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Since for spontaneous and low-energy induced fission, compound fissile nuclei and primary fission fragments in the vicinity of the scission point are in cold nonequilibrium states [1], when constructing the spin distributions of these fragments, it is necessary to take into account [2,3] only zero transverse bending- and wriggling-vibrations of the indicated fissile nuclei. Expressing the normalized distribution function of $W\left(\mathbf{J_1}\mathbf{,}\mathbf{J_2}\right)$ fission fragments over spins $\mathbf{J_1}$ and $\mathbf{J_2}$ in terms of the product of the squared moduli of the wave functions of zero bending- and wriggling-vibrations, one can obtain [4]:
$ W\left( {{\mathbf{J}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{J}}_{\mathbf{2}}} \right)=\frac{4{{J}_{1}}{{J}_{2}}}{\pi {{C}_{b}}{{C}_{w}}}\exp \left[ -\frac{1}{2}\left( \frac{1}{{{C}_{b}}}+\frac{1}{{{C}_{w}}} \right)\left( J_{1}^{2}+J_{2}^{2} \right)+\left( \frac{1}{{{C}_{b}}}-\frac{1}{{{C}_{w}}} \right)J_{1}^{{}}J_{2}^{{}}\cos \phi \right], (1)$
where $\phi \left(0\le \phi \le 2\pi \right)$ is the angle between the two-dimensional spin vectors of fragments $\mathbf{J_1}$ and $\mathbf{J_2}$ lying in plane $XY$. By integrating in (1) over variables $J_2$ and $\phi$, one can obtain [4] the normalized distribution of spin $J_1$ of the first fission fragment and estimate the average value $\bar{J}_1$ of spin $J_1$:
$
W({{J}_{1}})=\frac{4{{J}_{1}}}{{{C}_{b}}+{{C}_{w}}}\exp \left[ -\frac{2J_{1}^{2}}{{{C}_{b}}+{{C}_{w}}} \right],\bar{J}{}_{1}=\int\limits_{0}^{\infty }{{{J}_{1}}W({{J}_{1}})}d{{J}_{1}}=\frac{1}{2}\sqrt{\frac{\pi }{2}}{{\left( {{C}_{b}}+{{C}_{w}} \right)}^{{1}/{2}\;}}.(2)
$
For a fissile nucleus $^{236}$U at values [4] of parameters $M_w=1.6\cdot10^6\text{MeV}\cdot\text{Fm}^2\cdot\text{s}^2$; $M_b=2.0\cdot 10^6\text{MeV}\cdot\text{Fm}^2\cdot\text{s}^2$; $K_w=295\text{MeV}\cdot\text{rad}^{–2}$; $K_b=52\text{MeV}\cdot\text{rad}^{–2}$; $\hbar\omega_w=2.3\text{MeV}$; $\hbar\omega_b=0.9\text{MeV}$; $C_w=132\hbar^2$ and $C_b=57\hbar^2$, it follows that the energies of vibrational quanta $\hbar\omega_w$ and coefficients $C_w$ for wriggling-vibrations turn out to be noticeably larger than those for bending-vibrations. This means that the main contribution to $\bar{J}_1$ (2) comes from wriggling vibrations. Then the calculated value $\bar{J}_1=8.6$ correlates well with the experimental [5] average values of the spins of fission fragments $\bar{J}_1=7–9$.
This means that the spin distribution of fission fragments is determined with a good degree of accuracy by taking into account zero wriggling and bending vibrations of a composite fissile system. This confirms the assumption [6] about the inequality of the statistical Gibbs distribution with temperature $T$ for the spin distribution of fragments, which is used in [1].
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