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 withoutmagnetic field and with different magnetic field strengths

• Investigation of the position of the maximum of the particle flow density as a function of themagnetic field inhomogeneity

• Determination of the Bohr’s magneton µB

Contents

1 Introduction 3

2 Theoretical background 32.1 Magnetic momentum and force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Two-wire field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Influence of the velocity distribution of the particles and the beam profile . . . . . . . . . 52.5 Ideal case, method A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 Real beam cross-section, method B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6.1 Beam cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.2 Asymptotic behaviour for large fields . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Exective Instructions 103.1 Construction of the electric circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Vacuum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Heating of furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 The detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Recording of the measurement series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Analysis 13

A Mathematical background 14A.1 Two-wire field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14A.2 Real beam cross-section, method B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

A.2.1 Beam cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15A.2.2 Asymptotic behaviour for large fields . . . . . . . . . . . . . . . . . . . . . . . . . . 16

B Figures 17B.1 Calibration curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

References

[1] Bohm, M.: Hohere 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

[7] Old instructions for”Stern-Gerlach experiment“ by RWTH

[8] Petertill, M.: wall chart”Operating of the Stern-Gerlach apparatus“ in the lab

2

1 Introduction

In 1922 Otto Stern and Walther Gerlach for the first time observed the quantisation of the direction of theatom’s magnetic momentum (which in classical physics originates in the orbital angular momentum of the5s-electron circulating around the nucleus). For this they used electrically neutral silver atoms (electronicconfiguration: [Kr]4d105s1). Thus, the experiment provided the possibility of direct measurement of themagnetic momentum of atoms in addition to an indication of the theory of quantum mechanics developedsince 1925. In modern quantum mechanics (with Fermi-Dirac statistics and electron spin, postulated in1925 by Samuel Goudsmit and George Uhlenbeck) the Stern-Gerlach experiment proves the quantisationof the electron spin and enables direct measurement of the Bohr magneton µB . Therefore, the Stern-Gerlach 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 inorder to understand its physics. The derivations can be found in the appendix and are referenced withinthe text. In addition, various parts should be developed by the reader as exercise. For this purpose, thebibliography 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 inhomogeneousmagnetic field to the atomic magnetic momentum, which in the case of potassium is induced by the spinof a single electron.

This is expressed by the equation

Fz = µz ·∂B

∂z= −msgs ·µB ·

∂B

∂z(1)

for the force Fz, where the z-axis is chosen as quantisation axis. It depends on the magnetic momentumµz in z-direction as well as the deviation of the magnetic field. Here the variables are the spin quantumnumber ms for electrons, the Lande factor gs ≈ 2.00232 for electrons, and the Bohr magneton µB . Thederivation 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 itsspecial form it creates a field like the one generated by two parallel wires within a distance 2a and oppositecurrent directions. Here a is the radius of the convex pole shoe. This two-wire field is particularly usefulfor the splitting of the atom beam. In fact, with a proper adjustment of the atomic beam a field gradient∂B∂z can be achieved at a distance of about

√2a from the fictitious wires, which is almost constant over

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 beused (derivation in Appendix A.1):

∂B

∂z= 0.9661 · B

a. (2)

The magnetic induction B is determined from the applied coil current by a calibration curve, which isspecific 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 inFigure 2. The propagation direction of the particles is the x-direction, whilst the magnetic field acts alongthe 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 = Lv . The

deflection ∆z in z-direction of the particle is given by

∆z = 12 z (∆t)

2

= 12 ·

Fz

m · (∆t)2.

Due to ∆z � L one can assume ~v ≈ ~vx. When leaving the magnetic field, the velocity z in z-directionis given by the momentum mz = Fz ·∆t of the particle. At this speed, it moves towards the detector atthe position x = l uniformly and rectilinearly . Thus, the total time ttot, which is needed from entering

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the magnetic field until reaching the detector is ttot = lv . Now one can calculate the impact point u of

the particle as a function of its speed and the field inhomogeneity:

u = z + ∆z + z · (tges −∆t)

= z +L · lmv2

(1− L

2l

)µz∂B

∂z

= z − L · lmv2

(1− L

2l

)msgs ·µB

∂B

∂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 todescribe 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 thermalequilibrium at the temperature T . This means that the potassium atoms satisfy the Maxwell velocitydistribution 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

dN ∼ e−mv2

2kT v2dv.

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 formfor 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 movingparallelly to the beam direction when leaving the furnace enter the beam. Considering the atoms ofa given velocity v, which leave the furnace in a sufficiently small time interval ∆t, they can only beoriginating from a volume of a length v ·∆t in front of the window (see figure 3). The size of this volumeis directly proportional to the particle velocity v. Since the particle density n is the same in the entirefurnace, the number of atoms of a certain velocity v in the beam scales with v (in addition to the Maxwellvelocity distribution).

Figure 3: For the derivation of the geometric proportionality in the velocity distribution

As a result, one obtains

dN ∼ e−mv2

2kT v3dv.

Since we are interested in the number of particles as a function of the impact point u, dv has to besubstituted 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

|u− z| =

∣∣∣∣− L · lmv2

(1− L

2l

)msgs ·µB

∂B

∂z

∣∣∣∣=

L · lmv2

(1− L

2l

)|ms|gs ·µB

∂B

∂z

=L · l

2mv2

(1− L

2l

)gs ·µB

∂B

∂z. (4)

In this experiment, the inhomogeneity of the magnetic field ∂B∂z is positive by definition. This leads to

v =

√L · l2m

(1− L

2l

)gs ·µB

∂B

∂z

1

|u− z|

and

dv =

√L · l2m

(1− L

2l

)gs ·µB

∂B

∂z·(−1

2

)|u− z|− 3

2 · u− z|u− z|

du

= −1

2

√L · l2m

(1− L

2l

)gs ·µB

∂B

∂z

1

|u− z|· u− z|u− z|2

du

= − v

2(u− z)du. (5)

As only positive solutions are allowed for dv, this results in

dv =v

2|u− z|du (6)

as well as

v3dv =v4

2|u− z|du

=1

2

(L · l2m

(1− L

2l

)gs ·µB

∂B

∂z

1

|u− z|

)21

|u− z|du

=1

8

(L · lm

(1− L

2l

)gs ·µB

∂B

∂z

)21

|u− z|3du. (7)

Therefore, taking into account the normalisation, the total number of particles as a function of the impactpoint is:

dN =a · e−

b|u−z|

|u− z|3du, (8)

with

a =18

(L · lm

(1− L

2l

)gsµB

∂B∂z

)22∞∫−∞

18

(L · lm

(1− L

2l

)gsµB

∂B∂z

)2e−

b|u−z| 1

|u−z|3 du

=1

2∞∫−∞

e−b

|u−z| 1|u−z|3 du

(9)

b =L · l4kT

(1− L

2l

)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 ofthe 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 theindex m = ± 1

2 denotes the orientation of the magnetic momentum. Thus, the final distribution functionis

d2N = Φm(z)a · e−

b|u−z|

|u− z|3dudz (11)

2.5 Ideal case, method A

By means of the previous considerations, it is now possible to specify the particle current density I(u) asa function of the impact point u on the detector plane. To consider all particles of the beam, one has totake an integral of z and sum over all possible spin orientations:

I(u) =1

du

∑m

D∫−D

d2N

=

∑m

D∫−D

d2N

du

=

∑m

D∫−D

Φm(z)a · e−

b|u−z|

|u− z|3dz

. (12)

Since the distribution of the number of particles in the beam is equivalent for both orientations ms = ± 12 ,

one defines

Φ+ 12(z) ≡ Φ− 1

2(z) :=

I0(z)

2.

Therefore,

I(u) = a

D∫−D

I0(z)e−b

|u−z|dz

|u− z|3. (13)

Let us now consider the approximation of an infinitesimal box for the beam profile:

I0(z) = 2DI0δ(z).

Here δ(z) is the Dirac function, for which (if x0 is within the limits of integration) the following applies:∫f(x)δ(x− x0)dx = f(x0).

This results in

I(u) = 2aDI0e− b|u|

1

u3. (14)

Now, the maxima of the intensities should be determined. For this purpose the equation (14) is derivedand set equal to zero:

dI(u)

du= 2aDI0

b− 3|u|u5

e−b|u| !

= 0,

⇒ u(0)0 = ± b

3= ±

L · l(1− L

2l

)12kT

gs ·µB∂B

∂z. (15)

Therefore, it is immediately visible that the position of the maxima u0 increases linearly with the fieldinhomogeneity, i.e.

u(0)0 = const. · ∂B

∂z. (16)

In the evaluation part, this method is referred to as method A.

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2.6 Real beam cross-section, method B

2.6.1 Beam cross-section

To get to a better adaptation of the theory to the experiment, we will now look at a beam cross-sectionwith a finite width 2D. Thereby, the beam profile should be described by two steep, straight slopes anda parabolic peak (as shown in Figure 4), ie

I0(z) = i0 ·

D + z fur −D ≤ z ≤ −pD − 1

2p−12z2

p fur − p ≤ z ≤ pD − z fur p ≤ z ≤ D

dI0(z)

dz= i0 ·

1 fur −D ≤ z ≤ −p− z

p fur − p ≤ z ≤ p−1 fur p ≤ z ≤ D

(17)

d2I0(z)

dz2= i0 ·

0 fur −D ≤ z ≤ −p− 1

p fur − p ≤ z ≤ p0 fur p ≤ z ≤ D

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 byperforming a Taylor expansion of the solution function of the real beam cross section (cf. Appendix A.2,setting equation (22) to 0).

0 =

(D2 − 1

3p2)(

b

u0− 3

)+D4 − 1

5p4

u20

(5

(b

u0− 1

)+

1

12

b2

u20

(b

u0− 15

))Here, the first term is the solution in first order, while the second term represents a higher order correctionterm. Neglecting the higher order correction one receives the already known solution for u00 = b

3 . Since

8

the second term is a correction term of higher order, it is possible to replace u0 by u00 because the resultingvariations will be of an even higher order. This yields the following approximation:

0 =

(D2 − 1

3p2)(

b

u0− 3

)+D4 − 1

5p4

u20

⇒ b = 3u0 −D4 − 1

5p4

D2 − 13p

2

1

u0= 3u0 −

C

u0. (18)

The correction term − Cu0

takes into account the different degrees of force effect at different positions inthe beam cross section. In the analysis part, the Bohr magneton µB will also be determined by MethodB in addition to method A.

Exercise Use the equations (10) and (18) to deduce an expression, in which µB only depends onmensurable quantities.

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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 match-ing 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 table2. Initially, the power source for the solenoid is to remain turned off.

device operation mode settingfurnace heating (ca. 10 min) max 1 A"

continuous operation ca. 0.5 A"

detector normal operation 4.2 A∼flashing (max. 45 s) Umax (ca. 18 V∼)

magnet max 1.2 A∼

Table 1: settings

length of the pole shoes L = 7 · 10−2 mradius convex pole shoe a = 2.5 · 10−3 mfrom start of magnetic field to detector 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 wouldnot be possible at normal ambient pressure due to the interaction of the potassium atoms with gas atomsin the air. Before you start, insure yourself with the help of a cold cathode vacuum gauge whether thepressure is below the required limit of p ≈ 4 · 10−6 mbar. When heating the furnace or flashing thedetector (see section 3.4), the vacuum may briefly get worse (up to 2 orders of magnitude), but thenshould recover within a few minutes. If it does not recover, please report it to the supervisor! If thepressure rises above 10−4 mbar at any point of time, immediately turn off the detector andreport 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, highertemperatures 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 sufficientlydistinguishable from the background.The precise temperature is required for the analysis. Note this value and make sure that it remainsconstant throughout the whole experiment.Afterwards, the magnetic field can be decreased slowly. At the same time, the field should be invertedseveral times in order to minimise the remanence. The iron yokes in the coils can be screwed out forthe measurement at B = 0 and have to be carefully turned back in for the later measurements withmagnetic 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 neu-tral atoms. It consists of a tungsten wire that is heated to a temperature of T = 800 ◦C and is surroundedby a nickel cylinder with an entrance window for the potassium atom beam. By means of the match-ing transformer which is operated via a variable transformer, a voltage of about 50 V is applied at thewire. Without this bias voltage, the detector would provide a current due to its glow discharge. Thiscurrent would increase exponentially with the temperature of the wire. The cylinder is grounded. Whenpotassium atoms hit the glowing wire they are ionised (ionisation energy lower than the work functionfor electrons in tungsten) and accelerated towards the cylinder due to the potential difference betweenthe wire and the cylinder. The resulting ion current is of the order of ∼ pA and must therefore be ampli-fied. If the tungsten wire has the correct temperature, the measured current is proportional the numberof detected potassium atoms. In case of too low heating current, not all atoms can be ionised. If thetemperature 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 willevaporate 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 reducethe noise.Another problem is caused by the small value of the currents that should be measured: If the coaxialcable, which is used to read out the detector, is touched or moved, currents can be induced for a shorttime which are much higher than the signal itself. Therefore, this cable should not be touched and one

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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 currentin the detection plane should be measured at vanishing magnetic field and magnetic field strengths upto 1.2 A. The measurements should be performed in small steps, since an accurate determination of theposition of the maxima with magnetic field turned on as well as the beam parameters p and D withB = 0 has to be performed. In order to keep magnetic remanence effects low, one should take care tocarried 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. Thedetector 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 displace-ment of the detector.

• Determine the magnetic moment µB

– using the theory of an arbitrarily narrow beam box (by plotting u(0)0 versus b

3 , cf. equation15)

– using equation 18 for a real beam box (by plotting u(0)0 − C

3u(0)0

versus b3 - therefore, the real

beam profile has to be parametrised as shown in Figure4).

• Therefore, the positions u(0)0 of both maxima have to be determined for different field inhomo-

geneities.

• The conversion of the coil current into the field inhomogeneities can be found in Appendix B.1. Donot 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 mathemat-ically more precise theory is. Discuss all other influences on your measurement.

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A Mathematical background

A.1 Two-wire field

The magnetic induction of two anti-parallel electrical currents at the distance 2a is obtained by addingup their individual components

~Bi =~Ii × ~ri2π · r2i

.

Knowing ~I1 = −~I2 = ~I, one has

B(~r) =µ0Ia

πr1r2.

In order to let the atomic beam undergo a substantially constant force, it has to traverse the magneticanalyser at a point with almost constant field inhomogeneity. Therefore, it has to be determined at whichdistance 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.

Figure 9: Definition of a coordinate system

The following relationships for the distances r1 and r2 can be extracted from figure 9:

r21 = (a− y)2 + (z + z0)2

r22 = (a+ y)2 + (z + z0)2

By inserting into the equation above, this results in

B(y, z) =µ0 · I · a

π·[(a2 − y2

)2+ 2 (z + z0)

2 (a2 + y2

)+ (z + z0)

4]− 1

2

(19)

and by taking the derivative of this with respect to z the field inhomogeneity is obtained to

∂B

∂z= −2 ·µ0 · I · a · (z + z0)

π· a2 + y2 + (z + z0)2[

(a2 − y2)2

+ 2 (z + z0)2

(a2 + y2) + (z + z0)4] 3

2

.

At z = 0, the inhomogeneity’s dependence of y should disappear. To find this plane, ∂B∂z is expanded up

to first order in y2 at y = 0. The approximation yields

∂B

∂z(y, z) ≈ − 2µ0I · a · (z + z0)

π(a2 + (z + z0)

2)2 ·

1 + 2y2 · 2a2 − (z + z0)2(

a2 + (z + z0)2) .

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Thus, the dependence of y vanishes for z0 =√

2a. This is the distance of the atom beam from thefictitious wires.For the calculation of the magnetic moment, it is important to know the field inhomogeneity. It cannot be measured directly. However, it can be calculated from the magnetic field strength under theassumption of a linear relationship between B and ∂B

∂z for small deflections z. Now, the proportionality

factor ε =∣∣∂B∂z

∣∣ · aB should be determined in the area around y = 0. Therefore, ε(y, z) is expanded up tofirst order in y2 at y = 0 like above.

ε (y, z) ≈ 2a (z + z0)

a2 + (z + z0)2 ·(

1 +3a2 − (z + z0)

2(a2 + (z + z0)

2)2 · y2).The system of cover plates of the apparatus is designed in a way that the atomic beam has a width ofapproximately 4

3a in y-direction. In the area of this beam box ε(y, 0) changes slightly with y, thereforethe average of the highest possible value epsilon ε

(23a, 0

)≈ 0.9894 and the lowest value ε(0, 0) ≈ 0.9428

is used to calculate the inhomogeneity. Thus, the conversion factor is ε ≈ 0.9661.

A.2 Real beam cross-section, method B

A.2.1 Beam cross-section

Now let us discuss the model for the real beam cross section, described in section 2.6. Looking at theequations (17), we see that I0(z) is defined as two-times-differentiable. Based on this, the determination ofthe particle current density I(u) depends on the inhomogeneity of the magnetic field, i.e. of b (equation10). The maxima of I(u) are located at the positions u0(b), which now more or less differ from thepositions u00 = ± b

3 , i.e. from the approximation for an infinitesimal beam box. To determine the functionu0(b), we assume

dI

du(u0) = 0.

The derivative with respect to u can now be taken into the integral, as can be seen in the followingcalculation

dIdu = d

dua+D∫−D

I0(z) e− b|u−z|

|u−z|3 dz

= a+D∫−D

I0(z) ∂∂u

e− b|u−z|

|u−z|3 dz.

As one can easily verify, the replacement of ∂∂u by − ∂

∂z leaves the integrand unchanged.

dI

du= −a

+D∫−D

I0(z)∂

∂z

e−b

|u−z|

|u− z|3dz.

By partial integration (reminder:b∫a

f(x) · g′(x)dx = (f(x) · g(x))|ba−b∫a

f ′(x) · g(x)dx), it is now possible to

move the differentiation to I0. Please check that here the first term from the partial integration vanishes.

dI

du= a

+D∫−D

dI0(z)

dz

e−b

|u−z|

|u− z|3dz.

Inserting equation (17) returns

dI

du= ai0

−p∫

+D

−+p∫−p

z

p−

+D∫+p

e−b

|u−z|

|u− z|3dz.

Both occurring integrals can be solved analytically (e.g. with Maple) and result in

dI

du=ai0pb2·F (u)

15

with the solution function

F (u) = −|u+ p|e−b

|u+p| + |u− p|e−b

|u−p| + pb+ |u+D|u+D

e−b

|u+D| + pb+ |u−D|u−D

e−b

|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 posi-tion of the maxima, is thus mirror symmetric. By this simplification, we can restrict to the determinationof 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 infinitesi-mally narrow beam box.If we now assume

u0p,u0D,b

p,b

D� 1, (20)

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 whichwe need the following function:

f(u) = u · e− bu . (21)

The relevant derivatives are

f (3)(u) = b2

u4

(bu − 3

)e−

bu

f (5)(u) = 12 b2

u6

(5(bu − 1

)+ 1

12b2

u2

(bu − 15

))b−

bu .

Up to the sixth derivation, only the third and fifth derivative are of interest, because only in this casethe coefficients do not vanish.Thus, abortion after the sixth element results in:

F (u) = p

(D2 − 1

3p2)· f (3)(u) +

p

12

(D4 − 1

5p4)· f (5)(u) + . . . . (22)

From this we obtain, once more by zeroing, the equation for u0

0 =

(D2 − 1

3p2)(

b

u0− 3

)+D4 − 1

5p4

u20

(5

(b

u0− 1

)+

1

12

b2

u20

(b

u0− 15

)). (23)

The further way to determine u0 is explained in the theory section (section 2.6.2).

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B Figures

B.1 Calibration curves

Figure 10: Calibration curve of the magnet No. 52 (set-up A)

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Figure 11: Calibration curve of the magnet No. 77 (set-up B)

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