrons and gamma radiation.

In general, the X-rays shielding is more complex and difficult than the alpha or beta radiation shielding. In order to understand comprehensively the way how a X-ray loses its initial energy, how can be attenuated and how can be shielded we must have detailed knowledge of the its interaction mechanisms.

See also more theory: Interaction of X-rays with Matter

See also calculator: Gamma activity to dose rate (with/without shield)

See also XCOM – photon cross-section DB: XCOM: Photon Cross Sections Database

See also:

Radiation

Discovery of X-Rays – Wilhelm Conrad Röntgen

X-rays were discovered on November 8, 1895 by German physics professor Wilhelm Conrad Röntgen at the University of Würtzburg in Germany. He was studying electric discharges in glass tubes filled with various gases at very low pressures. In these experiments, Röntgen had covered the tube with some black paper and had darkened the room. He then discovered that a piece of paper painted with a fluorescent dye, at some distance from the tube, would glow when he turned on the high voltage between the electrodes in the tube. He realized that he had produced a previously unknown “invisible light,” or ray, that was being emitted from the tube and a ray that was capable of passing through the heavy paper covering the tube. Röntgen referred to the radiation as “X”, to indicate that it was an unknown type of radiation.

Realizing the importance of his discovery, Röntgen focused all his attention on the study of this new radiation that had the unusual property of passing through black paper. Through additional experiments, he also found that the new ray would pass through most substances casting shadows of solid objects such as blocks of wood, books and even his hand. He found that X-rays propagate in straight lines from which they are deflected neither by electric nor magnetic fields. The first x-ray image was a picture of his wife’s hand on a photographic plate formed due to X-rays. His discovery spread quickly throughout the world and Wilhelm Conrad Röntgen received the first Nobel Prize in Physics for his discovery.

See also:

X-rays

X-rays, also known as X-radiation, refers to electromagnetic radiation (no rest mass, no charge) of high energies. X-rays are high-energy photons with short wavelengths and thus very high frequency. The radiation frequency is key parameter of all photons, because it determines the energy of a photon. Photons are categorized according to the energies from low-energy radio waves and infrared radiation, through visible light, to high-energy X-rays and gamma rays.

Most X-rays have a wavelength ranging from 0.01 to 10 nanometers (3×10¹⁶ Hz to 3×10¹⁹ Hz), corresponding to energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.

Since the X-rays (especially hard X-rays) are in substance high-energy photons, they are very penetrating matter and are thus biologically hazardous. X-rays can travel thousands of feet in air and can easily pass through the human body.

Characteristics of X-rays – Properties

Key characteristics of X-rays are summarized in following few points:

• X-rays are high-energy photons (about 100 – 1 000 times as much energy as the visible photons), the same photons as the photons forming the visible range of the electromagnetic spectrum – light.
• X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. It may range from about 20 kV up to 300 kV. Radiation with low voltage is called “soft” – and radiation with high voltage is called “hard”.
• Photons (gamma rays and X-rays) can ionize atoms directly (despite they are electrically neutral) through the Photoelectric effect and the Compton effect, but secondary (indirect) ionization is much more significant.
• X-rays ionize matter via indirect ionization.
• Although a large number of possible interactions are known, there are three key interaction mechanisms  with matter.
  • Photoelectric effect
  • Compton scattering
  • Rayleigh scattering
• X-rays travel at the speed of light and they can travel hundreds of meters in air before spending their energy.
• Since the hard X-rays are very penetrating matter, it must be shielded by very dense materials, such as lead or uranium.
• The distinction between X-rays and gamma rays is not so simple and has changed in recent decades.  According to the currently valid definition, X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus.
• For X-rays generated by X-ray tube, there are two different types of X-rays spectra:
  • Bremsstrahlung
  • Characteristic X-rays
• Characteristic X-rays frequently accompany some types of nuclear decays, such as internal conversion and electron capture.

See also:

X-rays

X-rays, also known as X-radiation, refers to electromagnetic radiation (no rest mass, no charge) of high energies. X-rays are high-energy photons with short wavelengths and thus very high frequency. The radiation frequency is key parameter of all photons, because it determines the energy of a photon. Photons are categorized according to the energies from low-energy radio waves and infrared radiation, through visible light, to high-energy X-rays and gamma rays.

Most X-rays have a wavelength ranging from 0.01 to 10 nanometers (3×10¹⁶ Hz to 3×10¹⁹ Hz), corresponding to energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.

X-ray – Production

Since X-rays are high-energy photons, which have electromagnetic nature, they can be produced whenever charged particles (electrons or ions) of sufficient energy hit a material. It is similar to the photoelectric effect, where photons can be annihilated when they strike the metal plate, each one surrendering its kinetic energy to an electron.

X-rays can be generated by an X-ray tube, a vacuum tube that uses a high voltage to accelerate the electrons released by a hot cathode to a high velocity. The cathode must be heated in order to emit electrons. Electrons, accelerated by potential differences of tens of thousands of volts, are aimed at a metal target (usually made of tungsten or another heavy metal) in a vacuum tube. The larger the voltage between the electrodes the higher energy will the electrons attain. Upon striking the target, the accelerated electrons are abruptly stopped and X-rays and heat are generated. Most of the energy is transformed into heat in the anode (which must be cooled). Just 1% of the kinetic energy of the electrons is converted into X-rays. X-rays are usually generated perpendicular to the path of the electron beam.

A specialized source of X-rays which is becoming widely used in research is particle accelerator, which generates radiation known as synchrotron radiation. When ultra-relativistic charged particles move through magnetic fields they are forced to move along a curved path. Since their direction of motion is continually changing, they are also accelerating and so emit bremsstrahlung, in this case it is referred to as synchrotron radiation.

X-rays can also be produced by fast protons or other positive ions. The proton-induced X-ray emission or particle-induced X-ray emission is widely used as an analytical procedure.

Soft and Hard X-rays

X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. X-rays with high photon energies (above 5–10 keV) are called hard X-rays, while those with lower energy (and longer wavelength) are called soft X-rays. Due to their penetrating ability, hard X-rays are widely used to image the inside of visually opaque objects. The most often seen applications are in medical radiography. Since the wavelengths of hard X-rays are similar to the size of atoms, they are also useful for determining crystal structures by X-ray crystallography. By contrast, soft X-rays are easily absorbed in air. The attenuation length of 600 eV X-rays in water is less than 1 micrometer.

X-Ray Spectrum – Characteristic and Continuous

For X-rays generated by X-ray tube, the part of energy that is transformed into radiation varies from zero up to the maximum energy of the electron when it hits the anode. The maximum energy of the produced X-ray photon is limited by the energy of the incident electron, which is equal to the voltage on the tube times the electron charge, so an 100 kV tube cannot create X-rays with an energy greater than 100 keV. When the electrons hit the target, X-rays are created by two different atomic processes:

• Bremsstrahlung. The bremsstrahlung is electromagnetic radiation produced by the acceleration or deceleration of an electron when deflected by  strong electromagnetic fields of target high-Z (proton number) nuclei. The name bremsstrahlung comes from the German. The literal translation is ‘braking radiation’. From classical theory, when a charged particle is accelerated or decelerated, it must radiate energy. The bremsstrahlung is one of possible interactions of light charged particles with matter (especially with high atomic numbers). These X-rays have a continuous spectrum. The intensity of the X-rays increases linearly with decreasing frequency, from zero at the energy of the incident electrons, the voltage on the X-ray tube. Changing the material from which the target in the tube is made has no effect on the spectrum of this continuous radiation. If we were to switch from a molybdenum target to a copper target, for example, all features of the x-ray spectrum would change except the cutoff wavelength.
• Characteristic X-ray emission. If the electron has enough energy it can knock an orbital electron out of the inner electron shell of a metal atom. Since the process leaves a vacancy in the electron energy level from which the electron came, the outer electrons of the atom cascade down to fill the lower atomic levels, and one or more characteristic X-rays are usually emitted. As a result, sharp intensity peaks appear in the spectrum at wavelengths that are a characteristic of the material from which the anode target is made. The frequencies of the characteristic X-rays can be predicted from the Bohr model.

See also:

X-rays

X-rays, also known as X-radiation, refers to electromagnetic radiation (no rest mass, no charge) of high energies. X-rays are high-energy photons with short wavelengths and thus very high frequency. The radiation frequency is key parameter of all photons, because it determines the energy of a photon. Photons are categorized according to the energies from low-energy radio waves and infrared radiation, through visible light, to high-energy X-rays and gamma rays.

Most X-rays have a wavelength ranging from 0.01 to 10 nanometers (3×10¹⁶ Hz to 3×10¹⁹ Hz), corresponding to energies in the range 100 eV to 100 keV. X-ray wavelengths are shorter than those of UV rays and typically longer than those of gamma rays.

Soft and Hard X-rays

X-rays are usually described by their maximum energy, which is determined by the voltage between the electrodes. X-rays with high photon energies (above 5–10 keV) are called hard X-rays, while those with lower energy (and longer wavelength) are called soft X-rays. Due to their penetrating ability, hard X-rays are widely used to image the inside of visually opaque objects. The most often seen applications are in medical radiography. Since the wavelengths of hard X-rays are similar to the size of atoms, they are also useful for determining crystal structures by X-ray crystallography. By contrast, soft X-rays are easily absorbed in air. The attenuation length of 600 eV X-rays in water is less than 1 micrometer.

X-ray – Production

Since X-rays are high-energy photons, which have electromagnetic nature, they can be produced whenever charged particles (electrons or ions) of sufficient energy hit a material. It is similar to the photoelectric effect, where photons can be annihilated when they strike the metal plate, each one surrendering its kinetic energy to an electron.

X-rays can be generated by an X-ray tube, a vacuum tube that uses a high voltage to accelerate the electrons released by a hot cathode to a high velocity. The cathode must be heated in order to emit electrons. Electrons, accelerated by potential differences of tens of thousands of volts, are aimed at a metal target (usually made of tungsten or another heavy metal) in a vacuum tube. The larger the voltage between the electrodes the higher energy will the electrons attain. Upon striking the target, the accelerated electrons are abruptly stopped and X-rays and heat are generated. Most of the energy is transformed into heat in the anode (which must be cooled). Just 1% of the kinetic energy of the electrons is converted into X-rays. X-rays are usually generated perpendicular to the path of the electron beam.

A specialized source of X-rays which is becoming widely used in research is particle accelerator, which generates radiation known as synchrotron radiation. When ultra-relativistic charged particles move through magnetic fields they are forced to move along a curved path. Since their direction of motion is continually changing, they are also accelerating and so emit bremsstrahlung, in this case it is referred to as synchrotron radiation.

X-rays can also be produced by fast protons or other positive ions. The proton-induced X-ray emission or particle-induced X-ray emission is widely used as an analytical procedure.

X-Ray Spectrum – Characteristic and Continuous

For X-rays generated by X-ray tube, the part of energy that is transformed into radiation varies from zero up to the maximum energy of the electron when it hits the anode. The maximum energy of the produced X-ray photon is limited by the energy of the incident electron, which is equal to the voltage on the tube times the electron charge, so an 100 kV tube cannot create X-rays with an energy greater than 100 keV. When the electrons hit the target, X-rays are created by two different atomic processes:

• Bremsstrahlung. The bremsstrahlung is electromagnetic radiation produced by the acceleration or deceleration of an electron when deflected by  strong electromagnetic fields of target high-Z (proton number) nuclei. The name bremsstrahlung comes from the German. The literal translation is ‘braking radiation’. From classical theory, when a charged particle is accelerated or decelerated, it must radiate energy. The bremsstrahlung is one of possible interactions of light charged particles with matter (especially with high atomic numbers). These X-rays have a continuous spectrum. The intensity of the X-rays increases linearly with decreasing frequency, from zero at the energy of the incident electrons, the voltage on the X-ray tube. Changing the material from which the target in the tube is made has no effect on the spectrum of this continuous radiation. If we were to switch from a molybdenum target to a copper target, for example, all features of the x-ray spectrum would change except the cutoff wavelength.
• Characteristic X-ray emission. If the electron has enough energy it can knock an orbital electron out of the inner electron shell of a metal atom. Since the process leaves a vacancy in the electron energy level from which the electron came, the outer electrons of the atom cascade down to fill the lower atomic levels, and one or more characteristic X-rays are usually emitted. As a result, sharp intensity peaks appear in the spectrum at wavelengths that are a characteristic of the material from which the anode target is made. The frequencies of the characteristic X-rays can be predicted from the Bohr model.

See also:

X-rays

One of the most useful terms for estimating how quickly a nuclide will decay is the radioactive half-life (t_1/2). The half-life is defined as the amount of time it takes for a given isotope to lose half of its radioactivity. As was written, radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. In other words, a nucleus of a radionuclide has no “memory”. A nucleus does not “age” with the passage of time. Thus, the probability of its breaking down does not increase with time, but stays constant no matter how long the nucleus has existed.

Therefore, the rate of nuclear decay can be also measured in terms of half-lives. Each radionuclide has its own particular half-life that never changes, regardless of the quantity or form of the material (i.e., solid, liquid, gas, element or compound) or its past history. If a radioisotope has a half-life of 14 days, half of its atoms will have decayed within 14 days. In 14 more days, half of that remaining half will decay, and so on.

Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). After five half-lives have elapsed, only 1/32, or 3.1%, of the original number of atoms remains. After seven half-lives, only 1/128, or 0.78%, of the atoms remains. The number of atoms existing after 5 to 7 half-lives can usually be assumed to be negligible.

The fraction of the original activity remaining after succeeding half-lives is:

Activity after 1 half-life = ½ of the original

Activity after 2 half-lives = ½ x ½ = ¼ of the original

Activity after 3 half-lives = ½ x ½ x ½ = (½)³ = 1/8 of the original

Activity after 4 half-lives = (½)⁴ = 1/16 of the original

Activity after 5 half-lives = (½)⁵ = 1/32 of the original

Activity after 6 half-lives = (½)⁶ = 1/64 of the original

Activity after 7 half-lives = (½)⁷ = 1/128 of the original

Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N.e^-λt

The rate of nuclear decay is also measured in terms of half-lives. The half-life is the amount of time it takes for a given isotope to lose half of its radioactivity. If a radioisotope has a half-life of 14 days, half of its atoms will have decayed within 14 days. In 14 more days, half of that remaining half will decay, and so on. Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

The radioactive decay law can be derived also for activity calculations or mass of radioactive material calculations:

(Number of nuclei) N = N.e^-λt     (Activity) A = A.e^-λt      (Mass) m = m.e^-λt

, where N (number of particles) is the total number of particles in the sample, A (total activity) is the number of decays per unit time of a radioactive sample, m is the mass of remaining radioactive material.

Half-Life and Decay Constant

In calculations of radioactivity one of two parameters (decay constant or half-life), which characterize the rate of decay, must be known. There is a relation between the half-life (t_1/2) and the decay constant λ. The relationship can be derived from decay law by setting N = ½ N_o. This gives:

where ln 2 (the natural log of 2) equals 0.693. If the decay constant (λ) is given, it is easy to calculate the half-life, and vice-versa.

Half-Life and Radioactivity

The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Radioactive Decay Law

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

Biological Half-Life

In general, the biological half-life is the time taken for the amount of a particular element in the body to decrease to half of its initial value due to elimination by biological processes alone, when the rate of removal is roughly exponential. The biological half-live (t_biological) can be defined for  metabolites, drugs, and other substances. It is also very important in radiation protection, when considering internal exposure.

If the source of radiation is inside our body, we say, it is internal exposure. The intake of radioactive material can occur through various pathways such as ingestion of radioactive contamination in food or liquids, inhalation of radioactive gases, or through intact or wounded skin. Most radionuclides will give you much more radiation dose if they can somehow enter your body, than they would if they remained outside. The biological half-life depends on the rate at which the body normally uses a particular compound of an element. Radioactive isotopes that were ingested or taken in through other pathways will gradually be removed from the body via bowels, kidneys, respiration and perspiration. This means that a radioactive substance can be expelled before it has had the chance to decay.

As a result, the biological half-life significantly influences the overall dose from internal contamination. If a radioactive compound with radioactive half-life (t_1/2) is cleared from the body with a biological half-life t_b, the “effective’’ half-life (t_e) is given by the expression:

As can be seen, the biological mechanisms always decreases the overall dose from internal contamination.  Moreover, if t_1/2 is large in comparison to t_b, the effective half-life is approximately the same as t_b. For example, tritium has the biological half-life about 10 days, while the radioactive half-life is about 12 years.

See also: Biological Half-Life

See also: Effective Half-Life

See also:

Decay Law

Decay Constant and Radioactivity

The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Calculation of Radioactivity

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

 

See also:

Decay Constant

In nuclear physics and nuclear chemistry, the various species of atoms whose nuclei contain particular numbers of protons and neutrons are called nuclides. Nuclides are also characterized by its nuclear energy states (e.g. metastable nuclide ^242mAm). Each nuclide is denoted by chemical symbol of the element (this specifies Z) with the atomic mass number as superscript. Isotopes are nuclides that have the same atomic number and are therefore the same element, but differ in the number of neutrons.

There are nuclides that are unstable and radioactive. These nuclides are  known as radionuclides (radioactive nuclides) or radioisotopes (radioactive isotopes). These unstable isotopes decay through various radioactive decay pathways, most commonly alpha decay, beta decay, gamma decay or electron capture. Many other rare types of decay, such as spontaneous fission or neutron emission are known.

As was written, radioactive decay of radionuclides is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. In other words, a nucleus of a radionuclide has no “memory”. A nucleus does not “age” with the passage of time. Thus, the probability of its breaking down does not increase with time, but stays constant no matter how long the nucleus has existed.

Therefore, the rate of nuclear decay can be also measured in terms of half-lives. Each radionuclide has its own particular half-life that never changes, regardless of the quantity or form of the material (i.e., solid, liquid, gas, element or compound) or its past history. If a radioisotope has a half-life of 14 days, half of its atoms will have decayed within 14 days. In 14 more days, half of that remaining half will decay, and so on.

Half lives range from millionths of a second for highly radioactive fission products to billions of years for long-lived materials (such as naturally occurring uranium). After five half-lives have elapsed, only 1/32, or 3.1%, of the original number of atoms remains. After seven half-lives, only 1/128, or 0.78%, of the atoms remains. The number of atoms existing after 5 to 7 half-lives can usually be assumed to be negligible.

The fraction of the original activity remaining after succeeding half-lives is:

Activity after 1 half-life = ½ of the original

Activity after 2 half-lives = ½ x ½ = ¼ of the original

Activity after 3 half-lives = ½ x ½ x ½ = (½)³ = 1/8 of the original

Activity after 4 half-lives = (½)⁴ = 1/16 of the original

Activity after 5 half-lives = (½)⁵ = 1/32 of the original

Activity after 6 half-lives = (½)⁶ = 1/64 of the original

Activity after 7 half-lives = (½)⁷ = 1/128 of the original

Notice that short half lives go with large decay constants. Radioactive material with a short half life is much more radioactive (at the time of production) but will obviously lose its radioactivity rapidly. No matter how long or short the half life is, after seven half lives have passed, there is less than 1 percent of the initial activity remaining.

The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.

Radioactive decay law: N = N.e^-λt

The radioactive decay law can be derived also for activity calculations or mass of radioactive material calculations:

(Number of nuclei) N = N.e^-λt     (Activity) A = A.e^-λt      (Mass) m = m.e^-λt

, where N (number of particles) is the total number of particles in the sample, A (total activity) is the number of decays per unit time of a radioactive sample, m is the mass of remaining radioactive material.

Half-Life and Radioactivity of Radionuclides

The relationship between half-life and the amount of a radionuclide required to give an activity of one curie is shown in the figure. This amount of material can be calculated using λ, which is the decay constant of certain nuclide:

The following figure illustrates the amount of material necessary for 1 curie of radioactivity. It is obvious, that the longer the half-life, the greater the quantity of radionuclide needed to produce the same activity. Of course, the longer lived substance will remain radioactive for a much longer time. As can be seen, the amount of material necessary for 1 curie of radioactivity can vary from an amount too small to be seen (0.00088 gram of cobalt-60), through 1 gram of radium-226, to almost three tons of uranium-238.

Example – Radioactivity of Radionuclide

A sample of material contains 1 mikrogram of iodine-131. Note that, iodine-131 plays a major role as a radioactive isotope present in nuclear fission products, and it is a major contributor to the health hazards when released into the atmosphere during an accident. Iodine-131 has a half-life of 8.02 days.

Calculate:

1. The number of iodine-131 atoms initially present.
2. The activity of the iodine-131 in curies.
3. The number of iodine-131 atoms that will remain in 50 days.
4. The time it will take for the activity to reach 0.1 mCi.

Solution:

1. The number of atoms of iodine-131 can be determined using isotopic mass as below.

N_I-131 = m_I-131 . N_A / M_I-131

N_I-131 = (1 μg) x (6.02×10²³ nuclei/mol) / (130.91 g/mol)

N_I-131 = 4.6 x 10¹⁵ nuclei

2. The activity of the iodine-131 in curies can be determined using its decay constant:

The iodine-131 has half-live of 8.02 days (692928 sec) and therefore its decay constant is:

Using this value for the decay constant we can determine the activity of the sample:

3) and 4) The number of iodine-131 atoms that will remain in 50 days (N_50d) and the time it will take for the activity to reach 0.1 mCi can be calculated using the decay law:

As can be seen, after 50 days the number of iodine-131 atoms and thus the activity will be about 75 times lower. After 82 days the activity will be approximately 1200 times lower. Therefore, the time of ten half-lives (factor 2¹⁰ = 1024) is widely used to define residual activity.

 

See also:

Atom