1 Overview

In order to measure forces accurately with an Atomic Force Microscope (AFM), it is important to measure the cantilever spring constant. The force exerted on the cantilever can then be deduced from it's deflection via Hooke's law $F = -kx$.

The basic idea is to use the equipartition theorem[1],

$$\frac{1}{2} k \avg{x^2} = \frac{1}{2} k_BT,$$ (1)

where $k_B$ is Boltzmann's constant, $T$ is the absolute temperature, and $\avg{x^2}$ denotes the expectation value of $x^2$ as measured over a very long interval $t_T$,

$$\avg{A} \equiv \ensuremath{\lim_{{t_T}\rightarrow \infty}}\frac{1}{t_T} \ensuremath{\int_{-t_T/2}^{t_T/2} {A} \dd{t}}.$$ (2)

Solving the equipartition theorem for $k$ yields

$$k = \frac{k_BT}{\avg{x^2}},$$ (3)

so we need to measure (or estimate) the temperature $T$ and variance of the cantilever position $\avg{x^2}$ in order to estimate $k$.

1.1 Related papers

Various corrections taking into acount higher order modes [2,8], and cantilever tilt [9] have been proposed and reviewed [5,6,7], but we will focus here on the derivation of Lorentzian noise in damped simple harmonic oscillators that underlies all frequency-space methods for improving the basic $k\avg{x^2} = k_BT$ method.

Roters and Johannsmann describe a similar approach to deriving the Lorentizian power spectral density[3].

WARNING: It is popular to refer to the power spectral density as a ``Lorentzian''[1,3,6,5] even though eq. 53 differs from the classic Lorentzian[4].

$$L(x) = \frac{1}{\pi}\frac{\frac{1}{2}\Gamma} {(x-x_0)^2 + \p({\frac{1}{2}\Gamma})^2}$$ (4)

It is unclear whether the references are due to uncertainty about the definition of the Lorentzian or to the fact that eq. 53 is also peaked. In order to avoid any uncertainty, we will leave eq. 53 unnamed.

Download cantilever_calib.pdf
View the source files or my .latex2html-init configuration file

1 Overview
Copyright © 2009-10-12, W. Trevor King (contact)
Released under the GNU Free Document License, Version 1.2 or later
Drexel Physics