## Introduction and Background

The advent of nuclear reactors as a means of electric power generation trace back to the mid 1950s. Basically, akin to coal-fire powered electric plants nuclear reactors generate electric power thanks to steam-driven turbine. However, for nuclear reactors the source of heat emanates from fission of ^{235}U. Essentially, whatever triggers fission within a reactor fuel are low-energy neutrons which cause ^{235}U to discharge 2-3 high-energy neutrons as well as energetic fission units that manifest as heat energy. To this end, the heat energy generated is vital in heating a coolant essential in liberating steam and hence electric power. Of note, the low-energy neutrons trigger a chain reaction that recurs once a high-energy neutron decelerates within the reactor.

Significantly, a flawlessly performing commercial nuclear reactor ought to produce electric energy at a steady-rate (power level) and for a sustainably long time. As such, the power level is a function of the rate of fission and neutron population in a reactor. At a steady-rate the initial and the current populations of neutrons should be constant hence this is what engineering designs strive at, sustaining the same notwithstanding. The degree of sustainability is quantified by multiplication factor, k_{eff}. This is given by:

When the ratio k_{eff} is equivalent 1 then this is referred to as critical. Otherwise, when k_{eff }is less than1 then this is referred to as sub-critical. On the other hand, when k_{eff} is greater than 1, then this is referred to as super-critical reactor (Lehmann 45).

Basically, a reactor is considered sub-critical when it is in a shutdown condition where the numbers of fission reduce with time. However, a constant increase of a neutron source (k_{eff}<1) reverses the trend towards achieving a critical reactor. As a result, the k_{eff }multiplies by a factor M (subcritical multiplication factor) which correlates with the same by the formula below:

- M = 1/ (1 – k
_{eff}) (2)

With respect to the neutron level (N), the source (S) is connected to the same using the formula below:

- N = S/ (1 – k
_{eff}) (3)

From the above equations the below equations can be derived:

- N
_{2}/N_{1}= (1 – k_{1})/ (1 – k_{2}) (4) - CR
_{2}/CR_{1}= (1 – k_{1})/ (1 – k_{2}) (5)

Where; CR represents the count rate. The subscripts 1 and 2 represent initial and current status (either counts or neutron levels) (Lehmann 46).

The approach to criticality of a subcritical reactor can be monitored through a graph of: 1/M against control rods positions. This takes an inverse trend.

## Description and Experimental Setup

The experimental setup was such that an Oregon State TRIGA reactor was used. This reactor which is purposed for experimental analyses is as shown below:

## List of Equipment Used

The materials to be used include: TRIGA Mark II Reactor, Counter and Detector and three student volunteers.

## Procedure

Four control rods were fully inserted in the reactor which was on a shutdown mode thanks to the operator. This was then started up by the operator. One volunteer student was then chosen to press the button vital in raising the transient rod to 50% its height. The other two students were responsible for pressing respective buttons to raise the safety and the shim rods to 50% their original heights. The neutron populations were then left to stabilize before a neutron detector counter was started. The corresponding time was recorded and hence the count rate. The regulating control rod was then raised 10% its original height. Again the neutron populations were then left to stabilize before a neutron detector counter was restarted and thus; the count rate. The current count rate was then divided by the initial count rate, and a graph of 1/M plotted against % rod withdrawal. The same procedure was then repeated for 15, 20, 25 and 30% rod withdrawals. Later, the graph was then used to predict the control rods position for a critical reactor.

## Data

*Table 1: of 1/M and the count rates.*

## Analysis of Data

## Discussion of Results

The objective of this experiment was to investigate the correlation between 1/M and the % rod withdrawal of a reactor. From the graph plotted (Graph1) it is evident that the graph traces an inverse trend. As manifested by the graph the actual critical height of the rod is 80% the actual height. This is the x-axis intercept which is basically the point where the value of M approaches infinite. This value contradicts the predicted value of 100% that the actual height owing to experimental discrepancies resulting from human error. This includes misjudged timing when operating the reactor.

Just like the critical rod, if we were to shift interests to the other rods then the same procedure would be used because these rods affect the k_{eff} the same way. With regards to energy released, 1 gram of ^{235}U would produce 5.1 X 10^{23} MeV of power in fission (refer to appendix 1).

## Conclusion

From the experimental analyses the objective of the experiment was met since it was established that the graph of 1/M versus % rod withdrawal follow an inverse trend. Hence; these parameters are vital in controlling fission within a reactor.

## Works Cited

Lehmann, Walter. *Atomic and Molecular Structure: the development of our concepts*, San Francisco: W.H. Freeman & Co., 1972. Print.