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

\(g_{sour}^\alpha\)

\(b^\alpha \varphi \mathop {\sum}\limits_\beta {\sqrt {\rho _{im}^\beta } }\)

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

\(l_cf_0\mathop {\sum}\limits_\beta {\sqrt {a_{\alpha \beta }} } \left[ {\frac{{\rho _m^\beta }}{{\rho _m^\alpha b^\alpha }} + \frac{{\dot \gamma ^\beta }}{{\dot \gamma ^\alpha b^\beta }}} \right]\)

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

\(\frac{{l_cf_0}}{{\sqrt {\rho _{im}^\alpha } }}\mathop {\sum}\limits_\beta {\sqrt {a_{\alpha \beta }} \rho _{im}^\beta }\)

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

\(\frac{{l_cf_0}}{{\dot \gamma ^\alpha \rho _m^\alpha }}\mathop {\sum}\limits_{\beta ,\gamma } {n_\alpha ^{\beta \gamma }} \sqrt {a_{\beta \gamma }} \left[ {\frac{{\rho _m^\gamma \dot \gamma ^\beta }}{{b^\beta }} + \frac{{\rho _m^\beta \dot \gamma ^\gamma }}{{b^\gamma }}} \right]\)

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

\(\frac{{l_cf_0}}{{\dot \gamma ^\alpha \sqrt {\rho _{im}^\alpha } }}\mathop {\sum}\limits_\beta {n_\alpha ^{\beta \gamma }\sqrt {a_{\beta \gamma }} \rho _{im}^\gamma } \dot \gamma ^\beta\)

\(g_{re\;{\mathop{{{\rm{cov}}}}} }^\alpha\)

\(\frac{{l_cf_0}}{{\dot \gamma ^\alpha }}\left( {\mathop {\sum}\limits_\beta {\sqrt {a_{\alpha \beta }} } \frac{{\dot \gamma ^\beta }}{{b^\beta }}} \right)e^{\left( {\frac{{ - H_0\left( {1 - \sqrt {\frac{{\rho _{im}^\alpha }}{{\rho _s}}} } \right)}}{{kT}}} \right)}\)