# 2.3.5 Approach-retraction curves

Consider a cantilever oscillations near a sample surface. As shown in **chapter 2.2.1**, the tip-sample interaction potential has a characteristic appearance depicted in **Fig. 1**. As the cantilever touches the sample and deforms its surface, the **force of elastic repulsion** prevails. At the tip-sample separation on the order of a few tens of angstrom, the intermolecular interaction called the **Van der Waals force** predominates.

**Fig. 1. Typical appearance of the tip-sample interaction potential.**

As shown in **chapter 2.3.4**, the presence of external force dependent on spatial coordinates, gives rise to the change in resonance properties of the cantilever-sample oscillating system.

(1)

(2)

(3)

where - cantilever stiffness, – oscillating system Q-factor, – cantilever oscillations amplitude in the absence of external force.

Thus, measuring dependence of the oscillations resonant frequency, phase or amplitude on the tip-sample separation, one can render the derivative appearance and, in some cases, the interaction force itself. The corresponding experimental curves are called the **approach curves (Fig. 2)**.

**Fig. 2. The tip-to-sample approach curves.
– amplitude of the cantilever oscillations at resonant frequency,
– resonant frequency in the absence of the external force gradient.**