磁共振成像原理（理论）9：射频回波 (RF Echoes)-三脉冲回波（2）

接着上一篇文章继续推导。

$t=({\tau }_1+{\tau }_2{)}_{+}$：横向磁化分量演变：

在第三个脉冲 (${\alpha }_3$, 假设沿 $y^{\prime }$ 轴施加) 之后瞬间 ($t=({\tau }_1+{\tau }_2{)}_{+}$)，磁化矢量的表达式如下：

$x^{\prime }$方向磁化分量:
$$\begin{array}{rl}{M}_{x^{\prime }}\left[\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+}\right]& =M_{x^{\prime }}(\omega ,({\tau }_1+{\tau }_2{)}_{-})\cos {\alpha }_3-M_{z^{\prime }}(\omega ,({\tau }_1+{\tau }_2{)}_{-})\sin {\alpha }_3\\ & =M_{x^{\prime }}(\omega ,({\tau }_1+{\tau }_2))\cos {\alpha }_3-M_{z^{\prime }}(\omega ,({\tau }_1+{\tau }_2))\sin {\alpha }_3\\ & =-M_z^0(\omega )\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}\cos {\alpha }_3\cos \omega \left({\tau }_1+{\tau }_2\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & +M_z^0(\omega )\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}\cos {\alpha }_3\cos \omega \left({\tau }_2-{\tau }_1\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & -M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_1\right)e^{-{\tau }_1/T_1}\right]\sin {\alpha }_2\cos {\alpha }_3\cos \omega {\tau }_2{e}^{-{\tau }_2/T_2}\\ & -M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_2\right)e^{-{\tau }_2/T_1}-\left(1-\cos {\alpha }_1\right)\cos {\alpha }_2{e}^{-\left({\tau }_1+{\tau }_2\right)/T_1}\right]\sin {\alpha }_3\\ & +M_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3\cos \omega {\tau }_1{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}(4.41)\end{array}$$
$y^{\prime }$方向磁化分量保持不变（因为脉冲是$y^{\prime }$方向），仍为（4.39），重写如下
$$\begin{array}{rl}{M}_{y^{\prime }}\left[\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+}\right]=& M_z^0(\omega )\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}\sin \omega \left({\tau }_1+{\tau }_2\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & -M_z^0(\omega )\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}\sin \omega \left({\tau }_2-{\tau }_1\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & +M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_1\right)e^{-{\tau }_1/T_1}\right]\sin {\alpha }_2\sin \omega {\tau }_2{e}^{-{\tau }_2/T_2}\end{array}\text{\hspace{1em}(4.39)}$$

$z^{\prime }$方向磁化分量:
$$\begin{array}{rl}{M}_{z^{\prime }}\left[\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+}\right]=& M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_2\right)e^{-{\tau }_2/T_1}-\left(1-\cos {\alpha }_1\right)\cos {\alpha }_2{e}^{-\left({\tau }_1+{\tau }_2\right)/T_1}\right]\cos {\alpha }_3\\ & -M_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3\cos \omega {\tau }_1{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}\\ & +M_z^0(\omega )\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}\sin {\alpha }_3\cos \omega \left({\tau }_1+{\tau }_2\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & -M_z^0(\omega )\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}\sin {\alpha }_3\cos \omega \left({\tau }_2-{\tau }_1\right)e^{-\left({\tau }_1+{\tau }_2\right)/T_2}(4.42)\\ & +M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_1\right)e^{-{\tau }_1/T_1}\right]\sin {\alpha }_2\sin {\alpha }_3\cos \omega {\tau }_2{e}^{-{\tau }_2/T_2}\end{array}\text{\hspace{1em}(4.42)}$$

将$x^{\prime }$方向磁化分量和$y^{\prime }$方向磁化分量合并成复数形式，得到

Mx′y′[ω,(τ1+τ2)+]=Mx′(ω,(τ1+τ2)+)+iMy′(ω,(τ1+τ2)+)={−Mz0(ω)sin⁡α1cos⁡2α22cos⁡α3cos⁡ω(τ1+τ2)e−(τ1+τ2)/T2+Mz0(ω)sin⁡α1sin⁡2α22cos⁡α3cos⁡ω(τ2−τ1)e−(τ1+τ2)/T2−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2cos⁡α3cos⁡ωτ2e−τ2/T2−Mz0(ω)[1−(1−cos⁡α2)e−τ2/T1−(1−cos⁡α1)cos⁡α2e−(τ1+τ2)/T1]sin⁡α3+Mz0(ω)sin⁡α1sin⁡α2sin⁡α3cos⁡ωτ1e−τ1/T2e−τ2/T1}+i{Mz0(ω)sin⁡α1cos⁡2α22sin⁡ω(τ1+τ2)e−(τ1+τ2)/T2−Mz0(ω)sin⁡α1sin⁡2α22sin⁡ω(τ2−τ1)e−(τ1+τ2)/T2+Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2sin⁡ωτ2e−τ2/T2}=−Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2[cos⁡α3cos⁡ω(τ1+τ2)−isin⁡ω(τ1+τ2)]++Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2[cos⁡α3cos⁡ω(τ2−τ1)−isin⁡ω(τ2−τ1)]+−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2[cos⁡α3cos⁡ωτ2−isin⁡ωτ2]+−Mz0(ω)[1−(1−cos⁡α2)e−τ2/T1−(1−cos⁡α1)cos⁡α2e−(τ1+τ2)/T1]sin⁡α3+Mz0(ω)sin⁡α1sin⁡α2sin⁡α3cos⁡ωτ1e−τ1/T2e−τ2/T1} \begin{aligned} &M_{x'y'}[\omega,\left(\tau_{1}+\tau_{2}\right)_{+}] = \textcolor{red}{M_{x'}(\omega,\left(\tau_{1}+\tau_{2}\right)_{+})}+i \textcolor{blue}{{M_{y'}(\omega,\left(\tau_{1}+\tau_{2}\right)_{+})}}\\ =&\{ \textcolor{red}{-M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}\cos\alpha_{3}\cos\omega\left(\tau_{1}+\tau_{2}\right) e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}} \\ & \textcolor{red}{+M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}\cos\alpha_{3}\cos\omega\left(\tau_{2}-\tau_{1}\right) e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}} \\ & \textcolor{red}{-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}\cos\alpha_{3}\cos\omega\tau_{2} e^{-\tau_{2} /T_{2}}} \\ & \textcolor{red}{-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{2}\right) e^{-\tau_{2} /T_{1}}-\left(1-\cos\alpha_{1}\right)\cos\alpha_{2} e^{-\left(\tau_{1}+\tau_{2}\right) /T_{1}}\right]\sin\alpha_{3}} \\ & \textcolor{red}{+M_{z}^{0}(\omega)\sin\alpha_{1}\sin\alpha_{2}\sin\alpha_{3}\cos\omega\tau_{1} e^{-\tau_{1} /T_{2}}e^{-\tau_{2} /T_{1}}} \} \\ &+i\{ \textcolor{blue}{M_{z}^{0}(\omega) \sin \alpha_{1} \cos ^{2} \frac{\alpha_{2}}{2} \sin \omega\left(\tau_{1}+\tau_{2}\right) e^{-\left(\tau_{1}+\tau_{2}\right) / T_{2}} }\\ & \textcolor{blue}{-M_{z}^{0}(\omega) \sin \alpha_{1} \sin ^{2} \frac{\alpha_{2}}{2} \sin \omega\left(\tau_{2}-\tau_{1}\right) e^{-\left(\tau_{1}+\tau_{2}\right) / T_{2}}} \\ & \textcolor{blue}{+M_{z}^{0}(\omega)\left[1-\left(1-\cos \alpha_{1}\right) e^{-\tau_{1} / T_{1}}\right] \sin \alpha_{2} \sin \omega \tau_{2} e^{-\tau_{2} / T_{2}} }\}\\ \\ =& -M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[\cos\alpha_{3}\cos\omega\left(\tau_{1}+\tau_{2}\right)-i\sin\omega(\tau_{1}+\tau_{2})] + \\ &+M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[\cos\alpha_{3}\cos\omega\left(\tau_{2}-\tau_{1}\right) -i\sin\omega\left(\tau_{2}-\tau_{1}\right)]+\\ &-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}[\cos\alpha_{3}\cos\omega\tau_{2} -i\sin\omega \tau_2]+\\ & -M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{2}\right) e^{-\tau_{2} /T_{1}}-\left(1-\cos\alpha_{1}\right)\cos\alpha_{2} e^{-\left(\tau_{1}+\tau_{2}\right) /T_{1}}\right]\sin\alpha_{3} \\ & +M_{z}^{0}(\omega)\sin\alpha_{1}\sin\alpha_{2}\sin\alpha_{3}\cos\omega\tau_{1} e^{-\tau_{1} /T_{2}}e^{-\tau_{2} /T_{1}} \} \\ \end{aligned} ==​Mx′y′​[ω,(τ1​+τ2​)+​]=Mx′​(ω,(τ1​+τ2​)+​)+iMy′​(ω,(τ1​+τ2​)+​){−Mz0​(ω)sinα1​cos22α2​​cosα3​cosω(τ1​+τ2​)e−(τ1​+τ2​)/T2​+Mz0​(ω)sinα1​sin22α2​​cosα3​cosω(τ2​−τ1​)e−(τ1​+τ2​)/T2​−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​cosα3​cosωτ2​e−τ2​/T2​−Mz0​(ω)[1−(1−cosα2​)e−τ2​/T1​−(1−cosα1​)cosα2​e−(τ1​+τ2​)/T1​]sinα3​+Mz0​(ω)sinα1​sinα2​sinα3​cosωτ1​e−τ1​/T2​e−τ2​/T1​}+i{Mz0​(ω)sinα1​cos22α2​​sinω(τ1​+τ2​)e−(τ1​+τ2​)/T2​−Mz0​(ω)sinα1​sin22α2​​sinω(τ2​−τ1​)e−(τ1​+τ2​)/T2​+Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​sinωτ2​e−τ2​/T2​}−Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​[cosα3​cosω(τ1​+τ2​)−isinω(τ1​+τ2​)]++Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​[cosα3​cosω(τ2​−τ1​)−isinω(τ2​−τ1​)]+−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​[cosα3​cosωτ2​−isinωτ2​]+−Mz0​(ω)[1−(1−cosα2​)e−τ2​/T1​−(1−cosα1​)cosα2​e−(τ1​+τ2​)/T1​]sinα3​+Mz0​(ω)sinα1​sinα2​sinα3​cosωτ1​e−τ1​/T2​e−τ2​/T1​}​

其中第一项为：
−Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2[cos⁡α3cos⁡ω(τ1+τ2)−isin⁡ω(τ1+τ2)]=−Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2[(cos⁡2α32−sin⁡2α32)cos⁡ω(τ1+τ2)−i(cos⁡2α32+sin⁡2α32)sin⁡ω(τ1+τ2)]=−Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2{cos⁡2α32[cos⁡ω(τ1+τ2)−isin⁡ω(τ1+τ2)]−sin⁡2α32[cos⁡ω(τ1+τ2)+isin⁡ω(τ1+τ2)]}=−Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2cos⁡2α32e−iω(τ1+τ2)+Mz0(ω)sin⁡α1cos⁡2α22e−(τ1+τ2)/T2sin⁡2α32eiω(τ1+τ2) \begin{aligned} &-M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[\cos\alpha_{3}\cos\omega\left(\tau_{1}+\tau_{2}\right)-i\sin\omega(\tau_{1}+\tau_{2})] \\ =&-M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[(\cos^2 \frac {\alpha_3}{2}-\sin^2 \frac {\alpha_3}{2})\cos\omega\left(\tau_{1}+\tau_{2}\right)\\ &-i(\cos^2 \frac {\alpha_3}{2}+\sin^2 \frac {\alpha_3}{2})\sin\omega(\tau_{1}+\tau_{2})] \\ =&-M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\{\cos^2 \frac {\alpha_3}{2}[\cos\omega\left(\tau_{1}+\tau_{2}\right)-i\sin\omega\left(\tau_{1}+\tau_{2}\right)]\\ &-\sin^2 \frac {\alpha_3}{2}[\cos\omega\left(\tau_{1}+\tau_{2}\right)+i\sin\omega\left(\tau_{1}+\tau_{2}\right)]\} \\ =&-M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\cos^2 \frac {\alpha_3}{2}e^{-i\omega(\tau_1+\tau_2)} \\ & +M_{z}^{0}(\omega)\sin\alpha_{1}\cos^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\sin^2 \frac {\alpha_3}{2}e^{i\omega(\tau_1+\tau_2)} \end{aligned} ===​−Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​[cosα3​cosω(τ1​+τ2​)−isinω(τ1​+τ2​)]−Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​[(cos22α3​​−sin22α3​​)cosω(τ1​+τ2​)−i(cos22α3​​+sin22α3​​)sinω(τ1​+τ2​)]−Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​{cos22α3​​[cosω(τ1​+τ2​)−isinω(τ1​+τ2​)]−sin22α3​​[cosω(τ1​+τ2​)+isinω(τ1​+τ2​)]}−Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​cos22α3​​e−iω(τ1​+τ2​)+Mz0​(ω)sinα1​cos22α2​​e−(τ1​+τ2​)/T2​sin22α3​​eiω(τ1​+τ2​)​

第二项为：
Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2[cos⁡α3cos⁡ω(τ2−τ1)−isin⁡ω(τ2−τ1)]=Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2[(cos⁡2α32−sin⁡2α32)cos⁡ω(τ2−τ1)−i(cos⁡2α32+sin⁡2α32)sin⁡ω(τ2−τ1)]=Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2{cos⁡2α32[cos⁡ω(τ2−τ1)−isin⁡ω(τ2−τ1)]−sin⁡2α32[cos⁡ω(τ2−τ1)+isin⁡ω(τ2−τ1)]}=Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2cos⁡2α32e−iω(τ2−τ1)−Mz0(ω)sin⁡α1sin⁡2α22e−(τ1+τ2)/T2sin⁡2α32eiω(τ2−τ1) \begin{aligned} &M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[\cos\alpha_{3}\cos\omega\left(\tau_{2}-\tau_{1}\right) -i\sin\omega\left(\tau_{2}-\tau_{1}\right)]\\ =&M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}[(\cos^2 \frac {\alpha_3}{2}-\sin^2 \frac {\alpha_3}{2})\cos\omega\left(\tau_{2}-\tau_{1}\right)\\ &-i(\cos^2 \frac {\alpha_3}{2}+\sin^2 \frac {\alpha_3}{2})\sin\omega(\tau_{2}-\tau_{1})]\\ =&M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\{\cos^2 \frac {\alpha_3}{2}[\cos\omega\left(\tau_{2}-\tau_{1}\right)-i\sin\omega\left(\tau_{2}-\tau_{1}\right)]\\ &-\sin^2 \frac {\alpha_3}{2}[\cos\omega\left(\tau_{2}-\tau_{1}\right)+i\sin\omega\left(\tau_{2}-\tau_{1}\right)]\} \\ =&M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\cos^2 \frac {\alpha_3}{2}e^{-i\omega(\tau_2-\tau_1)} \\ & -M_{z}^{0}(\omega)\sin\alpha_{1}\sin^{2}\frac{\alpha_{2}}{2}e^{-\left(\tau_{1}+\tau_{2}\right) /T_{2}}\sin^2 \frac {\alpha_3}{2}e^{i\omega(\tau_2-\tau_1)} \end{aligned} ===​Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​[cosα3​cosω(τ2​−τ1​)−isinω(τ2​−τ1​)]Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​[(cos22α3​​−sin22α3​​)cosω(τ2​−τ1​)−i(cos22α3​​+sin22α3​​)sinω(τ2​−τ1​)]Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​{cos22α3​​[cosω(τ2​−τ1​)−isinω(τ2​−τ1​)]−sin22α3​​[cosω(τ2​−τ1​)+isinω(τ2​−τ1​)]}Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​cos22α3​​e−iω(τ2​−τ1​)−Mz0​(ω)sinα1​sin22α2​​e−(τ1​+τ2​)/T2​sin22α3​​eiω(τ2​−τ1​)​

第三项为：
−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2[cos⁡α3cos⁡ωτ2−isin⁡ωτ2]=−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2[(cos⁡2α32−sin⁡2α32)cos⁡ωτ2−i(cos⁡2α32+sin⁡2α32)sin⁡ωτ2]=−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2{cos⁡2α32(cos⁡ωτ2−isin⁡ωτ2)+−sin⁡2α32(cos⁡ωτ2+isin⁡ωτ2)}=−Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2cos⁡2α32e−iωτ2+Mz0(ω)[1−(1−cos⁡α1)e−τ1/T1]sin⁡α2e−τ2/T2sin⁡2α32eiωτ2 \begin{aligned} &-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}[\cos\alpha_{3}\cos\omega\tau_{2} -i\sin\omega \tau_2]\\ =&-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}[(\cos^2 \frac {\alpha_3}{2}-\sin^2 \frac {\alpha_3}{2})\cos\omega\tau_{2}\\ &-i(\cos^2 \frac {\alpha_3}{2}+\sin^2 \frac {\alpha_3}{2})\sin\omega\tau_{2}]\\ =&-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}\{\cos^2 \frac {\alpha_3}{2}(\cos\omega\tau_2-i\sin\omega\tau_2)+ \\ &-\sin^2 \frac {\alpha_3}{2}(\cos\omega\tau_2+i\sin\omega\tau_2) \}\\ =&-M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}\cos^2 \frac {\alpha_3}{2}e^{-i\omega\tau_2}\\ &+M_{z}^{0}(\omega)\left[1-\left(1-\cos\alpha_{1}\right) e^{-\tau_{1} /T_{1}}\right]\sin\alpha_{2}e^{-\tau_{2} /T_{2}}\sin^2 \frac {\alpha_3}{2}e^{i\omega\tau_2}\\ \end{aligned} ===​−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​[cosα3​cosωτ2​−isinωτ2​]−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​[(cos22α3​​−sin22α3​​)cosωτ2​−i(cos22α3​​+sin22α3​​)sinωτ2​]−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​{cos22α3​​(cosωτ2​−isinωτ2​)+−sin22α3​​(cosωτ2​+isinωτ2​)}−Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​cos22α3​​e−iωτ2​+Mz0​(ω)[1−(1−cosα1​)e−τ1​/T1​]sinα2​e−τ2​/T2​sin22α3​​eiωτ2​​

第五项为：
$$\begin{array}{rl}& M_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3\cos \omega {\tau }_1{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}\\ =& \frac{1}{2}{M}_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}{e}^{i\omega {\tau }_1}+\\ & \frac{1}{2}{M}_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}{e}^{-i\omega {\tau }_1}\end{array}$$

将上面的式子全部合并，得到如下表达式：
$$\begin{array}{rl}{M}_{xy}[\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+}]=& M_x(\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+})+i{M}_y(\omega ,{\left({\tau }_1+{\tau }_2\right)}_{+})\\ =& -M_z^0(\omega )\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}{\cos }^2\frac{{\alpha }_3}{2}{e}^{-i\omega \left({\tau }_1+{\tau }_2\right)}{e}^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & +M_z^0(\omega )\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}{\sin }^2\frac{{\alpha }_3}{2}{e}^{i\omega \left({\tau }_1+{\tau }_2\right)}{e}^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & +M_z^0(\omega )\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}{\cos }^2\frac{{\alpha }_3}{2}{e}^{-i\omega \left({\tau }_2-{\tau }_1\right)}{e}^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & -M_z^0(\omega )\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}{\sin }^2\frac{{\alpha }_3}{2}{e}^{i\omega \left({\tau }_2-{\tau }_1\right)}{e}^{-\left({\tau }_1+{\tau }_2\right)/T_2}\\ & +\frac{1}{2}{M}_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3{e}^{-i\omega {\tau }_1}{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}(4.43)\\ & +\frac{1}{2}{M}_z^0(\omega )\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3{e}^{i\omega {\tau }_1}{e}^{-{\tau }_1/T_2}{e}^{-{\tau }_2/T_1}\\ & -M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_1\right)e^{-{\tau }_1/T_1}\right]\sin {\alpha }_2{\cos }^2\frac{{\alpha }_3}{2}{e}^{-i\omega {\tau }_2}{e}^{-{\tau }_2/T_2}\\ & +M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_1\right)e^{-{\tau }_1/T_1}\right]\sin {\alpha }_2{\sin }^2\frac{{\alpha }_3}{2}{e}^{i\omega {\tau }_2}{e}^{-{\tau }_2/T_2}\\ & -M_z^0(\omega )\left[1-\left(1-\cos {\alpha }_2\right)e^{-{\tau }_2/T_1}-\left(1-\cos {\alpha }_1\right)\cos {\alpha }_2{e}^{-\left({\tau }_1+{\tau }_2\right)/T_1}\right]\sin {\alpha }_3\end{array}\text{\hspace{1em}(4.43)}$$

$t>{\tau }_1+{\tau }_2$：横向磁化分量演变：

$t\ge {\tau }_1+{\tau }_2$，横向磁化分量演变：
$$M_{x^{\prime }{y}^{\prime }}(\omega ,t)=M_{x^{\prime }{y}^{\prime }}[\omega ,({\tau }_1+{\tau }_2{)}_{+}]e^{-(t-{\tau }_1-{\tau }_2)/T_2}{e}^{-i\omega (t-{\tau }_1-{\tau }_2)}\text{\hspace{1em}(4.44)}$$

• $M_{x^{\prime }{y}^{\prime }}[\omega ,({\tau }_1+{\tau }_2{)}_{+}]$: 这是初始条件，代表第三个脉冲施加后瞬间 ($t=({\tau }_1+{\tau }_2{)}_{+}$) 的复横向磁化。其具体形式由公式 (4.43) 给出，包含了所有由前三个脉冲共同作用产生的磁化分量。
• $e^{-(t-{\tau }_1-{\tau }_2)/T_2}$: $T_2$ 弛豫衰减因子。表示从第三脉冲结束时刻 ($t_0={\tau }_1+{\tau }_2$) 开始，横向磁化以时间常数 $T_2$ 进行指数衰减。
• $e^{-i\omega (t-{\tau }_1-{\tau }_2)}$: 进动因子。表示在旋转坐标系中，磁化矢量以角频率 $\omega$ 继续绕$z^{\prime }$轴顺时针进动。$(t-{\tau }_1-{\tau }_2)$ 表示时间从 $t_0={\tau }_1+{\tau }_2$ 开始计算。

将公式 (4.43) 代入 (4.44)，将得到五类回波信号的时域表达式
$$\begin{array}{rlr}{S}_1(t)& =A_1{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega (t-2{\tau }_1)}d\omega & (4.45a)\\ S_2(t)& =A_2{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega (t-2{\tau }_1-{\tau }_2)}d\omega & (4.45b)\\ S_3(t)& =A_3{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega (t-2{\tau }_2)}d\omega & (4.45c)\\ S_4(t)& =A_4{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega [t-({\tau }_1+2{\tau }_2)]}d\omega & (4.45d)\\ S_5(t)& =A_5{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega [t-2({\tau }_1+{\tau }_2)]}d\omega & (4.45e)\end{array}$$
其幅值满足：
$$\begin{array}{rlr}{A}_1& =\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}co{s}^2\frac{{\alpha }_3}{2}& (4.46a)\\ A_2& =\frac{1}{2}\sin {\alpha }_1\sin {\alpha }_2\sin {\alpha }_3{e}^{-{\tau }_2/T_1}& (4.46b)\\ A_3& =-\sin {\alpha }_1{\sin }^2\frac{{\alpha }_2}{2}{\sin }^2\frac{{\alpha }_3}{2}& (4.46c)\\ A_4& =[1-(1-\cos {\alpha }_1)e^{-{\tau }_1/T_1}]\sin {\alpha }_2{\sin }^2\frac{{\alpha }_3}{2}& (4.46d)\\ A_5& =\sin {\alpha }_1{\cos }^2\frac{{\alpha }_2}{2}{\sin }^2\frac{{\alpha }_3}{2}& (4.46e)\end{array}$$

• $A_1$: 第一自旋回波 (SE1) 的幅度。$T_E=2{\tau }_1$，源于脉冲1和脉冲2对横向磁化的直接作用。
• $A_2$: 受激回波 (STE) 的幅度。$T_E=2{\tau }_1+{\tau }_2$，这是最关键的一个系数！
  • $e^{-{\tau }_2/T_1}$: 这个因子是受激回波的标志。它表明该回波的磁化在 ${\tau }_2$ 期间是以纵向磁化的形式“存储”的，因此其强度受 $T_1$ 弛豫（而非 $T_2$）调制。这使得受激回波在长延迟时间下可能比其他回波更显著。
• $A_3$: 次级回波的幅度。$T_E=2{\tau }_2$，负号表示该回波与主自旋回波相位相反。
• $A_4$: 第二自旋回波 (SE2) 的幅度。$T_E={\tau }_1+2{\tau }_2$，源于脉冲2产生的FID被脉冲3重聚焦。
• $A_5$: 第三自旋回波 (SE3) 的幅度。$T_E=2({\tau }_1+{\tau }_2)$，源于脉冲1产生的FID被脉冲3重聚焦。

回波信号的通用形式

将所有回波信号统一为一个简洁、通用的数学表达式：
$$S(t)=A_E{\int }_{-\infty }^{\infty }\rho (\omega )e^{-t/T_2(\omega )}{e}^{-i\omega (t-T_E)}d\omega \text{\hspace{1em}(4.47)}$$
其中：

• $A_E$: 回波幅度系数。一个复数，包含了所有与脉冲序列参数（翻转角 ${\alpha }_i$、时间间隔 ${\tau }_i$）和弛豫时间 ($T_1,T_2$) 相关的调制因素。公式 (4.46a-e) 是 $A_E$ 的具体例子。
• $e^{-t/T_2(\omega )}$: $T_2$ 弛豫衰减因子。描述了信号幅度随时间的指数衰减，提供了组织的 $T_2$ 对比度。
• $e^{-i\omega (t-T_E)}$: 相位演化因子。这是回波形成的核心。
  • 当 $t=T_E$ 时，此项为 $e^0=1$，所有频率成分同相位，信号达到最大值（回波峰）。
  • 当 $t\ne T_E$ 时，质子群失相，信号衰减。