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

## 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.