// -*- coding: utf-8 -*-
//
// spin_transition_function.h
//
// @copyright Deepansh J. Srivastava, 2019-2021.
// Created by Deepansh J. Srivastava, Apr 11, 2019.
// Contact email = srivastava.89@osu.edu
//
// =====================================================================================
// Single nucleus spin transition functions
// =====================================================================================
/**
* @brief Single nucleus spin transition function from the irreducible spherical tensor
* of rank @f$L=1@f$ is given as
* @f[
* \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,
* @f]
* where @f$\hat{T}_{10}@f$ is the irreducible 1st-rank spherical tensor operator in the
* rotating tilted frame.
*
* @param mi The quantum number associated with the quantized initial energy level.
* @param mf The quantum number associated with the quantized final energy level.
* @returns The spin transition function @f$\mathbb{p}@f$.
*/
static inline double STF_p(const double mf, const double mi) { return (mf - mi); }
/**
* Single nucleus spin transition function from the irreducible spherical tensor of rank
* @f$L=2@f$ is given as
* @f[
* \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),
* @f]
* where @f$\hat{T}_{20}@f$ is the irreducible 2nd-rank spherical tensor operator in the
* rotating tilted frame.
*
* @param mi The quantum number associated with the quantized initial energy level.
* @param mf The quantum number associated with the quantized final energy level.
* @returns The spin transition function @f$\mathbb{d}@f$.
*/
static inline double STF_d(const double mf, const double mi) {
return 1.2247448714 * (mf * mf - mi * mi);
}
/**
* Single nucleus spin transition function from the irreducible spherical tensor of rank
* @f$L=3@f$ is given as
* @f[
* \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)],
* @f]
* where @f$\hat{T}_{30}@f$ is the irreducible 3rd-rank spherical tensor operator in the
* rotating tilted frame.
*
* @param mi The quantum number associated with the quantized initial energy level.
* @param mf The quantum number associated with the quantized final energy level.
* @param spin The spin quantum angular momentum number.
* @return The spin transition function @f$\mathbb{f}@f$.
*/
static inline double STF_f(const double mf, const double mi, const double spin) {
double f_value = 1.0 - 3.0 * spin * (spin + 1.0);
f_value *= (mf - mi);
f_value += 5.0 * (mf * mf * mf - mi * mi * mi);
f_value *= 0.316227766;
return f_value;
}
/**
* The following single nucleus composite spin transition functions corresponding to
* rank @f$L=[0,2,4]@f$ irreducible tensors results from the second-order corrections to
* the quadrupole frequency. The functions are defined as
* @f[
* \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),
* @f]
* where @f$\mathbb{p}(m_f, m_i)@f$ and @f$\mathbb{f}(m_f, m_i)@f$ are single nucleus
* spin transition functions described before, and @f$I@f$ is the spin quantum number.
*
* @param mi The quantum number associated with the quantized initial energy level.
* @param mf The quantum number associated with the quantized final energy level.
* @param spin The spin quantum number, @f$I@f$.
* @param cl_value A pointer to an array of size 3 where the spin transition functions,
* @f$\mathbb{c}_{L}@f$, will be stored ordered according to @f$L=[0,2,4]@f$.
*/
static inline void STF_cL(double *restrict cl_value, const double mf, const double mi,
const double spin) {
double f_value = STF_f(mf, mi, spin);
double p_value = STF_p(mf, mi);
double temp = spin * (spin + 1.0) - 0.75;
temp *= p_value;
*cl_value++ = 0.3577708764 * temp + 0.8485281374 * f_value;
*cl_value++ = 0.1069044968 * temp + -1.0141851057 * f_value;
*cl_value++ = -0.1434274331 * temp + -1.2850792082 * f_value;
}
// =====================================================================================
// Two weakly coupled nuclei spin transition functions
// =====================================================================================
/**
* @brief The @f$\mathbb{d}_{IS}@f$ spin transition symmetry function.
*
* Two weakly coupled nuclei spin transition function from the irreducible spherical
* tensors is given as
* @f[
* \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},
* @f]
* where @f$\hat{T}_{10}(I)@f$ and @f$\hat{T}_{10}(S)@f$ are the irreducible first-rank
* spherical tensor operators in the rotating tilted frame for spin I and S,
* respectively.
*
* @param mIf The quantum number associated with the quantized final energy state
* corresponding to spin I.
* @param mSf The quantum number associated with the quantized final energy state
* corresponding to spin S.
* @param mIi The quantum number associated with the quantized initial energy state
* corresponding to spin I.
* @param mSi The quantum number associated with the quantized initial energy state
* corresponding to spin S.
* @return The spin transition symmetry function @f$\mathbb{d}_{IS}@f$.
*/
static inline double STF_dIS(const double mIf, const double mIi, const double mSf,
const double mSi) {
return mIf * mSf - mIi * mSi;
}
