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\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}

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%\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}
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\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}

