Physics Experiment Design of critical mass Essay Example

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Simulation of Critical Mass 8

Simulation of Critical Mass



Critical mass is a point where a chain reaction becomes self-sustaining. It is also referred to as the smallest mass of a fissionable material in which a self-sustaining chain reaction can occur. The smallest size of the uranium nucleus cause neutrons emitted during nuclear fission to travel at a high speed on a distance in the order of centimeters before interacting with other nuclei. There are several factors which determine the amount of critical mass in a fissionable material which include; shape of the material, composition and density of the material and also the material’s level of purity (Strieber, 2009) 

Simulation is the representation of elements in a system by arithmetic and logical process which is executable on a computer in order to predict its behavior. The purpose of a simulation is to predict the behavior of the real system under conditions which it has never experienced. Critical mass is the central idea behind nuclear weapons. Use of time independent models in estimating critical mass is highly preferred. Many simulation processes are done in line with the Monte Carlo method. In this experiment, I sought to use the nuclear fission of a uranium isotope U235 in simulation and calculation of its critical mass

Nuclear fission; Uranium isotope U235 has an unstable nucleus. Its nucleus hence spontaneously divides or splits into a number of fragments. The half-life of a radioactive substance is defined as the time required for half the number of nuclei to divide or disintegrate. The U235 is estimated to have a half-life of about 700 million years(Tushingham, 1996). Due to such small rates of disintegration only a small part of the nuclei splits at any particular time. This slow rate of disintegration generates only slight warmth on the uranium when felt.Nuclear weapons and nuclear power plants need U235 because it is the only isotope fissile with thermal neutrons.

Chain reaction; Chain reaction occurring in uranium is a random process. The rate of release of energy is increased due to chain reaction. In this process, the neutrons emitted during one spontaneous fission process collide with other uranium nuclei. These nuclei absorb the energy and become highly unstable. They undergo fission emitting more neutrons and triggering more fission. This process is referred to as generation. Assuming that two neutrons are emitted in each fission process, there will be 2N fissions at the end of the first generation, 4N after second and so on. This continuous process produces 2kN induced fissions after k generations. The number grows exponentially toapproximately billion times the original. This, however, is after only about 30 generations of the induced fission(Strieber, 2009).

PURPOSE; the main purpose of the experiment is to conduct and design experiments to illustrate critical mass in the radioactive material uranium U235.

Hypothesis on critical mass; The smallest mass of uranium in which a self-sustaining chain reaction occurs is known as the critical mass. For a small piece of uranium, in case of emission of two or more neutrons, the neutrons may leave the piece before coming across another uranium neutron. The average number of induced fissions caused by another will be less than two. The survival fraction, f, is determined by the mass, purity and shape of the uranium. In a situation where the survival fraction, f, is greater than unity (1), there occurs exponential growth in the release energy between generations. If the survival fraction is equal to 1, it is said to have a critical mass. Furthermore, if the survival fraction is greater than unity, there occurs spontaneous chain reaction and a nuclear explosion occurs (Tushingham, 1996).This spontaneous chain reaction can be obtained by bringing together two pieces of uranium whose combined mass is greater than the critical mass.

Subcritical: Does not sustain chain reactions.

Critical: Sustains chain reactions.

Supercritical: Sustainschain reactions and ishard to control.

MATERIAL: Rectangular block.

Physics Experiment Design of critical mass


Calculating the critical mass; Critical mass of a block of uranium can be calculated by use of the survival fraction for a range of masses and shapes(Strieber, 2009). It is noted that a block of uranium has a critical mass if the survival fraction, f, is equal to 1.0. In this design I sought to find the value of f for a rectangular block of mass M. This involves generating a large number of a simulated random fission (N) while monitoring the number of induced fissions which cause emitted neutrons (Nin).

Simulation of fission process; First, a location of the nucleus undergoing fission is chosen to be a random point. For a rectangular block of a*a*b the random points are subjected to the following conditions.

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Heavy nuclear fragments are not used in the experiment. The neutrons generated travel in numerous different directions. If there is isotropic distribution in the neutrons, then the probability of a neutron hitting an area on the sphere depends on the size of the area(Martini, 1998). Likewise, the probability of the generated neutron hitting another before leaving the surface depends on the distance along line of flight. In this simulation, it is assumed that a neutron generated can hit another nucleus after travelling a distance between 0 and 1cm.

Calculating survival fraction; A high number of random neutrons are generated. The neutron endpoints are recorded. These endpoints need to lie within the block. Nine random numbers are needed and formulas are applied to ensure the parameters lie within the required range. The endpoint is calculated for each neutron. This endpoint is then tested to verify its location within the block range. Value of fis calculated as the ratio Nin/N.

Survival fraction; The block is assumed to have a unit density so that the dimensions a and b are easily calculated.

Input; M = mass of the block

S = ratio of dimensions a/b

N = number of random points

Output; 1st set: M = 1, S = 1, N = 100, F= 0.85

2nd set: M = 1, S = 16, N = 100, F = 0.50

Critical Mass of a rectangular block;for a rectangular block, the values of S are taken to be S = 0.25, 0.5, 0.75, 1, 1.25, 1.5, and 1.75 and the mass of the block taken as M. The value of N is taken as 100 in each calculation. The critical mass is calculated by plotting a graph of the survival fraction f against mass M. These points are then joined on a smooth curve. These points do not lie exactly on the curve. The point where the curve crosses f=1 is then taken as the critical mass. The value of uncertainty in the critical mass is also estimated by bracketing the hand-drawn curve with a curve on either side. This includes all the data points. The value of M of which the bracketing curves cross the f=1 is noted.Once critical points corresponding to each shape have been drawn and established, the values are then used to plot a graph of critical masses against the dimensions S (0.25, 0.5, 0.75, 1, 1.25, 1.5,1.75).

Calculating critical mass of a cube; for a cube, the ratio is taken as S = a/b = 1. The procedure done for the rectangular block is then repeated. The critical mass is found by making a plot of values of survival fraction f against mass M.


Disadvantages of using simulations

Use of simulations in analysis has various limitations. The amount of time and resources spent in developing and completion of a simulation process is a major drawback (Rubinstein, 1981). Simulations require several processes to be implemented in order to obtain output.These stages of simulations require to be developed with much keenness. They consume much time in analysis. They also require numerous resources in development. The process is quiet cumbersome and involving, hence a limitation.

The results obtained in this simulation can be misleading in case wrong values are input in the system (Rubinstein, 1981). Errors can occurwheninputting values to the system, and also when taking values from the output of the simulation. Great care needs to be taken in the simulation process in order not to alter the input values to the system.Incase inaccurate results are output, the values need to be adjusted. Assumptions in the simulation need to be carefully noted and applied in order to avoid errors.

In case the model is lacking in appropriate network, the output of the system will not be accurate to reflect the real world situation. Simulations should hence be developed with great attention and should reflect the real world model(Rubinstein, 1981).  Each model in the simulation working independently should clearly be joined to other models in the system for proper functioning.

Lastly,in situations where the simulation duration is inadequate, the values output from the system will also be inaccurate. Estimating the appropriate period for a simulation processtherefore requires expertise knowledge to calculate.


This report seeks to design a simulation to calculate the critical mass of uranium U235. The simulation design and the expected results have been discussed. Greater improvements can be done in the simulation in order to obtain accurate results.

Projecting the future is a cumbersome task. While simulations can provide possible outcomes, these results are not always accurate. Great care need to be taken in choosing the inputs to a system because the quality of output is directly related to the input of the simulation.


Korn, G. A.(2007). Advanced dynamic-system simulation: model-replication techniques and

MonteCarlo simulation.Hoboken, N.J, Wiley-Interscience. Pg. 25-45

Lilley, J. S. (2013). Nuclear Physics Principles and Applications, Hoboken, Wiley. Pg. 56-75

Martini, S. (1998). Critical mass, New York, G.P. Putnam’s Sons. Pg. 67-86

Rubinstein, R. Y. (1981). Simulation and the montecarlo method, New York, Wiley. Pg. 32-76

Strieber, W. (2009). Critical mass, New York, Forge, Pg. 77-92

Tushingham, D. (1996). Critical mass, London, Methuen, pg. 13-65

Vandenbosch R. & Huizenga J. R. (1973), Nuclear fission, New York, Academic 71-93