Counting Errors and Statistics ; Interactions of Radiation with Matter
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Five General Sources of Errors
- Assuming the counting system is calibrated correctly, there are five general sources of error associated with counting a sample:
- a. Self-absorption
- b. Backscatter
- c. Resolving time
- d. Geometry
- e. Random disintegration of radioactive atoms (statistical variations).
- Resolving time is the time interval which must elapse after a detector pulse is counted before another full-size pulse can be counted.
- Any radiation entering the detector during the resolving time will not be recorded as a full size pulse; therefore,the information on that radiation interaction is lost. As the activity, or decay rate, of the sample increases, the amount of information lost during the resolving time of the detector is increased.As the losses from resolving time increase, an additional error in the measurement is introduced.
How to determine true count rate knowing the resolving time.
- R is the true count rate
- R0 is the observed count rate
- t is the resolving time of the detector
/ 1- R0 t
DETERMINATION OF MODE, MEDIAN, AND MEAN
- •Determination of the Mode: In a set of numbers the number that is repeated more often than any other is the Mode.
- • Determination of the Median: In the same set numbers where one half are below and the other half are above is the Median.
- • Determination of the Mean (Average): This is found by adding all of the numbers in a set together, and dividing by the number of values in the set.
VARIANCE AND STANDARD DEVIATION
- Using the Gaussian distribution we need to define the terms "variance" and "standard deviation“.
- The amount of scatter of data points around the mean is defined as the sample variance.
- It tells how much the data "varies" from the mean.
- Mathematically the standard deviation is the square root of the variance.
- A term more precise than the variance is standard deviation, represented by σ (“sigma”).
- The standard deviation of a population is defined mathematically as:
Standard Deviation curve properties
- If most of the data points are located close to the mean(average), the curve will be tall and steep and have a low numerical value for a standard deviation.
- If data points are scattered, the curve will be lower and not as steep and have a larger numerical value for a standard deviation.
- In a Gaussian distribution, it has been determined mathematically that:
- 68.2% of the area under the curve falls within the data point located at the mean ± (plus or minus) one standard deviation (1σ);
- 95.4% of the area under the curve falls between the data point located at ± two standard deviations (2σ);
- 99.97% of the area under the curve falls between the data point located at ± three standard deviations (3σ).
- The Chi-squared test (pronounced "ki") is used to determine the precision of a counting system.
- Precision is a measure of exactly how a result is determined without regard to its accuracy.
- It is a measure of the reproducibility of a result, or in other words, how often that result can be repeated, or how often a "success" can be obtained.
- This test results in a numerical value, called the Chi-squared value (Χ2), which is then compared to a range of values for a specified number of observations or trials.
- This range represents the expected (or predicted) probability for the chosen distribution.
- If the Χ2 value is lower than the expected range, this tells us that there is not a sufficient degree of randomness in the observed data.
- If the value is too high, it tells us that there is too much randomness in the observed data.
- The Chi-squared test is often referred to as a "goodness of-fit" test.
- It answers the question: How well does this data fit a distribution curve?
- If it does NOT fit a curve indicating sufficient randomness, then the counting instrument may be malfunctioning.
- A detector intercepts and registers only a fraction of the total number of radiations emitted by a radioactive source.
- The major factors determining the fraction of radiations emitted by a source that are detected include:
- • The fraction of radiations emitted by the source which travel in the direction of the detector window
- • The fraction emitted in the direction of the detector window which actually reach the window.
- • The fraction of radiations incident on the window which actually pass through the window and produce an ionization
- • The fraction scattered into the detector window
- The detector efficiency gives us the fraction of counts detected per disintegration, or c/d.
- Since activity is the number of disintegrations per unit time, and the number of counts are detected in a finite time, the two rates can be used to determine the efficiency if both rates are in the same units of time.
- Counts per minute (cpm) and disintegrations per minute (dpm) are the most common with counts per second (cps) and Bq (disintegrations per second) becoming more common in the US.
- Percent efficiency = cpm/dpm * 100
PROPAGATION OF ERROR
- The error present in a measurement includes the error present in the sample count, which contains both sample and background, and the error present in the background count.
- Rules for propagation of error preclude merely adding the two errors together.
- The total error in the measurement is calculated by squaring the error in the background and adding that to the square of the error in the sample count, and taking the square root of the sum.
= sqrt of (e2s+b
- es = error present in measurement(sample)
- es+b = error in sample count(sample + bkg)
- eb = error present in bkg count
How to minimize count errors.
- Some reduction in error can be obtained by increasing the background counting time.
- TB and TS are the background and sample counting times, respectively.
- These are the factors that we have absolute control over.
- In the previous section we talked about the reliability of the count itself.
- We have been able to state that a count under given circumstances may be reproduced with a certain confidence level, and that the larger the number of counts the greater the reliability.
- The total counting time required depends upon both the sample and background count rates.
- For high sample activities the sample count time can be relatively short compared to the background count time.
- For medium count rates we must increase the sample count time in order to increase precision.
- As the sample activity gets even lower, we approach the case where we must devote equal time to the background and source counts.
- By counting low activity samples for the same amount of time as that of the background determination, we increase the precision of our sample result.
- The detection limit of a measurement system refers to the statistically determined quantity of radioactive material (or radiation) that can be measured (or detected) at a preselected confidence level. This limit is a factor of both the instrumentation and technique/procedure being used.
- The two parameters of interest for a detector system with a background response greater than zero are:
- LC Critical detection level: the response level at which the detector output can be considered "above background"
- LD Minimum significant activity level, i.e., the activity level that can be seen with a detector with a fixed level of certainty
Two types of statistical counting errors
- Two types of statistical counting errors must be considered quantitatively in order to define acceptable probabilities for each type of error:
- Type I - occurs when a detector response is considered above background when in fact it is not (Type I errors are associated with LC)
- Type II - occurs when a detector response is considered to be background when in fact it is greater than background (Type II errors are associated with LD)
Interactions of Radiation with Matter -Ionization
- Ionization - Ionization is any process which results in the removal of abound electron from an electrically neutral atom or molecule by adding enough energy to the electron to overcome its binding energy.
- This leaves the atom or molecule with a net positive charge.
- The result is the creation of an ion pair made up of the negatively charged electron and the positively charged atom or molecule.
- A molecule may remain intact or break-up, depending on whether an electron that is crucial to molecular bonds is affected by the event.
Interactions of Radiation with Matter - Excitation
- Electron excitation is any process that adds enough energy to an electron of an atom or molecule so that it occupies a higher energy state than its lowest bound energy state.
- The electron remains bound to the atom or molecule, but depending on its role in the bonds of the molecule, molecular break-up may occur.
- No ions are produced and the atom remains electrically neutral.
- Nuclear Excitation is any process that adds energy to a nucleon in the nucleus of an atom so that it occupies a higher energy state (smaller binding energy).
- The nucleus continues to have the same number of nucleons and can continue in its same chemical environment
Interactions of Radiation with Matter - Bremsstrahlung
- Bremsstrahlung is the radiative energy loss of moving charged particles as they interact with the matter through which they are moving.
- Significant bremsstrahlung results from the interaction of a high speed charged particle with the nucleus of an atom (positive charge) via the electric force field.
- In the case of a negatively charged electron, the attractive force slows down the electron, deflecting it from its original path.
- The kinetic energy that the particle loses is emitted as a photon.
- Bremsstrahlung has also been referred to as "braking radiation", "white radiation", and "general radiation".
- Bremsstrahlung production is enhanced with high Z materials (larger coulomb forces) and high energy electrons (more interactions occur before all energy is lost).
Directly Ionizing Radiation
- Charged particles do not require physical contact with atoms to interact with them.
- The "Coulomb force" will act over a distance to cause ionization and excitation in the absorber medium.
- Charged particles (such as alpha and beta) that lose energy in this way are called directly ionizing radiation.
- The strength of this force depends on:
- – Energy (speed) of the particle
- – Charge of the particle
- – Density and atomic number (number of protons) of the absorber
- The "Coulomb force" for even a singly charged particle (an electron) is significant over distances greater than atomic dimensions.
- For all but very low physical density materials, the loss of kinetic energy for even an electron is continuous because the "Coulomb force" is constantly "pushing" on electrons of at least one atom and possibly many atoms at the same time.
- Specific ionization is the number of ion pairs formed by the particle per unit path length.
- The number of ion pairs produced is dependent on the type of ionizing particle and the material being ionized.
- An alpha particle traveling through air has a specific ionization of 80,000 ion pairs per cm of travel.
- A beta particle has a specific ionization of about 5,000 ion pairs per cm of travel in air.
- Specific ionization is a macroscopic quantity that accounts for all energy losses that occur before an ion pair is produced.
Linear Energy Transfer
- The LET is the average energy locally deposited in an absorber resulting from a charged particle per unit distance of travel (keV/cm).
- The LET is a measure of the local concentration of energy per path length resulting from ionization effects.
- Biological damage from radiation results from ionization;
- the LET is used for determining quality factors in the calculation of dose equivalent.
- Stopping power of an absorber is its ability to remove energy from a beam of charged particles.
- Stopping power is measured as the average energy lost by a charged particle per unit distanced travelled (keV/cm).
- Stopping power and LET may have the same units but are not equal.
- LET ≤ stopping power because some electron ions may interact via Bremsstrahlung or excitation and the resulting photons escape the local area.
- Materials having higher stopping power values cause the particle to lose its energy over shorter distances.
- Inversely related to the stopping power of the absorber is the range of the charged particle.
- The range of a charged particle in an absorber is the average depth of penetration of the charged particle into the absorber before it loses all its kinetic energy and stops.
- If a particle has a high range, the absorber has a low stopping power.
- If the particle has a short range, the absorber has a high stopping power.
- Specific Ionization, Linear Energy Transfer, Stopping Power, and Range can all be related to each other if one knows the average amount of energy needed to ionize a material.
- The average amount of energy needed to create an ion pair in a given medium is called the W-Value for the medium.
Summary of Energy Loss Terms and Units
- An alpha particle is made up of two protons and two neutrons, strongly bound together by nuclear forces.
- As an alpha particle approaches an electron, it experiences a strong electrostatic attraction, if it approaches an atomic nucleus it will tend to be repelled.
- Alpha particles have a mass about 8,000 times that of an electron.
- They have velocities on the order of 1/20 the speed of light.
- An alpha particle will have a high probability of interacting with orbital electrons and atomic nuclei.
- When speaking of "collisions" between subatomic particles, it should be understood that the particles need approach each other only sufficiently close for Coulomb forces to interact.
Ionization by alpha particle
- Alpha collisions may result in energy transfer by:
- –ionization and/or
- The kinetic energy of the alpha particle is gradually dissipated by ionizing or exciting atoms until it captures two electrons and becomes a helium atom.
- Since the average amount of energy to ionize most materials is much less than the initial energy of most alpha particles, many ionizations will occur before the alpha particle is stopped.
- Due to the high probability of interaction between an alpha particle and the orbital electrons of the absorbing medium and because of the +2 charge, a large number of ion pairs are formed per unit path length.
- This type of radiation loses its energy over a relatively short distance.
- For these reasons, the range of alpha particles is much less than the range of other forms of radiation.
- It is, a highly ionizing, weakly penetrating radiation.
- Alpha particles from a given radionuclide are all emitted with the same energy, consequently those emitted from a given source will have approximately the same range in a given material
- .Alpha particle range is usually expressed in centimeters of air.The relationship between range and energy has been expressed as:
- The number of ion pairs formed per centimeter of path in any given medium is called the specific ionization for that particular ionizing radiation.
- Approximately 34 electron volts of energy is lost for each primary ion pair formed in air.
- Depending on the energy of the alpha particle, the number of ion pairs formed per centimeter of path in air will range from 5,000 to 80,000.
- A beta particle is a free (unbound) electron with kinetic energy.
- The rest mass and charge of a beta particle are the same as that of an orbital electron.
- The negatively charged electron has an anti-particle, which has the same mass but a positive charge called a positron.
- Their masses are very much smaller than the mass of the nuclei of the atoms making up the absorbing medium.
- An interaction between a positively charged beta particle or a negatively charged beta particle and an orbital electron is therefore an interaction between two charged particles of similar mass.
- Negatively charged beta particles and orbital electrons have like charges;
- they experience an electrostatic repulsion when in the vicinity of one another. Positively charged beta particles (positrons really) and orbital electrons have opposite charges, so they experience an electrostatic attraction when in the vicinity of one another.
- Because the rest masses are equal, the interaction between either of these two beta particles and an orbital electron is similar to the collision between billiard balls.
- A beta particle may lose all of its energy in a single collision.
- In such an interaction, the target electron acquires such high kinetic energy it becomes a particle similar to the beta particle.
- Normally, however, a beta particle of either charge loses its energy in a large number of ionization and excitation events in a manner analogous to the alpha particle.
Beta Absorption (continued)
- Due to the smaller size and charge of the electron, there is a lower probability of beta radiation interacting in a given medium -- the range of a beta particle is greater than an alpha particle of comparable energy.
- A negatively charged beta particle has a charge opposite to that of the atomic nucleus, therefore an electrostatic attraction will be experienced as the beta approaches the nucleus.
- A positively charged beta particle has the same charge as the atomic nucleus, therefore an electrostatic repulsion will be experienced as the beta approaches the nucleus.
- Since the mass of either beta particle is small compared with that of a nucleus, large deflections of the beta can occur in collisions, particularly when electrons of low energies are scattered by high atomic number elements.
- A beta particle usually travels a random path in an absorbing medium.
- A beta particle may transfer energy through ionization and excitation.
- A beta may have a Bremsstrahlung interaction with an atom which results in the production of X-rays.
- The figure schematically shows a Bremsstrahlung interaction.
- A high energy beta penetrates the electron cloud surrounding the nucleus of the atom.
- This results in a change in velocity/kinetic energy of the particle and the emission of a Bremsstrahlung X-ray.
- The energy of the X-ray emitted depends on how much deflection of the beta particle occurred, which depends on how close the electron came to the nucleus.
- Therefore, a spectrum of different energy X-rays are observed from the many different Bremsstrahlung encounters an electron will have before it loses all of its energy.
- Because it is much less likely for a close encounter with the nucleus than a distant encounter, there are more low energy X-rays than high energy X-rays (maximum energy is the energy of the beta particle).
- Bremsstrahlung becomes an increasingly important mechanism of energy loss as the initial energy of the beta increases, and the atomic number of the absorbing medium increases.
- Beta particles resulting from radioactive decay may be emitted with an energy varying from practically zero up to a maximum energy.
- Each beta particle will have a range in an absorber based on its energy.
- Determining the number of beta particles found at a given depth in an absorber and the number of X-rays produced is complex and a function of the energy distribution of the beta particles.
Indirectly Ionizing Radiation
- Electromagnetic radiation and neutrons (no charge) have no coulomb force field extending beyond their physical dimensions to interact with the fundamental particles of matter.
- They must come sufficiently close and their physical dimensions contact these particles in order to interact.
- Radiation and neutrons have physical dimensions much smaller than atomic dimensions.
- They move freely through the largely empty space of matter and have a small probability of interacting with matter.
- Uncharged radiation does not continuously lose energy by constantly interacting with the absorber.
- Uncharged radiation may penetrate material and move "through" many atoms or molecules before its physical dimension contacts that of an electron or nucleus.
- This type of radiation is called indirectly ionizing radiation.
- The probability of interaction is dependent upon the energy of the radiation and the density and atomic number of the absorber.
- When indirectly ionization particles do interact, they produce directly ionizing particles (charged particles) that cause secondary ionizations.
- X- and gamma rays differ only in their origin.
- Both are electromagnetic waves, and differ from radio waves and visible light waves only in having much shorter wavelengths.
- The difference in name is used to indicate a different source:
- gamma rays are of nuclear origin, while
- X-rays are of extra-nuclear origin.
- Both X-rays and gamma rays have zero rest mass, no net electrical charge, and travel at the speed of light.
- They can be viewed as packets of energy (quanta) that interact with atoms to produce ionization even though they themselves possess no net electrical charge.
Three major mechanisms by which gamma rays lose energy
- There are three major mechanisms by which gamma rays lose energy by interacting with matter:
- –Photoelectric Effect
- –Compton Scattering
- –Pair Production
- The photon imparts all of its energy to an orbital electron of some atom.
- The photon simply vanishes.
- The photoelectric effect is only significant for initial photon energies less than 1 MeV.
- The energy is imparted to the orbital electron in the form of kinetic energy, overcoming the binding energy, causing the electron to leave its orbit, creating an ion pair.
- The high velocity electron, called a photoelectron, is a directly ionizing particle and typically has sufficient energy to knock other electrons from the orbits of other atoms, and it goes on its way producing secondary ion pairs until all of its energy is expended.
- The probability of photoelectric effect is maximum when the energy of the photon is equal to the binding energy of the electron.
- The tighter an electron is bound to the nucleus, the higher the probability of photoelectric effect so most photoelectrons are inner-shell electrons.The photoelectric effect is seen primarily as an effect o flow energy photons with energies near the electron binding energies of materials and high Z materials whose inner-shell electrons have high binding energies.
- In Compton scattering there is a partial energy loss for the incoming photon.
- The photon interacts with an orbital electron of some atom and only part of the energy is transferred to the electron.
- A photon continues on with less energy and in a different direction to conserve momentum in the collision.
- The high velocity electron, referred to as a Compton electron, produces secondary ionization, and the "scattered" photon continues on until it loses more energy in another photon interaction.
- By this mechanism of interaction, photons in a beam may be randomized in direction and energy, so that scattered radiation may appear around corners and behind shields where there is no direct line of sight to the source.
- The probability of a Compton interaction increases for loosely bound electrons.
- Most Compton electrons are valence electrons.
- Compton scattering is primarily seen as an effect of medium energy photons.
- Compton scattering is the dominant interaction for most materials for photon energies between 200 keV and 5 MeV.
- Pair production (the third mechanism) occurs when all of energy of the photon is converted to mass.
- This conversion of energy to mass only occurs in the presence of a strong electric field, which can be viewed as a catalyst.
- Such strong electric fields are found near the nucleus of atoms and is stronger for high Z materials.
- In pair production a gamma photon simply disappears in the vicinity of a nucleus, and in its place appears a pair of electrons:
- one negatively charged and one positively charged.
- These anti-particles are also called an electron and a positron respectively.
- The mass of these electrons has been created from the pure energy of the photon.
- Pair production is impossible unless the gamma ray possesses greater than 1.022 MeV of energy to make up the rest mass of the particles.
- Practically speaking, it does not become important until 2 MeV or more of energy is possessed by the incident photon.
- Any excess energy in the photon above the 1.022 MeV required to create the two electron masses, is shared between the two electrons as kinetic energy of motion.
- The probability of pair production is lower than photoelectric and Compton interactions because the photon must be close to the nucleus.
- The probability increases for high Z materials and high energies.
- The negative electron behaves in the same way as any electron with kinetic energy, producing secondary ion pairs until it loses all of its energy of motion.
- The positive electron (positron) also produces secondary ionization as long as it is in motion, but when it has lost its energy and slowed almost to a stop, it encounters a free (negative) electron somewhere in the material.
- The two are attracted by their opposite charges, and upon contact, because they are antiparticles, they annihilate each other, converting the mass of each back into pure energy.
- Thus, two gamma rays of 0.511 MeV each arise at the site of the annihilation.
- The ultimate fate of the "annihilation gammas" is either photoelectric absorption or Compton scattering followed by photoelectric absorption.
Pair Production and Annihilation
- The neutron has a mass number of 1 and no charge.
- Because it has no charge the neutron can penetrate relatively easily into a nucleus.
- Free unbound neutrons are unstable (radioactive) and disintegrate by beta emission with a half-life of approximately 10.6 minutes.
- The resultant decay product is a proton which eventually combines with a free electron to become a hydrogen atom.
- During the time when free neutrons exist, they can interact with the material they are in and lose energy.
- Neutron interactions with the nucleus are very energy dependent so neutrons are classified on the basis of their kinetic energies.
- When a neutron is in "thermal equilibrium" with a material, it has kinetic energies appropriate for the kinetic energies of the atoms of the material.The velocity of free neutrons in various substances at room temperature is approximately 2,200 meters per second.
- Neutrons with this average kinetic energy at 20° C are called thermal neutrons.
- When neutrons are classified by their kinetic energies into various categories, frequently the energy ranges and names given to each neutron energy range is determined by the materials being used or research being conducted.
- For example, reactor physics, weapons physics, accelerator physics, and radiobiology each have generated a classification system that serves their needs.
Neutron Energy Categories
- Typically the only category common to all the classification categories is thermal.
- The classification used for neutron interaction in tissue is important in radiation dosimetry.
- One should be familiar with the classification of neutrons by energy that applies to the area where they are working so no confusion arises when using terminology.
- Classification of neutrons according to kinetic energy is important from two standpoints:
- – the interaction of neutrons with the nuclei of atoms differs with the neutron energy
- – the methods of producing, detecting and shielding against the various classes of neutrons are different
- Detection of neutrons is relatively difficult, due to:
- -the lack of ionization along their paths;
- -negligible response to externally applied electric, magnetic or gravitational fields;
- -the fact that they interact primarily with atomic nuclei, which are exceedingly small.
Neutron Interactions described
- When describing neutron reactions with a nucleus, the standard notation is (n,Y) where n is the initial neutron and Y is the resulting emissions following the interaction with the nucleus.
- Radiative capture with gamma emission is the most common type of reaction for slow neutrons.
- This (n,γ) reaction often results in product nuclei which are radioactive.
- A second type of general reaction for slow neutrons is that giving rise to charged particle emission.
- Typical examples include (n,p), (n,d), and (n,α) reactions, i.e., reactions in which a proton, a deuteron, or an alpha particle is ejected from the target nucleus.
- A third type of neutron-induced nuclear reaction is fission.Typically, fission occurs following the absorption of aslow neutron by several of the very heavy elements.
- This process of converting a stable nucleus to its radioactive counterpart by neutron bombardment is called "neutron activation."
Neutron Reaction (Fission)
- When 235-U nuclei undergo fission by neutrons, an average of 2 to 3 neutrons are expelled along with associated gamma radiation.
- The nucleus splits into two smaller nuclei which are called primary fission products or fission fragments.
- These products usually undergo radioactive decay to form secondary fission product nuclei.
- As an example, if one neutron fissions a 235-U nucleus, it could yield yttrium-95, xenon-139, two neutrons and fission energy.
- There are some 30 different ways that fission may take place with the production of about 60 primary fission fragments.
- These fragments and the atoms which result from their decay are referred to as fission products, and they number between 400 and 600, according to the type and number of nucleons their nuclei possess.
- Many fission products have found application in medicine, industry, and research.
- A well known example is 131-I which is used extensively in medicine as both a diagnostic and therapeutic agent.
- The fission process is the source of energy for nuclear reactors and some types of nuclear weapons.
- Neutrons generated from the fissioning of the fuel in a reactor are used to activate stable materials to a radioactive form.
- Many radioisotopes used in medicine are produced by neutron activation.
Elastic and Inelastic Scattering
- Scattering is the dominant process for fast neutrons when the neutron is moving too fast to become absorbed by a nucleus.
- The original free neutron continues to be a free neutron following the interaction.
- Multiple scattering by a neutron is the mechanism of slowing down or "moderating" fast neutrons to thermal energies.
Elastic and Inelastic Scattering
- Elastic scattering occurs when a neutron strikes a nucleus.
- The neutron can transfer much of its kinetic energy to that nucleus which recoils off with the energy lost by the neutron.
- The recoil nucleus can be knocked away from its electrons and, being positively charged, can cause ionization and excitation.
- Hydrogen causes the greatest energy loss to the neutron because the single proton in the nucleus is approximately the same mass as the neutron.
- During elastic scattering reactions, no gamma radiation is given off by the nucleus.
occurs when a neutron strikes a large nucleus.
- The neutron penetrates the nucleus for a short period of time, transfers energy to a nucleon inside, and then exits with a small decrease in energy.
- The nucleus is left in an excited state, emitting gamma radiation, which can cause ionization and/or excitation.
Inelastic Scattering image
Neutron Reactions in Biological Systems
- Fast neutrons lose energy in soft tissue mainly by repeated scattering interactions with hydrogen nuclei.
- The hydrogen nuclei are themselves scattered in the process.
- The scattered hydrogen nuclei have been "knocked"free of their electron and are thus moving protons (called recoil protons) which cause ionization.
- Slow neutrons are captured in soft tissue and release energy by two principal mechanisms
- The gamma and proton energies may be absorbed in the tissue and cause damage.
- Whenever charged particles, neutrons or photons are able to penetrate the nucleus and have sufficient energy, transmutations may be caused which often result in radioactivity.
- The bombarding projectile can be a neutron, proton, deuteron, alpha particle, electron, or gamma photon.
- When shielding against X-rays and gamma rays, photons are removed from the incoming beam on the basis of the probability of an interaction such as photoelectric effect, Compton scatter, or pair production.
- This process is called attenuation and can be described using the "linear attenuation coefficient", μ, which is the probability of an interaction per path length x through a material.
- The linear attenuation coefficient varies with photon energy, type of material, and physical density of material.
- Mathematically the attenuation of photons is given by:
Photon attenuation equation explained
- Intensity is reduced exponentially with shield thickness and only approaches zero for large thicknesses.
- I(x) never actually equals zero.
- Shielding for X-rays and gamma rays becomes an ALARA issue and not an issue of shielding to zero intensities.
- The formula is used to calculate the radiation intensity from a narrow beam behind a shield of thickness x, or to calculate the thickness of absorber necessary to reduce radiation intensity to a desired level.
- Tables and graphs are available which give values of μ determined experimentally for all radiation energies and many absorbing materials.
- The larger the value of μ the greater the reduction in intensity for a given thickness of material.
- The fact that lead has a high μ for X- and gamma radiation is why it is widely used as a shielding material.
- Attenuation of the initial beam of photons occurs by photoelectric, Compton, and pair production interactions, additional photons can be produced by subsequent interactions.
Photon attenuation build up equation
- If the beam is narrow, these additional photons are "randomized" and are no longer part of the narrow beam of radiation.
- If the beam is broad, photons can be "randomized" and scattered into the area one is trying to shield.
- The secondary photons are accounted for by a build up factor, B, in the attenuation equation as follows:
- I = B I0e-ux
- where B is the buildup factor.
- Tables of dose build-up factors can be found in the Radiological Health Handbook.
- The buildup is mostly due to scatter.
- Scattered radiation is present to some extent whenever an absorbing medium is in the path of radiation.
- The absorber then acts as a new source of radiation.
- Frequently, room walls, the floor, and other solid objects are near enough to a source of radiation to make scatter appreciable.
- When a point source is used under these conditions, the inverse square law is no longer completely valid for computing radiation intensity at a distance.
Shielding Fast Neutrons
A complete shield for a capsule type neutron source may consist of, first, a thick layer of paraffin to slow down the neutrons, then a surrounding layer of cadmium to absorb the slow neutrons, and finally, an outer layer of lead to absorb both the gammas produced in the cadmium and those emanating from the capsule.
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