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\title{The Model Calculation of Angular Dependence of CESR Linewidth
in Aluminium}
\author{{\it \ Andrei B. Bondarenko, Yurii N. Proshin} \\
Kazan State University, Kremlevskaya, 18, Kazan 420008, Russia\\
{\em E-mail: Yurii.Proshin@ksu.ru}
}
\date{{\small Received June 30, 1997\\ Revised July 15, 1997\\ Accepted
July 25, 1997}}

\begin{document}

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\markright{{\sc Magnetic Resonance in Solids.  Electronic Journal  }
{\bf 1}, {\sc 1 (1997)}\hfill}
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%{\Large \bf Vol. 1, \ \  No. 1, \ \ 1997}
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\begin{center}
{\bf Abstract}
\end{center}

A simple model of the Fermi surface for the interpretation of the conduction
electron spin resonance (CESR) results in aluminium is developed. According
to the real aluminium Fermi surface 48 small circles with the large amount of
the $g$-factor shift are arranged in five layers on the Fermi sphere, as
distinct from the random distribution of them in the previous model proposed
by Silsbee and Beuneu (SB). The present model reproduces the known
experimental CESR linewidth dependencies versus the frequency at
different temperature region as well as the SB model.
Additionally to the SB results the CESR linewidth is found to be
independent from the magnetic field direction in the high temperature
approximation. In both other cases (intermediate and
low temperatures) the $g$-value anisotropy over Fermi surface is predicted
to lead to the essential angular CESR linewidth dependence with five peaks.
Some applications of the model to various systems with conduction electrons
are discussed.


\begin{description}
  \item[PACS]: 76.30.Pk; 71.18.+y
  \item[Keywords]: conduction electrons, Fermi surface, spin
resonance, aluminium
\end{description}

\newpage

\section{Introduction.}

The angular dependence of the conduction electron spin resonance (CESR) can
be caused by the influence of sample boundaries. It is often considered as
the sole cause in the treatment of experimental data, especially for thin
metallic films \cite{films,Rev?}. The appearance of the CESR angular
dependence may be also connected with the Fermi surface anisotropy and (or)
conduction electrons $g$-factor spread over the Fermi surface \cite{Walker}.
Together with magnetic breakdown \cite{pr-mb,ftt-mb,UFN} this must be
taken into account for the metallic specimens with monocrystal or oriented
polycrystal structure.

In this paper we advance a simple model of such anisotropy and we
believe that the model can work not only for usual polyvalent metals with
complicated Fermi surface (Al, Be, Mg, Zn, {\it etc.}), but it also may be
useful for the interpretation of the CESR data for modern materials (organic
conductors \cite{org-cond}, for example). The model is appreciable founded on
the one proposed by Silsbee and Beuneu (SB) \cite{sb} to explain the CESR
linewidth linear dependence on frequency in terms of the motional narrowing
within the intermediate temperature range.

It is known \cite{UFN, Jafet, Bir-Pik} that the conduction electrons with
different quasimomenta ${\bf p}$ have different $g$-factor: $g({\bf p})$%
. The difference between the conduction electron $g$-factor and the free
electron value $g_{0}$ arises due to the spin-orbit coupling. Beuneu \cite
{beun} calculated explicitly the conduction electron $g$-factor for
aluminium in more than 8000 points on the Fermi surface, thus he determined
the ${\bf p}$-dependence of the $g$-shift
\begin{equation}
\delta g({\bf p})=g({\bf p})-\langle g({\bf p})\rangle \equiv g({\bf p})-g%
\text{.}  \label{eq1}
\end{equation}
Here $\left\langle ...\right\rangle $ means standard average on the Fermi
surface.

A remarkable feature of this calculation is the existence of very long tails
in the spread of $g$-shift, which extend to values of $\delta g$ as large as
plus and minus several hundreds.

The appearance of these very large $g$-shifts is a natural consequence of
the existence of 24 degeneracy points on the free electron approximation
Fermi surface where the second and third zones contact \cite{FS}. Al is a
metal with a face-centered cubic lattice and a cubical octahedron Brillouin
zone. These points ($W$-points) are localised on the intersections of
quadrangular and hexangular faces (see Fig.\ref{FigZB}). Here the $g$-factor
can achieve the value of several hundreds. This circumstance was taken into
account in the following way \cite{sb}. Spin-orbit interaction lifts the
degeneracy and small spin-orbit gap gives rise to locally very small
effective masses and hence to exceedingly large orbital moments which, in
turn, are strongly coupled to the spin \cite{Jafet}. There are 48
small spots with large $g$-shifts on the Fermi surface of aluminium in the
vicinity of the $W$-points, 24 ones on each of the second- and third-zone
surfaces, after lifting of the degeneracy by the spin-orbit splitting. The
role of $g$ anisotropy in clarifying experimental CESR linewidth
dependencies versus the frequency at different temperatures for Al has been
explored theoretically by Beuneu \cite{beun}, SB \cite{sb}. Freedman and
Fredkin \cite{FF}, Silsbee and Long \cite{SL} have investigated combined
effects of an exchange and the $g$ anisotropy.

\begin{figure}[htbp]
\centerline{\psfig{figure=fig1.eps,clip=}}%
%\end{center}
\caption{The real Fermi surface and the Brillouin zone for aluminium.}
\label{FigZB}
\end{figure}

As regard to the magnetic breakdown, it provides a contribution to the full
CESR linewidth with the $g$-anisotropy existence only \cite{pr-mb}, \cite
{ftt-mb}. The calculations \cite{ftt-mb,UFN} have showed that this
contribution is directly proportional to the magnetic breakdown
probabilities. In turn, these probabilities are known to depend on the
magnetic field inclination angle. Hence this dependence is extended to the
CESR linewidth.

To conclude this section let us mention that the number of references
incorporating the real Fermi surface structure of metals for the CESR
research may be found in the recent review \cite{UFN}.

\section{Model discussion.}

Assuming that all mechanisms of a broadening and a shift of the
CESR line give additive contributions, we will consider in more
detail the contribution of indicated anisotropy $g({\bf p})$
distribution over the Fermi surface to
the angular dependence of the CESR linewidth. We will neglect all other
causes of spin lifetime shortening due to the scattering on other electrons,
phonons, boundaries, impurities, dislocations and inhomogeneities of other
kinds leading to homogeneous linewidth of CESR.

The simplest interpretation of the observed line width is as follows. The
conduction electrons with different quasi-momenta (and different $g$-factors
) should be in resonance with different fields. Hence, if it were possible to
observe, the experiment would exhibit an inhomogeneous CESR line profile
corresponding to the real spread of the electron $g$-factor over the Fermi
surface
\begin{equation}
\Delta \omega _g\simeq \frac 1\hbar \sigma _g\mu _{\text{B}}H.  \label{eqDwg}
\end{equation}
Here $\sigma _g=\left\langle \left( \delta g\right) ^2\right\rangle ^{1/2}$
is the mean square deviation of the $g$-factor over the Fermi surface, $\mu
_{\text{B}}$ is the Bohr magneton. In this case $\Delta \omega _g$ is the
CESR linewidth due to the $g$ anisotropy.

In reality, such an inhomogeneously broadened line is not observable. The
reason is that the conduction electrons do not ''stand still'' at different
places on the Fermi surface. In a magnetic field the conduction electrons
move over the Fermi surface either along cyclotron orbits (high fields, low
temperatures, pure metals), or in a diffusive mode because of all kinds of
scattering mechanisms (by convention we will call this case a ''high
temperature'' one). In the second case the magnetic field influence on the
electrons motion over the Fermi surface is negligible.

In the high temperature case the conduction electron is able to ''visit''
many points over the Fermi surface during spin lifetime $\tau _s$ because of
$\tau _s\gg \tau $ where $\tau $ is scattering momentum time, characterising
the transition from one ${\bf p}$ state on Fermi surface to another one.
These various points are characterised by different $g$-factors $g({\bf p})$
and different local resonance fields $H$, correspondingly (as a rule the
CESR spectrometers operate at a fixed frequency $\omega _s$). As a result
the conduction electrons experience the action of a somewhat averaged field,
and the CESR line becomes narrower. The corresponding contribution of the $g$%
-factor anisotropy to the linewidth is given by the known formula of
''motional narrowing''
\begin{equation}
\Delta \omega \sim \frac{\sigma _g^2}{g^2}\omega _s^2\tau .  \label{eq2}
\end{equation}
Here $\omega _s$ is the resonance frequency.

In rather pure metals the temperature decreasing causes an increase in the
conduction electron momentum relaxation time $\tau $. The contribution to
the linewidth (\ref{eq2}) also rises. This rise can go on until $\tau
^{-1} $ becomes of the order of $\Delta \omega _g$. However before the full
breakdown of motional narrowing the conduction electrons start to move along
cyclotron orbits. In this situation an electron can manage to accomplish
several revolutions around the Fermi surface between two scattering events
because the cyclotron frequency $\omega _c$ becomes larger than $\tau ^{-1}$%
. Under this condition the $g$-factor must be averaged over the cyclotron
orbits at first.

For each orbit characterised by $p_z$ there is its $g$-factor shift \cite
{beun,FF} (magnetic field ${\bf H}$ is directed parallel to $z$ axis)
\[
\Delta g(p_z)=\left\langle g({\bf p})\right\rangle _{p_z=\text{const}%
}-\left\langle g({\bf p})\right\rangle .
\]
This shift significantly reduces the $g$-factor mean square spread $\sigma
_g $ which is equal to $\left\langle \left( \Delta g(p_z)\right)
^2\right\rangle ^{1/2}$ for the considered case. For instance, in Al the $g$%
-factor averaging over all the Fermi surface due to the conduction electron
diffusive motion yields $\sigma _g=0.469$, while the initial averaging over
cyclotron orbits can reduce this spread down to $\sigma _g=0.067$ \cite{beun}%
. In the most interesting intermediate case ($\omega _c\tau \sim 1$) the
averaging procedure should be very sophisticated due to the mixed nature of
conduction electron motion in magnetic field (see below) and the real Fermi
surface computations have not been carried out \cite{beun}.

The present research is devoted to the development of a model proposed by SB
\cite{sb} for the CESR linewidth behaviour interpretation in aluminium.
Briefly we will point out the main features of the SB model. The conduction
electrons were taken as a free electron gas. Instead of the real complicated
Fermi surface with actual $g$-spread they suggested to consider the Fermi
sphere lying in several Brillouin zones. The radius of Fermi sphere is $p_{%
\text{F}}$. This procedure may be named inverse reconstruction of Harrison.
Almost anywhere on the Fermi sphere the $g$-factor was taken to be equal to $%
g$ (see Eq.(\ref{eq1})), that was calculated by Beuneu \cite{beun}.
The $48$ small circular areas (named as $R$-disks) with
the large $g$-shift were corresponded to the
nearest neighbourhoods of $W$-points
\[
\left| \delta g_0\right| =1.1\cdot 10^3.
\]
This shift was taken positive on one-half of the $R$-disks, negative on the
rest, to leave a mean $g$-shift of zero. The radius of the $R$-disks was
appropriately taken to give a disk area $\pi p^2$ equal to the area, near
the $W$-points of the real Fermi surface in aluminium over which the energy
gap is dominantly determined by the spin-orbit interaction
\[
p=0.7\cdot 10^{-3}p_{\text{F}}.
\]

Thus, $\sigma _q$ calculated on the sphere was equal to $2.7$. To simplify
the calculations SB supposed that the $48$ $R$-disks were RANDOMLY
distributed over the sphere. Naturally, owing to this assumption the
possible linewidth anisotropy was omitted from the model.

Our model differs from the above mentioned one by a REGULAR ARRANGEMENT of
the $R$-disks on the Fermi sphere. This disposition of the $R$-disks
corresponds most closely to the arrangement of the $W$-points
neighbourhoods on the real Fermi surface.

\begin{figure}
\centerline{\psfig{figure=fig2.eps,clip=}}%
\caption{The schematic view of the free-electron Fermi surface in aluminium
with $R$-disks (the shaded areas). The direction of conduction electron
movement is shown by arrow. The characteristic angles used in our calculations
are indicated.}
\label{FigFS}
\end{figure}

Using the reverse reconstruction of Harrison we obtain that $48$ $R$-disks
are arranged in $5$ layers (see Fig.\ref{FigFS}): $16$ disks are in
equatorial one and $8$ disks are in each of the remaining ones.
The appropriate angles on the $R$-disks levels are equal to $\theta _i$
\begin{equation}
\begin{array}{c}
\theta _{1,\ldots ,8}=\arctan (1/2) \\
\theta _{9,\ldots ,16}=\pi /2-\arctan (1/2) \\
\theta _{17,\ldots ,32}=\pi /2 \\
\theta _{33,\ldots ,40}=\pi /2+\arctan (1/2) \\
\theta _{41,\ldots ,48}=\pi -\arctan (1/2).
\end{array}
\label{eqQ48}
\end{equation}
Individual electrons are assumed to be scattered randomly from point to
point on the Fermi sphere at rate $1/\tau $. Between scattering events
they move along cyclotron orbits on the Fermi sphere with a linear velocity
in quasimomentum space,

\[
\left| {\bf \dot p}\right| =\omega _cp_{\text{F}}\sin \theta \text{, }
\]
where $\theta $ is the angle between the electron quasimomentum ${\bf p}$ and
the applied field ${\bf H}(0,0,H)$.

Now we can use the analytical ''motional narrowing'' procedure of the
initial $g$-factor distribution with reasonable facility.

\section{The CESR linewidth calculations for model Fermi surface.}

The main parameter of the proposed theory is the magnitude of the typical
orbital segment length in ${\bf p}$-space, $\omega _{c}\tau p_{\text{F}}$,
traversed by an electron along cyclotron orbit between scattering events.
This parameter determines the manner in which motional narrowed linewidth
arises. There are three different temperature regimes defined by $\omega
_{c}\tau p_{\text{F}}$. In the high-temperature region (case $A$) the
conduction electrons are scattered on and off the $R$-disks more rapidly
than they move on and off the disks because of their cyclotron
motion, and hence the effects of cyclotron motion are negligible. An
electron has no time to cross completely the $R$-disk along cyclotron orbit
before the next scattering event
\[
\omega _{c}\tau p_{\text{F}}\ll p\text{.}
\]

In the low temperature regime (case $C$) the conduction electron, on the
contrary, can make few revolutions along a single cyclotron orbit before
scattering to a new orbit
\[
\omega _c\tau \gg 1\text{.}
\]
The cyclotron motion now gives effective averaging over the full orbit of
the $g$ variation associated with all traversed $R$-disks.

\begin{figure}
\centerline{\psfig{figure=fig3.eps,clip=}}%
\caption{The $R$-disk (the shade area). $\theta _{i}$ is the angle on the
R-disk, $\delta p$ is the impact parameter, the line with arrows indicates the
direction of the conduction electrons movement.}
\label{FigR-disk}
\end{figure}

The virtue of the present model is the possibility of averaging in an
intermediate temperature regime (case $B$)
\[
p\ll \omega _c\tau p_{\text{F}}\ll p_{\text{F}}\text{.}
\]
Under these conditions the conduction electron passes along cyclotron orbit
a distance larger than a size of the $R$-disk. It may cross either no or one
or several $R$-disks completely, but it has no time ($\tau < \omega _c$)
to make even one revolution along a single cyclotron orbit.

The correlation time for the narrowing is determined not by the time between
collisions, but the average time spent on the $R$-disks, which is now
determined only by the speed of the cyclotron orbital motion,
not the collision rate. The time $\tau $ in Eq.(\ref{eq2}) must
be replaced by the disk transit time, which is approximately
equal to $p/(p_{\text{F}}\omega _c)$.
As noted by SB, in this regime the linewidth varies as $\omega _s^2/\omega
_c $ and is therefore linearly proportional to the field at which the
experiment is performed, and a natural explanation is obtained for the
experimentally observed linear dependence of linewidth upon frequency.

Note the conclusions of SB about the frequency and temperature dependence of
the CESR linewidth in aluminium are valid for our model. Therefore we focus
our attention on those new outcomes, which could not be obtained in indicated
work. To clarify this difference we will not change all parameters
introduced by SB: $\delta g_{0}, p, g, p_{\text{F}}, \omega _c$, etc.

In the high-temperature regime (case $A$) the conventional result for
motional narrowing is appropriate (\ref{eq2}). The linewidth does not depend
upon the magnetic field inclination angle $\beta $. We have the SB result
\[
\left\langle \left( \delta g\right) ^2\right\rangle =\left( \delta
g_0\right) ^2\left( 48\pi p^2/4\pi p_{\text{F}}^2\right)
\]
\begin{equation}
A)\text{\medskip }\ \Delta \omega ^{(A)}=\Delta \omega _{\text{SB}%
}^{(A)}=12\left( \frac{\delta g_0}g\frac p{p_{\text{F}}}\omega _s\right)
^2\cdot \tau \equiv 12F^2\tau  \label{eqA}
\end{equation}
with parameter $F=(\delta g_0/g)(p/p_{\text{F}})\omega _s$. Here and below $%
\Delta \omega $ denotes the contribution of the motional averaged $g$
anisotropy to the full CESR linewidth $1/T_2$, assuming a scalar $g$-shift
rather than the tensor. Taking into account $g$-tensor properties leads to an
appearance of a correction factor of 2 \cite{sb, Walker-tens} in all the
linewidth expressions, see Eqs.(\ref{eqA}), (\ref{eqB}) and (\ref{eqC}).

Firstly we examine our system for a case of ${\bf H}\Vert {\bf c.}$

In intermediate temperature region (case $B$) we have to take into
account the peculiar character of the conduction electrons movement.

Following SB the spin precession phase of an electron passed through the $R$%
-disk relative to the mean phase of all of the spins is written as
\begin{equation}
\phi (\theta ,\delta p)=\left[\frac{\delta g_0}g\omega _s\right]\frac{2\left(
p^2-\delta p^2\right) ^{\frac 12}}{\omega _cp_{\text{F}}\sin \theta }
\label{eqA2}
\end{equation}
where $\theta $ is angle of orbit, $\delta p$ is impact parameter (see Fig.%
\ref{FigR-disk}).

Let us determine the rate at which any given electron moving along the orbit
with polar angle $\theta $ meets $R$-disk

\begin{equation}
R(\theta )=\left\{
\begin{array}{c}
 \omega _{c}p_{\text{F}}\sin \theta\times 2p_{\text{F}}/({4\pi p
\sin \theta _{i}^{^{\prime }}}) \text{,\ if\ }\theta \in \left( \theta
_{i}-\Delta \theta ,\theta _{i}+\Delta \theta \right) \\
0,\text{\ if\ }\theta \notin \left( \theta _{i}-\Delta \theta ,\theta
_{i}+\Delta \theta \right)
\end{array}
\right. ,  \label{eqA3}
\end{equation}
where the angle $\theta _{i}^{^{\prime }}$ is equal to $\arcsin \left(
\left( \sin \left( \theta _{i}-\Delta \theta \right) +\sin \left( \theta
_{i}+\Delta \theta \right) \right) /2\right) ,$ $\theta _{i}$ is a mean
angle on the $R-$disk level, $\Delta \theta =2\arcsin \left( p/p_{\text{F}%
}\right) ,$ 2$p$ is the collision cross section.

Using the fact that the rate at which the electrons gain the mean-square
precession phase error is then given by
\begin{equation}
\frac d{dt}\left\langle \phi ^2(t)\right\rangle =\frac 1{4\pi }
\int\limits_{0}^{\pi}2\pi R(\theta )\sin \theta \left\langle \phi
^2(\theta ,\delta p)\right\rangle _{\delta p}d\theta  \label{eqA4}
\end{equation}
and with the result of SB
\begin{equation}
\Delta \omega =\frac 12\frac d{dt}\left\langle \phi ^2(t)\right\rangle ,
\label{eqA5}
\end{equation}
where $\left\langle ...\right\rangle _{\delta p}$is the average $\phi
(\theta ,\delta p)$ over impact parameters ($-p\leq \delta p\leq p),$ we get

\begin{equation}
B)\medskip\ \Delta \omega ^{(B)}=\frac{F^2\Delta \theta }{24\omega _c}%
\sum_{i=1}^{48} \frac 1{\sin \theta _i}  \label{eqB}
\end{equation}
with a parameter $\Delta \theta \simeq 2p/p_{\text{F}}$. The SB result
differs from expression (\ref{eqB}) and may be written as
\begin{equation}
\Delta \omega _{\text{SB}}^{(B)}=\frac{12F^2}{\omega _c}\left( \frac p{p_{%
\text{F}}}\right) .  \label{eqB-sb}
\end{equation}

As a next step, let us consider the low temperature case. The CESR linewidth
is given by Eq.(\ref{eq2}) and
\begin{equation}
\delta g(\theta ,\delta p)=\delta g_0\frac{2\left( p^2-\delta p^2\right)
^{\frac{1}{2}}}{2\pi p_{\text{F}}\sin \theta } .\label{eqBB2}
\end{equation}

The probability that the orbit with polar angle $\theta $ in this case
traverses a $R$-disk is
\begin{equation}
P(\theta )=\left\{
\begin{array}{c}
1\text{,\ if\ }\theta \in \left( \theta_i-\Delta \theta ,\theta_{i}+\Delta
\theta \right) {, \ }i=1,2,...,48 \\
0,\text{\ if\ }\theta \notin \left( \theta _{i}-\Delta \theta ,\theta
_{i}+\Delta \theta \right) {,\ }i=1,2,...,48
\end{array}
\right. .  \label{eqBB3}
\end{equation}

Defining the mean-square deviation of $g$($p$) from its mean value $g$

\begin{equation}
\left\langle \lbrack \delta g(\theta ,\delta p)]^2\right\rangle =\frac
1{4\pi }\int\limits_0^\pi 2\pi P(\theta )\sin \theta
d\theta \left\langle [\delta g(\theta ,\delta p)]^2\right\rangle _{\delta p},
\label{eqBB4}
\end{equation}
we obtain by a straightforward calculation

\begin{equation}  \label{eqC}
C)\text{\medskip}\ \Delta \omega ^{(C)}=\frac{F^2\tau }{6\pi ^2}
\sum_{i=1}^{48} \ln \frac{\tan \left( \theta _{i^{^{\prime
}}}/2\right) }{\tan \left( \theta _{i^{^{\prime \prime }}}/2\right) }\simeq
\frac{F^2\tau \Delta \theta }{3\pi ^2}\sum_{i=1}^{48}%
\frac 1{\sin \theta _i}
\end{equation}
with $\theta _{i^{^{\prime }}(i^{^{\prime \prime }})}=\theta _i\pm \Delta
\theta $. SB have obtained
\begin{equation}  \label{eqC-sb}
\Delta \omega _{\text{SB}}^{(C)}=\frac{12F^2}\tau \left( \frac p{p_{\text{F}%
}}\right) .
\end{equation}

The expressions (\ref{eqB}), (\ref{eqC}) are valid for the case of ${\bf H}%
\Vert {\bf c}$ or $\beta =0$ (${\bf c}$ is the symmetry axis of crystal, see
Fig.\ref{FigFS}). For arbitrary angle of the magnetic field tilt it is
necessary to substitute $\xi _i$ instead of $\theta _i$ into equations (\ref
{eqA3})--(\ref{eqC}), where $\xi _i$ are found from next relationship
\begin{equation}
\cos \xi _i=\cos \theta _i\cos \beta -\sin \theta _i\sin \beta \cos \gamma _i
\label{eqKsi}
\end{equation}
here $\gamma _i=\gamma ^s+m\Delta \gamma _s$ is the angle in the
plane of the $R$-disks layer with number $s=s(i)$
(see Fig.\ref{FigFS}). $\gamma ^s$ is the polar angle
defining the magnetic field direction relative to the $R$-disk
with $m=0$ ($\gamma ^2=\gamma ^4=\gamma ^1+\pi /8=\gamma ^3+\pi /8=\gamma
^5+\pi /8$), $m=m(i)=0,1,2,\ldots $ is the number of $R$-disk on a given
layer, $\Delta \gamma _3=\pi /8$ for equatorial level and $\Delta \gamma
_s=\pi /4$ for remaining ones.

Note, the expressions (\ref{eqB}) and (\ref{eqC}) in the case 0$\leq \xi
_i\leq \Delta \theta $ do not give a good result, because expression (\ref
{eqA2}) cannot be used. In this case we must use the simple expression (\ref
{eq2}) with $g=\delta g_0$.

The linewidth dependence upon the magnetic field inclination angle $\beta $
is calculated for typical values of $\omega _c\tau $ and corresponding
curves are represented in Fig.\ref{FigTemp}.

\begin{figure}
\centerline{\psfig{figure=fig4.eps,clip=}}%
\caption{The theoretical CESR linewidth of aluminium versus angle $\beta$
calculated with Eqs. (\protect{\ref{eqA}}), (\protect{\ref{eqB}}),
(\protect{\ref{eqC}}) a) The low
temperature regime. On the inset there is the high temperature regime line.
b) The intermediate temperature regime.}
\label{FigTemp}
\end{figure}

\section{Discussion and conclusions.}

Let us summarise.

In the high temperature region the linewidth is independent from the
magnetic field direction. The conduction electrons diffusion over Fermi
surface is caused by various mechanisms of scattering. It makes insufficient
a consideration of comprehensive structure of the Fermi surface and our
results coincide completely with the SB ones. Both results are shown in the
inset at the right upper corner of the Fig.\ref{FigTemp}).

Two other cases are more interesting.

In the intermediate temperature region there are characteristic peaks at
values of $\beta $ coinciding with appropriate angles determined by Eq. (\ref
{eqQ48}), (see Fig.\ref{FigTemp}-b). If we average expression (\ref{eqB}) on
the angle $\beta $, we can obtain approximately the same result that
could be directly derived from the SB expression (\ref{eqB-sb})
\begin{equation}
\frac 1\pi \int\limits_0^\pi \Delta \omega (\beta
)d\beta \approx \Delta \omega _{\text{SB}}.  \label{eq-aver}
\end{equation}

In the low temperature region the angle dependence is represented on the Fig.%
\ref{FigTemp}a). The mean value (\ref{eq-aver}) coincides numerically with
corresponding value of the SB.

\begin{figure}
\centerline{\psfig{figure=fig5.eps,clip=}}%
\caption{The CESR linewidth as a function of angle $\beta $. Our model
predicts various behaviours for different values of the
angle $\gamma =\gamma _1$.}
\label{FigGamma}
\end{figure}

On the Fig.\ref{FigGamma} we represent the calculated CESR linewidth of
aluminium versus angle $\beta $ for different values $\gamma _i$. Obviously,
the linewidth dependence on $\gamma _i$ is symmetric relatively $\pi /8$ and
periodic with the period of $\pi /4$. We also observe
a symmetry of the shape of curves relatively the angle $\beta =\pi /2$.
There are five peaks within the $\beta $ range (0, $\pi )$ on angles
coinciding with ones determined by right sides expression (\ref{eqQ48}).
The peaks, which correspond angles $\beta =\pi /2\pm \arctan (1/2)$,
are weakly expressed on angles $\gamma _1\simeq 0$, $\pi /8$ $,$ the ones
have a maximum for angle $\gamma _1=\pi /4.$
The intensity of the peaks corresponding angles $\beta =\arctan (1/2)$ and $%
\beta =\pi -\arctan (1/2)$ have inverse dependence.

To evaluate the temperature dependence of the CESR linewidth at high
temperatures we have to take into account the phonon-induced relaxation.
Since we have no difference with the model of SB in this regime, we will
reproduce their estimation of the phonon-induced contribution to the
CESR linewidth, $\Delta \omega _{ph}$ below.

More detail estimation using the real Fermi surface structure near the $W$%
-points yields $\Delta \omega _{ph}=1.5\cdot 10^{-4}/\tau $. The resultant
expression for $\Delta \omega _{ph}$ determined from the fit with
experimental data \cite{sb} is
\begin{equation}  \label{eqDwph}
\Delta \omega _{ph}=\frac{1.5\cdot 10^{-4}}\tau
\end{equation}

Thus, at high temperatures we have two contributions to the CESR linewidth
caused by $g$ anisotropy (\ref{eqA}) and (\ref{eqDwph})
\begin{equation}  \label{eqDwA}
\Delta \omega _g^{(A)}=\Delta \omega ^{(A)}+\Delta \omega _{ph}.
\end{equation}
The scattering rate, $\tau $, is determined by the phonon one. It is well
known, that the last value varies as $T^3$%
\[
\frac 1\tau =pT^3\text{.}
\]
The theoretical fit \cite{sb} has yielded $p=2.1\cdot 10^7$sec$^{-1}$K$^{-3}$%
.

The temperature dependence of the CESR linewidth (or dependence upon $\omega
_c\tau $) is calculated using the equations (\ref{eqB}), (\ref{eqC}), (\ref
{eqDwA}) and it has qualitatively the same character as the dependence of
SB. Naturally, we have the close agreement for the average values of Eq. (%
\ref{eq-aver}) and we must use this dependence for polycrystal nonoriented
samples.

There are the following main experimental features
(see corresponding figure from the SB work).

1) At high temperatures the experimental dependence of linewidth ($\sim T^3$%
) is reproduced by (\ref{eqDwA}). An apparent frequency dependence of the
phonon spin-flip scattering can be understood as a consequence of the
motional narrowing model, with the phonon-induced relaxation remaining
independent of frequency.

2) There is the linewidth minimum at the intermediate temperatures. And the
minimum linewidth depends linearly on frequency, as indicated by (\ref{eqB}).

The linewidth behaviour with temperature in our model
agrees qualitatively to experimental data too.
For an extended discussion see mentioned paper of SB \cite{sb}.

Note a gap between estimated and observed linewidth is essentially in the
low-temperature region. At such conditions there is no quantitative
agreement, but the qualitative accordance occurs.

The pointed disagreement is easily explained by the simplicity of the
proposed model. The real Fermi surface of aluminium has several sheets
pertaining to different bands. At low temperatures in high purity metals the
conduction electrons move on each Fermi surface sheet separately. Hence we
must consider the real $g$ anisotropy over the real Fermi surface to obtain
adequate results. The same holds true not only for the temperature and
frequency dependence of the CESR linewidth, but angular one as well.

Note, in case of conduction electron small-angle scattering predominance the
equilibrium would be established separately within each of the Fermi surface
hulls. This would require that the characteristic time of scattering leaving
the conduction electron in its hull would be less than that of the
interhull scattering. In this situation one could take advantage of a model
involving several conduction electron groups.

To each of the electron groups one can ascribe such averaged characteristics
as the $g$-factor, the spin relaxation time $\tau _s$, the momentum
relaxation time $\tau $, and so on --- depending on the model sophistication
degree (see for instance a model proposed for Al in work \cite{SL}).
This model involves three mechanisms of conduction electron connection: (1)
conduction electron intergroup scattering with frequency 1/$\tau _{ij}$, (2)
exchange interaction, and (3) conduction electrons diffusive reflection from
the boundary.

Note that in paper \cite{pr-mb}, dedicated to the CESR experimental research
in Zn, the magnetic breakdown was pointed as a possible intergroup
connection mechanism (see too \cite{ftt-mb, UFN}).

Nevertheless, we think that the present model may be useful for qualitative
understanding the spin relaxation processes connected with $g$ anisotropy.
The clear physical ideas lie in the model origin.

The model of reconstructed Fermi surface is possible to use in organic
conductors in which intensive CESR investigations are carried out \cite
{org-cond} and their Fermi surfaces consist of several sheets too.
These materials are interesting for explorers from their
superconducting properties with the transition temperature $T_c$ up to $15$
K \cite{org-cond-Tc}.

We show, that in low and intermediate temperature the $g$-value anisotropy
over Fermi sphere leads to the angular dependence of CESR linewidth.
Unfortunately, as far as we know, there is no experimental data
of angular dependence of CESR linewidth in Al. In works \cite{Lub71}--%
%,Jan72, Lub73, Dun74, Dun75, Sam76, Mei77,
\cite{Sam78} only temperature
dependence of CESR linewidth in mono- and polycrystal or the characteristics
of the thin films were investigated.

Nevertheless, we believe that our model is qualitativy relevant to reality.

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\end{document}
