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.
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):
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.
Practical aspects of ion-trap mass spectrometry; V.1:
fundamentals of Ion trap mass spectrometry
ed. by Raymond E March, John F J Todd
CRC Press, Boca Raton, 1995.
Ion Traps
by Pradip K. Ghosh
Oxford University Press, New York, 2007