J.Va’vra, Detector Lecture in Rome, January 25, 2002
Lecture in Rome
January 25,2002
The drift
of electrons and ions
in gases
or,
how to design a good TPC
1
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Content
- The Langevin theory of electron drift.
- The Boltzman equation to solve the electron drift:
a) Simple Shultz & Gresser alghoritm,
b) B. Schmidt modification,
c) S. Biagi MAGBOLTZ program,
d) S. Biagi MAGBOLTZ -MONTE program.
- Monte Carlo method by B. Franek.
- Two examples from the real world:
a) PEP-N TPC
- Example of a TPC with very large distortions
b) CRID TPC
- In this example we want to show how much the old
Langevin’s theory is wrong compared to the most
recent method to calculate this problem.
2
J.Va’vra, Detector Lecture in Rome, January 25, 2002
No one doubts an extraordinary capability of TPC to
handle a very high multiplicity of tracks:
A typical STAR TPC event:
and pull physics signals out of it:
3
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The drift of electrons and ions
in gases (macroscopic view)
_________________________________________________________
r
r
A single electron moving in electric and magnetic fields, E and B, and under
an influence of a frictional force, can be described by a system of linear
differential equations:
m
r
r
r r
r
dv
= e E + e[ vx B] − K v
dt
(1)
where
m
- is the mass of the electron,
e
r
v
- is the electric charge of a particle,
- is drift velocity vector,
r
K v - "Langevin" frictional force, K is a constant,
m/K - has a dimension of time (in next chapter we will prove
that this is an average time between collisions: τ ≡ m/K).
_________________________________________________________
We are interested in a steady state solution of equation (1),
r
d v /dt = 0, which occurs for t >> τ . From the equation (1) we get:
r
e r e r r
K r
dv
E + [ vx B] −
v,
=0=
dt
m
m
m
e r K r e r r
E=
v − [ vx B]
m
m
m
r
e
1r e r r
E = v − [ vx B]
m
τ
m
4
(2)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Let's define the new variables ( ω is the cyclotron frequency):
r
r e r
e r
e
B, ε =
E, µ =
ω=
τ
m
m
m
Equation (2) changes to:
where
r r
r 1 r
ε = v − [ vx ω ]
τ

i
j
r r 
vx ω =  v x v y

ω
ω
 x
y
(4)


vz 
ω 
z
k
(5)
Expressing equations (4) and (5) explicitly:
1
ε = v − ω v +ω v
x
z y
y z
τ x
1
ε =ω v + v −ω v
y
z x τ y
x z
1
ε = −ω v + ω v + v
z
y x
x y τ z
This can be rewritten in a matrix form:
r r
Mv=ε
where
 1

τ

M =  ωz

 −ω
y

−ω
z
1
τ
ω
x
(3)
ω
(6)
(7)

y 

−ω 
x
1 
τ 
(8)
The solution of the drift velocity is obtained by inverting matrix M:
r
r
v = M−1 ε
5
(9)
J.Va’vra, Detector Lecture in Rome, January 25, 2002

1 + ω 2 τ2

x
τ

−1
−ω z τ + ωx ω y τ2
M =
1 + ω2 τ2 
 ω τ + ω ω τ2
 y
x z
ω τ + ω ω τ2
z
x y
1 + ωy2 τ2
−ω
x
τ + ω ω τ2
y z
τ + ω ω τ2 
x z 
ωx τ + ωy ωz τ2 

−ω
y
1+ ω 2 τ2
z
(10)
where
e
ω2 = ω x2 + ωy2 + ωz2 = ( )2 B2
m
(11)
is the square of the cyclotron frequency of the electron.
The final solution can be rewritten after some algebra in a form:
r
v=
r r
r ωτ r r
E ⋅B r
[E +
[E × B] + (ω τ)2
B]
B
1 + (ω τ)2
B2
µ
(12)
where the drift direction is governed by the dimensionless parameter ω τ.
This equation is one of the most famous equations of the TPC detector
physics. It appeared in every TPC proposal I know about. It is unfortunately
not quite right. Before I will tell you a better method, let’s play with it.
r
r
For ω τ = 0, v is parallel to E , and equation (12) yields:
r
r
v = µE
where
(13)
is the electron mobility, which is proportional to the average time
between collisions - see eq. (3). From equation (11) we obtain an expression for
ω τ:
ω τ=
e
Bτ
m
6
(14)


J.Va’vra, Detector Lecture in Rome, January 25, 2002
For
ωτ=0
==>
ω τ large
==>
r r
ω τ large & E . B = 0
==>
r
r
r
r
v = µ E , i.e. v is aligned with E ,
r
r
v tends to be aligned along B,
r r
r
v tends to be aligned along Ex B.
_______________________________________________________________
In practical chambers we have these conditions typically:
µ ~ 104 cm2 V-1 s-1 for electrons,
µ ~ 1 cm2 V-1 s-1 for ions,
B≤
1 T = 10-4 V s cm-2,
ω τ = B µ ≈ 10-4 for ions,
ω τ = B µ ≈ 1 for electrons.
τ ≈ 2-5 psec for electrons,
1
τ
≈ (2-5) x 1011 Hz collision rate for electrons,
The effect of typical magnetic fields on ion drift is negligible.
_______________________________________________________________
r
r
Example #1 ( E is perpendicular to B):
We assume:
r r
r
r
E . B = 0 , E = (E ,0,0) , B = (0,0,B )
x
z
From equation (12) we obtain:
r
µ
µ
vx =
Ex ≡
E
2
2
1 + (ω τ)
1 + (ω τ)
r
µ
ωτ
µ
vy = −
E x Bz ≡ −
ω τ E
2
2
B
1 + (ω τ)
1 + (ω τ)
z
(15)
v =0
z
Lorentz angle θ xy :
tan θ
xy
=
v
y
v
=− ω τ
x
In practice, by measuring the Lorentz angle we determine ω τ .
7
(16)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The drift velocity magnitude:
v(E,B)=
v x2 + v y2 =
µ
1 + (ω τ)2
r
r
E = µ E cos θ =
xy
= v ( E , B= 0 ) c o sθxy = v(E cos θ xy ,B = 0)
(17)
This is known as Tonk's theorem (L. Tonks, Phys. Rev. 97(1955)1443):
whatever the drift direction of the electron, the component of the electric field
along this direction determines the drift velocity magnitude for any magnetic
field value.
Experimental verification of the Tonk's theorem is methane drift velocity
data (T. Kunst, B. Goetz and B. Schmidt, NIM, A324(1993)127) :
Fig.1 - Experimental verification of the Tonk's theorem, where
α ≡ θxy is Lorentz angle, E is Electric field, N is density
of the gas, Td is Townsend (1 Td = 10-21 Vm2 ~ 250 V/cm),
Hx is Huxley (1 Hx = 10-27 Tm3 ~ 250 G).
8
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Note:
N = N0
p 273
760 T
N0
the Loschmidt number,
the Loschmidt number defined at 0oC
-
= 2.687 x 1025 molecules/m3.
___________________________________________________________
r
r
Example #2 ( E is parallel to B):
We assume:
r r
r
r
Ex B = 0 , E = (0,0,E ), B = (0,0,B )
z
z
From equation (12) we obtain:
vx = 0
vy = 0
(18)
r
E ⋅B
µ
vz =
[Ez + (ω τ)2 z z Bz ] ≡ µ E
1 + (ω τ)2
B2
___________________________________________________________
r
r
Example #3 ( E is nearly parallel to B):
We assume: r
r
r
B ≈ Bz , E = (0,0,Ez ), B = (0,By ,Bz ), By << Bz
First we evaluate:

i
j
r r 
Ex B =  E
E
x
y

B
B
 x
y


k
i
j
 
E  = 0 0
z
B   0 By
z
k

Ez  = i Ez By
Bz 

r r
E. B = E B + E B + E B = E B
x x
y y
z z
z z
9
J.Va’vra, Detector Lecture in Rome, January 25, 2002
From equation (12) we obtain:
B
ωτ
ω τ
y
E B ≈
v(B = 0)
B z y 1 + (ω τ)2 B
z
B
E ⋅B
(ω τ)2
µ
y
vy =
(ω τ)2 z z By ≈
v(B = 0)
2
2
2
B
1 + (ω τ)
B
1 + (ω τ)
z
E ⋅B
µ
vz =
[Ez + (ω τ)2 z z Bz ] ≈ µ Ez = v(B = 0)
2
1 + (ω τ)
B2
vx =
µ
1 + (ω τ)2
(19)
where v(B = 0) = µ Ez is drift velocity for B = 0 . We can define two Lorentz
angles:
v
(ω τ)2 By
y
=
2 B
v
z 1 + (ω τ)
z
v
y
tan θ xy =
=ωτ
v
x
tan θ yz =
(20)
SLD CRID example:
______________________________________________
10
J.Va’vra, Detector Lecture in Rome, January 25, 2002
. 40 TPC's
. Drift box gas: C2H6 + TMAE (~0.1 %)
. Maximum drift length: 1.2 m
______________________________________________
1
r2 − 2 z2 0
a) Bz (r,z) = B0
+ B0
, Bz = 0.6 T
z 2 r
r z
0 0
Br (r,z) = B0
r
r z
= κ z , r0= 1.2 m, z0= 1.5 m, B0
r = 0.0214 T.
r z
0 0
b) For simplicity we assume that Br is parallel with y-axis ( Br ≈ By ),
c) CRID TPC has operates with electric field E ~ 400 V/cm,
d) Average drift velocity of 4.3cm/µs,
e) ωτ ~ 0.87 and θ xy ~ 41o for C2H6 gas (see SLAC-PUB-4403),
f) TPC active length is between z1 = 0.1 m and z2 = 1.2 m.
y
θ
θ
xy
x
z
1
11
yz
z
2
TPC length
z ↑↑ E
z
(TPC axis)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Expected distortion in x-direction:
t2
t2
v(B = 0)
δx =
v x dt ≈
Br dt =
2
B
1 + (ω τ)
z
t1
t1
ωτ
∫
=
=
∫
z2
ωτ
v(B = 0)
1
Br dz =
B
v(B = 0)
1 + (ω τ)2
z
z1
ωτ
∫
z2
1
ω τ
Br dz =
κ [z 22 − z 12 ] ≈ 9.9 mm (21)
2 B
2
1 + (ω τ) z
1 + (ω τ)
z1
∫
where κ is a constant:
κ=
0
1 Br
r
.
2 B r z
z 0 0
Similarly for y-direction:
t2
δy =
∫
v y dt ≈
t1
(ω τ)2
t2
(ω τ)2
v(B = 0)
Br dt =
2
B
1 + (ω τ)
z
t1
∫
z2
1
(ω τ)2
2
2
=
Br dz =
κ [z 2 − z 1 ] ≈ 8.6 mm (22)
2 B
2
1 + (ω τ) z
1 + (ω τ)
z1
∫
These calculations were verified by the measurement using the UV fiducials
fibers (J.V.’s calculation in K. Abe et al., NIM A343 (1994) 74):
12
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Fiducials points with UV fibers
______________________________________________
1) Magnet off
c) Detector end
d) High voltage end
2) Magnet on
a) Detector end
b) High voltage end
______________________________________________
. UV fibers - extremely useful feature of TPCs.
13
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Comparison with theory
______________________________________________
14
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The drift of electrons in gases
(simple microscopic view)
____________________________________________________________
The simple macroscopic theory, based on a concept of the friction force,
cannot predict ω τ, which has to be obtained by measuring the Lorentz angle.
Can we do better by introducing the more detailed picture of the electronmolecule collisions ?
1) As electron moves in the gas it suffers random collisions with molecules of
the gas. We assume that there is no correlation in the direction before
and after the collision.
2) Number of collisions n in drift distance x is related to the average drift
velocity v as follows:
v)(1 τ )
n = (x
(23)
where
τ - is an average time between collisions,
1 τ - is average rate of collisions.
3) The average time between collisions τ is related to the electron
instantaneous velocity vinst, the collision cross-section σ and the density
of the gas N as follows:
1
τ
= Nσ v
inst
(24)
3) The differential probability P of having the next collision between
time t and t + dt is:
P=
1
τ
e − t τ dt
15
(25)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
We will show that this "new picture" will confirm results obtained using the
macroscopic concept of the friction force. However, as we will see, this new
picture is still too simplistic.
__________________________________________________________
Example #1 (a uniform electric field E and no magnetic field):
An electron between collisions moves accordingly to the equation of motion:
dv
m
=e E
(26)
dt
Its solution is the electron displacement as a function of time:
x(t) =
1 e
E t2
2m
The average displacement < x > at average time between collisions τ is
obtained by averaging x(t) over time t, using the probability distribution of t, i.e.
eq. (25):
∞
<x>=
1 e
∫ 2 m Et
2 1 e−t
0
τ
τ dt = e E τ2
m
(27)
The average electron drift velocity is defined as:
< v >=
< x>
τ
=
e
Eτ=µE
m
(28)
___________________________________________________________
r
r
Example #2 ( E is perpendicular to B):
We assume: r r
r
r
E . B = 0 , E = (Ex ,0,0) , B = (0,0,Bz )
An electron between collisions
accordingly to the equation of motion:
r moves
r
r r
dv
m
= e E + e [v × B]
(29)
dt
16
J.Va’vra, Detector Lecture in Rome, January 25, 2002
which can be rewritten as a system of differential equations:
m
m
m
d v (t)
x
= e E x + e v y Bz
y
= −e v x Bz
dt
d v (t)
dt
d v (t)
z
dt
(30)
=0
e
B and obtain the solution for the
m z
initial conditions v x (0)= v y (0)= vz (0)= 0 :
We introduce the cyclotron frequency ω =
e
1
v x (t)= (
E
) s i nω t
m x ω
e
1
v y (t)= (
E
)(cos ω t −1)
m x ω
vz (t)= 0
(31)
The drift velocity is then given by the following averages:
e
1
< v x (t) > =
E
m x ω
∞
∫ sin ωt τ e
1
− t τ dt =
0
E τ
e
µ
x
=
=
E
m 1 + (ω τ)2 1 + (ω τ)2 x
< v (t) > =
y
e
1
E
m x ω
∞
∫ (cos ωt − 1) τ e
0
1
2
(µ =
− t τ dt =
µ
ω τ
e Ex ω τ
= −
=
E B
x z
m 1 + (ω τ)2 1 + (ω τ)2 B
z
< vz (t) > = 0
17
e
τ)
m
(32)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
We have obtained the same results as in equations (15) which was derived
from the friction force model. This also proves that our assumption that τ ≡
m/K was correct. The Lorentz angle θ xy is obtained using equations (32) and
(13):
tan θ xy =
v
v
y
= − ω (τE,B) ≡ −
x
e
v ( E , B= 0 ) B
B τ = −µ B = −
(33)
m
E
In practice, the equation (33) is only approximate because our modeling of
electron collisions with the gas is still too simple minded. The correct equation
is :
tan θ xy = ψ
v ( E , B= 0 ) B
E
(34)
where ψ = ψ (E N ,B N) is the magnetic deflection factor.
Measurement of the magnetic deflection factor ψ was done, for example, by T.
Kunst, B. Goetz and B. Schmidt, NIM A324(1993)127) :
18
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Fig. 2 - Magnetic deflection factor ψ for (a) CH4, (b) 90%
Ar+10% CH4, (c) 95% Ar + 5% CH4 gases.
Note: The fact that ψ is not really a constant casts a doubt that we are
dealing with a real theory so far.
19
J.Va’vra, Detector Lecture in Rome, January 25, 2002
r
r
Example #3 :( E is nearly parallel to B)
We assume: r
r
r
B ≈ Bz , E = (0,0,E ), B = (0,B ,B ), By << Bz
z
y z
An electron between collisions
accordingly to the equation of motion:
r moves
r
r r
dv
m
= e E + e [v × B]
(35)
dt
which can be written as a system of differential equations:
m
m
m
d v (t)
x
= e ( vy Bz − vz By )
y
= −e v x Bz
dt
d v (t)
dt
d v (t)
z
dt
(36)
= e Ez + e v x By
e
B and obtain the solution for the
m z
initial conditions v x (0)= v y (0)= vz (0)= 0 :
We introduce the cyclotron frequency ω =
v x (t)=
v y (t)=
B E
y
z
(cos ω t −1)
2
B
B B E
y
z
3
B
z
(ω t − sin ωt)
E
vz (t)= z (Bz2 ω t + By2 sin ωt)
B3
20
(37)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The drift velocity is then given by the following averages:
B E ∞
y z
1
µ2
< v x (t) > =
(cos ωt − 1) e − t τ dt = −
B E
2
τ
1 + (ω τ)2 y z
B
∫
(38)
0
< v y (t) > =
< vz (t) > =
B B E ∞
y
z
3
B
E
z
3
B
∞
∫
z
∫ (ωt − sin ω t) τ e
1
− t τ dt =
0
(Bz2 ωt + By2 sin ω t)
0
µ3
1+ (ω τ)2
2
By Bz Ez
2
1 + µ Bz
e − t τ dt =
τ
1 + (ω τ)2
1
µ Ez
Again, we have obtained the same results as in equations (19)
obtained using the friction force model.
___________________________________________________________
However, in all these examples, ωτ and µ had to be obtained at
the end from the experiment, and -not- from the theory !!
=> The exercise so far was very useful,
but not sufficient....
___________________________________________________________
. Before we describe the real theory, we have to say few words
about the electron diffusion
21
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Electron diffusion
__________________________________________________________
Drifting electrons scatter on the gas molecules. Their motion can be
described by their random motion, which is characterized by the mean energy
ε and gives rise to diffusion, and by their collective motion, which is
characterized by the average drift velocity. The motion follows the continuity
equation (the total electron current is given by the sum of the drift current and
r
r
r
the diffusion current: J = n v − D ∇n ). Its solution has in the simplest case the
isotropic distribution, i.e. the point-like cloud of electrons at time t = 0 will
create a Gaussian density distribution at time t :
1
r2
)3 exp(−
)
4Dt
4 π Dt
n= (
(39)
where
D
-
is the diffusion coefficient
From equation (39) follows that the diffusion width of an electron cloud σ x ,
after starting point-like and traveled time interval t, is:
σx =
2Dt
(40)
One can show that the diffusion coefficient is related to electron energy ε as
follows:
D=
ε
2 ε
τ= k τ
3m
m
where
22
(41)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
ε
εk
-
electron energy
so called "characteristic energy"
m
-
mass of the electron
τ
-
average time between collisions
Recalling the expression for electron mobility:
r
r
e
v = µE,
µ=
τ,
(42)
m
we obtain expression for the characteristic electron energy in terms of the D/ µ
ratio :
εk =
Dm Dm De
=
=
.
µm
τ
µ
e
(43)
The diffusion width σ x of an electron cloud width, after starting point-like
and traveled over a distance x:
σx =
2 D t=
2Dx
=
µE
2ε x
k
e E
.
(44)
The smallest diffusion corresponds to a thermal energy εk = k T,
3
(ε = k T ~
0.04 eV at 24oC) resulting in the smallest possible diffusion width
2
σ x of an electron cloud (so called "thermal limit"):
σx =
2 D t=
23
2k T x
.
e E
(45)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
However the reality is more complicated unfortunately :
1. Electric field alters the diffusion so that it is necessary to introduce
two
diffusion coefficients DL and DT , one for the
longitudinal
and
one transverse direction in respect to electric
field. Example in
methane is due to B. Schmidt, Dissertation, Univ. of
Heidelberg, 1986:
Fig.3 - Diffusion coefficient in methane
2. Magnetic field alters the diffusion so that the transverse diffusion
coefficient DT (B) in respect to its direction gets smaller (neglect
the effect of the electric field effect for the moment):
D (B)
T
D (B = 0)
T
=
1
1 + (ω τ)2
,
while the longitudinal diffusion coefficient remains the same
DL (B) = DL (0).
24
(46)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
How to predict the drift velocity,
diffusion and Lorentz angle ?
___________________________________________________________
1. Boltzmann equation method.
A full theory of electron transport in gases can get rather complicated. We
will follow more simple path of Schultz and Gresser, NIM 151 (1978) 413 and G.
Schultz, Ph.D. thesis, 1976. For more complete description see L. G. H. Huxley
and R.W. Crompton, "The diffusion and drift of electrons in gases." For a theory
which includes the ionization and attachment processes see K.F. Ness and R.E.
Robson, Phys. Rev A, 34(1986)2185.
Drifting electrons scatter on the gas molecules. This motion follows the
Boltzmann transport equation, which expresses the conservation of number of
electrons. If f(v,r,t) is the distribution function of electrons at r, v of the phase
space at time t , the simplest 1-dimensional form of the Boltzmann equation is:
∂ f ∂f ∂ r ∂ f ∂ v ∂f
+
+
−
=0
∂t ∂ r ∂ t ∂ v ∂t ∂ t Coll.
where
∂f
∂t
∂f ∂r
∂ r ∂t
∂f ∂ v
∂ v ∂t
represents time evolution of f(v,r,t),
represents loss of electrons in interval dr due
to diffusion,
represents loss of electrons in interval dv due
to acceleration caused by field E,
25
(47)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
∂f
- represents loss of electrons in interval dv due to
∂t Coll.
collisions of electrons with molecules of gas.
To solve the equation (47) we introduce the following simplifications:
a) Expand the distribution function f in terms of the Legendre polynomial
expansion, and use in our case two terms only:
f = f0 (ε)+ f1 (ε) cos θ +. . .
(48)
b) Assume no ionization and no attachment processes in this example,
c) Express the distribution function f using electron energy ε =
1
2
m v , and the
2
mean free path l (ε) between two elastic collisions,
d) Assume that the electric field E is parallel with x axis (no magnetic field for
now),
e) Assume a stationary case, i.e. no x or t dependence.
One gets two coupled equations :
m vf 1
∂f
∂f
eE 0 + m v
=−
∂v
∂x
l (ε)
eE ∂
m2 3 ∂ v4 f 0
1
∂f
(v2 f1 )+ m v2 1 =
(
)
2 v∂ v
2
∂ x M 2 v ∂ v l (ε)
(49)
(50)
where M is mass of the molecule.
∂f
We now assume
= 0 , i.e. uniform distribution along x direction. This
∂x
yields these two equations:
eE
m vf 1
∂ f0
=−
∂v
l (ε)
eE ∂
m2 3 ∂ v4 f 0
(v2 f1 )=
(
)
2 v∂ v
M 2 v ∂ v l (ε)
26
(51)
(52)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
One can now eliminate f1 and solve for f0 :
2 ( e E 2) ∂
∂(f 0 v)
2m ∂ ε v f 0
[ ε l (ε)
]+
[
]= 0
3 m ∂ε
∂ε
M ∂ ε l (ε)
(53)
Equation (53) can be easily solved (first, we assume l (ε) = const.) :
f0 (ε ) = C ε exp[ −
3m
ε
2
(
) ]
M e El (ε )
ε max
where a constant C is obtained from a normalization :
∫f
0
(54)
( ε ) d ε = 1.
0
A fraction of energy lost by electron scattering elastically from molecule of
mass M can be approximated as follows :
∆ε
ε
=
2m
(1 − cos θ ) ,
M
(55)
while the mean fraction of energy lost is:
Λ = 2m M
(56)
However, the solution (54) must be changed if l (ε) is not a constant and if
the mean fraction of energy lost in the collision is not equal to Λ = 2m M, but
it is Λ = Λ(ε). In this case equation (53) becomes :
ε
f0 (ε ) = C ε exp[ −
∫
0
3 Λ (ε, ) ε,
(eE l ( ε, ))2
d ε, ]
(57)
Unfortunately, one has to introduce several complications:
a) If the energy of electrons is similar as the thermal energy of the molecules
27
J.Va’vra, Detector Lecture in Rome, January 25, 2002
( ε = kT ~ 0.025 eV), it is necessary to introduce an additional term in equation
(53), which changes to :
2 ( e E 2) ∂
∂(f 0 v)
2m ∂ ε v f 0
[ ε l (ε)
]+
[
]+
3 m ∂ε
∂ε
M ∂ ε l (ε)
2
∂ 2 Λ(ε) ε
∂(f 0 v)
+
[
]
kT
=0
∂ε
m
l (ε)
∂ε
This changes solution (57) to:
f0 (ε ) = C ε exp[ −
ε
∫
0
(58)
3 Λ ( ε, ) ε,
,
d ε ] (59)
,
[ ( e El ( ε ))2 + 3 Λ ( ε, ) ε, kT ]
b) If we wish to add inelastic collisions, we must add to equation (53) an
additional term, which changes equation (53) to :
2 ( e E 2) ∂
∂(f 0 v)
2m ∂ ε v f 0
[ ε l (ε)
]+
[
]+
3 m ∂ε
∂ε
M ∂ ε l (ε)
2
∂ 2 Λ(ε) ε
∂(f 0 v)
+
[
]
kT
∂ε
m
l (ε)
∂ε
+
∑
[
k
2 m ( ε + εk )
f0 (ε + ε k )−
l k ( ε +ε k )
(60)
2 m ε
f0 ( ε)] = 0 ,
l k (ε)
where
εk
-
is the excitation energy of the k-th state
lk
-
is the mean free path between two collisions
which gives rise to the excitation.
In practice, the inelastic collisions are vibrational and rotational
molecular excitations caused by electrons of sufficient energy. The last term
in equation (60) can be approximated by :
∑ε
k
k
∂ v f 0 (ε)
[
]
∂ ε l k (ε)
28
(61)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
and the solution (59) is still valid provided we use :
ε k l e (ε)
2m
Λ(ε) =
+
M
ε l k (ε)
k
∑
(61)
The last equation can be expressed in terms of cross sections:
Λ(ε) =
2m
+
M
∑ εε σ (ε)
k
σ k (ε)
(62)
e
k
where
1
N l (ε)
p 273
N = N0
760 T
σ (ε ) =
N0
the cross section,
the Loschmidt number,
the Loschmidt number defined at 0oC
-
= 2.687 x 1025 molecules/m3,
p
-
pressure in Torr,
T
-
absolute temperature in K.
c) Finally, one includes magnetic field. This results in the following
r
r
modification of the solution (59) ( E is perpendicular to B):
ε
f0 (ε ) = C ε exp[ −
∫
0
3 Λ ( ε, )G(B) ε,
d ε, ] (63)
[ ( e El ( ε, ))2 + 3 Λ ( ε, ) ε, k T G ( B])
where
2
G(B) = 1 +
e2 Bz le ( ε )2
2m ε
(64)
d) Gas mixtures are calculated as follows:
∑δ σ
σ Λ= ∑ δ σ Λ
σ =
i
(65)
i
i
i
i
i
29
i
(66)
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Once we obtain the function f0 (ε), the electron transport coefficients are
r
r
calculated as follows ( E is perpendicular to B):
2 eE
1. Drift velocity : v x = −
3 m
ε max
∫
0
ε max
v
y
=
∫
e E e B
3m
ε le (ε) ∂ (f0 (ε) v(ε ))
dε
G(B)
∂ε
le2 (ε) v (ε)
G(B)
0
v=
v
2
x
+v
2. Lorentz angle : θ = tan −1 (
3. The diffusion coefficient :
v
v
∂(f 0 (ε) v( ε ))
dε
∂ε
(63)
2
y
y
)
(64)
x
DT =
1
3
ε max
∫
le (ε) v (ε ) f (ε)d ε
0
G(B)
(65)
0
Note: From the Lorentz angle we determine the ω τ term needed in
many equation in the earlier part of the lecture.
The theory works with 5-10% accuracy.
________________________________________________________________
30
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The following three examples are my calculations using a code written
by P. Coyle following the Schultz and Gresser theory:
Example #1 ( f
0
(ε) function in methane) :
Electron energy distribution in methane
(B=0 kG, 760 Torr, 298 degC)
Distribution function
f0(E)
16
14
100 V/cm (V-drift ~ 1.9 cm/us)
12
200 V/cm (V-drift ~ 3.8 cm/us)
10
400 V/cm (V-drift ~ 7.17 cm/us)
600 V/cm (V-drift ~ 9.11 cm/us)
8
800 V/cm (V-drift ~ 9.86 cm/us)
6
1000 V/cm (V-drift ~ 9.97 cm/us)
4
2
0
0
0.2
0.4
0.6
0.8
Electron energy [eV]
Example #2 ( f
0
(ε) function in various gases) :
Electron energy distribution
at 1 kV/cm in various gases
(B=0 kG, 760 Torr, 298 degC)
Distribution function
f0(E)
18
16
CO2
CH4
Ar
iC4H10
C2H6
He
N2
Xe
Ne
14
12
10
8
6
4
2
0
0
0.2
0.4
0.6
Electron energy [eV]
31
0.8
1
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Example #3 (Calculated Lorentz angles):
Lorentz angle in CO2/C2H6 mix as a
function of electric field gradient
(B=6 kG, 760 Torr, 298 degC)
Lorentz angle [deg]
50
45
40
35
400 V/cm
30
1000 V/cm
25
100 V/cm
20
ECRID data (400 V/cm)
15
10
5
0
0
10
20
30
40
50
60
70
80
90
100
CO2 fraction in the mix [%]
Great !! We can predict the Lorentz angle, and therefore
32
ω τ.
J.Va’vra, Detector Lecture in Rome, January 25, 2002
However, the presented theory in this lecture
is not sufficiently precise:
___________________________________________________________________________
1. It does not calculate the longitudinal diffusion coefficient DL.
2. B. Schmidt (Dissertation, Univ. of Heidelberg, 1986) introduced (a) six terms,
(b) an anisotropy in the elastic scattering, and (c) higher order of vibration
cross-sections to explain the methane data:
Fig.4 - Drift velocity in methane
Fig.5 - Diffusion coefficient in methane
33
J.Va’vra, Detector Lecture in Rome, January 25, 2002
3. S. Biagi (NIM A273(1988)533) included (a) three terms, and (b) the
ionization and attachment processes (this Schmidt did not include).
His computer program, MAGBOLTZ, is available from the CERN
library. A review of noble gases Helium, Neon, Argon, and Xenon, mixed
with the popular quenchers, namely Methane, Ethane, Isobutane, DME,
and Carbon dioxide is available at:
http://wwwcn.cern.ch/writeup/garfield/examples/gas/trans2000.html,
and it is being updated regularly (see A. Sharma, ICFA Instrumentation
Bulletin, Vol.16, http://www.slac.stanford.edu/pubs/icfa/).
_______________________________________________________________
Example #1 (comparison of my data and MAGBOLTZ program):
J.Va’vra, NIM A324 (1993) 113.
Fig.6 - Measurements and calculation of drift and diffusion for heliumethane mixtures.
___________________________________________________________________
4. S. Biagi is now supporting MAGBOLTZ-MONTE program only, which
has improved convergence compared to the MAGBOLTZ program.
34
J.Va’vra, Detector Lecture in Rome, January 25, 2002
2. Monte Carlo method.
I will mention a recent attempt by H. Pruchova and B. Franek
(RAL Report RAL-95-034; Fall 1996 Issue of ICFA Bulletin, and a web address:
http://www.slac.stanford.edu/pubs/icfa/.
They proceed as follows:
1. All electrons are traced in small time steps ∆ tsmp (sampling step). These
steps are few ps long.
2. In each sampling step the electron is traced in elementary steps ∆ telm . The
elementary step ∆ telm is chosen to be small fraction of mean free path,
calculated from the electron-molecule cross-section at the given energy. In
each ∆ telm the electron is traced according to a drift velocity at a given electric
field; it is also decided whether there is a collision between the electron and a
gas molecule by generating a random number and the interaction process is
chosen according to the corresponding cross section tables. There is a number
of possible electron collisions with molecules:
a) elastic scattering (electron does not lose energy),
b) inelastic scattering (electron loses energy),
c) ionization scattering (a new electron is produced),
d) attachment scattering (electron is absorbed by a molecule),
e) etc.
3. When the parent electron reaches the time t + ∆ tsmp , all electrons created during
∆ tsmp are traced in the above described manner from their time of creation until
they all reach the time t + ∆ tsmp .
35
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Fig.5 - Electron molecule cross-sections in methane
Fig.6 - Calculated electron paths in xy-plane.
36
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Fig.7 - Calculated electron paths in xz-plane.
Fig.8 - Prediction of the 1-st Townsend ionization coefficient
using the Monte Carlo and the Boltzmann equation methods.
37
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Fig.8 - Prediction of the gas gain using the Monte Carlo.
___________________________________________________
One should mention that similar attempts were made by J. Groh, Interner
Bericht DESY FH1T-89-03 May 1989, and by M. Matobe et al., IEEE Trans.
Nucl. Sci. NS-32,541(1985).
Difficulties of the Monte Carlo method:
1. Very demanding on the CPU time.
2. It is difficult to obtain the "correct" cross-sections. Methane is about the best
studied gas of all. Some cross-sections in literature are altered to obtain the best
match between theory and data using the Boltzmann equation procedure. There
is some risk that the incorrectly "tuned" cross-sections would yield
inconsistencies if used in the Monte Carlo method.
3. Still some difficulties to predict the practical quantities such as drift velocity,
diffusion, etc., even in methane. Nevertheless, the results are impressive.
More work is needed in this area !
38
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Example #1 - PEP-N TPC
- Distortions in dipole magnet as a function of:
a) gas choice,
b) magnetic field uniformity,
c) drift field.
- Field cage design.
PEP-N tracking design concept:
Dipole
magnet
z
E-drift
Detector
Bz
Br
x
Beam pipe
39
y
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Magnetic field uniformity
Field parameterization provided by Mario Posocco.
1) Initial dipole design (field map: DV02):
Angle between the Β-field and Ε−drift directions
Radial component of magnetic field in PEP-N
0
0
-0.02
-2
-4
-0.04
-6
-8
-0.06
r = 0 cm
-0.08
r = 10 cm
r = 20 cm
-0.1
r = 30 cm
-0.12
r = 40 cm
r = 50 cm
-0.14
0
10
20
30
40
50
60
-10
-12
r = 0 cm
-14
-16
-18
-20
r = 20 cm
r = 30 cm
r = 10 cm
r = 40 cm
r = 50 cm
0
10
Vertical distance z [cm]
20
30
40
50
60
Vertical distance z [cm]
2) The first iteration to improve the magnetic field uniformity (DV03):
Angle between the Β-field and Ε−drift directions - Modif_1
Radial component of magnetic field in PEP-N -Modif_1
0
0
-0.01
-2
-4
-0.02
-6
-8
-0.03
r = 0 cm
-0.04
r = 10 cm
-0.05
r = 20 cm
r = 30 cm
-0.06
r = 40 cm
-0.07
r = 50 cm
-0.08
0
10
20
30
40
50
60
-10
-12
r = 0 cm
-14
-16
r = 20 cm
r = 30 cm
r = 10 cm
r = 40 cm
-18
-20
r = 50 cm
0
10
Vertical distance z [cm]
20
30
40
50
60
Vertical distance z [cm]
3) The second iteration to improve the uniformity (DV06b):
Angle between the Β-field and Ε−drift directions
Radial component of magnetic field in PEP-N
10
0.1
r = 0 cm
0.08
5
r = 10 cm
r = 20 cm
0.06
0
r = 30 cm
0.04
r = 40 cm
-5
r = 50 cm
0.02
r = 0 cm
r = 10 cm
0
-10
-0.02
-15
r = 20 cm
r = 30 cm
r = 40 cm
r = 50 cm
-0.04
0
10
20
30
40
50
60
Vertical distance z [cm]
-20
0
10
20
30
40
Vertical distance z [cm]
40
50
60
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Distortions
The Langevin theory, which has been used for many TPC
designs, is only an approximate method; it does not predict ωτ.
In fact, it turns out that there is no single ωτ, which would
explain all three drift velocity components.
r
v=
r rr
r ω τr r
µ
2 E ⋅B
[E +
[E × B]+(ωτ)
B]
B
B2
1+(ω τ)2
I will use the MAGBOLTZ-MONTE program to calculate
the drift velocity components vx,y,z (E,B) gas. Once I know the
drift velocity components as a function of z-vertical, I calculate
the distortions in the detecting plane as follows:
t2
d x=
∫
t1
t2
d y=
∫
t1
(dz)
i
v x dt = ∑ v x
i v
i
z
(dz)
i
vy d t = ∑ vy
i
v
i
z
In the following, I calculate the worst case distortion at r = 50cm,
and for the total drift of 50cm.
41
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Distortion calculation using the
Magboltz-Monte program
J.V., 4.18.2001
Vx and Vy = f(z-vertical)
(Field map DV.02, E-drift = 400V/cm)
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
Vx(E=400V/cm, B)
Vy(E=400V/cm, B)
Linear ( Vx(E=400V/cm, B))
Linear ( Vy(E=400V/cm, B))
80% He + 20% CO2
Vx = -6.03E-05*Z-vertical
Vy = -7.28E-04*Z-vertical
0
5
10
15
20
25
30
35
40
45
z-vertical [cm]
J.V., 4.18.2001
Vz = f(z-vertical)
(Field map DV.02, E-drift = 400V/cm)
1.045
80% He + 20% CO2
1.043
Vz(E=400V/cm, B) at 24 C
1.041
1.039
1.037
1.035
0
10
20
30
40
50
z-vertical [cm]
J.V., 4.18.2001
Distortions in x&y = f (z-vertical)
0
-0.5
-1
80% He + 20% CO2
-1.5
dx (based on Magboltz-MONTE)
-2
dy (based on Magboltz-MONTE)
-2.5
0
5
10
15
20
25
30
35
40
45
50
Starting drift point in Z-vertical [cm]
- It takes ~8 hours on my Linux box to do this calculation…
This is because vx&vy are small fraction of vz, and we want
to achieve a good precision in distortion calculation.
42
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Distortion with the best field map (DV06b)
J.V., 4.30.2001
Vx and Vy = f(z-vertical)
E-drift =400V/cm,Field map: DV06b
0.03
Vx(E=400V/cm, B)
Vy(E=400V/cm, B)
Poly. ( Vx(E=400V/cm, B))
Poly. ( Vy(E=400V/cm, B))
80% Ar + 20% CH4
-0.01
-0.05
-0.09
Vx = 4.21E-06*Z 3 - 1.01E-04*Z2 - 6.82E-03*Z
-0.13
-0.17
-0.21
Vy = 6.36E-06*Z 3 - 1.76E-04*Z 2 - 7.28E-03*Z
-0.25
0
5
10
15
20
25
30
35
40
45
50
z-vertical [cm]
J.V., 4.30.2001
Vz = f(z-vertical)
E-drift =400V/cm,Field map: DV06b
6.2
80% Ar + 20% CH4
6.18
Vz(E=400V/cm, B) at 24 C
6.16
6.14
6.12
6.1
0
10
20
30
40
50
z-vertical [cm]
J.V., 4.30.2001
Distortions in x&y = f (z-vertical)
E-drift =400V/cm,Field map: DV06b
0
80% Ar + 20%CH4
-0.5
-1
-1.5
dx (based on Magboltz-MONTE)
dy (based on Magboltz-MONTE)
-2
-2.5
0
10
20
30
40
50
Starting drift point in Z-vertical [cm]
This means that even the fast gas is the candidate. With a slow
gas, the distortion is only few mm in the latest map.
43
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The worst case distortion dx & dy in the PEP-N TPC:
I. Distortions = f(gas choice) at r = 50cm & 50cm drift:
Gas
80%Ar+20%CH4
80%He+20%CO2
80%He+19%CO2+1%CH4
80%He+15%CO2+5%CH4
80%He+15%CO2+5%iC4H10
80%He+20%iC4H10
Field
map
DV.02
DV.02
DV.02
DV.02
DV.02
DV.02
E-drift
[V/cm]
400
400
400
400
400
400
dx
[cm]
-4.2
-.07
-0.1
-.12
-0.1
-0.3
dy
[cm]
-4.9
-0.9
-0.91
-1.03
-1.0
-1.5
Vz-ave
[cm/us]
6.15
1.039
1.072
1.225
1.182
1.72
II. Distortions = f(E-drift) at r = 50cm & 50cm drift:
Gas
80%Ar+20%CH4
80%Ar+20%CH4
80%He+20%CO2
80%He+20%CO2
Field
map
DV.02
DV.02
DV.02
DV.02
E-drift
[V/cm]
400
200
400
200
dx
[cm]
-4.2
-6.8
-.07
-0.1
dy
[cm]
-4.9
-4.2
-0.9
-0.9
III. Distortions = f(field map) at r = 50cm & 50cm drift:
Gas
Field
E-drift dx dy
map
[V/cm] [cm] [cm]
80%Ar+20%CH4
DV.02 400
-4.2 -4.9
80%Ar+20%CH4
DV.03 400
-2.7 -2.9
80%He+20%CO2
DV.02 400
-.07 -0.9
80%He+20%CO2
DV.03 400
-.04 -0.5
80%Ar+20%CH4
DV.06b 400
-1.0 -1.04
80%He+20%CO2
DV.06b 400
-.08 -0.25
44
Vz-ave
[cm/us]
6.15
6.9
1.039
0.53
Vz-ave
[cm/us]
6.15
6.16
1.039
1.039
6.15
1.039
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Field cage design
(Schematic picture only – nothing to scale)
45
J.Va’vra, Detector Lecture in Rome, January 25, 2002
The STAR TPC is using the wire & pads to detect
the drifting electron charge:
. There are other possibilities, but that would take another lecture...
46
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Example #2:
Temperature effect on the Lorentz angle.
Reference:
J. Va’vra, SLAC Detector lectures, Gaseous detectors I, 8.11.1998, and
Bari RICH conference paper: K. Abe et al., Nucl.Instr.&Meth., A343(1994)74.
________________________________________________________________
The final solution of the Langevin’s theory can be rewritten in a form:
r
v=
r r
r ωτ r r
E ⋅B r
B]
[E +
[E × B] + (ω τ)2
B
1 + (ω τ)2
B2
µ
(12)
where the drift direction is governed by the dimensionless parameter ω τ,
which cannot be predicted by this theory. One should also point out that
Langevin’s theory is a simplification, and it is not obvious how well it should
work. We will try anyway to check this.
________________________________________________________________
r
r
Example #3 ( E is nearly parallel to B):
We assume: r
r
r
B ≈ B , E = (0,0,E ), B = (0,B ,B ), B << B
z
y z
z
y
z
First we evaluate:



j
k
i
j
k
r r  i
 

Ex B =  E
E
E  =  0 0 E  = i Ez By
x
y
z
z

B
B
B   0 By Bz 
 x
y
z
r r
E . B = E x Bx + E y By + Ez Bz = Ez Bz
From equation (12) we obtain:
vx =
µ
1 + (ω τ)2
B
ωτ
y
ω τ
Ez By ≈
v(B = 0)
2
B
B
1 + (ω τ)
z
47
J.Va’vra, Detector Lecture in Rome, January 25, 2002
E ⋅B
(ω τ)2
(ω τ)2 z z By ≈
1 + (ω τ)2
B2
1 + (ω τ)2
E
⋅
B
µ
vz =
[E + (ω τ)2 z z Bz ] ≈ µ Ez
1 + (ω τ)2 z
B2
vy =
µ
B
y
v(B = 0)
B
z
(19)
= v(B = 0)
where v(B = 0) = µ Ez is drift velocity for B = 0 . We can define two Lorentz
angles:
B
(ω τ)2
y
y
=
(20)
2
B
v
1
+
(ω
τ)
z
z
v
y
tan θ xy =
=ωτ
v
x
_____________________________________________
tan θ yz =
v
Let’s consider the SLD CRID example:
. 40 TPC's
. Drift box gas: C2H6 + TMAE (~0.1 %)
48
J.Va’vra, Detector Lecture in Rome, January 25, 2002
. Maximum drift length: 1.2 m
y
θ xy
θ yz
x
z
1
z
2
z ↑ ↑ Ez
(TPC axis)
TPC length
_____________________________________________
a) Bz (r,z) = B0
+
z
1 0 r2 − 2 z2 0
B
, Bz = 0.6 T
2 r
r z
0 0
r z
Br (r,z) = B0
= κ z , r0= 1.2 m, z0= 1.5 m, B0
r r z
r = 0.0214 T.
0 0
rTPC middle = 1.218 m - this is where I calculate Br.
b) For simplicity we assume that Br is parallel with y-axis ( Br ≈ By ),
c) CRID TPC has operates with electric field E ~ 400 V/cm,
d) Average drift velocity of Vz ~ 4.3cm/µs at 24 C,
e) ωτ ~ 0.87 and θ xy ~ 41o for C2H6 gas (see SLAC-PUB-4403),
This was calculated by Paschal Coyle using a program Lorenz,
which was based on the Schultz-Gresser alghoritm, which is considered
incorrect !! I will use instead the Steve Biagi’s MAGBOLTZ-MONTE
program. (Note: this is a Monte Carlo version of the old MAGBOLTZ
program, which he considers no longer correct either, and Steve does not
support it any more).
f) TPC active length is between z1 = 0.1 m and z2 = 1.2 m.
49
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Example:
Using the Langevin’s theory and
t2
~ 0.87, the expected distortion in x-direction:
t2
v(B = 0)
δx =
v x dt ≈
Br dt =
2
B
1 + (ω τ)
z
t1
t1
ωτ
∫
=
=
∫
z2
ωτ
v(B = 0)
1
Br dz =
B
v(B = 0)
1 + (ω τ)2
z
z1
ωτ
∫
z2
1
ω τ
Br dz =
κ [z 22 − z 12 ] ≈ 9.9 mm (21)
2 B
2
1 + (ω τ) z
1 + (ω τ)
z1
∫
where κ is a constant:
κ=
0
1 Br
r
.
2 B r z
z 0 0
Similarly for y-direction:
t2
t2
(ω τ)2
v(B = 0)
δy =
v y dt ≈
Br dt =
2
B
1 + (ω τ)
z
t1
t1
∫
=
(ω τ)2
∫
z2
1
(ω τ)2
Br dz =
κ [z 22 − z 12 ] ≈ 8.6 mm (2))
2 B
1 + (ω τ) z
1 + (ω τ)2
z1
∫
As we said and also will see in the next, this particular method is
not correct. Instead, I will now use the MAGBOLTZ-MONTE
program to calculate vx,y,z(E = 400V/cm, B) at two different
temperatures of 24o and 35o.
50
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Important point:
GEOMETRY OF THE MAGBOLTZ-MONTE PROGRAM:
THE ELECTRIC FIELD IS TAKEN ALONG THE Z-AXIS AND THE
MAGNETIC FIELD IS TAKEN IN THE Z-X PLANE AT AN ANGLE,
BTHETA, TO THE ELECTRIC FIELD.
This means that Br is aligned with x-axis, as opposed to a case of
the Langevin’s theory, where it was aligned with y-axis.
Input parameters are:
Drift of electrons in SLD CRID
r-SLD is in the middle of TPC
C2H6, 1 bar, 400V/cm
z-SLD r-SLD Br
Bz
B-tot Angle Ez
[cm] [m]
[Tesla] [Tesla] [Tesla] [deg] [V/cm]
0 1.218
0 0.6088 0.6088
0
400
10 1.218 0.0014 0.6087 0.6087 0.1363
400
20 1.218 0.0029 0.6083 0.6084 0.2727
400
30 1.218 0.0043 0.6077 0.6078 0.4094
400
40 1.218 0.0058 0.6069 0.6069 0.5467
400
50 1.218 0.0072 0.6058 0.6059 0.6845
400
60 1.218 0.0087 0.6045 0.6046 0.8232
400
70 1.218 0.0101 0.603 0.6031 0.9628
400
80 1.218 0.0116 0.6012 0.6013 1.1036
400
90 1.218 0.013 0.5992 0.5993 1.2457
400
100 1.218 0.0145 0.5969 0.5971 1.3893
400
110 1.218 0.0159 0.5944 0.5946 1.5346
400
120 1.218 0.0174 0.5917 0.592 1.6817
400
51
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Results from the Magboltz-Monte program are:
a) 24oC
Vx - 24C Error Vx-fit at 24C Vy - 24C Error Vy-fit at 24C Vz - 24C Error Omega*tau Theta(x,y)
[cm/usec]
[cm/usec]
[cm/usec]
[cm/usec]
[cm/usec]
at 24 C
at 24C [deg]
0
0
0.006 0.002
0.00486
0.002 0.002
0.00374
4.297 0.002 0.7695473
0.009 0.002
0.00972
0.009 0.002
0.00748
4.296 0.002 0.7695473
45
0.014 0.002
0.01458
0.009 0.002
0.01122
4.292 0.002 0.7695473 32.73522627
0.018 0.002
0.01944
0.016 0.002
0.01496
4.299 0.002 0.7695473 41.63353934
0.022 0.002
0.0243
0.019 0.002
0.0187
4.299 0.002 0.7695473 40.81508387
0.031 0.002
0.02916
0.021 0.002
0.02244
4.291 0.002 0.7695473 34.11447295
0.028 0.002
0.03402
0.026 0.002
0.02618
4.299 0.002 0.7695473 42.8789036
0.035 0.002
0.03888
0.032 0.002
0.02992
4.294 0.002 0.7695473 42.43622979
0.044 0.002
0.04374
0.038 0.002
0.03366
4.301 0.002 0.7695473 40.81508387
0.047 0.002
0.0486
0.035 0.002
0.0374
4.295 0.002 0.7695473 36.67434967
0.059 0.002
0.05346
0.038 0.002
0.04114
4.304 0.002 0.7695473 32.78428087
0.061 0.002
0.05832
0.046 0.002
0.04488
4.295 0.002 0.7695473 37.01988625
Average Vz(E,B,24 C) =
4.2968
38.809732
+- 0.0038
+4.3250476
a) 35oC
Vx - 35C Error Vx-fit at 35C Vy - 35C Error Vy-fit at 35C Vz - 35C Error Omega*tau Theta(x,y)
[cm/usec]
[cm/usec]
[cm/usec]
[cm/usec]
[cm/usec]
at 35 C
at 35C [deg]
0
0
0.004 0.002
0.00476
0.005 0.002
0.00393
4.351 0.002 0.8256303
0.009 0.002
0.00952
0.008 0.002
0.00786
4.35 0.002 0.8256303 41.63353934
0.018 0.002
0.01428
0.014 0.002
0.01179
4.349 0.002 0.8256303 37.87498365
0.017 0.002
0.01904
0.016 0.002
0.01572
4.351 0.002 0.8256303 43.26429541
0.02 0.002
0.0238
0.023 0.002
0.01965
4.348 0.002 0.8256303 48.9909131
0.031 0.002
0.02856
0.028 0.002
0.02358
4.345 0.002 0.8256303 42.08916217
0.038 0.002
0.03332
0.033 0.002
0.02751
4.349 0.002 0.8256303 40.97173633
0.038 0.002
0.03808
0.032 0.002
0.03144
4.348 0.002 0.8256303 40.10090755
0.049 0.002
0.04284
0.033 0.002
0.03537
4.345 0.002 0.8256303 33.95905982
0.049 0.002
0.0476
0.038 0.002
0.0393
4.352 0.002 0.8256303
37.793943
0.048 0.002
0.05236
0.04 0.002
0.04323
4.344 0.002 0.8256303 39.80557109
0.053 0.002
0.05712
0.045 0.002
0.04716
4.35 0.002 0.8256303 40.33314163
Average Vz(E,B,35 C) =
4.3485
40.648411
+- 0.0026
+3.7587069
52
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Lorentz angle in x,y plane:
tan θ xy =
v
v
y
=ωτ
x
Lorentz angle Theta(x,y) = f(z-SLD, temperature)
(using the MAGBOLTZ-MONTE program)
J.V., S.B., 7.4.2000
60
50
40
30
24 C
20
35 C
10
0
0
10
20
30
40
50
60
70
80
90
100
110
120
z-SLD [cm]
Average Lorentz angle : 38.8o 4.3 at 24oC, and 40.6o 3.8 at 35oC
The Lorentz angles are the same within the errors of this calculation.
Vx and Vy = f(z-SLD, temperature)
(using the MAGBOLTZ-MONTE program)
0.07
Vx(E=400V/cm, B) at 24 C
Vy(E=400V/cm, B) at 24 C
Vx(E=400V/cm, B) at 35 C
Vy(E=400V/cm, B) at 35 C
Linear ( Vx(E=400V/cm, B) at 35 C)
Linear ( Vy(E=400V/cm, B) at 35 C)
Linear ( Vx(E=400V/cm, B) at 24 C)
Linear ( Vy(E=400V/cm, B) at 24 C)
0.06
0.05
0.04
0.03
J.V., S.B., 7.4.2000
Vy(35C) = 3.93E-04*Z-SLD
Vy(24C) = 3.74E-04*Z-SLD
Vx(35C) = 4.76E-04*Z-SLD
0.02
Vx(24C) = 4.86E-04*Z-SLD
0.01
0
0
20
40
60
z-SLD [cm]
53
80
100
120
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Vz = f(z-SLD, temperature)
(using the MAGBOLTZ-MONTE program)
J.V., S.B., 7.4.2000
4.45
Vz(E=400V/cm, B) at 24 C
Vz(E=400V/cm, B) at 35 C
4.4
4.35
4.3
4.25
0
20
40
60
80
100
120
z-SLD [cm]
. There is a large effect on the Vz velocity component (going from 24 C to 35 C),
o
and a small effect on components Vx and Vy.
54
o
J.Va’vra, Detector Lecture in Rome, January 25, 2002
I then integrate numerically to get the total distortions, and compare the predictions
based on the Magboltz-Monte program (use fits to Vx and Vy and average Vz) with the
Langevin’s theory using P. Coyle’s omega*tau = 0.87:
Distortions in x&y = f (z-SLD, temperature)
(using the MAGBOLTZ-MONTE program)
0.9
dx(24C) = -6E-05*(z-SLD)^2 - 6E-17*z-SLD + 0.8144
dx(35C) = -5E-05*(z-SLD)^2 + 6E-17*z-SLD + 0.7881
0.8
dx or dy [cm]
0.7
0.6
J.V., 7.8.2000
Choose the omega*tau in the
Langevin's theory according to
P. Coyle's prediction (SchultzGresser alghoritm)
dx(35C) = -6E-05*(z-SLD)^2 + 0.0002*z-SLD + 0.8444
0.5
dy(35C) = -5E-05*(z-SLD)^2 + 3E-17*z-SLD + 0.6507
dy(35C) = -5E-05*(z-SLD)^2 + 0.0002*z-SLD + 0.7346
dy(35C) = -4E-05*(z-SLD)^2 + 0.6267
0.4
dx at 24 C (J.V.'s prediction using Magboltz-Monte)
dy at 24 C (J.V.'s prediction using Magboltz-Monte)
dx at 35 C (J.V.'s prediction using Magboltz-Monte)
dy at 35 C (J.V.'s prediction using Magboltz-Monte)
dy at 35C (Langevin's theory with omega*tau = 0.87)
dx at 35C (Langevin's theory with omega*tau = 0.87)
0.3
0.2
0.1
0
0
20
40
60
80
100
120
z-SLD [cm]
Tune omega*tau. There is no single omega*tau constant, which
would make a good agreement with the predictons of the MagboltzMonte program:
Distortions in x&y = f (z-SLD, temperature)
(using the MAGBOLTZ-MONTE program)
0.9
Tune omega*tau in the Langevin's
theory for the "best" agreement
with the Magboltz-Monte program
(a single omega-tau cannot do it !)
0.8
0.7
dx or dy [cm]
J.V., 11.7.2000
0.6
dx(35C) = -6E-05*(z-SLD)^2 + 0.0002*z-SLD + 0.8185
0.5
dy(35C) = -4E-05*(z-SLD)^2 + 0.0002*z-SLD + 0.6139
0.4
dx at 24 C (MAGBOLTZ-MONTE)
dy at 24 C (MAGBOLTZ-MONTE)
0.3
dx at 35 C (MAGBOLTZ-MONTE)
0.2
dy at 35 C (MAGBOLTZ-MONTE)
dy at 35C (Langevin's theory with omega*tau = 0.75)
0.1
dx at 35C (Langevin's theory with omega*tau = 0.75)
0
0
20
40
60
z-SLD [cm]
55
80
100
120
J.Va’vra, Detector Lecture in Rome, January 25, 2002
Conclusions:
1. I calculated the effect of temperature on the Lorentz angle, drift
velocity components and x and y distortions, using the
MAGBOLTZ-MONTE program.
2. If I use ωτ = 0.78 and the old Langevin’s naive theory, I get a
reasonable agreement with the MAGBOLTZ-MONTE prediction
for the x, y distortions. If I use the old P. Coyle prediction, which
was 0.87 and was based on the old Schultz-Gresser algorithm to
solve the Boltzman equation (the solution uses only two terms and
this is known to be incorrect), I get a disagreement with the
Magboltz-Monte.
56