Vibrations and Relaxations in a Soft Sphere Glass: Boson Peak and Structure Factors
H. R. Schober
The dynamics of a soft sphere model glass, studied by molecular dynamics, is investigated. The vibrational density of states divided by $\omega^2$ shows a pronounced boson peak. Its shape is in agreement with the universal form derived for soft oscillators interacting with sound waves. The excess vibrations forming the boson peak have mainly transverse character. From the dynamic structure factor in the Brillouin regime pseudo dispersion curves are calculated. Whereas the longitudinal phonons are well defined up to the pseudo zone boundary the transverse ones rapidly get over-damped and go through the Ioffe-Regel limit near the boson peak frequency. In the high $q$ regime constant-$\omega$ scans of the dynamic structure factor for frequencies around the boson peak are clearly distinct from those for zone boundary frequencies. Above the Brillouin regime, the scans for the low frequency modes follow closely the static structure factor. This still holds after a deconvolution of the exact harmonic eigenmodes into local and extended modes. Also the structure factor for local relaxations at finite temperatures resembles the static one. This semblance between the structure factors mirrors the collective motion of chain like structures in both low frequency vibrations and atomic hopping processes, observed in the earlier investigations.
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\title{Vibrations and Relaxations in a Soft Sphere Glass: Boson Peak and 
Structure Factors} 
\author{H. R. Schober}

\affiliation{Institut f\"ur
  Festk\"orperforschung, Forschungszentrum J\"ulich, D-52425 J\"ulich,
  Germany}
\date{\today} 
\begin{abstract}
The dynamics of a soft sphere model glass, studied by molecular dynamics, is
investigated. The vibrational density of states divided by $\omega^2$ 
shows a pronounced boson peak. Its shape is in agreement with the
universal form derived for soft oscillators interacting with
sound waves. The  excess vibrations forming the boson peak have
mainly transverse character. From the dynamic structure factor in the
Brillouin regime pseudo dispersion curves are calculated.
Whereas the longitudinal phonons are well
defined up to the pseudo zone boundary the transverse ones rapidly get 
over-damped and go through the Ioffe-Regel limit near the boson peak 
frequency. In the high $q$ regime constant-$\omega$ scans of the
dynamic structure factor  for frequencies
around the boson peak are clearly distinct from those for zone boundary 
frequencies. Above the Brillouin regime, the scans for the low 
frequency modes follow  closely the static structure factor. This still holds
after a deconvolution of the exact harmonic eigenmodes into local and extended modes. 
Also the structure factor
for local relaxations at finite temperatures resembles the static one.
This semblance between the structure factors mirrors the collective
motion of chain like structures in both low frequency vibrations and atomic
hopping processes, observed in the earlier investigations.
\end{abstract}
\pacs {PACS number(s): 61.43.Fs, 63.50.+x, 66.30.Fq} 
\maketitle

\section{Introduction}

Despite large efforts, the vibrational dynamics of amorphous materials and glasses is 
still not fully understood and subject to a 
controversial debate. At a first sight there are many similarities
between glasses and amorphous materials and their
ordered counterparts, the crystals. The vibrational densities of states 
are  normally similar and typical
average frequencies or Debye temperatures
do not change dramatically from one state to another. This reflects the the
similarity of the short range order of both states, primarily the average nearest 
neighbour distance is similar.
At higher, but still low, frequencies the vibrational density 
of states (DOS), $Z(\omega)$, in glasses exceeds the Debye value and a maximum is
seen in the inelastic scattering intensity, the so called boson peak (BP). 
Due to disorder the sound waves are increasingly damped with increasing frequency 
and eventually the damping exceeds the
Ioffe-Regel limit: The mean free path of the phonons diminishes below
their wavelengths, the phonons are over-damped. The vibrations can
still be understood by harmonic theory but their eigenvectors will
have a complicated structure reflecting the structural disorder.
The frequency, $\omega_{\rm IR}$, where the Ioffe-Regel limit is reached,
is normally near the BP frequency, $\omega_{\rm BP}$.
As in defect crystals, one finds in addition to the ``band'' modes localized 
vibrational modes above the maximum frequency of the band modes or in gaps of the
spectrum. Additionally, particularly in molecular glasses, there can be the
equivalent of the optical vibrational modes of crystals, albeit often broken up by
disorder.

The dispute centers mainly on two issues: first,the nature of the excess 
of low frequency
modes above the Debye value, i.e. the origin of the BP, and secondly on the properties
of the phonons at higher frequencies. The term phonon is used here for an 
extended vibrational
mode which can be classified by a $q$-value. The latter question has gained 
great interest by the advance of modern synchrotron sources which allow to measure
structure factors in Brillouin scattering.

The frequency domain, typically around $\nu = 1$~THz, of 
the BP is accessible to a variety of
spectroscopic methods like
Raman \cite{winterling:75} and neutron scattering \cite{buchenau:88}.  From
the temperature dependence of the measured intensities it has been
deduced that the excitations are essentially harmonic vibrations.
The boson peak is, therefore, a maximum in $Z(\omega)/\omega^2$.
Mostly it is not a maximum of the vibrational spectral density, $Z(\omega)$,
itself. 

Over the years a large number of explanations of the excess intensity and the BP
have been proposed. First, there are some materials where some molecular unit has
an inherent low frequency mode, such as in plastic crystals. These modes will
trivially show as ``excess'' intensity in $Z(\omega)$. We will not consider these.
Secondly, there are materials where the BP is a real effect of disorder effect. 
The model, studied here, belongs to this class it has further no
molecular vibrations or low lying optic modes in its crystalline form.
Some authors attribute the BP to
vibrations of clusters of atoms of typical sizes
\cite{malinovsky:91,pang:92}. The origin of these clusters is unclear and they
have not been identified in numerical simulations.
Another popular qualitative explanation of the Boson peak is a
softening of acoustic phonons by static disorder
\cite{elliott:92,masciovecchio:99} due to elastic Rayleigh
scattering. However, estimates show that the
Rayleigh contribution is too small to explain the
experimental data on thermal conductivity in
glasses~\cite{elliott:92}.
Sometimes the BP is related to low lying optic modes of parental
crystals~\cite{freitas:85,dove:97,sagaev:99}. Whereas the BP is a general feature
of glasses such crystal structures with soft optic modes cannot be
identified always.  Such a mechanism is possible in some cases. 
A distinction optic modes, which disorder has broken into segments, and resonant 
vibrations is not very clear in all cases.   

Recent work on harmonic lattice models demonstrated that softening of
disordered force constants can smear and push to low frequencies peaks
which exist in the crystalline
DOS~\cite{schirmacher:98,taraskin:01,kantelhardt:01}. 
In another approach the vibrations of a random distribution of atoms,
interacting  with a Gaussian-shaped pair potential, was studied
\cite{grigera:02} in a harmonic scalar approximation. 

A different explanation is offered by the soft potential model (SPM) 
\cite{KKI:83,IKP:87}. This model gives a unified description of the
glassy low energy dynamics: tunneling, local relaxation and excess in low 
frequency vibrations. For the latter it postulates low frequency
oscillators which interact bilinearly with the sound waves and hybridize
with them. Thus quasi-local vibrations, also called
resonant vibrations, are created \cite{maradudin:71}. 
Fitting the model to the
experimental data, one finds effective masses of 20--100 atomic
masses for the entities moving in these effective soft potentials
\cite{BGGS:91}.The sound waves
in turn will be damped by resonant scattering with these modes 
\cite{BGGPRS:92}. For low frequencies the SPM predicts an increase of the 
excess of the DOS as $Z_{\rm excess}(\omega) \propto \omega^4$. 
This rapid increase of 
$Z_{\rm excess}(\omega)$
will lead to level repulsion and an eventual cross over to
$Z_{\rm excess}(\omega) \propto \omega$ \cite{GPPS:93}.
Recently it
has been shown that the interaction between the soft oscillators
in harmonic approximation causes unstable vibrations, negative eigenvalues. 
The structural configuration becomes unstable. The
response of the glass is then a local relaxation to a new stable configuration
which always exist due to anharmonicity. The harmonic spectrum in this new stable
configuration exhibits a BP with a universal shape as it is observed in 
experiment. In particular the shape of the BP is independent of the actual
magnitude of the anharmonicity \cite{GPS:03}.  
  
Computer simulations of a soft sphere glass (SSG) \cite{LS:91,SL:91} 
confirmed the existence
of low frequency quasi-localized vibrations with effective masses
ranging from ten atomic masses upwards. 
The modes were
centered at structural irregularities with large local strains \cite{LMNS:00}.
Regions of local strain have also been observed in earlier computer
simulations \cite{chen:88}.  Similar effective masses 
have since been observed in many different 
simulations, e.g. SiO${}_2$ \cite{jin:93}, Se \cite{OS:93} 
in Ni-Zr \cite{hafner:94}, Pd-Si \cite{ballone:95},
Ni-B \cite{ee:96},  
amorphous ice \cite{cho:94} and 
amorphous and quasi-crystalline Al-Zn-Mg \cite{hafner:93}.  In an
earlier simulation of amorphous silicon low frequency localized
vibrations have been observed at coordination defects \cite{biswas:88}. 
It should be noted that computer simulations with the usual periodic
boundary conditions have a lower cutoff, $\omega_{\rm min-sound}$ 
for the sound waves, which is given 
by the maximal wavelength fitting into the periodicity volume. Quasi localized
vibrations with frequencies $\omega < \omega_{\rm min-sound}$ appear, therefore, in
simulations as truly localized vibrations which can be used as an easy way to
identify them. For $\omega > \omega_{\rm min-sound}$ hybridisation with the sound
waves occurs and the local vibrations are delocalized, they become    
quasi-localized. This will lead to a small shift in $Z_{\rm excess}(\omega)$ but
does not affect the number of excess modes. For a more detailed discussion see
\cite{SO:96,SR:03}. For frequencies $\omega > \omega_{\rm min-sound}$ it is still 
possible
to change from the basis of exact harmonic eigenvectors to a basis of sound waves
and local vibrations with a small bilinear interaction \cite{SO:96}. 
The latter basis is the one used in the SPM.

High resolution synchrotron scattering made it possible to observe the dynamic
structure factor, $S(q,\omega)$, in Brillouin scattering 
\cite{benassi:96}. For a recent review see \cite{ruocco:01}. As in crystals,  
At the lowest frequencies glasses have well defined
longitudinal and transverse sound waves. Only the longitudinal one is observed
in Brillouin scattering at low $q$. 
At the lowest frequencies the inelastic scattering function or the dynamic structure
factor shows, as function of frequency for a given $q$, only one peak 
which is ascribed to the longitudinal sound wave. At higher frequencies a second peak, 
ascribed to the transverse sound is observed \cite{scopigno:03}. This can be
understood considering that
in a disordered system a transverse
sound wave acquires a longitudinal component and vice versa. 
Fitting a damped harmonic oscillator to $S(q,\omega)$ in 
amorphous silica \cite{pilla:00} and
glycerol \cite{masciovecchio:00}, one group concluded in that propagating collective
excitations exist in glasses at high $q$-values, i. e. frequencies far above the 
BP. From measurements of densified silica another group \cite{foret:02,courtens:03}     
reach the opposite conclusion that the BP marks the end of the propagating acoustic 
modes.  


This question can at present  not be answered unambiguously from experiment.
Additional insight can be gained by computer simulation.
Molecular dynamics simulations of amorphous silica gave structure factors in good
agreement with experiment. A fit with a damped harmonic oscillator function
gave a damping of the longitudinal mode $\gamma(q) \propto q^2$ for frequencies
above the BP \cite{dellanna:98,taraskin:00}.

The aim of the present paper is to use a simple model glass, representative for
dense packed metallic glasses, to see what information can be gained from
the dynamic structure both in the Brillouin regime and for higher $q$-values.
For this we extend our previous study of the soft sphere glass \cite{LS:91,SL:91,SO:96}.
This glass has the advantage that there is no parent crystal with optical modes
and no torsional motion as one has in silica which is built from more or less rigid 
SiO$_4$-tetrahedra.
In the earlier studies \cite{SO:96} we found 
in a normal mode analysis that the Ioffe-Regel limit
is reached for the transverse acoustic modes near the BP. For the lowest
frequency in the calculation
the width of the phonon
line could be split into two approximately equal parts attributed to static
disorder and to resonance scattering with QLV.


\section{Computational details}

The soft sphere glass (SSG) is described by an inverse sixth-power potential
\begin{equation}
u(r) = \epsilon \left ( \frac{\sigma}{r} \right )^{6}
     + A \left( \frac{r}{\sigma} \right )^4 + B \; .
\label{eq_pot}
\end{equation}
To simplify the computer simulation and normal-mode analysis, the
potential is cut off at $r/\sigma = 3.0$, and then shifted by
a polynomial, $A(r/\sigma)^4 + B$, where $A = 2.54\cdot{10}^{-5}\epsilon$
and $B=-3.43\cdot{10}^{-3}\epsilon$
were chosen so that
the potential and the force are zero at the cutoff. This form of the
shifting function was chosen so that its effect is negligible near
$r/\sigma = 1.0$. Quantities such as the pressure and average
potential energy will be changed by a few percent as a result of this
truncation, but any changes in the equilibrium structure will be small.
Without loss of
generality one can set $\epsilon = \sigma = m = 1$. Where we do not
explicitly state the units we infer these ``system units''. Note that
this soft sphere potential is much softer than the often used one with
$u_{12} \propto (\sigma / r)^12$.

The calculations were done using the samples prepared earlier 
for investigations of vibrations \cite{SO:96} and relaxations \cite{OS:99}.
These had been prepared by the following procedure.
Liquid configurations of 5488 soft spheres  were produced
via constant-energy molecular dynamics (MD)
simulation with cubic periodic
boundary conditions at a density, $\rho \sigma^{3} = 1.0$,
and temperature, $kT/\epsilon \approx 0.54$
(about 2.5 times the melting temperature at this
density\cite{hoover:72}). For the simulation, we used the velocity-Verlet
algorithm with a time step of
0.04 - in units of $(m\sigma^2/\epsilon)^{1/2}$.
The liquid was first quenched within the
MD simulation by velocity rescaling to a reduced temperature of about
$0.005 T_g$. The quench rate was about
0.015$k/(m \sigma^2 \epsilon)^{1/2}$.
After the MD quench, each sample was 
heated to $0.05 T_g$ and aged for several 1000 
further MD time steps
to stabilize the potential energy and to avoid spurious minima. 
Each system was finally quenched to zero temperature using a
combination the steepest-descent/conjugate-gradient algorithm 
of our version of the Harwell code DEVIL, the metals version of the HADES code.
To improve statistics 11
different configurations of 5488 atoms,
created in this way, were analyzed.

\section{vibrations}
The vibrational dynamics was calculated in classical harmonic approximation
from the force constant matrix of the $T=0$
minimum configuration. The numerically exact minimization of the potential
energy prevents the occurrence of spurious unstable modes. The elements of
the force constant matrix are given by
\begin{equation}
D^{mn}_{\alpha\beta} = \frac{\partial^2 u(|{\bf R}^m - {\bf R}^n|)}
{\partial R^m_\alpha \partial R^n_\beta}, \hspace{1cm} m \ne n.
\end{equation}
Different from our previous work \cite{SO:96} all eigenvalues, $(\omega^\sigma)^2$, 
and eigenvectors, ${\bf e}^n(\sigma)$, 
were calculated by direct diagonalisation. 

To localize the BP we calculate 
the vibrational density of states from the
frequencies of the $3N-3$ vibrational modes $\sigma$ as
\begin{equation}
Z(\omega) = \left< \frac{1}{3N-3} \sum_\sigma \delta (\omega -\omega^\sigma)
         \right>
\end{equation}
where $\delta $ is the discretized $\delta$ function
and $\left< \right>$ stands for the averaging over configurations.
Within the numerical accuracy the resulting spectrum is unchanged from Fig.~1 of 
\cite{SO:96}. The maximal frequency is $\omega_{\rm max} \approx 12.6$. 
There is a slight
dip around $\omega = 9.5$ which is characteristic for the chosen soft sphere glass.
It reflects the factor three between the longitudinal and transverse sound
velocities. It disappears for harder models, i. e. a larger exponent of the
$ \sigma / r$-term in Eq.~\ref{eq_pot}.  

Fig.~\ref{fig_BP} (solid line) shows the calculated
$Z(\omega)/\omega^2$, corresponding to the BP. From the size
of our simulation cell we calculate from  the elastic constants 
$\omega_{\rm min-sound} =$ 0.52 and 1.76 for the longitudinal and transverse sound,
respectively. The simulated density of states has no sound waves below these
cutoffs.
To compensate for this we added for the lower frequencies a Debye contribution 
to $Z(\omega)$, 
calculated from the elastic constants. 
The dashed line shows the theoretical estimate for the model of interacting soft
oscillators and sound waves \cite{GPS:03} 
\begin{equation}
Z_{\rm theor}(\omega)  =  Z_{\rm D}(\omega) + Z_{\rm excess}(\omega)
\end{equation}
where $Z_{\rm D}(\omega)$ is the Debye spectrum and $Z_{\rm excess}(\omega)$ the
excess causing the BP. There is very good agreement between the two curves up to
$\omega = 2\omega_{\rm BP}$. The
deviation at higher frequencies has to be expected. We arrive, in agreement with
the earlier work,  at an estimate
$\omega_{\rm BP} = 0.5$ which is near to $\omega_{\rm min-sound}$ of the 
transverse branch and well below the corresponding value of the longitudinal branch.
The considered soft sphere glass, therefore, represents an interesting example of a 
material where one has a low lying pronounced BP and a strong separation of the
two branches of sound waves. This distinguishes it from the Lennard-Jones glass.

\begin{figure}
\includegraphics[bb=40 110 470 470,totalheight=6cm,keepaspectratio]{plots/govom2.eps}
\caption{Vibrational spectrum divided by $\omega^2$ versus $\omega$. The solid line 
gives the simulation results, corrected by a Debye contribution at the lowest
frequencies. The dashed line is a fit with the model of Ref.~\cite{GPS:03}.} 
\label{fig_BP}
\end{figure}

In classical harmonic approximation the dynamic structure factor can be written 
as
\begin{equation}
S(q,\omega) = \frac{1}{N\omega}\left<\sum_\sigma \left| \sum_n 
 ({\bf q} \cdot {\bf e}^n(\sigma)) \exp(i{\bf q} \cdot {\bf R}^n)\right|^2 
\delta (\omega -\omega^\sigma) \right>.
\label{eq_dynS}
\end{equation}
 where $\delta $ is the discretized $\delta$ function
and $\left< \right>$ stands for the averaging over configurations and directions of
${\bf q}$. Here we omitted the temperature dependent factor $kT/\hbar\omega$, the
high temperature limit of the phonon occupation number.  The inelastic scattering 
intensity is given by
\begin{equation}
I(q,\omega) \propto \frac{kT}{\omega}S(q,\omega).
\label{eq_intens}
\end{equation}
Often the structure factor is defined including the temperature factor, it is then 
directly
proportional to the inelastic scattering intensity. 
In our previous paper \cite{SO:96} we used a definition excluding the 
factor $1/\omega$ in Eq.~\ref{eq_dynS}:
\begin{equation}
S_{\rm FT}(q,\omega) = \omega S(q,\omega).
\label{eq_SFT}
\end{equation} 
This corresponds to a projection of the 
Fourier transform of the eigenvectors. 
For broad distributions in 
$\omega$ the different definitions
will lead to shifts in the maximum and with.

In analogy with the usual structure factor which projects out the longitudinal
components of the vibrations one can define a transverse one as
\begin{equation}
S_{\rm tr}(q,\omega) = \frac{1}{N\omega}\left<\sum_\sigma \left| \sum_n 
 ({\bf q} \times {\bf e}^n(\sigma)) \exp(i{\bf q} \cdot {\bf R}^n)\right|^2 
\delta (\omega -\omega^\sigma) \right>.
\label{eq_dynS_tr}
\end{equation}
It gives, for small $q$, access to the transverse vibrations which are not 
observable in the scattering experiments. We study the structure
factors both in the Brillouin regime, i.e. below the first peak of the
static structure factor (in our system at $q_{\rm FP} = 7\sigma^{-1}$ see 
Fig.~\ref{fig_const_om}), and above for 
large $q$ values. 

\subsection{Brillouin regime}

As done by other authors we do constant-$q$ scans of the dynamic
structure factor and define the maxima in $\omega$ as phonon frequencies 
and derive this way a pseudo dispersion
curve. This is shown in Fig.~\ref{fig_ph} for both the longitudinal and 
transverse branches 
determined from Eqs.~\ref{eq_dynS} and \ref{eq_dynS_tr}, respectively.
The error bars indicate the full width half maximum.
\begin{figure}
\includegraphics[bb= 80 150 510 590,totalheight=10cm,keepaspectratio]{plots/omega_q_bw.eps}
\caption{Pseudo dispersion curves in the soft sphere glass (longitudinal:top 
transverse:bottom) calculated from the structure factor without the phonon occupation
factor. The error bars indicate the full width at half height. The dashed line gives 
the linear dispersion given by the sound velocities, calculated from the elastic
constants of the glass.} 
\label{fig_ph}
\end{figure}
 
One sees a very different behaviour of the two branches. Whereas the longitudinal one
is well defined well beyond the pseudo zone boundary 
$q_{\rm ZB} = 0.5 \cdot q_{\rm FP}$,
the transverse one becomes rapidly over-damped. The different broadenings of the 
two branches 
indicate a large difference in their coupling to the BP-modes. This is seen even
more clearly when we define the phonons from the scattering intensity, i.e.
include the factor $kT/\omega$. The longitudinal branch is
only slightly affected whereas the transverse one broadens even more and above
$q = 1.8 \sigma^{-1}$ the maximal intensity is no longer at some ill defined
phonon line but drops to the BP frequency. This indicates a transverse nature of the
BP. It should be noted that the frequencies of both branches at the 
pseudo zone boundary are 
much higher than the BP-frequency $\omega_{\rm BP}$. This certainly does not justify
an explanation of the BP in terms of soft zone edge phonons. 
For low $q$, both branches show a linear
dispersion according to $\omega = c q$. The transverse branch shows above
$q=1,5 \sigma^{-1}$ an apparent slightly positive dispersion as has been seen for the
longitudinal branch in SiO$_2$ \cite{dellanna:98}.

\begin{figure}
\includegraphics[bb= 80 150 510 590,totalheight=10cm,keepaspectratio]{plots/width_bw.eps}
\caption{Halfwidth-halfmaximum divided by $\omega$ versus $omega$, calculated from the 
dynamic structure factor with different $\omega$-weighting. Open and full symbols 
longitudinal and transverse branches, respectively; squares: weighting according
Eq.~\protect{\ref{eq_dynS}}, circles: Eq.~\protect{\ref{eq_intens}}, triangles:
Eq.~\protect{\ref{eq_SFT}}. The dashed line indicates the Ioffe-Regel limit 
$\Gamma(q) = \omega(q) / 2\pi$.} 
\label{fig_width}
\end{figure}

Fig.~\ref{fig_width} gives the broadening of the pseudo dispersion for different
weightings with $1/\omega$, corresponding to the definitions via the scattering
intensity (Eq.~\ref{eq_intens}), the correlation function (Eq.~\ref{eq_dynS}) or
a Fourier transform of the eigenstates (Eq.~\ref{eq_SFT}). As to be expected the
definition does not change the broadening at the low $q$-values but is 
essential above the Ioffe-Regel limit when $\Gamma(\omega) > \omega/2\pi$. 
The longitudinal branch goes through $\omega_{\rm IR}$ at $q = 4 \sigma^{-1}$ 
beyond which value it rises rapidly as $q \to q_{\rm FP}$. The transverse branch
goes through $\omega_{\rm IR}$ already at $q = 0.5 \sigma^{-1}$. There are
three apparent regions in $\Gamma(q)$, best seen in the values derived 
$S_{\rm FT}(q,\omega)$. 
At the lowest $q$ the width, $\Gamma(q)$,
increases with a power of $q^2$ or larger, followed by 
$\Gamma(q) \propto q \propto \omega(q)$ and finally there is an upturn for 
$q \to q_{\rm FP}$. 

The soft potential model predicts $\Gamma(q) \propto q^4$   
for $\omega(q) <  \omega_{\rm BP}$ which is derived from the initial increase of
the excess vibrational DOS $Z_{\rm excess}(\omega) \propto \omega^4$ \cite{BGGPRS:92}. 
Unfortunately,
the low value of $\omega_{\rm BP}$ in the investigated soft sphere glass
excludes this regime from our simulation. The
simulation cell would have to be large enough not only to allow small $q$-values
it would also have to accommodate several mean free paths of these phonons. 
Otherwise the phonons would be scattered by a periodic repetition of equal 
defects instead of a distribution of different defects. For $q < 1\sigma^{-1}$ 
we observe
for the transverse branch an increase $\Gamma(q) \propto q^2 $ as has been
observed earlier for strongly frustrated systems \cite{angelani:00}.
The section where  
$\Gamma(q) \propto \omega{q}$ is broadly compatible with the
increase of the total $Z(\omega) \propto \omega $ for $\omega > \omega_{\rm BP}$.
Above the BP the transverse modes are strongly mixed with the quasi-localized
vibrations and amongst each other. Therefore all transverse modes can be considered as
scattering centers.      

Our results for the pseudo dispersion curves agree with the
earlier calculation by Caprion {\it et al.} \cite{caprion:98}. There is,
however, a marked difference in the calculated widths. This might be due both
to the better resolution in the present work and also to the different procedure
used to calculate $\Gamma(\omega)$.
We define the width directly from the width at half maximum whereas Caprion
uses a Lorentzian fit. It was noted earlier that the peaks in $S(q,\omega)$ are 
asymmetric \cite{SO:96}. This makes a fit dependent on details of the adopted
fitting procedure.

\subsection{Structure factor at high $q$-values}

The dynamic structure factor in the Brillouin regime provides only rather limited 
information on the structure of the vibrational modes, even if one includes the
experimentally hardly measurable transverse branch. Additional information can
be gained if one includes $q$-values beyond the first diffraction peak 
where $S(q,\omega)$ can be measured
by inelastic neutron scattering. As seen from Fig.~\ref{fig_dynS_high}
the dynamic structure factor exhibits even for this simple material
a rich structure. For the sake of clarity
we plotted $S_{\rm FT}(q,\omega)/q^2Z(\omega)$. Taking constant $\omega$ cuts
of $S(q,\omega)$
one gets distinct patterns for the average of the modes with that frequency. 

\begin{figure}
\includegraphics[bb= 140 90 550 450,totalheight=10cm,keepaspectratio]{plots/sofqnu_3d.eps}
\caption{Dynamic structure factor $S_{FT}(q,\omega)/q^2Z(\omega)$. For $\omega = 0$
the static structure factor was taken.} 
\label{fig_dynS_high}
\end{figure}

\begin{figure}
\includegraphics[bb= 60 50 530 480,totalheight=10cm,keepaspectratio]{plots/Sofq_boson_peak_a.eps}
\caption{Constant frequency scans of the dynamic structure factor. Solid line: static
structure factor, a: dashed line $\omega = 0.6 \approx \omega_{\rm BP}$, 
long dashed line $\omega = 3.8$, dotted line
$\omega = 8$, dash dotted line $\omega = 12$; b: $\omega = 0.6 \approx \omega_{\rm BP}$,
dashed line averaged as in a, dotted line: extended mode, dash dotted line: local
mode.} 
\label{fig_const_om}
\end{figure}

In Fig.~\ref{fig_const_om}a we show four such constant omega scans: for 
$\omega = \omega_{\rm BP}$, for the top of the apparent acoustic phonon in 
Fig.~\ref{fig_ph}, for $\omega$ near the Einstein average and for localized
vibrations. 
For comparison the static structure factor, $S_{\rm stat}(q)$, is also shown. 
Comparing the low frequency and high frequency 
vibrations one
observes a clear shift of phase with respect to the static structure factor.
The low frequency vibrations are in phase but the localized ones are in 
anti-phase to $S_{\rm stat}(q)$. 
In simple densely packed materials, e.g. metals, the nearest neighbour coupling
dominates. Localized vibrations are vibrations of neighbouring atoms 
against each other and this is seen as phase shift against $S_{\rm stat}(q)$.
Low frequency vibrations on the other hand must not involve significant changes
of the nearest neighbour distance and therefore show no phase shift. The 
intermediate frequency modes ($\omega =$3.8 and 8) show a mixed behaviour. 
The lower of the two frequencies corresponds to the high frequency edge of the
transverse mode. In the constant-$omega$ scan one sees a narrow line at low $q$
from the longitudinal phonon, a flat maximum around $q_{\rm FP}$ and a weak
modulation, clearly distinct from $S_{\rm stat}(q)$ at larger $q$. For  the other 
scan with $\omega =$8,
which is above the apparent maximal transverse phonon frequencies, one sees again 
the narrow
peak of the longitudinal phonon, now at a higher $q$-value, and additionally 
a broader peak marking the
descent of the pseudo-dispersion for $q \to q{\rm FP}$. The oscillations for
larger $q$ are out of phase with $S_{\rm stat}(q)$ and damped compared to the
curve for the localized vibrations. Clearly the character of the modes with
$\omega \approx \omega_{\rm BP}$ is distinct from the one of the zone boundary
modes. Using our previous results, we take a closer look at the modes responsible 
for the BP. 

In Ref.~\cite{SO:96} we deconvoluted the exact low frequency eigenvectors 
of the present system into
extended sound wave like modes and local modes with a weak bilinear interaction.  
This was done by rotating the basis of modes in a narrow energy range. This way
we extracted optimally localized vibrational modes and automatically gained the
proper number of extended phonons. The total number of modes is preserved and
the number of modes in a given frequency interval is only marginally
altered. The resulting extended and localized modes, are no longer eigenvectors
of the dynamic matrix but have a weak bilinear interaction. This is the picture
used in the soft potential model.  
In Fig.~\ref{fig_const_om}b we compare $S_{\rm stat}(q)$ with
constant-$\omega$ scans at $\omega \approx \omega_{\rm BP}$ for the 
the true harmonic eigenmodes and the deconvoluted extended
and local parts. It is not surprising that the curves of the deconvoluted extended modes
follows $S_{\rm stat}(q)$ even more closely than the one of the eigenmodes. 
This property of sound was was suggested 
by Buchenau {\it et al.} \cite{buchenau:96} as a distinguishing feature between them
and local vibrations. This difference is clearly not very pronounced in 
the present case. For the deconvoluted local low frequency modes the first peak is 
certainly reduced but still
quite pronounced. For larger $q$ the scans of the extended and local modes are 
hardly distinguishable. This somewhat surprising result is a consequence of the
strongly collective nature of the local vibrations, which form the cores of the
quasi-localized vibrations seen in $Z_{\rm excess}$. These modes are collective
motions of chains of 10 and more atoms \cite{LS:91,SOL:93,SO:96}. Such modes are
typical fore densely packed metallic glasses. 

\section{Relaxations}

Besides vibration one observes in a glass aperiodic jumps of groups of atoms,
local relaxations. Such jumps give e. g. rise to the telegraph noise in point
contacts and are thought to be the elementary process in diffusion \cite{RMP}.
In our previous study of relaxations in the soft sphere glass we found them
to be closely correlated to the quasi-localized vibrations. They are
again collective motions of chains of atoms \cite{SOL:93,OS:99}. At low temperatures
any given atom moves in such a jump only a fraction of the nearest neighbour
distance. 
One can observe these collective jumps in a molecular dynamics simulation by 
monitoring the total displacement of all atoms as function of time
\begin{equation}
\Delta R(t) = \sqrt{ \sum_n \left({\bf R}^n(t)-{\bf R}^n(0)\right)^2}
\label{delta_r}
\end{equation}
where ${\bf R}^n(t)$ is the position vector of particle, $n$, at time, $t$, and
${\bf R}^n(0)$ is the one at the starting or reference configuration. 
$\Delta R(t)$ oscillates due to the vibrations and changes
due to relaxations, i.e. due to the transitions from one local energy
minimum to  another. At low temperatures $T \approx 0.05 T_{\rm g}$,
the maximal distance
an individual atom travels in these jumps is only $0.3 \sigma$, about
a quarter of  $R_{NN}$. 
Increasing the temperature by a factor of four the jumps seen at the 
lower temperature can no longer
be resolved and new jumps over larger distances are observed.

\begin{figure}
\includegraphics[bb= 50 100 530 410,totalheight=6cm,keepaspectratio]{plots/SofQ_rel.eps}
\caption{Structure factor of local relaxations at temperatures up to $0.15 T_{\rm g}$.
Full line: static structure factor, dashed line: average over relaxations with
$\Delta R < 5\sigma$, dash dotted line: average over relaxations with
$\Delta R < 10\sigma$.} 
\label{fig_relS}
\end{figure}

All observed relaxations were collective jumps localized to 10 and more atoms
forming twisted chain-like structures with some side branching \cite{OS:99}.
This collectivity is again reflected in the relaxational structure factor, defined in
analogy with the vibrational (Eq.~\ref{eq_SFT}) one
\begin{equation}
S_{\rm rel}(Q) = \frac{1}{Q^2} \left\langle \left| 
             \sum_n {\bf Q}\Delta {\bf R}^n \exp(i{\bf Q}{\bf R}^n) 
                               \right|^2 \right\rangle
\label{eq_rel}
\end{equation}
where $\Delta {\bf R}^n$ is the jump vector of atom $n$ and $\langle \dots \rangle$ 
denotes
averaging over angle and relaxations. The relaxational structure factor
is similar to the corresponding curve for the low frequency vibrations. It
follows its static counterpart closely, independent of the jump lengths.
The smoothing out of the higher peaks is more pronounced for the longer
jumps occurring predominantly at higher temperatures. Comparing 
Figs.~\ref{fig_const_om}a and \ref{fig_relS} one sees the close correlation 
between quasi-localized vibrations and atomic jumps expressed which we
established earlier by correlating vibrational eigenvectors an jump vectors
\cite{OS:99}.

\section{Conclusions}

The vibrational spectrum of the $1/r^6$-soft-sphere glass shows a pronounced
boson peak. The boson peak frequency is very low at about 5\% of
the maximal frequency. The shape of the boson peak is in accordance with the
one obtained from a model of interacting soft oscillators and sound waves.
Calculating pseudo dispersion curves one finds a different behaviour for
longitudinal and transverse phonons. Whereas the longitudinal ones are well
developed in the whole of the first pseudo Brillouin zone the transverse 
ones become over-damped near the boson peak frequency. Perhaps not surprisingly,
but often forgotten, this shows that depending on the underlying atomic structure
the two branches of sound waves vary in their behaviour as function of $q$. 
In the studied model glass, the boson peak vibrations have predominantly
transverse character.  
 
From the structure factor at higher $q$ we find that the boson peak vibrations
are distinct from the zone boundary modes. They closely follow the structure
of the static structure factor. Splitting the exact eigenmodes into extended
phonons and localized modes one finds that both components again follow the
large $q$ part of the static structure factor. The larger extent of the phonons
is seen as a larger amplitude at the first peak. This semblance of the structure
factors can be understood from the collectivity of the localized vibrations,
the cores of the quasi-localized vibration. As we found earlier these modes are
collective vibrations of chains of ten and more atoms. These chains move in such
way that the nearest neighbour distances do not change markedly. 

The structure factors of local relaxations again resembles the static one and is
closely related to the one of the boson peak vibrations. This is seen as a 
consequence of of the similar structure of the low frequency localized vibrations
which produce the boson peak in inelastic scattering and the collective jump
processes which are the elementary step in diffusion and relaxation in metallic
glasses. 



%\eject
%\bibliography{md}
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%\bibliographystyle{prsty}
\end{document}
The elastic constants and vibrational densities of states (DOS) 
of glasses and
amorphous materials are similar to the ones of their crystalline
counterparts. Both classes of materials have at the lowest frequencies
well defined sound waves. At slightly higher frequencies one observes in
glasses additional vibrational excitations. The dynamic structure factor
observed in neutron scattering shows a small maximum near frequencies
of a few meV corresponding to a maximum of the DOS divided by the squared 
frequency, the boson peak, which is also seen in Raman scattering.
The temperature dependence of the inelastic scattering intensities 
shows that these extra vibrations are largely harmonic with a crossover
to anharmonicity at the lowest frequencies. 
The boson peak is reflected in a maximum of the specific heat, normalized
to the Debye value, near 10~K. The 
There is still a controversy about the nature of the modes responsible for
the boson peak.

experiments  
The dynamics of glasses and amorphous materials is characterized by some
distinct features. 
%Whereas the overall shape of the vibrational density
%of states in general resembles the one of the crystalline
%counterparts glasses exhibit a number of interesting low energy excitations.
At the lowest frequencies one finds normal sound waves 
corresponding to the
usual Debye behavior. At higher frequencies
additional vibrational excitations are observed. These lead to the
so called Boson peak in the dynamic scattering function, typically 
around 1 THz. 
Additionally to these vibrational excitations which obey approximately
Bose-statistics at temperatures two level excitations
are observed  below a few Kelvin. \cite{phillips:81,gunterodt:81}
These can be envisaged as quantum mechanical tunneling transitions 
between adjacent glassy configurations of nearly equal energy. They
dominate at the lowest temperatures properties such as specific heat,
sound attenuation and internal friction. A model
description is provided by the standard tunneling model 
\cite{phillips:72,anderson:72}. In this temperature range glasses show a 
quasi-universal behavior.

At higher temperatures experiments indicate the existence of wide
distributions of relaxations 
\cite{scott:70,gilroy:81,hunklinger:86,buchenau:88,keil:93,topp:96}. 
These can be envisaged as thermally activated transitions
over barriers again separating local minima in configurational space.
These barriers will be higher, the configurations further apart and
their energies more different than in the tunneling case. The properties
of glasses vary much more between materials in this temperature regime.
These ``relaxations'' from one local minimum into another
one can lead to diffusion or merely to local rearrangements. This depends
on the interconnecting paths between the local minima in the 
configurational 
space. Taking the grossly oversimplified picture of a multidimensional
mountain range, representing the potential energy as function of 
all atomic
positions, for a given maximal altitude, representing the temperature, 
some
minima will be connected while others cannot be reached. 
A mountaineer, our
point in configurational space, can either be confined to a single 
minimum or
a small number of minima or he can wander off to far away places.
Since jumps over barriers are in general only weakly influenced by
the structure of the second minima diffusional jumps are expected 
to be similar to non-diffusional ones.
%The very low isotope effect in diffusion 
%\cite{faupel:90,heesemann:95} shows that diffusion is by a highly
%cooperative process.

A semi-empirical description of the behavior up to temperatures of
about 20K was given by the soft potential model (SPM). \cite{KKI:83,IKP:87}
A basic assumption of this model is a strong correlation between soft
quasi-local vibrations and local relaxations.
A distribution of soft potentials for the movement
of some reaction coordinate is assumed. Depending on the parameters
these potentials can be either one-well or two-well thus describing either 
soft local vibrations or
depending on the barrier height, tunneling or activated
hopping. The modes described by these effective potentials
coexist with the ``normal'' vibrational modes such as sound waves.
This model encompasses the standard tunneling
model as its low-temperature limit.
It does, however, give no direct insight
into the nature of the modes.
By fitting this model to the
experimental data, one finds effective masses of 20--100 atomic
masses for the entities moving in these effective soft potentials.
\cite{BGGS:91,BGGPRS:92}
The model can be extended to higher temperatures and energies using plausible
assumptions on the distribution of the model parameters. \cite{gil:93}
The validity of the assumtion of a strong correlation between the different
excitations has been questioned recently.\cite{sokolov:97}


Computer simulations of the soft sphere glass (SSG) confirmed the 
existence
of low frequency quasi-localized vibrations with effective masses
ranging from ten atomic masses upwards, Ref.~\onlinecite{LS:91,SL:91. 
The modes were
centered at structural irregularities with large local strains.
Regions of local strain have also been observed in earlier computer
simulations \cite{chen:88}.  Similar effective masses 
have since been observed in many different 
simulations, e.g. SiO${}_2$ \cite{jin:93}, Se \cite{OS:93} 
in Ni-Zr \cite{hafner:94}, Pd-Si \cite{ballone:95},
Ni-B \cite{ee:96},  
amorphous ice \cite{cho:94} and 
amorphous and quasi-crystalline Al-Zn-Mg \cite{hafner:93}.  In an
earlier simulation of amorphous silicon low frequency localized
vibrations have been observed at coordination defects \cite{biswas:88}. 
Strong anharmonic effects in these modes have 
been observed
both in computer simulations and in experiment \cite{fabian:97}.

The concept of localized low frequency modes was disputed by 
Mazzacurati {\it et al.} \cite{Mazzacurati:96}. They find in a computer
simulation that the low frequency modes are composed of a uncorrelated
random component sinusoidal waves. This is, however just what is expected
from the interaction between quasi-localized and extended modes as descibed
above. For the soft sphere glass it has been shown that the exact eigenmodes
can be decomposed into localized vibrations and extended phonons even
well above the boson peak \cite{SO:96}.

The resulting modes can no longer be described
as either quasi-local or as extended phonons similar to crystalline ones.
To show their different behavior they have been called diffusons 
\cite{fabian:96}. The soft local vibrations of the soft potential model 
have to be understood
as idealized modes with this interaction ``switched off''\cite{SO:96}(II).
% These modes are
%well described by the SPM \cite{SO:96} (II).
 
%Truly localized vibrations exist in defect crystals for frequencies larger
%than the maximum frequency of the host, or in band gaps. In 
%amorphous materials
%this maximum frequency is not well defined nevertheless the localization
%of vibrations at high enough frequencies is still observed. At 
%low frequencies
%in the ordered as well as in the amorphous structures sound waves exist.
%Their interaction with the quasi-localized modes causes these to 
%delocalize.
%They are not exponentially localized but only with some power law of 
%distance. This interaction with the sound waves implies also an 
%interaction
%among the QLV. 
%The interaction
%increases with frequency due to the increase in the densities of state
%of both sound waves and QLV. At frequencies around the Boson peak 
%both types
%of vibrations have become so strongly mixed that the resulting modes are
%undistinguishable, Ref.~\onlinecite{SO:96} (II).

Computer simulations of relaxations in amorphous structures are more 
difficult. One approach is to search the potential-energy hyper-surface for
adjacent minima.
Heuer and Silbey \cite{heuer:93} searched systematically in a 
Lennard-Jones type glass for pairs of local
minima of the potential energy. A parameterization in terms of the 
soft potential model again gives a qualitative
agreement with the assumptions of that model. The effective masses of the 
jump-modes from one well into the other ranged from two atomic 
masses upwards.

Monitoring the atomic displacements in molecular dynamics simulations
just above the glass transition in many different materials
single atom jumps over nearest neighbor distances have been observed, 
e.g. in a soft sphere system 
\cite{miyagawa:88} and a two component Lennard-Jones system 
\cite{wahnstrom:91}. In the strongly supercooled regime these jumps
have been found to be strongly cooperative. In metallic systems one
observes typically chains of jumping atoms. 
\cite{sanyal:96,teichler:97,schuler:97,SGO:97,donati:98}
Additionally to these jumps a motion involving shorter distances per atom 
is seen
in  molecular dynamics which is mostly referred to as cage or stray motion.
This motion does, however, contribute to diffusion \cite{GS:98}.  

In the glassy state 
strongly collective jumps were observed in 
molecular dynamics studies at temperatures ranging up to
15\% and 30\% of the glass temperature  for the SSG
\cite{SOL:93} and for Se \cite{OS:95}, respectively. In these jumps
the total jump length, added over all atoms, is of the order of the
atomic distance but, other than for the above jumps, in the liquid the
single atoms jump only a fraction of this distance. The numbers of
atoms involved is similar to the one in the quasi-localized vibrations.

 
The mechanism of diffusion in metallic glasses is still highly controversial,
for reviews see \cite{limoge:90,frank:96,frank:97}. Experiments are
often interpreted in terms of the free volume model \cite{cohen:79}. In
its most simple interpretation one assumes that an atom can jump to an 
adjacent site whenever this opens up due to fluctuations. Essentially this
corresponds to a ``single-atom-jump model''. An alternative approach to
to the concept of a free volume   was given in Ref.~\onlinecite{sietsma:95} 
in terms of
holes which are more spread and where atoms surrounding the holes jump
collectively.  

Related to the free volume concept is the concept of 
diffusion via thermally created vacancies.
Such a mechanism was e.g. postulated
for Hf diffusion in amorphous Ni$_{54}$Zr$_{46}$  where 
from pressure experiments an activation volume
for diffusion of the order of the atomic volume was deducted
\cite{grandjean:97} similar to earlier results for Co diffusion 
\cite{hofler:92}.
The opposite result was found for Co in 
Co$_{81}$Zr$_{19}$ where this mechanism was ruled out due to the vanishing
activation volume \cite{klugkist:98}. No pressure dependence of Co and Fe
diffusion has also been observed in the metal-metalloid glasses 
Co$_{76.7}$Fe$_2$Nb$_{14.3}$B$_7$ and 
Fe$_{40}$Ni$_{40}$B$_{20}$ glasses \cite{faupel:90,rummel:94,ratzke:95}.

Another possibility of a defect mediated diffusion is via an 
interstitial mechanism. This will certainly hold for diffusion of
small atoms such as H \cite{wipf:97b}
but has also been assumed for larger atoms such as
Co \cite{hofler:92}.

The correlation between prefactor and activation
energy for a large number of amorphous metals differs 
qualitatively to the one 
found in crystals
\cite{naundorf:98}. This points to a different diffusion mechanism, i.e.
one inherent to the disordered structure. This view is supported by
the very small isotope effect observed experimentally 
\cite{faupel:90,heesemann:95}. This would be in agreement with the
collective jumps observed in the simulation \cite{SOL:93}.

The situation is complicated by the large aging effects in amorphous
metals. The above results refer to ``fully relaxed'' materials. In
unrelaxed materials a tendency to single atom jumps is observed
\cite{ratzke:92,frank:96}. However, there is strong evidence that in
relaxed metallic glasses at least the structure forming component
diffuses by a collective mechanism intrinsic to the disorder. Our
computer study of a one-atomic metallic glass should shed some light
on the properties of these jumps without the complications introduced
by the experimentally necessary alloying.
  
In this paper we
present a comprehensive report of our investigation of relaxations in
a SSG at low and moderate temperatures. The aim is first to get some
insight into the structural relaxations in a glass as such. To extend
the model description beyond the ``universal'' low temperature regime
the validity of the assumption of correlations between vibrations and
relaxations needs to be tested and the change with temperature studied.
This can be best done with a simple model glass which has other, than
the hard sphere glass, well defined phonons at all temperatures.
Learning about the single relaxations will give insight into the 
diffusion mechanism at these temperatures. A direct observation of diffusion
is of course ruled out by the orders of magnitude higher computer time
needed.

We describe the soft sphere glass
by a purely repulsive $1/R^6$-interaction. This interaction is much softer
than the often taken $1/R^{12}$ law. The exponent is chosen to be near the
phase transition between fcc and bcc crystalline phases. This condition
should give better glass formation than the harder potentials. E.g. it was
found that the number of QLV is strongly enhanced compared to the
harder glasses \cite{SL:91}.
This model glass has been used to study the behavior of instantaneous
modes \cite{bembenek:95} and of the local structure \cite{jund:97}
across the glass transition and of the influence of the quenching rate
on the dynamics \cite{jund:97b}.

The advantage of a monatomic model glass is that the dynamics will not
depend on parameters such as the size difference between the constituents.
One pays for the simplicity by a high instability against crystallization.
This limits the observation time at finite temperatures. In binary
systems local chemical order hinders long range diffusion. This is
already observed for interstitial atoms in metallic crystals. Self
interstitial atoms are highly mobile which is connected with their low
frequency resonant vibrations. These might even be at lower frequencies
for impurity interstitials. The related atomic jump motion is, however,
locally confined. Diffusion necessitates either an exchange with an host
atom or an additional different jump with higher activation energy 
\cite{ERS:86}. The local dynamics is in both cases rather similar
but the effect on diffusion differs grossly. One compensation for the
instability of our one-atomic model glass is that aging effects are
speeded up.
  
The paper is organized as follows: In the next section we describe the
potential and give details of the simulation. In section III we present
some basic definitions, discuss the the time evolution of the glasses 
and show how the relaxations are monitored. The properties of these
relaxations are discussed in section IV. To round these results off
we then discuss the short time dynamics at higher temperatures. Finally
we discuss our results in the context of previous work and experiments.

\section{Computational details}

The soft sphere glass (SSG) is described by an inverse sixth-power potential
 \begin{equation}
u(r) = \epsilon \left ( \frac{\sigma}{r} \right )^{6}
     + A \left( \frac{r}{\sigma} \right )^4 + B \; .
\end{equation}
To simplify the computer simulation the
potential is cut off at $r/\sigma = 3.0$, and then shifted by
a polynomial, $A(r/\sigma)^4 + B$, where the choice of 
$A = 2.54\cdot{10}^{-5}\epsilon$
and $B=-3.43\cdot{10}^{-3}\epsilon$
ensure that
the potential and the force are zero at the cutoff. This form of the
shifting function was chosen so that its effect is negligible near
$r/\sigma = 1.0$. Quantities such as the pressure and average
potential energy will be changed by a few percent as a result of this
truncation, but any changes in the equilibrium structure will be small.
At any rate, our interest is in the existence and characterization
of relaxations in general, and not in
a quantitative description for any specific potential. Without loss of
generality one can set the scale factors for energy, length and mass, 
$\epsilon = \sigma = m = 1$. Where we do not
explicitly state the units we infer these ``system units''.

The inverse sixth-power potential
is a well-studied theoretical model that qualitatively mimics
many of the structural and thermodynamic properties of bcc forming
metals including the existence, in its bcc crystal form, of
very soft shear modes\cite{hoover:71}.
It gives
a high concentration of low-frequency
resonant (quasi-localized) modes in the glass \cite{LS:91,SL:91}.
The glass-transition temperature was estimated from the vanishing of the
diffusion constant as
$kT_{g} \approx 0.085 \epsilon$.\cite{LS:91} 

We studied systems with 500, 1024 and 5488 particles, respectively. 
They were produced
by quenching well equilibrated
liquid configurations produced
via constant-energy molecular dynamics (MD)
simulation with cubic periodic
boundary conditions again at a density, $\rho \sigma^{3} = 1.0$,
and temperature, $kT/\epsilon \approx 0.54$
(about 2.5 times the melting temperature at this
density\cite{hoover:72}). All calculations were done at constant volume
(density). The average nearest neighbor distance is, independent of temperature,
$R_{NN} \approx 1.1 \sigma$.
For the simulation, we used the velocity-Verlet
algorithm\cite{swope:82} with a time step of
0.04 - in units of $(m\sigma^2/\epsilon)^{1/2}$.
The liquid is first quenched within the
MD simulation by velocity rescaling to a reduced temperature of about
$0.005 T_g$. The quench rate was about
0.015$k/(m \sigma^2 / \epsilon)^{1/2}$.
After the MD quench, each sample was 
heated to $0.05 T_g$ and aged for several 1000 
further MD time steps
to stabilize the potential energy and to avoid spurious minima. 
Each system is then quenched to zero temperature using a
combination steepest-descent/conjugate-gradient algorithm \cite{fletcher:64}.
The resulting potential energy equals roughly the one of the fcc
crystal of the same density with about 1.5 \%\ Frenkel defects.
In all 60, 21 and 15
different configurations of 500, 1024 and 5488 atoms, respectively,
were created in this way and analyzed.
Structure factors and radial distribution functions have been shown
previously \cite{SO:96}.
To illustrate the glassy nature of our zero-temperature sample and to
aid comparison of this model glass to real materials
the configurationally averaged structure factor, $S(k)$, for our system
has been shown in (I).

To study the glasses at elevated temperatures the quenched glasses 
of 5488 atoms were heated 
in stages to 5\%, 10\% and 15\% of the glass temperature. At the two higher
temperatures
the glasses were observed for 90000 time-steps, corresponding to about 5000
vibrations of an average frequency. Taking an average frequency 
in a simple metal of a few THz this corresponds to observation times of 
the order of ns. At $T=0.05 T_g$ the glasses were
observed for 81000 time steps. At $T=0.05 T_g$ some of the glasses were observed
for 90000 additional time steps. The smaller samples were observed at four
different temperatures between 5\% and 12.5\% of $T_g$. Due to their greater
instability, see next section, they were only observed for 9000 time steps at
each temperature.
 
\section{Time evolution and relaxations}

\subsection{Definitions}

During the runs the total kinetic
and potential energies as well as the total displacement $\Delta R$ from the
starting configuration were monitored. We define

\begin{equation}
\Delta R(t) = \sqrt{ \sum_n \left({\bf R}^n(t)-{\bf R}^n(0)\right)^2}
\label{delta_r}
\end{equation}
where ${\bf R}^n(t)$ is the position vector of particle $n$ at time $t$ and
${\bf R}^n(0)$ is the one at the starting or reference configuration. 
$\Delta R(t)$ exhibits both oscillations due to the vibrations and changes
due to relaxations, i.e. due to the transition from one local
minimum to another. In order to separate the effect of the
latter ones, the subject of this paper, it is useful to look at the
time averaged displacement
\begin{equation}
\Delta \overline{R(t)} = 
\sqrt{ 
\sum_n \langle\left({\bf R}^n(t)-{\bf R}^n(0)\right)^2\rangle_{\Delta t}}
\label{delta_r_av}
\end{equation}
where $ \langle ....\rangle_{\Delta t}$ indicates averaging over a time
interval $\Delta t$, typically a few hundred time steps. 
By quenching the time averaged positions to $T=0$ the configuration of the
underlying minima can be found, similar to the procedure of 
Ref.~\onlinecite{weber:85}. 
At low temperatures where the sample only
visits a few minima this procedure is unambiguous. At higher temperatures,
let us say $T=0.5 T_g$ it depends strongly on the averaging  and quench
procedure into which minimum the system relaxes. Similar to the above
one can define a $\Delta R^0(t)$ for the minimum configurations
\begin{equation}
\Delta {R^0(t)} = 
\sqrt{ 
\sum_n \left({\bf R}^{0n}(t)-{\bf R}^{0n}(0)\right)^2} =
\sqrt{ \sum_n (\Delta R^{0n})^2}
\label{delta_r_0}
\end{equation}
where ${\bf R}^{0n}(t)$ is now the minimum ($T=0$) position of 
particle $n$.
$\Delta {R^0(t)}$ has steps whenever the system moves from one 
local minimum
to another and is constant otherwise. To show the influence of aging on
the energy we define $\Delta E^0(t)$ in analogy to Eq. (\ref{delta_r_0})
\begin{equation}
\Delta {E^0(t)} = E^0(t) - E^0(0)
\label{delta_e_0}
\end{equation}
where $E^0(t)$ is the total potential energy of the configuration given by
the ${\bf R}^{0n}(t)$.

To classify the relaxations we introduce in analogy to the vibrations
a relaxational participation ratio
\begin{equation}
p_{\Delta R} = \left\{ N \left[ \sum_m | {\bf R}^{(i)m} -  {\bf R}^{(f)m} |^4
\right] /(\Delta R^0)^4 \right\}^{-1}
\label{p_delta_r}
\end{equation}
where $(i)$ and $(f)$ stand for initial and final configuration, respectively
and $\Delta R^0$ is given by Eq.~(\ref{delta_r_0}). For truly local
relaxations this participation ratio scales with the system size $N$ as
$p_{\Delta R} \propto 1/N$. We will see, that similar to the findings
for the quasi-localized vibrations, this proportionality does not hold 
exactly. As alternative to the participation ratio an effective mass
\begin{equation}
m_{\Delta R} = m (\Delta R^0)^2 /|\Delta {\bf R}^{\rm max}|^2
\label{m_eff}
\end{equation}
can be defined to classify localization. Here 
$|\Delta {\bf R}^{\rm max}|$ stands
for the maximal atomic displacement.
The two definitions of Eqs. (\ref{p_delta_r}) and (\ref{m_eff}) cannot be
converted into each other. As a rough guide one can take for our system
and $N=5488$ that $p_{\Delta R} \approx 0.02$ corresponds to $m_{\Delta R}
\approx 20$.

\subsection{Time evolution of the glasses}

Figs.~\ref{fig_e0(t)} and \ref{fig_r0(t)} (full lines)
show the changes of the ensemble averages of the potential energies 
and displacements of the minima configurations, 
Eqs.~(\ref{delta_e_0}) and (\ref{delta_r_0}) with
time. The average was taken over 14 samples of
glasses with $N=5488$. (One sample out of 15 partially crystallized 
at the highest temperature during
the observation time).
The samples were quenched to $T=0$ and then heated to 
$T=0.004$ aged
for a time $\Delta t=3240$, heated to $T=0.008$ aged for 
$\Delta t=3600$ and
once more heated to $T=0.012$ and aged for  $\Delta t=3600$. 
Instead of the
final heating stage four of the glasses were kept at $T=0.008$ for an 
additional $\Delta t=3600$ (dashed line). The dotted lines indicate the
heating stages. The first 9000 time steps ($\Delta t= 360$) have been 
omitted
from the plots in order to avoid effects of only spuriously stable
configurations. One of our configurations showed toward the end of the
last time interval large relaxations accompanied
by a large energy reduction. In order to exclude with certainty a partial 
crystallization, we excluded this configuration from the two Figs.

As expected the potential energy, Fig.~1, drops during aging. After each
heating there is a rapid drop with a subsequent leveling off, at least at
the lower two temperatures. If one takes the formation energy of a Frenkel
defect in the fcc lattice as measure ($E^{\rm FP} = 6.05 \epsilon$) the
total drop in energy corresponds to the annihilation of only about seven 
Frenkel defects. The final configurations show no trace of crystallization.

Also the average displacement of the atoms is very small, less than 
15\% of
the nearest neighbor distance. The largest part of this displacement 
follows
immediately the heating.
Striking is the similarity between the energy
relaxation, Fig. 1, and the spatial relaxation, Fig. 2. At the relatively
low temperatures considered the corresponding curves using the time averaged
quantities, Eq.~\ref{delta_r_av}, are very similar to the ones for the
underlying ground states.

In our previous investigation of the
vibrational dynamics (II) we observed for glasses which had run over the 
first two
time segments in Figs.~\ref{fig_e0(t)} and \ref{fig_r0(t)} only a small
drop in the number of quasi-local soft modes was observed.
 
The tendency to crystallize depends strongly on the size of the sample
used in the MD. Whereas all the samples of $N=5488$ atoms were stable up
to temperatures $T=0.008$, three out of 21 samples with $N=1024$
and 30\% of the ones with $N=500$ started to crystallize within one tenth
of the time.

\subsection{Local relaxations}

In the following we want to study single relaxation processes in the glass.
Different from the usual practise we monitor the total
displacement according to Eq. (\ref{delta_r}). This allows the observation
of collective jumps of large groups of atoms with total jump lengths of
the order of the nearest neighbor distance.
In previous 
studies, e.g. Refs.~\onlinecite{miyagawa:88,wahnstrom:91}, in the 
supercooled region
similar plots for the relaxations of single atoms have been shown. Small
distance collective jumps cannot be discerned from the fluctuations in
these single atom plots.

Fig.~\ref{fig_pir}
shows an example for a glass which had been aged for $\Delta t=5000 $ at 
$kT=0.004 \epsilon$ (corresponding to the dashed line in the third sector
of Figs.~\ref{fig_e0(t)} and \ref{fig_r0(t)}). The glass was heated during 500
time steps to the respective temperature and subsequently observed for 72000
time steps, i.e. $\Delta t \approx 3000$. This way it was assured that the
glass has the same starting configuration at all temperatures. The 
configuration had been in its lowest energy minimum, $A$,
immediately before quenching and the shown displacement are relative to this 
configuration, ${\bf R}^{0n}(t=0) = {\bf R}^{An}(T=0)$ in Eq.~\ref{delta_r_0}.

At $ kT=0.002 
\epsilon$ the glass clearly jumps between three configurations. Let us denote
the configurations $A,B,C$. The jump sequence is 
$B \to C \to B \to C \to A \to C \to A$.
By quenching to $T=0$~K we find the potential 
energy minima shifted  relative to each other by
each other $\Delta E^B - \Delta E^A = 1.74 \cdot {10}^{-3}\epsilon$ and
$\Delta E^C - \Delta E^A = 1.81 \cdot {10}^{-3}\epsilon$. These energy 
differences are of the order of the temperature. The corresponding 
spatial distances between the minimal configurations, Eq.~(\ref{delta_r_0}),
are $\Delta R^{A-B} = 1.63 \sigma$, $\Delta R^{A-C} = 0.96 \sigma$ and
$\Delta R^{B-C} = 1.0 \sigma$, respectively, i.e. they are of the order
of the nearest neighbor distance $R_{NN} = 1.1 \sigma$. The maximal distance
an individual atom travels in these jumps is only $0.3 \sigma$, about
a quarter of  $R_{NN}$.

The average offset $\overline{\Delta R} \approx 1.5 \sigma$ during the 
residence
time in $A$ corresponds to a vibrational mean square displacement (VMSD)
$\langle u^2(T)\rangle_{\rm vib }\approx 8.2 \cdot {10}^{-4} \sigma^2$ per 
atom.
Doubling the temperature the  VMSD doubles raising the mean offset to 
$2.1 \sigma$. The three minima
can still be seen but some of the excursions to them are so short-lived
that they cannot be discerned from anharmonic vibrations. Doubling the
temperature again the signature of transitions between the minima $A, B$ 
and $C$ has more or less vanished in the noise but additional, larger
relaxations appear. The same story repeats raising the temperature again
by 50\%. 

From Fig.~\ref{fig_pir} we have seen that at a given temperature one
observes small local relaxations (rearrangements) where groups of atoms
move collectively. Raising the temperature these relaxations merge into
the vibrational background and new relaxations become observable. 
In the following we will take a closer and more systematic look at the 
properties.

\section{Properties of single relaxations}

\subsection{Procedure}

For a systematic study of the single relaxations the glasses undergo during
their history, Figs.~\ref{fig_e0(t)} and \ref{fig_r0(t)} we have to 
analyze their $\Delta R(t)$ as shown in Fig.~\ref{fig_pir} as example. 
We calculate
average displacements according to Eq. (\ref{delta_r_av}) averaging over
900 time steps which corresponds to somewhat over three periods of the
quasi-localized vibrations. For ${\bf R}^n(0)$ we take the positions
in the starting minimum. If $\overline{\Delta R}$ exceeds 
a value of $0.2\sigma$
the new averaged configuration is considered a possible minimum. 
The possible candidates for new configurations are then quenched to
$T=0$ to see whether they really belong to a local minimum of the potential
energy. Configurations which thus relax to their predecessor are discarded.
Using this procedure the three different minima are obviously recorded at
$T = 0.002 \epsilon /k$ but not for the shortest excursions at 
$T = 0.012 \epsilon /k$. At $T = 0.008 \epsilon /k$ the small 
maximum around $t=140$ 
is caused by an underlying local minimum of the potential energy. The large 
jump around $t=550$, on 
the other hand, is not resolved into smaller sub-jumps,
the possible intermediate steps seen as tiny spikes
are neglected.

For the smaller sizes ($N=500$ and $N=1024$) the relaxations
were taken at temperatures ranging between 0.05 T${}_g$ and
0.125 T${}_g$, whereas
for $N=5488$ the shown relaxations were observed during the aging
shown in Figs.~\ref{fig_e0(t)} and \ref{fig_r0(t)}.

\subsection{Structure of relaxations}

We observed on average 
in each of the 15 samples ($N=5488$)
6, 15 and 15 relaxations at the temperatures
$T/T_g = 0.05, 0.1$ and $0.15$ respectively. 
In Fig.~\ref{fig_p_delta_r} the participation ratios of the observed jumps
are plotted against the jump length, independent of the observation 
temperature. Shown are
the results for the three system sizes. The lines are
drawn as guides to the eye to show the approximate
scaling of the participation ratio with the system size. We discuss the
apparent deviation for the larger jump distances further down.

Looking, particularly, at the larger jump distances one sees that
the scaling of the participation ratio with system size does not
strictly hold. This resembles the similar behavior observed for the 
quasi-localized vibrations (II). With increasing system size
the probability to find more than one relaxation center in the system
increases. Under suitable conditions the relaxations will interact
which in turn increases their participation ratio. Since the interaction
strength increases with the total displacement long distance jumps
will interact more strongly than short distance ones. Depending on
the energetics different scenarios can evolve.
Interaction can
lead to a growth of single relaxations with temperature if two relaxations
attach to each other and the activation energy of one of the two
constituents is lower than both the ones of the partner and the pair.
This situation was illustrated in Ref.~\cite{SOL:93}. Another possibility
is that the pair is most favored. This is the case for the low temperature
relaxations of Fig.~\ref{fig_pir}. There, configuration $B$ is obtained 
from $A$ by flipping a pair of relaxators whereas only one of the two
relaxators is flipped in the transition $A \to C$. An isolated flipping
of the second relaxator needs higher energies. It causes the little
blip in $\Delta R$ for $kT=0.008 \epsilon$ at $t\approx 150$. 
Different from our work on the vibrations we have 
not tried to systematically separate the observed relaxations into
their local constituents. 

Taking only the jumps
with energy gain less than $k T$,
the average jump length increases in this
temperature range from $\Delta R \approx R_{NN} \approx 1.1 \sigma$ to
$\Delta R \approx 1.5 R_{NN}$.
The maximum distance a single atom moves in these jumps
increases from about
$0.2R_{NN}$ to $0.3R_{NN}$.
The average participation ratio grows
from approximately $0.015$ to $0.021$. These numbers correspond
to localizations to about 18 to 22 atoms.
The increase in the total jump length $\Delta R$ with temperature
is a combined effect
of longer jumps of the single atoms and an increase in the number of
atoms participating.
The strongest localization we observed is
to about 6 atoms, which is still more than the typical value found by
Heuer and Silbey \cite{heuer:93}.
We cannot say for certain whether this is due to the
different interaction model or to the finite temperature in our
simulation. If we do not restrict the averaging to small energy gains
the average values for $\Delta R$ and $p_{\Delta R}$ approximately double. 
This
might be due to a larger number of multiple jumps thus included.

As observed already earlier \cite{SOL:93} the relaxations are collective 
motions
along closely packed directions which gives them chain-like structures.
This structure is more pronounced at the higher temperatures. As an
illustration Fig.~\ref{fig_c21m2} shows the most active atoms of a relaxation
at $T\approx 0.1 T_g$. The total jump length is $\Delta R = 1.98 \sigma$
the participation ratio $p_{\Delta R} = 0.018$ and the effective mass
$m_{\Delta R} = 23$. Shown are all atoms whose squared displacement is
more than 30\% of the maximal one. The displacements of the individual
atoms are shown true to scale by the shaded spheres. 

To classify the structure more quantitatively we 
define, again following (II), a dimensionality of the relaxations and
calculate for each relaxation $j$ the tensor
\begin{equation}
G_{\alpha\beta}(j ) = 
\frac{\sum_n \left| {\Delta {\bf R}}^n(j ) \right|^\mu
                           (R^n_\alpha - R^{\rm {CM}}_\alpha )
                           (R^n_\beta - R^{\rm {CM}}_\beta )}
         {\sum_n \left| {\Delta {\bf R}}^n(j )\right|^\mu} .
\label{g_tensor}
\end{equation}
where we take the exponents $\mu = 2$ and $\mu = 4$,
corresponding to the effective mass and participation ratio, respectively.
${\bf R}^{\rm CM}$ is the corresponding center of mass coordinate
of the relaxation:
\begin{equation}
{\bf R}^{\rm CM} = \frac{\sum_n \left| {\Delta {\bf R}}^n(j ) 
\right|^\mu {\bf R}^n}
        {\sum_n \left| {\Delta {\bf R}}^n(j ) \right|^\mu }.
\label{center_of_m}
\end{equation}
Diagonalizing $G$ we obtain three eigenvalues $\rho^i(j ,\mu)$ 
and from these an  
average  gyration radius \cite{jin:93,yonezawa:80}
\begin{equation}
R_{\rm{gyr}}(j ,\mu ) = 
\sqrt{\frac{1}{3} \sum_i \rho^i(j ,\mu )}.
\label{gyration}
\end{equation}
If a relaxation is localized on a single atom $R_{\rm{gyr}}= 0$, 
and for an extended
relaxation  it is the root-mean square distance with the weight 
determined by $\mu $.
An effective dimension of the relaxation can be defined as
\begin{equation}
d(j ,\mu) = \sum_i \rho^i(j ,\mu )  \ / \ 
\max{\rho^i(j ,\mu )}
\label{dimension}
\end{equation}
Averaging over all relaxations in the systems with $N=5488$ we get weighting
with the 
effective mass a gyration radius $\overline{R}_{\rm gyr}(2) = 5.8~\sigma$ and
a dimension 
$\overline{d}(2) = 2.1$.
Taking the participation ratio as weight the values are slightly
reduced to
$\overline{R}_{\rm gyr}(4) = 4.1~\sigma$ and 
$\overline{d}(4) = 1.6$. These values are rather similar to the ones
found for the quasi-localized vibrations in the same system 
\cite{SO:96}. If we take the average only over relaxations with
$\Delta R < 2~\sigma$ the gyration radii are somewhat reduced to
$\overline{R}_{\rm gyr}(2) = 4.4~\sigma$ and  
$\overline{R}_{\rm gyr}(4) = 2.2~\sigma$ whereas the dimension stay nearly
the same. The effective mass related values are always higher than 
the ones related to the participation ratio reflecting the different
weight of the long range displacements. 

Fig.~\ref{fig_dim} summarizes the results for $\overline{d}(4)$. We find a
considerable spread of values. This is related to bends and side-branching
of the relaxational chain structures. On the other hand,
the dimensionality is nearly
independent of the jump length $\Delta R$, see Fig.~\ref{fig_dim}.
The slight increase with $\Delta R$ could be due to unresolved 
multiple jumps.

\subsection{Correlations}

We have seen that in an individual relaxation chains of atoms jump
collectively. These jumps are not uncorrelated events but
successive jumps tend to involve the same atoms. As a quantitative
measure we introduce
the correlation 
\begin{equation}
c_{\Delta R}[j,j'] = \frac{1}{\Delta R[j]\Delta R[j']}
 \sum_n \Delta R^n[j] \cdot \Delta R^n[j']
\end{equation}
where $j$ and $j'$ denote successive relaxations. Other than with a definition
in terms of the product of the jump vectors $\Delta {\bf R}^n[j]$ 
successive jumps centered on the same atoms contribute also if they are
orthogonal to each other. Direct return jumps give $c_{\Delta R}[j,j'] = 1$.
The correlation between jumps $A\to B$ and $B \to C$, of Fig.~\ref{fig_pir},
is $c_{\Delta R}[A\to B, B \to C] = 0.87$. For completely uncorrelated jumps
one would get values of the order of $1/N$.

Fig.~\ref{fig_corr_delta_r} (bottom) shows the observed distribution of the
correlation values for the $N=5488$ system averaged over all 
relaxations
observed either at the lowest temperature $T \approx 0.05 T_g$ (solid bars)
or in either the whole time range of Fig.~\ref{fig_e0(t)} (dotted bars).
It is evident that all observed jumps are correlated with their
immediate neighbors in time.
We observe two maxima in the distributions, around 0.4 and at 1. 
The latter value
stems from immediate return jumps ($A \to B \to A$). At the
lowest temperature this accounts for 20\% of the observed jumps. Raising
the temperature this value drops rapidly, reflecting the greater freedom
of the material to cross barriers between potential minima. The broad
distribution around $c_{\Delta R}[j,j'] \approx 0.4$ originates from
both, jumps which cannot be reversed due to a too strong drop in potential
energy and reversible jumps involving more than two minima. 

The observed correlation values depend non-trivially on the system size.
This is exemplified by the upper part of Fig.~\ref{fig_corr_delta_r}
where 
the corresponding distribution for the smaller samples with $N=1024$
is shown.
Striking is the shift of the main peak to higher correlations for the
smaller system size. With increasing size more and more only weakly
coupled relaxations in distant regions become possible. In the limit
$N \to \infty$ the correlation between successive jumps will decrease to
zero
since there will always be a multitude of distant jumps intervening. In
fact already with our small samples we frequently observe higher correlations
to later jumps than to the next one.

The correlation between jumps  holds at
the investigated temperatures over considerable times, i.e.
the relaxations remain
centered on the same ``active groups'' of atoms. This is illustrated in
Fig.~\ref{fig_e30m3} which shows the initial positions of all atoms
which have moved by more than $0.35~\sigma$ during a time interval 
$\Delta t = 3600 \ (m\sigma^2/\epsilon)^{1/2}$ at $T\approx 0.15T_g$.
The total displacement is $\Delta R = 8~\sigma$ the participation
ratio $p_{\Delta R} = 0.077$ and the effective mass $m_{\Delta R}=107$
still quite small compared to the total mass 5488. Compared to the
single relaxation in Fig.~\ref{fig_c21m2} this combined one is quite
extended. One can envisage it as being built up of some more chain-like
constituents. More or less independent of the large relaxation there is an
additional strongly localized one.

Another correlation is postulated by the soft potential model
\cite{KKI:83,IKP:87} where local relaxations are
related
to soft local vibrations. We have seen that relaxations interact and
the same holds for the soft vibrations (II). These interactions
are, however, not included in the SPM. In (II) we extracted the
``non-interacting'' soft modes and showed that the SPM is applicable
to these. To do the same procedure for the relaxations seems rather
tedious. Instead we restrict ourselves to the samples with $N=500$
where due to the size and hence the lack of partners this interaction is 
strongly reduced for both vibrations and relaxations.
We calculate the projection of the normalized jump vector onto the
eigenvectors ${\bf e}_{\rm soft}^n$ of the soft vibrations
\begin{equation}
P = \frac{1}{\Delta R} \sum_{\rm soft} \sum_n \left( \Delta {\bf R}^n
  {\bf e}_{\rm soft}^n \right).
\end{equation}
where we use  in each glass between 1 and 3 soft modes as 
determined in (I).
$P$ should vary between 1 for total correlation and $1/N$ if jumps
and soft vibrations are unrelated. Fig.~\ref{fig_corr_e_delta_r}
shows that the truth lies in between. There is a strong correlation
but the value of one assumed in the SPM is only a first approximation. In 
Ref.~\onlinecite{OS:95} we have shown for amorphous Se that $P$ approaches
its limit quite fast if one increases the subspace of soft modes.
We expect the same for our soft sphere glass. The highest values of $P$
are obtained for the most local relaxations which do not lead to a
large energy change.

\section{Higher temperatures}
Due to the tendency to crystallize the SSG can at higher temperatures only
be observed on short time scales. It is however possible to monitor the
glass for short times after a rapid heating to a given temperature.
As throughout the paper the sample volume is kept constant. Compared to
constant pressure conditions this means for the soft sphere potential a
distortion of the temperature axis.
We define the average mean square displacement (msqd) as usual
\begin{equation}
 \langle u^2(\Delta t) \rangle = \langle \; \sum_n 
|{\bf R}^n(t_0+\Delta t) - {\bf R}^n(t_0)|^2 \rangle .
\end{equation} 
$ \langle u^2 \rangle$ is made up from two approximately independent 
contributions from the vibrations and relaxations or jump processes, 
respectively.
\begin{equation}
\langle u^2 \rangle = \langle u^2_{\rm vib} \rangle +  
\langle u^2_{\rm rel} \rangle.
\label{u2_sep}
\end{equation}
In classical harmonic approximation and for averaging times larger than 
the typical vibration period
\begin{equation}
\langle u^2_{\rm vib}\rangle = \langle {\frac{3 k T}{m}
\int d\omega \ \frac{1}{\omega^2}g^n(\omega)} \rangle
\end{equation}
where $g^n(\omega)$ is the local vibrational spectrum of atom $n$.
$ \langle u^2_{\rm vib} \rangle$ increases linearly with temperature and
is independent of the time difference $\Delta t$, 
except for $\Delta t$ values
smaller than the typical vibration period. In the liquid state and
for sufficiently long observation times one has for the second part
\begin{equation}
\langle u^2_{\rm rel} \rangle \sim 6 D(T) \Delta t.
\label{msqd_liq}
\end{equation}

At the highest temperatures the diffusion constant increases linearly with
temperature in accordance with simulations on liquid Rb \cite{balucani:94}.
In the solid and in the undercooled melt the
behavior will be much more complex. We do not reach the asymptotic
time regime and the diffusion constant 
will have an exponential or stretched exponential dependence
on temperature.
With increasing temperature the number
of relaxation processes will increase and at the same time the displacement
in an individual relaxation will also increase. We thus still
expect an increase of
$ \langle u^2_{\rm rel} \rangle$  with temperature as well as with 
averaging
time. The latter will, however, not be linear since during the observation
time some of the relaxations will be reversed which would lead ultimately
to a time independent contribution similar to the vibrations. Counterbalancing
this effect somewhat is the triggering of relaxations by relaxations which
would be observed more likely in the longer times. These effects can be
clearly observed in Fig.~\ref{fig_msqd}. 

We used the glasses
with $N=5488$ after aging to the end of the second temperature
interval $T \approx 0.1 T_g $ of Fig.~\ref{fig_e0(t)} as starting 
configurations.
These were then rapidly heated to the observation temperature and after a
short equilibration lasting 200 time-steps observed during $\Delta t = 10$
(broken line) and $\Delta t = 40$ (solid line),
respectively. If one fixes the maximal frequency around
5~THz, typical for a soft metal, this corresponds to a few ps, 
i.e. observation times of neutron 
scattering experiments. 
At the lowest temperatures, $kT < 0.01\epsilon $ 
the total msqd $\langle u^2 \rangle$ is nearly entirely
vibrational. The extrapolation to higher temperatures is indicated
by the dotted line. 
Above $kT \approx 0.12 \epsilon$ the msqd exhibits liquid behavior.
The intermediate temperature range comprises the glass as well as the 
undercooled liquid which we cannot distinguish from this measurement.
Apart from the lowest temperatures, where the statistics is very poor,
the factor between the relaxational contributions for the two time-intervals
$\Delta t$ varies between 4 and 5. 

To see the effects of the inherent tendency to crystallization we also
calculated the msqd averaging for $\Delta t = 80$
after aging the samples for $\Delta t_{\rm aging} = 800$ at
each temperature. The result is shown by the dash-dotted line in 
Fig.~\ref{fig_msqd}. For $k T < 0.04 \epsilon$ no dramatic effects occur.
At $k T = 0.05$ the msqd drops to the previous value for half the observation
time and finally above $k T = 0.07$ to the vibrational contribution.
This clearly shows crystallization. The little maximum around $k T = 0.1$
is then the trace of some defect annihilation. The sudden increase at
$k T = 0.12$ shows the melting of the samples. At the higher temperatures
the curve follows the behavior of the other two ones.

An important feature characterizing the dynamics in glasses is the deviation
from Gaussian behavior (non-Gaussianity). Following Raman \cite{rahman:64}
we introduce the non-Gaussianity parameter $\alpha_n$ as the normalized ratio
of the moments of the displacements. For the lowest moments this gives
\begin{equation}
\alpha_2(\Delta t) = \frac{3\langle u^4\rangle (\Delta t)}
{5 \left[\langle u^2 \rangle (\Delta t) \right]^2}
   - 1.
\label{non_gauss}
\end{equation}  
The solid and dashed lines in Fig.~\ref{fig_non_gauss} show the temperature
variation of $\alpha_2(\Delta t)$ for two time intervals $\Delta t =60$ and
$\Delta t = 15$, respectively. For the short time interval the value stays
fairly constant around 0.3 before it drops to zero above the melting
temperature. For the longer time interval $\alpha_2(\Delta t)$ grows in
the amorphous state to much higher values around 1, drops to about 0.5 and
then drops to zero faster than the short time value. This reflects the
different weighting of the vibrational and relaxational contributions.
At finite temperatures the latter always dominates for long enough times.
The maximum of $\alpha_2(\Delta t)$ around $kT = 0.03 \epsilon$ indicates
that during $\Delta t$ considerable relaxation have taken place which,
however, involve only a limited number of atoms, as was illustrated in 
Fig.~\ref{fig_e30m3}. 

Analogous to Eq.~(\ref{u2_sep}) the fourth moments can be decomposed into
vibrational and relaxational parts 
\begin{equation}
\langle u^4 \rangle = \langle u^4_{\rm vib} \rangle + 
\langle u^4_{\rm rel} \rangle +
\frac{10}{3}\langle u^2_{\rm vib} \rangle \langle u^2_{\rm rel} \rangle .
\label{u4_sep}
\end{equation}
The total non-Gaussianity can then be separated into a vibrational and a
relaxational part \cite{zorn:97}
\begin{equation}
\alpha_2(\Delta t) = 
\frac{ \alpha_2^{\rm vib}(\Delta t)[ \langle u^2_{\rm vib} \rangle]^2 +
\alpha_2^{\rm rel}(\Delta t)[ \langle u^2_{\rm rel} \rangle]^2 }
{ [\langle u^2_{\rm vib} \rangle  +\langle u^2_{\rm rel} \rangle ]^2 }
\label{nongauss_sep}
\end{equation}
Neglecting anharmonicity we can extrapolate the moments of the vibrational
displacements from low temperatures and using Eqs.~(\ref{u2_sep}), 
(\ref{u4_sep}) and (\ref{nongauss_sep}) calculate the non-Gaussianity of
the relaxations alone. The results for the two values of $ \Delta t $ are
shown by the dash-dotted and dotted lines in Fig.~\ref{fig_non_gauss}.
As to be expected the non-Gaussianity $\alpha_2^{\rm rel}(\Delta t)$ of
the relaxations is larger for the smaller time interval and increases
rapidly upon lowering the temperature below the glass transition.

Due to the short lifetime of the soft sphere glass at finite temperatures
these short time effects cannot directly be observed in real experiments.
Our computer simulation compresses the effects observed in real materials
into short time spans. The qualitative behavior corresponds to the one
observed in real materials. \cite{zorn:97}   

\section{Discussion and conclusion}
In the present paper we have studied relaxations in a monatomic 
soft sphere
glass. Using sufficiently large samples in the simulation the glasses can
be kept at temperatures up to 15\% of the glass transition 
temperature in
a meta-stable state for several thousand vibrational periods. At higher
temperatures the stability is drastically reduced and the glass can only
be observed for short times after rapid heating to the desired 
temperature.
This disadvantage is somewhat compensated by the concomitant rapid aging
which is a notorious problem in the more stable binary glasses. At low
temperatures we can identify a number of local jump processes. These can
be envisaged as collective motions of chain like structures of ten and
more atoms where a single atom moves only a fraction of a nearest neighbor
distance. We observe a number of reversible jumps, i.e. jumps
connecting a limited number of configurations belonging to local energy
minima of similar value. The majority of jumps, especially after heating,
leads to minima of lower energy, corresponding to aging. No qualitative
difference between the jumps was observed. This is in accordance with
the assumptions of the soft potential model and the results of 
Heuer and Silbey using different simulation techniques. 
\cite{heuer:93,heuer:98} Our earlier findings \cite{SOL:93} are
confirmed by the more detailed study.

Increasing the temperature the total jump length as well as the number
of strongly participating atoms grow. A similar increase has also
been observed in simulations of Se. \cite{oligschleger:thesis,SGO:97b}
In the instantaneous mode spectrum it is observed as an increase
of the participation ratio of the modes with imaginary frequency.
\cite{bembenek:95} 

The observed jumps are correlated with the  quasi-local vibrations
discussed earlier \cite{LS:91,SO:96}. The correlation is, however,
weaker than postulated in the soft potential model. At the lowest
temperatures the number of atoms participating in the jumps is clearly
smaller than one would expect from the corresponding vibrations. The
growth of the participation ratio with temperature is clearly outside
the scope of the soft potential model in its present form and the model
should be extended to account for these effects.


In regard to diffusion our results clearly indicate a collective mechanism 
as opposed to a  vacancy one. We find low dimensional chain-like 
structures for the jumping atoms. This is in disagreement to the
assumption of collectivity due to holes \cite{sietsma:95}. Whereas the
total displacement of such a chain is of the order of the atomic distance
the single atoms move only a fraction of this distance. 
We neither observed a qualitative
difference between reversible and irreversible jumps nor did we find any
single atom jumps. 

Collectivity does not only hold for a single jump process but there
is also a marked correlation between subsequent jump events. This 
collectivity seems to hold well into the liquid state 
\cite{teichler:97,schuler:97,SGO:97}. The ``collective regions'' do not
form compact structures whence it is difficult to associate them with
``frozen in unrelaxed regions'' as done e.g. in Ref.~\onlinecite{frank:96}.
Such regions had been neither seen in the vibrational structures studied in
(II).


The quasi-localized low frequency vibrations are centered at atoms an
local environment distinct from the average one. \cite{LS:91}
The shell of nearest
neighbors is compressed but less dense. The deficiency is made up
at distances between the first and second shell. Due to the correlation
this holds also for the local relaxations (jumps). If one wants to
talk of defects this corresponds more to the signature of an interstitialcy
than a vacancy. Speaking of free volume one should keep in mind that excess
volumes due to a  vacancy or a  self-interstitial in a metals are similar.
\cite{ERS:86} The difference is merely in its localization. A reduction
of volume upon aging does not give clear information on the nature of
the annealed ``defects''. We have reported earlier (II) a small reduction
of soft centers upon aging.

The results of the computer simulation agree well with experimental 
results. Low temperature thermal relaxations in glasses
are observed e.g. by internal friction and neutron scattering experiments.
Measuring the isotope effect it has been shown that diffusion
in metallic glasses is via collective motion involving numbers
of particles similar to our results. \cite{faupel:90,heesemann:95}


In the liquid state, upon cooling towards the glass transition,
the distribution of the atomic
mean square displacements deviates 
increasingly from a Gaussian distribution in agreement with earlier 
simulations. \cite{bernu:87} 
The ``rapid
quench values'' for the glassy state show for sufficient long observation
times a further increase upon cooling. This is due a strong
non-Gaussianity of the relaxations. For low temperatures this 
contribution eventually diminishes due to the rare occurrence of
relaxations. The non-Gaussianity drops to the value given by the
vibrations.
   
\acknowledgements
We are grateful for many discussions to U. Buchenau, A. Heuer and R. Zorn.
This work was partially supported by the Deutsche Forschungsgemeinschaft in the
Schwerpunkt ``Unterk\"uhlte Metallschmelzen: Phasenselektion und
Glasbildung''

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{md}
%%%%%%%%%%%%%%%cp%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   
\end{document}
\begin{figure}
%\epsfig{figure=plots/r6gll_history_E.eps,width=15cm,angle=00}
\caption{Change of the configurational part $ \Delta E^0/N$ of the
potential energy per atom during aging and heating (full
line) of a glass with $N=5488$ atoms. The
different temperature intervals ($kT=0.004, 0.008$ and $0.012 \epsilon $) 
are indicated
by the dotted lines. The dashed line gives the average over some configurations
kept at $kT=0.008\epsilon$ instead of further heating.}
\label{fig_e0(t)}
\end{figure}
\begin{figure}
%%\epsfig{figure=plots/r6gll_history_R.eps,width=15cm,angle=00}
\caption{Average displacement per atom during aging and heating (full
line) of a glass with $N=5488$ atoms. The
different temperature intervals ($kT=0.004, 0.008$ and $0.012 \epsilon$) 
are indicated
by the dotted lines. The dashed line gives the average over some configurations
kept at $kT=0.008 \epsilon$ instead of further heating.}
\label{fig_r0(t)}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/pir6era35_all.eps,width=15cm,angle=0}
\caption{Displacement relative to a local minimum configuration as function
of time for a glass with $N=5488$ at different temperatures. Please note the
different scales for $\Delta R$.}
\label{fig_pir}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/particip_delta_r.eps,width=10cm,angle=90}
\caption{Participation ratio of relaxations against jump distance for 
configurations with $N=500$ ($\star$), $N=1024$ ($\circ$) and $N=5488$ 
($\times$) at
temperatures up to $T\approx 0.15 T_g$. The lines are guides to the eye 
which give the ideal scaling of the participation ratio with system size.}
\label{fig_p_delta_r}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/c21m2_d21m2.ps,width=8cm,angle=0}
\caption{Relaxation at $T\approx 0.1 T_g$ in a glass with $N=5488$
($\Delta R = 1.98~\sigma, p_{\Delta R} = 0.018$) . Shown are
all atoms whose squared displacement is more than 30\% of the maximal one, 
full 
circles: initial position, shaded circles: final position. }
\label{fig_c21m2}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/dimension_delta_r.eps,width=8cm,angle=90}
\caption{Dimension of jumps related to participation ratio in a glass 
with $N=5488$ at
temperatures up to $T\approx 0.15 T_g$}
\label{fig_dim}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/correlation_r1_r2_new.eps,width=8cm,angle=00}
\caption{Bottom: Distribution of correlations between successive jumps for 
$N = 5488$ at $T\approx 0.05 T_g$ (solid bar) and for all
temperatures up to $T\approx 0.15 T_g$ (dotted bar). Top: Distribution of 
correlations between successive jumps for 
$N=1024$ at temperatures up to $T\approx 0.125 T_g$.}
\label{fig_corr_delta_r}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/cre30m3_crj30m4.eps,width=10cm,angle=0}
\caption{Initial positions of atoms displaced by more than $0.35~\sigma$
in a time interval $\Delta t = 3600 \ (m\sigma^2/\epsilon)^{1/2}$ at 
$T\approx 0.15T_g$. Total displacement $\Delta R = 8.02~\sigma$,
$p_{\Delta R} = 0.077$. }
\label{fig_e30m3}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/correlation_e_delta_r.eps,width=10cm,angle=0}
\caption{Distribution of the projections of the normalized jump vectors
onto the eigenvectors of the 
quasi-localized soft vibrations for $N=500$.}
\label{fig_corr_e_delta_r}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/msqd_temp.eps,width=10cm,angle=0}
\caption{Mean square displacement after $ \Delta t = 40 $ (solid line)
and $ \Delta t = 10 $ (dashed line). The dotted line shows the extrapolated
vibrational contribution. The dash dotted line shows the mean square 
displacement after $ \Delta t = 80 $ after the material was aged at
the given temperature for $ \Delta t = 800$. The density was kept constant
at $\rho \sigma^{3} = 1$.}
\label{fig_msqd}
\end{figure}
\begin{figure}
%\epsfig{figure=plots/msqd_temp_nongauss.eps,width=10cm,angle=0}
\caption{Non-Gaussianity parameter for $\Delta t = 60$ (solid line) and 
$\Delta t = 15 $ (dashed line). The non-Gaussianity parameters of the
relaxational contributions alone are shown by the dash-dotted and dotted 
lines. The density was kept constant
at $\rho \sigma^{3} =1 $.}
\label{fig_non_gauss}
\end{figure}
\end{document}


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\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
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\end{thebibliography}