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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.

Contents

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 [1].

Reference

[1] J. R. Cooper, K. Randle, R. S. Sokhi, Radioactive Releases in the Environment: Impact and Assessment, Wiley, 2003.

Absorbed Dose

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 D_T. The absorbed dose to the mass \delta m_T , is defined as the imparted energy \delta E_T per unit mass of the tissue or organ, i.e.


D_T= {\delta E_T \over \delta m_T}

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.


Recommended radiation weighting factors from ICRP 103 (2007)
  Radiation type   Radiation weighting factor, wR
  Photons   01
  Electronsa and muons     01
  Protons and charged pions   02
  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).


Older 1991 quality or weighting factors for different types of radiation [2]


Reference

[2] ICRP Publication 72. Annals of the ICRP 26, 1996, Pergamon Press

Equivalent Dose

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.


H_{T,R} = w_R \cdot D_{T,R}


where H_{T,R} is the equivalent dose in tissue T and w_R 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.


H_{T} = \sum_{R} (w_R \cdot D_{T,R})


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.


{dH_{T} \over dt} = w_R \cdot {dD_{T,R} \over dt}


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

Effective Dose

1991 (and 2007 in brackets) weighting factors for individual organs [ICRP]. Courtesy SRP


Recommended tissue weighting factors (ICRP 103 (2007)).
  Tissue wT ΣwT
  Bone marrow (red), Colon, Lung, Stomach, Breast, Remainder tissues* 0.12 0.72
  Gonads 0.08 0.08
  Bladder, Oesophagus, Liver, Thyroid 0.04 0.16
  Bone Surface, Brain, Salivary glands, Skin 0.01 0.04
Total 1.00
      * 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:


E = \sum_{T} (w_T \cdot H_{T})


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.

Committed effective dose.jpg


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.

Effects of radiation exposure to man [3]


Reference

[3] 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


Radiotoxicity = E(\tau)


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


Radiotoxicity = A \cdot e(\tau )


where A is the activity of the radionuclide at the moment of intake.

Annual Limits of Intake (ALI) for ingestion which results in a dose of 0.02 Sv for the main radioactive by-products of nuclear waste. The values are given in both Becquerel and mass units.


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


ALI = {0.02 \cdot Sv \over e(50)}


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


DAC(Bq/m^3) = {ALI{inh}(Bq) \over 2400m^3}


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


DWC(Bq/litre) = {ALI_{ing}(Bq) \over 913  litre}


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

Hypothetical curves depicting (a) threshold, (b) linear non-threshold, and (c) hormetic dose-response models using cancer (number of tumours per animal) as the endpoint. The reduction in number of tumours per animal at the lower doses (1–6) compared to the number of tumours per animal (5 tumours per animal) in the control indicates a reduced risk of cancer. (Reprinted by permission from Nature [5]. © 2003 Macmillan Publishers Ltd.)

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 [4].

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 [5] 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 [6]. 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].


References

[4] T. D. Luckey, Radiation Hormesis, CRC Press, Boca Raton, 1991.

[5] E. J. Calabrese, et al., Nature 421, 691 (2003).

[6] R. E. Mitchel, D. R. Boreham, Proc. International Radiation Protection Association, 10th Quadrennial Meeting, Hiroshima, Japan, 15-19 May 2000.

[7] P. C. Kesavan, in: High Levels of Natural Radiation, L. Wei, T. Sugahara, Z. Tao (eds.), Elsevier, Amsterdam p. 111, 1996.


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 Fig. 1, a straight-line relationship between the logarithm of the intensity versus the thickness d of the shield is obtained, i.e.


ln{I \over I_0} = - \mu d

or

{I \over I_0} = e^{ - \mu d}


where

I is the gamma radiation intensity transmitted through an absorber of thickness d,

I_0 is the gamma radiation intensity at zero absorber thickness,

d is the absorber thickness,

\mu slope of the absorption curve – the attenuation coefficient.

Fig.1. Measurement of the attenuation of gamma radiation under conditions of good geometry. Ideally, the beam should be well collimated, and the source should be as far away as possible from the detector. The absorber should be midway between the source and the detector, and it should be thin enough so that the likelihood of a second interaction between a photon already scattered by the absorber and the absorber is negligible. In addition, there should be no scattering material in the vicinity of the detector


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:


\mu_l (cm^{-1}) = \mu_m (cm^2/g) \cdot \rho (g/cm^3)


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:


\mu = \left [ {(\Delta I/I) \over \Delta d} \right ]_{\Delta d \rightarrow 0}


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:


I = I_0 \cdot e^{-\mu_l d}


or


I = I_0 \cdot e^{-(\mu_l/\rho) \cdot (\rho d)}


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 Fig. 2, 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.

Fig.2. Gamma radiation attenuation under conditions of broad beam geometry showing the effect of photons scattered into the detector


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.


I = B \cdot I_0 \cdot e^{-(\mu_l/\rho) \cdot (\rho d)}


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


I = B(\mu d,E) \cdot I_0 \cdot e^{-(\mu_l/\rho) \cdot (\rho d)}



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.


\mu = \mu_{pe} + \mu_{cs} + \mu_{pp}


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 the Equivalent 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


 {dD \over dt} = I \cdot (\mu_l/\rho)^{tis}


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.


 {dH \over dt} = I \cdot (\mu_l/\rho)^{tis}


In the case of no shielding, the gamma radiation intensity (energy per unit area per second), I , can be written:


 I = I_0 /(4\pi R^2)


where  I_0 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:


I_0 = A \cdot \sum_{i} (E_i \cdot P_i)


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 \sum_{i} (E_i \cdot P_i) must include all lines. Combining the above relations, one obtains:


{dH \over dt} =  A/(4 \pi R^2) \cdot \sum_{i} E_i \cdot P_i \cdot (\mu_l/\rho)_i^{tis}


where the energy dependence of the mass absorption coefficient has been accounted for. Inserting numerical values, one obtains:


{dH \over dt}(\mu Sv/h) = (5.77 \cdot 10^{-4}) \cdot A /(4 \pi R^2) \cdot \sum_{i} E_i(keV) \cdot P_i \cdot {(\mu_l/\rho)}_i^{tis}

where the activity A is expressed in Bq, R in cm, Ei in keV and μl in cm2/g. In the case where a shield is used, the corresponding relation is


{dH \over dt} = A/(4 \pi R^2) \cdot \sum_{i} E_i \cdot P_i \cdot B_i \cdot e^{-(\mu_l/\rho)_i^{shield}\cdot (\rho d)} \cdot 
{(\mu_l/\rho)}_i^{tis}


again inserting numerical values one obtains:


{dH \over dt}(\mu Sv/h) = (5.77 \cdot 10^{-4}) \cdot A/(4 \pi R^2)  \cdot \sum_{i} E_i(keV) \cdot P_i \cdot B_i \cdot 
e^{-(\mu_l/\rho)_i^{shield}\cdot (\rho d)} {(\mu_l/\rho)}_i^{tis}


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.


Absorption in Tissue

The dependence of (μ/ρ)tis on energy is shown in Fig. 3 and Table 3. 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

Fig. 3. Mass absorption coefficient for tissue
Table 3. Table of mass absorption coefficient for tissue



Example Co-60

As an example, consider the evaluation of the gamma dose rate from 1 MBq of 60Co at im. The six gamma energies and their emission probabilities are shown in Table 4.

Table 4. Spectral data for 60Co


It follows that...

{dH \over dt}}(\mu Sv/h) = (5.77 \cdot 10^{-4}) \cdot A/(4 \pi R^2)  \cdot \sum_{i} E_i(keV) \cdot P_i \cdot B_i \cdot e^{-(\mu_l/\rho)_i^{shield}\cdot (\rho d)} {(\mu_l/\rho)}_i^{tis}


When there is no shield, this expression reduces to


{dH \over dt} }(\mu Sv/h) = (5.77 \cdot 10^{-4}) \cdot A /(4 \pi R^2) \cdot \sum_{i} E_i(keV) \cdot P_i \cdot {(\mu_l/\rho)}_i^{tis}


Hence for A = 1 MBq, R = 100 cm,

{dH \over dt}(\mu Sv/h) = 5.77 \cdot 10^{-4} \cdot {10^6\over(4\cdot3.14159 \cdot 10^4)} \cdot 73.37 = 0.337 \mu Sv/h


These reults can of course be obtained easily and directly in the gamma dosimetry and Shieldimg module as explained in the section Modes of Operation.


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. The data is shown in the diagrams and tables below.

References

Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients

Data Tables


Lead

Fig. 3a. Mass attenuation coefficient for Lead
Table 3a. Table of mass attenuation coefficient for Lead


Concrete

Fig. 3b. Mass attenuation coefficients for Concrete
Table 3b. Table of mass attenuation coefficients for Concrete


Iron

Fig. 3c. Mass attenuation coefficients for Iron
Table 3c. Table of mass attenuation coefficients for Iron


Tin

Fig. 3d. Mass attenuation coefficients for Tin
Table 3d. Table of mass attenuation coefficients for Tin


Tungsten

Fig. 3e. Mass attenuation coefficients for Tungsten
Table 3e. Table of mass attenuation coefficients for Tungsten


Uranium

Fig. 3f. Mass attenuation coefficients for Uranium
Table 3f. Table of mass attenuation coefficients for Uranium


Water

Fig. 3g. Mass attenuation coefficients for Water
Table 3g. Table of mass attenuation coefficients for Water


Aluminum

Fig. 3h. Mass attenuation coefficients for Aluminium
Table 3h. Table of mass attenuation coefficients for Aluminium


Air

Fig. 3i. Mass attenuation coefficients for Air
Table 3i. Table of mass attenuation coefficients for Air


Tissue

Fig. 3j. Mass attenuation coefficients for Tissue
Table 3j. Table of mass attenuation coefficients for Tissue


Compare the mass attenuation coefficients of tissue with the mass absorption coefficients for tissue. 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 [1] 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) and are given below. Due to the complexity of the data and their double dependence on energy and mean free path lengths, tabulated values (bins) are used. The buildup factors have been set to 1 when tissue is selected as the shield medium. This option is of use for investigating the attenuation of low energy X-rays in the outside layer of the skin (a few mm).

Reference

[1] “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

[2] B. Schlein, L. A. Slaback, Jr., B. Kent Birky: The Health Physics and Radiological Health Handbook, 3rd edition. Scinta, Silver Spring, MD 1998

[3] G. E. Chabot, Shielding of Gamma Radiation

[4] D. K. Trubey New Gamma-Ray Buildup Factor Data for Point Kernel Calculations: ANS-6.4.3 Standard Rference Data

[5] Buildup factor animation


see also Shielding Equations and Buildup Factors Explained

Concrete and Iron

Table 3m. Build-up factors for concrete
Table 3n. Build-up factors for iron



Tin and Tungsten

Table 3o. Build-up factors for tin
Table 3p. Build-up factors for tungsten



Uranium and Water

Table 3q. Build-up factors for uranium
Table 3r. Build-up factors for water



Aluminium and Air

Table 3s. Build-up factors for aluminium
Table 3t. Build-up factors for air



Lead

Table 3u. Build-up factors for lead



The Dosimetry & Shielding Module

Fig. 5. Dosimetry & Shielding Module Interface

The dosimetry and shielding module can be accessed from the main application page. The gamma dosimetry and shielding interface is shown in Fig. 5. 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.


The Dosimetry and Shielding page 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 three 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, shild material and thickness are known


The main tab allows the user to select the nuclide, its source strength (in mass, becquerels, curies, or number of atoms), source/detector distance, shield material and material thickness. The results of the calculations are given in a results table, which in addition to the gamma dose rate (or shield thickness), gives the half- and tenth-value thicknesses of shield material and the specific gamma dose rate constant. By using the Result Details button, 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 can be viewed by using the Graph button. The user can print the results (including graph) by using the Print button.


User Interface

Source/Detector Distance (cm)

The source/detector distance is specified by entering the value in the input box. The default value is 100 cm.

Source Strength

Source Strength menu

The source strength is set by specifying first the unit i.e. Activity(Bq), Activity(Ci), Mass(g), or the Number of atoms, and then the amount. The amount should be entered in scientific notation in the form 1, 10, 1E2, 1E-6 etc. The source strength boxes function also as an activity calculator which allows the conversion of one unit into another. As an example, for the nuclide Co-60 select the unit Activity(Ci) and amount to 1. On changing the entry in the unit list box, one obtains Activity(Bq)= 3.70E10, Mass(g) = 8.84E–4, Number of atoms = 8.88E18 respectively.


Shielding Material

Shield material drop-down menu

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),

• tin (Sn),

• tungsten (W),

• uranium (U),

• water,

• 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).

Start

On pressing the Start button, the results of the calculation, with the above parameters set, are shown in the Results window.

Reset

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.

Options Window

Fig. 6. The Options window

The mode of operation can be set in the Options window shown in Fig. 6. The program has three modes of operation for calculation of the a) Gamma Dose Rate, B) Shield Thickness and c) the Source Strentgh. 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 calcluated for a given dose rate at the detector and a goven shield material and thickness.

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 fig. 6 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 EVEN MENTIONED!

The reason for the introduction of a thresholed 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!

Modes of Operation

The mode of operation can be selected in the Options Window. There are three modes of operation: gamma dose rate, shield thickness, and source strength modes and these are described in detail in this section.


Gamma Dose Rate mode

Calculation of the gamma dose rate

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 "air" 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 0.337 µSv/h.


Shield Thickness mode

Calculation of the shield thickness

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 calulation, it was shown that the gamma dose rate at 1m from an unshielded source in 0.337 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

Calculation of the source strength

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.


Results Window

In the Dose Rate mode of operation, the main result is the gamma dose rate as shown in Fig. 7 for 1 Ci Co-60. Subsidiary results are the specific gamma dose rate constant, and the 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. In the bottom panel the number of gamma rays, X-rays and gamma + X-rays is given. In addtion the total photon energy emitted per disntegration is shown ∑E.P (in eV per disintegration)

Fig. 7. The Results summay for 1 Ci Co-60 (Dose Rate Mode)




In the Shield Thickness mode of operation, the main result is the shield thickness required to give a gamma dose rate at the detector. In this case the gamma dose rate at the detector from a 1 Ci Co-60 source was set to 2 µSv/h. As can be seen the resulting thickness of a Pb shield is 15.9 cm. Subsidiary results are again the specific gamma dose rate constant and the HVL and TVL values.

Fig. 8. The Results summay for 1 Ci Co-60 (Shield Thickness Mode)



In the Source Strength mode of operation, the main result is the source strength required to give a gamma dose rate at the detector. In this case the gamma dose rate at the detector was set to 2 µSv/h. The thickness of the Pb shield was set to 15.9 cm. As expected, the source strenght was calculated a approx. 1 Ci. Subsidiary results are again the specific gamma dose rate constant and the HVL and TVL values.

Fig. 8. The Results summay for 1 Ci Co-60 (Source Strength Mode)




Detector

At the detector, the resulting gamma dose rate is given in units of μSv/h in the Dose Rate mode. If the mode of operation is the thickness mode, the user must enter the required dose rate at the detector (in this mode the dose rate is an input quantity and as such the input box is white. In addition, under Detector, some general information is given. This includes the required shield thickness to reduce the γ dose rate by 50% (in cm) HVL and the required medium thickness to reduce the γ dose rate by 90% (in cm) TVL.

Equivalent Dose Rate Constant Γ

The equivalent dose rate constant Γ is a source property and is defined through the relation

H = Γ · A/r2

where H is the equivalent dose rate, A the activity and r the distance from the point source. The quantity Γ does not itself depend on A or r. It is a useful quantity since from a knowledge of the dose rate constant Γ, the activity and the distance, one can readily obtain the dose rate. The unit of the dose rate constant is Sv ·m2 · s-1 · Bq-1. Usually, however, the dose rate constant is expressed in the unit mSv · m2 · h-1 · GBq-1.

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.


Results Details

The Results Details are shown directly under the main results as shown in Fig.9. The Results Details consists of the the energies, the corresponding emission probabilities, the mass attenuation coefficients, the B factors, etc.

Fig. 9. The Results Details: Double clicking on the column header will rearrange the records in ascending/descending order


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.

Graph Window

The gamma energies and emission probabilities for the selected nuclides are shown in the Graph window as shown in Fig.10 (for Co-60). From this window the image can be printed or the data downloaded for use in other applications.

Fig. 10. The Results Graph showing the emission energies and the emission probabilities


Input/Output Restrictions

To avoid various under/overflow problems etc., restrictions on the input/ output data have been set. These are shown in Fig. 11.

Fig. 11. Input/Output restrictions to avoid under/overflow


Validation of the Results/Benchmarking

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

Fig. 12. Shielding of Na24
Fig. 12b. Shielding of Na24

An example of a shielding calculation is given in Cember (ref. [1] 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 consdirably 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 [1].

Reference:

[1] H. Cember, Introduction to Health Physics, 3rd Edition, McGraw Hill, 1996, page 430.


Case Study: Shielding Co-60

Fig. 13. Shielding of Co60

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

location of 192Ir on the nuclide chart

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.


Fig. 14. Gamma spectrum of 192Ir

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.



Fig. 15. Gamma dose rate at 1 m for a 104 Ci source of 192Ir

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

Fig. 16. Results of the database search for nuclides with gamma energies in the range 10(±1) MeV

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

Extract from the nuclide chart showing Rb-81, Rb-82m, and Kr-81m. The Rb-81/Kr-81m generator is produced from proton irradiation of Kr-82

Rb-81/Kr-81m generators are becoming increasingly of interest for use in pulmonary ventilation studies [1]. 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.


Reference

[1] N. R. Williams et al, Eur. J. Nucl. Med. (1985) 10:33-38.


Fig. 17. A nuclide mixture consisting of the parent Rb-81 (100 MBq), the impurity Rb-82m (5 MBq) and Kr-81m (100 MBq). Due to the short half-life of Kr-81m, this daughter is in radioactive equilibrium with the parent Rb-81

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.


Fig. 19. Dosimetry & Shielding calculation with Rb-81/Kr-81m generator

Thereafter, a dosimetry & shielding calculation is made with this mixture. The results are shown in Figs. 18 and 19.


Fig. 20. Dose contributions from the various emission lines

The main contribution to the dose rate arise from the 511 keV annihilation X-rays.


Fig. 21. Gamma spectrum of the Rb-81/Kr-81m generator

The gamma spectrum is shown in Fig. 21.



Summary

The Dosimetry & Shielding module in Nucleonica is a very user-friendly and reliable tool for dosimetry and shielding calculations. The key features of the program can be summarised as follows:

Fig. 17. Summary of the main features of the Dosimetry & Shielding module in Nucleonica


• very user friendly and reliable

• point source geometry

• accurate interpolation of the mass absorption coefficient, rather than using energy bins

• for shield materials, accurate interpolation of the mass attenuation coefficient,B factors are determined by energy bins

• large number of nuclides available – spectral data from the JEFF3.1 library and the 8th Table of the Isotopes

• two modes of operations: a) dose rate calculation for a shield thickness and b) thickness calculations for a given dose rate

• user definable threshold energy

References

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

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