Radioactive Decay Law Lecture PDF

Radioactive Decay Law  This law is given by: Nt = No∙exp(-λt) where No = initial number of radioactive nuclei Nt = number of radioactive nuclei at time t λ = decay constant (s-1)  It tells us that the number of radioactive nuclei will decrease in an exponential fashion...

Radioactive Decay Law  This law is given by: Nt = No∙exp(-λt) where No = initial number of radioactive nuclei Nt = number of radioactive nuclei at time t λ = decay constant (s-1)  It tells us that the number of radioactive nuclei will decrease in an exponential fashion with time with the rate of decrease being controlled by the Decay Constant.  The decay constant depends on nothing but the nuclear properties.  the number of radioactive nuclei which will decay during the time interval from t to t+dt must be proportional to N and to dt. In symbols therefore: the minus sign indicating that N is decreasing. Turning the proportionality in this equation into an equality λ, is called the Decay Constant integratet he above equation  This final expression is known as the Radioactive Decay Law where ln x represents the natural logarithm of x  The Law is shown in graphical form in the figure below: Number of radioactive nuclei at any time, Nt, against time, t. classic exponential manner All three curves here are exponential in nature, only the Decay Constant is different. Notice that when the Decay Constant has a low value the curve decreases relatively slowly and when the Decay Constant is large the curve decreases very quickly  In other words from our analysis above by plotting the expression: Notice that this expression is simply an equation of the form y = mx + c where m = -l and c = ln N0.  Half-Life  Half Life is an indicator that expresses the length of time it takes for the radioactivity of a radioisotope to decrease by a factor of two.  Some of the radioisotopes have a relatively short half life.  These tend to be the ones used for medical diagnostic purposes because they do not remain radioactive for very long following administration to a patient and hence result in a relatively low radiation dose. This indicator is called the Half Life and it expresses the length of time it takes for the radioactivity of a radioisotope to decrease by a factor of two. From a graphical point of view we can say that when: Technetium-99 (99Tc) is an isotope of technetium which decays with a half-life of 211,000 years to stable ruthenium-99, emitting beta particles, but no gamma rays. Technetium-99m sodium pertechnetate - This variant is used for diagnostic imaging of the thyroid, salivary gland, urinary bladder, vesicoureteral reflux, and nasolacrimal drainage imaging. At temperatures below 7.46 K, pure, metallic, single- crystal technetium becomes a type-II superconductor. Technetium has a very high magnetic penetration depth below this temperature, greater than any other element, except for niobium. Technetium-99m is used to image the skeleton and heart muscle in particular, but also for brain, thyroid, lungs, liver, spleen, kidney, gall bladder, bone marrow, salivary and lachrymal glands, heart blood pool, infection and numerous specialized medical studies. Technetium-99m is the decay product of molybdenum-99 and undergoes gamma decay to form the ground state of technetium-99. Ground-state technetium-99 can further undergo decay to form ruthenium (element 44). Doctors or trained nuclear medicine health professionals will administer Tc- 99m radiotracers to patients before a diagnostic test, usually by injection, to help diagnose medical conditions. As the half-life of Tc-99m is only six hours, it does not stay in the human body long. Relationship between the Decay Constant and the Half Life applying it to the Radioactive Decay Law. when These last two equations express the relationship between the Decay Constant and the Half Life. Question 1 (a) The half-life of 99mTc is 6 hours. After how much time will 1/16th of the radioisotope remain? (b) Verify your answer by another means. we can calculate the Decay Constant as follows Now applying the Radioactive Decay Law So it will take 24 hours until 1/16th of the radioactivity remains. Relationship between Decay Constant and Half Life 0.693  t1 2 Mathematical Model of Attenuation  Ix and Io are related by:  This final expression tells us that the radiation intensity will decrease in an exponential fashion with the thickness of the absorber with the rate of decrease being controlled by the Linear Attenuation Coefficient (μ). Influence of the Linear Attenuation Coefficient  Half Value Layer  An indicator is usually derived from the exponential attenuation equation which helps us think more clearly about what is going on.  This indicator is called the Half Value Layer, and it expresses the thickness of absorbing material which is needed to reduce the incident radiation intensity by a factor of two.  We can say that when:  the thickness of absorber is the Half Value Layer. Relationship between the Linear Attenuation Coefficient and the Half Value Layer the minus sign indicating that the intensity is reduced by the absorber. Turning the proportionality in this equation into an equality, we can write: where the constant of proportionality, μ, is called the Linear Attenuation Coefficient.  when and inserting it in the exponential attenuation equation, that is:  Question 1 How much aluminium is required to reduce the intensity of a 200 keV gamma-ray beam to 10% of its incident intensity? Assume that the Half Value Layer for 200 keV gamma-rays in Al is 2.14 cm. We are told that the Half Value Layer is 2.14 cm. Therefore the Linear Attenuation Coefficient is Now combining all this with the exponential attenuation equation:  So the thickness of aluminium required to reduce these gamma-rays by a factor of ten is about 7 cm. This relatively large thickness is the reason why aluminium is not generally used in radiation protection - its atomic number is not high enough for efficient and significant attenuation of gamma- rays. Half - life The time taken for the activity of a radionuclide to decay by half its value Half - life Isotope Half-Life Tritium 12.4 y Carbon 14 5730 y Sulphur 35 87.4 d Phosphorus 33 25.6 d Phosphorus 32 14.3 d Iodine 125 60.1 d If we use three half-thickness' of absorber then this will reduce the intensity by: 1/2+1/2+1/2 = 1/8 Barriers of lead, concrete or water can stop radiation or reduce radiation intensity.  Radiation Hazards....neutrons, x-rays & gamma rays are more hazardous for the entire body......alpha & beta emitters are more hazardous When they are ingested or inhaled..  The Absorber  Energy is deposited in the absorber when radiation interacts with it.  It is usually quite a small amount of energy but energy nonetheless.  The quantity that is measured is called the Absorbed Dose and it is of relevance to all types of radiation be they X- or γ-rays, α- or β- particles.  The SI unit of absorbed dose is called the gray (Gy). The gray is defined as the absorption of 1 J of radiation energy per kilogram of material (J∙kg-1).  The traditional unit of absorbed dose is called the rad, which supposedly stands for Radiation Absorbed Dose. It is defined as the absorption of 10-2 J of radiation energy per kilogram of material.  1 Gy = 100 rad  Equivalent Dose  Often the effectiveness with which different types of radiation produce a particular chemical or biological effect varies with a Quality factor that is characteristic of the type of radiation.  The dose-equivalent (DE) in sievert (Sv) is the product of the dose in Gy and that quality factor.  The dose-equivalent (DE) in rem is the product of the dose in rad and that quality factor.  Recently, the newly defined radiation weighting factors, WR have been adopted to represent the Quality factor  Units: The disintegration rate of a radioactive nuclide is called its Activity. The unit of activity is the becquerel named after the discoverer of radioactivity. 1 Bq = 1 disintegration per second this is a small unit, activity more usually measured in: kilobecquerel (kBq) = 103 Bq Megabecquerel (MBq) = 106 Bq Gigabecquerel (GBq) = 109 Bq Terabecquerel (TBq)= 1012Bq ANTOINE HENRI BECQUEREL 1852-1908 Units: Old units still in use: Curie (Ci) = 3.7 x 1010 disintegration per second therefore: 1 Ci = 3.7 x 1010 Bq = 37 GBq 1 mCi = 3.7 x 107 Bq = 37 MBq 1 Ci = 3.7 x 104 Bq = 37 kBq 1 MBq ~ 27 Ci  Radioactive Materials The rate at which the is radiation emitted is called the activity Becquerel (Bq) OR Curie (Ci) 1 Bq = 1 disintegration per second (dps) 1 Ci = 3.7 x 1010 Bq Half-life The TIME taken for one half the nuclei in the sample to decay  Radioactive Materials The rate at which the is radiation emitted is called the activity Becquerel (Bq) OR Curie (Ci) Cs-137 ~ 30 years I-131 ~ 8 days 1 Bq = 1 disintegration per second (dps) Sr-90 ~ 28 yrs 1 Ci = 3.7 x 1010 Bq Half-life The TIME taken for one half the nuclei in the sample to decay  Interventional Radiology Radiography /Fluoroscopy Computed Tomography Mammography BMD C-Arm Dental OPG/CBCT Dental-IOPA Average Effective Dose (mSv) for diagnostic radiology procedures Exam Dose (mSv) Dental x-rays 0.01 Mammogram 0.04 Chest x-ray 0.10 Abdomen x-ray 0.7 Lumbar spine 1.5 Chest CT 7.0 Abdominal CT 8.0 cf: Mettler et al. Radiology 2008, 248(1):254-263 Natural Background 3.1 mSv/ year Computed Tomography Equipment Patient Effective Dose: 2-20 mSv  Interventional Radiology (Angiography & Angioplasty procedures) Patient Effective Dose: 1to100 mSv  General Radiography Patient Effective Dose : 0.1 – 1 mSv Patient Effective Dose : ~ 0.4 mSv 15 /9 8 Dental Radiography (IOPA) Patient Effective Dose : 0.005 – 0.01 mSv Basic Factors for Radiation Protection Time Distance I1 d12 = I2 d2 2 (INVERSE SQUARE LAW)) Shielding