# Simulation of the CP-1 model

and the excerpts in E. Fermi, Note e Memorie (Collected Papers), Volume II (United States, 1939-1954). Overall, a good agreement was found. For all TRIPOLI-4 simulations, otherwise specified, we have used the ENDF/B-VIII nuclear data library.

## The first divergence of CP-1

Prior to performing the crucial experiment of December 2nd, E. Fermi and his group had monitored the neutron amplification factor (the inverse of the neutron counting rate) at the center of the pile, as a function of the number of layers. The sub-critical approach to criticality, which is now customary in the reactor start-up procedures, has been carefully noted in report CP 413. We have tried to reproduce the divergence curve with TRIPOLI-4: the resulting plot is reported below and is in good agreement with measurements. It is to be noted that in TRIPOLI-4 we have decided to approximate the neutron amplification factor by using the formula M = 1/(1-k), whereas in reality M resulted from the amplification of the source. In CP-1 there was no external neutron source: the neutrons were provided by the spontaneous fissions in U238. We expect some systematic discrepancies to be apparent, especially at lower reactivity, ie, for a small number of built layers of the pile. The overall agreement between the model and experimental data is fairly good.

Comparison between TRIPOLI-4 results and experimental data for the sub-critical approach to reactor divegence.

On December 2nd, 1942, George Weil under the direction of Enrico Fermi removed the last inches of the control rod "number 21" (a rudimentary wooden stick covered with a cadmium sheet) which made CP-1 slightly supercritical, all the safety rods having been previously withdrawn: the reactor was left in this state for about thirty minutes, after which Fermi ordered the safety rods to be inserted again, and the reactor went to shut-down. The estimated reactivity excess of CP-1 (all rods withdrawn) was about 60 pcm, according to report CP 413.

We have run several simulations in order to estimate the reactivity of CP-1 as a function of the control rod positions. In particulard, based on the pieces of information available in report CP 413 and in the Note e Memorie, it is known that CP-1 was about critical with the control rod "21" inserted by 8.5 feet (all other rods out) and with the automatic rod "38" inserted by 7 feet (again, all other rods out). The corresponding TRIPOLI-4 simulations yield the values reported in the table below, which are in remarkably good agreement with respect to the data given in the original reports. In particular, the systematic bias with respect to measurements seems approximately constant and can presumably be attributed to the impact of the nuclear data library and the overall effect of technological uncertainties.

Comparison between TRIPOLI-4 results and experimental data for the critical rod positions.

The critical buckling of CP-1 was estimated in CP-1 using the diffusion theory in report CP 413. Based on this value, and supposing an approximately spherical shape of the neutron flux, the radial flux profile computed with TRIPOLI-4 for the fundamental k-eigenmode can be compared with the expected shape predicted by the report.

Comparison of the radial flux profile for the fundamental eigenmode computed with TRIPOLI-4 and the shape predicted in in report CP 413 using diffusion theory.

Control rod worth, kinetics parameters and reactor period

During the operation of CP-1, the control rod worth was carefully estimated for both the manual control rod "21" and the automatic control/regulation rod "38". The former relies on cadmium, whereas the latter relies on borated steel. The reactivity worth close to the critical position is reported in the following table.

Control rod worths for rod "21" (cadmium) and "38" (borated steel).

To further substantiate this analysis, we have computed the reactivity worth of the two control rods as a function of the rod position within the core of CP-1. From the figure below it is immediately apparent that the critical positions predicted by the TRIPOLI-4 model are fairly well consistent with the positions estimated experimentally.

Comparison of the control rod worth of rod "21" and "38" as a function of the rod length instered into CP-1.

The full set of estimated kinetics parameters of CP-1 is not directly provided in the report CP 413. However, based on the excerpts of other reports written by Fermi and co-authors concerning CP-2, it is possible to have a rough idea of the kinetics parameter of the pile. We have correspondingly computed the adjoint-weighted kinetics parameters with TRIPOLI-4 using the Iterated Fission Probability method. The resulting values are shown in the following table.

At the time the orginal reports were written, the delayed neutron precursor families were only 5, to be compared to the set of 6 (ENDF library) or 8 (JEFF library) in the modern nuclear data evaluations. Thus, the values per family are not directly comparable. The average values are overall in agreement, although some discrepancies are present.

The reactor period as a function of the control rod positions was also estimated during the operation of CP-1. Based on the in-hour formula, and assuming small reactivity insertions, we have estimated the reactor period with TRIPOLI-4 and compared it to the measured values. The resulting values are displayed in the following table for the control rod "21" and the control/regulation rod "38".

Reactor period based on the hypothesis of small reactivity insertions: comparison between TRIPOLI-4 simulations and experimental values of the exponential relaxation time.

Energy deposition and dose calculations

We have computed the energy deposition due to neutrons and photons within CP-1. The simulation results obtained with TRIPOLI-4 are shown in the following figure. The major contribution is due to neutrons.

The energy deposition due to neutrons and photons in CP-1, as computed by TRIPOLI-4.

Based on the normalization factor corresponding to the energy deposited in the pile, and the conversion factor leading the power measured in CP-1, we have also computed the dose within the room where CP-1 was operated. The dose was estimated using the H*10 response function. The resulting dose profile is shown in the following figure.

Spatial distribution of the dose within the room where CP-1 was operated, as computed by TRIPOLI-4.