18.7 fitting two and three together g10

\[ F_{3}^{iso}(E,\vec{P},L)=-1/{ K}_{df}^{iso}(E^*) \]

\[ K_{df}^{iso}=\frac{P[0] M_0^2 }{E^2-M_r^2}+ P[2] \]

\[ F_{3}^{iso}(E,\vec{P}=0,L)=\frac{1}{L}\left[ \frac{\tilde F^s}{3}- \tilde F^s\frac{1}{1/(2\omega K_2^s)+\tilde F^s+\tilde G^s}\tilde F^s\right]_{kp}\\ \left[\frac{1}{2\omega K_2^s}\right]_{kp}=\delta_{kp}\left[(k\cot\delta )+|q^*_{2,k}(1-H(\vec{k}))|\right]\frac{1}{32\pi\omega_k E_{2,k}^*} \] and \[ \frac{k}{m} \cot \delta= \frac{1}{a_0m}+\frac{r_0m}{2}\frac{k^2}{m^2}\\ P[3]=am\\ P[4]=r_0m \] The best fit:

\[\begin{gather} \chi^2/d.o.f.=1.70598 \\ P[0]=74.0805\pm (11) \\ P[1]=9.12899\pm (0.00017) \\ P[2]=-706.844\pm (6e+02) \\ P[3]=-0.159792\pm (0.0024) \\ P[4]=-31.5321\pm (7.5) \\ \end{gather}\] {\[\begin{gather} C=\begin{pmatrix} 130& 0.581& -0.449& -0.119& -0.17\\ 0.581& 2.87e-08& -0.572& -0.481& -0.414\\ -0.449& -0.572& 3.57e+05& 0.194& 0.715\\ -0.119& -0.481& 0.194& 5.8e-06& 0.666\\ -0.17& -0.414& 0.715& 0.666& 56.7\\ \end{pmatrix} \end{gather}\]}

18.7.1 kcot 1 par

\[ K_{df}^{iso}=\frac{P[0] M_0^2 }{E^2-M_r^2}+ P[2] \]

\[ \frac{k}{m} \cot \delta= \frac{1}{a_0m} \] The best fit:

\[\begin{gather} \chi^2/d.o.f.=1.46817 \\ P[0]=81.5648\pm (12) \\ P[1]=9.12952\pm (0.00059) \\ P[2]=1455.65\pm (8.9e+02) \\ P[3]=-0.155717\pm (0.0021) \\ \end{gather}\] {\[\begin{gather} C=\begin{pmatrix} 137& -0.356& -0.168& -0.00778\\ -0.356& 3.5e-07& 0.374& -0.255\\ -0.168& 0.374& 7.96e+05& -0.788\\ -0.00778& -0.255& -0.788& 4.48e-06\\ \end{pmatrix} \end{gather}\]}

18.7.2 kcot 1 kis0 3 par covariance

We minimise the correlated \(\chi^2\) \[ \chi^2= \sum_{i,j} ( E^{predicted} - E^{latt})_i C_{i,j}^{-1}( E^{predicted} - E^{latt})_j \]

\[ K_{df}^{iso}=\frac{P[0] M_0^2 }{E^2+M_r^2}+ P[2] \]

\[ \frac{k}{m} \cot \delta= \frac{1}{a_0m} \] The best fit:

\[\begin{gather} \chi^2/d.o.f.=1.44987 \\ P[0]=96.629\pm (16) \\ P[1]=9.12853\pm (0.0017) \\ P[2]=1773.18\pm (9.8e+02) \\ P[3]=-0.15631\pm (0.0027) \\ \end{gather}\] {\[\begin{gather} C=\begin{pmatrix} 1& -0.463& 0.0674& 0.0289\\ -0.463& 1& 0.105& -0.123\\ 0.0674& 0.105& 1& -0.9\\ 0.0289& -0.123& -0.9& 1\\ \end{pmatrix} \end{gather}\]}

18.7.3 kcot 1 kis0 2 par covariance

We minimise the correlated \(\chi^2\) \[ \chi^2= \sum_{i,j} ( E^{predicted} - E^{latt})_i C_{i,j}^{-1}( E^{predicted} - E^{latt})_j \]

\[ K_{df}^{iso}=\frac{P[0] M_0^2 }{E^2+M_r^2} \]

\[ \frac{k}{m} \cot \delta= \frac{1}{a_0m} \] The best fit:

\[\begin{gather} \chi^2/d.o.f.=1.62602 \\ P[0]=80.4844\pm (14) \\ P[1]=9.12735\pm (0.0016) \\ P[2]=-0.153147\pm (0.0013) \\ \end{gather}\] {\[\begin{gather} C=\begin{pmatrix} 1& 0.0558& 0.184\\ 0.0558& 1& 0.5\\ 0.184& 0.5& 1\\ \end{pmatrix} \end{gather}\]}