Spin transition functions (STF), \(\xi_L^{(k)}(i,j)\)

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Single nucleus spin transition functions

The single spin transition functions for \(\left|m_i\right> \rightarrow \left|m_f\right>\) transition, where \(m_j\) is the spin quantum number, and the subscripts \(i\) and \(f\) refer to the initial and final energy states.

double STF_p(const double mf, const double mi)

Single nucleus spin transition function from the irreducible spherical tensor of rank \(L=1\) is given as.

\[\begin{split} \mathbb{p}(m_f, m_i) &= \left< m_f | \hat{T}_{10} | m_f \right> - \left< m_i | \hat{T}_{10} | m_i \right> \\ &= m_f - m_i, \end{split}\]
where \(\hat{T}_{10}\) is the irreducible 1st-rank spherical tensor operator in the rotating tilted frame.

Return

The spin transition function \(\mathbb{p}\).

Parameters
  • mi: The quantum number associated with the quantized initial energy level.

  • mf: The quantum number associated with the quantized final energy level.

double STF_d(const double mf, const double mi)

Single nucleus spin transition function from the irreducible spherical tensor of rank \(L=2\) is given as

\[\begin{split} \mathbb{d}(m_f, m_i) &= \left< m_f | \hat{T}_{20} | m_f \right> - \left< m_i | \hat{T}_{20} | m_i \right> \\ &= \sqrt{\frac{3}{2}} \left(m_f^2 - m_i^2 \right), \end{split}\]
where \(\hat{T}_{20}\) is the irreducible 2nd-rank spherical tensor operator in the rotating tilted frame.

Return

The spin transition function \(\mathbb{d}\).

Parameters
  • mi: The quantum number associated with the quantized initial energy level.

  • mf: The quantum number associated with the quantized final energy level.

double STF_f(const double mf, const double mi, const double spin)

Single nucleus spin transition function from the irreducible spherical tensor of rank \(L=3\) is given as

\[\begin{split} \mathbb{f}(m_f, m_i) &= \left< m_f | \hat{T}_{30} | m_f \right> - \left< m_i | \hat{T}_{30} | m_i \right> \\ &= \frac{1}{\sqrt{10}} [5(m_f^3 - m_i^3) + (1 - 3I(I+1))(m_f-m_i)], \end{split}\]
where \(\hat{T}_{30}\) is the irreducible 3rd-rank spherical tensor operator in the rotating tilted frame.

Return

The spin transition function \(\mathbb{f}\).

Parameters
  • mi: The quantum number associated with the quantized initial energy level.

  • mf: The quantum number associated with the quantized final energy level.

  • spin: The spin quantum angular momentum number.

Composite single nucleus spin transition functions

The composite single spin transition functions are linear combinations of the single spin transition functions.

void STF_cL(double *restrict cl_value, const double mf, const double mi, const double spin)

The following single nucleus composite spin transition functions corresponding to rank \(L=[0,2,4]\) irreducible tensors results from the second-order corrections to the quadrupole frequency. The functions are defined as

\[\begin{split} \mathbb{c}_{0}(m_f, m_i) &= \frac{4}{\sqrt{125}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(m_f, m_i) + \sqrt{\frac{18}{25}} \mathbb{f}(m_f, m_i), \\ \mathbb{c}_{2}(m_f, m_i) &= \sqrt{\frac{2}{175}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(m_f, m_i) - \frac{6}{\sqrt{35}} \mathbb{f}(m_f, m_i), \\ \mathbb{c}_{4}(m_f, m_i) &= -\sqrt{\frac{18}{875}} \left[I(I+1) - \frac{3}{4}\right] \mathbb{p}(m_f, m_i) - \frac{17}{\sqrt{175}} \mathbb{f}(m_f, m_i), \end{split}\]
where \(\mathbb{p}(m_f, m_i)\) and \(\mathbb{f}(m_f, m_i)\) are single nucleus spin transition functions described before, and \(I\) is the spin quantum number.

Parameters
  • mi: The quantum number associated with the quantized initial energy level.

  • mf: The quantum number associated with the quantized final energy level.

  • spin: The spin quantum number, \(I\).

  • cl_value: A pointer to an array of size 3 where the spin transition functions, \(\mathbb{c}_{L}\), will be stored ordered according to \(L=[0,2,4]\).

Two weakly coupled nuclei spin transition functions

The weakly coupled spin transition function for \(\left|m_{i_I}, m_{i_S}\right> \rightarrow \left|m_{f_I}, m_{f_S}\right>\) transition. Here, the subscript \(I\) and \(S\) denotes the two weakly coupled spins.

double STF_dIS(const double mIf, const double mIi, const double mSf, const double mSi)

The \(\mathbb{d}_{IS}\) spin transition symmetry function.

Two weakly coupled nuclei spin transition function from the irreducible spherical tensors is given as

\[\begin{split} \mathbb{d}_{IS}(m_{f_I}, m_{f_S}, m_{i_I}, m_{i_S}) &= \left<m_{f_I}m_{f_S}|\hat{T}_{10}(I)~\hat{T}_{10}(S)|m_{f_I} m_{f_S}\right> -\left<m_{i_I}m_{i_S}|\hat{T}_{10}(I)~\hat{T}_{10}(S)|m_{i_I} m_{i_S}\right> \\ &= m_{f_I} m_{f_S} - m_{i_I} m_{i_S}, \end{split}\]
where \(\hat{T}_{10}(I)\) and \(\hat{T}_{10}(S)\) are the irreducible first-rank spherical tensor operators in the rotating tilted frame for spin I and S, respectively.

Return

The spin transition symmetry function \(\mathbb{d}_{IS}\).

Parameters
  • mIf: The quantum number associated with the quantized final energy state corresponding to spin I.

  • mSf: The quantum number associated with the quantized final energy state corresponding to spin S.

  • mIi: The quantum number associated with the quantized initial energy state corresponding to spin I.

  • mSi: The quantum number associated with the quantized initial energy state corresponding to spin S.