26
E. Wilson and B. J. Holzer
and ϕ to become μ, the phase advance per cell. The matrix for one period is
now:
M =
cos μ − ww sin μ
w 2 sin μ
−
1+w 2 w
w 2
sin μ cos μ + ww sin μ
.
(2.24)
Next we invent some new functions of β:
β = w 2 ,
α = −ww = −
β
2 ,
γ =
1+(ww )
2
w 2
=
1+α 2
β .
(2.25)
These functions (which are not the same as the parameters used in special
relativity!) are a complete and compact description of the dynamics. The matrix
now becomes even simpler:
M =
cos μ + α sin μ
βsin μ
− γ sin μ cos μ − α sin μ
=
a b
c d
.
(2.26)
This is the Twiss matrix. It is the basic matrix for periodic lattices and should be
memorized.
We can imagine that if we can only find an independent way of computing the
numerical values of the four elements we can solve and find:
cos μ = (Tr M) /2 = (a + d) /2,
β = b/ sin μ > 0,
α = (a − b) / (2 sin μ) ,
γ = −c/ sin μ.
(2.27)
These Twiss parameters, μ, β, α, and γ , are therefore rigorously determined
by the overall effect of the focusing properties of the lattice elements. Still, they
vary around the ring and apply to the point chosen in the period as a starting and
finishing point. We shall see that each individual component, quadrupole, dipole, or
drift space in the ring has its own matrix and this provides the independent method of
calculation. We must first choose the starting point, the location, s, where we wish
to know β and the other Twiss parameters. By starting there and multiplying the
element matrices together for one turn we are able to find a, b, c, d numerically for
that location. We can then apply the above four equations to find the Twiss matrix.
If the machine has a natural symmetry in which there are a number of identical
periods, it is sufficient to do the multiplication up to the corresponding point in the
next period. The values of α, β, and γ would be the same if we went on for the
whole ring. By choosing different starting points we can trace β(s) and α(s). We
now give the matrices for the three basic lattice elements.
E. Wilson and B. J. Holzer
and ϕ to become μ, the phase advance per cell. The matrix for one period is
now:
M =
cos μ − ww sin μ
w 2 sin μ
−
1+w 2 w
w 2
sin μ cos μ + ww sin μ
.
(2.24)
Next we invent some new functions of β:
β = w 2 ,
α = −ww = −
β
2 ,
γ =
1+(ww )
2
w 2
=
1+α 2
β .
(2.25)
These functions (which are not the same as the parameters used in special
relativity!) are a complete and compact description of the dynamics. The matrix
now becomes even simpler:
M =
cos μ + α sin μ
βsin μ
− γ sin μ cos μ − α sin μ
=
a b
c d
.
(2.26)
This is the Twiss matrix. It is the basic matrix for periodic lattices and should be
memorized.
We can imagine that if we can only find an independent way of computing the
numerical values of the four elements we can solve and find:
cos μ = (Tr M) /2 = (a + d) /2,
β = b/ sin μ > 0,
α = (a − b) / (2 sin μ) ,
γ = −c/ sin μ.
(2.27)
These Twiss parameters, μ, β, α, and γ , are therefore rigorously determined
by the overall effect of the focusing properties of the lattice elements. Still, they
vary around the ring and apply to the point chosen in the period as a starting and
finishing point. We shall see that each individual component, quadrupole, dipole, or
drift space in the ring has its own matrix and this provides the independent method of
calculation. We must first choose the starting point, the location, s, where we wish
to know β and the other Twiss parameters. By starting there and multiplying the
element matrices together for one turn we are able to find a, b, c, d numerically for
that location. We can then apply the above four equations to find the Twiss matrix.
If the machine has a natural symmetry in which there are a number of identical
periods, it is sufficient to do the multiplication up to the corresponding point in the
next period. The values of α, β, and γ would be the same if we went on for the
whole ring. By choosing different starting points we can trace β(s) and α(s). We
now give the matrices for the three basic lattice elements.