Add CoulombTwoBody

ohno · Apr 20, 2024 · 334e930 · 334e930
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diff --git a/README.md b/README.md
@@ -34,7 +34,6 @@ E(H)
 Helium cation has symbol $\mathrm{He}^+$ and atomic number 2 ($Z=2$). Therefore the ground state ($n=1$) energy is $-2 E_\mathrm{h}$.
 
 ```julia
-using Antique
 He⁺ = HydrogenAtom(Z=2)
 E(He⁺)
 # output> -2.0
@@ -51,6 +50,7 @@ There are more examples on each model page.
 - [Morse Potential](https://ohno.github.io/Antique.jl/stable/MorsePotential/) `MorsePotential`
 - [Rigid Rotor](https://ohno.github.io/Antique.jl/stable/RigidRotor/) `RigidRotor`
 - [Hydrogen Atom](https://ohno.github.io/Antique.jl/stable/HydrogenAtom/) `HydrogenAtom`
+- [Coulomb 2-Body System](https://ohno.github.io/Antique.jl/stable/HydrogenAtom/) `CoulombTwoBody`
 
 ## Future Works
 

diff --git a/developer/dev.ipynb b/developer/dev.ipynb
diff --git a/docs/make.jl b/docs/make.jl
@@ -21,16 +21,17 @@ makedocs(;
   ),
   pages=[
     "Home" => "index.md",
-    "Delta Potential"         => "DeltaPotential.md"       ,
-    "Infinite Potential Well" => "InfinitePotentialWell.md",
-    "Harmonic Oscillator"     => "HarmonicOscillator.md"   ,
-    "Morse Potential"         => "MorsePotential.md"       ,
-    "Pöschl-Teller Potential" => "PoschlTeller.md"         ,
-    "Hydrogen Atom"           => "HydrogenAtom.md"         ,
-    "Rigid Rotor"             => "RigidRotor.md"           ,
-    "Infinite Potential Well 3D" => "InfinitePotentialWell3D.md",
-    # "Harmonic Oscillator 3D"  => "HarmonicOscillator3D.md" ,
-    # "API reference"           => "API.md"                  ,
+    "Delta Potential"              => "DeltaPotential.md"         ,
+    "Infinite Potential Well"      => "InfinitePotentialWell.md"  ,
+    "Harmonic Oscillator"          => "HarmonicOscillator.md"     ,
+    "Morse Potential"              => "MorsePotential.md"         ,
+    "Pöschl-Teller Potential"      => "PoschlTeller.md"           ,
+    "Rigid Rotor"                  => "RigidRotor.md"             ,
+    "Hydrogen Atom"                => "HydrogenAtom.md"           ,
+    "Coulomb 2-Body System"        => "CoulombTwoBody.md"         ,
+    # "Infinite Potential Well 3D" => "InfinitePotentialWell3D.md",
+    # "Harmonic Oscillator 3D"     => "HarmonicOscillator3D.md"   ,
+    # "API reference"              => "API.md"                    ,
   ],
 )
 

diff --git a/docs/src/CoulombTwoBody.md b/docs/src/CoulombTwoBody.md
@@ -0,0 +1,274 @@
+```@meta
+CurrentModule = Antique
+```
+
+# Coulomb 2-Body System
+
+This is the model of two particles interacting through Coulomb forces such as positronium, muonium, hydrogen atoms, deuterium atoms, etc.
+
+## Definitions
+
+This model is described with the time-independent Schrödinger equation
+```math
+  \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),
+```
+and the Hamiltonian
+```math
+  \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + V(r),
+```
+where $\mu=\left(\frac{1}{m_1}+\frac{1}{m_2}\right)^{-1}$ is the reduced mass of particle 1 and particle 2. The potential includes only Coulomb interaction and it does not include fine or hyperfine interactions in this model. Parameters are specified with the following struct.
+
+#### Parameters
+```@docs; canonical=false
+Antique.CoulombTwoBody
+```
+
+#### Potential
+```@docs; canonical=false
+Antique.V(::CoulombTwoBody, ::Any)
+```
+
+#### Eigen Values
+```@docs; canonical=false
+Antique.E(::CoulombTwoBody)
+```
+
+#### Eigen Functions
+```@docs; canonical=false
+Antique.ψ(::CoulombTwoBody, ::Any, ::Any, ::Any)
+```
+
+#### Radial Functions
+```@docs; canonical=false
+Antique.R(::CoulombTwoBody, ::Any)
+```
+
+#### Associated Laguerre Polynomials
+```@docs; canonical=false
+Antique.L(::CoulombTwoBody, ::Any)
+```
+
+#### Spherical Harmonics
+```@docs; canonical=false
+Antique.Y(::CoulombTwoBody, ::Any, ::Any)
+```
+
+#### Associated Legendre Polynomials
+```@docs; canonical=false
+Antique.P(::CoulombTwoBody, ::Any)
+```
+
+## Usage & Examples
+
+[Install Antique.jl](@ref Install) for the first use and run `using Antique` before each use. The energy `E()`, wavefunction `ψ()`, potential `V()` and some other functions are suppoted. In this system, the model is generated by `CoulombTwoBody` and several parameters `z₁`, `z₂`, `m₁`, `m₂`, `mₑ`, `a₀`, `Eₕ` and `ℏ` are set as optional arguments.
+
+```@example CTB
+using Antique
+Ps = CoulombTwoBody(z₁=-1, z₂=1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)
+; #hide
+```
+
+#### Parameters
+
+```@repl CTB
+Ps.z₁
+Ps.z₂
+Ps.m₁
+Ps.m₂
+Ps.mₑ
+Ps.a₀
+Ps.Eₕ
+Ps.ℏ
+```
+
+#### Eigen Values
+
+Examples of [positronium](https://en.wikipedia.org/wiki/Positronium#Energy_levels):
+
+```@repl CTB
+E(Ps, n=1)
+E(Ps, n=2)
+```
+
+#### Mass and Charge Dependence
+
+The values of masses are cited from the 2018 CODATA recommended values, [E. Tiesinga, et al., Rev. Mod. Phys. 93, 025010 (2021)](https://doi.org/10.1103/RevModPhys.93.025010).
+
+```@example CTB
+me = 1.0           # me # 
+mµ = 206.7682830   # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mmusme
+mp = 1836.15267343 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mpsme
+md = 3670.48296788 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mdsme
+mt = 5496.92153573 # me # https://physics.nist.gov/cgi%2Dbin/cuu/Value?mtsme
+mh = 5495.88528007 # me # https://physics.nist.gov/cgi-bin/cuu/Value?mhsme
+ma = 7294.29954142 # me # https://physics.nist.gov/cgi-bin/cuu/Value?malsme
+
+Ps = CoulombTwoBody(m₁=me, m₂=me)
+Mu = CoulombTwoBody(m₁=me, m₂=mµ)
+H  = CoulombTwoBody(m₁=me, m₂=mp)
+D  = CoulombTwoBody(m₁=me, m₂=md)
+T  = CoulombTwoBody(m₁=me, m₂=mt)
+BO = CoulombTwoBody(m₁=me, m₂=Inf)
+
+He3⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh)
+He4⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=ma)
+He∞⁺ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=Inf)
+
+pμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp)
+dμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=md)
+tμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mt)
+bμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=Inf)
+hμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=mh)
+αμ = CoulombTwoBody(z₁=-1, z₂=+2, m₁=mµ, m₂=ma)
+
+println("    \tE / Eₕ")
+println("Ps  \t", E(Ps))
+println("Mu  \t", E(Mu))
+println("H   \t", E(H))
+println("D   \t", E(D))
+println("T   \t", E(T))
+println("∞H  \t", E(BO))
+println("³He⁺\t", E(He3⁺))
+println("⁴He⁺\t", E(He4⁺))
+println("∞He⁺\t", E(He∞⁺))
+println("pμ  \t", E(pμ))
+println("dμ  \t", E(dμ))
+println("tμ  \t", E(tμ))
+println("bμ  \t", E(bμ))
+println("hμ  \t", E(hμ))
+println("αμ  \t", E(αμ))
+```
+
+```@example CTB
+println("   \t<δ³(r)> / a₀⁻³")
+println("1/8π =\t", 1/8/π)
+println("Ps    \t", abs(ψ(Ps,0,0,0))^2)
+println("Mu    \t", abs(ψ(Mu,0,0,0))^2)
+println("H     \t", abs(ψ(H ,0,0,0))^2)
+println("D     \t", abs(ψ(D ,0,0,0))^2)
+println("T     \t", abs(ψ(T ,0,0,0))^2)
+println("∞H    \t", abs(ψ(BO,0,0,0))^2)
+println("1/π = \t", 1/π)
+```
+
+#### Lifetime of Positronium
+
+The lifetime $\tau$ of positronium (Ps, $\mathrm{e}^+\mathrm{e}^-$) is written as
+
+```math
+\tau = \frac{1}{\Gamma},
+```
+
+```math
+\Gamma = 4 \pi \alpha^4 c {a_0}^2 \langle\delta^3(\pmb{r})\rangle.
+```
+
+where $\langle\delta^3(\pmb{r})\rangle = \langle\psi|\delta^3(\pmb{r})|\psi\rangle = |\psi(\pmb{0})|^2 = \frac{1}{8\pi} a_0^{-3} \simeq 2.685\times10^{29}~\mathrm{m}^{-3}$ is the value of probability density at the origin ($r=0$). Reference:
+- (7.169) in D. J. Griffiths, Introduction to Elementary Particles (John Wiley & Sons, Inc. 1987) ISBN 0-471-60386-4
+- [S. Berko, H. N. Pendleton, Annual Review of Nuclear and Particle Science, 30, 543 (1980)](https://doi.org/10.1146/annurev.ns.30.120180.002551)
+- [A. M. Frolov, S. I. Kryuchkov, and V. H. Smith, Jr., Phys. Rev. A, 51, 4514 (1995)](https://doi.org/10.1103/PhysRevA.51.4514)
+
+```@example CTB
+α  = 7.2973525693e-3    #       # https://physics.nist.gov/cgi-bin/cuu/Value?alph
+c   = 299792458         # m s-1 # https://physics.nist.gov/cgi-bin/cuu/Value?c
+a₀  = 5.29177210903e-11 # m     # https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0
+
+Ps = CoulombTwoBody(z₁=1, z₂=-1, m₁=1.0, m₂=1.0, mₑ=1.0, a₀=1.0, Eₕ=1.0, ℏ=1.0)
+δ = abs(ψ(Ps,0,0,0))^2 * a₀^(-3)
+Γ = 4 * π * α^4 * c *  a₀^2 * δ
+τ = 1/Γ
+println("<δ> = ", abs(ψ(Ps,0,0,0))^2, " a₀⁻³")
+println("    = ", δ, " m⁻³")
+println("Γ   = ", Γ / 1e9, " GHz")
+println("τ   = ", τ / 1e-12, " ps")
+```
+
+#### Hyperfine Splitting
+
+The hyperfine splitting of hydrogen atoms is given as
+
+```math
+\Delta E (\mathrm{H}) = -\frac{2}{3} \mu_0 \gamma_\mathrm{p} \gamma_\mathrm{e} \hbar^2 \langle\delta^3(\pmb{r})\rangle
+```
+
+in [Griffiths(1982)](https://doi.org/10.1119/1.12733). This fomula is not available for deuterium (D) and positronium (Ps). Because of the different spin between the proton and the deuteron for D, the contribution of positron-electron pair annihilation for Ps. Note the definition of gyromagnetic ratio. The mass of protons is used for all nucleons and nuclei:
+
+```math
+\begin{aligned}
+&\gamma_\mathrm{e}   = \frac{-e}{2 m_\mathrm{e}} g_\mathrm{e},     &
+&\gamma_\mathrm{e^+} = \frac{+e}{2 m_\mathrm{e}} g_\mathrm{e},     &
+&\gamma_\mathrm{\mu} = \frac{-e}{2 m_\mathrm{\mu}} g_\mathrm{\mu}, \\
+&\gamma_\mathrm{p}   = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{p},     &
+&\gamma_\mathrm{d}   = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{d},     &
+&\gamma_\mathrm{t}   = \frac{+e}{2 m_\mathrm{p}} g_\mathrm{t},     &
+&\gamma_\mathrm{h}   = \frac{+2e}{2 m_\mathrm{p}} g_\mathrm{h}.    &
+\end{aligned}
+```
+
+The value of probability density at the origin is $\langle\delta^3(\pmb{r})\rangle = \langle\psi|\delta^3(\pmb{r})|\psi\rangle = |\psi(\pmb{0})|^2 \simeq \frac{1}{\pi} a_0^{-3} \simeq 2.148\times10^{30}~\mathrm{m}^{-3}$ in Mu, H, D and T. This values are very different in Ps, $^3\mathrm{He}^+$ and muonic hydrogen ($\mathrm{p\mu}$) due to the difference of reduced masses and charges. The energy can be converted to frequency (Hz) by $v = \Delta E / h$.
+
+```@example CTB
+a₀ = 5.29177210903e-11   # m  # https://physics.nist.gov/cgi-bin/cuu/Value?bohrrada0
+Eₕ = 4.3597447222071e-18 # J  # https://physics.nist.gov/cgi-bin/cuu/Value?hr
+ℏ  = 1.054571817e-34     # J s # https://physics.nist.gov/cgi-bin/cuu/Value?hbar
+
+me = 9.1093837015e-31  # kg # https://physics.nist.gov/cgi-bin/cuu/Value?me
+mµ = 1.883531627e-28   # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mmu
+mp = 1.67262192369e-27 # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mp
+md = 3.3435837724e-27  # kg # https://physics.nist.gov/cgi-bin/cuu/Value?md
+mt = 5.0073567446e-27  # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mt
+mh = 5.0064127796e-27  # kg # https://physics.nist.gov/cgi-bin/cuu/Value?mh
+
+e  = 1.602176634e-19  # C      # https://physics.nist.gov/cgi-bin/cuu/Value?e
+µ₀ = 1.25663706212e-6 # N A-2  # https://physics.nist.gov/cgi-bin/cuu/Value?mu0
+h  = 6.62607015e-34   # J Hz-1 # https://physics.nist.gov/cgi-bin/cuu/Value?h
+eV = 1.602176634e-19  # J      # https://physics.nist.gov/cgi-bin/cuu/Value?evj
+
+ge = 2.00231930436256 # https://physics.nist.gov/cgi-bin/cuu/Value?gem
+gµ = 2.0023318418     # https://physics.nist.gov/cgi-bin/cuu/Value?gmum
+gp = 5.5856946893     # https://physics.nist.gov/cgi-bin/cuu/Value?gp
+gd = 0.8574382338     # https://physics.nist.gov/cgi-bin/cuu/Value?gdn
+gt = 5.957924931      # https://physics.nist.gov/cgi-bin/cuu/Value?gtn
+gh = -4.255250615     # https://physics.nist.gov/cgi-bin/cuu/Value?ghn
+
+Ps = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=me, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+Mu = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mµ, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+H  = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+D  = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=md, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+T  = CoulombTwoBody(z₁=-1, z₂=+1, m₁=me, m₂=mt, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+he = CoulombTwoBody(z₁=-1, z₂=+2, m₁=me, m₂=mh, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+pμ = CoulombTwoBody(z₁=-1, z₂=+1, m₁=mµ, m₂=mp, mₑ=me, a₀=a₀, Eₕ=Eₕ, ℏ=ℏ)
+
+ΔE_H  = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gp / mp * abs(ψ(H,0,0,0))^2
+ΔE_D  =   1 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gd / mp * abs(ψ(D,0,0,0))^2
+ΔE_T  = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gt / mp * abs(ψ(H,0,0,0))^2
+ΔE_Ps = 7/6 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * ge / me * abs(ψ(Ps,0,0,0))^2
+ΔE_Mu = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gµ / mµ * abs(ψ(Mu,0,0,0))^2
+ΔE_he = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * ge / me * gh / mp * abs(ψ(he,0,0,0))^2
+ΔE_pµ = 2/3 / 4 * µ₀ * ℏ^2 * e^2 * gµ / mµ * gp / mp * abs(ψ(pµ,0,0,0))^2
+
+println("H \t", ΔE_H  / h / 1e6,  " MHz (Antique.jl + CODATA2018)")
+println("  \t", "1420.405751768(1)  MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)")
+println("D \t", ΔE_D  / h / 1e6,  " MHz (Antique.jl + CODATA2018)")
+println("  \t", "327.384352522(2)   MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)")
+println("T \t", ΔE_T  / h / 1e6,  " MHz (Antique.jl + CODATA2018)")
+println("  \t", "1516.701470773(8)  MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)")
+println("Ps\t", ΔE_Ps / h / 1e6,  " MHz (Antique.jl + CODATA2018)")
+println("  \t", "203391.7(6)        MHz (https://doi.org/10.48550/arXiv.hep-ph/0310099)")
+println("Mu\t", ΔE_Mu / h / 1e6,  "  MHz (Antique.jl + CODATA2018)")
+println("  \t", "4463.30278(5)      MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)")
+println("³He⁺\t", ΔE_he / h / 1e6,  " MHz (Antique.jl + CODATA2018)")
+println("  \t", "-8665.649867(10)   MHz (https://doi.org/10.48550/arXiv.hep-ph/0109128)")
+println("µp\t", ΔE_pµ / h / 1e12, "  THz (Antique.jl + CODATA2018)")
+println("  \t", 0.182725*eV / h / 1e12 ,"  THz (https://doi.org/10.1119/1.12733, https://doi.org/10.1016/j.nimb.2012.04.001)")
+```
+
+## Testing
+
+Unit testing and Integration testing were done using computer algebra system ([Symbolics.jl](https://symbolics.juliasymbolics.org/stable/)) and numerical integration ([QuadGK.jl](https://juliamath.github.io/QuadGK.jl/stable/)). The test script is [here](https://github.com/ohno/Antique.jl/blob/main/test/CoulombTwoBody.jl).
+
+```@eval
+using Markdown
+using Antique
+Markdown.parse(Antique.load("../../test/result/CoulombTwoBody.log"))
+```
diff --git a/docs/src/HydrogenAtom.md b/docs/src/HydrogenAtom.md
@@ -136,7 +136,7 @@ Potential energy curve:
 using CairoMakie
 
 f = Figure()
-ax = Axis(f[1,1], xlabel=L"$r~/~a_0$", ylabel=L"$V(r)~/~E_\mathrm{h}0$",  limits=(0.0,15.0,-2.0,0.2))
+ax = Axis(f[1,1], xlabel=L"$r~/~a_0$", ylabel=L"$V(r)~/~E_\mathrm{h}$",  limits=(0.0,15.0,-2.0,0.2))
 lines!(ax, 0.1:0.01:20, r -> V(H, r))
 f
 ```

diff --git a/docs/src/index.md b/docs/src/index.md
@@ -34,7 +34,6 @@ E(H)
 Helium cation has symbol $\mathrm{He}^+$ and atomic number 2 ($Z=2$). Therefore the ground state ($n=1$) energy is $-2 E_\mathrm{h}$.
 
 ```julia
-using Antique
 He⁺ = HydrogenAtom(Z=2)
 E(He⁺)
 # output> -2.0
@@ -46,12 +45,6 @@ There are more examples on each model page.
 
 ```@raw html
 <div class="catalog">
-  <div class="item">
-    <a target="_blank" href="./DeltaPotential">
-      <img src="assets/fig/DeltaPotential.png" alt="DeltaPotential"/>
-    </a>
-    <code>DeltaPotential</code>
-  </div>
   <div class="item">
     <a target="_blank" href="./InfinitePotentialWell">
       <img src="assets/fig/InfinitePotentialWell.png" alt="InfinitePotentialWell"/>
@@ -76,21 +69,18 @@ There are more examples on each model page.
     </a>
     <code>MorsePotential</code>
   </div>
-  <div class="item">
-    <a target="_blank" href="./RigidRotor">
-      <img src="assets/fig/RigidRotor.png" alt="RigidRotor"/>
-    </a>
-    <code>RigidRotor</code>
-  </div>
-  <div class="item">
-    <a target="_blank" href="./HydrogenAtom">
-      <img src="assets/fig/HydrogenAtom.png" alt="HydrogenAtom"/>
-    </a>
-    <code>HydrogenAtom</code>
-  </div>
 </div>
 ```
 
+- [Delta Potential](https://ohno.github.io/Antique.jl/stable/DeltaPotential/) `DeltaPotential`
+- [Infinite Potential Well](https://ohno.github.io/Antique.jl/stable/InfinitePotentialWell/) `InfinitePotentialWell`
+- [Harmonic Oscillator](https://ohno.github.io/Antique.jl/stable/HarmonicOscillator/) `HarmonicOscillator`
+- [PoschlTeller](https://ohno.github.io/Antique.jl/stable/PoschlTeller/) `PoschlTeller`
+- [Morse Potential](https://ohno.github.io/Antique.jl/stable/MorsePotential/) `MorsePotential`
+- [Rigid Rotor](https://ohno.github.io/Antique.jl/stable/RigidRotor/) `RigidRotor`
+- [Hydrogen Atom](https://ohno.github.io/Antique.jl/stable/HydrogenAtom/) `HydrogenAtom`
+- [Coulomb 2-Body System](https://ohno.github.io/Antique.jl/stable/HydrogenAtom/) `CoulombTwoBody`
+
 ## Future Works
 
 [List of quantum-mechanical systems with analytical solutions](https://en.wikipedia.org/wiki/List_of_quantum-mechanical_systems_with_analytical_solutions)

diff --git a/src/Antique.jl b/src/Antique.jl
@@ -10,7 +10,8 @@ module Antique
     :PoschlTeller,
     :HarmonicOscillator3D,
     :RigidRotor,
-    :InfinitePotentialWell3D
+    :InfinitePotentialWell3D,
+    :CoulombTwoBody,
   ]
 
   # for Julia 1.1