Help:Dosimetry & Shielding H*(10)
Level: Introductory, Intermediate
In this section the formalism for dosimetry and shielding calculations is developed. In the following sections a brief description of the interaction of radiation with matter is given together with the physical basis of radiation dosimetry and shielding.
Biological Effects of Ionising Radiation
When ionising radiation passes through tissue, the component atoms may be ionised or excited. As a result the structure of molecules may change and result in damage to the cell. In particular, the genetic material of the cell, the DNA (deoxyribonucleic acid) may be changed. Two categories of radiation-induced injury are recognised: deterministic effects and stochastic effects. Deterministic effects are usually associated with high doses and are characterised by a threshold. Above this threshold the damage increases with dose. Stochastic effects are associated with lower doses and have no threshold. The main stochastic effect is cancer.
The radiation dose depends on the intensity and energy of the radiation, the exposure time, the area exposed and the depth of energy deposition. Various quantities such as the absorbed dose, the equivalent dose and the effective dose have been introduced to specify the dose received and the biological effectiveness of that dose .
 J. R. Cooper, K. Randle, R. S. Sokhi, Radioactive Releases in the Environment: Impact and Assessment, Wiley, 2003.
Usually the interaction of radiation with matter involves a transfer of energy from the radiation to the matter. Ultimately, the energy transferred either to tissue or to a radiation shield is dissipated as heat. The radiation dose depends on the intensity and energy of the radiation, the exposure time, the area exposed and the depth of energy deposition. Various quantities such as the absorbed dose, the effective dose and the equivalent dose have been introduced to specify the dose received and the biological effectiveness of that dose.
One of the earliest established phenomena regarding radiation is its ability to ionize a gas. On this basis, the unit called the roentgen (R) was introduced. The roentgen is defined as the amount of exposure that will create 2.58 × 10−4 C of singly charged ions in 1 kg of air at STP. Since about 34 eV of energy is needed to produce one ion pair in air, 1 R corresponds to an energy absorption per unit mass of 0.0088 J kg−1. Today it is more usual to use the quantity called the absorbed dose (D) which specifies the amount of radiation absorbed per unit mass of material. The modern SI unit of absorbed dose is the gray (Gy) where one gray is one joule per kilogram 1Gy = 1 J kg−1. In dosimetry, it is useful to define an average dose for a tissue or organ . The absorbed dose to the mass , is defined as the imparted energy per unit mass of the tissue or organ, i.e.
The absorbed dose rate is the rate at which an absorbed dose is received. The units are Gy s−1, mGy hr−1, etc. Biological effects depend not only on the total dose to the tissue but also on the rate at which this dose was received. In organisms, mechanisms exist which enable molecules such as deoxyribonucleic acid (DNA) to recover if they have not been too badly damaged. Hence it is possible for organs to recover from a potentially lethal dose provided that the dose was supplied at a sufficiently slow rate. This phenomena can be exploited in cancer radiotherapy.
For more information see Effective dose calculation with examples
Quality or Weighting Factor
The biological effect of radiation is not directly proportional to the energy deposited by radiation in an organism. It depends, in addition, on the way in which the energy is deposited along the path of the radiation, and this in turn depends on the type of radiation and its energy. Thus the biological effect of the radiation increases with the linear energy transfer (LET) defined as the mean energy deposited per unit path length in the absorbing material (units keV μm−1). Thus for the same absorbed dose, the biological effect from high LET radiation such as α particles or neutrons is much greater than that from low LET radiation such as β or γ rays.
The quality or weighting factor, wR, is introduced to account for this difference in the biological effects of different types of radiation. The weighting factors for the various types of radiation and energies is given in the table.
|Radiation type||Radiation weighting factor, wR|
|Electronsa and muons||1|
|Protons and charged pions||2|
|Alpha particles, fission fragments, heavy ions||20|
|Neutrons|| A continuous function of neutron energy
See Radiation weighting factors
| All values relate to the radiation incident on the body or, for internal radiation sources,
emitted from the incorporated radionuclide(s).
a Note the special issue of Auger electrons discussed in ICRP 103 (2007).
 ICRP Publication 72. Annals of the ICRP 26, 1996, Pergamon Press
The absorbed dose does not give an accurate indication of the harm that radiation can do. Equal absorbed doses do not necessarily have the same biological effects. An absorbed dose of 0.1 Gy of alpha radiation, for example, is more harmful than an absorbed dose of 0.1 Gy of beta or gamma radiation. To reflect the damage done in biological systems from different types of radiation, the equivalent dose is used. It is defined in terms of the absorbed dose weighted by a factor which depends on the type of radiation i.e.
where is the equivalent dose in tissue T and is the radiation weighting factor. The ICRP weighting factors are given in the previous section.
Equal equivalent doses from different sources of radiation delivered to a point in the body should produce approximately the same biological effects. However, a given equivalent dose will in general produce different effects in different parts of the body. A dose to the hand is, for example, considerably less serious than the same dose to blood forming organs. If there are several types of radiation present, then the equivalent dose is the weighted sum over all contributions, i.e.
The SI unit of dose is the sievert, Sv (1 Sv = 1 J kg-1, the old unit is the rem, 1 Sv = 100 rem). This is the equivalent dose arising from an absorbed dose of 1 Gy. Hence for γ rays, where wR = 1, an absorbed dose of 1 Gy gives an equivalent dose of 1 Sv. The same absorbed dose for α particles, where wR = 20, gives an equivalent dose of 20 Sv. The equivalent dose rate is the rate at which an equivalent dose is received, i.e.
The equivalent dose rate is expressed in Sv/s or mSv/hr.
The sievert, Sv, is the unit describing the biological effect of radiation deposited in an organism. The biological effect of radiation is not just directly proportional to the energy absorbed in the organism but also by a factor describing the quality of the radiation. An energy deposition of, for example, 6 J per kg due to gamma radiation (quality factor = 1) i.e. 6 Sv is lethal. This same energy deposited in the form of heat (quality factor = 0) will only increase the body temperature by 1 mK and is therefore completely harmless.The difference between the two types of radiation is due to the fact that biological damage arises from ionisation.
For more information see Effective dose calculation with examples
|Bone marrow (red), Colon, Lung, Stomach, Breast, Remainder tissues*||0.12||0.72|
|Bladder, Oesophagus, Liver, Thyroid||0.04||0.16|
|Bone Surface, Brain, Salivary glands, Skin||0.01||0.04|
|* Remainder tissues: Adrenals, Extrathoracic (ET) region, Gall bladder, Heart, Kidneys, Lymphatic nodes, Muscle, Oral mucosa, Pancreas, Prostate(♂), Samll intestine, Spleen, Thymus, Uterus/cervix(♀).|
In general, cells which undergo frequent cell division, and organs and tissue in which cells are replaced slowly, exhibit high radiation sensitivity. This is why different tissues show different sensitivities to radiation. The thyroid, for example, is much less sensitive than bone marrow. In order to take these effects into account, equivalent doses in different tissues must be weighted. The resulting effective dose is obtained using:
The values for tissue weighting factors are given in the table.
For more information see Effective dose calculation with examples
Committed Effective Dose, E(τ)
A person irradiated by gamma radiation outside the body will receive a dose only during the period of irradiation. However, following an intake by ingestion or inhalation, some radionuclides persist in the body and irradiate the various tissues for many years. The total radiation dose in such cases depends on the half-life of the radionuclide, its distribution in the body, and the rate at which it is expelled from the body. Detailed mathematical models allow the dose to be calculated for each year following intake. The resulting total effective dose delivered over a lifetime (70 years for infants, 50 y for adults) is called the committed effective dose. The name arises from the fact that once a radionuclide has been taken up into the body, the person is “committed” to receiving the dose. The ICRP has published values for committed doses following intake of 1 Bq of radionuclide via ingestion and inhalation. These are known as the effective dose coefficients e(τ) and have been calculated for intake by members of the public at six standard ages, and for intake by adult workers. The unit of the effective dose coefficient is Sv/Bq.
Collective Effective Dose
On the assumption that radiation effects are directly proportional to the radiation dose without a threshold, then the sum of all doses to all individuals in a population is the collective effective dose with unit manSv. As an example, in a population consisting of 10,000 persons, each receives a dose of 0.1 mSv. The collective dose is the 10 000 × 0.0001 = 1 manSv. The effects of various doses to man are listed below.
 K. H. Lieser, Nuclear and Radiochemistry: Fundamentals and Applicationss. VCH/Wiley 1997.
Radiotoxicity and Annual Limits of Intake (ALI)
Radiotoxicity of an isotope refers to its potential capacity to cause damage to living tissue as the result of being deposited inside the body. This damage potential is governed by the type and energy of the radioactive disintegration, the physical halflife, the rate at which the body excretes the material, and the radio-sensitivity of the critical organ. The radiotoxicity is defined here in terms of dose received by a population ingesting all the radioactive materials present at a given time, taking into account the nature and energy of the emitted radiation and its effect on biological organisms. For this purpose it is suitable to use the Committed Effective Dose E(τ) – see inset – as a measure of the radiotoxicity, hence
The committed effective dose of a radionuclide is given by the effective dose coefficient emultiplied by the activity of the radionuclide at the time of intake, hence
where A is the activity of the radionuclide at the moment of intake.
It should be noted that many radionuclides decay to nuclides that are themselves radioactive (radioactive daughters). The effective dose coefficients take into account the ingrowth of daughters in all regions of the body following an intake of unit activity of the parent nuclide. They do not take into account any activity of daughter nuclides in the initial intake. This is in line with current and previous ICRP dose compendia. The activity is just the number of disintegrations per second and is measured in units of Becquerel, Bq (1Bq = 1 disintegration per second). The effective dose coefficient e is a measure of the damage done by ionising radiation associated with the radioactivity of an isotope. It accounts for radiation and tissue weighting factors, metabolic and biokinetic information. It is measured in units of Sievert per Becquerel (Sv/Bq) where the Sievert is a measure of the dose arising from the ionisation energy absorbed.
The Annual Limit of Intake (ALI) of an isotope is defined as the activity required to give a particular annual dose. Publication 60 of the ICRP recommends a committed effective dose limit of 20 mSv per year, hence
The ALI is a calculated value based on the primary dose limit and gives only the annual limit of intake. It is sometimes more useful to establish the limits on the concentration of a radionuclide in air orwater whichwould lead to this intake. For this purpose the derived air concentration (DAC) is introduced for airborne contaminants. The DAC is the average atmospheric concentration of the radionuclide which would lead to the ALI in a reference person as a consequence of exposure at the DAC for a 2000 h working year. A reference person inhales 20 litres of air per minute or 2400 m3 during the working year. The derived air concentration is
137Cs, for example, has an ALIinh = 3.0 × 106 Bq. It follows that the DAC = 1.2 Bq/m3. Similarly the derived water concentration (DWC) is given by
based on a water intake of 2.5 litre per day. For members of the public, the values obtained for the DAC and DWC should be further reduced by a factor 20 correcponding to a dose limit of 1 mSv per year.
Radiation Hormesis and the Linear Non-Threshold (LNT) Model
Although it is generally believed that low doses arising from chemicals, pharmaceuticals, radiation, etc. produce effects proportional to high doses, there is evidence to suggest this is incorrect and that low doses may have a beneficial effect to biological systems. This positive effect arising from low doses is referred to as “hormesis” from the Greek word “hormaein” which means “to excite”. Radiation hormesis refers to the stimulation of biological functions by low doses of radiation.
The first observation of hormesis dates to the 1940s where it was reported that low doses of Oak bark extract stimulated fungi growth (in contrast to inhibiting growth at high doses). In the 1980s, the first complete report on radiation hormesis was published .
Toxicology, and in particular the dose response relation, is very important in many medical and public-health issues. Predictions based on this relationship have major implications for risk assessment and risk communication to the public. At issue here is the known hormetic (beneficial or positive), response of cells and organisms to radiation dose.
It has been claimed recently  that the toxicological models in current use by regulatory authorities to extrapolate dose response at low doses of carcinogens are incorrect. Traditionally, the dose-response relationship used for risk assessment to obtain the risk from low doses of carcinogens is the so-called “linear non-threshold model” (LNT) shown in the figure. There is increasing evidence, however, that the dose-response relation is actually “U” shaped or “J” shaped. This “U” shape is a manifestation of hormesis where a response stimulation occurs at low doses.
Current radiation protection standards are based on the assumption that all doses, no matter how small, can result in health detriment and the likelihood is directly proportional to dose received; i.e. the accepted dose response relationship for estimating harm is the so-called linear no-threshold (LNT) model. According to the Health Physics Society, there is increasing scientific evidence that this model represents an oversimplification of the biological mechanisms involved and that it results in an overestimation of health risks in the low dose range. The Health Physics Society notes that radiogenic health effects (primarily excess cancers) are observed in human epidemiology studies only at doses in excess of 0.1 Sv delivered at high dose rates. Below this dose, estimation of adverse health effects is speculative. UNSCEAR is also showing increasing reservation toward the use of dose commitment (individual dose integrated over infinite time) and collective dose. Both are consequences of the linear-non-threshold model of radiation effects. Recent radiobiological and epidemiological studies suggest that this model has lost credibility . The organisation is proposing to spend more time and resources to learn the effect of anthropogenic radiation on individual plants and animals. It is well known, for eample, that in Kerala, India, where the natural radiation level (up to about 400 millisieverts per year) is much higher than the average global one (2.4 mSv), black rats for 800 to 1000 generations have shown no adverse biological effects [6, 7].
 T. D. Luckey, Radiation Hormesis, CRC Press, Boca Raton, 1991.
 E. J. Calabrese, et al., Nature 421, 691 (2003).
 R. E. Mitchel, D. R. Boreham, Proc. International Radiation Protection Association, 10th Quadrennial Meeting, Hiroshima, Japan, 15-19 May 2000.
 P. C. Kesavan, in: High Levels of Natural Radiation, L. Wei, T. Sugahara, Z. Tao (eds.), Elsevier, Amsterdam p. 111, 1996.
Gamma Dose Rate
Attenuation of Gamma Radiation
Gamma radiation cannot be completely absorbed, but only reduced in intensity, when passing through matter. If mono-energetic gamma radiation attenuation measurements are made under conditions of good geometry, i.e. with a well-collimated, narrow beam of radiation, as shown in the figure below, a straight-line relationship between the logarithm of the intensity versus the thickness d of the shield is obtained, i.e.
is the gamma radiation intensity transmitted through an absorber of thickness d,
is the gamma radiation intensity at zero absorber thickness,
is the absorber thickness,
slope of the absorption curve – the attenuation coefficient.
Since the product μd in the above relation must be dimensionless, if the absorber thickness is measured in cm, then the attenuation coefficient is called the linear attenuation coefficient μl and has dimension cm-1. If the thickness d is in g/cm2 then the attenuation coefficient is called the mass attenuation coefficient μm and has units of cm2/g. The relationship between these coefficients is:
where ρ is the density of the absorber. The attenuation coefficient is the fraction of the gamma radiation beam attenuation per unit thickness of absorber and is defined as:
where ΔI/I is the fraction of the gamma radiation attenuated by an absorber of thickness Δd. The attenuation coefficient thus defined is sometimes called the total attenuation coefficient.
Generally, for energies between about 0.75 and 5 MeV, almost all materials have, on a mass basis, about the same gamma radiation attenuation properties. To a first approximation, therefore, shielding properties are approximately proportional to the density of the shielding material. Under conditions of good geometry, the attenuation of a beam of gamma radiation is given therefore by:
However, under conditions of poor geometry, i.e. for a broad beam or for a very thick shield, the above relation underestimates the required shield thickness. It assumes that every photon that interacts with the shield will be removed from the beam and thus will not be available for counting in the detector. Under conditions of poor geometry, as shown in the figure below, this assumption is not valid; a significant number of photons may be scattered by the shield into the detector, or photons that had been scattered out of the beam may be scattered back in after a second collision.
The shield thickness for conditions of poor geometry may be estimated by modification of the basic attenuation relation given above through the use of a build-up factor B, i.e.
The build-up factor, which is always greater than 1, may be defined as the ratio of the intensity of the radiation, including both the primary and scattered radiation, at any point in a beam, to the intensity of the primary radiation only at that point. Build-up factors have been calculated for various gamma energies and for various absorbers. The build-up factor is in general a function of the total attenuation coefficient, the thickness of the shielding material d, and the energy of the gamma radiation, i.e. B = B(μd,E), hence
Absorption of Gamma Radiation
The attenuation coefficient discussed above is a measure of how photons are removed from the beam under conditions of good geometry. Attenuation is a result of three basic processes: the photoelectric effect (pe), Compton scattering (cs), and pair production (pp) and the total attenuation coefficient is a sum of the attenuation coefficients for these processes, i.e.
Photoelectric absorption results when a photon interacts with a bound electron. If the energy of the photon is greater than or equal to the binding energy of the electron, the electron is released with kinetic energy equal to any excess energy of the photon over the binding energy. This photoelectron then dissipates its energy to the medium mainly by excitation and ionisation.
Compton scattering results from elastic scattering of the photon with weakly bound or “free” electrons. In this process, the scattered photon has less energy than the incident photon. Since the collision is elastic, the electron gains this loss in photon energy.
Pair production results when the energy of the photon exceeds 1.02 MeV. In the neighbourhood of a heavy nucleus, such a photon can spontaneously disappear and results in the formation of an electron-positron pair. Photon energy in excess of that needed to form the pair appears as kinetic energy of the particles. The positron and electron are projected in the forward direction (relative to that of the initial photon) and lose their kinetic energy by excitation, ionisation, Bremsstrahlung etc. Finally, when the positron has lost its kinetic energy, it will combine with an electron to produce annihilation radiation consisting of two 0.51 MeV photons. The photons may then be lost from the medium or may undergo Compton scattering or photoelectric absorption.
The total attenuation coefficient μ given above is the fraction of the energy of the beam that is removed per unit distance in the medium. The energy absorbed in the medium is determined by the energy absorption coefficient μen. The difference between μ and µen results from the fact that energy may be lost from the medium through Compton scattering and by annihilation radiation. For dose calculations in tissue for example, the energy absorption coefficient µen must be used. For shielding calculations, the attenuation coefficient should be used.
Calculation of Dose Rates
Absorbed Dose Rate in Tissue
From the above discussion, the energy deposition rate (for mono-chromatic radiation) per unit mass of tissue is given by dI/d(ρd) or
where I is the radiation intensity at the detector and (μl/ρ)tis is the mass energy absorption coefficient in tissue. Since the quality factor for gamma radiation is 1, D and H are equal, i.e.
In the case of no shielding, the gamma radiation intensity (energy per unit area per second), I , can be written:
where is the intensity at the source (energy per unit time) and R is the distance from the source to the detector. The source strength can be written:
where A is the activity of the source and E and P the gamma emission energy and emission probability per disintegration respectively. Where there is more than a single emission line, the summation must include all lines. Combining the above relations, one obtains:
where the energy dependence of the mass absorption coefficient has been accounted for. Inserting numerical values, one obtains:
where the activity A is expressed in Bq, R in cm, Ei in keV and μl /ρ in cm2/g. In the case where a shield is used, the corresponding relation is
again inserting numerical values one obtains:
In the evaluation of dH/dt , inaccuracies are introduced due to the fact that (μl/ρ)itis, (μl/ρ)ishield, and Bi are usually only tabulated for discrete energies and interpolation must be used. The main source of inaccuracy here is in the (μl/ρ)ishield value since this is contained inside the exponential function. To avoid this problem, fitting functions are used in the evaluation of (μl/ρ)itis and (μl/ρ)ishield as discussed in the following section. In the case of the build-up factors, due to double dependency on μd and E, the simpler procedure of energy bin allocation is used.
Ambient Dose Rate, Air Kerma Rate, Exposure Rate
A detailed description of the formulae used for these quantities is described in the Photon Dose Rate Constants Application.
Shielding and Buildup
A detailed description of the formulae used when shielding and buildup are important are given in the Appendix: Shielding and Buildup
Absorption in Tissue
The dependence of (μ/ρ)tis on energy is shown in the figure and table below. This data has been taken from the NIST database. In the calculations, a linear interpolation is carried out (actually the linear interpolation is carried out on the log(mass-absorption coefficient) vs. log(energy) plot). For energies lower than the minimum energy (0.001 MeV), an extrapolation is performed.
Data for Tissue
As an example, consider the evaluation of the absorbed dose rate in tissue from 1 MBq of 60Co at im. The six gamma energies and their emission probabilities are shown in the table below.
It follows that...
When there is no shield, this expression reduces to
Hence for A = 1 MBq, R = 100 cm,
Attenuation in Shield Materials
Data from the NIST database were linearly interpolated (again the linear interpolation is carried out on the log(mass-attenuation coefficient) vs. log(energy) plot. For energies lower than the minimum energy (0.001 MeV), an extrapolation is performed. Data for air, aluminium, concrete, iron, lead, tin, tungsten, uranium, water etc. are given in the application (see Attenuation and Buildup tab).
For an explanation of the difference between the mass attenuation and mass absorption coefficients see the section on Absorption of Gamma Radiation
Build-up Factors (B) for Shield Materials
The B values  have been taken from “American National Standard for Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Materials”, ANSI/ANS-6.4.3-1991 (1991). Due to the complexity of the data and their double dependence on energy and mean free path lengths, tabulated values are used. An example of the buildup factors for concrete are shown below. These and many buildup factors for other material (e.g. iron, tin, tungsten, uranium, water, aluminium, air, lead, etc.) are available in the Dosimetry & Shielding H*(10) application in Nucleonica (see Attenuation and Buildup).
 “American National Standard for Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Materials”, ANSI/ANS-6.4.3-1991 (1991)
See also New Photon Exposure Buildup Factors
 B. Schlein, L. A. Slaback, Jr., B. Kent Birky: The Health Physics and Radiological Health Handbook, 3rd edition. Scinta, Silver Spring, MD 1998
 G. E. Chabot, Shielding of Gamma Radiation
Dosimetry & Shielding H*(10) Application
The Dosimetry & Shielding H*(10) application can be accessed from the nuclear science page. The gamma dosimetry and shielding interface is shown in the figure below. The basic geometric arrangement of the source, shield and detector is shown schematically. The lines shown indicate some of the paths of photons which lead to a contribution in the detector. Associated with source, shield and detector are a number of input boxes in which one can specify the source and its strength, the shield material and thickness, and the source detector distance. These input boxes are considered in more detail below.
The Dosimetry & Shielding H*(10) application allows the user to calculate gamma dose rates from point sources of single nuclides and nuclide mixtures. All known gamma lines and emission probabilities for the nuclide(s) are accounted for in the calculation. There are four main modes of operation:
- Calculation of the dose rate for a given shield material and thickness.
- Calculation of the thickness of shield material required to obtain a given dose rate
- Obtain the source strength when the dose rate, shield material and thickness are known
- Calculation of the source/detector distance when the dose rate, shield material and thickness, dose rate are known
The main tab allows the user to select the nuclide, its source strength (in mass, becquerel, curie, number of atoms, or photons/s), source/detector distance, shield material and material thickness. The results of the calculations are given in a results grids i.e.
- the half- and tenth-value thicknesses of shield material
- the ambient dose rate, air kerma rate, and exposure rate.
- the integral ambient dose during an exposure time following cooling.
By using the Show radiation details checkbox, a list of all energy lines and emission probabilities used in the calculation are given. In addition subsidiary quantities used in the calculations, such as the absorption coefficient, number of mean free path in the shield material, and the build-up factor for each energy line are given. The gamma dose rate contribution from each energy line is also listed. The threshold energy for contributions to the dose rate can be set by the user in the Options tab. Finally, a spectrum of the lines used in the calculations are shown in the Graph.
Ambient Dose Rate
The source strength is set by specifying first the unit i.e. Activity(Bq),Activity(Ci), Mass(g), Number of atoms, or photons/s and then the amount. The amount should be entered in scientific notation in the form 1.0, 10.0, 1.0E2, 1.0E-6 etc.
It can be seen that the source strength can be set in photons/s. This is a useful feature for calculating the dose rate, etc. for photons with a given energy. If the photons/s item is selected, the user is prompted to enter the number of photons/s and the energy (keV) of the photons.
The shield can be set by selecting from the shield material input box. The default value is lead. It is assumed that the shield materials are at standard temperature and pressure: T = 300K and P = 0.1MPa. The user has a choice of 10 materials:
• lead (Pb),
• concrete (dry ρ = 2.4 × 103 kg · m−3),
• iron (Fe),
• tin (Sn),
• tungsten (W),
• uranium (U),
• aluminium (Al),
• air (dry air at near sea level – this option has been included in order to compare the “vacuum” calculations (i.e. no shield) with more realistic case where the space between the source and detector is filled with air.)
• tissue (this option allows the user to investigate the absorption of low energy gammas and X-rays in the outer layers of the skin, i.e. in the epidermis and dermis, which act as a natural shield for the body).
Source/Detector Distance (cm)
The source/detector distance is specified by entering the value in the input box. The default value is 100 cm.
On pressing the Start button, the results of the calculation, with the above parameters set, are shown in the Results grids.
The program can be reset by pressing the Reset button directly. The program can be terminated using the File command in the taskbar or by closing the browser window.
The main result is the gamma dose rate as shown in the figure below for 1 MBq Co-60. Te results include:
- half-value layer (HVL) and tenth-value layer thicknesses (TVL) of shield material required to reduce the gamma dose rate to 50% and 10% respectively of the initial value.
- ambient dose rate, air kerma rate, and exposure rate.
- integral ambient dose during an exposure time following cooling.
If the Show radiation details box is checked, a list of all energy lines and emission probabilities used in the calculation are given. In addition subsidiary quantities used in the calculations, such as the absorption coefficient, number of mean free path in the shield material, and the build-up factor for each energy line are given. The gamma dose rate contribution from each energy line is also listed. The threshold energy for contributions to the dose rate can be set by the user in the Options tab. Finally, a spectrum of the lines used in the calculations are shown in the Graph.
In the bottom panel the number of gamma rays, X-rays and gamma + X-rays is given. In addition, the total photon energy emitted per disintegration is shown ∑E.P (in eV per disintegration)
Half- and Tenth-Value Layers
It is useful to evaluate the half-value layer (HVL) and the tenth-value layer (TVL), i.e. the thickness of shield required to reduce the photon intensity to one half or one tenth of its initial value. The half-value layer is obtained from I(x)/I0 = 1/2 = exp(−μx1/2) or x1/2 = HVL = ln2/μ. Similarly, the tenth value layer TVL = ln10/μ.
Ambient Dose Rate, Air Kerma Rate, and Exposure Rate
A detailed description of the formulae used for these quantities is described in the Photon Dose Rate Constants Application.
The radiation details are shown directly under the main results as shown in the figure above. The radiation details consist of the the energies, the corresponding emission probabilities, the mass attenuation coefficients, the B factors, etc. The entries in each of the columns can be arranged by clicking on the column header caption. For example, clicking on the Gamma Energy, the entries are rearranged in decreasing order with the highest energy at the top. Clicking again will rearrange the entries such that the smallest energy is at the top. Clicking on the Emission Probability will arrange the emission probabilities is ascending/descending order. It is also useful here to click on the Gamma Dose Rate so that one can see the most important lines which contribute to the dose.
In addition to the tabular information shown above, the energies and emission probabilities are given in the graph below the grid.
Half and Tenth value shield thicknesses
It is useful to evaluate the half-value layer (HVL) and the tenth-value layer (TVL), i.e. the thickness of shield required to reduce the photon intensity to one half or one tenth of its initial value. The half-value layer is obtained from I(x)/I0 = 1/2 = exp(−μx1/2) or x1/2 = HVL = ln2/μ. Similarly, the tenth value layer TVL = ln10/μ. It should be note that the calculation of the HVL and TVL values depends on whether the geometry in "narrow beam" or "broad beam".
Narrow, well collimated beams
The normal formula for the intensity reduction in shield materials is…
I = I0 · exp(-µx) (1)
where x is the shield material thickness, µ is the attenuation coefficient, I0 the original intensity and I the intensity after traversing the shield thickness x. It then follows that
µx = ln(I0/I)
For the half-value and tenth-value shield thicknesses
x = ln(I0/I)/µ
for Cs 137 (the dominant emission at 661.6 keV (from Ba137m)), µ = 8.57x10-2 cm-1. Hence
x1/2 = ln2 / 8.57x10-2 = 8.09 cm
x1/10 = ln10 / 8.57x10-2 = 26.87 cm
These values agree with what you find in the literature. However, it should be noted that they apply for “well collimated beams” or “narrow beams”.
Broad, uncollimated beams
In general, one usually has broad radiation beams as shown schematically in the main user interface above.
With broad beams, radiation can be scattered into the detector by so-called multiple scattering. This multiple scattering is shown in the above diagram. The formula governing broad beam situations is given by
I = I0 · B(µx,E) · exp(-µx) (2)
where B is the scattering coefficient and depends on the shield thickness x and the attenuation coefficient µ. For well collimated beans, B=1. In the more general case of broad beams, it follows
µx = ln(B·I0/I)
For the half-value and tenth-value shield thicknesses
x = ln(B·I0/I)/µ
for Cs 137 (the dominant emission at 661.6 keV (from Ba137m)), µ = 8.57x10-2 cm-1. The B values have to be interpolated from tables of data given in Nucleonica. For the half-value thickness B = 5.22, hence
x1/2 = ln(10.44) / 8.57x10-2 = 27.37 cm x1/10 = ln(141) / 8.57x10-2 = 57.75 cm
Note that the results for the half and tenth value thicknesses for Cs 137 radiation in water shields is strongly dependent on multiple scattering as the results above show. Stated alternatively, the half and tenth value thicknesses for water shields depend strongly on the geometrical setup i.e. on whether narrow radiation beams or broad radiation beams are involved
In order to calculate the attenuation of gamma radiation in shield materials, using the formula I = I0 · B(µx) · exp(-µx), one needs first to evaluate the interpolated values of the mass attenuation coefficient and the B factor since the mass attenuation coefficients and B factors are listed in tables for various materials in Nucleonica. The mass attenuation factors for water are shown in the figures below.
Interpolation of the mass attenuation coefficient
Consider making a calculation for Cs-137 at the energy 661.6 keV (from the Ba137m daughter in equilibrium). Consider also a water shield. The first step is to do an interpolation (using the data highlighted in the red box below) to obtain the attenuation coefficient at 661.6 keV.
A linear interpolation would give
m (gradient) = (µ2-µ1)/(E2-E1) = (µ-µ)/(E-E1)
This relation can then be rearranged to give the µ value at energy E. It follows that
µ(661,6 keV)= 8.624x10-2
A better result, however, is to use a linear interpolation of the logarithmic values i.e. on a log µ vs. log E plot i.e.
m (gradient) = (log(µ2)-log(µ1))/(log(E2)-log(E1)) = (log(µ)-log(µ1))/(log(E)-log(E1))
log(µ) = log(µ1) + (log(E) –log(E1))*(log(µ2)-log(µ1))/(log(E2)-log(E1))
µ (661,6 keV)= 8.57x10-2 cm-1.
(note µ is the same as µ/ρ since the density of water is 1 g cm-3). This is exactly the value obtained in Nucleonica’s D&S application as shown below.
Interpolation of the B factor
Interpolations of the B factors are more complicated since the B factors depend both on the E and µd where µd is the number of mean free paths of the gamma photon in the material i.e. B = B(µd,E)). For each value of µd and E there is a corresponding B value as shown in the figure below. In the previous example, we have calculated the mass attenuation coefficient for the 661.6 keV line of Cs-137 in water i.e. µ = 8.642x10-2 cm-1. Here we would like to calculate the B factors for various shield thicknesses. The table of B factors for various energies and µd values is shown below for water. We demonstrate the interpolation details for a water shield of 10 cm. The resulting µd value is µd = 8.642x10-1. This lies between the first two lines in the diagram below between R= 0.5 and R=1. Similarly the energy of 661.6 keV lies between the two columns at 0.8 and 0.6 MeV. A red box in the table below shows the region of interest.
The red area is shown magnified in the diagram below using the data for water in the Nucleonica database. The data are from the “American National Standard for Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Materials”, ANSI/ANS-6.4.3-1991 (1991). The values of B at R (or µd) at 0.5 and 1.0 and energies of 600 keV and 800 keV are shown. The task here is to calculate B at µd = 0.86 and E = 661.6 keV. This is described in more detail below.
Step 1: at µd=0.5, linear interpolation of B vs. E
m (gradient) = (B2-B1)/(E2-E1) = (B-B1)/(E-E1)
B = B1 + (E-E1)(B2-B1)/E2-E1)
For E = 661.6 keV, B = 1.54
Step 2: at µd= 1.0, linear interpolation of B vs. E
m (gradient) = (B2-B1)/(E2-E1) = (B-B1)/(E-E1)
B = B1 + (E-E1)(B2-B1)/E2-E1)
For E = 661.6, B = 2.29
Step 3: at E = 661.6 keV, linear interpolation of B vs. µd
m (gradient) = (B2-B1)/(µd2-µd1) = (B-B1)/(µd-d1)
B = B1 + (µd-µd1)(B2-B1)/(µd2-µd1)
For µd = 0.86, B = 2.08
Integral Gamma Dose
The integral gamma dose (µSv) is the total gamma dose accumulated during the given exposure time which is 4 hours by default. For a source in secular equilibrium this is the gamma dose rate times the exposure duration in hours e.g. 4. For short-lived isotopes the photon emission varies over the time and the dose is given by an integral over the time taking the decay of the source into account:
is the equivalent gamma dose rate (µSv/h)
is the cooling time (hour)
is the exposure time (hour)
This quantity is evaluated using the number of disintegrations occurring during the exposure duration for each nuclide present in the source.
Effective buildup factor
The effective buildup factor is given by:
is the gamma dose rate(µSv/h) due to the uncollimated source and
is the gamma dose rate (µSv/h) due to the collimated source
Effective number of mean free paths
The effective number of mean free paths MFPeff is obtained with:
is the gamma dose rate (µSv/h) due to the unshielded source and
is the gamma dose rate (µSv/h) due to the shielded source
Dose rate/Thickness graph
The Dose rate/Thickness graph has been extended to include the half and tenth values of the shield material. The graph also show the results for narrow beam and broad beam (including the B-factor) geometries.
Attenuation & Buildup
The Attenuation & Buildup tab gives details on the mass attenuation coefficients and buildup factors. For the energies associated with the selected nuclide/mixture, the selected shield material, and the thickness of the shield, the calculated mass attenuation coefficients and buildup factors are shown below. The energies given in the drop down box are those for the selected nuclide or mixture. Any other energy can also be used in the calculation.
The default values for the shield material and the shield thickness are those set in the main tab. However any other shield material and thickness can be selected/input. The shield thickness can also be specified in units of µd (mean free paths). Based on the values selected for the energy, shield, and shield thickness, the mass attenuation coefficient and buildup factor are calculated and also shown in the (green) input mask.
Below the (green) input mask, detailed information on the buildup factors can be viewed using the appropriate check boxes. In the figures below the buildup factors are shown in both graphical and tabular form.
In the table above the buildup factors are given as a function of energy and number of mean free paths (µd). The column headers µd0.5 means µd = 0.5, µd1 mean µd = 1, etc. Data in the energy range 0.03 - 15 MeV and µd 0.5 - 40 are given. Columns highlighted in green give the µd and B values at different energies for the values (shield material, shield thickness) set in the input mask.
Below the (green) input mask, detailed information on the mass coefficients can be viewed using the appropriate check boxes. In the figures below the mass coefficients are shown in both graphical and tabular form.
In the table above the mass coefficients (for attenuation and energy absorption) are given as a function of energy and number of mean free paths (µd). Data in the energy range 0.001 - 20 MeV are given.
The mode of operation can be set in the Options tab shown below. The program has four modes of operation for calculation of the
- Gamma Dose Rate,
- Shield Thickness,
- Source Strength, and
In mode a) the gamma dose rate is calculated for a shield material and thickness. In mode b) the shield thickness is calculated for a given gamma dose rate at the detector. In mode c) the source strength is calculated for a given dose rate at the detector and a given shield material and thickness. In mode d) the distance between the source and detector is calculated corresponding to a given source and source strength, a shield and shield thickness, and a gamma dose rate. These are described further in the section Modes of Operation.
In the Energy range option, the user can choose to include only gammas, X-rays, or both in the calculation. In addition the user can set the minimum (threshold) energy of gamma and X-rays to be included in the calculation. The default value is 15 keV – photons with lower energy are absorbed by the outer layers of human tissue. See the following section for a more detailed discussion of the threshold energy.
The "Threshold Energy"
Many users have commented that one of the most confusing points in the literature with regard to the evaluation of the specific gamma dose rate constant and indeed any dose rate calculations is that different literature sources quote different results. This is very very confusing. The reason for the differences is due to the fact that different "threshold energies" are used in the calculations. The threshold energy (see figure above) is the energy below which one ignores the contribution from the gammas and x-rays to the dose rate. This is sometime taken as 15 keV, sometimes 10 keV and sometimes even 0 keV (i.e. all lines are counted. In many books, computer programs etc, this threshold energy is not mentioned!
The reason for the introduction of a threshold is due to the fact that the outer layers of the skin absorb low energy gamma radiation (usually energies below 15 keV) and thus do not contribute to the whole bode dose. An important advantage with Nucleonica is that the user can change this threshold energy (in the Options tab in fig 6 above) and see how sensitive the results are to the change. The results can be very sensitive!
When a nuclide mixture has been selected, the Mixture details tab becomes activated with typical contents shown below.
Modes of Operation
The mode of operation can be selected in the Options Window. There are four modes of operation: gamma dose rate, shield thickness, source strength, and distance modes and these are described in detail in this section.
Gamma Dose Rate mode
The basic setup is shown in the figure. For a given nuclide or nuclide mixture, a known activity, and a known shield thickness, the gamma dose rate is calculated.
Example: In the example shown, the nuclide Co60 has been selected from the drop-down menus. The source detector distance is 1m. In this mode of operation, the gamma dose rate is calculated for a known activity, shield and shield thickness. In this case the activity has been set to A = 1 MBq (note 1 MBq is the default activity for all nuclides). The shield material "lead" has been selected with a thickness of 0 cm (i.e. effectively no shield). On pressing the Start button the gamma dose rate is shown highlighted in red - in this case with a value of 0.3539 µSv/h.
Shield Thickness mode
The setup is shown. For a given nuclide or nuclide mixture, a known activity, and a known shield material, and known gamma dose rate, the thickness of the shield material is calculated is calculated.
Example: In the previous calculation, it was shown that the gamma dose rate at 1m from an unshielded source in 0.3539 µSv/h. In this example, one would like to know how much lead shielding is required to reduce the gamma dose rate to 0.1 µSv/h. In the dose rate box the value 0.1 is set. In the shielding drop down menu the shield material lead is selected. On pressing the start button, the shield thickness is calculated and highlighted in red. The resulting lead shield thickness is 3.04 cm.
Source Strength mode
The setup is shown. For a given nuclide or nuclide mixture, a known activity, and a known shield material, and known gamma dose rate, the source strength is calculated.
Example: Based on the previous two calculations, it is of interest to calculate the source strength (activity) of a Co60 source, when the dose rate at 1m behind a 3.04cm thick lead shield is 0.1µSv/h. In the dose rate box, the value 0.1 is entered. In the shield drop-down menu, the lead shield is selected. A thickness of 3.04cm is also entered. On pressing the Start button the calculated activity of 1E6 (Bq) is shown highlighted in red.
The basic setup is shown in the figure. For a given nuclide or nuclide mixture, a known activity, known shield and shield thickness, and gamma dose rate, the distance is calculated.
Example: Based on the above calculations, it is of interest to calculate the source detector distance corresponding to :
- Source Strength Co60 = 1MBq
- Shield material/thickness = Pb / 1 cm
- Gamma dose rate = 2 µSv/h
On pressing the Start button the corresponding distance is 36.6 cm is shown highlighted in red.
Validation and Case Studies
Shielding Calculations, comparison with literature results: There are relatively few literature results with which one can make a direct comparison. The examples given below are of direct interest.
Case Study: Shielding Na-24
An example of a shielding calculation is given in Cember (ref.  below): Design a spherical lead storage that will attenuate the exposure rate from 1 Ci of Na24 to 10 mR/h at a distance of 1 m from the source. The given answer is 13.17 cm of lead.
With the Dosimetry & Shielding module in Nucleonica (and taking 1R ≈ 104 μSv), the thickness required is 12.8 cm. In view of the approximate nature of the calculation by Cember, this result is close enough to Cember's value of 13.17 cm. The user interface with summary results is shown in Fig. 12. Here the half-value (2.47cm) and tenth value thicknesses (6.28cm) can be seen. It can also be seen that the build-up factor 4.45 is considerably greater than one indicating that buildup in the shield is important. More detail results can be seen in Fig. 12b where the individual dose rate and buildup factor for each gamma line of Na24 are shown. Finally a spectrum is shown of all gamma lines accounted for in the calculation. This problem is discussed in detail by Cember .
 H. Cember, Introduction to Health Physics, 3rd Edition, McGraw Hill, 1996, page 430.
Case Study: Shielding Co-60
Another example of shielding calculation is given in the booklet coming with the Karlsruhe Chart of the Nuclides.
For a source strength of 1 Ci of Co60 through 5 cm of lead at a distance of 10 cm from the source the resulting dose is 11.9 R/h. The same calculation with the Dosimetry and Shielding module gives 12.7 × 104 μSv/h ≈ 12.7 R/h.
Case Study: Gamma Radiography with Ir-192
The isotope 192Ir is a beta emitter with a half-life of 73.8 d (see DataSheets). From the inset it can be seen that 192Ir can be produced by neutron bombardment of iridium metal and decays to the stable 192Pt. With a specific activity of 3.4×1014 Bq/g (from the Derived Data) the material is highly radioactive. The main radiation hazard arises not through the particle emission, which can be easily shielded, but through the associated gamma emission.
The gamma spectrum is shown in Fig. 14. For this reason, 192Ir is used in industrial radiography. Typical transports of the isotope involve 10000 Curies (corresponding to a mass of about 1.1 g) and clearly the material has to be strongly shielded.
From the Dosimetry and Shielding module it can be seen that the gamma dose rate at one metre from such a source is in excess of 40 Sv per hour as shown in Fig. 15. It can also be seen that the tenth value thickness of lead is 1.22 cm. Clearly, strong shielding of this material is required in order to ensure that transport handlers do not receive doses over the public limit of 1 milliSievert per year. With a 7 cm shield of uranium, the gamma dose rate at 1m reduces to a few μSv/h.
Case Study: Thickness of Neptunium Targets
For a series of planned experiments on laser irradiation of neptunium, it is required to know what thickness of neptunium is required to absorb a reasonable fraction of the gamma radiation. The energy of the gamma photons in such laser experiments is around 10 MeV. An estimate of the required thickness can be made as follows.
In a first step, a search of the database is made to find any nuclide with gamma emission in this range. The result of a search for nuclides with gamma energies in the range 10(±1) MeV is shown in Fig. 16. From Fig.16, it can be seen that Al24 emits gamma photons with an energy of 9.94MeV. This energy is close enough to the 10 MeV photons produced in the laser experiments. The Al24 isotope is therefore chosen as the source in a shielding calculation.
In the second step, a shielding calculation is made. Neptunium is, however, not one of the standard shield materials in the database. The heavy metal uranium can, however, be selected to “simulate” the neptunium shield. The results give the half-value shield thickness (HVL) for 9.94 MeV photons to be 1.3 cm. Clearly, relatively thick samples are required for the planned experiments.
To test the sensitivity of the method, other shield materials can be used. The HVL for lead and tungsten are 2.1 cm and 1.4 cm respectively. For lower energy photons, for example around 5 MeV, the nuclide I138 can be selected. The HVL values for uranium, lead and tungsten shields are 0.8 cm, 1.5 cm, and 1.0 cm respectively. The results therefore show that for gamma photons in the energy range 5–10 MeV, heavy metal shield thicknesses of around 1 cm are required.
Case Study: Rb-81/Kr-81m Generator
Rb-81/Kr-81m generators are becoming increasingly of interest for use in pulmonary ventilation studies . During such studies, the Kr-81m gas can be continuously delivered to patients from the generator. The Kr-81m gamma emission (190 keV) enables it to be used concurrently with the perfusion agent to obtain lung ventilation and perfusion images. In the following case study, we are interested in the dose rate obtained due to the handling of such generators. The Rb-81/Kr-81m generator is produced by the proton irradiation of krypton-82 (see insert from the Karlsruhe Nuclide Chart, 7th Edition).
The Rb-81 thus produced contains around 5% of Rb-82m and it is this nuclide which dominates the shielding problem. Since the daughter product Kr-81m has a much shorter halflife than the parent Rb-81, it is in secular equilibrium with the parent. However Rb-81 and the Rb-82m are much more important than Kr-81m, especially for shielding. Kr-81m has only a single gamma line at 190 keV, whereas Rb-81 and Rb-82m have many lines that are more energetic.
 N. R. Williams et al, Eur. J. Nucl. Med. (1985) 10:33-38.
In the first step, a nuclide mixture is created with Rb-81 (100 MBq), Rb-82m (5 MBq)and Kr-81m (100 MBq) as shown in fig.17.
Thereafter, a dosimetry & shielding calculation is made with this mixture. The results are shown in Figs. 19 and 20. The main contribution to the dose rate arise from the 511 keV annihilation X-rays.
The gamma spectrum is shown in Fig. 21.
The Dosimetry & Shielding H*(10) is a very user-friendly and reliable tool for dosimetry and shielding calculations. It is a further development of Nucleonica’s Dosimetry & Shielding++ with the following new/extended features:
- The list of calculated quantities now covers a) air kerma rates Kair; b) exposure rates X; and c) ambient dose equivalent rates H*(10) for approximately 1500 gamma and x-ray emitting radionuclides (depending on the database used).
- The threshold energy used in the calculations for dose quantities can be set by the user to investigate the effect of low energy photons on the dose calculations.
- A key feature is the inclusion of short-lived daughters in the calculations as indicated by an asterisk * before the physical quantities and following the nuclide name e.g. Cs137*.
- The underlying dataset used in the calculations can be selected from a list of international nuclear datafiles (JEFF3.1, ENDF/B-VII.1, 8th TORI)
- The application has four modes of operation. In mode a) the gamma dose rate is calculated for a shield material and thickness. In mode b) the shield thickness is calculated for a given gamma dose rate at the detector. In mode c) the source strength is calculated for a given dose rate at the detector and a given shield material and thickness. In mode d) the distance between the source and detector is calculated corresponding to a given source and source strength, a shield and shield thickness, and a gamma dose rate. These are described further in the section Modes of Operation.
To avoid various under/overflow problems etc., restrictions on the input/ output data have been set. These are shown below.
1. J. H. Hubbell and S. M. Seltzer: Radiation Research 136, 147 (1993) See also the NIST website at: http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html
2. “American National Standard for Gamma-Ray Attenuation Coefficients and Buildup Factors for Engineering Materials”, ANSI/ANS-6.4.3-1991 (1991)
3. B. Schlein, L. A. Slaback, Jr., B.Kent Birky: The Health Physics and Radiological Health Handbook, 3rd ed. Scinta, Silver Spring, MD, 1998
4. Safety of Source Transports in Question after Iridium Leak. NucleonicsWeek Jan. 17, 9–11 (2002)
5. MC. Limacher et al., ACC expert consensus document. Radiation safety in the practice of cardiology, J. Am. Coll. Cardiol. 1998;31;892-913. pdf
Appendix: Shielding and Buildup
If a shield material is present, the attenuation and buildup in the shield have to be accounted for. The photon fluence rate for each photon is then
where is the buildup factor for photons of energy ,
is the attenuation coefficient for photons of energy ,
and d is the shield thickness.
This relation can also be expressed as
However in the following sections, the simplified version is used.
Air kerma rate constant and air kerma rate
It follows from
the air kerma rate is given by
Exposure rate constant and exposure rate
Similarly, it follows
The exposure rate is given by
Ambient dose rate constant and ambient dose rate
From a previous section
The ambient dose rate is given by
Photon dose equivalent rate, HX
The photon dose equivalent Hx (measured in Sv) is a quantity introduced in Germany in 1980. It became the legal quantity in Germany on 1 January 1986. Hx was an interim solution because at that time no international agreement on dose equivalent quantities had been achieved. In Germany Hx was replaced by SI quantities such as H*(10) on 1 August 2001. Hx was not accepted internationally. Nevertheless, it is interesting to describe the background behind Hx.
Most instruments available up to the early 1980s were designed for Exposure (rate) and calibrated in R(/h). The question arose whether these instruments could still use them to measure dose equivalent. The answer was yes, because Exposure is a good estimate for the dose equivalent of photons in tissue. Therefore Hx was defined as
Using the relation, 1 R = 2.58 x 10-4 C/kg of air,
Since this conversion does not depend on photon energy, Hx and Exposure X are strongly related quantities; they just differ by the factor 100. This is why some instruments allow the user to select either R or Sv as the unit. Basically Hx was not really a new quantity. It was more the old quantity Exposure in a new wrapping.
Using the relation gives
Nucleonica (previous) photon dose equivalent rate, Htis
The dose rate in tissue calculated in Nucleonica is
is the energy fluence (energy per unit area per unit time),
is the mass energy absorption coefficient in tissue.
The above formula is value for a thin layer of tissue in which there is no attenuation and no backscattering from other layers. It follows
Since the ratio of the mass absorption coefficients in air to tissue is approx. 1.10, then