Quantities for Describing Radiation Interactions

Ho Kyung Kimhokyung@pusan.ac.kr

Pusan National University

Radiation DosimetryAttix 2

References

F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley and Sons, Inc., 1986

G. F. Knoll, Radiation Detection and Measurement, 4th ed., John Wiley and Sons, Inc., 2010

2

Nonstochastic quantities for describing the interactions of the radiation field with matter (in terms of expectation values for the infinitesimal sphere at the point of interest)

1. Kerma 𝐾𝐾– The first step in energy dissipation by indirectly ionizing radiation (i.e., energy transfer to charged particles)

2. Absorbed dose 𝐷𝐷– The energy imparted to matter by all kinds of ionization radiations, but delivered by the charged particles

3. Exposure 𝑋𝑋– X- & 𝛾𝛾-ray fields in terms of their ability to ionize air

3

Kerma

Relevant only for• Fields of indirectly ionizing radiations (e.g., photons or neutrons)• Any ionizing radiation source distributed within the absorbing medium

Energy transferred

𝜖𝜖𝑡𝑡𝑡𝑡 = (𝑅𝑅𝑖𝑖𝑖𝑖)𝑢𝑢− 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑢𝑢𝑖𝑖𝑜𝑜𝑖𝑖𝑡𝑡 + �𝑄𝑄

– 𝜖𝜖𝑡𝑡𝑡𝑡 = energy transferred (stochastic quantity)– (𝑅𝑅𝑖𝑖𝑖𝑖)𝑢𝑢 = radiant energy† of uncharged particles entering 𝑉𝑉– 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑢𝑢

𝑖𝑖𝑜𝑜𝑖𝑖𝑡𝑡= radiant energy of uncharged particles leaving 𝑉𝑉, except that which originated from radiative losses‡ of KE by charged particles while in 𝑉𝑉

– ∑𝑄𝑄 = net energy derived from rest mass in 𝑉𝑉• positive when Δ𝑚𝑚 ↓ (𝑚𝑚 → 𝐸𝐸)• negative when Δ𝑚𝑚 ↑ (𝐸𝐸 → 𝑚𝑚)

4‡ Conversion of charged-particle KE to photon energy through either bremsstrahlung or in-flight annihilation of positrons

† The energy of particles (excluding rest energy) emitted, transferred, or received

𝐾𝐾 ≡d𝜖𝜖𝑡𝑡𝑡𝑡d𝑚𝑚

• The expectation value of the energy transferred to charged particles per unit mass at a point of interest, including radiative-loss energy but excluding energy passed from one charged particle to another

• Simply, the KE received by charged particles in the specified finite volume 𝑉𝑉• The kerma for x- or 𝛾𝛾-rays consists of the energy transferred to electrons & positrons per unit

mass of medium• 1 Gy = 1 J/kg = 102 rad = 104 erg/g

For monoenergetic photons

𝐾𝐾 = Ψ𝜇𝜇𝑡𝑡𝑡𝑡𝜌𝜌 𝐸𝐸,𝑍𝑍

– 𝜇𝜇𝑡𝑡𝑡𝑡𝜌𝜌

= mass energy-transfer coefficient (depending on 𝐸𝐸 & 𝑍𝑍)

– 𝜇𝜇𝑡𝑡𝑡𝑡 = linear energy-transfer coefficient

5

For spectral photons

𝐾𝐾 = �0

𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚Ψ′(𝐸𝐸)

𝜇𝜇𝑡𝑡𝑡𝑡𝜌𝜌 𝐸𝐸,𝑍𝑍

d𝐸𝐸

An average value of mass energy-transfer coefficient for the spectrum Ψ′(𝐸𝐸) is given by

𝜇𝜇𝑡𝑡𝑡𝑡𝜌𝜌 Ψ′ 𝐸𝐸 ,𝑍𝑍

=𝐾𝐾Ψ

=∫0𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 Ψ′(𝐸𝐸) 𝜇𝜇𝑡𝑡𝑡𝑡

𝜌𝜌 𝐸𝐸,𝑍𝑍d𝐸𝐸

∫0𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚 Ψ′(𝐸𝐸) d𝐸𝐸

Kerma rate

�̇�𝐾 =d𝐾𝐾d𝑡𝑡

=dd𝑡𝑡

d𝜖𝜖𝑡𝑡𝑡𝑡d𝑚𝑚

6

Components of kerma𝐾𝐾 = 𝐾𝐾𝑐𝑐 + 𝐾𝐾𝑡𝑡

– 𝐾𝐾𝑐𝑐 = kerma due to collision interactions (local or nearby the charged-particle track)• Coulomb-force interactions with atomic electrons (ionization & excitation)

– 𝐾𝐾𝑡𝑡 = kerma due to radiative interactions (remote or far away from the charged-particle track)• radiative interactions with the Coulomb force field of atomic nuclei• bremsstrahlung, in-flight annihilation

Net energy transferred

𝜖𝜖𝑡𝑡𝑡𝑡𝑖𝑖 = (𝑅𝑅𝑖𝑖𝑖𝑖)𝑢𝑢− 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑢𝑢𝑖𝑖𝑜𝑜𝑖𝑖𝑡𝑡 − 𝑅𝑅𝑢𝑢𝑡𝑡 + �𝑄𝑄 = 𝜖𝜖𝑡𝑡𝑡𝑡 − 𝑅𝑅𝑢𝑢𝑡𝑡

– 𝑅𝑅𝑢𝑢𝑡𝑡 = radiant energy emitted as radiative losses by the charged particles (which themselves originated in 𝑉𝑉)

7

Collision kerma

𝐾𝐾𝑐𝑐 =d𝜖𝜖𝑡𝑡𝑡𝑡𝑖𝑖

d𝑚𝑚

• The expectation value of the net energy transferred to charged particles per unit mass at a point of interest, excluding both radiative-loss energy and energy passed from one charged particle to another

Radiative kerma

𝐾𝐾𝑡𝑡 =d𝑅𝑅𝑢𝑢𝑡𝑡

d𝑚𝑚

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For monoenergetic photons

𝐾𝐾𝑐𝑐 = Ψ𝜇𝜇𝑒𝑒𝑖𝑖𝜌𝜌 𝐸𝐸,𝑍𝑍

– 𝜇𝜇𝑒𝑒𝑒𝑒𝜌𝜌

= mass energy-absorption coefficient (depending on 𝐸𝐸 & 𝑍𝑍)

• 𝜇𝜇𝑒𝑒𝑒𝑒𝜌𝜌≈ 𝜇𝜇𝑡𝑡𝑡𝑡

𝜌𝜌for low 𝑍𝑍 and 𝐸𝐸 (where radiative losses are small)

– 𝜇𝜇𝑒𝑒𝑖𝑖 = linear energy-absorption coefficient

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𝐸𝐸𝛾𝛾 (MeV)

𝜇𝜇𝑡𝑡𝑡𝑡 − 𝜇𝜇𝑒𝑒𝑖𝑖𝜇𝜇𝑡𝑡𝑡𝑡

× 100

𝑍𝑍 = 6 29 82

0.1 0 0 0

1 0 1.1 4.8

10 3.5 13.3 26

Absorbed dose

Relevant for• All types of ionizing radiation fields• Any ionizing radiation source distributed within the absorbing medium

Energy imparted

𝜖𝜖 = (𝑅𝑅𝑖𝑖𝑖𝑖)𝑢𝑢− 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑢𝑢 + 𝑅𝑅𝑖𝑖𝑖𝑖 𝑐𝑐 − (𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡)𝑐𝑐+�𝑄𝑄

– 𝜖𝜖 = energy imparted (stochastic quantity)– 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑢𝑢 = radiant energy of all the uncharged radiation leaving 𝑉𝑉– 𝑅𝑅𝑖𝑖𝑖𝑖 𝑐𝑐 = radiant energy of the charged particles entering 𝑉𝑉– 𝑅𝑅𝑜𝑜𝑢𝑢𝑡𝑡 𝑐𝑐 = radiant energy of the charged particles leaving 𝑉𝑉– ∑𝑄𝑄 = net energy derived from rest mass in 𝑉𝑉

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𝐷𝐷 =d𝜖𝜖d𝑚𝑚

• The expectation value of the energy imparted to matter per unit mass at a point• The same dimension & units with 𝐾𝐾• Simply, the energy per unit mass to produce any effects attributable to the radiation (the most

important quantity in radiological physics)• Impossible to write a relationship 𝐷𝐷 and Ψ of indirect radiation• 1 Gy = 1 J/kg = 102 rad = 104 erg/g

Absorbed dose rate

�̇�𝐷 =d𝐷𝐷d𝑡𝑡

=dd𝑡𝑡

d𝜖𝜖d𝑚𝑚

11

Comparative example 1 of 𝜖𝜖, 𝜖𝜖𝑡𝑡𝑡𝑡, & 𝜖𝜖𝑡𝑡𝑡𝑡𝑖𝑖

12Attix Fig. 2.1a

Ex

Comparative example 2 of 𝜖𝜖, 𝜖𝜖𝑡𝑡𝑡𝑡, & 𝜖𝜖𝑡𝑡𝑡𝑡𝑖𝑖

13Attix Fig. 2.1b

Ex

Ex

If the positron in Attix Fig. 2.1b had been annihilated in flight when its remaining KE was 𝑇𝑇3 , what are the values of 𝜖𝜖, 𝜖𝜖𝑡𝑡𝑡𝑡, & 𝜖𝜖𝑡𝑡𝑡𝑡𝑖𝑖 ?

14

Ex

Exposure

Nonstochastic quantity defined only for x-ray & 𝛾𝛾-ray photons

𝑋𝑋 =d𝑄𝑄d𝑚𝑚

– The absolute value of the total charge d𝑄𝑄 of the ions (of one sign) produced in air when all the electrons (negatrons & positrons) liberated by photons in air of mass d𝑚𝑚 are completely stopped in air

– The ICRU says that "the ionization arising from the absorption of bremsstrahlung emitted by the electrons is not to be included in d𝑄𝑄"

• Also, the radiative losses through in-flight annihilation positrons– Simply, the ionization equivalent of the 𝐾𝐾𝑐𝑐 in air for x- & 𝛾𝛾-rays

15

W-value �𝑊𝑊

• See Attix 12 for more details• The mean energy expended in a gas per ion pair formed

�𝑊𝑊 =∑𝑇𝑇𝑖𝑖(1 − 𝑔𝑔𝑖𝑖)∑𝑁𝑁𝑖𝑖(1 − 𝑔𝑔𝑖𝑖′)

– 𝑇𝑇𝑖𝑖 = initial KE of the 𝑖𝑖th electron (or positron)– 𝑔𝑔𝑖𝑖 = fraction of 𝑇𝑇𝑖𝑖 that is spent by the particle in radiative interactions along its full path in air– 1 − 𝑔𝑔𝑖𝑖 = fraction spent in collision interactions– 𝑁𝑁𝑖𝑖 = total number of ion pairs that are produced in air by the 𝑖𝑖th electron of energy 𝑇𝑇𝑖𝑖– 𝑔𝑔𝑖𝑖′ = fraction of the ion pairs that are generated by the photons resulting from radiative loss interactions– 1 − 𝑔𝑔𝑖𝑖′ = fraction of the ion pairs produced by collision interactions that occur along the particle track

• Not count the energy going into radiative losses, nor the ionization produced by the resulting photons

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All the KE spent by electrons in collision interactions

All the ion pairs produced in collision interactions by electrons

• eV/ion pair– 33.97 eV/ip for x- & 𝛾𝛾-rays in air

•�𝑊𝑊𝑚𝑚𝑎𝑎𝑡𝑡𝑒𝑒

=33.97 eV

ip (or electron)

1.602×10−19 C/electron× 1.602 × 10−19 J

eV= 33.97 J

eV

• Constant values for each gas, independent of photon E, for x- & 𝛾𝛾-ray energies above a few keV• Convenient for relating (𝐾𝐾𝑐𝑐)𝑎𝑎𝑖𝑖𝑡𝑡 and 𝑋𝑋

Exposure rate

�̇�𝑋 =d𝑋𝑋d𝑡𝑡

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𝑋𝑋 to Ψ• For monoenergetic photons

𝑋𝑋 = Ψ𝜇𝜇𝑒𝑒𝑖𝑖𝜌𝜌 𝐸𝐸,𝑎𝑎𝑖𝑖𝑡𝑡

𝑒𝑒�𝑊𝑊 𝑎𝑎𝑖𝑖𝑡𝑡

= (𝐾𝐾𝑐𝑐)𝑎𝑎𝑖𝑖𝑡𝑡𝑒𝑒�𝑊𝑊 𝑎𝑎𝑖𝑖𝑡𝑡

=(𝐾𝐾𝑐𝑐)𝑎𝑎𝑖𝑖𝑡𝑡33.97

– 1 R = 1 esu0.001293 g

× 1 C2.998×109 esu

× 103 g1 kg

= 2.580 × 10−4 C/kg

– Conversion factors• 𝑋𝑋 (in C/kg) = 2.58 × 10-4𝑋𝑋 (in R)• 𝑋𝑋 (in R) = 3876 𝑋𝑋 (in C/kg)

• For spectral photons

𝑋𝑋 = �0

𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚Ψ′(𝐸𝐸)

𝜇𝜇𝑒𝑒𝑖𝑖𝜌𝜌 𝐸𝐸,𝑎𝑎𝑖𝑖𝑡𝑡

𝑒𝑒�𝑊𝑊 𝑎𝑎𝑖𝑖𝑡𝑡

d𝐸𝐸 ≈𝑒𝑒�𝑊𝑊 𝑎𝑎𝑖𝑖𝑡𝑡

�0

𝐸𝐸𝑚𝑚𝑚𝑚𝑚𝑚Ψ′(𝐸𝐸)

𝜇𝜇𝑒𝑒𝑖𝑖𝜌𝜌 𝐸𝐸,𝑎𝑎𝑖𝑖𝑡𝑡

d𝐸𝐸

18

Significance of exposure

1. Ψ is proportional to 𝑋𝑋 for any given photon energy or spectrum2. The measurement of 𝑋𝑋 may estimate the effects of x- or 𝛾𝛾-ray in tissue because air is an

approximately "tissue-equivalent" material (𝑍𝑍𝑎𝑎𝑖𝑖𝑡𝑡 ≈ 𝑍𝑍𝑚𝑚𝑢𝑢𝑚𝑚𝑐𝑐𝑚𝑚𝑒𝑒)3. The value of 𝐾𝐾𝑐𝑐 in muscle, per unit 𝑋𝑋, is nearly independent of photon E4. One can characterize a photon field at a point

19Attix Figs. 2.2a & b

Other quantities for radiation protection

Quality factor 𝑄𝑄• Weighting factor to provide an estimate of the relative human hazard of different types &

energies of ionizing radiations• Dimensionless• Determined from the experimental relative biological effectiveness (RBE) & the unrestricted linear

energy transfer (𝐿𝐿∞) or the collision stopping power

20Attix Fig. 2.3

Dose equivalent 𝐻𝐻𝐻𝐻 ≡ 𝐷𝐷𝑄𝑄

– Defined at a point (i.e., a point quantity)– Sievert, 1 Sv = 1 J/kg– 1 rem = 10-2 J/kg (equivalently to "rad")– Not strictly a physical quantity

Equivalent dose 𝐻𝐻𝑇𝑇,𝑅𝑅𝐻𝐻𝑇𝑇,𝑅𝑅 = 𝐷𝐷𝑇𝑇,𝑅𝑅𝑤𝑤𝑅𝑅

– Equivalent dose in an organ or in tissue 𝑇𝑇 due to radiation 𝑅𝑅– Not a point quantity but an average over a tissue or organ– 𝐻𝐻𝑇𝑇 = ∑𝑅𝑅𝐻𝐻𝑇𝑇,𝑅𝑅 = ∑𝑅𝑅𝐷𝐷𝑇𝑇,𝑅𝑅𝑤𝑤𝑅𝑅– Not a measurable quantity

Effective dose

𝐸𝐸 = �𝑇𝑇

𝐻𝐻𝑇𝑇𝑤𝑤𝑇𝑇

– Not a measurable quantity

21Knoll 2