Space-Dependent Calculation of the Multiplicity Moments for Shells With the Inclusion of Scattering

Figures & data

Fig. 1. Illustration of the neutron paths for $\mu >{\mu }_{cr}$ and for $\mu <{\mu }_{cr}$ in the shell with a central cavity.

Fig. 2. First, second, and third moments of the number of neutrons emitted from a solid sphere for a distributed source (dashed lines, blue) and a central point source (solid lines, red) as functions of the outer radius $R$.

TABLE I Input Parameters Used in the Calculations

	First Moment	Second Moment	Third Moment
Spontaneous fission (^252Cf)	${\nu }_{sf,1}=3.757$	${\nu }_{sf,2}=11.962$	${\nu }_{sf,3}=31.812$
Induced fission (93% ^235U + 7% ^238U at 2 MeV)	${\nu }_{r,1}=2.637$	${\nu }_{r,2}=5.623$	${\nu }_{r,3}=9.476$

Fig. 3. First, second, and third moments of the number of neutrons emitted from a solid sphere (blue lines, marked with circles) and a shell with an inner radius $r_0=0.1$ mean free path (red lines, marked with triangles), both with a central source. $R_{eff}$ is the outer radius in optical units for both cases.

Fig. 4. The first three factorial moments as functions of the radius of a pure ²³⁵U sphere, with varying degrees ${\beta }_s$ of elastic scattering being included. Left column: full range ($R$ between 0 and 7 cm); right column: half range ($R$ between 0 and 3.5 cm).

Fig. 5. Comparison of measured and calculated first, second, and third moments of the number of neutrons emitted from the Rocky Flats Shells for four different outer radii and with an inner radius of 2.0126 cm.

TABLE II Macroscopic Cross Sections of the Rocky Flats Shells at 1 MeV*

	$93\mathrm{\%}$ ^235U	$7\mathrm{\%}$ ^238U	93% ^235U + 7% ^238U
${\mathrm{\Sigma }}_f$	${\mathrm{\Sigma }}_f^{235}=0.05351$	${\mathrm{\Sigma }}_f^{238}$=0.0000482	${\mathrm{\Sigma }}_f=0.05355$
${\mathrm{\Sigma }}_{el}$	${\mathrm{\Sigma }}_{el}^{235}=0.17225$	${\mathrm{\Sigma }}_{el}^{238}=0.014068$	${\mathrm{\Sigma }}_{el}=0.18632$
${\mathrm{\Sigma }}_T$			${\mathrm{\Sigma }}_T$ = 0.23988

*In units of cm⁻¹.

TABLE III Factorial Moments of the Rocky Flats Shells at 1 MeV

	First Moment	Second Moment	Third Moment
Induced fission	${\nu }_{f,1}=2.52359$	${\nu }_{f,2}=5.10119$	${\nu }_{f,3}=8.00135$
Fission plus elastic scattering	${\nu }_{r,1}=1.34015$	${\nu }_{r,2}=1.13887$	${\nu }_{r,3}=1.78635$

Fig. 6. Comparison of measured and calculated first, second, and third moments of the number of neutrons emitted from the Rocky Flats Shells for four different outer radii and with an inner radius of 2.0126 cm. In the calculated values, the material properties correspond to those of the isotopic composition of the Rocky Flats Shells at a neutron energy of 1 MeV.