\documentstyle[psfig]{article} \textwidth=16cm \textheight=24.5 cm \hoffset=-1.0cm \voffset=-2.5cm \newcommand{\text}{\rm} \begin{document} \title{Determination of NMR-spectrum moments of $F^{19}$ in $CaF_2$ from the line shape parameters.} \author{{\it \ B.M. Bolotov} \\ Institute of Physics and Technology, Ibragimova, 1, Almaty 480082,\\ Kazakhstan\\ {\em E-mail: bolotov@satsun.sci.kz} } \date{{\small Received April 28, 2000\\ Accepted May 17, 2000}} \maketitle \pagestyle{myheadings} %\markright{{\sc Magnetic Resonance in Solids. Electronic Journal } %{\bf 3}, {\sc 2 (1999)}\hfill} \thispagestyle{empty} \markright{{\sl Magnetic Resonance in Solids. Electronic Journal. Vol. 4, No. 1 (2000)}\hfill} %\markright{{ Magnetic Resonance in Solids. Electronic Journal } %{ 3}, { 2 (1999)}\hfill} \thispagestyle{empty} \bigskip\bigskip \hbox{ \vbox{\psfig{figure=logo25a.eps,clip=}} \hspace{.5in} %\vbox{\parbox{2.51in}{\LARGE \bf {\it \ \ \ Volume} 3 \\ {\it No.} 2, 1999} %\vbox{\parbox{2.51in}{ \bf { \ \ \ Volume} 3 \\ { No.} 2, 1999} \vbox{\parbox{2.51in}{\LARGE\bf \ \ Volume 4 \\ No. 1, 2000} \vspace{.2in}}} \newpage \vspace*{1in} \begin{center} {\bf Abstract} \end{center} Results of calculation of NMR line moments of $F^{19}$ in $CaF_2$ compared with experimental data are presented.\\ \begin{description} \item[PACS]: 76.60.-k;75.20.Ck \item[Keywords]: nuclear magnetic resonance, line shape, moments \end{description} \newpage For determination of suitability degree of applied line shape function in NMR experiments, it is convenient to compare the experimental line moments with those calculated. We carried out such a comparison for the free induction decay (FID) signal of $F^{19}$ in $CaF_2$. In this compound fluorine has a simple cubic lattice, high Debye temperature and spin S=1/2, so we can neglect temperature contributions to line moments and exclude quadrupolar interactions. Previously \cite{1} we suggested to treat the FID signal as a convolution of two functions. The firat function \begin{equation} g_i=I_{\nu}(bt)/(bt)^{\nu} \end{equation} is a solution of equation of motion of damped harmonic oscillator, and the second one \begin{equation} g_r=exp\{-\alpha^2\tau_c^2(\sqrt{1+t^2/\tau_c^2}-1)\} \end{equation} determines a summation law for spin-system non-correlated responses on disturbing action of hf-impulse. Here b is the local field value determined by the first two zeros of FID oscillation; $I_{\nu}$ is the Bessel function of the first kind; ${\tau_c}$ is the correlation time of z-component of local fields; ${\alpha^2}$ is the the second moment of $g_r$(${\omega}$). Accoprding to the model \cite{2}, $g_i$ describes a signal from the spins going into a zone of correlated dipole - dipole (d-d) interaction, and $g_r$ - from all the rest. Here the correlation is meant to be full phase coherence of superposition of local fields from environmental nuclei on chosen nucleus. In this publication on the basis of experimental data of precise FID measurements of $F^{19}$ in $CaF_2$ made by Engelsberg and Lowe \cite{3}, we present results of calculation of line moments (up to $14^{th}$) with the help of line form function as \begin{equation} G(t)=g_i*g_r \end{equation} where $g_i$ and $g_r$ are determined by the expressions (1) and (2), and b, ${\nu}$, ${\alpha}$ and ${\tau_c}$ have been determined from the data \cite{3} according to procedure \cite{1}. It is known that an arbitrary even moment of NMR symmetric line may be described by means of line shape function \cite{4}: \begin{equation} M_{2n}(G(\omega))=\mid\frac{d^{2n}G(t)}{dt^{2n}}\mid_{t=0}=G^{(2n)}(0). \end{equation} An arbitrary full moment of $G(t)$ function, as a product of functions $g_i$ and $g_r$, is easily defined according to Leibniz`s formula \cite{5}: \begin{equation} M_{2n}=G^{(2n)}(t)_{t=0}=\sum_{k=0}^{2n}C_{2n}^{k}g_{i}^{(k)}(0)g_{r}^{(2n-k)}(0)= \sum C_{2n}^{k}M_{k}^{i}M_{2n-k}^r \end{equation} where $C_{2n}^{k}=\frac{2n!}{k!(2n-k)!}$ . Thus, knowing a derivative of functions $g_i$ and $g_r$, we learn, according to (4), their corresponding moments and consequently $n^{th}$ full moment of relaxation function $G(\omega)$, too. Taking into account that for symmetric line odd moments are equal zero, we determine all even moments $G(\omega)$ up to $14^{th}$ one with the formula (5). The $n^{th}$ derivative of $g_r(t)$ is defined from recurrent formula \cite{3} \begin{eqnarray} &&g_{r}^{(2n)}(0)=(-\alpha^{2})[(2n-1)g_{r}^{(2n-2)}(0)+ \\ \nonumber &&\sum_{k=1}^{n-1}{\frac{(2n-2k-1)!!(2k-1)(2n-2k)!(-1)^{n-k}} {(2n-2k)!!\tau_{c}^{2n-2k}}} C_{2n-2k}^{2n-1}g_{r}^{(2k-2)}(0)]. \end{eqnarray} From equation (6), the following expression have been obtained for $12^{th}$ and $14^{th}$ derivatives $g_{r}(t)$ \begin{equation} g_{r}^{(12)}(0)=10395(945\frac{\alpha^2}{\tau_{c}^{10}}+945\frac{\alpha^4}{\tau_c^{8}}+420\frac{\alpha^6}{\tau_{c}^{6}} +105\frac{\alpha^8}{\tau_{c}^{4}}+15\frac{\alpha^{10}}{\tau_{c}^{2}}+\alpha^{12}) \end{equation} \begin{equation} g_{r}^{(14)}(0)=-135135(10395\frac{\alpha^2}{\tau_{c}^{12}}+ 10395\frac{\alpha^4}{\tau_{c}^{10}}+ 4725\frac{\alpha^6}{\tau_{c}^{8}}+ 1260\frac{\alpha^8}{\tau_{c}^{6}} +210\frac{\alpha^10}{\tau_{c}^{4}}+ 21\frac{\alpha^{12}}{\tau_{c}^{2}}+\alpha^{14}). \end{equation} \begin{figure}[tb] %[tbph] \centerline{\psfig{figure=fig1ppe.eps,height=3in,clip=} } %width=4.75 in,height=3.in \caption{Calculated and experimental \protect{\cite{3}} values of square roots from FID signal moments of F$^{19}$ in CaF$_2$ for magnetic field directions along the basic crystallographic axes. } \label{} \end{figure} The expressions for derivatives of $g_r(t)$ from $2^{th}$ to $10^{th} $ are presented in \cite{3}. In \cite{1} the simple expression is given for arbitrary even moment of a signal from the cell \begin{equation} M_{2n}^{i}=\frac{(2n-1)b^{2n}}{2n\prod_{k=1}^{n}(\nu+k)}. \end{equation} Proceeding with calculations we supposed that $$C_{0}^{k}=M_{0}^{i}=M_{0}^{r}=C_{k}^{k}\equiv 1 $$. In Table 1 (see Appendix) we present the comparative data of moment values for NMR spectra of $F^{19}$ in Ca$F_2$ obtained by means of analysis of experimental data \cite{3}, and theoretical values from works \cite{6,7} calculated on the basis of parameters of the line form function suggested in \cite{1}. The figure 1 illustrates the table data. Thus it has been shown that analytical equation for a signal shape in crystalline diamagnetics not only describes the real spectrum but makes possible to determine the highest moments without labour-consuming procedures \cite{6,7}. The suggested approach permits to describe NMR signal in the case of quadrupole nuclei too. In this case it is sufficient to set correctly a distribution function of quadrupolar frequencies \cite{8}. \begin{thebibliography}{9} \bibitem{1} B.M. Bolotov, D.A. Aksenov, A.I. Polyakov and S.P. Sen'shin {\it Phys. Stat. Sol. (b)} 139, K137 (1987) \bibitem{2} A.A. Lundin, B.N. Provotorov {\it Zh.Eksp.Teor.Fiz.} {\bf 70}, 2201 (1976) (in Russian) \bibitem{3} M. Engelsberg, I.J. Lowe {\it Phys.Rev.} {\bf B10}, 822 (1974) \bibitem{4} M. Goldman, Spin temperature and nuclear magnetic resonance in solids, Oxford, 1970 \bibitem{5} I.N.Bronshtein, K.A.Semendyaev, Spravochnik po matematike, Moskva, Nauka, 1981 (in Russian) \bibitem{6} S.J. Knack Jensen, Kgaersgaard Hansen E. {\it Phys.Rev.} {\bf B7}, 2930 (1973) \bibitem{7} G.W. Canters, C.S. Johnson {\it J. Mag. Resonance} {\bf 6}, 1 (1972) \bibitem{8} D.A.Aksenov, B.M. Bolotov, A.I. Polyakov, H.V. Chasnikova {\it Phys. Stat. Sol.(b)} {\bf 140}, K51 (1987) \end{thebibliography} \newpage \centerline{\bf Appendix} \centerline{\bf Table 1.} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \cline{1-5} \multicolumn{5}{|c|}{FID form parameters} \\ \hline $B_0$&\multicolumn{4}{|c|}{\cite{1}}& n & \multicolumn{3}{|c|}{$(M_n)^{1/n}gauss$} \\ \hline direct & b & $\nu$ & $\tau_c(\mu sec)$ & $\alpha$ & Order of & Experiment & Theory & Calculation \\ & gauss &&& gauss & moment & \cite{3} & \cite{6,7} & (this work) \\ \hline [100] & 6.037 & 0.585 & 68.55 & 1.379 & 2 & $3,614\pm 0,036$ & 3,603 & 3.661 \\ \hline & & & & &4 & $4,352\pm 0,046$ & 4,349 & 4.352 \\ \hline & & & & &6 & $4,882\pm 0,055$ & 4,85 & 4.830 \\ \hline & & & & &8 & $5,326\pm 0,068$ & 5,37 & 5.212 \\ \hline & & & & &10 & $5,736\pm 0,078$ & & 5.646 \\ \hline & & & & &12 & $6,141\pm 0,090$ & & 6.134 \\ \hline & & & & &14 & $6,55\pm 0,11$ & & 6.664 \\ \hline & & & & & & & & \\ \hline [111] & 2.713 & 0.93 & 132.44 & 0.722 & 2 & $1,508\pm 0,019$ & 1,493 & 1.558 \\ \hline & & & & &4 & $1,865\pm 0,025$ & 1,854 & 1.865 \\ \hline & & & & &6 & $2,134\pm 0,028$ & 2,11 & 2.174 \\ \hline & & & & &8 & $2,367\pm 0,033$ & 2,36 & 2.384 \\ \hline & & & & &10 & $2,585\pm 0,041$ & & 2.597 \\ \hline & & & & &12 & $2,797\pm 0,045$ & & 2.819 \\ \hline & & & & &14 & $3,004\pm 0,049$ & & 3.050 \\ \hline & & & & & & & & \\ \hline [110]& 4.084& 1.025& 155.84 & 0.9 & 2 & $2,191\pm 0,029$ & 2,218 & 2.220 \\ \hline & & & & &4 & $2,679\pm 0,039$ & 2,715 & 2.679 \\ \hline & & & & &6 & $3,026\pm 0,052$ & 3,07 & 3.011 \\ \hline & & & & &8 & $3,304\pm 0,068$ & 3,42 & 3.248 \\ \hline & & & & &10 & $3,544\pm 0,078$ & & 3.494 \\ \hline & & & & &12 & $3,762\pm 0,083$ & & 3.757 \\ \hline & & & & &14 & $3,967\pm 0,090$ & & 4.037 \\ \hline \end{tabular} \end{center} \end{document}