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The following apply for the nuclear reaction: a + b ↔ R → c. in the centre of mass frame, where a and b are the initial species about to collide, c is the final species, and R is the resonant state.
Nuclear reactions may be shown in a form similar to chemical equations, for which invariant mass must balance for each side of the equation, and in which transformations of particles must follow certain conservation laws, such as conservation of charge and baryon number (total atomic mass number). An example of this notation follows:
the equation indicates that the decay constant λ has units of t −1, and can thus also be represented as 1/ τ, where τ is a characteristic time of the process called the time constant. In a radioactive decay process, this time constant is also the mean lifetime for decaying atoms.
The decay scheme of a radioactive substance is a graphical presentation of all the transitions occurring in a decay, and of their relationships. Examples are shown below. It is useful to think of the decay scheme as placed in a coordinate system, where the vertical axis is energy, increasing from bottom to top, and the horizontal axis is the proton number, increasing from left to right.
The generic equation is: A Z X → A Z+1 X′ + e − + ν e [1] where A and Z are the mass number and atomic number of the decaying nucleus, and X and X′ are the initial and final elements, respectively. Another example is when the free neutron (1 0 n) decays by β − decay into a proton (p): n → p + e − + ν e.
In nuclear physics, the Bateman equation is a mathematical model describing abundances and activities in a decay chain as a function of time, based on the decay rates and initial abundances. The model was formulated by Ernest Rutherford in 1905 [ 1 ] and the analytical solution was provided by Harry Bateman in 1910.
When describing a nuclear reactor, where neutron population is directly proportional to thermal power, the following equation is used: P = P 0 e t / τ {\displaystyle P=P_{0}e^{t/\tau }} where P {\displaystyle P} is the reactor power at time t {\displaystyle t} , given an initial power P 0 {\displaystyle P_{0}} , and τ {\displaystyle \tau ...
Ernest Rutherford, working in Canada and England, showed that radioactive decay can be described by a simple equation (a linear first degree derivative equation, now called first order kinetics), implying that a given radioactive substance has a characteristic "half-life" (the time taken for the amount of radioactivity present in a source to ...