snompy.fdm.eff_pol_n_taylor

fdm.eff_pol_n_taylor(sample, A_tip, n, z_tip=None, r_tip=None, L_tip=None, g_factor=None, d_Q0=None, d_Q1=None, d_Qa=None, n_lag=None, method=None, n_trapz=None, n_tayl=None)

Return the effective probe-sample polarizability using the finite dipole model, demodulated at harmonics of the tapping frequency, using a Taylor series representation of the bulk FDM.

Note

This function primarily exists to check the validity of the Taylor approximation to eff_pol_n which is used by other functions. For best performance it is recommended to use eff_pol_n.

Parameters:

samplesnompy.sample.Sample

Object representing a layered sample with a semi-infinite substrate and superstrate.

A_tipfloat

The tapping amplitude of the AFM tip.

nint

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

z_tipfloat

Height of the tip above the sample.

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.

d_Qafloat

Depth of a single representative charge within the tip. Specified in units of the tip radius. Used by the “Q_ave” implementation of the finite dipole model to calculate the effective quasistatic reflection coefficient for the tip.

n_lagint

The order of the Gauss-Laguerre integration used by the “multi” and “Q_ave” methods.

method{“bulk”, “Q_ave”}

The method of the finite dipole model to use. See eff_pol() for descriptions of the different methods.

n_trapzint

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

n_taylint

Maximum power index for the Taylor series in beta.

Returns:

alpha_effcomplex

Effective polarizability of the tip and sample, demodulated at n.

See also

eff_pol_n

The non-Taylor series version of this function.

snompy.demodulate.demod

The function used here for demodulation.

Notes

This function is valid only for reflection coefficients, beta, with magnitudes less than around 1. For a more generally applicable function use eff_pol_n()

This function implements \(\alpha_{eff, n} = \sum_{j=0}^{J} a_{j,n} \beta^j\), where \(\beta\) is beta, \(j\) is the index of the Taylor series, \(J\) is n_tayl and \(a_{j,n}\) is the Taylor coefficient, implemented here as taylor_coef().