Calculation of Nuclear Stopping in the Semiclassical Approximation

Abstract

The results of calculating nuclear stopping in the semiclassical approximation for the H–Be, H–C, H–W, O–C, O–Be, and O–Al systems are presented. It was found that, in the presence of a well in the interatomic-interaction potential, an additional maximum appears in the dependence of the nuclear stopping on the energy of the bombarding particles. When using the universal potential without a well, this feature is absent. It is shown that, by means of scaling, the data obtained for systems with hydrogen are recalculated for collisions with the participation of D and T hydrogen isotopes. The results obtained are in good agreement with classical calculations, which is explained by the fact that large scattering angles make the main contribution to the nuclear stopping and the applicability criterion changes to the condition that angular momentum $\ell$ ≫ 1.

APPENDIX

Derivation of the Formula for the Transport Section Q _tr

$${{Q}_{{{\text{tr}}}}} = 2\pi \mathop \smallint \limits_0^\pi d\theta {\text{sin}}\theta \left( {1 - {\text{cos}}\theta } \right){{\left| {f\left( \theta \right)} \right|}^{2}}.$$

(A1)

Here, θ is the scattering angle in the center of mass system and f(θ) is the scattering amplitude, which, without any assumptions and simplifications, can be written as

$$f = \frac{1}{k}\mathop \sum \limits_{\ell = 0}^\infty {{e}^{{i{{\delta }_{\ell }}}}}\left( {2\ell + 1} \right){{P}_{\ell }}\left( {{\text{cos}}\theta } \right){\text{sin}}{{\delta }_{\ell }},$$

(A2)

where ${{\delta }_{\ell }}$ is the scattering phase and ${{\delta }_{\ell }}$ in the semiclassical approximation is determined by the expression

$$\begin{gathered} {{Q}_{d}} = 2\pi \mathop \smallint \limits_0^\pi d\theta {\text{sin}}\theta \left( {1 - {\text{cos}}\theta } \right) \\ \times \,\,\left| {\frac{1}{{4{{k}^{2}}}}\left( {\mathop \sum \limits_{l = 0}^\infty {{e}^{{i{{\delta }_{l}}}}}\left( {2l + 1} \right){{P}_{l}}\left( {{\text{cos}}\theta } \right){\text{sin}}{{\delta }_{l}}} \right)} \right. \\ \left. { \times \,\,\left( {\mathop \sum \limits_{m = 0}^\infty {{e}^{{i{{\delta }_{m}}}}}\left( {2m + 1} \right){{P}_{m}}\left( {{\text{cos}}\theta } \right){\text{sin}}{{\delta }_{m}}} \right)} \right|. \\ \end{gathered} $$

(A3)

For a term with unity, we obtain a term corresponding to the elastic-scattering cross section. For a member with cosθ, we use the properties of the Legendre polynomials:

$${\text{cos}}\theta {{P}_{l}}\left( {{\text{cos}}\theta } \right) = \frac{{\left( {l + 1} \right){{P}_{{l + 1}}}\left( {{\text{cos}}\theta } \right)}}{{2l + 1}} + \frac{{l{{P}_{{l - 1}}}\left( {{\text{cos}}\theta } \right)}}{{2l + 1}}.$$

(A4)

In the product of two sums, due to the orthogonality of the Legendre polynomials in the sum over $\ell$, there will be contributions from terms with m = $\ell$ – 1 and m = $\ell$ + 1.

Consider the term with m = $\ell$ – 1.

$$\begin{gathered} \left( {2l - 1} \right){\text{cos}}\theta {{P}_{{l - 1}}}\left( {{\text{cos}}\theta } \right) \\ = \left( l \right){{P}_{l}}\left( {{\text{cos}}\theta } \right) + \left( {l - 1} \right){{P}_{{l - 2}}}\left( {{\text{cos}}\theta } \right). \\ \end{gathered} $$

(A5)

Consider the term with m = $\ell$ + 1.

$$\begin{gathered} \left( {2l + 3} \right){\text{cos}}\theta {{P}_{{l + 1}}}\left( {{\text{cos}}\theta } \right) \\ = \left( {l + 2} \right){{P}_{{l + 2}}}\left( {{\text{cos}}\theta } \right) + \left( {l + 1} \right){{P}_{l}}\left( {{\text{cos}}\theta } \right). \\ \end{gathered} $$

(A6)

If we take the integral, then the terms with the index $\ell$ and terms with the indices $\ell$ – 2 and $\ell$ + 2 contribute to the corresponding terms of the sum when $\ell$ = $\ell$ – 2 and $\ell$ = $\ell$ + 2.

Let us rewrite the sum:

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {2l + 1} \right){\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}} \\ - \,2\pi \frac{1}{{{{k}^{2}}}}\mathop \smallint \limits_0^\pi d\theta {\text{sin}}\theta \left| {\left( {\mathop \sum \limits_{l = 0}^\infty {{e}^{{i{{\delta }_{l}}}}}\left( {2l + 1} \right){{P}_{l}}\left( {{\text{cos}}\theta } \right){\text{sin}}{{\delta }_{l}}} \right)} \right.~ \\ \times \,((l){{P}_{l}}({\text{cos}}\theta ){\text{sin}}{{\delta }_{{l - 1}}}{{e}^{{ - i{{\delta }_{{l - 1}}}}}} \\ \left. {{{\,}_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}}} + \,(l + 1){{P}_{l}}({\text{cos}}\theta ){\text{sin}}{{\delta }_{{l + 1}}}{{e}^{{ - i{{\delta }_{{l + 1}}}}}})} \right|. \\ \end{gathered} $$

(A7)

Given the normalization of the Legendre polynomials,

$$\mathop \smallint \limits_0^\pi d\theta {\text{sin}}\theta {{P}_{l}}\left( {{\text{cos}}\theta } \right){{P}_{l}}\left( {{\text{cos}}\theta } \right) = \frac{2}{{2l + 1}},$$

(A8)

we obtain

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {2l + 1} \right){\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}} \\ - \,\,\frac{{4\pi }}{{{{k}^{2}}}}\left| {\mathop \sum \limits_{l = 0}^\infty ({\text{lsin}}{{\delta }_{l}}{{\sin}}{{\delta }_{{l - 1}}}{{e}^{{i\left( {{{\delta }_{l}} - {{\delta }_{{l - 1}}}} \right)}}}} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}}}}}} + \,\,\left( {l + 1} \right){\text{sin}}{{\delta }_{l}}{{\sin}}{{\delta }_{{l + 1}}}{{e}^{{i\left( {{{\delta }_{l}} - {{\delta }_{{l + 1}}}} \right)}}})~} \right|. \\ \end{gathered} $$

(A9)

At $\ell$ = 0, the term in the second sum does not contribute. Shifting the summation index, we obtain

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {2l + 1} \right){\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}} \\ - \,\,\frac{{4\pi }}{{{{k}^{2}}}}\left| {\mathop \sum \limits_{l = 0}^\infty (l\, + \,1)\,{\text{sin}}{{\delta }_{{l + 1}}}\,{\text{sin}}{{\delta }_{l}}({{e}^{{i({{\delta }_{{l + 1}}} - {{\delta }_{{l - 1}}})}}}\, + \,{{e}^{{i({{\delta }_{l}} - {{\delta }_{{l + 1}}})}}})} \right|. \\ \end{gathered} $$

(A10)

Splitting the first sum into two sums with weights $\ell$ and $\ell$ + 1 and shifting the summation in the first term by 1, we have

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {l + 1} \right)({\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}} + {\text{si}}{{{\text{n}}}^{2}}{{\delta }_{{l + 1}}}) \\ - \,\,\frac{{4\pi }}{{{{k}^{2}}}}\left| {\mathop \sum \limits_{l = 0}^\infty \left( {l + 1} \right){\text{sin}}{{\delta }_{{l + 1}}}{\text{sin}}{{\delta }_{l}}2{\text{cos}}\left( {{{\delta }_{{l + 1}}} - {{\delta }_{l}}} \right)} \right|. \\ \end{gathered} $$

(A11)

Combining the amounts, we obtain

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {l + 1} \right)\{ {\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}} + {\text{si}}{{{\text{n}}}^{2}}{{\delta }_{{l + 1}}} \\ \, - 2{\text{sin}}{{\delta }_{{l + 1}}}{\text{sin}}{{\delta }_{l}}\left( {{\text{cos}}{{\delta }_{{l + 1}}}{\text{cos}}{{\delta }_{l}} + {\text{sin}}{{{{\delta }}}_{{l + 1}}}{\text{sin}}{{\delta }_{l}}} \right)\} ~ \\ \end{gathered} $$

(A12)

or

$$\begin{gathered} {{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty (l + 1)\{ {\text{si}}{{{\text{n}}}^{2}}{{\delta }_{l}}{\text{co}}{{{\text{s}}}^{2}}{{\delta }_{{l + 1}}} \\ - {\kern 1pt} \,\,2{\text{sin}}{{\delta }_{{l + 1}}}\,{\text{sin}}{{\delta }_{l}}\,{\text{cos}}{{\delta }_{{l + 1}}}\,{\text{cos}}{{\delta }_{l}}\, + \,{\text{si}}{{{\text{n}}}^{2}}{{\delta }_{{l + 1}}}\,{\text{co}}{{{\text{s}}}^{2}}{{\delta }_{l}}\} . \\ \end{gathered} $$

(A13)

Having carried out trigonometric transformations, we obtain the Mott–Massey formula [7]:

$${{Q}_{d}} = \frac{{4\pi }}{{{{k}^{2}}}}\mathop \sum \limits_{l = 0}^\infty \left( {l + 1} \right){\text{si}}{{{\text{n}}}^{2}}\left( {{{\delta }_{l}} - {{\delta }_{{l + 1}}}} \right).$$

(A14)