Outline • General dose calculation considerations, Absorbed Dose in Radioactive absorbed fraction Media • Radioactive disintegration processes and associated dose deposition – Alpha disintegration Chapter 5 – Beta disintegration – Electron-capture transitions F.A. Attix, Introduction to Radiological – Internal conversion Physics and Radiation Dosimetry • Summary

Introduction Radiation equilibrium • We are interested in calculating the absorbed dose in a. The atomic composition radioactive media, applicable to cases of of the medium is – Dose within a radioactive organ homogeneous – Dose in one organ due to radioactive source in another b. The density of the organ medium is • If conditions of CPE or RE are satisfied, dose homogeneous c. The radioactive source calculation is straightforward is uniformly distributed • Intermediate situation is more difficult but can be d. No external electric or handled at least in approximations magnetic fields are present

Charged-particle equilibrium Limiting cases • Emitted radiation typically includes both • Each charged particle of a given type and photons (longer range) and charged energy leaving the volume is replaced by an particles (shorter range) identical particle of the same energy entering • Assume the conditions for RE are satisfied the volume • Consider two • Existence of RE is sufficient condition for CPE limited cases based • Even if RE does not exist CPE may still exist on the size of the (for a very large or a very small volume) radioactive object

1 Limiting cases: small object Limiting cases: large object • A radioactive object V having a mean radius not much • A radioactive object V with mean radius » l/m for greater than the maximum charged-particle range d the most penetrating g-rays • CPE is well approximated at any internal point P that is at least a distance d from the boundary of V • RE is well approximated at any internal point P • If d« l/m for the g-rays, the absorbed dose D at P that is far enough from the boundary of V so g-ray approximately equals to the energy per unit mass of penetration through that distance is negligible medium that is given to the charged particles in radioactive decay (less their radiative losses) • The dose at P will then equal the sum of the • The photons escape from the object and are assumed not energy per unit mass of medium that is given to to be scattered back by its surroundings charged particles plus g-rays in radioactive decay

Limiting cases: large object Absorbed fraction • Deciding upon a maximum • An intermediate-size radioactive object V g-ray “range” L for this case requires quantitative • Dose at P will then equal the sum of the energy criterion per unit mass of medium that is given to – For primary beam only (m) the charged particles plus dose from some g-rays penetration is < 0.1% through L~7 mean free paths • To estimate the dose from g-rays define – Taking into account scatter, absorbed fraction: (broad beam geometry, m  ) g - ray radiant energy absorbed in target volume will increase the required AF  object size to satisfy the g - ray radiant energy emitted by source attenuation objectives

Attenuation of 1MeV g-rays in water

Intermediate-size radioactive object Case 1

• Reciprocity theorem: energy spent • Consider volume V filled by a homogeneous medium and a in dv due to the source in dv’:

uniformly distributed g-source  d v ,dv   dv ,d v  and  dv ,V   V ,dv • The volume may be surrounded by • No source in dv” Case 1: infinite homogeneous medium identical to V, but non- • Define Rdv as the expectation radioactive (an organ in the value of the g-ray radiant energy body) emitted by the source in dv, and Case 2: infinite vacuum (object in dv,V the part of that energy that is air) spent in V (source dv and target V)

2 Case 1 continued Case 1 continued • The absorbed fraction with respect • Reduction in the absorbed dose due to g-rays to source dv and target V is : energy escaping from V • Can estimate AF using mean effective attenuation  dv ,V  V ,dv m  AF dv ,V   coefficient for g-rays energy fluence through a R dv R dv distance r in the medium source, target  2  • Estimates reduction in absorbed 1  m r AF  1  e sin  d  d  dv ,V     dose relative to RE condition 4 • For very small radioactive objects   0   0 (V  dv) this absorbed fraction • For poly-energetic sources have to find an average approaches zero; for an infinite value of the absorbed fraction radioactive medium it equals unity

Case 1 example Case 2 Radioactive object V surrounded by vacuum

• More difficult to calculate radius of sphere • The reciprocity theorem is only approximate due to the lack of backscattering • Energy dependence (max ~80keV) • Dose is lower than in Case 1 (ratios >1) • Dose calculations published in MIRD (Medical Internal Radiation Dosimetry) reports • The larger the radius, the lower the g-ray energy – the closer to RE condition (AF=1)

Case 2 MIRD tables

Pamphlet 11, 1975 • To obtain a crude estimate of the dose at some • AF is incorporated in “S-value” point P within a uniformly g-active homogeneous tabulated for each radionuclide object, it may suffice to obtain the average and source-target configuration distance r from the point to the surface of the used in nuclear medicine object by • Dose absorbed in any organ Pamphlet 1  2  (target) due to a source in some r    r sin  d  d  other organ is calculated based 5rev, 1978 4   0   0 on cumulative activity, D=AxS

• Then one may employ men = m in the straight- • S values are typically calculated ahead approximation to obtain by Monte Carlo technique • Straightforward approach, but  m en r AF dv ,V  1  e accuracy is limited

3 MIRD tables MIRD tables

Bouchet et al., MIRD Pamphlet No. 19, The Journal of Nuc. Med., Vol. 44, p. 1113, 2003 • The accuracy is continuously improved through revisions and updates • Example: kidneys represent a frequent source of radiopharmaceutical uptake in both diagnostic and therapeutic nuclear medicine Bouchet et al., MIRD Pamphlet No. 19, THE JOURNAL OF NUC. MED., Vol. 44, p. 1113, 2003

Radioactive disintegration processes Alpha disintegration • Radioactive nuclei undergo transformations to a more stable state through expulsion of energetic particles • Occurs mainly in heavy nuclei • Example: decay of radium to radon, presented by the mass-energy balance equation

226 222 4 • The total mass, energy, momentum, and electric 88 Ra  86 Rn  2 He  4 .78 MeV  1602 y charge are conserved parent 1 / 2 daughter • Energy equivalent of the rest mass: • Each of the elemental terms represents the rest 12 mass of a neutral atom of that element – 1 amu=1/12 of the mass of 6C nucleus=931.50 MeV – 1 electron mass=0.51100 MeV • Energy term represents kinetic energy released

Absorbed dose from Alpha disintegration - disintegration • Take into account both branches through the average branching ratios • For radium average kinetic energy given to charged particles per disintegration

E   0 .946  4 .78 MeV  0.054  4.6 MeV  4.77 MeV • For CPE condition in a small (1 cm) radium-activated object: D=4.77 x n MeV/g, where n – number of • Two branches are available for disintegration disintegrations per gram of the matter • Kinetic + quantum energy released is 4.78MeV • For RE condition (large object): D=4.78 x n MeV/g, • After the -particle slows down it captures two electrons from includes g-ray energy its surroundings, becoming a neutral He atom

4 Beta disintegration Beta disintegration • Nuclei having excess of neutrons typically emit an electron, - particle; atomic number Z is increased by 1 • Nuclei having excess of protons typically emit positron, + particle; atomic number Z is decreased by 1 • Nucleus is left in an exited state, emitting one or more g-rays • Example:

32 32 - 0 • Kinetic energy 1.71 MeV is shared between - and neutrino 15 P  16 S    0   1 .71 MeV  1 / 2 14 .3 d 32 + kinetic energy • Charge balance is realized through initially ion of 16S capturing an electron

Beta disintegration Beta disintegration

• Average kinetic energy of the - or + particle is typically • In + disintegration valence electron is emitted simultaneously 0.3-0.4 Emax; 1/3 Emax is often used for purposes of roughly 15 15 estimating absorbed dose • Example: 8O -> 7N 15 15 -  0 • Neutrino is not included in dose estimates due to almost zero 8 O  7 N  e    0   1 .73 MeV  122 s rest mass energy and no charge 1 / 2 2 positron annihilation yields 2m0c =1.022 MeV

Absorbed dose from Example 5.1 - disintegration • Uniformly distributed - and g-ray source • The rest-mass loss is spent • Under CPE condition D=nEavg MeV/g for n – half in 1 -MeV g -ray production and disintegrations per gram of medium - – half in  -decay, for which Emax=5 MeV and Eavg=2 MeV • Any additional contributions to energy • The point of interest P is located >5 cm inside_ the deposition due to g-rays must be included boundary of the object, at an average distance r = 20 for RE condition cm from the boundary. -1 -1 • Radiative losses by -rays, such as • men=0.0306 cm and m = 0.0699 cm for the g -rays bremsstrahlung and in-flight annihilation, • A total energy of 10-2 J converted from rest mass in are ignored each kg of the object • Estimate the absorbed dose at P

5 Example 5.1 Example 5.2 • For -ray Emax=5 MeV corresponds to maximum particle range (Appendix E) d ~2.6 cm << 1/m=14.3 cm • CPE exists at P, therefore dose due to -rays: • What is the absorbed dose rate (Gy/h) at the

1  2  2 center of a sphere of water 1 cm in radius, D   E avg   10 J/kg  10 J/kg 2 32 homogeneously radioactivated by 15P, 1/m • For g-ray 20cm is not >> 1/m=14.3 cm with 6 x 105 disintegrations per second d L>5cm • RE does not exists at P, therefore have occurring per gram of water? (Assume time r  20 cm P to use absorbed fraction: constancy)  m r AF  1  e en  0 .46 32 32 - 0  2  3 P  S      1 .71 MeV D g  0 .46  (1 / 2 )  10 J/kg  2 .3  10 J/kg 15 16 0  1 / 2 14 .3 d  3  2 D tot  ( 2.3  10 )  10 J/kg  1.23  10 J/kg

Example 5.2 Electron-capture transitions

• Emax=1.71 MeV corresponds to maximum • Parent nucleus captures its own atomic particle (-) range of ~0.8 cm < 1 cm electron from K-shell (~90% probability) or L-shell (~10% probability) and emits • CPE condition monoenergetic neutrino D  N  E • Absorbed dose rate: avg • Resulting shell vacancy is filled with

5 dis MeV electron from a higher orbit, leading to D  6  10  0 .694 g sec dis emission of a fluorescence x-ray + 5 MeV s 10 Gy  4 .164  10  3600  1 .602  10 • Process competing with  disintegrations g sec hr MeV/g  2 .4 Gy/h

Electron-capture transitions Electron-capture transitions

22 22 • Example: 11Na -> 10Ne with half-life for both branches of 2.60 years – + branch

22 22 -  0 11 Na  10 Ne  e    0   0 .546 MeV (k.e.)  1 / 2  2 .60 y 2m0=1.022 MeV

– EC branch • Binding energy for K-shell Eb~ 1 keV + 22 22 0 • For  to occur the minimum atomic mass decrease of 2m0 11 Na  10 Ne  0   E b  1 .275 MeV (E g ) between the parent and daughter nuclei is required to supply   2 .60 y 1 / 2 + 1.568 MeV  with kinetic energy; EC does not have this requirement

6 Absorbed dose for EC process Internal conversion

• An excited nucleus can impart its energy • Most of the energy is carried away by directly to its own atomic electron, which neutrino then escapes with the net kinetic energy of • The only available energy for dose hv-Eb (hv is the excitation energy) deposition comes from electron binding • No photon is emitted in this case term Eb, which is very small compare to that • Process competing with g-ray emission of neutrino • Internal conversion coefficient is the ratio of Ne/Ng

Internal conversion Absorbed dose for internal conversion • If IC occurs in competition with g-ray emission, it results in increase in absorbed dose in small objects (CPE condition) due to release of electron locally depositing the energy

E IC  hv  E b

• In addition electron binding energy is contributed

137 137 to the dose unless it escapes as a fluorescence x-ray • Example: 55Cs -> 56Ba

Absorbed dose for internal Fluorescence data conversion

• If the fraction p = 1 – AF of these fluorescence x- rays escape, then the energy contributed to dose per IC event under CPE condition

f IC  hv  p K Y K h v K  p L Y L h v L

 m r • Using straight-ahead approximation p  e en

• Values of fluorescence yield YK,L and the mean emitted x-ray energies hvK,L are tabulated Electron binding energies and Fluorescence yield for K mean fluorescence x-ray and L shells energies, K and L shells

7 Example 5.5 Example 5.5 • A sphere of water 10 cm in diameter contains a uniform source of 137Cs undergoing 103 dis/(g s). • First check RE condition: 1/m=30.6 cm >> r=5 What is the absorbed dose at the center, in grays, for • Find absorbed fraction AF  1  e  0 .0327  5  0 .151 137m 5 a 10-day period, due only to the decay of 56Ba? • Dose in 10 days=8.64x10 s is Use the mean-radius straight-ahead approximation. 3 dis g  rays MeV D g  10  0 .85  0 .662 g s dis g  ray Gy For g-ray of 0.662 MeV  1 .602  10 10  8 .64  10 5 s  AF -1 MeV / g in water men=0.0327 cm  1 .17  10  2 Gy

Example 5.5 Example 5.5

• For the K-shell conversion process we need • Similarly, for the L+M+…-shell conversion process L h v -1 h v  E -1 YK=0.90, K = 0.032 MeV, and men=0.13 cm for we need YK=0.90, L b = 6 keV, and men=24 cm

 m en r  0 .13  5  24  5 0.032 MeV. Then p K  e  e  0 . 52 and the for 6 keV. Then p L  e  0 and the corresponding dose contribution is dose contribution

L 3 dis IC ( L ) MeV K 3 dis IC ( K ) MeV D  10  0 .078  ( hv  p L Y L h v L ) D  10  0 .078  ( hv  p K Y K h v K ) IC IC g s dis IC ( K ) g s dis IC ( L )

Gy 10 Gy 5  1 .602  10 10  8 .64  10 5 s  1 .602  10  8 .64  10 s MeV / g MeV / g

-3  1 .080 (0 .662  0 .52  0 .90  0 .032 )  10  2 Gy  1 .65  10 Gy

 6.99  10 -3 Gy K L  2 • The total absorbed dose is D tot  D g  D IC  D IC  2 .03  10 Gy

Summary • General approach to dose calculation within and outside of distributed radioactive source Radioactive decay – Absorbed fraction • Radioactive disintegration processes and calculation of absorbed dose Chapter 6 – Alpha disintegration – Beta disintegration F.A. Attix, Introduction to Radiological – Electron-capture transitions Less important for Physics and Radiation Dosimetry – Internal conversion dose deposition

8 Outline Introduction • Decay constants • Particles inside a nucleus are in constant motion • Mean life and half life • Natural radioactivity: a particle can escape from a nucleus if it acquires enough energy • Parent-daughter relationships; removal of • Most lighter atoms with Z≤82 (lead) have at least daughter products one stable isotope • Radioactivation by nuclear interactions • All atoms with Z > 82 are radioactive and disintegrate until a stable isotope is formed • Exposure-rate constant • Artificial radioactivity: nucleus can be made unstable • Summary upon bombardment with neutrons, high energy protons, etc.

Total decay constants Total decay constants

• Consider a large number N of identical • The product of t (for a time interval radioactive atoms t<<1/) is the probability that an individual • The rate of change in N at any time atom will decay during that time interval dN • The expectation value of the total number of    N dt atoms in the group that disintegrate per unit • We define  as the total radioactive decay of time (t<<1/) is called the activity of the (or transformation) constant, it has the group, N dimensions reciprocal time (usually s-1)

Total decay constants Partial decay constants

• Integrating the rate of change in number of • If a nucleus has more than one possible mode atoms we find of disintegration (i.e., to different daughter products), the total decay constant can be N   t  e written as the sum of the partial decay N 0 constants i: • The ratio of activities at time t to that at t = 0 0    A   B  

 N   t • The total activity is N   N   N     e A B

 N 0

9 Partial decay constants Units of activity

• The partial activity of the group of N nuclei with • The old unit of activity was the Curie (Ci), originally respect to the ith mode of disintegration can be defined as the number of disintegrations per second written occurring in a mass of 1 g of 226Ra  N   N e   t i i 0 • When the activity of 226Ra was measured more 10 -1 • Each partial activity iN decays at the rate accurately the Curie was set equal to 3.7  10 s determined by the total decay constant  since the • More recently it was decided by an international stock of nuclei (N) available at time t for each type standards body to establish a new special unit for of disintegration is the same for all types, and its -1 depletion is the result of their combined activity activity, the becquerel (Bq), equal to 1 s 10 • The fractions iN/N are constant 1 Ci  3 .7  10 Bq

Mean life and half life Mean life and half life • The expectation value of the time needed for an initial • The half-life  is the expectation value of population of N0 radioactive nuclei to decay to 1/e of their 1/2 original number is called the mean life =1/ the time required for one-half of the initial N 1 number of nuclei to disintegrate, and hence   0 .3679  e  

N 0 e for the activity to decrease by half: ln e 1   1     N    0 .5  e 1 / 2

•  represents the average lifetime of an individual nucleus  N 0 •  is also the time that would be needed for all the nuclei to 0 .6391    0 .6391  disintegrate if the initial activity of the group, N , were 1 / 2 0  maintained constant instead of decreasing exponentially

Radioactive parent-daughter Mean life and half life relationships

• Consider an initially pure large population (N1)0 of parent nuclei, which start disintegrating with a decay constant 1 at time t = 0 • The number of parent nuclei remaining at time t is - t N1 = (N1)0e 1 • Simultaneously the daughter will disintegrate with a nd decay constant of 2 (2 generation doing the decaying )

• The rate of removal of the N2 daughter nuclei which • Exponential decay characterized in terms of mean exist at time t0 will -2N2 life and half life

10 Radioactive parent-daughter Radioactive parent-daughter relationships relationships • Thus the net rate of accumulation of the daughter • The ratio of daughter to parent activities vs. time: nuclei at time t is  N  2 2 2    2   1  t dN 2  1  e    N   N 1 1 2 2  N    dt 1 1 2 1

  t   N e 1   N 1  1  0 2 2 • If 1 is composed of partial decay constants 1A, 1B, and so on, resulting from disintegrations of A, • The activity of the daughter product at any time t, B, … types, then the ratio for a particular type A is assuming N2 = 0 at t = 0, is

   t   t  2 N 2  1 A  2       t  N   N 2 e 1  e 2 2 1 2 2 1  1  0     1  e     2 1  1 N 1  1  2   1

Equilibria in parent-daughter Equilibria in parent-daughter activities activities • The activity of a daughter resulting from an initially • This maximum occurs at the same time that the pure population of parent nuclei will be zero both at activities of the parent and daughter are equal, but t = 0 and  only if the parent has only one daughter (1A = 1) • We can find the time tm when 2N2 reaches a maximum • The specific relationship of the daughter’s activity

d  N  to that of the parent depends upon the relative 2 2   1t m   2 t m  0    1 e   2 e  dt magnitudes of the total decay constants of parent (1) and daughter (2) ln  /   t  2 1 giving m  2   1

Daughter longer-lived than Daughter longer-lived than

parent, 2 < 1 parent, 2 < 1 • For a single daughter product the ratio of activities

 N       t 2 2  2 e 1 2  1   1 N 1  1   2 • This ratio increases continuously with t for all times - t • Since 1N1 = 1 (N1)0e 1 one can construct the activity curves vs. time for the representative case of metastable tellurium-131 decaying to its only daughter iodine-131; and then to xenon-131: - -   131 m Te  131 I  131 Xe 52 53 54 Qualitative relationship of activity vs. time for Te-131m as parent   30 h  193 h 1/2 1/2 and I-131 as daughter

11 Daughter shorter-lived than Daughter shorter-lived than

parent, 2 > 1 parent, 2 > 1

• For t >> tm the value of the daughter/parent • If the decay scheme is branching to more than one activity ratio becomes a constant, assuming N2 = 0 daughter (1=1A+1B+…) at t = 0:  2 N 2 1 A  2  2 N 2  2    N      N    1 1 1 2 1 1 1 2 1 • For the special case of transient equilibrium where

• The existence of such a constant ratio of activities  1  2 

is called transient equilibrium, in which the  1 A  2   1 daughter activity decreases at the same rate as that the activity of the Ath daughter is equal to its of the parent parent’s – secular equilibrium condition

Daughter shorter-lived than Daughter shorter-lived than

parent, 2 > 1 parent, 2 > 1 • To estimate how close is the daughter to • Example of approaching a transient equilibrium with its parent transient 99 we evaluate the ratio of activities at a time t = ntm equilibrium: Mo 99m to that of the equilibrium time te: to Tc • Two branches: 86%   N   2 2  99m   decays to Tc,   1 N 1  nt m  n ln   /   14% to other  1  e 2 1  1 for large n   N  excited states of  2 2  99   Tc   1 N 1  t e

Only daughter much shorter- Only daughter much shorter-

lived than parent, 2 >> 1 lived than parent, 2 >> 1 226 • For long times (t >> 2) the ratio of activities • An example of this is the relationship of Ra 222 218  N  parent, decaying to Rn daughter, then to Po: 2 2  2  1  ,g   1 N 1  2   1 226 222 218 88 Ra  86 Rn  84 Po  1 / 2 1602 y  1/2  3 .824 d the activity of the daughter very closely  6 -1 -1  1 .1845 10 d   0 .18125 d approximates that of the parent 1 2 • Such a special case of transient equilibrium, where • The ratio of activities the daughter and parent activities are practically  N 0 .18125 equal, is called secular equilibrium (typically, with 2 2   1 .000007  1  N 0 .18125  1 .1845  10  6 a long-lived parent “lasting through the ages”) 1 1

12 Only daughter much shorter- Removal of daughter products lived than parent, 2 >> 1 • For diagnostic or therapeutic applications of short-lived radioisotopes, it is useful to remove the daughter product from its relatively long-lived parent, which continues producing more daughter atoms for later removal and use • The greatest yield per milking will be obtained at time tm since the previous milking, assuming complete removal of the daughter product each time • Waiting much longer than t results in loss of activity • It can be shown that in a case 2 >> 1 all the progeny m atoms will eventually be nearly in secular equilibrium with due to disintegrations of both parent and daughter a relatively long-lived ancestor

Radioactivation by nuclear Removal of daughter products interactions • Assuming that the initial daughter activity is zero at • Stable nuclei may be transformed into radioactive time t = 0, the daughter’s activity at any later time t is species by bombardment with suitable particles, or obtained from photons of sufficiently high energy

 2      t • Thermal neutrons are particularly effective for this  N   N 1  e 2 1  2 2 1 1 purpose, as they are electrically neutral, hence not  2  1 • This equation tells us how much of daughter activity repelled from the nucleus by Coulomb forces, and exists at time t as a result of the parent-source are readily captured by many kinds of nuclei disintegrations, regardless of whether or how often • Tables of isotopes list typical reactions which give the daughter has been separated from its source rise to specific radionuclides

Radioactivation by nuclear Radioactivation by nuclear interactions interactions

-1 -2 • Let Nt be the number of target atoms present in the • If  is the particle flux density (s cm ) at the sample, and  is sample to be activated: the interaction cross section (cm2/atom) for the activation process, then the initial rate of production (s-1) of activated N m A atoms is dN N   act  t     N  A t  dt  0 where NA = Avogadro’s constant (atoms/mole) • The initial rate of production of activity of the radioactive -1 A = gram-atomic weight (g/mole), and source being created (Bq s ) is given by

 d  N act   m = mass (g) of target atoms only in the      N t   dt  0 sample here  is the total radioactive decay constant of the new species

13 Radioactivation by nuclear Radioactivation by nuclear interactions interactions

• If we may assume that  is constant and that Nt is • After an irradiation time t >> =1/, the rate of decay equals not appreciably depleted as a result of the activation the rate of production, reaching the equilibrium activity level process, then the rates of production given by these  N    N  equations are also constant act e t • Assuming N = 0 at t = 0, the activity in Bq at any time t • As the population of active atoms increases, they act -1 after the start of irradiation, can be expressed as decay at the rate Nact (s )  N   N  1  e   t   N  1  e   t • Thus the net accumulation rate can be expressed as act act e   t   • If no decay occurs during the irradiation period t (which will be dN act approximately correct if t << )   N t    N act dt  N act    N t  t

Radioactivation by nuclear interactions Exposure-rate constant

• The exposure-rate constant  of a radioactive 2 nuclide emitting photons is the quotient of l (dX/dt) by A, where (dX/dt) is the exposure rate due to photons of energy greater than , at a distance l from a point source of this nuclide having an activity A: if no decay 2 l  dX      A  dt   • Units are R m2 Ci-1 h-1 or R cm2 mCi-1 h-1 • It includes contributions of characteristic x-rays and Growth of a radionuclide of decay constant  due to a constant internal bremsstrahlung rate of nuclear interaction

Exposure-rate constant Exposure-rate constant

• We would like to calculate specific g-ray constant  at a

given point source; the exposure-rate constant  may be calculated in the same way by taking account of the additional x-ray photons (if any) emitted per disintegration • At a location l meters (in vacuo) from a g-ray point source having an activity A Ci, the flux density of photons of the

single energy Ei is given by

1 Ak • The exposure-rate constant  was defined by the ICRU to replace the 10 9 i  E  3 .7  10 Ak i  2 .944  10 earlier specific gamma-ray constant , which only accounts for the exposure i 4 l 2 l 2 rate due to g-rays where k is the number of photons of energy E emitted per •  is greater than  by 2% or less, with except for Ra-226 (12%) and I-125 i i  disintegration (in which case  is only about 3% of  because K-fluorescence x-rays following electron capture constitute most of the photons emitted)

14 Exposure-rate constant Exposure-rate constant

• Flux density can be converted to energy flux density, • For photons of energy Ei the exposure rate is given by 2 expressed in units of J/s m (expressing Ei in MeV):

 4 Ak i E i 2  dX  1  m   d   1  m    4 .717  10 (J/s m ) en en E 2            i     E i l dt 33 .97  dt 33 .97    E   E ,air   E   E ,air • And related to the exposure rate by recalling i i i i and the total exposure rate for all of the g-ray energies d  d  dR       Ei present is dt dt  da  n dX 1  m en      E  m  e e    i en     dt 33 .97  X        K    K  / 33 .97 i 1   E ,air     c air   c air i  W W   E ,air   air   air

Exposure-rate constant Exposure-rate constant

• Substituting the expression for the energy flux • The specific g-ray constant for this source is defined as the density, we obtain exposure rate from all g-rays per curie of activity, normalized to a distance of 1 m by means of an inverse- n   dX  5 A  m   1 .389  10  k E  en   C/kg s square-law correction: 2  i i   dt l  i 1    E ,air   i  2 n   dX l  m  2     193 .8  k E  en   R m /Ci h • This can be converted into R/h, remembering that  i i   dt A  i 1    E ,air  1 R = 2.58  10-4 C/kg and 3600 s = 1 h:  i  2 where Ei is expressed in MeV and men/ in m /kg dX A n   m    193 .8  k E  en   R/h 2 2  i i   • If (men/)Ei,air is given instead in units of cm /g, the constant dt l  i 1    E ,air   i  in this equation is reduced to 19.38

Exposure-rate constant Exposure-rate constant

• Applying this to, for example, 60Co, we note first • The exposure rate (R/hr) at a distance l meters that each disintegration is accompanied by the from a point source of A curies is given by emission of two photons, 1.17 and 1.33 MeV dX  A • Thus the value of k is unity at both energies  i 2 • Using the mass energy absorption coefficient values dt l for air at these energies are, we find where  is given for the source in R m2/Ci h, and attenuation and scattering by the surrounding   193 .8(1 .17  0 .00270  1 .33  0 .00262 ) medium are assumed to be negligible  1 .29 R m 2 /Ci h which is close to the value given in the table

15 Summary • Decay constants • Mean life and half life • Parent-daughter relationships; removal of daughter products • Radioactivation by nuclear interactions • Exposure-rate constant

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