164
An Introduction to Beam Physics
Similarly, all aberrations involving positions and angles simultaneously have
to vanish, and hence it is necessary to have
(x|x
ix y
iy a
ia b
i b ) = (y|x
ix y
iy a
ia b
i b ) = 0;
if any of them prevail, they will entail a position dependent fuzziness that
becomes stronger with an increase of the positions x and y.
Interestingly enough, all higher order aberrations depending on a and b only
linearly can be corrected by a reshaping of the focal plane; in fact, (x|xa),
etc., produce a tilt of the image, and (x|xxa), etc., produce a curvature of the
image. Fig. 7.2 shows how the matrix element (x|xxa) can be corrected by
shaping the image position parabolically.
At any given position, due to the matrix element (x|xxa), any ray with a
given a is moved up or down in proportion to a, where the amount of deflection depends quadratically on x; so the rays arrive at the x plane as shown.
However, tracing the rays backwards shows that they in fact all intersect before the plane, and the point where this happens depends quadratically on x.
In similar ways, (x|x
4 a), etc., can be corrected.
7.3 Spectrometers and Spectrographs
Spectrometers and spectrographs are devices for the purpose of measuring momentum, energy, or mass of charged particles. Momentum spectrometers are mainly used in nuclear physics for the determination of the
momentum distribution of nuclear reaction products. Most of the momentum
spectrometers known are magnetic because of the fact that the energies that
need to be analyzed are too high to allow sufficient deflection by electric fields.
In addition, magnets have two more advantages. They automatically preserve
the momentum, and can be built big enough to achieve large acceptance.
The mass spectrometer is mainly used for the analysis of masses of
molecules, and they can be operated at much lower energies. They have
a long history, and their applications pervade many disciplines from physics
and chemistry to biology, environmental sciences, etc. Mass spectrometers are
also more diverse; among the major types, there are sector field, quadrupole,
accelerator, energy loss, time-of-flight, Fourier transform ion cyclotron resonance and ion trap mass spectrometers.
For all the different types of spectrometers, the goal is to achieve high
resolution, and in many cases large acceptance at the same time. As
resolution improves, the need for better understanding and correction of high
order aberrations increases. In the following, the linear theory of various types
of spectrometers will be discussed, followed by the studies of aberrations and
their correction.
An Introduction to Beam Physics
Similarly, all aberrations involving positions and angles simultaneously have
to vanish, and hence it is necessary to have
(x|x
ix y
iy a
ia b
i b ) = (y|x
ix y
iy a
ia b
i b ) = 0;
if any of them prevail, they will entail a position dependent fuzziness that
becomes stronger with an increase of the positions x and y.
Interestingly enough, all higher order aberrations depending on a and b only
linearly can be corrected by a reshaping of the focal plane; in fact, (x|xa),
etc., produce a tilt of the image, and (x|xxa), etc., produce a curvature of the
image. Fig. 7.2 shows how the matrix element (x|xxa) can be corrected by
shaping the image position parabolically.
At any given position, due to the matrix element (x|xxa), any ray with a
given a is moved up or down in proportion to a, where the amount of deflection depends quadratically on x; so the rays arrive at the x plane as shown.
However, tracing the rays backwards shows that they in fact all intersect before the plane, and the point where this happens depends quadratically on x.
In similar ways, (x|x
4 a), etc., can be corrected.
7.3 Spectrometers and Spectrographs
Spectrometers and spectrographs are devices for the purpose of measuring momentum, energy, or mass of charged particles. Momentum spectrometers are mainly used in nuclear physics for the determination of the
momentum distribution of nuclear reaction products. Most of the momentum
spectrometers known are magnetic because of the fact that the energies that
need to be analyzed are too high to allow sufficient deflection by electric fields.
In addition, magnets have two more advantages. They automatically preserve
the momentum, and can be built big enough to achieve large acceptance.
The mass spectrometer is mainly used for the analysis of masses of
molecules, and they can be operated at much lower energies. They have
a long history, and their applications pervade many disciplines from physics
and chemistry to biology, environmental sciences, etc. Mass spectrometers are
also more diverse; among the major types, there are sector field, quadrupole,
accelerator, energy loss, time-of-flight, Fourier transform ion cyclotron resonance and ion trap mass spectrometers.
For all the different types of spectrometers, the goal is to achieve high
resolution, and in many cases large acceptance at the same time. As
resolution improves, the need for better understanding and correction of high
order aberrations increases. In the following, the linear theory of various types
of spectrometers will be discussed, followed by the studies of aberrations and
their correction.