566
R. Frühwirth et al.
If the momentum p is given in GeV/c, the lower and upper limits are approximately
equal to
θ min ≈
2.66 · 10 −6 Z 1/3
p
, θ max ≈
0.14
A 1/3 p
.
The average number of scattering processes in a layer of thickness d (in cm) is given
by
N(d) = dσ
N A ρ
A
,
where σ is the integrated elastic cross section, N A is the Avogadro constant, ρ is
the density of the material (in g/cm 3 ), and A is the atomic mass of the nucleus. In
track reconstruction it is convenient to work with the projected scattering angles
in two perpendicular planes. The projected multiple scattering angle θ P is equal to
the sum of the projected single scattering angles, and its variance can be obtained
by multiplying the variance of the projected single scattering angle by the average
number of scatters, the projected single scattering angles being uncorrelated. With
increasing thickness d the distribution of the projected scattering angle approaches
a normal distribution, and the two projected angles become independent. For thin
scatterers, however, the width of the Gaussian core is notably narrower than is
indicated by the variance [42]. This is taken into account by Highland’s formula
for the standard deviation of the projected scattering angle [44]:
σ P = E(θ
2
P )
1/2
=
0.0136
βp
d/X 0 [1 + 0.038 ln(d/X 0 )] ,
where X 0 is the radiation length of the material in cm, β = v/c is the particle
velocity in units of c, and p is the particle momentum in GeV/c. The logarithmic
correction ceases to be applicable above d ≈ X 0 .
If a scatterer is sufficiently thin, the transverse offset of the track due to multiple
scattering can be neglected. Only the track direction is affected in this case. If the
direction is represented by the polar angle θ and the azimuthal angle ϕ, their joint
covariance matrix is given by
var((θ ) = σ
2
P , var((ϕ) = σ
2
P / sin
2 θ, cov((θ, ,ϕ) = 0.
If the direction is represented by the direction tangents t x = dx/dz and t y = dy/dz,
the covariance matrix is [45]
Var[((t x , ,t y )
T
] = σ
2
P (1 + t
2
x + t
2
y )
1 + t 2
x t x t y
t x t y 1 + t 2
y
.
R. Frühwirth et al.
If the momentum p is given in GeV/c, the lower and upper limits are approximately
equal to
θ min ≈
2.66 · 10 −6 Z 1/3
p
, θ max ≈
0.14
A 1/3 p
.
The average number of scattering processes in a layer of thickness d (in cm) is given
by
N(d) = dσ
N A ρ
A
,
where σ is the integrated elastic cross section, N A is the Avogadro constant, ρ is
the density of the material (in g/cm 3 ), and A is the atomic mass of the nucleus. In
track reconstruction it is convenient to work with the projected scattering angles
in two perpendicular planes. The projected multiple scattering angle θ P is equal to
the sum of the projected single scattering angles, and its variance can be obtained
by multiplying the variance of the projected single scattering angle by the average
number of scatters, the projected single scattering angles being uncorrelated. With
increasing thickness d the distribution of the projected scattering angle approaches
a normal distribution, and the two projected angles become independent. For thin
scatterers, however, the width of the Gaussian core is notably narrower than is
indicated by the variance [42]. This is taken into account by Highland’s formula
for the standard deviation of the projected scattering angle [44]:
σ P = E(θ
2
P )
1/2
=
0.0136
βp
d/X 0 [1 + 0.038 ln(d/X 0 )] ,
where X 0 is the radiation length of the material in cm, β = v/c is the particle
velocity in units of c, and p is the particle momentum in GeV/c. The logarithmic
correction ceases to be applicable above d ≈ X 0 .
If a scatterer is sufficiently thin, the transverse offset of the track due to multiple
scattering can be neglected. Only the track direction is affected in this case. If the
direction is represented by the polar angle θ and the azimuthal angle ϕ, their joint
covariance matrix is given by
var((θ ) = σ
2
P , var((ϕ) = σ
2
P / sin
2 θ, cov((θ, ,ϕ) = 0.
If the direction is represented by the direction tangents t x = dx/dz and t y = dy/dz,
the covariance matrix is [45]
Var[((t x , ,t y )
T
] = σ
2
P (1 + t
2
x + t
2
y )
1 + t 2
x t x t y
t x t y 1 + t 2
y
.