In [1]:
%pylab inline
from ipywidgets import interact
from pyatoms.J.SingleAtom import SingleAtom
import mpmath as mp
In [2]:
Ho = SingleAtom(8, 3)
Ho.CF.setSymmetry("C3v")
The crystal field operator is a sum of Stevens' operators corresponding to the desired symmetry (Stevens, 1952). We are going to use crystal field parameters published by Donati et al. (2014). The energy units are millielectronvolts by default.
In [3]:
Ho.CF.setCoefficient(2, 0, -140e-3)
Ho.CF.setCoefficient(4, 0, 1.06e-3)
We plot the energy spectrum as function of the expectation value of the $J_z$ operator ($\left<\hat{J}_z\right>$). As we have only defined uniaxial anisotropy so far, all the states are $J_z$ eigenstates and form dimers (except for $J_z=0$ state).
In [4]:
plot(Ho.Js, Ho.Es, 'ro')
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
Out[4]:
Adding a transversal term leads to creation of singlets at every third state (C3v).
In [5]:
Ho.CF.setCoefficient(4, 3, 1e-3)
plot(Ho.Js, Ho.Es, 'ro')
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
Out[5]:
In [6]:
Ho.ZT.setBz(1e-3)
plot(Ho.Js, Ho.Es, 'ro')
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
Out[6]:
Magnetic field can be applied in any direction. One can observe e.g. the effect of rotating the field vector from z to x axis.
In [7]:
@interact(theta=(0, pi/2, pi/12))
def _plt(theta=0):
Ho.ZT.setBtheta(theta)
plot(Ho.Js, Ho.Es, 'ro')
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
In [8]:
Ho.ZT.setBxyz(0,0,0)
colors = log(abs(asarray(Ho.transitions(Ho.Jp)[0][0,:].tolist(), dtype=float)))
scatter(Ho.Js, Ho.Es, s=100, c=colors, cmap='Reds', vmin=-20, vmax=0, edgecolors="k")
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
title("Matrix element of $J_+$ operator\nbetween the ground state and the rest (on log scale)")
Out[8]:
To calculate transition probabilities between states through an operator of the form $\vec{\sigma}\vec{J}$, we use the function J_transitions, that uses the formula $$p_{i\to f} = \frac1{J(J+1)}\left[|\left<f\middle|J_z\middle|i\right>|^2 + \frac12\left(|\left<f\middle|J_+\middle|i\right>|^2 + |\left<f\middle|J_-\middle|i\right>|^2\right)\right]$$
(see Hirjibehedin et al., 2007)
In [9]:
Ho.ZT.setBxyz(0,0,0)
colors = log(abs(asarray(Ho.J_transitions()[0,:].tolist(), dtype=float)))
scatter(Ho.Js, Ho.Es, s=100, c=colors, cmap='Reds', vmin=-20, vmax=1, edgecolors="k")
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
title("Transition probability from the ground state to the other states (on log scale)")
Out[9]:
In [10]:
Ho.CF.setCoefficient(2,0,-0.239068)
Ho.CF.setCoefficient(4,0,8.59023e-5)
Ho.CF.setCoefficient(4,3,2.93446e-5)
Ho.CF.setCoefficient(6,0,1.86782e-7)
Ho.CF.setCoefficient(6,3,-1.96786e-6)
Ho.CF.setCoefficient(6,6,6.30483e-7)
plot(Ho.Js, Ho.Es, 'ro')
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5);
Note that the probability of a transition to the other side of the parabola is very low.
In [11]:
colors = log(abs(asarray(Ho.J_transitions()[0,:].tolist(), dtype=float)))
scatter(Ho.Js, Ho.Es, s=100, c=colors, cmap='Reds', vmin=-20, vmax=1, edgecolors="k")
ylabel("Energy (meV)")
xlabel(r"$\left<\hat{J}_z\right>$")
xlim(-8.5, 8.5)
title("Transition probability from the ground state to the other states (on log scale)")
print("Transition probability within the ground doublet: {}".format(Ho.J_transitions()[0,1]))
print("Transition probability to the first excited state: {}".format(Ho.J_transitions()[0,2]))
The actual probability of transitions within the doublet should be zero. We can increase the precision to see the probability decrease.
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print("Transition probability within the ground doublet: {}".format(mp.nstr(Ho.J_transitions()[0,1], 3)))
with mp.workdps(200):
Ho = SingleAtom(8, 3)
Ho.CF.setSymmetry("C3v")
Ho.CF.setCoefficient(2, 0, -0.239068)
Ho.CF.setCoefficient(4, 0, 8.59023e-5)
Ho.CF.setCoefficient(4, 3, 2.93446e-5)
Ho.CF.setCoefficient(6, 0, 1.86782e-7)
Ho.CF.setCoefficient(6, 3, -1.96786e-6)
Ho.CF.setCoefficient(6, 6, 6.30483e-7)
print("Transition probability within the ground doublet: {}".format(mp.nstr(Ho.J_transitions()[0,1], 3)))
The transition probability within the ground doublet will increase with magnetic field.
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Bfield = logspace(-8, -1)
probability = []
for B in Bfield:
Ho.ZT.setBz(B)
probability.append(Ho.J_transitions()[0,1])
loglog(Bfield, probability)
ylabel("B_z (T)")
xlabel("Transition probability")
xlim(-8.5, 8.5);
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