Sci. Rep. Kanazawa Univ.
Vol.53, p.55-80, 200955

On the analysis of collective motions of atoms
in mineral structures, and the mechanism of
the α-β polymorphic transition in quartz
Kuniaki KIHARA
Department of Earth Sciences, Kanazawa University. Kanazawa 920-1192
Abstract : Computer programs were prepared for analyzing the atom trajectories generated inmolecular dynamics simulation (MDS), and applied to reinterpret the mechanism of the α-βpolymorphic transition in quartz (SiO2). The programs contain the parts for calculating atomic
quantities such as bond distances, angles and mean square displacements, and for collective ones
such as X-ray scattering intensities, power spectral densities and normal modes. With the aid of asimplified structure model composed of atom sites of double-minimum potentials, the atomic
mechanism of the α-β transition in quartz generated in the simulations was successfullyinterpreted. A transfer mode, in which atoms can move over the energy barriers between the
double-minimum wells, is excited as an extra mode with an optic character even in the lowtemperature phase (α), and the α-β transition is driven by the softening of this mode. The strong
softening of this mode near the transition point also causes, via mode coupling, the softening of
the two transversal acoustic branches with low frequencies in some ranges of small wave vector
q in Γ-M, which cause the characteristic diffuse scattering and may cause the incommensurate
satellite reflections observed when temperature approaches to the transition point.

1. Introduction
Since the discovery of X-ray diffraction by Max von Laue, structural analyses have been
performed for a large number of minerals. So far, structural data (lattice constants, atomic
coordinates, atomic temperature factors and so on) and their temperature or pressure
dependencies have been available with enough accuracy. Now it is important to ask
ourselves whether or not we sufficiently understand the true mechanism of phenomena inminerals. Atomic coordinates, displacement parameters and so on obtained in usual
diffraction analyses are time and space averages in principle, where the details of atom
dynamics are lost in the averaging processes. For example, the strange thermal-expansion
in quartz is obviously caused by the collective behaviors of atoms. The knowledge of atom
dynamics is crucially important in discussing every property of this and many minerals.
In molecular dynamics simulation (MDS) of solid structure, we can obtain atomic
trajectories around the equilibrium positions in hypothetical structure characterized withgiven interaction parameters for atoms. Our crystals in MDS are of small atomic ensembles,much smaller than the practical crystals; in this sense, we may distinguish the crystals in
MDS from the real ones referring to “MDS crystal”. However, MDS has been applied to alarge number of solid structures, and succeeded to produce the trajectory data of atoms,

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

whose averages are in good agreement with the atomic positions experimentally
determined. Our stance is basically on reproducing reasonable structural data such as
atomic positions, mean square displacements, spectroscopic data and so on in MDS, whichmust be comparable with those obtained in usual measurements. With the success of MDS
in obtaining reasonable structural data, we infer dynamics in real crystals from analyzing
the atom trajectories generated in MDS. Our purposes in this article are firstly to prepare
computer programs to analyze the trajectory data in MDS crystal, and secondly to apply
those to MDS quartz. As stated later, our MDS quartz was in harmony with the real crystalof quartz in most structural data. This article contains nine sections: sections 2 to 5 describe
how the data are prepared for the comparison with experimental data. Sections from 6
explain those data obtained from MDS quartz, and the α-β transition mechanism and the
dynamical aspect of internal structure at high temperatures are discussed. Computer
programs developed in the present study are listed with brief explanations in section 8.
Any detail about MDS calculations is not explained here.
2. Calculation of structural data
2.1. Lattice constants and atomic coordinates
In usual crystal analyses with diffraction data, lattice constants and point group are
preliminary determined before intensity measurements. Symmetrically independent atomic
parameters are determined with least-squares method applied to structure factors or theirsquares obtained from such diffraction intensities. The quantities obtained are time andspace averages. To compare with the structural data, such as atom coordinates, atom meansquare displacements (MSD) and so on, obtained in experiments, the time-dependent
quantities in the MDS crystal must be averaged over time or space. (We must be careful
whether or not the related processes are stationary.) The following analyses are all done
without assuming any point group symmetry, but assuming constant translational
symmetry. It may be possible to examine the point group and space group symmetries in
the MDS crystal in that way.
In MDS for constant pressure, the size of crystal containing N basic cells, each ofwhich contains n atoms, are given at every step of run. (The word “basic cell” is used here
to represent the cell, whose multiple along their three edges is the MDS crystal.) The sizesof MDS crystal is represented with vectors A, B and C, whose components in Cartesian
coordinates are arranged vertically to form a matrix of (3 × 3) lines and rows, which wecall as M matrix. Lattice constants of Bravais lattice are calculated from <M>, the time
average of matrix M. Time-averaged <rκ!> is used to calculate space and time-averaged
atomic positions as

which is ready to calculate fractional coordinates in the crystallographic unit-cell.

Kuniaki KIHARA

57

2.2. Atomic mean square displacements and probability density functions
The matrix of MSD of atom κ(= 1~n) in a basic cell is calculated as time-average as(2.1)
where super-script T means the transposed matrix, and(2.2)
This gives the time average of atom κ in the basic cell. Matrix B is diagonallized to provide
MSD, <uii2(κ)>, of thermal ellipsoid of atom κ, which is ready to be compared to quantities
directly obtained from anisotropic atomic temperature factors βij in structure analysis. The
probability density function (pdf) of atom κ is calculated by accumulating uκ! over all
atoms κ in the MDS crystal. The pdf obtained in this way may correspond to that obtainedin a generalized structure analysis with structure factors developed in power series tohigher order terms.
2.3. Atomic distances and angles
Atomic distances and angles resulted from usual structure analysis are |<ri> - <rj>| and <ri>- <rj> - <rk>, respectively. It is further possible to calculate the averages of time dependent
atomic distances and angles, i.e., |<ri - rj>| and <ri - rj - rk> with MDS results. In general,these two kinds of distances or angles are not identical, depending on the symmetries ofrelated atomic sites; i.e.,

The right side terms represent the distances and angles for averaged atomic positions, andnot necessarily represent real distances and angles of bonds, respectively. The left side
terms give the averages of distances or angles, and therefore provide the time-averaged
values of true bond distances and angles in principle. In our X-ray structure analyses ofquartz (Kihara, 2001 a), the Si-O distance (i.e., |<rSi>-<rO>|) shows decreasing tendencywith increasing temperature. In our MDS quartz, we obtain the two kinds of values, |<rSi> <rO>| showing a similar tendency to the X-ray analysis and |<rSi - rO>| showing a rather
positive temperature dependence. These results led us to conclude that the real bonddistance for Si-O expands with increasing temperature.
2.4. Order parameter and time-dependent displacement parameter of atom
A number of structural transitions from higher temperature phases to lower ones appear to

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

be driven by atomic departures from the positions of high symmetry to of low symmetry.
In such a case, a thermodynamic function may behave as a function of such displacement
or quantity termed as “order parameter” in general. Among a variety of possibilities, weemploy the magnitude of departure of structure from that of high symmetry phase as orderparameter. The temperature dependence of order parameter may depend on the nature andmechanism of the phase transition. We define a time-dependent “atomic displacement
parameter” for MDS crystal. The temperature dependence of vibrational mode, which
drives phase transition, could be identified to the power spectral densities generated for thetime-dependent displacement parameters at atom sites. We define below the orderparameter and displacement parameter for the case where atomic sites are characterizedwith double minimum potentials.
2.4.1. Model structure with atom-sites of double minimum potentials
Let a crystal has two kinds of atoms A (circles) and B (squares) in its orthogonal primitive
unit-cell as shown in Fig. 1. The mean potential at each site has double minima, say α1(solid circles and squares) and α2 (open circles and squares), separated by the energy
barrier at the middle of minima, which may be denoted as “β-position” (open dotted circles
and squares). This model may ease explaining atomic behaviors in the structure with atomsites of double-minimum potentials. We below explain the structural behaviors expected orassumed for this model structure. The structure is assumed such that the relative
arrangement of atoms is nearly kept in both the low (α) and high temperature (β) phases.
At lower temperatures of the low temperature phase, all the atoms vibrate in one ofthe double wells, α1 (or α2), i.e., the crystal is perfectly ordered, and one of the three optic

Fig. 1. A model structure of double-minimum potential energies.
Circles represent atom A and squares atom B. At each atom site, energy minimum α1 is indicatedwith solid circle (atom A) or solid square (atom B), and α2 is open circle (atom A) or open square(atom B).

Kuniaki KIHARA

59

modes at Γ-point (wave vector q=0) is represented with the displacement pattern shownwith arrows in Fig. 1. This mode drives all the atoms to move toward the β-positions
between α1 and α2, and, therefore, could be the soft mode (SM) of the transition. However,the situation is not so simple as in the case, where atomic sites are of single-minimum
potentials. With increasing temperature, the kinetic energies of atoms increase: in thepresent case, some atoms may be energized to go beyond the barrier to another well, α2 (orα1). At low temperatures, this kind of motions occurs separately in the crystal, and theatoms transferred may be restored to the original sides. In this way, a “transfer mode” isactivated as an extra mode (to the normal modes). The point is that the transfer mode withlarger amplitude has the same symmetry as the normal mode with smaller amplitude, inwhich atoms are confined either in the α1 or α2 wells. That is, the latter (normal) mode is
not the true soft mode, but only boosts the transfer mode via coupling between them; inthis sense, this mode is called QSM (quasi soft mode) hereafter. With increasing
temperature, more atoms can transfer from α1 to α2, and the restoring force from α2 to α1
increases, depending on the ratio between the numbers of atoms staying in respective wells.
Approaching to the transition temperature, the total driving force of the transfer mode, i.e.,the difference between the restoring forces from both the sides decreases toward zero. Thatis, the frequency of the transfer mode decreases with temperature approaching to thetransition point. When both the wells α1 and α2 are equally occupied by atoms, thepositions averaged over their displacements become equal to those of the middle positionsof α1 and α2, i.e., the high symmetry β-positions on the symmetry planes generated normal
to the a-axis. The β-phase is thus resulted from the freezing out of the transfer mode, andin this sense, this mode is called as TSM (true soft mode) hereafter. In the higher symmetry
β-phase, all the atoms are to be enough energized to move almost freely over the barriers,
giving rise to a new normal mode with greater amplitude over the α1 and α2 wells, insteadof QSM, which disappears or fades.
2.4.2. Atomic displacement parameter and order parameterIn the case of the model structure characterized with double-minimum potentials, we may
represent the displacements (u1, u2, u3) of atoms from their corresponding β-positions, such
as

(2.3)
where a0 is the time-averaged a-axis dimension. A displacement may be normalized todimensionless displacements (termed here “atomic displacement parameter”) with theseparations ΔAA and ΔBB for α1-α2 at sites A and B, respectively,
(2.4)

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

The normalizing process is necessary to relate the displacements to the macroscopic orderparameter. Equation (2.4) represents vibrations confined in the potential wells in QSM at
low temperatures, but even the transfer motions over α1 and α2 in TSM at hightemperatures. It may be noted that the displacements contain two components ofvibrational modes, QSM and TSM. The behaviors as represented with equation (2.3) are
observed only in QSM and TSM, and therefore the components of atomic displacements
parallel with the α1-α2 lines (the a-axis in this case) must be strongly contributed by QSM
or TSM (δ1A and δ1B). The spectral densities of the space Fourier components of δ1A and δ1B
at wave vector q = 0 are to exaggerate the effects of QSM or TSM.
The normalized displacement parameters δ1A and δ1B can be generalized as δjκ for the
directions j and atoms κ. Time-average <δjκ> is directly calculated for each of Z = nN
atoms, and then its average over all Z atoms is calculated in(2.5)
to represent the total order parameter of the crystal. A parameter is also defined in(2.6)
to represent displacements for each of n atoms in a basic cell, where δp,jκ represents thedisplacement of atom κ in the p-th basic cell. Both ηj and ηjκ are the macroscopic orderparameters, ready to compare with observed quantities in structure analyses.
ηj and ηjκ are expected to be zero in the cases of j = 2 and 3, even in the doubleminimum cases as suggested in Fig. 1. In the β-phase, η1 is null, but shows strong
temperature dependence in the α-phase, because of strong asymmetry of the transferring
mode. That is to say, it is possible to represent the structural change in the crystal taking
the unique parameter η1, the order parameter. In general, the lines joining two minima arein different orientations from atom to atom, but a similar treatment to this case may be
applicable, if appropriate internal coordinate systems can be chosen for all the atom sites.
The α-β transition in quartz is this case.
3. X-ray scattering intensities
3.1. Total scattering intensities
The scattering amplitude of X-rays by an MDS crystal at time t is directly calculated as(3.1)
where fκ(Q) is the scattering factor of atom κ, rκ(t) atom position at time t, Q scatteringvector (Q = 4 πsinθ/λ with θ, half of scattering angle). The total scattering intensity is thenKuniaki KIHARA

61

calculated with
(3.2)
where <.......> means to take long time-average of quantity embodied in parentheses. The
resolution of the scattering vector Q depends on the size of the crystal (A, B, C). The small
size of an MDS crystal may cause some extensions of diffraction spots.
3.2. Diffuse scattering
The calculations of scattering intensities at arbitrary Q are just straightforward, andapplicable either to crystal or non-crystal. Here scattering at Q away from reciprocal lattice
points is calculated for thermal vibrations in crystals.
Bragg scattering (0-th order scattering) is that occurring for atoms vibrating aroundtheir equilibrium positions without interacting themselves. It occurs when scattering vector
Q/2π is superposed on reciprocal lattice vector H (=ha*+kb*+!c*). On the other hand,vibrating atoms in crystals interact each other, causing higher order scattering at non-zerowave vector q. The number of wave vector q, equal to the number of primitive unit-cells in
crystal, is usually enormously large in real crystals, and therefore their distribution covers
almost continuously the reciprocal space, the Brillouin zone (BZ). Since scattering
intensities from a mode at wave vector q depends on its frequency and displacement
pattern, its distribution in the BZ may provide useful information for dynamics of atoms in
the crystal.
The total scattering appears as the addition of higher-order scattering to the 0-th order,

It may be practical to omit the second- and higher-order terms, since the higher-order terms
are usually so small. The first-order intensity I1(Q) is contributed by the first-orderscattering,where I1(Q, jq) is calculated in the equation,(3.3)
Ejq and ωjq are the mean energy and the angular frequency, respectively, of mode jq. F1(Q,jq) is the first-order structure factor expressed in

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz(3.4)
where rκ is the position vector of atom κ in the basic cell, averaged for time or latticetranslations in the crystal, eκ(jq) the polarization vector of atom κ in mode jq, mκ the massof atom κ. For deriving (3.4), we are indebted to Willis and Pryor (1975). It is possible tocalculate each term in (3.3) and (3.4) for the MDS crystal, and therefore enable us tocalculate the contribution from the first-order term separately. Ejq, ωjq and eκ(jq) arecalculated from the atomic trajectory data as described later in section 5. In such process,the modes causing the diffuse scattering could be identified.
4. Spectral analysis of MDS crystal
Power spectral densities are calculated with the FFT method applied to the trajectory dataof MDS. The calculated spectra contain all the effects from thermal disturbances, and someof them are compared with the results of spectroscopic measurements such as Raman and
IR. Some peaks in the spectral densities may be attributed to appropriate normal modes
with the aid of lattice dynamical calculations or normal mode analysis of the trajectory dataof MDS as described in section 5. Our chief concern is on the collective motions of atoms
and their modes of vibrations. Spectra are calculated directly from FFT applied to the
trajectory data. The FFT routines of Hino (1977) are employed in our package to calculate
the power spectral density of MDS crystal.
4.1. Power spectrum
A random process x(t), which is stationary and ergodic, is expressed by taking a standard
Fourier integral for it sampled in time span -T/2 ! t ! T/2 as(4.1)
where θω is the phase of constituent wave of frequency ω, and x(t) is 0 outside the given
time span (-T/2, T/2). The Fourier transform of (4.1) is(4.2)
On the other hand autocorrelation function C (τ) is given as(4.3)
Replacing x(t) in (4.3) with (4.1) and reminding that x(T ) is 0 outside (-T/2,T/2), the next
expression is obtained,Kuniaki KIHARA

63

(4.4)
where X *(ω) is the complex conjugate of X (ω). When τ=0, we obtain(4.5)
On the other hand, time average of x 2, <x 2>, is given using spectral density S (ω) as(4.6)
Consequently the spectral density, i.e., mean energy per unit time is expressed as(4.7)
Returning to (4.4), we have two relations,
(4.8)
and
(4.9)
That is, power spectral density and correlation function is in a Fourier transform each other.
We assume that the time-dependent systems in the following applications are stationary
and ergodic random processes.
4.2. Power spectral densities of geometrical scattering amplitude
Now, we define our spectral densities. Atomic positions in crystals are time-dependent
around their mean positions, and therefore Y (Q, t) in (3.1) is also time-dependent aroundthe mean amplitude. Omitting time-independent terms from (3.1), we define the timedependent geometrical scattering amplitude at wave vector q as
(4.10)
where rκ(t) is the position vector of atom κ (1 to Z =nN ) at time t. The Fourier transform ofG (q, t) is

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

and its spectral density is given in(4.11)
In our calculation, the spectral density is first obtained from (4.11) applying FFT. If
necessary, auto correlation function is calculated with (4.8). At q = 0, G (q=0, t) contains
no time-dependent term, and only an enormous peak appears at ω = 0.
4.3. Power spectral density of atomic displacement parameter
The spectrum of atomic displacement parameter is expected to exaggerate that for specified
atomic displacements. The normalized atomic displacement parameter as defined in (2.4)(Fig. 1) becomes the order parameter if averaged over time or space. Its spectral density
must show that of QSM or TSM in the phase transition. Equation (2.4) is generalized to thecase, where all the α1 - α2 joins are not in the single direction, but in unique-directions forsite to site as(4.12)
where uκ (t) is the displacement of atom κ at time t from the high symmetry β-positions
toward the low symmetry α1 (or α2). The normalizing constant Δκ is the α1-α2 separation
at 0 K or its approximation. (The normalizing constant is not essential for spectrum, as
noted before, but normalized δκ(t) is used for the order parameter.) The space Fouriertransform of δκ(t) is defined in
1−2

1−2

(4.13)
where Rκ is the mean position vector of atom κ. The spectral density of δ(q, t) is calculatedwith equation,(4.14)The normalized atomic displacement parameter in (4.12) is defined for the case of doubleminimum potentials, but equation (4.12) could be applicable in more general cases,
specifying displacement vectors; the normalizing constant is not serious, can be replacedby an arbitrary non-zero value, for the purpose of obtaining spectrum. This spectrum is freefrom the appearance of enormous 0-frequency peak when q = 0, and superior for this point
to other types of spectrum.
5. Normal mode analysis of MDS crystal
In the standard lattice dynamical calculation, the dynamical matrix is formed for the static

Kuniaki KIHARA

65“averaged” structure concerned. The normal mode frequencies and polarization vectors arecalculated from the eigenvalues and eigenvectors, respectively, of the dynamical matrix. In
MDS, atom trajectories are available; their analysis may be more straightforward, in asense, than preparing the dynamical matrix of static structure with assumed force
parameters. Dove (1993) described the possible procedure for the normal mode analysis of
the atom trajectories generated in MDS. However, the method has been not common so far
for MDS crystal, and its reliability or usefulness has been not established. As already statedabout the power spectral density of MDS crystal, the problem is how the spectra areassigned. We have attempted to apply the method to the trajectory data in MDS quartz.
An instantaneous displacement, uκ!(t), of atom κ in the primitive cell ! from its
equilibrium position is given as the sum of all contribution from 3nN normal modes,(5.1)
where q is the wave vector (|q| = 2 π/λ, λ is the wave length of mode jq), mκ mass of atom
κ, Ejq, and ωjq are average energy and angular frequency of mode jq, respectively. The
displacement in (5.1) is simplified using normal mode coordinate, Θ(q, ωjq), as(5.2)
Polarization vector eκ(jq) represents the direction of displacement of atom κ in that mode,and orthonormalized as

Normal coordinate Θ(q, ωjq) is just the mode jq vibrating with amplitude (Ejq / ω2jq)1/2 andwith frequency ωjq, and simply expressed as(5.3)
Let assume that the MDS crystal contains N primitive unit-cells, each contains n
atoms. We define 3n space Fourier components of uκ!(t) as(5.4)
The preceding factor of the summation is to convert the displacement uκ!(t) to the Fouriercomponents for each atom. All terms in right hand side in (5.4) are available in MDS. It isnoted that Uκ(q, t) in (5.4) is expressed on the three components in the orthogonal
coordinate system employed in MDS. Now let make a column matrix arranging the threecomponents for an atom, and join them for all n atoms in a unit cell to make a big column

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

matrix T(q, t) of (3n×1) as(5.5)

T(q, t), expressed on the orthogonal coordinate of the MDS crystal, is transformed to thatof normal coordinates as shown below.
First, a (3n×3n) Hermite matrix is prepared taking transposed matrix of complex
conjugate of T(q, t), T(q, t)* as(5.6)
Assuming that T(q, t) is transformed to time-dependent normal mode vector Θ(q, t) with atransformation matrix A(q), i.e.,(5.7)
The time-average of column vector Θ(q, t) and its transposed conjugate complex vector
Θ(q, t)* (＝T(q, t)* A(q)*) is given as(5.8)

Since the normal mode coordinates are orthogonal each other, the left side of (5.8) must be
diagonalized. That is, the right side of (5.8) is the process for diagonalizing matrix<Ψ(q, t)>. Each of 3n elements of left side of (5.8) may correspond to the normal
coordinates <Θ(q, ωjq)>, and we can relate the frequency and the average energy of mode
with
(5.9)
Since the diagonal elements of the diagonalized matrix on the left side of (5.8) is the
eigenvalues of <Ψ(q, t)>, it is not necessary to find the transformation matrix A(q),because 3n eigenvalues are directly determined applying standard routine such as

Kuniaki KIHARA

67

Householder method to <Ψ(q, t)>. Putting the eigenvalue as δjq2, the harmonic frequency
ωjq is calculated as(5.10)
and 3n polarization vectors of mode jq are given with 3n eigenvectors. For reference,transformation matrix A(q) is given by arranging (3n×1) eigenvectors Vjq| for j-th
eigenvalue in the square matrix such as(5.11)
In our computer program, total energy Ejq is evaluated taking the high temperatureapproximation, i.e., equal partition of energy over mode, and the diagonalization of<Ψ(q, t)> is done without symmetry restraints. In application to MDS quartz, someobviously inconvenient results occurred in polarization vectors. We consult with grouptheoretical analysis to overcome this problem.
6. Examples of application to MDS quartz
6.1. Brief review for the structure of quartz and its change
The structure of quartz is constituted of a framework of SiO4 units, which are linked toneighbor units by sharing all the corner O atoms. The space group is either P3221 or P3121
for α-quartz, which is known to transform to the hexagonal polymorph with space group
P6222 or P6422, respectively, at around 846 K. The atom positions in a α-phase crystal of
single domain at low temperatures such as room temperature are usually referred as “α1positions” (Young, 1962). There is another set of positions, “α2-positions”. The position
corresponding to the middle of α1 and α2 positions are sometime called as “β-position”.
The SiO4 units are believed rather rigid, and the atom positions are usually expressed withthe orientations of the units, such as “atoms in α1-orientations or in α2-orientations”. The
two orientations or sets of positions are related by the rotational operation of corner-linked
SiO4 units around the <100> axes (2-fold axes) passing the central Si atoms, being
associated by the small translational operation along the axes. That is, the α1 and α2orientations are related by the screw operation of corner-linked SiO4 via the corresponding
β-orientations. The angle is calculated rather great as about 30°at room temperature, but
the atomic separations are not so; about 0.8 Å at the O-site and 0.34 Å at the Si site
(Young, 1962; Kihara, 1990). On the other hand, in the diffraction studies, the atomic
positions for Si and O atoms in hexagonal β-quartz are assigned either to the corresponding
β-positions (single-minimum model) or to both the α1 and α2 site with equal weight (50-to50) (double-minimum model or split-site model). These different views of β-quartz are
directly related to the different interpretations for the α-β polymorphic transition. One isrepresented by Young (1962): the atom equilibrium positions continuously change from

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

those in the α-phase, say the α1-positions, to those at the β-positions with increasing
temperature. This may imply that the minima of potentials at atom sites continuously shift
toward the corresponding β-positions with increasing temperature in α-quartz, and finallyreach the β-positions of single-minimum in the β-phase. Another is based on the idea of
domains in the α1 and α2 orientations (for example, Wright and Lehmann, 1981). In thedomain idea, the structure of quarts is the average of the domains in the two orientations:the ratio of α1 and α2 domains varies with temperature in the α-phase, but kept equal in the
β-phase. The domains may possibly be dynamic, changing time to time, or place to place.
Four A1 normal modes can occur theoretically at Γ in α-quartz, see (7.1), but it isreported that an extra band appears in Raman measurements (Shapiro et. al., 1967; Scott,1968). Figure 2 of Shapiro et. al. (1967) shows the temperature dependence of A1 Raman
bands of quartz. A small 147 cm−1 band at 306 K, considered as the extra band (satelliteband), grows, but softens quickly with increasing temperature. The 206 cm−1 band,
believed as the soft mode of the α-β transition (for example, Ercombe, 1967), shows rather
weak softening, and appears not frozen at the transition point.
The methods shown in sections 3, 4 and 5 were applied to analyze the collective
motions of atoms in MDS quartz in expecting to draw a determinative conclusion for themechanism of the structural changes in quartz, together with the associated phenomena
such as the strange thermal expansion behaviors, the A1 satellite band, characteristic diffuse
scattering and so on. Here some selected results are described, focusing mainly on the atom
displacement parameter, power spectral densities and normal mode analysis. It was
confirmed that the α- and β-phases of MDS quartz belong to space groups P3221 andP6222, respectively. (Our MDS was started with α-quartz in P3221.)6.2. Time dependent atomic displacement parameter and order parameter
Time dependent atomic displacements calculated with equation (2.4) for selected atoms in
our MDS quartz (Kihara 2001 a) suggest for TSM to be exited at some low temperatures
below the transition point, likely for the model structure in 2.4.1. The order parameter was
calculated for the displacements of atoms from the β-positions taking the α1−α2components. The normalizing constant Δκ of (4.12) was approximated to thecorresponding separations between the α1 and α2 positions at 300 K. This one-dimensional
order parameter η is thus unity at 300 K, and expected to remain nearly unity below thistemperature. The order parameters calculated with (2.6) for each atom in the unit-cell andcalculated with (2.5) for the unique η representing the crystal are all almost equal eachother, suggesting that the structural change is driven by the displacements of atoms at all
the sites activated by QSM and/or TSM.
Fig. 2 shows the temperature evolution of η calculated with (2.5). It is well known
that the temperature evolution of thermodynamic function in the α-β transition in quartz is
first-order in a rigorous sense, but appears as second-order-like. A thermodynamic function
1−2

Kuniaki KIHARA

69

Fig. 2. Temperature dependence of order parameter for quartz in a molecular dynamics simulation. Solidcircles represent values for molecular dynamics simulation, which are fitted as the curve of solidline with equation (6.2) in text.

in the first-order change may be well approximated by the Landau expansion including theterms up to the 6-th order,(6.1)
where the parameters a, b and c are restrained as b<0, a, c>0 to guarantee the transition to
be the first-order, and To is a temperature lower than the critical temperature Tc. The orderparameter η is 0 in the β-phase, but changes with temperature in the α-phase as(6.2)
The conditions for the minimum and maximum of thermodynamic function Φ guarantee
five real roots, if temperature T satisfies a condition To ! T ! To + b 2/3ac. At Tc, Φcontacts with the η axis (Φ=0) at η=0 and ±η. In Fig. 2, η is plotted for all the points of
MDS quartz in the range from 300 to 1400 K. Fitting the data points between 300 and 900
K with (6.2) provides Tc " To " 920 K, and η(Tc) = 0.02. These values suggest that thistransition is nearly the second-order, or otherwise the fourth-order term in (6.1) disappears.
The result is in a weak contrast to the value of Tc - To (" 7°K) for real quartz (Kihara, 2001c). We consider that our MDS quartz is so small, and our values represent for that small
quartz with given energy parameters.
The behavior of the time-dependent atomic displacement parameter, normalized as(4.12), suggests that TSM as shown in the model structure composed of atom sites with

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

double-minimum potentials is already excited at some low temperatures (around 600 K)
below the transition point in MDS quartz. So in MDS quartz, there may be two A1 modes
corresponding to QSM for small amplitude motions confined in the wells and TSM forlarge amplitude oscillations over the two corresponding wells.
6.3. Power spectral density
Fig. 3 illustrates the power spectral densities along Γ-M for the geometrical scattering
amplitudes in (4.10) for the MDS at 300 K. (M is the points on the zone boundaries
perpendicular to a* and its symmetry equivalent directions.) The case for q = 0 is notshown, because it has no oscillatory component. There are 27 peaks expected at q=[0 ξ 0],i.e., along Γ-M , for both the phases of quartz. We can see about 20 peaks in Fig. 3,
showing some characteristic features of distribution: the first is that all the peaks appear
below 40 THz, the second that three to four peaks appear between 30 and 40 THz, which
are known to the spectra for the deformation of SiO4, the third that no peak is seen in 25 30 THz,the fourth that the dispersion is small for peaks above 10 THz, which may becharacteristic for the optic modes, and the fifth that the dispersion relations for the spectra
below 10 THz are complicated, probably due to the acoustic branches. These features are
all quite in good harmony with the observed ones for quartz, that is, our MDS quartz
successfully represents the frequency distribution of phonon spectrum of quartz.
The power spectral density was then calculated for the components of atomic
displacements in the α1 - α2 lines uκ (t) in (4.12). Fig. 4 illustrates the temperature
1−2Fig. 3. Power spectral densities S (ν) (sec) in an arbitrary logarithmic scale along Γ-M calculated withatomic trajectories in a molecular dynamics simulation of quartz at 300 K. Wave vectors areindicated with the aid of hexagonal reciprocal lattice vectors such as [0 0.1 0] for q = 0.1 b* andso on.

Kuniaki KIHARA

71

Fig. 4. Temperature evolution of power spectral densities of time-dependent atomic displacement
parameter at Γ(q = 0). Spectra below 10 THz are indicated in arbitrary linear scales, whose
maximum values are indicated in parentheses following temperature values.

dependence of the spectra below 10 THz, obtained for δ(q=0, t) with (4.13). The spectrum
in the higher frequency zone appears to share most common features with that observed forgeometrical scattering amplitudes. Through all the temperatures studied, a peak of low
frequency is predominant, as expected. That band appeared at about 3 THz at 300 K
quickly shifts toward the low frequency side with increasing temperature; it becomes ahuge peak nearly at ν = 0 at 900 K, and begins to shift toward the higher frequency sidewith further increase of temperature. The temperature dependence of this band is wellcomparable with the A1-satellite band at 147 cm−1 (4.41 THz) observed in the Raman
experiments of α-quartz at room temperature (Figure 2 of Shapiro et al., 1967). Later in thenormal mode analysis, this band is assigned to QSM or/and TSM, which drives the α-βtransition in quartz.
6.4. Normal mode analysis
6.4.1. Evaluation of performance
The normal modes in MDS quartz were analyzed assuming the ordered structure, that is,

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

27 normal modes occur at each wave vector q. In the analysis of MDS quartz, the transfer
mode (TSM) excited as an extra mode, may cause a difficulty in assignment. If TSM
belongs to the same symmetry as the A1 normal mode (QSM), and both are excited, thepresent analysis would result in an erroneous frequency (probably the averaged value forboth). Irrespective of this expected difficulty, the normal mode analysis was performed
taking the part of 82 ps from a long time-series. The frequencies and the atomic
displacement patterns of modes were compared with the corresponding observations forquartz to examine the present performance.
Figs. 5 a, b, c illustrate the displacement patterns of atoms in MDS quartz, which are
the exaggerated polarization vectors in the QSM or/and TSM at q = 0: a. 300 K, b. 895 K,
c. 1200 K. Since TSM is not to be excited at 300 K, the pattern at 300 K may be of QSM.
However, the patterns at 895 and 1200 K must be dominated by TSM or its successor
mode in the β-phase. The displacement patterns for the β-phase (Fig. 5 c) comply with the
(a)

(c)

(b)

Fig. 5. Polarization vectors of atoms of an MDS
quartz at (a) 300, (b) 895 and (c) 1200 K ina strongly temperature dependent mode atΓ-point, obtained in normal mode analyses.
Integers from 1 to 9 are the running
numbers; numbers 1 to 3 for Si, andnumbers 4 to 9 for O atoms. The
orthonormalized vectors (arrows) areprojected on (001), whose relative
magnitudes are arbitrarily scaled, but Z coordinates shown nearby are in absolutevalues.

Kuniaki KIHARA

73

Table 1. Character tables for point groups of quartz
a. At Γ-point of low quartz
2 C3[001]

3 C2[100]A132 (D3)

1

1

1

A2

1

1

−1

E

2

−1

0

2 C6[001]

2 C3[001]

C2[001]

3 C2[120]

3 C2[100]

1

1

1

1

I

b. At Γ-point of high quartz
622(D6)

I

A1

1

1

A2

1

1

1

1

−1

−1

E1

2

1

−1

−2

0

0

E2

2

−1

−1

2

0

0

B1

1

−1

1

−1

1

−1

B2

1

−1

1

−1

−1

1

C2[100]C2[120]c. In Γ-M of high quartz
2(C2)

E

C2[120]

Σ1(A)

1

1

Σ2(B )

1−1

d. At M-point in high quartz
222(D2)I

C2[001]

M1(A)

1

1

1

1

M2(B1)

1

1

−1

−1

M3(B3)

1

−1

1

−1

M4(B2)

1

−1

−1

1

B1 representation (Tab. 1), well known for the soft mode in β-quartz. It is also seen for αMDS quartz that the displacement pattern nearly satisfies the A1 symmetry at hightemperatures (Fig. 5 b), but not well at low temperatures (Fig. 5 a). Beside theseinconsistencies, the obtained frequencies appear comparable with those estimated from thepeak positions in the power spectrum density.(In Fig. 4, the frequencies of QSM or/and
TSM in the present normal mode analysis are indicated with arrows.)
As stated, the performance of the normal mode analysis of MDS quartz appears
different at low temperatures and at high temperatures. It has not been answered so far

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

what cause the difference. Since our main concern is on the low frequency modes at hightemperatures near the phase transition, the normal mode analysis is performed mainly forthe low frequency modes.
6.4.2. Low frequency Γ-M branches
Fig. 6 illustrates the dispersion relations of low frequency Γ-M branches, an optic andthree acoustic branches, of the β-phase of MDS quartz at 1400 K: #1 and 2 are thetransversal acoustic branches in antisymmetric Σ2 (Table 1), #3 the longitudinal acoustic
one in totally symmetric Σ1, and #4 the optic branch in Σ2. The dispersion relations andtheir temperature dependences for our MDS quartz are well comparable and rather in good
agreement with those for inelastic neutron scattering measurements by Dolino et al. (1992).
The characteristic features in these low-frequency Γ-M branches obtained in the normal
mode analysis of MDS quartz are briefly viewed below taking the case of β-phase, with theaid of Fig. 7 illustrating the displacement patterns in the zone boundary M -point modes of
the #1 and 2 branches at two temperatures just below and above the transition point.
The #4 optic branch (Σ2) is connected to the SM (B1) at Γ. The #1 branch is dominatedby transversal acoustic strains in Σ2(B ) with dominant component in X (denoted as TAX),
containing some amounts of the SM (B1) components. The #2 branch shows the clear TAZ
character in Σ2 at small q from 0 to about 0.2 b*, but that character disappears at q larger

Fig. 6. Dispersion relations in four low-lying branches of MDS quartz in Γ-M at 1400 K.The branches
are numbered in the sequence from the lowest to the higher branches. M1 to M3 are the symmetry
representations at zone boundary M -points, and Σ1 and Σ2 at Γ-M . See Table 1. The eigenvectors
are exchanged between #2 and #4 branches in the range 0.2 ~ 0.3 b*.

Kuniaki KIHARA

(a)

(c)

(b)

(d)

75

Fig. 7. Atomic displacement patterns at zone-boundary M -point deduced from harmonic latticedynamical approaches to MDS quartz. The orthonormalized vectors (arrows) are projected on(001), whose relative magnitudes are arbitrarily scaled, but the Z -coordinates shown nearby are in
absolute values.
(a) Branch #1 at 975 K, (b) Branch #2 at 975 K, (c) Branch #1 at 1400 K, and (d) Branch #2 at
1400 K.

than 0.3 b*. On the other hand, the #4 branch in Σ2 loses the SM character at q from 0.2 to
0.3 b*, but then gains the TAZ characters to the M -point. These unexpected changes inbranches #2 and #4 appear correlated each other, suggesting the exchange of theeigenvectors between them. The optic SM and TA1 in quartz are considered to couple

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

strongly with each other via strain and its space gradient (Aslanyan and Levanyuk, 1979).
In the present MDS quartz, branch #4 can couple not only with #1 (TAX), but also with #2(TAZ), both in the same symmetry. Taking the suggested eigenvector exchange between #2
and #4 into consideration, it may be reasonable to conclude that, at q smaller than about
0.2 b*, the softening of the SM (#4) at q near 0 causes #2 (TAZ) to be softened, and also #1(TAX) via the mode coupling. The #3 acoustic branch is free from such coupling with othermodes, and keeps the longitudinal character (LA) in Σ1
The exchange or mixing of the polarization vectors between the #2 (TAZ) branch andthe optic soft branch in Γ-M are also observed at high temperatures in the α-phase of MDS
quartz. Such exchange does not occur at 300 K. We consider that the eigenvector exchange
is associated by the excitation of the large-amplitude transfer modes, i.e., A1 TSM (opticmode) in the α-phase and B1 SM in the β-phase. The #4 optic branch strongly softens at Γwith temperature approaching to the transition point from below, causing the two acoustic
branches #1 and #2 to soften via mode coupling at small q around 0.1 b*. These low
frequency branches are shown to cause the diffuse scattering observed in diffraction
experiments (Kihara and Matsui, 2003). If the MDS crystal were large enough, thebranches could give incommensurate satellite reflections on a* (b*) near the α-β transition
point.
7. β-phase structure and α-β transition in quartz
7.1. Structure of β-quartz
There have been a number of discussions about the β-phase structure of quartz since
Young (1962), who preferred the single-minimum model of potentials in the X-ray
diffraction study of the β-phase. On the contrary, Wright and Lehmann (1981) succeeded
to refine an α1−α2 split-atom model with neutron diffraction data, leading them to prefer
the α1-α2 domain model. We consider that the successful refinement of the latter authors is
not essential to support the superiority of the split-atom model to the single-minimum
model. Actually Kihara (1990) succeeded to refine the single-minimum model assuming
generalized structure factors in a better agreement factor between observed and calculated
structure amplitudes than the split-atom model. For example, Kihara (1990) obtained
extended, but unimodal probability density functions (pdf) either site for O or Si. The atom
pdfs in our MDS quartz were all unimodal both in the α and β phases (Kihara, 2001 a). On
the other hand, the time-series of δκ(t) and its temperature dependence for arbitrarily
chosen atom sites show the double-minimum characteristic (Kihara, 2001 a). Both theresults obtained with our MDS quartz prove that the unimodal pdfs in the β-phase areresulted from the double-minimum potentials. The success of the split-atom refinement
only shows that the pdf in quartz is not so simple as expressed by a normal Gaussian
distribution. If atoms can move almost freely over the energy barriers of double-minimum
potentials, the pdfs obtained in usual diffraction experiments could be extended, but
unimodal, depending on the form of potential energies.

Kuniaki KIHARA

77

7.2. Soft mode in quartz
The point groups of α- and β-quartz are 32 and 622, respectively. The 27 modes at Γ areassigned asΓ = 4 A1(R)+4 A2(IR)+8 E (R,IR)+1 A2(A)+1 E (A) for α-quartz, andΓ=1 A1(R)+2 A2(IR)+3 B1+2 B2+4 E1(R,IR)+4 E2(R)+1 A2(A)+1 E1(A) for β-quartz,(7.1)

where Raman and IR active representations are denoted with R and IR in parentheses,
respectively, and acoustic modes are with A in parentheses. In (7.1), B1 and B2 modes are
silent mode. Shapiro et al. (1967) found an extra A1 band of 147 cm−1 (4.4 THz at room
temperature) in a Raman measurement of α-quartz. In usual lattice dynamical calculationsof α-quartz at room temperature, the band of 207 cm−1 (6.21 THz) is assigned as the softmode of A1 symmetry (For example, Elcombe, 1967). However, the frequency of the A1band shows only weak temperature dependence with no sign of freezing out at thetransition point and even in β-quartz. The band strongly softens is the satellite band of 147
cm−1. Scott (1968, 1974) explained the possibility of the satellite band and its temperature
evolution with Fermi resonance. We stressed here another possibility such that the strongly
temperature dependent band, i.e., the transfer mode TSM observed in MDS quartz
corresponds to the satellite mode observed in the Raman measurement.
TSM and QSM may possibly be mixed up as a single mode in the normal mode
analysis. Because TSM is not excited at 300 K in our MDS quartz, the dominant peakobserved at 3 THz in the power spectral density (Fig. 4) may correspond to QSM. On theother hand, the behaviors seen in the atomic displacement parameters (Kihara, 2001 a)
suggest that TSM is excited at some higher temperature, probably already excited at 600 K.
The widening of the peak at 700 K (Fig. 4) is probably due to the partial superposition of
QSM and TSM. The amplitude of TSM is expected greater than that of QSM, and therefore
the frequency of TSM is expected lower than that of QSM. TSM rapidly softens to thehuge peak of ν !0 at 900 K, the transition point in our MDS quartz. On the other hand,
QSM is only weakly softened with increasing temperature. There is no peak corresponding
to QSM observed at 900 K and 1000 K (Fig. 4 in linear scale): this may arise from thedecrease of relative intensity of QSM compared to TSM.
Now consider the temperature dependence of the frequency of TSM, νTSM (q=0), in the
α-phase of MDS quartz. Taking the observed temperature dependence of νTSM, whichsoftens to 0 when temperature approaches to Tc, we assume that νTSM2 (q=0) is proportional
to (ΔE - kBT ) α, where ΔE is the barrier height for the mode (probably equivalent with theaverage of heights at all the sites), and kB is the Boltzmann constant. Putting ΔE as kBTc, we
obtain a relation containing the order parameter η,
(7.2)

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

We obtain 6.5 for α/β in fitting with (7.2). By analogy with a simple lattice dynamics, νTSM2
is expected to be proportional with the restoring force associated with the mode, suggestingthat the restoring force is proportional to η to about 6-th power. This may suggest that thepotentials extend with only weak slopes in the vicinity of η = 0.(Let remind that η varies
between 0~1.)This is another evidence to show that the atoms in MDS quartz can movealmost freely over the energy barriers at high temperatures to result to the unimodal atomic
distributions.
7.3. Structural changes in quartz
The atomic motions in MDS quartz are summarized here. At low temperatures (below 600
K), our MDS quartz is perfectly ordered with atoms undergoing thermal vibrations
confined in, say, the α1-wells of potentials. In the A1 Γ-point normal mode (denoted as
QSM) with frequency about 3 THz, atoms oscillate around the α1-minima, being confinedin the α1-wells; the motions of atoms can be collected as the screw oscillations of SiO4
units around <100>. With increase of temperature, some atoms are energized to move overthe energy barriers (β-positions) to the α2-wells. The transfer α1->α2 or α2->α1 occurs
equally at every site. This is confirmed by knowing that η obtained with (2. 6) is equal foreach atom. This is the result of collective motions of atoms, and shows that a vibrationalmode, denoted as TSM, other than the normal modes is excited as an extra mode. Thismode also belongs to the A1 rep, and is driven by QSM of frequency about 3 THz, via
strong coupling through the structural deformations with the same symmetry. The
frequency of TSM can vary from 0 to the maximum of that of the A1 QSM. With increasing
temperature, QSM changes to less significant, while TSM becomes remarkable. At thetemperature, where both the wells α1 and α2 are occupied equally by atoms, the restoring
force of TSM becomes null, and the α-β transition is completed. With further increase of
temperature, the mode average energy (kBT in the high temperature approximation) rises,
and therefore the effective mean potentials become to of single-minimum-like, and thetransfer mode becomes to be identified as the B1 normal mode. Inversely when the modeenergy approaches the level of the barrier from above, the effects of the double-minimum
appear in the atomic motions, i.e., the α1! α2 restoring force is recovered, and instead of
the B1 mode, the A1 TSM appears.
8. Computer programs
Our MDS is calculated with computer program CTPMD (Matsui, 1988), partially modified
for the parts writing out the results for the analysis. In our CTPMD, the time-series of all
the atoms are stored in sequential-unformatted file “save 9.dat”, and the necessary data forfurther calculations of simulations are saved in formatted “save 4.dat”. The program
package MDPROC contains four programs, listed below, all written in FORTRAN.

Kuniaki KIHARA

79(1) MDSTR
Calculate the quantities ready to compare with those of usual structure analyses.
Atomic fractional coordinates and size of MDS crystal, both in time and ensemble
averaged
Atomic mean square displacements and probability density functions in contours
Atomic distances and angles in time and ensemble averages(2) MDOPSPECT
Order parameters (macroscopic)
Power spectrum densities for atomic displacement parameters(3) MDNORMODE
Frequencies and polarization vectors of normal modes(4) MDSPECT
X-ray scattering intensities
X-ray diffuse scattering
Power spectrum densities for geometrical scattering amplitudes
9. Summary
In this article, the procedures for analyzing time-series data of atoms generated inmolecular dynamics simulation were described for the calculations of bond distances andangles, mean square displacements and probability density functions of atoms, power
spectral densities and normal mode analysis. The methods were applied to the atom
trajectories in MDS quartz to discuss the dynamic nature of atoms in quartz, which haveremained controversial for long time.
The power spectral densities of space Fourier components of time-dependent atomic
displacement parameters calculated for the α1-α2 directions at atom sites showed a single
dominant band behaving like the soft mode of the α-β transition. The polarization vectors
and frequencies of the 27 possible normal modes for quartz were calculated at different
values of q. At high temperatures, the dispersion curves of low frequency branches were
well comparable with the measured ones in experiments (inelastic neutron scattering)particularly in the β-phase region. These results were used to discuss the causes of thestrong softening near the transition point occurring in the low frequency branches in Γ-M .
The two lowest frequency acoustic branches containing transversal characters and an optic
branch share the same symmetry and therefore can couple themselves, and also eigenvector
exchange occurs between the optic mode and the second lowest acoustic branch at q inrange 0.2 to 0.3 b*. The present analysis also showed that the temperature dependences of
the three low frequency branches (two transversal acoustic and one optic branches) areessentially important to generate the incommensurate satellite reflections on a* (b*) via
coupling.
The model based on double-minimum potentials at atom sites proposed with thepresent results of analysis of MDS quartz is also successful to explain the thermal

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On the analysis of collective motions of atoms in mineral structures,and the mechanism of the α-β polymorphic transition in quartz

expansion behaviors of Si-O bond distances and cell dimensions. The subject appears
somewhere else (Kihara, 2001 b), and was not discussed here.
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Dove M T. (1993) Introduction to lattice dynamics. In Cambridge topics in mineral physics
and chemistry. Cambridge University Press
Elcombe M. M. (1967) Some aspects of the lattice dynamics of quartz. Proc. Phys. Soc. 91,
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Hino M. (1977) Spectral analysis (in Japanese). Asakura, Tokyo
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Shapiro, S M, D. C. O’Shea, H. Z. Cummins (1967) Raman scattering study of the α-βtransition in quartz. Phys. Rev. Letters 19, 361-364
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