Table 1 Coefficients for dislocation-density equations (Eqs. 1–2).

\(g_{sour}^{\alpha }\)

\(b^{\alpha } \varphi \sum\limits_{\beta } {\rho_{im}^{\beta } }\)

\(g_{mnter - }^{\alpha }\)

\(l_{c} f_{0} \sum\limits_{\beta } {\alpha_{\alpha \beta } } \left[ {\frac{{\rho_{m}^{\beta } }}{{\rho_{m}^{\alpha } b^{\alpha } }} + \frac{{\dot{\gamma }^{\beta } }}{{\dot{\gamma }^{\alpha } b^{\beta } }}} \right]\)

\(g_{immob - }^{\alpha }\)

\(\frac{{l_{c} f_{0} }}{{\sqrt {\rho_{im}^{\alpha } } }}\sum\limits_{\beta } {\sqrt {\alpha_{\alpha \beta } } } \rho_{im}^{\beta }\)

\(g_{mnter + }^{\alpha }\)

\(\frac{{l_{c} f_{0} }}{{\dot{\gamma }^{\alpha } \rho_{m}^{\alpha } }}\sum\limits_{\beta ,\gamma } {n_{\alpha }^{\beta \gamma } \sqrt {\alpha_{\beta \gamma } } } \left[ {\frac{{\rho_{m}^{\gamma } \dot{\gamma }^{\beta } }}{{b^{\beta } }} + \frac{{\rho_{m}^{\beta } \dot{\gamma }^{\gamma } }}{{b^{\gamma } }}} \right]\)

\(g_{immob + }^{\alpha }\)

\(\frac{{l_{c} f_{0} }}{{\dot{\gamma }^{\alpha } \sqrt {\rho_{im}^{\alpha } } }}\sum\limits_{\beta } {n_{\alpha }^{\beta \gamma } \sqrt {\alpha_{\beta \gamma } } } \rho_{im}^{\gamma } \dot{\gamma }^{\beta }\)

\(g_{{re{\text{cov}} }}^{\alpha }\)

\(\frac{{l_{c} f_{0} }}{{\dot{\gamma }^{\alpha } }}\left( {\sum\limits_{\beta } {\sqrt {\alpha_{\alpha \beta } } } \frac{{\dot{\gamma }^{\beta } }}{{b^{\beta } }}} \right)e^{{\left( {\frac{{ - H_{0} \left( {1 - \sqrt {\frac{{\rho_{im}^{\alpha } }}{{\rho_{s} }}} } \right)}}{kT}} \right)}}\)