The Grüneisen parameter, γ, named after
Eduard Grüneisen
Eduard Grüneisen (26 May 1877 – 5 April 1949) was a German physicist and the co-eponym of Mie–Grüneisen equation of state.
Grüneisen was born in Giebichenstein, near Halle (Saale).
The Grüneisen parameter was named after him.
Since ...
, describes the effect that changing the volume of a
crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
has on its
vibrational properties, and, as a consequence, the effect that changing temperature has on the size or dynamics of the
crystal lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n ...
. The term is usually reserved to describe the single thermodynamic property , which is a weighted average of the many separate parameters entering Grüneisen's original formulation in terms of the
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
nonlinearities.
Thermodynamic definitions
Because of the equivalences between many properties and derivatives within thermodynamics (e.g. see
Maxwell Relations
file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...
), there are many formulations of the Grüneisen parameter which are equally valid, leading to numerous distinct yet correct interpretations of its meaning.
Some formulations for the Grüneisen parameter include:
where is volume,
and
are the principal (i.e. per-mass) heat capacities at constant pressure and volume, is energy, is entropy, is the volume
thermal expansion coefficient
Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions.
Temperature is a monotonic function of the average molecular kinetic ...
,
and
are the adiabatic and isothermal
bulk moduli,
is the
speed of sound
The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends strongly on temperature as w ...
in the medium, and is density. The Grüneisen parameter is dimensionless.
Grüneisen constant for perfect crystals with pair interactions
The expression for the Grüneisen constant of a perfect crystal with pair interactions in
-dimensional space has the form:
where
is the
interatomic potential
Interatomic potentials are mathematical functions to calculate the potential energy of a system of atoms with given positions in space.M. P. Allen and D. J. Tildesley. Computer Simulation of Liquids. Oxford University Press, Oxford, England, 1989 ...
,
is the equilibrium distance,
is the space dimensionality. Relations between the Grüneisen constant and parameters of
Lennard-Jones
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical science at the University of Cambridge. He was an imp ...
,
Morse
Morse may refer to:
People
* Morse (surname)
* Morse Goodman (1917-1993), Anglican Bishop of Calgary, Canada
* Morse Robb (1902–1992), Canadian inventor and entrepreneur
Geography Antarctica
* Cape Morse, Wilkes Land
* Mount Morse, Churchi ...
, and Mie potentials are presented in the table below.
The expression for the Grüneisen constant of a 1D chain with Mie potential exactly coincides with the results of MacDonald and Roy.
Using the relation between the Grüneisen parameter and interatomic potential one can derive the simple necessary and sufficient condition for
Negative Thermal Expansion in perfect crystals with pair interactions
A proper description of the Grüneisen parameter represents a stringent test for any type of interatomic potential.
Microscopic definition via the phonon frequencies
The physical meaning of the parameter can also be extended by combining thermodynamics with a reasonable
microphysics model for the vibrating atoms within a crystal.
When the restoring force acting on an atom displaced from its equilibrium position is
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
in the atom's displacement, the frequencies ω
i of individual
phonon
In physics, a phonon is a collective excitation in a periodic, Elasticity (physics), elastic arrangement of atoms or molecules in condensed matter physics, condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phon ...
s do not depend on the volume of the crystal or on the presence of other phonons, and the thermal expansion (and thus γ) is zero. When the restoring force is non-linear in the displacement, the phonon frequencies ω
i change with the volume
. The Grüneisen parameter of an individual vibrational mode
can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency
:
Relationship between microscopic and thermodynamic models
Using the
quasi-harmonic approximation The quasi-harmonic approximation is a phonon-based model of solid-state physics used to describe volume-dependent thermal effects, such as the thermal expansion. It is based on the assumption that the harmonic approximation holds for every value of ...
for atomic vibrations, the macroscopic Grüneisen parameter () can be related to the description of how the vibrational frequencies (
phonons
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
) within a crystal are altered with changing volume (i.e. 's).
For example, one can show that
if one defines
as the weighted average
where
's are the partial vibrational mode contributions to the heat capacity, such that
Proof
To prove this relation, it is easiest to introduce the heat capacity per particle
; so one can write
This way, it suffices to prove
Left-hand side (def):