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Collections

The current EquationOfStates contain

EquationOfState
├─ AntonSchmidt
├─ BreenanStacey
├─ FiniteStrainEquationOfState
│  ├─ BirchMurnaghan2nd
│  ├─ BirchMurnaghan3rd
│  ├─ BirchMurnaghan4th
│  ├─ PoirierTarantola2nd
│  ├─ PoirierTarantola3rd
│  └─ PoirierTarantola4th
├─ Murnaghan
└─ Vinet

Usage

Construct an EquationOfState

We will use BirchMurnaghan3rd as an example.

A BirchMurnaghan3rd can be constructed from scratch, as shown above. It can also be constructed from an existing BirchMurnaghan3rd, with Setfield.jl @set! macro:

julia> using Setfield

julia> eos = Murnaghan(1, 2, 3.0)
Murnaghan{Float64}(1.0, 2.0, 3.0, 0.0)

julia> @set! eos.v0 = 4
Murnaghan{Float64}(4.0, 2.0, 3.0, 0.0)

julia> eos
Murnaghan{Float64}(4.0, 2.0, 3.0, 0.0)

To modify multiple fields (say, :v0, :b′0, :b′′0, :e0) at a time, use @batchlens from Kaleido.jl:

julia> using Setfield, Kaleido

julia> lens = @batchlens(begin
           _.v0
           _.b′0
           _.b′′0
           _.e0
       end)
IndexBatchLens(:v0, :b′0, :b′′0, :e0)

julia> eos = BirchMurnaghan4th(1, 2.0, 3, 4)
BirchMurnaghan4th{Float64}(1.0, 2.0, 3.0, 4.0, 0.0)

julia> set(eos, lens, (5, 6, 7, 8))
BirchMurnaghan4th{Float64}(5.0, 2.0, 6.0, 7.0, 8.0)

Users can access BirchMurnaghan3rd's elements by "dot notation":

julia> eos = BirchMurnaghan3rd(1, 2, 3, 4.0)
4-element BirchMurnaghan3rd{Float64}:
 1.0
 2.0
 3.0
 4.0

julia> eos.v0
1.0

Calculate energy on an EquationOfState

The $E(V)$ relation of equations of state are listed as below:

  1. Murnaghan:

    \[E(V) = E_{0}+K_{0} V_{0}\left[\frac{1}{K_{0}^{\prime}\left(K_{0}^{\prime}-1\right)}\left(\frac{V}{V_{0}}\right)^{1-K_{0}^{\prime}}+\frac{1}{K_{0}^{\prime}} \frac{V}{V_{0}}-\frac{1}{K_{0}^{\prime}-1}\right].\]

  2. BirchMurnaghan2nd:

    \[E(V) = E_{0} + \frac{9}{8} B_{0} V_{0} \left(\left( V / V_0 \right)^{-2 / 3}-1\right)^{2}.\]

  3. BirchMurnaghan3rd:

    \[E(V) = E_{0}+\frac{9}{16} V_{0} B_{0} \frac{\left(x^{2 / 3}-1\right)^{2}}{x^{7 / 3}}\left\{x^{1 / 3}\left(B_{0}^{\prime}-4\right)-x\left(B_{0}^{\prime}-6\right)\right\}.\]

    where $x = V / V_0$, and $f = \frac{ 1 }{ 2 } \bigg[ \bigg( \frac{ V_0 }{ V } \bigg)^{2/3} - 1 \bigg]$.

  4. BirchMurnaghan4th:

    \[E(V) = E_{0}+\frac{3}{8} V_{0} B_{0} f^{2}\left[\left(9 H-63 B_{0}^{\prime}+143\right) f^{2}+12\left(B_{0}^{\prime}-4\right) f+12\right].\]

    where $H = B_0 B_0'' + (B_0')^2$.

  5. PoirierTarantola2nd:

    \[E(V) = E_{0}+\frac{1}{2} B_{0} V_{0} \ln ^{2} x.\]

  6. PoirierTarantola3rd:

    \[E(V) = E_{0}+\frac{1}{6} B_{0} V_{0} \ln ^{2} x\left[\left(B_{0}^{\prime}+2\right) \ln x+3\right].\]

  7. PoirierTarantola4th:

    \[E(V) = E_{0}+\frac{1}{24} B_{0} V_{0} \ln ^{2} x\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{2} x\right. \left.+4\left(B_{0}^{\prime}+2\right) \ln x+12\right\}.\]

    where $H = B_0 B_0'' + (B_0')^2$.

  8. Vinet:

    \[E(V) = E_{0}+\frac{9}{16} V_{0} B_{0} \frac{\left(x^{2 / 3}-1\right)^{2}}{x^{7 / 3}}\left\{x^{1 / 3}\left(B_{0}^{\prime}-4\right)-x\left(B_{0}^{\prime}-6\right)\right\}.\]

  9. AntonSchmidt:

    \[E(V)=\frac{\beta V_{0}}{n+1}\left(\frac{V}{V_{0}}\right)^{n+1}\left[\ln \left(\frac{V}{V_{0}}\right)-\frac{1}{n+1}\right]+E_{\infty}.\]

Calculate pressure on an EquationOfState

The $P(V)$ relation of equations of state are listed as below:

  1. Murnaghan:

    \[P(V) = \frac{B_{0}}{B_{0}^{\prime}}\left[\left(\frac{V_{0}}{V}\right)^{B_{0}^{\prime}}-1\right].\]

  2. BirchMurnaghan2nd:

    \[P(V) = \frac{3}{2} B_{0}\left(x^{-7 / 3}-x^{-5 / 3}\right).\]

  3. BirchMurnaghan3rd:

    \[P(V) = \frac{3}{8} B_{0} \frac{x^{2 / 3}-1}{x^{10 / 3}}\left\{3 B_{0}^{\prime} x-16 x-3 x^{1 / 3}\left(B_{0}^{\prime}-4\right)\right\}.\]

  4. BirchMurnaghan4th:

    \[P(V) = \frac{1}{2} B_{0}(2 f+1)^{5 / 2}\left\{\left(9 H-63 B_{0}^{\prime}+143\right) f^{2}\right.\left.+9\left(B_{0}^{\prime}-4\right) f+6\right\}.\]

  5. PoirierTarantola2nd:

    \[P(V) = -\frac{B_{0}}{x} \ln x.\]

  6. PoirierTarantola3rd:

    \[P(V) = -\frac{B_{0} \ln x}{2 x}\left[\left(B_{0}^{\prime}+2\right) \ln x+2\right].\]

  7. PoirierTarantola4th:

    \[P(V) = -\frac{B_{0} \ln x}{6 x}\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{2} x+3\left(B_{0}^{\prime}+6\right) \ln x+6\right\}.\]

  8. Vinet:

    \[P(V) = 3 B_{0} \frac{1-\eta}{\eta^{2}} \exp \left\{-\frac{3}{2}\left(B_{0}^{\prime}-1\right)(\eta-1)\right\}.\]

  9. AntonSchmidt:

    \[P(V) = -\beta\left(\frac{V}{V_{0}}\right)^{n} \ln \left(\frac{V}{V_{0}}\right).\]

Calculate bulk modulus on an EquationOfState

The $B(V)$ relation of equations of state are listed as below:

  1. BirchMurnaghan2nd:

    \[B(V) = B_{0}(7 f+1)(2 f+1)^{5 / 2}.\]

  2. BirchMurnaghan3rd:

    \[B(V) = \frac{B_{0}}{8 x^{10 / 3}}\left\{x^{5 / 3}\left(15 B_{0}^{\prime}-80\right)-x\left(42 B_{0}^{\prime}-196\right)\right.\left.+27 x^{1 / 3}\left(B_{0}^{\prime}-4\right)\right\}.\]

  3. BirchMurnaghan4th:

    \[B(V) = \frac{1}{6} B_{0}(2 f+1)^{5 / 2}\left\{\left(99 H-693 B_{0}^{\prime}+1573\right) f^{3}\right.\left.+\left(27 H-108 B_{0}^{\prime}+105\right) f^{2}+6\left(3 B_{0}^{\prime}-5\right) f+6\right\}.\]

  4. PoirierTarantola2nd:

    \[B(V) = \frac{B_{0}}{x}(1-\ln x).\]

  5. PoirierTarantola3rd:

    \[B(V) = -\frac{B_{0}}{2 x}\left[\left(B_{0}^{\prime}+2\right) \ln x(\ln x-1)-2\right].\]

  6. PoirierTarantola4th:

    \[B(V) = -\frac{B_{0}}{6 x}\left\{\left(H+3 B_{0}^{\prime}+3\right) \ln ^{3} x-3\left(H+2 B_{0}^{\prime}+1\right) \ln ^{2} x\right.\left.-6\left(B_{0}^{\prime}+1\right) \ln x-6\right\}.\]

  7. Vinet:

    \[B(V) = -\frac{B_{0}}{2 \eta^{2}}\left[3 \eta(\eta-1)\left(B_{0}^{\prime}-1\right)+2(\eta-2)\right]\times \exp \left\{-\frac{3}{2}\left(B_{0}^{\prime}-1\right)(\eta-1)\right\}.\]

  8. AntonSchmidt:

    \[B(V) = \beta\left(\frac{V}{V_{0}}\right)^{n}\left[1+n \ln \frac{V}{V_{0}}\right].\]

Public interfaces

EquationsOfState.Collections.Energy — Type
Energy()
(::EquationOfState)(::Energy)(v)
(::EquationOfState)(::Energy)

Return the energy of an EquationOfState on volume v. If eos has units, v must also has.

Return a function-like object that takes a volume as a variable, suitable for mapping onto an array.

Examples

julia> f = Vinet(1, 2, 3)(Energy());

julia> map(f, 1:1:10)
10-element Array{Float64,1}:
 0.0
 0.367905230584308
 0.7652477289745814
 1.0516459435179233
 1.2560420090256408
 1.405149833626178
 1.5165867441792136
 1.6017034530570884
 1.6679539823686644
 1.7203642945516917

However, these methods are preserved for special cases (see #52). In most cases, the Julia do block syntax is preferred:

julia> map(1:1:10) do v
           Vinet(1, 2, 3)(Energy())(v)
       end
10-element Array{Float64,1}:
 0.0
 0.367905230584308
 0.7652477289745814
 1.0516459435179235
 1.2560420090256412
 1.405149833626178
 1.5165867441792138
 1.6017034530570884
 1.6679539823686644
 1.7203642945516917
source
EquationsOfState.Collections.Pressure — Type
Pressure()
(::EquationOfState)(::Pressure)(v)
(::EquationOfState)(::Pressure)

Return the pressure of an EquationOfState on volume v. If eos has units, v must also has.

Examples

julia> f = Vinet(1, 2, 3)(Pressure());

julia> map(f, 1:1:10)
10-element Array{Float64,1}:
  0.0
 -0.45046308428750254
 -0.3384840350043251
 -0.24010297221667418
 -0.17314062272722755
 -0.12795492664586872
 -0.09677154467733216
 -0.07468060255179591
 -0.05864401631176751
 -0.04674768462396211
source
EquationsOfState.Collections.BulkModulus — Type
BulkModulus()
(::EquationOfState)(::BulkModulus)(v)
(::EquationOfState)(::BulkModulus)

Return the bulk modulus of an EquationOfState on volume v. If eos has units, v must also has.

Examples

julia> f = BirchMurnaghan3rd(1, 2, 3)(BulkModulus());

julia> map(f, 1:1:10)
10-element Array{Float64,1}:
 2.0
 0.9216086833346415
 0.444903691617472
 0.2540009203153288
 0.16193296566524193
 0.11130584492987289
 0.08076305569984538
 0.06103515625
 0.047609811583958425
 0.03808959181078831
source
EquationsOfState.Collections.Murnaghan — Type
Murnaghan(v0, b0, b′0, e0)

Create a Murnaghan equation of state. The elements' type will be handled automatically.

This equation of state can have units. The units are specified in Unitful.jl's @u_str style.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

Examples

julia> Murnaghan(1, 2, 3.0)
Murnaghan{Float64}(1.0, 2.0, 3.0, 0.0)

julia> Murnaghan(Int8(1), 2//1, 3.0, 4)
Murnaghan{Float64}(1.0, 2.0, 3.0, 4.0)

julia> Murnaghan(1u"nm^3", 2u"GPa", 3, 3.0u"eV")
Murnaghan{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0, 3.0 eV)
source
EquationsOfState.Collections.BirchMurnaghan2nd — Type
BirchMurnaghan2nd(v0, b0, e0)

Create a Birch–Murnaghan 2nd order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: BirchMurnaghan3rd, BirchMurnaghan4th

Examples

julia> BirchMurnaghan2nd(1, 2.0)
BirchMurnaghan2nd{Float64}(1.0, 2.0, 0.0)

julia> BirchMurnaghan2nd(Int8(1), 2//1, 0.0)
BirchMurnaghan2nd{Float64}(1.0, 2.0, 0.0)

julia> BirchMurnaghan2nd(1u"nm^3", 2u"GPa", 3.0u"eV")
BirchMurnaghan2nd{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0 eV)
source
EquationsOfState.Collections.BirchMurnaghan3rd — Type
BirchMurnaghan3rd(v0, b0, b′0, e0)

Create a Birch–Murnaghan 3rd order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: BirchMurnaghan2nd, BirchMurnaghan4th

Examples

julia> BirchMurnaghan3rd(1, 2.0, 3)
BirchMurnaghan3rd{Float64}(1.0, 2.0, 3.0, 0.0)

julia> BirchMurnaghan3rd(Int8(1), 2//1, 4, 0.0)
BirchMurnaghan3rd{Float64}(1.0, 2.0, 4.0, 0.0)

julia> BirchMurnaghan3rd(1u"nm^3", 2u"GPa", 4.0, 3u"eV")
BirchMurnaghan3rd{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 4.0, 3.0 eV)
source
EquationsOfState.Collections.BirchMurnaghan4th — Type
BirchMurnaghan4th(v0, b0, b′0, b′′0, e0)

Create a Birch–Murnaghan 4th order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • b′′0: the second-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: BirchMurnaghan2nd, BirchMurnaghan4th

Examples

julia> BirchMurnaghan4th(1, 2.0, 3, 4)
BirchMurnaghan4th{Float64}(1.0, 2.0, 3.0, 4.0, 0.0)

julia> BirchMurnaghan4th(Int8(1), 2//1, 4, 5.0, Float16(6))
BirchMurnaghan4th{Float64}(1.0, 2.0, 4.0, 5.0, 6.0)

julia> BirchMurnaghan4th(1u"nm^3", 2u"GPa", 3.0, 4u"1/GPa", 5u"eV")
BirchMurnaghan4th{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0, 4.0 GPa⁻¹, 5.0 eV)
source
EquationsOfState.Collections.PoirierTarantola2nd — Type
PoirierTarantola2nd(v0, b0, e0)

Create a Poirier–Tarantola order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: PoirierTarantola3rd, PoirierTarantola4th

Examples

julia> PoirierTarantola2nd(1, 2.0)
PoirierTarantola2nd{Float64}(1.0, 2.0, 0.0)

julia> PoirierTarantola2nd(Int8(1), 2//1, 3.0)
PoirierTarantola2nd{Float64}(1.0, 2.0, 3.0)

julia> PoirierTarantola2nd(1u"nm^3", 2u"GPa", 3.0u"eV")
PoirierTarantola2nd{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0 eV)
source
EquationsOfState.Collections.PoirierTarantola3rd — Type
PoirierTarantola3rd(v0, b0, b′0, e0)

Create a Poirier–Tarantola 3rd order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: PoirierTarantola2nd, PoirierTarantola4th

Examples

julia> PoirierTarantola3rd(1, 2.0, 3)
PoirierTarantola3rd{Float64}(1.0, 2.0, 3.0, 0.0)

julia> PoirierTarantola3rd(Int8(1), 2//1, 3.0, Float16(4))
PoirierTarantola3rd{Float64}(1.0, 2.0, 3.0, 4.0)

julia> PoirierTarantola3rd(1u"nm^3", 2u"GPa", 3, 4.0u"eV")
PoirierTarantola3rd{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0, 4.0 eV)
source
EquationsOfState.Collections.PoirierTarantola4th — Type
PoirierTarantola4th(v0, b0, b′0, b′′0, e0)

Create a Birch–Murnaghan 4th order equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • b′′0: the second-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

See also: PoirierTarantola2nd, PoirierTarantola3rd

Examples

julia> PoirierTarantola4th(1, 2.0, 3, 4)
PoirierTarantola4th{Float64}(1.0, 2.0, 3.0, 4.0, 0.0)

julia> PoirierTarantola4th(Int8(1), 2//1, 3.0, Float16(4), 5)
PoirierTarantola4th{Float64}(1.0, 2.0, 3.0, 4.0, 5.0)

julia> PoirierTarantola4th(1u"nm^3", 2u"GPa", 3, 4u"1/GPa", 5.0u"eV")
PoirierTarantola4th{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0, 4.0 GPa⁻¹, 5.0 eV)
source
EquationsOfState.Collections.Vinet — Type
Vinet(v0, b0, b′0, e0)

Create a Vinet equation of state. The elements' type will be handled automatically.

Arguments

  • v0: the volume of solid at zero pressure.
  • b0: the bulk modulus of solid at zero pressure.
  • b′0: the first-order pressure-derivative bulk modulus of solid at zero pressure.
  • e0: the energy of solid at zero pressure. Its default value is 0u"eV" (0), if other parameters have (no) units.

Examples

julia> Vinet(1, 2.0, 3)
Vinet{Float64}(1.0, 2.0, 3.0, 0.0)

julia> Vinet(Int8(1), 2//1, 3.0, Float16(4))
Vinet{Float64}(1.0, 2.0, 3.0, 4.0)

julia> Vinet(1u"nm^3", 2u"GPa", 3, 4.0u"eV")
Vinet{Quantity{Float64,D,U} where U where D}(1.0 nm³, 2.0 GPa, 3.0, 4.0 eV)
source