Download Advanced lab-class for bachelor students in physics

Advanced lab-class for bachelor students in physics
Experiment T14
Stern-Gerlach
February 17, 2014
Requirements
• spin of electrons
• movement of atoms in magnetic fields
Aim of the experiment
• Record data to obtain the distribution of the particle flow density in the detection plane without
magnetic field and with different magnetic field strengths
• Investigation of the position of the maximum of the particle flow density as a function of the
magnetic field inhomogeneity
• Determination of the Bohr’s magneton µB
Contents
1 Introduction
3
2 Theoretical background
2.1 Magnetic momentum and force . . . . . . . . . . . . . .
2.2 Two-wire field . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Influence of the velocity distribution of the particles and
2.5 Ideal case, method A . . . . . . . . . . . . . . . . . . . .
2.6 Real beam cross-section, method B . . . . . . . . . . . .
2.6.1 Beam cross-section . . . . . . . . . . . . . . . . .
2.6.2 Asymptotic behaviour for large fields . . . . . . .
. .
. .
. .
the
. .
. .
. .
. .
. . . . . . . .
. . . . . . . .
. . . . . . . .
beam profile
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
3
3
4
5
7
8
8
8
3 Exective Instructions
3.1 Construction of the electric circuit .
3.2 Vacuum system . . . . . . . . . . . .
3.3 Heating of furnace . . . . . . . . . .
3.4 The detector . . . . . . . . . . . . .
3.5 Recording of the measurement series
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
10
10
11
11
12
13
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Analysis
13
A Mathematical background
A.1 Two-wire field . . . . . . . . . . . . . . . . .
A.2 Real beam cross-section, method B . . . . .
A.2.1 Beam cross-section . . . . . . . . . .
A.2.2 Asymptotic behaviour for large fields
.
.
.
.
14
14
15
15
16
B Figures
B.1 Calibration curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
17
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
References
[1]
Böhm, M.: Höhere Experimentalphysik, Bd 272
[2]
Mayer-Kuckuk, T.: Atomphysik, Bo 137
[3]
Bergmann, Schaefer: Lehrbuch der Experimentalphysik, Band 4, Teil 1, Bd 15
[4]
Haken, H., Wolf, H. C.: Atom- und Quantenphysik, Bo 164
[5]
T. Hebbeker: Vorlesungsskript zur Experimentalphysik IV, Kapitel 8
[6]
Phywe, Instructions for Stern-Gerlach experiment“ LEP 5.1.11-00
”
Old instructions for Stern-Gerlach experiment“ by RWTH
”
Petertill, M.: wall chart Operating of the Stern-Gerlach apparatus“ in the lab
”
[7]
[8]
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
Introduction
In 1922 Otto Stern and Walther Gerlach for the first time observed the quantisation of the direction of the
atom’s magnetic momentum (which in classical physics originates in the orbital angular momentum of the
5s-electron circulating around the nucleus). For this they used electrically neutral silver atoms (electronic
configuration: [Kr]4d10 5s1 ). Thus, the experiment provided the possibility of direct measurement of the
magnetic momentum of atoms in addition to an indication of the theory of quantum mechanics developed
since 1925. In modern quantum mechanics (with Fermi-Dirac statistics and electron spin, postulated in
1925 by Samuel Goudsmit and George Uhlenbeck) the Stern-Gerlach experiment proves the quantisation
of the electron spin and enables direct measurement of the Bohr magneton µB . Therefore, the SternGerlach experiment is a fundamental experiment, which demonstrates the nature of quantum mechanics.
Later the experiment was repeated using hydrogen atoms. In our lab course, potassium atoms are used.
2
Theoretical background
In this first part the theoretical basics of the Stern-Gerlach experiment are explained, which are needed in
order to understand its physics. The derivations can be found in the appendix and are referenced within
the text. In addition, various parts should be developed by the reader as exercise. For this purpose, the
bibliography is helpful, especially [7] and [8] on which this guide is partly based on.
2.1
Magnetic momentum and force
As explained in the introduction, it is necessary to determine the effect of applying an inhomogeneous
magnetic field to the atomic magnetic momentum, which in the case of potassium is induced by the spin
of a single electron.
This is expressed by the equation
Fz = µz ·
∂B
∂B
= −ms gs · µB ·
∂z
∂z
(1)
for the force Fz , where the z-axis is chosen as quantisation axis. It depends on the magnetic momentum
µz in z-direction as well as the deviation of the magnetic field. Here the variables are the spin quantum
number ms for electrons, the Landé factor gs ≈ 2.00232 for electrons, and the Bohr magneton µB . The
derivation shall be left to you.
2.2
Two-wire field
For the creation of the inhomogeneous magnetic field the pole shoes shown in figure 1 is used. By its
special form it creates a field like the one generated by two parallel wires within a distance 2a and opposite
current directions. Here a is the radius of the convex pole shoe. This two-wire field is particularly useful
for the splitting of the atom beam. In fact, with
√ a proper adjustment of the atomic beam a field gradient
∂B
2a from the fictitious wires, which is almost constant over
∂z can be achieved at a distance of about
the beam cross-section (see Appendix A.1). In practice, the field gradient can not be measured directly,
but is determined by measuring the magnetic induction B. For this, the following relationship can be
used (derivation in Appendix A.1):
B
∂B
= 0.9661 · .
(2)
∂z
a
The magnetic induction B is determined from the applied coil current by a calibration curve, which is
specific for each experimental set-up (Appendix B.1).
3
Figure 1: Pole shoes to produce a two-wire field
2.3
Trajectory
Here, the path of a single particle with mass m is calculated. Different particle trajectories are shown in
Figure 2. The propagation direction of the particles is the x-direction, whilst the magnetic field acts along
the z-axis. Here, the particle detected at position u(2) is faster than the one detected at u(1) (∆u > ∆z).
The particle detected at u(3) has a different spin orientation with respect to the other two particles.
Figure 2: particle trajectories
First, the particle passes through the magnetic field of a length L, which takes the time ∆t =
deflection ∆z in z-direction of the particle is given by
∆z
=
=
2
1
2 z̈ (∆t)
2
1 Fz
2 · m · (∆t)
L
v.
The
.
Due to ∆z L one can assume ~v ≈ v~x . When leaving the magnetic field, the velocity ż in z-direction
is given by the momentum mż = Fz · ∆t of the particle. At this speed, it moves towards the detector at
the position x = l uniformly and rectilinearly . Thus, the total time ttot , which is needed from entering
4
the magnetic field until reaching the detector is ttot = vl . Now one can calculate the impact point u of
the particle as a function of its speed and the field inhomogeneity:
u = z + ∆z + ż · (tges − ∆t)
L·l
L
∂B
= z+
1
−
µz
mv 2
2l
∂z
L
L·l
∂B
1−
= z−
ms gs · µB
.
mv 2
2l
∂z
(3)
Faster particles are deflected less, because the time of their exposure to the magnetic field is shorter.
2.4
Influence of the velocity distribution of the particles and the beam profile
Now we want to find the velocity distribution function for many particles in the furnace, in order to
describe the spatial dependence of the particle flux in the detector plane.
When the potassium atoms are vaporized, they form a gas insides the furnace, which is in thermal
equilibrium at the temperature T . This means that the potassium atoms satisfy the Maxwell velocity
distribution everywhere in the furnace, and the particle density n is identical everywhere in the furnace.
Therefore, the number of particles dN with a velocity between v and v + dv is
mv 2
dN ∼ e− 2kT v 2 dv.
This thermal proportionality is also valid for the particles in the atomic beam, which leaves the furnace.
However, one still needs to take into account a geometrical proportionality, which has the following form
for a rectilinear uniform motion of the atoms
dN ∼ v dv.
This can be explained as follows: Due to a system of cover plates, only the potassium atoms moving
parallelly to the beam direction when leaving the furnace enter the beam. Considering the atoms of
a given velocity v, which leave the furnace in a sufficiently small time interval ∆t, they can only be
originating from a volume of a length v · ∆t in front of the window (see figure 3). The size of this volume
is directly proportional to the particle velocity v. Since the particle density n is the same in the entire
furnace, the number of atoms of a certain velocity v in the beam scales with v (in addition to the Maxwell
velocity distribution).
Figure 3: For the derivation of the geometric proportionality in the velocity distribution
As a result, one obtains
mv 2
dN ∼ e− 2kT v 3 dv.
Since we are interested in the number of particles as a function of the impact point u, dv has to be
substituted by du. As the impact point for identical particles with the same speed and the same z solely
5
differs by ms (according to the equation (3)), one has
L·l
L
∂B 1−
ms gs · µB
|u − z| = −
mv 2
2l
∂z L·l
L
∂B
=
1−
|ms |gs · µB
mv 2
2l
∂z
L·l
L
∂B
=
1−
gs · µB
.
2mv 2
2l
∂z
(4)
In this experiment, the inhomogeneity of the magnetic field ∂B
∂z is positive by definition. This leads to
s
L·l
L
∂B 1
v=
1−
gs · µB
2m
2l
∂z |u − z|
and
s
dv
3
L·l
u−z
L
∂B
1
=
1−
gs · µB
· −
|u − z|− 2 ·
du
2m
2l
∂z
2
|u − z|
s
L
∂B 1
u−z
1 L·l
du
1−
gs · µB
·
= −
2 2m
2l
∂z |u − z| |u − z|2
v
= −
du.
2(u − z)
(5)
As only positive solutions are allowed for dv, this results in
dv =
v
du
2|u − z|
(6)
as well as
v 3 dv
v4
du
2|u − z|
1 L·l
1−
2 2m
1 L·l
1−
8
m
=
=
=
2
1
∂B 1
du
gs · µB
∂z |u − z|
|u − z|
2
1
L
∂B
gs · µB
du.
2l
∂z
|u − z|3
L
2l
(7)
Therefore, taking into account the normalisation, the total number of particles as a function of the impact
point is:
b
a · e− |u−z|
dN =
du,
(8)
|u − z|3
with
1
8
a =
2
R∞
−∞
1
8
L·l
m
2
L
1 − 2l
gs µB ∂B
∂z
2 − b
L
1
1 − 2l
gs µB ∂B
e |u−z| |u−z|
3 du
∂z
1
=
2
R∞
b
− |u−z|
e
−∞
b =
L·l
m
L·l
4kT
L
1−
2l
(9)
1
|u−z|3 du
gs · µB
∂B
.
∂z
(10)
Here, the factor 2 in front of the integral is due to the fact that both flight directions (into and out of
the furnace) are equally likely for reasons of symmetry.
6
Additionally, the finite dimensions of the beam and the beam profile have an influence:
dN ∼ Φm (z)dz.
Φm (z) describes the number of particles that fly between z and z + dz into the magnetic field, where the
index m = ± 21 denotes the orientation of the magnetic momentum. Thus, the final distribution function
is
b
a · e− |u−z|
dudz
(11)
d2 N = Φm (z)
|u − z|3
2.5
Ideal case, method A
By means of the previous considerations, it is now possible to specify the particle current density I(u) as
a function of the impact point u on the detector plane. To consider all particles of the beam, one has to
take an integral of z and sum over all possible spin orientations:


ZD
1 X
I(u) =
d2 N 
du
m
−D

= 
X
m

= 
ZD
−D
D
XZ
m

d2 N 
du

b
a · e− |u−z| 
dz .
Φm (z)
|u − z|3
(12)
−D
Since the distribution of the number of particles in the beam is equivalent for both orientations ms = ± 12 ,
one defines
I0 (z)
Φ+ 21 (z) ≡ Φ− 21 (z) :=
.
2
Therefore,
ZD
b
dz
.
(13)
I(u) = a
I0 (z)e− |u−z|
|u − z|3
−D
Let us now consider the approximation of an infinitesimal box for the beam profile:
I0 (z) = 2DI0 δ(z).
Here δ(z) is the Dirac function, for which (if x0 is within the limits of integration) the following applies:
Z
f (x)δ(x − x0 )dx = f (x0 ).
This results in
1
.
(14)
u3
Now, the maxima of the intensities should be determined. For this purpose the equation (14) is derived
and set equal to zero:
b
I(u) = 2aDI0 e− |u|
(0)
⇒ u0
b
dI(u)
b − 3|u| − |u|
!
= 2aDI0
= 0,
e
du
u5 L
L · l 1 − 2l
b
∂B
=± =±
gs · µB
.
3
12kT
∂z
(15)
Therefore, it is immediately visible that the position of the maxima u0 increases linearly with the field
inhomogeneity, i.e.
∂B
(0)
u0 = const. ·
.
(16)
∂z
In the evaluation part, this method is referred to as method A.
7
2.6
2.6.1
Real beam cross-section, method B
Beam cross-section
To get to a better adaptation of the theory to the experiment, we will now look at a beam cross-section
with a finite width 2D. Thereby, the beam profile should be described by two steep, straight slopes and
a parabolic peak (as shown in Figure 4), ie


für − D ≤ z ≤ −p
D + z
2
I0 (z) = i0 · D − 12 p − 12 zp für − p ≤ z ≤ p


D−z
für p ≤ z ≤ D


für − D ≤ z ≤ −p
1
dI0 (z)
z
= i0 · − p für − p ≤ z ≤ p

dz

−1 für p ≤ z ≤ D


für − D ≤ z ≤ −p
0
d2 I0 (z)
1
= i0 · − p für − p ≤ z ≤ p

dz 2

0
für p ≤ z ≤ D
(17)
Figure 4: Mathematical approach for the particle current density in case of a vanishing magnetic field
In this case, the parameter p is defined by the transition from the parabolic to the straights.
2.6.2
Asymptotic behaviour for large fields
If one applies sufficiently large magnetic fields, it is possible to derive a conditional equation for u0 by
performing a Taylor expansion of the solution function of the real beam cross section (cf. Appendix A.2,
setting equation (22) to 0).
D4 − 15 p4
b
b
1
b
1 b2
−3 +
5
−
1
+
−
15
0 = D2 − p2
3
u0
u20
u0
12 u20 u0
Here, the first term is the solution in first order, while the second term represents a higher order correction
term. Neglecting the higher order correction one receives the already known solution for u00 = 3b . Since
8
the second term is a correction term of higher order, it is possible to replace u0 by u00 because the resulting
variations will be of an even higher order. This yields the following approximation:
D4 − 51 p4
b
1 2
2
−3 +
0 =
D − p
3
u0
u20
⇒b
=
3u0 −
D4 − 15 p4 1
C
1 2 u = 3u0 − u .
2
D − 3p 0
0
(18)
The correction term − uC0 takes into account the different degrees of force effect at different positions in
the beam cross section. In the analysis part, the Bohr magneton µB will also be determined by Method
B in addition to method A.
Exercise Use the equations (10) and (18) to deduce an expression, in which µB only depends on
mensurable quantities.
9
3
Exective Instructions
Figure 5: Construction of the experimental apparatus
3.1
Construction of the electric circuit
To prevent damage to the detector, it is essential to make sure that the matching transformer is connected between the voltage source and the detector.
For the operating voltages see table 1, the dimensions needed for the analysis can be obtained from table
2. Initially, the power source for the solenoid is to remain turned off.
device
furnace
detector
operation mode
heating (ca. 10 min)
continuous operation
normal operation
flashing (max. 45 s)
magnet
setting
max 1 A"
ca. 0.5 A"
4.2 A∼
Umax (ca. 18 V∼ )
max 1.2 A∼
Table 1: settings
length of the pole shoes
radius convex pole shoe
from start of magnetic field to detector
L = 7 · 10−2 m
a = 2.5 · 10−3 m
l = 0.455 m
Table 2: dimensions of the apparatus
10
Figure 6: Electric circuit for: (a) furnace; (b) magnetic analyser; (c) detector
3.2
Vacuum system
The complete experimental set-up is placed in a ultra-high vacuum system, since the experiment would
not be possible at normal ambient pressure due to the interaction of the potassium atoms with gas atoms
in the air. Before you start, insure yourself with the help of a cold cathode vacuum gauge whether the
pressure is below the required limit of p ≈ 4 · 10−6 mbar. When heating the furnace or flashing the
detector (see section 3.4), the vacuum may briefly get worse (up to 2 orders of magnitude), but then
should recover within a few minutes. If it does not recover, please report it to the supervisor! If the
pressure rises above 10−4 mbar at any point of time, immediately turn off the detector and
report to the supervisor. If the detector is not turned off, it can burn out within a few minutes!
Figure 7: Schematic set-up of the vacuum pump system
3.3
Heating of furnace
Potassium already evaporates at a temperature of just 70 ◦ C. To achieve sufficient beam intensities, higher
temperatures have to be achieved in the furnace. 180 ◦ C should, however, not be exceeded under any
11
circumstances. During the heating, roughly record the detector current and the furnace temperature (e.g.
from 100 ◦ C a short series of measurements for every 10 ◦ C) with the maximum magnetic field applied
(accordingly, flat peaks) and decide on the lowest temperature at which the peaks are still sufficiently
distinguishable from the background.
The precise temperature is required for the analysis. Note this value and make sure that it remains
constant throughout the whole experiment.
Afterwards, the magnetic field can be decreased slowly. At the same time, the field should be inverted
several times in order to minimise the remanence. The iron yokes in the coils can be screwed out for
the measurement at B = 0 and have to be carefully turned back in for the later measurements with
magnetic field.
3.4
The detector
Figure 8: Sketch of the Langmuir-Taylor detector
In the experiment, a Langmuir-Taylor-detector is used, which serves for the detection of electrically neutral atoms. It consists of a tungsten wire that is heated to a temperature of T = 800 ◦ C and is surrounded
by a nickel cylinder with an entrance window for the potassium atom beam. By means of the matching transformer which is operated via a variable transformer, a voltage of about 50 V is applied at the
wire. Without this bias voltage, the detector would provide a current due to its glow discharge. This
current would increase exponentially with the temperature of the wire. The cylinder is grounded. When
potassium atoms hit the glowing wire they are ionised (ionisation energy lower than the work function
for electrons in tungsten) and accelerated towards the cylinder due to the potential difference between
the wire and the cylinder. The resulting ion current is of the order of ∼ pA and must therefore be amplified. If the tungsten wire has the correct temperature, the measured current is proportional the number
of detected potassium atoms. In case of too low heating current, not all atoms can be ionised. If the
temperature is raised, a current from the glow discharge can also be measured despite the bias voltage.
This current will increase with increasing temperature. Additionally, contaminations on the wire will
evaporate and produce an additional signal, as they are partly ionised. Because of these contaminations,
the detector should be baked out at the beginning of the measurement (“flashing”) in order to reduce
the noise.
Another problem is caused by the small value of the currents that should be measured: If the coaxial
cable, which is used to read out the detector, is touched or moved, currents can be induced for a short
time which are much higher than the signal itself. Therefore, this cable should not be touched and one
12
should wait for a short period of time before recording the data after having changed the detector position.
3.5
Recording of the measurement series
For the following measurements, the furnace temperature has to be kept constant. The detector current
in the detection plane should be measured at vanishing magnetic field and magnetic field strengths up
to 1.2 A. The measurements should be performed in small steps, since an accurate determination of the
position of the maxima with magnetic field turned on as well as the beam parameters p and D with
B = 0 has to be performed. In order to keep magnetic remanence effects low, one should take care to
carried out the measurements in a way that the coil current is only increased.
It is important to consider that the range of the d.c. amplifier has to be adjusted if necessary. The
detector position is changed by turning the millimetre screw. The detector is moved 1.8 mm per turn.
4
Analysis
The aim of the evaluation is to review the two previously mentioned theories. Thereto, proceed as follows:
• Plot the current values that you recorded during the heating of the furnace versus the temperature.
Discuss the temperature at which the experiment should reasonably be carried out.
• For various coil currents, plot the detector current (∼ particle current density) versus the displacement of the detector.
• Determine the magnetic moment µB
(0)
– using the theory of an arbitrarily narrow beam box (by plotting u0
15)
(0)
– using equation 18 for a real beam box (by plotting u0 −
C
(0)
3u0
versus
versus
b
3
b
3,
cf. equation
- therefore, the real
beam profile has to be parametrised as shown in Figure4).
(0)
• Therefore, the positions u0
geneities.
of both maxima have to be determined for different field inhomo-
• The conversion of the coil current into the field inhomogeneities can be found in Appendix B.1. Do
not forget to write down which set-up you have used!!!
• Discuss the differences in your results and comment on how reasonable the usage of the mathematically more precise theory is. Discuss all other influences on your measurement.
13
A
A.1
Mathematical background
Two-wire field
The magnetic induction of two anti-parallel electrical currents at the distance 2a is obtained by adding
up their individual components
~
~ i = Ii × r~i .
B
2π · ri2
~ one has
Knowing I~1 = −I~2 = I,
µ0 Ia
.
πr1 r2
In order to let the atomic beam undergo a substantially constant force, it has to traverse the magnetic
analyser at a point with almost constant field inhomogeneity. Therefore, it has to be determined at which
distance from the wires (plane of wires z = −z0 ) the field inhomogeneity is as constant as possible, i.e.
∂B
∂z = const. This plane is chosen to be at z = 0.
B(~r) =
Figure 9: Definition of a coordinate system
The following relationships for the distances r1 and r2 can be extracted from figure 9:
r12 = (a − y)2 + (z + z0 )2
r22 = (a + y)2 + (z + z0 )2
By inserting into the equation above, this results in
i− 21
2
µ0 · I · a h 2
2
4
· a − y 2 + 2 (z + z0 ) a2 + y 2 + (z + z0 )
B(y, z) =
π
(19)
and by taking the derivative of this with respect to z the field inhomogeneity is obtained to
∂B
2 · µ0 · I · a · (z + z0 )
=−
·h
∂z
π
a2 + y 2 + (z + z0 )2
(a2
−
2
y2 )
2
+ 2 (z + z0 )
(a2
+
y2 )
+ (z + z0 )
At z = 0, the inhomogeneity’s dependence of y should disappear. To find this plane,
to first order in y 2 at y = 0. The approximation yields


2
2
∂B
2µ0 I · a · (z + z0 )
2a − (z + z0 ) 
2
.
(y, z) ≈ − 2 · 1 + 2y · 2
∂z
2
a2 + (z + z0 )
π a2 + (z + z0 )
14
4
∂B
∂z
i 23 .
is expanded up
√
Thus, the dependence of y vanishes for z0 = 2a. This is the distance of the atom beam from the
fictitious wires.
For the calculation of the magnetic moment, it is important to know the field inhomogeneity. It can
not be measured directly. However, it can be calculated from the magnetic field strength under the
assumption of a linear relationship between B and ∂B
∂z for small deflections z. Now, the proportionality
· a should be determined in the area around y = 0. Therefore, (y, z) is expanded up to
factor = ∂B
∂z
B
first order in y 2 at y = 0 like above.
(y, z) ≈
2
3a2 − (z + z0 )
2
.
·
1
+
·
y
2
2 2
a2 + (z + z0 )
a2 + (z + z0 )
2a (z + z0 )
The system of cover plates of the apparatus is designed in a way that the atomic beam has a width of
approximately 43 a in y-direction. In the area of this beam
box (y, 0) changes slightly with y, therefore
the average of the highest possible value epsilon 23 a, 0 ≈ 0.9894 and the lowest value (0, 0) ≈ 0.9428
is used to calculate the inhomogeneity. Thus, the conversion factor is ≈ 0.9661.
A.2
A.2.1
Real beam cross-section, method B
Beam cross-section
Now let us discuss the model for the real beam cross section, described in section 2.6. Looking at the
equations (17), we see that I0 (z) is defined as two-times-differentiable. Based on this, the determination of
the particle current density I(u) depends on the inhomogeneity of the magnetic field, i.e. of b (equation
10). The maxima of I(u) are located at the positions u0 (b), which now more or less differ from the
positions u00 = ± 3b , i.e. from the approximation for an infinitesimal beam box. To determine the function
u0 (b), we assume
dI
(u0 ) = 0.
du
The derivative with respect to u can now be taken into the integral, as can be seen in the following
calculation
b
+D
−
R
|u−z|
dI
d
= du
a
I0 (z) e|u−z|3 dz
du
=
−D
+D
R
a
−D
As one can easily verify, the replacement of
∂
∂u
dI
= −a
du
−
b
|u−z|
∂ e
I0 (z) ∂u
|u−z|3 dz.
∂
by − ∂z
leaves the integrand unchanged.
b
Z+D
∂ e− |u−z|
I0 (z)
dz.
∂z |u − z|3
−D
By partial integration (reminder:
Rb
Rb
f (x) · g 0 (x)dx = (f (x) · g(x))|ba − f 0 (x) · g(x)dx), it is now possible to
a
a
move the differentiation to I0 . Please check that here the first term from the partial integration vanishes.
dI
=a
du
Z+D
b
dI0 (z) e− |u−z|
dz.
dz |u − z|3
−D
Inserting equation (17) returns
 −p
Z
dI
= ai0

du
+D
Z+p
−
z
−
p
−p

Z+D
+p
b
e− |u−z|
dz.
 |u − z|3
Both occurring integrals can be solved analytically (e.g. with Maple) and result in
dI
ai0
= 2 · F (u)
du
pb
15
with the solution function
b
b
F (u) = −|u + p|e− |u+p| + |u − p|e− |u−p| + p
b
b
b + |u + D| − |u+D|
b + |u − D| − |u−D|
+p
.
e
e
u+D
u−D
It immediately yields the equation for determining the positions of the maxima
F (u0 ) = 0.
One can see now that F (u0 ) is point-symmetrical. The solution function u0 (b), which indicates the position of the maxima, is thus mirror symmetric. By this simplification, we can restrict to the determination
of the positive values only.
A.2.2
Asymptotic behaviour for large fields
If the field inhomogeneity is sufficiently large, u0 converges the solution which is given by the infinitesimally narrow beam box.
If we now assume
u0 u0 b b
, , ,
1,
(20)
p D p D
then the following calculations describes a more precise behaviour of the function u0 (b) for large fields.
With the help of the aforementioned approximation, F (u) can be developed as a Taylor series, for which
we need the following function:
b
(21)
f (u) = u · e− u .
The relevant derivatives are
f (3) (u)
=
b2
u4
f (5) (u)
=
12 ub 6 5
b
u−
2
b
3 e− u
b
u
−1 +
1 b2
12 u2
b
u
− 15
b
b− u .
Up to the sixth derivation, only the third and fifth derivative are of interest, because only in this case
the coefficients do not vanish.
Thus, abortion after the sixth element results in:
1 2
p
1 4
2
(3)
4
F (u) = p D − p
· f (u) +
D − p
· f (5) (u) + . . . .
(22)
3
12
5
From this we obtain, once more by zeroing, the equation for u0
D4 − 51 p4
1 2
b
b
1 b2
b
2
0= D − p
−3 +
5
−1 +
− 15
.
3
u0
u20
u0
12 u20 u0
The further way to determine u0 is explained in the theory section (section 2.6.2).
16
(23)
B
B.1
Figures
Calibration curves
Figure 10: Calibration curve of the magnet No. 52 (set-up A)
17
Figure 11: Calibration curve of the magnet No. 77 (set-up B)
18