Abstract: This page shows the Mathieu Stability Diagram for the Paul Trap. It is possible to set the operating parameters for the trap. The secular frequencies of ion motion are displayed. The user may click on the stability diagram to set the q and a parameters to desired values. A utility to accurately compute q and a to achieve a desired βz and βr pair is also provided.
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  • The diagram on the left shows the cross-section of a Paul trap.
  • The upper and lower endcaps are the two parts the hyperboloid of two sheets 2z2-x2-y2 = 2z02.
  • The ring is the hyperboloid of one sheet x2+y2-2z2 = r02.
  • The endcaps are grounded.
  • The ring is supplied with Udc + Vrf cos(2πft).
  • The coordinates x, y, and z of the ion satisfy differential equations which take the form of the Mathieu equation.
The Mathieu Stability Chart: The motion of the ion within the trap is governed by the Mathieu parameters qz and az, which can be expressed in terms of the parameters r0, z0, m (ion mass), Q (ion charge), f (drive frequency), Vrf (RF voltage), and Udc (DC voltage).

Let Ω = 2πf be the drive angular frequency. Then

qz = 8 Q Vrf / [m Ω2 (r02 + 2 z02)]

az = -16 Q Udc / [m Ω2 (r02 + 2 z02)]

The parameters qz and az govern ion motion in the z direction.
Ion motion in the x and y directions is governed by the parameters qr = -qz / 2 and ar = -az/2.

On this page we use q = qz and a = az and never use the r subscripted parameters. For details, see the references given later on this page.

Paul trap parameters:

Length unit: r0: z0: Ion mass in amu: Ion charge in e:
f (RF frequency): Udc(in V): Vrf(in V):

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You can save the results of the calculations by copying text from this text area.

Finding q and a from βz and βr: βz: βr:
Scroll up to see the updated chart after this computation.

References:

Acknowledgments:

We are grateful to Professors A. G. Menon and Anindya Chatterjee for introducing us to ion traps.