# Radiation Units and Dose Calcula

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```					RADIATION UNITS AND DOSE CALCULATIONS:

Three concepts are used in measuring ionizing radiation:

1. Measurement of the electronic charge by an ionizing particle or
ray.

2. Measurement of the energy imparted to matter by the ionizing
particle or ray.

3. Assessment of the biological damage caused by radiation
interaction.

Each concept has unique units and some use in radiation protection.
Terms which are related to measuring ionizing radiation are:

• exposure

• absorbed dose

• biological dose equivalent

• quality factor

• activity

1
EXPOSURE:

Exposure is defined as a measure of the charge produced in air by
gamma and x-rays. The roentgen, R, is the unit of exposure. Here:

1esu
1R                    @STP Dry Air
cm 3

where esu = electrostatic unit

The roentgen is applicable to the measurement of charge in air from
gamma and x-rays. The charge produced per unit time is the exposure rate
(R/hr.)

Example: A radiation worker has worked for two hours in a radiation
field where the exposure rate is 25 mR/hr. Calculate the total gamma and x-
ray exposure received by the worker.

 mR 
Exp.   25       2hr 
    hr 

 50mR

2
ABSORBED DOSE:

Absorbed does is the energy imparted per unit mass of matter by
ionization and excitation caused by ionizing radiation. The unit of absorbed

to any form of matter. For example, the RAD
applies to air, biological system, etc.

The unit useful for particles that have limited ranges (alpha and beta)

Absorbed Dose Rate= energy imparted to matter/unit time.

Example: an alpha source delivers 100 RAD/hr to the air surrounding
the source. Determines the absorbed dose, D, in m RAD imparted to the air
in one hour.

D   100      1hr 

3
RELATIVE BIOLOGICAL EFFECTIVENESS:

Biological effects depend not only on the total energy deposited per
gram (or per volume) but also on the way in which the energy is distributed

In particular the biological effect of any radiation increases with
linear energy transfer (LET) of the radiation. For example, given the same
absorbed dose, the biological effect damage due to alpha particles which
produce dense tracks of ionization is much greater than the damage from
gamma rays, which are less heavily ionized.

RBE = 1     200 kev x-rays for a given tissue organ

RBE depends upon the tissue, the biological effect under
consideration, the dose and dose rate.

4
QUALITY FACTOR:

The quality factor is a dimensionless factor that accounts for the
difference in biological damage to humans, caused by radiation of different
types and energies.

This factor, QF, is based on the concept of linear energy transfer. A LET is
measured for various energies and types of radiation using a sphere of water
that approximates the size of a human body and the constituents of soft
tissue. A QF is then assigned.

Table 3.2 Linear Energy Transfer (LET)
Quality Factor (QF) Relationship

Linear Energy Transfer (keV / 10-6 inch)           Quality Factor

3.5 or less                                        1

3.5 - 7.0                                          1-2

7.0 - 23.0                                         2-5

23.0 - 53.0                                        5 - 10

53.0 - 175.0                                       10 - 20

5
DOSE EQUIVALENT:

The term Dose Equivalent (D) is used to denote the damage done to
the human body by ionizing radiation.

the unit = REM

= Roetgen Equivalent Man

D = R x QF

Dose equivalent rate (DR) is provided in rem/hr or millirem/hr.

If radioactive material is deposited within the body then distribution
of the material must be known.

D = R x QF x Distribution Factor

= RAD x QF x Distribution Factor

Beyond the scope of study

6

NCRP-39

x-rays                           1                            1
Gamma Rays                       1                            1
Beta Particles                                                1
E>0.03MeV                 1
E<0.03 MeV                1.7
Neutrons                                                      10
Thermal - 1 keV           2
10 keV                    2.5
100 keV                   7.5
500 keV                   11
1 MeV                     11
2.5 MeV                   9
5 MeV                     8
7 MeV                     7
10 MeV                    6.5
14 MeV                    7.5
20 MeV                    8
Protons                                                       10
Alpha Particles                  10                           20
Heavy Recoil Nuclei              20                           20

7
For most purposes, external radiation hazards may be considered as
having a quality factor of one.

Exposure to external radiation is measured by dosimeter of some
type.

Neutron Flux (N/cm2/sec)

Dose  1 x109 thermal N/cm2

= 2.5x107 fast N/cm2

=1 rem

Table 3.4 Neutron Flux Dose Equivalents

Neutron Energy                         Neutrons/cm2 Equivalent To 1 rem
(N/cm2)

Thermal                                970x106
100 eV                                 720x106
5 keV                                  820x106
20 keV                                 400x106
100 keV                                120x106
500 keV                                43x106
1 MeV                                  25x106
2.5 MeV                                29x106
5 MeV                                  26x106
7.5 MeV                                24x106
10 MeV                                 24x106
>10 MeV                                14x106

8
PROBLEM:

A worker receives an absorbed dose of :

Calculate the dose equivalent.

SOLUTION:

D  2QF a  3QF    N
 4QF   

 2(20)  3(10)  4(1)
 74rem

9
PROBLEM:

A worker is exposed to 100 keV Nt flux of 2.5x105 Nt/cm2/sec for
three hours. Calculate the dose equivalent in rem.

SOLUTION:

t = 3 hrs(3600 sec/hr)

t = 10800 sec

D=DR x t

= 2.5x105 Nt/cm2/sec x 10800 sec

= 2.7x107 Nt/cm2

1 rem = 120x106 Nt/cm2

D = 0.225 rem

10
DOSE RATE CALCULATIONS:

      ergs 
DR   S    / A       
      g 

S = C (Ci)

 3.7 x1010 dps 
                 C Ci 
      Ci       

= 3.7x1010C dps or gamma/sec

= 3600 sec / hr x 3.7x1010C gamma/sec

= 1.33x1014C gamma/hr

E ( MeV ) 106 eV                ergs 
1.33x1014C gamma/hr                         1.6 x1012
gamma MeV                        eV 

erg
 2.13 x108 CE
hr

11
Radiation emitted by the source is distributed uniformly over the
surface area of a sphere of radius r.

A  4r 2 ft 2
2
 12in   2.54cm 
2

 4r ft 
2   2
       
 ft   in 
 1.17 x104 r 2 cm 2

For tissue over the gamma range 0.2 to 4.0 MeV an approximate
         cm 2
value of     of 0.03      .
          g

Now, a rem of gamma radiation is absorbed when 100ergs are
absorbed per gram of body tissue.

          
       1rem 
DR  S   / A              
       100ergs 

     g    

              erg   0.03cm 2 
  2.13 x108 CE                 
              hr  
      g     
     ergs / g 
                    
 1.17 x104 r 2 cm 2  100
      1rem   
CE
 5.46 2
r
6CE
 2
r

C - Ci

E - MeV

r - ft

12
Table 7.5 Gamma Decay for Selected Radionuclides

Percent     Dose Rate per Curie at
Radionuclide   Gamma Energy (MeV) Gamma Yield 1 Meter,G/10 (rem/hr)

Celsium-134       E1 0.57              23               0.84
E2 0.605             98
E3 0.796             99

Celsium-137       E1 0.662             85               0.31

Cobalt-58         E1 0.511             30               0.53
E2 0.810             99

Cobalt-60         E1 1.173             100              1.40
E2 1.332             100

Chromium-51       E1 0.315             9                0.016

Iodine-131        E1 0.364             82               0.20
E2 0.284             5.4
E3 0.637             6.8

Iron-59           E1 1.095             56               0.66
E2 1.295             44

Manganese-54      E1 0.835             100              0.47

Nickel-65         E1 1.48              25               0.31
E2 1.12              16

13
PROBLEM:

Calculate the dose rate, rem/hr, three ft. from a source containing one Ci
of Mn-54.

SOLUTION:

E = 0.835 MeV
C = 1 Ci
r = 3 ft

6CE
DR 
r2
MeV
 6  1Ci  0.835
 3 ft 
2

 0.557 rem / hr

14
PROBLEM:

What happens when it emits two ’s?

Calculate the does rate, rem/hr, 10 ft. from a 15 Ci Co-60 source.

SOLUTION:
E  1.173 1  1.332 1
 2.505 MeV
C  15Ci
r  10 ft
DR  6 15Ci  2.505 MeV  / 10 ft 
2

 2.25rem / hr

15
PROBLEM:

The dose rate 10 ft. from a pump is 10 rem/hr. Determine the dose
rate 25 ft. from the pump.

SOLUTION:

DR10  10rem / hr
2
r 
DR25  DR10  10 
 r25 
2
10rem  10 

hr  25 
 
rem
 1.6
hr

16

Internal exposures can be caused by beta, gamma, or neutron sources
that are ingested, inhaled or enter the body by way of wounds.

Internally deposited radionuclides are usually concentrated in
localized areas with in the body depending on the chemical characteristics

The exact length of time which it remains in the body (organs) depends to a
great extent on the organs involved and the form of the radionuclides.

Table 8.3 Critical Organs for Selected Radionuclides

Hydrogen-3 (tritium)                  Whole Body
Phosphorous-32                        Bone
Carbon-14                             Fat
Sulfur-35                             Testes, whole body
Chromium-51                           Lower Intestine
Cobalt-60                             Lower Large Intestine
Zinc-65                               Liver, prostate
Rubidium-87                           Pancreas
Technetium-99m                        Upper Large Intestine
Iodine-131 (and I-125)                Thyroid
Cesium-137                            Whole body, liver, spleen
Gold-198                              Lower Large Intestine
Mercury-203                           Kidney
Iridium-192                           Lower Large Intestine
Uranium-235                           Lower Large Intestine
Plutonium-239                         Bone

17
The biological half-life for a particular radionuclide is the time
required to reduce the concentration of a radionuclide to 1/2.

The effective half-life is the time during which a radionuclide may
cause damage, and is determined by combining the effects of physical and
biological half-lives:

1    1   1
                    T = half-life
Teff TP TB


TB  TP 
TBTP
TBTP
Teff 
TB  TP

18
Table 8.4 Effective Half-Lives of Common Radionuclides

T2r               T2B               T2eff
Physical         Biological          Effective
Half-Life   Half-Life          Half-Life
Hydrogen-3            4.5x103           8                   8
Carbon-14             2.0x106           12                  12
Phosphorus-32         14.3              1155                14.1
Sulfur-35             87.1              90                  44.3
Chromium-51           27.8              616                 26.6
Manganese-54          300               25                  23.1
Iron-59               45.1              600                 41.9
Cobalt-58             72                9.5                 8.4
Cobalt-60             1.9x103           9.5                 9.5
Zinc-65               245               933                 194
Rubidium-87           1.8x1013          60                  60
Strontium-90          1.0x10 4          1.8x10 4            6.4x104
Technetium-99m        0.25              20                  0.25
Iodine-131            8                 138                 7.6
Cesium-134            840               70                  64.6
Cesium-137            1.1x104           70                  70
Iridium-192           74.5              20                  15.8
Gold-198              2.7               120                 2.6
Mercury-203           45.8              14.5                11
Radon-222             3.83       None (Inert Gas)           3.83
Uranium-235           2.6x1011          300                 300
Plutonium-239         8.9x106           7.3x104             7.2x104

19
PROBLEM:

Calculate the Teff of Cs-137

SOLUTION:

T
1.1 x10 days   70days 
4

1.1 x104 days  70days
 70days

PROBLEM:

Calculate the effective half-life of S-35

SOLUTION:

T
 87.1days  90days 
87.1days  90days
 44.3days

20
PROBLEM 1:

Calculate the maximum critical dimensions of the core.
A thermal homogeneous reactor has a cylindrical bare core:

•height equals its diameter
•1.3% enriched uranium metal
•light water as a moderator

Possible Data
  1.0558
 TR  0.45
  0.830                                                    D
Given:                                       LM  2.88           L2       (on tables)
f  0.870                                                    a
LSM  5.74
  1.40

Table in Problem Statement

SOLUTION:

k  1.067
L2   2.88   8.2944cm 2
2

L2   5.74   32.95cm 2
2
S

M 2  L2  L2  41.24cm 2
S

d E   0.71 0.45cm   0.32cm
k  1 1.0673  1                v  f  a
B2                      0.00163 2 
M2      41.24                        D
2          2
    2.405 
B 
2
     
 H e   Re 
Re  R  d e
H e  H  2d e  2 R  2d  2 Re
2               2
    2.405 
0.00163            
 2 Re   Re 
Re  71.14cm
R  70.82cm
H  2 R  141.64cm

21
PROBLEM 2:

Calculate the maximum critical dimensions of the core.
A thermal homogeneous reactor has a cylindrical bare core:

•height equals its diameter
•1.3% enriched uranium metal
•light water as a moderator

Possible Data
  1.0558
 TR  0.45
  0.830                                               D
Given:                                       LM  2.88      L2       (on tables)
f  0.870                                               a
LSM  5.74
  1.40

Table in Problem Statement

SOLUTION:

k  1.067
L2   2.88  1  f   1.078cm 2
2

L2  1.10  5.74   36.242cm 2
2
S

M 2  L2  L2  37.320cm 2
S

d E   0.71 0.45cm   0.32cm
k  1 1.0673  1
B2   2                   0.0018cm 1
M         37.320
2           2
    2.405 
B 
2
     
 H e   Re 
2            2
    2.405 
0.0018            
 2 Re   Re 
Re  67.705cm
H  2 R  135.4cm

22
PROBLEM:

Calculate Slowing Down Length: LS2

SOLUTION:

1                                       D
L2      N f   S  f  TR  f   or
a
S
3

Initial E
1  E Ni     
Nf      ln                           Final E
  E Nf




E
ln     S  f
2
S  f
2

  
E
  L2  N f
3 1  Cos    3 1  Cos 
S

23
PROBLEM:

Determine k00 if the mass of U235 (f = 0.024 g/cc) in a homogenous
reactor ~6kg. The reactor core is a cube with water as moderator.

SOLUTION:

Assume- critical dimentions
Assume d = 0.71 to be small

2

Bg  3  
2

S
k  1  v  f   a 
Bm                       
m2          D      

Critcal----> Bg = Bm
2
k  1    
 3 
S
2
M
2

k  3   M 2  1
S

24
PROBLEM:

Calculate S and M2

First determine S.

Mass (g) = fS3

6, 000 gram
 S  63cm
3            g
0.024
cc

Next, M2.
33, Provided in Tables
M  L
2
  L
2
S
2

Dt       Dt          0.18cm
L2                   
 a  af   am 0.02cm  0.037cm
M 2  33  3  36
2

k  3    36   1
 63 
 1.27

25
TABLE 6.1

Moderator     Density           Dt       L               M2

Water         1.00g/cc          0.18cm   2.88cm   33cm2   41cm2

Heavy Water   1.10              0.85     100      120     10,120

Beryllium     1.84              0.61     23.6     98      655

Graphite      1.62              0.92     50.0     350     2,850

26
EXTERNAL EXPOSURE:
Neutrons - Energy Dependent
Tabulated graphically

Gamma’s

for a beam with a single energy

air
•  R                 8         a 
X    1. 83 *10             IE  
sec                            

photons                     's
where I - Intensity                                or
cm 2 * sec                 cm 2 * sec

E - Energy MeV

a                                    cm 2
mass absorption coefficient
                                     g
in mR/hr
air
•  mR             
X      0. 0659IE  a 
 h                

Dose from External Exposure

tis
•  mrad           
D      0. 0576I  a 
 h               
or
tis
  a 
 
•  mrad                                •
D         0. 0874                          X
 h                 a       
air
           
          

Equivalent Dose
•         •
HfD
where f energy dependent function which depends on the type of tissue.

27
INTERNAL EXPOSURE:

Gamma and Charged particle

•  rad         5. 92 *10 4 C  t  E
D   
 sec                     M

where C(t) Ci
M mass of the organ grams
E is energy in MeV

Equivalent dose
•       •
H  DQ

for a multi mode decay
•       •
H  D

Retention function

R  t   C 0 qe b t
q is describes mode of internalization i.e. ingestion or
inhalation

Single intake

51.1C d qe e t
•  rem 
H      
day          M

Continuous Intake

            
t
 e  t                                       Cd q
C t   Cd q  e                                              Ct          1  e e t
e
0

28
Then
•  rem 
H       

51.1C d q 1  e  et   
 day               M e

29

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