Overview
The spring constants of AFM cantilevers are often required in applications of
the AFM. For force measurements where normal deflection of the cantilever is
monitored, the normal spring constant, which relates the applied normal force
to the normal deflection of the cantilever, is required (see Fig. 1 for
a schematic).

Fig. 1: Schematic illustration of the normal spring constant.

In Lateral Force Microscopy, the torsional spring constant, which relates the
applied torque to the angle of twist of the cantilever is important (see Fig. 2
for a schematic).

Fig. 2: Schematic illustration of the torsional spring constant.

Here, we briefly outline the Sader method for
calibrating the normal spring constant of rectangular AFM cantilevers, as well
as the Torsional Sader method
which enables the calibration of the torsional spring constant of rectangular
AFM cantilevers.
We also describe the physics underpinning the method and its
extension to cantilevers of arbitrary shape.
First, though, we note that both techniques require the plan view dimensions
of the rectangular cantilever (length and width), which can be obtained with
sufficient accuracy using an optical microscope. Furthermore, the resonant
frequency and quality factor of the peaks are also needed. These are typically obtained
by measuring the thermal noise spectra of the unloaded AFM cantilever, and
fitting the response of a Simple Harmonic Oscillator (SHO) with added white noise floor,
to the fundamental power spectrum resonant peak, see Fig. 3.

Fig. 3: SHO fit to thermal noise power spectrum. The red line is the response of a
SHO, whilst the black circles are experimental data from the thermal noise
spectrum.

Once the required quantities are measured, the normal and torsional spring
constants of the cantilever can be determined using the Sader method [1], [4],
provided that the following conditions are met:
 That the length of the cantilever greatly exceeds its width, which in turn
greatly exceeds its thickness;
 That the quality factor is much greater than unity.
In practice, the first requirement is typically satisfied for practical
rectangular cantilever, whilst the second requirement is usually satisfied
when the cantilever is immersed in air.
Sader method
The normal spring constant of a rectangular AFM cantilever can be determined
using the Sader method [1], which states
where L and b are the length and width of the cantilever,
respectively, ρ is the density of the fluid, ω_{f} and
Q_{f} are the (radial) resonant frequency and quality factor of
the fundamental resonance peak, respectively, and Γ_{i}^{f}
is the imaginary part of the hydrodynamic function given by Eq. (20) of
Ref. [2].
In order to assist the user, the Sader method has been implemented in an
online calibrator, meaning that no code needs to
be written by AFM users.
Torsional Sader method
The Sader method was extended to allow
calibration of the torsional spring constant of rectangular AFM cantilevers [4],
using the result
where L and b are the length and width of the cantilever,
respectively, ρ is the density of the fluid, ω_{t} and
Q_{t} are the (radial) resonant frequency and quality factor of
the fundamental torsional resonance peak, respectively, and
Γ_{i}^{t}
is the imaginary part of the hydrodynamic function given by Eq. (20) of
Ref. [5].
Again, in order to assist the user, the Torsional Sader method has been
implemented in an online calibrator.
Cantilevers of arbitrary shape
The extension of the Sader method to cantilevers of arbitrary shape is reported in
[6], [7]. This includes an explanation of the underlying physical basis
of the method, its validity in the presence of nonidealities, such as added spheres
or tips, and its general applicability to small elastic bodies of arbitrary shape
and composition.
iPhone and web apps that implement the Sader method for rectangular and arbitrary cantilevers are available at the links at the top of this page.
Global Calibration Initiative (GCI)
The Sader Method GCI [7]  available via link at the top of this page and sadermethod.org  enables the standardisation of cantilever calibration, for cantilevers of any type and shape, utilising data from the worldwide AFM community.
References
 J. E. Sader et al., Review of Scientific Instruments, 70, 3967 (1999).
 J. E. Sader, Journal of Applied Physics, 84, 64 (1998).
 J. W. M. Chon et al., Journal of Applied Physics, 87, 3978 (2000).
 C. P. Green et al., Review of Scientific Instruments, 75, 1988 (2004).
 C. P. Green and J. E. Sader, Journal of Applied Physics, 92, 262 (2002).
 J. E. Sader et al., Journal of Applied Physics, 97, 124903 (2005).
 J. E. Sader, Review of Scientific Instruments, 83, 103705 (2012).
 J. E. Sader et al., Review of Scientific Instruments, 87, 093711 (2016).