DetailedBalance

Consider a transition from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \rightarrow f} between two states of the nucleus with spins Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_i } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_f} , respectively. The transition probability is proportional to the squared matrix element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{\lambda \mu}} is a hermitian tensor operator of rank Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} responsible for the process. Define the reduced transition probability

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(T_{\lambda}; i \rightarrow f)=\sum_{\mu M_f}|\langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle|^2}

as a sum of squared matrix elements over final projections Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_f} and operator projections Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} .

a) Express Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(T_{\lambda}; i \rightarrow f)} in terms of the reduced matrix element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f|| T_{\lambda}|| i)} and show that it does not depend on the initial projection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_i} .

b) Establish the detailed balance between the reduced transition probabilities of the direct Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \rightarrow f} , and inverse Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \rightarrow i} processes.

SOLUTION:

a) According to the Wigner-Eckert theorem, the entire dependence of the matrix element of a tensor operator on the magnetic quantum numbers is concentrated in the vector coupling coefficients,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle J_f M_f| T_{\lambda \mu}| J_i M_i \rangle = (-)^{J_f-M_f} \left( \begin{array}{lll} J_f & \lambda & J_i \\ -M_f & \mu & M_i \end{array} \right) (f|| T_{\lambda}|| i) }

We obtain the rate by squaring this and summing over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_f}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(T_{\lambda}; i \rightarrow f)= (f|| T_{\lambda}|| i)^2 \sum_{\mu M_f} \left( \begin{array}{lll} J_f & \lambda & J_i  \\  -M_f & \mu & M_i \end{array} \right) \left( \begin{array}{lll} J_f & \lambda & J_i  \\  -M_f & \mu & M_i \end{array} \right)}

Using the orthogonality condition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{m_1 m_2} \left( \begin{array}{lll} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{array} \right) \left( \begin{array}{lll} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{array} \right)=\dfrac{\delta_{j_3 j'_3}\delta_{m_3 m'_3}}{2j_3+1}}

Which leads us to our final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B(T_{\lambda}; i \rightarrow f)= \dfrac{(f|| T_{\lambda}|| i)^2}{2J_i+1}}

It is obvious that this result does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_i}