snompy.fdm.taylor_coef

fdm.taylor_coef(z_tip, j_taylor, A_tip, n, r_tip, L_tip, g_factor, d_Q0, d_Q1, n_trapz)

Return the coefficient for the power of reflection coefficient used by the Taylor series representation of the bulk FDM.

Parameters:

z_tipfloat

Height of the tip above the sample.

j_taylorinteger

The corresponding power of the reflection coefficient in the Taylor series.

A_tipfloat

The tapping amplitude of the AFM tip.

nint

The harmonic of the AFM tip tapping frequency at which to demodulate.

r_tipfloat

Radius of curvature of the AFM tip.

L_tipfloat

Semi-major axis length of the effective spheroid from the finite dipole model.

g_factorcomplex

A dimensionless approximation relating the magnitude of charge induced in the AFM tip to the magnitude of the nearby charge which induced it. A small imaginary component can be used to account for phase shifts caused by the capacitive interaction of the tip and sample.

d_Q0float

Depth of an induced charge 0 within the tip. Specified in units of the tip radius.

d_Q1float

Depth of an induced charge 1 within the tip. Specified in units of the tip radius.

n_trapzint

The number of intervals used by snompy.demodulate.demod() for the trapezium-method integration.

Returns:

a_jcomplex

Coefficient for the power of reflection coefficient used by the Taylor series representation of the bulk FDM

See also

snompy.demodulate.demod

The function used here for demodulation.

Notes

This function implements

\[\begin{split}a_{j,n} = \begin{cases} 1, & \text{if $j = 0$, $n = 0$}\\ 0, & \text{if $j = 0$, $n \neq 0$}\\ \frac{1}{2} \hat{F_n}[f_t(j)], & \text{if $j \neq 0$} \end{cases}\end{split}\]

where \(\hat{F_n}[f_t(j)]\) is the \(n^{th}\) Fourier coefficient of the function \(f_t(j)\), which is implemented here as geom_func_taylor().