Case Study: Design of Scanning Probe Microscope

In the last lecture, we have discussed about the Ashby process for material selection. In this lecture, we will try to elaborate this procedure through two examples. We will first consider the design of a Probe for Scanning Probe Microscope (SPM). Then we will consider the design of a Telescope Mirror.

Scanning Probe Microscopes (SPM) have contradictory specifications First, the scan range of an SPM should exceed the size of the largest features on the sample. This usually implies the use of large and thin walled piezo electric translators.

On the other hand, any SPM is, basically, a system of mechanical resonators. Hence, it is very important to optimize the design of the SPM for high resonance frequencies. This usually means to decrease the length of the probe. In this design exercise we will consider the maximisation of resonating frequency as our design objective.

The eigen-frequency of any single degree of freedom system is given by:

where k is the spring constant and m_eff is the effective mass. The spring constant of a cantilevered beam with uniform cross section is given by

For a rectangular cross section with a width b (perpendicular to the deflection) and a height h one obtains for I = bh³ /12

Combining the expressions we get the final result for f as :

Now, the effective mass can be calculated using Raleigh's method. If you recall, the general expression for the kinetic energy T using Raleigh's method is

For uniform beam with a constant cross section and length, the deflection may be expressed as:

Also, T can be given by:

Hence the effective mass = (9/20)m

Now, in the final form, the resonating frequency may be written as:

It is evident from this equation, that one way to increase the eigen- frequency is to choose a material with as high a ratio of E/ ρ as possible. In fact the Material Property index in this case is given by M=(E/ ρ ) ^1/2 . The table below shows the E/ ρ values of a few materials. As per the table both Aluminium and Steel are good candidates for this purpose.