Tests

Overview

This module is a collection of files that test the functions in the MrfmSim package. To run the unit tests, open up a terminal in the MrfmSim directory and run the following command:

python -m pytest

Signal verification

We can test the signal-calculation modules by computing the signal from a single spin. Consider a magnet-tipped cantilever operated in the hangdown geometry:

• a magnetic field is applied along the \(z\) direction to polarize the cantilever’s magnet and to define the quantization axis of the sample’s spins;

• the cantilever oscillates in the \(x\) direction;

• the long axis of the cantilever is parallel to the \(z\) direction;

• the magnet at the tip of the cantilever is a sphere of radius \(r\), polarized along the \(z\) direction, with saturation magnetization \(M_s\) and saturation field \(\mu_0 M_s\);

• the center of the spherical magnet lies at the origin, \((0,0,0)\);

• the sample lies in the \(xy\) plane; for simplicity, we further assume that the spin lies along the \(y = 0\) line;

• the distance between the center of the magnet and the sample plane is \(z\).

Some notes on the additional tests

Magnet Test

The RectMagnet is tested through its symmetry factors: it is symmetric in the \(z\) direction when x and y are 0. The gradient functions are tested against a numerical derivative calculation from the field. \(B_z\) is an even function in the \(x\) direction, \(B_{zx}\) is an odd function in the x-direction and :math: B_{zxx} is an even function in the \(x\) direction.

Polarization Test

The polarization test for irrad_periodic function is taken from John’s notebook

Experiment Test

Some of the tests are calculated against some of the old notebook examples from John and Corinne. Note to see the notebook and examples go to the MrfmSim-archived package’s git commit (c74d504).

• IBMCyclic dF_spin -> test_experiment from John Marohn.

• IBMCyclic dF2_spin -> value taken from test-ibmexpt-1.ipynb simulation 1, #4 - 9, matching with relative 5e-5 precision.

• CermitARP is taken from “test_cornellexptobject.py”

• CermitESR is tested against value from “test-cornellexpt-4.ipynb” (Moore experiment - local peak, # 0 - 5, Issac Experiment, # 24 - 28) the difference in value is expected because the min_abs_offset function fixed a previous issue.

• CermitARP_SmallTip is tested against “test-cornellexpt-8-Hickman_nanomagnet_simulations.ipynb”, first simulation.

• CermitARP_SmallTip is compared with CermitARP, in two cases using a rectangular magnet, one is when the amplitude is small, they should be the same, and when tip_sep is large, the effect of amplitude is also small (There is a comparison in “test-micrometertip_smallampapprox_vs_exactsol.ipynb” on small amplitude, however, the result is unchecked).

• CmeritSingleSpinESR_Approx is compared with CermitARP_SmallTip.

• SingleSpinESR

  • The approximation is tested against both SPAM and hangdown geometry of the analytical solution

Force detection

The force produced by a single spin is

\[\delta F_{\mathrm{spin}} = \mu_{\mathrm{spin}} \: G_{zx}(r_{\mathrm{spin}})\]

with \(\mu_{\mathrm{spin}}\) the spin magnetic moment and \(G_{zx} = \partial B_z / \partial x\) and \(r_{\mathrm{spin}}\) the location of the spin. The gradient experienced by a spin located in the \(y = 0\) plane is

\[G_{zx}(x,0,z) = 4 \mu_0 M_s r^3 \: \frac{x \: (x^2 - 4 z^2)}{(x^2 + z^2)^{7/2}}\]

We see that the gradient \(G_{zx}\) is zero for a spin directly below the tip at \((0,0,z)\). For the single-spin force experiment, we must therefore place the spin “off to the side”, at a location \((x,0,z)\). The gradient is maximized at (approximately) \(x_{\mathrm{opt}} = \pm 0.389295 \: z\). For a spin placed at this optimal location,

\[G_{zx}(x_{\mathrm{opt}}, 0, z) \approx 0.914269 \: \frac{\mu_0 M_s}{r} \left( \frac{r}{z} \right)^{4}\]

In the hangdown-geometry experiment delineated above, the force acting on a magnet-tipped cantilever from a single spin placed at an optimal location “off to the side” of the cantilever tip is

\[\delta F_{\mathrm{spin}} \approx 0.914269 \: \mu_{\mathrm{spin}} \: \frac{\mu_0 M_s}{r} \left( \frac{r}{z} \right)^{4}\]

With

• tip magnetization \(\mu_0 M_s = 1800 \: \mathrm{mT}\) (cobalt)

• tip radius \(r = 50 \: \mathrm{nm}\)

• tip-sample separation \(20 \: \mathrm{nm}\), that is, \(z = 70 \: \mathrm{nm}\)

• a single (fully polarized) electron spin placed at \((27.2507, 0.0000, 70.0000) \: \mathrm{nm}\)

the force is

\[\delta F_{\mathrm{spin}} = -79.231 \: \mathrm{aN}\]

Force-gradient detection

The force-gradient produced by a single spin is

\[\delta k_{\mathrm{spin}} = \mu_{\mathrm{spin}} \: G_{zxx}(r_{\mathrm{spin}})\]

with \(G_{zxx} = \partial^2 B_z / \partial x^2\). For a single spin directly below the tip at \((0,0,z)\),

\[G_{zxx}(0,0,z) = - \frac{4 \mu_0 M_s r^3}{z^5} = - \frac{4 \mu_0 M_s}{r^2} \left( \frac{r}{z} \right)^{5}\]

In the hangdown-geometry experiment delineated above, the force gradient acting on a magnet-tipped cantilever from a single spin placed directly below the cantilever tip is

\[\delta k_{\mathrm{spin}} = - \mu_{\mathrm{spin}} \: \frac{4 \mu_0 M_s}{r^2} \left( \frac{r}{z} \right)^{5}\]

With

• tip magnetization \(\mu_0 M_s = 1800 \: \mathrm{mT}\) (cobalt)

• tip radius \(r = 50 \: \mathrm{nm}\)

• tip-sample separation \(20 \: \mathrm{nm}\), that is, \(z = 70 \: \mathrm{nm}\)

• a single (fully polarized) electron spin placed at \((0.0, 0.0, 70.0) \: \mathrm{nm}\)

the force gradient is

\[\delta k_{\mathrm{spin}} = 4.95203 \: \mathrm{aN} \: \mathrm{nm}^{-1} = 4.95203 \: \mathrm{nN} \: \mathrm{m}^{-1}\]

Testing Lists

class tests.test_experiment.test_ibmcyclic.TestIBMCyclic

Bases: object

Test IBMCyclic Experimental method.

test_IBMCyclic_dF2_spin()

Test IBM cyclic force variance signal for nucleus.

Values are taken from “test-ibmexpt-1.ipynb” simulation 1 #4- #9 based on John’s calculation.

test_IBMCyclic_dF_spin()

Test IBM curie law signal.

class tests.test_formula.test_field.TestXTrapzFieldGradient

Bases: object

Test trapz field gradient.

test_xtrapz_field_gradient_large_distance()

Test gradient against a large tip-sample separation.

When the distance is very large, the change of Bzx in between grid points are small.

test_xtrapz_field_gradient_smallamp()

Test gradient against a small x0p.

When the 0 to the peak is very small compared to the magnet, the field is equivalent to Bzxx. Here we “fake” a small grid step for the x direction, because the grid step here is only used to calculate x_0p. The cantilever is 150 nm away from the sample.

class tests.test_formula.test_field.TestXTrapzFxDtheta

Bases: object

Test xtrapz_fxdtheta.

test_xtrapz_fxdtheta_cos()

Test xtrapz_fxdtheta against :math: cos{x}dx

Let’s consider the field method returns 1, the trapz integral should return the value of

\[\int_{-\pi}^{\pi}{x_0 \cos\theta d\theta}\]

The results: (-pi/2 -> 0): x0 (-pi -> 0): 0

test_xtrapz_fxdtheta_multidim()

Test xtrapz_fxdtheta against :math: x cos{x} for multi-dimensions.

test_xtrapz_xtrapz_fxdtheta_x2cos()

Test xtrapz_fxdtheta against :math: x2cos{x}.

Let’s consider a case where the field f(x) = x

\[\int_{-\pi}^{\pi}{(x-x_0 \cos\theta)x_0 \cos\theta d\theta}\]

The results: (-pi/2 -> 0): x^2*x0 - pi/2*x*x0^2 + 2/3 * x0^3 (-pi -> 0): - pi*x*x0^2

test_xtrapz_xtrapz_fxdtheta_xcos()

Test xtrapz_fxdtheta against :math: x cos{x}.

Let’s consider a case where the field f(x) = x

\[\int_{-\pi}^{\pi}{(x-x_0 \cos\theta)x_0 \cos\theta d\theta}\]

The results: (-pi/2 -> 0): x*x0 - x0^2*pi/4 (-pi -> 0): - x0^2 * pi/2

tests.test_formula.test_field.test_B_offset()

Test B_offset.

Test the nucleus.

tests.test_formula.test_field.test_min_abs_offset()

Test min_abs_x.

tests.test_formula.test_field.test_min_abs_offset_symmetry()

Test if min_abs_x result is symmetric. If a matrix is flipped, the result should also be flipped

tests.test_formula.test_math.test_as_strided_x()

Testing if as_strided_x maximum matches running max.

tests.test_formula.test_math.test_ogrid_sub()

Test ogrid_sub.

The sub results in a iterable mapping. Here tests if the dimension and values are correct.

tests.test_formula.test_math.test_slice_matrix()

Test slice matrix cuts the matrix in the center.

tests.test_formula.test_misc.test_convert_grid_pts()

Test convert_grid_pts.

Make sure the final value is an int and is the floor value.

tests.test_units.test_units()

Test if units are loaded correctly.