Experiments in Modern Physics Lab. · Elementary charge and Millikan experiment 10. Zeeman Effect . 3 Experiments in modern physics | Physics Dept. Experiment 1 Microwave Wavelength

Physics Dept.

Experiments in Modern

Physics Lab. FIRST SEMISTER

Name:

Level:

2014-2015

2 Experiments in modern physics | Physics Dept.

Experiments:

1. Microwave Wavelength measurement

2. Study of Stefan-Boltzmann’s law of radiation

3. Specific charge of electron (e/m0) measurement

4. The Study of Electron Diffraction

5. copper X-ray diffraction study of copper (Cu)

6. Single-slit diffraction and the Heisenberg's uncertainty principle

7. Balmer series / Determination of Rydberg’s constant

8. Study of Atomic spectra of two-electron systems: He

9. Elementary charge and Millikan experiment

10. Zeeman Effect


Experiment 1

Microwave Wavelength Measurements

Aim:

Measuring the wavelength of microwave

Theory:

Frequency and wavelength of an electromagnetic wave are related by the following equations:

𝒇𝝀 = 𝒄

Where c is the wave velocity

Where the wave travels in an open space, its velocity equals to speed of light,

C0=300,000Km/sec

For the general case of propagation in a path other than the open space the velocity is reduced according to the

formula:

𝒄 =𝒄𝟎µƐ

Where ɛ is the relative dielectric constant of a medium and μ is the relative magnetic permeability of the same

medium.

Task:

With this experiment we shall measure wavelength inside the wave guide and since our generator posses a know

frequency, we shall calculate the wave velocity by the formula above.

Method:

1. Use the setup shown in fig.1, so as to generate a stationary wave inside the waveguide, having the pattern shown in

fig.2. before turning on the power, note the following:

There are not harmful voltages involved in the trainer's operation, however this must not lower the level of

attenuation required to operate it.

The RF power level in the trainer are not harmful, however, the human eyes may be damaged by even low

levels of radiation. Do not look into any waveguide at any time when the units are on.

2. Turn on the equipment. Connect an oscilloscope to the detector put, in order to display the signal level. The

detected amplitude is negative. Adjust the attenuator for a suitable value, let's say 3dB.

3. Measure and record in a graph the stationary wave pattern, as already seen in the preceding worksheet.

- The wavelength is twice of distance between two consecutive points of max or min (indicated by D in fig.2).

- As a matter of facts, since a precise location of the max and min points is awkward, it is preferable to locate two

equal level points in the wave, such asY1 and Y2 of fig.2

- From the figure obtained for the wavelength and the known frequency of the generator, 10,75GHz, calculate the

wave velocity.

Note: λ=2Δd, why?

The second part of the experiment requires removing of the attenuator and shorting plate. The waveguide remains

therefore open at one end and radiates toward the exterior.

While proceeding with this experiment never glance into the open end of the guide, since this would expose your

eye to an EM radiation, which is weak but potentially hazardous.


Place a metal plate (you may use the same shorting plate held by hand) if front of the open waveguide end,

orthogonally with the direction of the propagation. Move the reflecting plate closer to and farther from the open mouth

of the guide while you monitor the signal level displayed by the oscillator.

You will notice alternate variation of level which reminds you of the stationary wave pattern already studied.

There are interesting consecrations descending from the phenomenon just observed:

- The signal out leaving the open guide and is being partially reflected and re-enters the guide where it generates a

stationary wave.

- Applying the principles using in the first part of this worksheet, it would be possible to measure the distance between

two consecutive max or min position of the reflecting plate and then calculate wavelength and velocity for propagation

in the open air.

- While the reflecting plate is being moved nearer and farther from the guide's mouth, the signal displayed at the

oscilloscope will not simply show an alternance of max and min level, but will have wider changes as the speed of the

plate increases. This is due to so called Doppler Effect: the speed of the plate adds or subtract (depending on the sense

of movement) to the speed of the wave. The reflected wave consequently re-enters the guide with a frequency slightly

different that the original. The direct and reflected wave max and the detector display a level waving t a rate equaling

the difference in frequency of the two waves.

Fig 1, set up for worksheet

Figure 2 wave length measurement


Experiment 2

Study of Stefan-Boltzmann’s law of radiation

In this experiment, you will learn:

Black body radiation, Thermoelectric e. m. f. And, Temperature dependence of resistances

Aim of the experiment:

Studying the Stefan-Boltzmann‘s low of black body radiation.

Theory:

According of Stefan-Boltzmann‘s law, the energy emitted by a black body per unit area and unit time is proportional

to the power “four” of the absolute temperature of the body. Stefan-Boltzmann‘s law is also valid for a so-called

―grey‖ body whose surface shows a wavelength independent absorption-coefficient of less than one. In the

experiment, the ―grey‖ body is represented by the filament of an incandescent lamp whose energy emission is

investigated as a function of the temperature.

If the energy flux density L of a gray body, e.g. energy emitted per unit area and unit time at temperature T and

wavelength λWithin the interval dλ, is designated by dL(T, λ)/dλ,

Planck‘s formula states:

where: c = velocity of light (3.00 * 10

8 [m/s]), h = Planck‘s constant (6.62 * 10

–34 [J ・ s]) and, k = Boltzmann‘s

constant (1.381 * 10–23

[J ・ K–1]) Integration of equation (1) over the total wavelength-range from λ = 0 to λ = ∞ gives the flux density L(T) (Stefan-

Boltzmann‘s law).

Where ϭ = 5.67 * 10

–8 [W ・ m–2 ・ K-4]

The proportionality L ≈ T4 is also valid for a so-called ―grey‖ body whose surface shows a wavelength-independent

absorption-coefficient of less than one.

To prove the validity of Stefan-Boltzmann‘s law, we measure the radiation emitted by the filament of an incandescent

lamp which represents a ―grey‖ body fairly well. For a fixed distance between filament and thermopile, the energy

flux ɸ which hits the thermopile is proportional to L(T).

ɸ ≈ L(T)

Because of the proportionality between ɸ and the thermoelectric e.m.f., Utherm of the thermopile, we can also write:

If the thermopile is at a temperature of zero degrees Kelvin. Since the thermopile is at room temperature ‗TR‘ it also

radiates due to the T4 law so that we have to write:

Under the present circumstances, we can neglect T

4R against T

4 so that we should get a straight line with slope “4”

when representing the function Utherm = f(T) double logarithmically.

1

2

3


The absolute temperature T = t + 273 of the filament is calculated from the measured resistances R(t) of the tungsten

filament (t = temperature in centigrade). For the tungsten filament resistance, we have the following temperature

dependence:

The resistance R0 at 0°C can be found by using the relation:

Solving R(t) with respect to t and using the relation T = t + 273 gives:

R(tR) and R(t) are found by applying Ohm‘s law, e. g. by voltage and current measurements across the filament.

Task and evaluation:

Equipment

Thermopile, molltype

Shielding tube

Universal measuring amplifier

Power supply var.15 VAC/12 VDC/5 A

Lamp holder

Filament lamp 6V/5A,

Connection box

Resistor in plug-in box 100 Ω

Optical profile bench l = 60 cm

Base f. opt. profile-bench, adjust.

Slide mount f. opt. pr.-bench, h = 30 mm 08286.01 2

Digital multimeter

Connecting cord, l = 500 mm, blue

Connecting cord, l = 500 mm, red

Set-up and procedure 1. The experiment is started by setting up the circuit of Fig. 2 to measure the filament‘s resistance at room

temperature. A resistor of 100 is connected in series with the lamp to allow a fine adjustment of the current. For 100

mADC and 200 mADC the voltage drops across the filament are read and the resistance at room temperature is

calculated. The current intensities are sufficiently small to neglect heating effects.

2. The experiment set-up of Fig. 1 is then built up. The 100 resistor is no longer part of the circuit. The filament is

now supplied by a variable AC-voltage source via an ammeter allowing measurement of alternating currents of up to 4

amperes. The voltmeter is branched across the filament and the alternating voltage is increased in steps of 1 volt up to

a maximum of 8 V AC.

Remark: the supply voltage of the incandescent lamp is 6 V AC. A voltage of up to 8 V AC can be applied if the

period of supply is limited to a few minutes.

3. Initially, a voltage of 1 V AC is applied to the lamp and the Moll-thermopile, which is at a distance of 30 cm from

the filament, is turned (slide-mount fixed) to the right and to the left the thermoelectric e.m.f. shows a maximum. The

axis of the cylindrical filament should be perpendicular to the optical bench axis. Since the thermoelectric e.m.f. is in

the order of magnitude of a few millivolts, an amplifier has to be used for accurate readings. The factor of

amplification will be 102 or 10

3 when using the voltmeter connected to the amplifier in the 10 V range.

4

5

6


Note: Before a reading of the thermoelectric e.m.f. is taken, a proper ―zero‖-adjustment has to be assured. This is

done by taking the lamp together with its slide-mount away from the bench for a few minutes. The amplifier is used in

the LOW DRIFT- mode (104 Ω) with a time constant of 1 s.

After the lamp has been put back onto the bench, the reading can be taken if the Moll-thermopile has reached its

equilibrium. This takes about one minute. Care must be taken that no background radiation disturbs the

measurement.

4. After adjusting the circuit, keep taking data on by increasing the current of the lump 0.5 A each until reaching to

4A.

5. Record your data from a table as below:

I(A) U (v) U thermal (mv) T (k)

0.5

1

....

.

4

Note: T can be calculated via equation 6.

6. Plot a diagram between U thermal (mv) and T (k) as below:

Fig. 3: Thermoelectric e.m.f. of thermopile as a function of the filament‘s absolute temperature.

7. The double logarithmic, graphical representation of the energy flux versus absolute temperature is shown in Fig. 3.

The slope S of the straight line is calculated (Use equation 3), by regression, to be:

S = 4.19 •± 0.265 & The true value of S, which is 4, is found to be within the limits or error.

8. Compare the extracted experimental and theoretical values, DISCUSS YOUR RESULT ACCURATELY.

NOTE: the theoretical value of the below parameters has reached as in a certain condition:

R(tR) = 0.165 [] and R0 = 0.15 []

Fig. 2: Circuit to measure the

resistance of the filament at room

temperature

Fig. 1: Set-up for experimental verification of Stefan-

Boltzmann‘s law of radiation.


Experiment 3

Specific charge of the electron (e/m0) measurement


Cathode rays, Lorentz force, Electron in crossed fields, Electron mass, Electron charge.


Determination of the specific charge of the electron (e/m0) from the path of an electron beam in crossed

electric and magnetic fields of variable strength. Electrons are accelerated in an electric field and enter a

magnetic field at right angles to the direction of motion. The specific charge of the electron is determined

from the accelerating voltage, the magnetic field strength and the radius of the electron orbit.

Theory and evaluation

If an electron of mass m0 and charge e is accelerated by a potential difference U it attains the kinetic energy:

Where 𝑣 is the velocity of the electron. In a magnetic field of strength B, the Lorentz force acting on an

electron with velocity 𝑣 is:

If the magnetic field is uniform, as it is in the Helmholtz arrangement the electron therefore follows a spiral

path along the magnetic lines of force, which becomes a circle of radius r if 𝒗 is perpendicular to B

Since the centrigugal force m0 ・𝑣 2

/r thus produced is equal to the Lorenth force, we obtain

Where B is the absolute magnitude of B From equation (1), it follows that

To calculate the magnetic field B, the first and fourth Maxwell equations are used in the case where no time

dependent electric fields exist.

We obtain the magnetic field strength BZ on the z-axis of a circular current I for a symmetrical arrangement

of 2 coils at a distance ‗a‘ from each other:

And R = radius of the coil

2

1


For the Helmholtz arrangement of two coils (a = R) with number of turns ‗n‘ in the center between the coils

one obtains:

For the coils used, R = 0.2 m and n = 154. The mean,

e/m0 = (1.84 ± 0.02) ・ 1011

As/kg

Literature value: e/m = 1.759 ・ 10-11 As/kg


Equipment: you need,

Narrow beam tube 1

Pair of Helmholtz coils 1

Power supply, 0...600 VDC 1

Power supply, universal 1

Digital multimeter 2

Connecting cord, l = 100 mm, red 1

Connecting cord, l = 100 mm, blue 1



Connecting cord, l = 750 mm, yellow

Set-up and procedure

1. The experimental set up is as shown in Fig. 1. The electrical connection is shown in the wiring diagram in

Fig. 2. The two coils are turned towards each other in the Helmholtz arrangement. Since the current must be

the same in both coils, connection in series is preferable to connection in parallel. ‗‘The maximum

permissible continuous current of 5 A should not be exceeded‘‘.

2. Set V=100 volts DC over the anode material- accelerate electron from the cathode material- and I=0.5 A

DC over the connected coils.

3. If the polarity of the magnetic field is correct, a curved luminous trajectory is visible in the darkened

room. By varying the magnetic field (current) and the velocity of the electrons (acceleration and focusing

voltage) the radius of the orbit can be adjusted, that it coincides with the radius defined by the luminous

traces. When the electron beams ciinudes with the luminous traces, only half of the circle is observable. The

radius of the circle is then 2, 3, 4 or 5 cm.

Note: If the trace has the form of a helix this must be eliminated by rotating the narrow beam tube around its

longitudinal axis.

Fig.1: Experimental set-up for determining the specific charge of the electron.

3


Fig. 2: Wiring diagram for Helmholtz coils.

4. Collect your data in a table as below:

V anode (v) I coil (A) B (mT) r (cm)

100 0.5 Use eq. 3 Corresponding to I coil

100 2

100 ....

100 4

Note: equation 3 can be employed to extract the B value with mT.

5. Plot a diagram between B (mT), calculated via equ. 3, and r (cm). When the slop is determined, the

experimental value of e/m0 can be easily found by using the below equation (eq 2):

𝑒

𝑚=

2𝑉

𝑠𝑙𝑜𝑝2

6. Repeat the above steps for V anode = 200 and 300V respectively.

7. Calculate theoretical value of e/m0, and compare it with the experimental result.

8. Discuss your result.

Fig. 3: Wiring diagram for Narrow beam tube.


Experiment 4

The Study of Electron Diffraction


Bragg reflection, Debye-Scherrer method, Lattice planes, Graphite structure, Material waves, De Broglie

equation


Fast electrons are diffracted from a polycrystalline layer of graphite: interference rings appear on a

fluorescent screen. The interplanar spacing (d) in graphite is determined from the diameter of the rings

and the accelerating voltage.

Fig. 1: Experimental set-up: electron diffraction, the connected set (left hand side) and the electronic circuit

of it (right hand side) are shown.

Fig. 2: Set-up and power supply to the electron diffraction tube.


To explain in the interference phenomenon, a wavelength λ, which depends on momentum, is assigned to

the electrons in accordance with the de Broglie equation:

where h = 6.625 · 10

–34 Js, Planck‘s constant.

1


The momentum can be calculated from the velocity 𝑣 that the electrons acquire under acceleration voltage

UA:

The wavelength is thus

Where e = 1.602 · 10

–19 As (the electron charge) and m = 9.109 · 10

–31 kg (rest mass of electron).

At the voltages UA used, the relativistic mass can be replaced by the rest mass with an error of only 0.5%.

The electron beam strikes a polycrystalline graphite film deposit on a copper grating and is reflected in

accordance with the Bragg condition:

2d sin ө = n ·λ, n = 1, 2, …

Fig. 3: Crystal lattice of graphite.

Where d is the spacing between the planes of the carbon atoms and θ is the Bragg angle (angle between

electron beam and lattice planes).

In polycrystalline graphite the bond between the individual layers (Fig. 3) is broken so that their orientation

is random. The electron beam is therefore spread out in the form of a cone and produces interference rings

on the fluorescent screen.

The Bragg angle ө can be calculated from the radius of the interference ring but it should be remembered

that the angle of deviation a (Fig. 2) is twice as great:

α= 20.

From Fig. 2 we read off

where R = 65 mm, radius of the glass bulb.

For small angles α (cos 10° = 0.985) can put

2

3

4

5

6

Fig. 4: Graphite planes for the first two interference

rings.


So that for small angles ө we obtain

With this approximation we obtain

The two inner interference rings occur through reflection from the lattice planes of spacing d1 and d2 (Fig.

4), for n = 1 in (7).


Equipment: you need,

Electron diffr. tube a. mounting. 1

High voltage supply unit, 0-10 kV. 1

High-value resistor, 10 MΩ. 1

Connecting cord, 30 kV, 500 mm. 1

Power supply, 0...600 VDC. 1

Vernier caliper, plastic 03014.00 1

Connecting cord, l = 250 mm, red. 2

Connecting cord, l = 250 mm, blue. 2

Connecting cord, l = 750 mm, red. 2

Connecting cord, l = 750 mm, yellow. 1

Connecting cord, l = 750 mm, blue. 1

Connecting cord, l = 750 mm, black


1. Set up the experiment as shown in Fig. 1. Connect the sockets of the electron diffraction tube to the power

supply as shown in Fig. 2. Connect the high voltage to the anode G3 through a 10 M protective resistor.

2. Set the Wehnelt voltage G1 and the voltages at grid 4 (G4) and G3 so that sharp, welldefined diffraction

rings appear. Read the anode voltage at the display of the HV power supply.

3. To determine the diameter of the diffraction rings, measure the inner and outer edge of the rings with the

vernier caliper (in a darkened room) and take an average. Note that there is another faint ring immediately

behind the second ring.

4. Collect the data in a table as below:

UA(kv) d1 (cm) d2 (cm) r (cm) D1 (cm) D2 (cm) R (cm) λ= 𝟏𝟓𝟎𝟎

𝑽(𝑲𝑽)

nm

Equation 3

5. Plot a diagram between both r and R versus λ for each of the rings. As below (fig 5):

6. The wavelength is calculated from the anode voltage in accordance with (equa.3) which is λ= 1500

𝑉(𝐾𝑉) nm

after substituting the constants.

6. To calculate the inter-planer distance‘d‘, Applying the regression lines expressed by Y = AX + B.

7. Find ‗d‘ from equ. 7. (S will be extracted 1st then, the remaining parameters are substituted)

8. Compare d for the both ring as well as to comparing the experimental value to the theoretical value.

Note: R = 65 mm, radius of the glass bulb

6a

7


Fig. 5: Radii of the first two interference rings as a function of the wavelength of the electrons.

An experimental value: according to fig 5

Slopes:

A1 = 0.62 (2) · 109

A2 = 1.03 (2) · 109

The lattice constants: d1 = 211 pm and d2 = 126 pm

in accordance with (7),

Notes

– The intensity of higher order interference rings is much lower than that of first order rings. Thus, for

example, the second order ring of d1 is difficult to identify and the expected fourth order ring of d1 simply

cannot be seen. The third order ring of d1 is easy to see because graphite always has two lattice planes

together, spaced apart by a distance of d1/3. (Fig. 6) In the sixth ring, the first order of ring of d4 clearly

coincides with the second order one of d2.

Radii (mm) calculated according to (4) for the interference rings to be expected when UA = 7 kV:

d1 = 213 pm d2 = 123 pm d3 = 80.5 pm d4 = 59.1 pm d5 = 46.5 pm.

Fig.6 The appearance of the inner and outer rings.

Fig. 6: Interplanar

spacing in

graphite


– The visibility of high order rings depends on the light intensity in the laboratory and the contrast of the

ring system which can be influenced by the voltages applied to G1 and G4.

– The bright spot just in the center of the screen can damage the fluorescent layer of the tube. To avoid this

reduce the light intensity after each reading as soon as possible.

Experiment 5

X-ray diffraction study of copper (Cu)

You will learn:

X-ray tube, Bremsstrahlung, Characteristic radiation, Energy levels, Crystal structures, Lattice constant,

Absorption, Absorption edges, Interference, The Bragg equation, Order of diffraction

Aim of this experiment,

Spectra of X-rays from a copper anode are to be analyzed by means of different monocrystals and the

results plotted graphically. The energies of the characteristic lines are then to be determined from the

positions of the glancing angles for the various orders of diffraction. It’s also an approach to find out

spacing (d) as well as the wavelength of the produced X-ray.

Equipment

X-ray basic unit, 35 kV 1

Goniometer for X-ray unit, 35 kV 1

Plug-in module with Cu X-ray tube. 1

Counter tube, type B. 1

Lithium fluoride crystal, 1

Potassium bromide crystal, 1

Recording equipment:

XYt recorder 1

Connecting cable, l = 100 cm, red 2

Connecting cable, l = 100 cm, blue 2

or

Software X-ray unit, 35 kV 1

RS232 data cable 1

PC, Windows 95 or higher

Tasks

1. The intensity of the X-rays emitted by the copper anode at maximum anode voltage and anode current is

to be recorded as a function of the Bragg angle, using an LiF monocrystal as analyzer.

2. Step 1 is to be repeated using the KBr monocrystal as analyzer.

3. The energy values of the characteristic copper lines are to be calculated and compared with the energy

differences of the copper energy terms.


Set up the experiment as shown in Fig. 1. Fix a diaphragm tube in the X-ray outlet tube (1 mm tube diameter

using the LiF crystal and 2 mm tube diameter using the KBr crystal). With the X-ray unit switched off,

connect the goniometer and the counter tube to the appropriate sockets in the base plate of the experimenting

area. Set the goniometer block with mounted analyzing crystal to the middle position and the counter tube to

the right stop.

The following settings are recommended for the recording of the spectra:

— Auto and Coupling mode

— Gate time 2 s; Angle step width 0.1°


— Scanning range 3°-55° using the LiF monocrystal, and 3°-75° using the KBr monocrystal

— Anode voltage UA = 35 kV; Anode current IA = 1 mA

When the spectra are to be recorded with an XY recorder, connect the Y axis to the analog output (Imp/s) of

the X-ray basic unit and, correspondingly, the X input to the analog output for the angular position of the

crystal (select the analog signal for the crystal angle with the selection button for this output). When a PC is

used for the recording of the spectra then follow this short instruction for easy operation:

1) Switch on the x-ray unit

2) Open the door of the unit (check the position of the goniometer)

3) Connect the X-ray unit via RS232 cable to the PC port COM1, COM2 or USB port (for USB computer

port use USB to RS232 Converter 14602.10)

Fig. 1: Experimental set-up for the analysis of X-rays

4) Start the "Measure" program and select "Gauge" -> "X-ray Device"

5) Select the parameters shown in Fig. 1a and press continue button (select the Crystal you are using: KBr or

LiF).

6) Close the door of the X-ray device

7) Start the measurement (see Fig. 1b)

Fig.1a: Measuring parameters for the recording software

Fig.1b: Graphical user interface of the "X-ray Device"-software


Note

Never expose the counter tube to primary radiation for a longer length of time.

8) When the set up time is finished, you must have an output peak (as shown in fig 4) and be able to do the

measurement.


When electrons of high energy impinge on the metallic anode of an X-ray tube, X-rays with a continuous

energy distribution (the so-called bremsstrahlung) are produced. X-ray lines whose energies are not

dependent on the anode voltage and which are specific to the anode materials, the so-called characteristic X-

ray lines, are superimposed on the continuum. They are produced as follows: An impact of an electron on an

anode atom in the K shell, for example, can ionize that atom. The resulting vacancy in the shell is then filled

by an electron from a higher energy level. The energy released in this de-excitation process can then be

transformed into an X-ray which is specific for the anode atom.

Fig. 2 shows the energy level scheme of a copper atom. Characteristic X-rays produced from either the L –>

K or the M –> K transitions are called Kα and Kβ lines respectively. M1 –> K and L1 –> K transitions do not

take place due to quantum mechanical selection rules.

Fig. 2: Energy levels of copper (Z = 29)

Accordingly, characteristic lines for Cu with the following energies are to be expected (Fig. 2):

Kα* is used as the mean value of the lines Kα1 and Kα2. The analysis of polychromatic X-rays is made

possible through the use of a monocrystal. When X-rays of wavelength λ impinge on a monocrystal under

glancing angle ϑ, constructive interference after scattering only occurs when the path difference Δ of the

partial waves reflected from the lattice planes is one or more wavelengths (Fig. 3).

1


Fig. 3: Bragg scattering on the lattice planes

This situation is explained by the Bragg equation:

(d = the interplanar spacing; n = the order of diffraction) If d is assumed to be known, then the energy of the

X-rays can be calculated from the glancing angle ϑ, which is obtainable from the spectrum, and by using the

following relationship:

On combining (3) and (2) we obtain:

Fig. 4: X-ray intensity of copper as a function of the glancing angle; LiF (100) monocrystal as Bragg

analyzer

2

3

4


Fig. 5: X-ray intensity of copper as a function of the glancing angle; KBr (100) monocrystal as Bragg

analyzer

Planck's constant h = 6.6256 ・ 10-34 Js

Velocity of light c = 2.9979 ・ 108 m/s

Lattice constant LiF (100) d = 2.014 ・ 10-10 m

Lattice constant KBr (100) d = 3.290 ・ 10-10 m

and the equivalent 1 eV = 1.6021 ・ 10-19 J

Fig. 4 shows that well-defined lines are superimposed on the bremsspectrum continuum. The angles at

which these lines are positioned remains unaltered on varying the anode voltage. This indicates that these

lines are characteristic copper lines. The first pair of lines belongs to the first order of diffraction (n = 1),

whilst the second pair belongs to n = 2. When the LiF monocrystal is replaced by the KBr monocrystal for

the analysis of the copper X-ray spectrum, Bragg scatterings are allowed up to an order of diffraction of 4 (n

= 4) (Fig. 5). The structures which are additional to those in Fig. 4 result from the higher lattice constant of

the KBr monocrystal. The energy values of the characteristic copper X-ray lines are listed in the Table, as

calculated using (4).

Table of results,


Taking the energy values from the Table, the mean values of the energies of the characteristic lines are: EKα

= 8.028 keV and EKβ = 8.867 keV. Both of these experimental values correspond to better then 1% with

literature values (see (1) and Fig. 2). A variation of the evaluation is posible by using the calculated

characteristic copper X-ray lines from one spectrum in order to derive the corresponding lattice constant

from the other spectrum. The bremsstrahlung spectrum in Fig. 6 is subject to a noticeable drop in intensity in

the direction of smaller angles at 8.0° and 16.3°. This drop coincides with the theoretically expected bromide

K absorption edge (EK = 13.474 keV) in the 1st and 2nd

order of diffraction. The K absorption edges of

potassium, lithium and fluorine cannot be observed, since the intensity of the bremsstrahlung spectrum is too

low in these energy regions. (For K and L absorption edges, refer to experiment 5.4.12.00).

Note

The atomic energy values were taken from the "Handbook of Chemistry and Physics", CRC Press Inc.,

Florida.

Experiment 6

Single-slit diffraction and the Heisenberg's uncertainty principle

What will you learn about:

Diffraction, Diffraction uncertainty, Kirchhoff‘s diffraction formula, Measurement accuracy, Uncertainty

of location, Uncertainty of momentum, Wave-particle dualism, De Broglie relationship

Aim:

‗‘confirm Heisenberg‘s uncertainty principle‘‘.

Equipment

Laser, He-Ne 1.0 mW, 220 V AC 1

Diaphragm, 3 single slits 1

Diaphragm, 4 double slits 1

Diaphragm, 4 multiple slits 1

Diaphragm holder 1

Photoelement f. opt. base plt. 1

Slide device, horizontal 1

Universal measuring amplifier 1


Optical profile-bench, l = 1500 mm 1

Base f. opt. profile-bench, adjust 2

Slide mount f. opt. pr.-bench, h = 30 mm 3



Screen 1

Tasks

1. To measure the intensity distribution of the Fraunhofer diffraction pattern of a single slit (e.g. 0.1 mm).

The heights of the maxima and the positions of the maxima and minima are calculated according to

Kirchhoff‘s diffraction formula and compared with the measured values.

2. To calculate the uncertainty of momentum from the diffraction patterns of single slits of differing widths

and to confirm Heisenberg‘s uncertainty principle.


1. Different screens with slits (0.1 mm, 0.2 mm and 0.05 mm) are placed in the laser beam one after the

other. The distribution of the intensity in the diffraction pattern is measured with the photo-cell as far


behind the slit as possible. A slit (0.3 mm wide) is fitted in front of the photocell. The voltage drop at the

resistor attached parallel to the imput of the universal measuring amplifier is measured and is approximately

proportional to the intensity of the incident light.

Important: In order to ensure that the intensity of the light from the laser is constant, the laser should be

switched on about half an hour before the experiment is due to start. The measurements should be taken in a

darkened room or in constant natural light. If this is not possible, a longish tube about 4 cm in diameter and

blackened on the inside (such as a cardboard tube used to protect postal packages) can be placed in front of

the photcell.

2. Another approach (to carry out the experiment) is a photo-cell replacement by a screen from the set up

(our experiment). This approach doesn‘t require neither a photo-cell nor an amplifier or a millimetre to

measure the intensity. The diffraction pattern is seen easily above the screen as you can measure the value of

(a (figure 2)) which represents the distance between the centre of the intense light and the next diffraction

pattern.

3. After the a, b and d values (as seen at fig 2) are measured, they can be substituted in equ.12.

4. Alter the position of each slit and screen and repeat the steep 2 and 3.

5. Discuss your result.

Caution: Never look directly into a non attenuated laser beam

Fig. 1: Experimental set-up for measuring the distribution of intensity in diffraction patterns. (note, you can use a screen instead of

a detector an a voltmeter)

Fig. 2: Diffraction (Fraunhofer) at great distance (Sp = aperture or slit, S = screen).


The principal maximum, and the first secondary maximum on one side, of the symmetrical diffraction

pattern of a slit 0.1 mm wide (for example) are recorded. For the other slits, it is sufficient to record the two

minima to the right and left of the principal maximum, in order to determine α (Fig. 2).


1. Observation from the wave pattern viewpoint

When a parallel, monochromatic and coherent light beam of wave-length M passes through a single slit of

width d, a diffraction pattern with a principal maximum and several secondary maxima appears on the

screen (Fig. 2).

The intensity, as a function of the angle of deviation α, in accordance with Kirchhoff‘s diffraction formula,

is

Where,

The intensity minima are at

The angle for the intensity maxima are

The relative heights of the secondary maxima are:

The measured values (Fig. 3) are compared with those calculated.

1

Fig. 3: Intensity in the diffraction pattern of a

0.1 mm wide slit at a distance of 1140 mm. The

photocurrent is plotted as a function of the

position.


Kirchhoff‘s diffraction formula is thus confirmed within the limits of error.

2. Quantum mechanics treatment

The Heisenberg uncertainty principle states that two canonically conjugate quantities such as position and

momentum cannot be determined accurately at the same time. Let us consider, for example, a totality of

photons whose residence probability is described by the function fy and whose momentum by the function

fp. The uncertainty of location y and of momentum p are defined by the standard deviations as follows:

where h = 6.6262x10

-34 Js, Planck‘s constant (―constant of action‖), the equals sign applying to variables

with a Gaussian distribution.

For a photon train passing through a slit of width d, the expression is

Whereas the photons in front of the slit move only in the direction perpendicular to the plane of the slit (x-

direction), after passing through the slit they have also a component in the y-direction. The probability

density for the velocity component Oy is given by the intensity distribution in the diffraction pattern. We use

the first minimum to define the uncertainty of velocity (Figs. 2 and 4).

where α1 = angle of the first minimum.

The uncertainty of momentum is therefore

where m is the mass of the photon and c is the velocity of light. The momentum and wavelength of a particle

are linked through the de Broglie relationship:

2

3

4

5


Fig. 4: Geometry of diffraction at a single slit a) path covered b) velocity component of a photon

The angle α1 of the first minimum is thus

according to (1).

If we substitute (8) in (7) and (3) we obtain the uncertainty relationship

If the slit width Δy is smaller, the first minimum of the diffraction pattern occurs at larger angles α1.

In our experiment the angle α1 is obtained from the position of the first minimum (Fig. 4a):

If we substitute (10) in (7) we obtain

Substituting (3) and (11) in (9) gives

after dividing by h.

The results of the measurements confirm (12) within the limits of error.

* The widths of the slits were measured under the microscope.

6

7

8

9

10

11

12


Experiment 7

Balmer series / Determination of Rydberg’s constant

What you can learn about …

Diffraction image of a diffraction grating, Visible spectral range, Single electron atom, Atomic model

according to Bohr, Lyman-, Paschen-, Brackettand Pfund-Series, Energy level, Planck‘s constant, Binding

energy, The wave

Aim:

1. Wave lengths of the visible lines of the Balmer series of H are measured.

2. Determination of Rydberg’s constant

Principle

The spectral lines of hydrogen and mercury are examined by means of a diffraction grating. The known

spectral lines of Hg are used to determine the grating constant. The wave lengths of the visible lines of the

Balmer series of H are measured.

Equipments

Spectrum tube, hydrogen 1

Spectrum tube, mercury 1

Holders for spectral tubes, 1 pair 1

Cover tube for spectral tubes 1

Connecting cord, 30 kV, l = 1000 mm 2

Object holder, 5_5 cm 1

Diffraction grating, 600 lines/mm 1

High voltage supply unit, 0-10 kV 1

Insulating support 2

Tripod base -PASS- 1

Barrel base -PASS- 1

Support rod -PASS-, square, l = 400 mm 1

Right angle clamp -PASS- 3

Stand tube 1

Meter scale, demo, l = 1000 mm 1

Cursors, 1 pair 1

Measuring tape, l = 2 m 1

Tasks

2. Determination of the visible lines of the Balmer series in the H spectrum, of Rydberg’s constant and of

the energy levels.


1. The experimental set-up is shown in Fig. 1. Hydrogen or mercury (Hg) spectral tubes connected to the

high voltage power supply unit are used as a source of radiation. The power supply is adjusted to about 5

kV. The scale is attached directly behind the spectral tube in order to minimize parallax errors. The

diffraction grating should be set up at about 50 cm and at the same height as the spectral tube. The grating

must be aligned so as to be parallel to the scale. The luminous capillary tube is observed through the

grating. The room is darkened to the point where it is still possible to read the scale. The distance 2 l

between spectral lines of the same color in the right and left first order spectra are read without moving

one‘s head.

2. The distance d between the scale and the grating is also measured. Three lines are clearly visible in the

Hg spectrum.

3. The grating constant ‗g‘ is determined by means of the wavelengths given in Table 1.

4. The energy levels in hydrogen, is determined from the measured wavelengths by means of Balmer‘s

formula.


5. The data can be sorted in a table as below:

N Colour l1 (cm) l2 (cm) Lav (cm) λ (A)

Equ 2.

Then, plot a diagram between {1/(nf2-ni

2)} and 1/λ (equ. 6).

Determine the Rydberg‘s constant.

Compare the theoretical and experimental value. Then, discuss.

Fig. 1: Experimental set-up to determine the spectral lines of the hydrogen atom.

Fig. 2: Diffraction at the grating.


1. Diffraction grating

If light of wavelength λ impinges on a grating with constant g, it is diffracted. Intensity peaks occur when

the angle of diffraction α fulfills the following condition:

Light is collected by the eye on the retina, therefore the light source is seen in the color of the observed

spectral line on the scale in the prolongation of the light beams. For the diffraction of the nth

order, the

following relation is deduced from the geometrical structure (Fig. 2):

1


In the examples given in Table 1, the average obtained for the three measurements of the grating constant is

g = 1.671 mm.

Tab. 1: Determination of the grating constant from the wavelength of the Hg spectrum

2. Hydrogen spectrum

Due to collision ionization, H2 is converted to atomic hydrogen in the spectral tube. Electrons from the H

atoms are exited to higher energy levels through collisions with electrons. When they return to lower energy

levels, the atoms emit light of frequency f given by the energy difference of the concerned states:

where h is Planck‘s constant.

Applying Bohr‘s atomic model, the energy En of a permitted electron orbit is given by:

where ε0 = 8.8542 ・ 10-34

As/Vm is the electric field constant, e = 1.6021 ・ 10-19 C is the electronic charge

and

me = 9.1091 ・ 10-31

kg is the mass of the electron at rest. The emitted light can therefore have the following

frequencies:

If the wave number N = λ-1

is used instead of the frequency f, substituting. c = λ ・ f. one obtains:

Here Rth is Rydberg‘s constant, which follows from Bohr‘s atomic model.

2

3

3

5

6


Fig. 3: Energy level diagram of the H atom.

Fig. 3 shows the energy level diagram and the spectral series of the H atom. For m →∞, one obtains the

limits of the series; the associated energy is thus the ionization energy (or the binding energy) for an electron

in the nth

permitted orbit. The binding energy can be calculated by means of the equation:

where c = 2.99795 ・ 108 m/s and h = 6.6256 ・ 10-34 J s = 4.13567 ・ 10-15 eV s. The ground state is found

to be 13.6 eV.

Tab. 2: Examples of measurements for the H spectrum (Balmer series) Distance d = 450 mm

Notices

– Next to the atomic hydrogen spectrum, the molecular H2 band spectrum may be observed if the room is

sufficiently darkened. The numerous lines, which are very close to each other, are due to the oscillations of

the molecule.

– The Hζ line is situated on the border of the visible spectral range and is too weak to be observed by simple

methods.

n = 1: Lyman series

Spectral range:

ultraviolet

n = 2: Balmer series

Spectral range:

ultraviolet till red

n = 3: Paschen series

Spectral range:

infrared

n = 4: Bracket series

Spectral range:

infrared

n = 5: Pfund series

Spectral range:

infrared


– The treatment of more complex atoms requires quantum mechanics. In this case, the energies of the states

are determined by the eigenvalues of the hamiltonian of the atom. For atoms similar to hydrogen,

calculations yield the same results as Bohr‘s atomic model.

Experiment 8

Study of Atomic spectra of two-electron systems: He

Note: in this experiment, He source is examined.

You can learn about …

Parahelium, orthohelium, exchange energy, spin, angular momentum, spin orbit interaction, singlet series,

triplet series, multiplicity, Rydberg series, selection rules, forbidden transitions, metastable state, energy

level, excitation energy.

Aim:

The spectral lines of He are examined by means of a diffraction grating. The wavelengths of the lines are

determined from the geometrical arrangement and the diffraction grating constants.

Equipment

Spectrum tube, mercury 1

Spectrum tube, helium 1

Holders for spectral tubes, 1 pair 1

Cover tube for spectral tubes 1

Connecting cord, 30 kV, l = 1000 mm 2

Object holder, 5_5 cm 1

Diffraction grating, 600 lines/mm 1

High voltage supply unit, 0-10 kV 1

Insulating support 2


Barrel base -PASS- 1

Support rod -PASS-, square, l = 400 mm 1

Right angle clamp -PASS- 3

Stand tube 1

Meter scale, demo, l = 1000 mm 1

Cursors, 1 pair 1

Measuring tape, l = 2 m 1

Tasks

1. Determination of the wavelengths of the most intense spectral lines of He.


1. The experimental set-up is shown in Fig. 1. Helium or mercury spectral tubes connected to the high

voltage power supply unit are used as a source of radiation. The power supply is adjusted to about 5 kV.

The scale is attached directly behind the spectral tube in order to minimize parallax errors. The diffraction

grating should be set up at about 50 cm and at the same height as the spectral tube. The grating must be

aligned so as to be parallel to the scale.

The luminous capillary tube is observed through the grating. The room is darkened to the point where it is

still possible to read the scale. The distance 2l between spectral lines of the same color in the right and left

first order spectra are read without moving one‘s head.

2. The distance ‗d‘ between the scale and the grating is also measured.

The individual lines (first order) of the spectral lamp are observed by means of the grating and the distance

2l between equal lines is determined with the metre scale.


3. The grating constant is determined.

4. The data can be sorted in a table as below:

colour l1 (cm) l2 (cm) Lav (cm) λ (A)

5. λ is determined by using the equ. 2.

Fig.1: Experimental set up for measuring the spectra of He


1. If light of wavelength λ falls on a grating having a grating constant k, it is diffracted. Intensity maxima

occur if the angle of diffraction which satisfies the condition:

From Fig. 2. we have:

and hence

for the first-order diffraction.

2. Excitation of the He and Hg atoms results from electron impact. The energy difference produced when

electrons revert from the excited state E1 to the ground state E0 is emitted as a photon with a frequency ƒ.

where h = Planck‘s constant = 6.63 ・ 10-34 joule-seconds.

1

2


The Hamiltonian operator (non-relativistic) for the two electrons 1 and 2 of the He atom is:

m and e represent the mass and charge of the electron respectively,

Fig. 2: Diffraction of light of wavelength l at the grating.

is the Laplace operator, and ri is the position of the i-th electron. The spin-orbit interaction energy

was ignored in the case of the nuclear charge Z = 2 of helium, because it is small when Z is small.

electron-electron interaction term, then the eigenvalues of the Hamiltonian operator without interaction are

those of the hydrogen atom:

n, m = 1, 2, 3, … .

As the transition probability for simultaneous two-electron excitation is very much less than that for one-

elecron excitation, the energy spectrum of the system without interaction is:


The interaction term remores out the angular momentum degeneracy of the pure hydrogen spectrum and the

exchange energy degeneracy. There results an energy adjustment:

in which are the antisymmetricated non-interacted 2-particle wave functions with symmetrical or

antisymmetrical position component, l* is the angular momentum quantum number, and α is the set of

the other quantum numbers required.

In the present case, the orbital angular momentum of the single electron l is equal to the total angular

momentum of the two electrons L, since only one-particle excitations are being considered and the second

electron remains in the ground state (l = 0).

Cnl and Anl are the coulomb and exchange energy respectively. They are positive. Coupling the orbital

angular momentum L with the total spin S produces for S = 0, i. e. ɸ+, a singlet series and for S = 1, i. e. ɸ

-, a

tirplet series. Because of the lack of spin-orbit interaction, splitting within a triplet is slight. As the disturbed

wave functions are eigenfunctions for S2 and as S

2 interchanges with the dipole operator, the selection rule

(which is characteristic for 2-electron systems with a low nuclear-charge number) results and forbids

transitions between the triplet and singlet levels.

In addition, independent of the spin-orbit interaction, the selection rule for the total angular momentum

Fig. 3: Helium spectrum. Fig. 4: Spectrum of mercury.


applies except where

If the spin-orbit interaction is slight, then

applies.

Detailed calculations produce the helium spectrum of Fig. 3.

The following table gives the measured lines:

Table 1: Measured spectral lines of He and the corresponding energy-level transitions.

The magnitudes of the exchange interaction and the Coulomb interaction of the two electrons can be

estimated by comparing the energies of the transitions:

3. Hg, likewise, is a 2-electron system and possesses the structure of 2 series.

The spin-orbit interaction, however, is relatively pronounced so that only the total angular momentum

is a ―good‖ conservation parameter. Splitting inside the triplet is pronounced.

Moreover, the selection principle

non longer applies since S is no longer a ―good‖ conservation parameter (transition from L–S for the j–j

coupling).

The table below gives the lines obtained by experiment:

Table 2: Measured spectral lines of Hg and the corresponding energy-level transitions.

Literature:

G. Herzberg, Atomic Spectra and Atomic Structure (DoverPubl.);

D.R. Bates, Quantum Theory II (Academic Press Inc.).


Experiment 9

Elementary charge and Millikan experiment

What will you learn about.

Electric field, Viscosity, Stokes‘ law, Droplet method and Electron charge.

Aim:

Charged oil droplets subjected to an electric field and to gravity between the plates of a capacitor are

accelerated by application of a voltage. The elementary charge is determined from the velocities in the

direction of gravity and in the opposite direction.

Equipment

Millikan apparatus 1

Multi-range meter w. overl. prot1

Power supply, 0…600 VDC 1

Stage micrometer, 1 mm - 100 div. 1

Stop watch, interruption type 2

Cover glasses 18_18 mm, 50 pcs. 1

Commutator switch 1


Stand tube 1

Connecting cord, l = 250 mm, black 1



Connecting cord, l = 750 mm, black 3

Optional accessories:

Radioactive source, Am-241, 74 kBq 1

Circular level 02122.00 1

FlexCam Scientific Pro II

Tasks

1. Measurement of the rise and fall times of oil droplets with various charges at different voltages.

2. Determination of the radii and the charge of the droplets.


1. The experimental set up is as shown in Fig. 1. The power unit supplies the necessary voltages for the

Millikan apparatus. The lighting system is connected to the 6.3 V a.c. sockets. First calibrate the eyepiece

micrometer with a stage micrometer. By connecting the fixed (300 V d.c.) and the variable (0 to 300 V

d.c.) outputs in series, a voltage supply of more than 300 V d.c. can be obtained. The commutator switch

will be used to invert the polarity of the capacitor.

Fig. 1: Experimental set up for determining the elementary

charge with the Millikan apparatus.


2. Set the capacitor voltage to a value between 300 V and 500 V.

3. Blow in the oil droplets.

4. Select an oil droplet and by operating the commutator switch move the droplet between the highest and

lowest graduations on the eyepiece micrometer. Correct the focusing of the microscope if necessary.

Note: the following criteria when selecting the droplet:

– The droplet must not move too fast, and then it has a small charge (it should need ca. 1…3 s for the way of

30 div.)

– The droplet must not move too slowly and should not exhibit any sqaying movements. Increase the

capacitor voltage if required.

4. Sum together some rise times using the first stopwatch.

5. Sum together some fall times using the second stopwatch.

Note: only one stopwatch can be used to operate the experiment.

6. The added times should be greater than 5 s in both cases.

7. Determine the Q and e value respectively by equalizing the three forces (Fg, Fe and Fr) means (eq 1, 2

and 4). You have to Calculate the unknown values from the equations before be able to determine the Q

value.


The falling and rising movement of a charged oil droplet in the electric field of the capacitor is obverserved

and the velocities are determined.

The force F experienced by a sphere of radius r and velocity 𝑣 in a viscous fluid of viscosity ɳ, is:

Stockes‘ law sometime‘s called ‗drag force‘‘

The sheric droplet of mass m, volume V and density ρ1 is located in the earth‘s gravitational field.

Force of buoyancy is given by

The Force of the electrical field is given by

From the sum of the forces affecting a charged particle, the fall and rise velocities of the droplets are

obtained.

1

2

3

4

5


Subtraction or addition of these equations gives the radius and the charge of the droplet.

Note: To find out r value,

Thus:

The apparent weight in air is the true weight minus the up thrust (which equals the weight of air displaced by the oil drop).

Calibrating of the eyepiece micrometer: Scale with 30 div. = 0.89 mm (means crossed

Distance by the droplet must be multiplied by 0.89 to extract its value in mm).

The measured falling and rising times of 20 droplets are given in table 1.

Fig. 2 shows that the charge of the droplets have certain values which are multiples of the elementary charge

‗e‘

As a mean value, the elementary charge is obtained as

Fig. 2: Measurements on various droplets for determining the elementary charge by the Millikan method.

6

7

8

http://en.wikipedia.org/wiki/Upthrust


Table 1: Measurements on various droplets for determining the elementary charge by the Millikan method.

t1 and t2 are the fall and rise times of the droplets.

Alteration of the charge

Using a radioactive source (e.g. Am-241, 74 kBq) the charge of the oil droplets in the capacitor chamber can

be altered. The radioactive source has to be positioned in front of the mica window of the Millikan Unit

which is transparent for α particles.

Observation with Video camera

A video camera, which can be used in place of the eye (not used in our experiment), is employed for the

demonstration of the movement of the droplet. The time measurements of the moving droplet becomes much

easier, and will even be more accurate, due to the better visibility. The intensity of the light from the

illumination device is sufficient for observation with a video camera.

Experiment 10

Zeeman Effect

You learn about....

Bohr‘s atomic model, quantisation of energy levels, electron spin, Bohr‘s magneton, interference of electromagnetic

waves, Fabry-Perot interferometer.

Aim:

The normal Zeeman Effect is studied using a cadmium spectral lamp as a specimen, Also, Determining the Bohr‘s

magneton.

Principle:

The “Zeeman effect” is the splitting up of the spectral lines of atoms within a magnetic field. The simplest is the

splitting up of one spectral line into three components called the “normal Zeeman effect”. The normal Zeeman

effect is studied using a cadmium spectral lamp as a specimen. The cadmium lamp is submitted to different magnetic

flux densities and the splitting up of the red cadmium line (643.8 nm) is investigated using a Fabry-Perot

interferometer. The evaluation of the results leads to a fairly precise value for Bohr‘s magneton.


Equipment

Fabry-Perot interferometer 1

Cadmium lamp for Zeeman effect 1

Electromagnet without pole shoes 1

Pole pieces, drilled, conical 1

Rotating table for heavy loads 1

Power supply for spectral lamps 1

Variable transformer, 25 V AC/20 V DC, 12 A 1

Capacitor, electrolytic, 22000 mF 1


Optical profile-bench, l = 1000 mm 1

Base for opt. profile-bench, adjust. 2

Slide mount for opt. profile-bench, h = 30 mm 5

Slide mount for opt. profile-bench, h = 80 mm* 2

Lens holder 4

Lens, mounted, f = +50 mm 2

Lens, mounted, f = +300 mm 1

Iris diaphragm 1

Polarising filter, on stem 1

Polarization specimen, mica 1

Connecting cord, l = 25 cm, 32 A, red 1

Connecting cord, l = 25 cm, 32 A, blue 1






CDC-Camera for PC

incl. measurement software* 88037.00 1

PC with USB interface, Windows 98SE/Windows

Me/Windows

2000/Windows XP

*Alternative to CCD-Camera incl. measurement

software, two slide mounts, h = 80 mm for classical

version of the Zeeman Effect:

Slide mount for optical profile-bench, 1

Sliding device, horizontal 1

Swinging arm 1

Plate holder with tension spring 1

Screen, with aperture and scale 1

Slide mount for opt. profile-bench, h = 80 mm 1

Tasks

1. Using the Fabry-Perot interferometer, a self-made telescope, a CCD-camera and measurement software, the

splitting up of the central line into two s-lines is measured in wave numbers as a function of the magnetic flux density.

In the classical version where the CCD-Camera is not available, a screen with scale and a sliding device are

used to measure the splitting.

2. From the results of point 1. a value for Bohr‘s magneton is evaluated.

3. The light emitted within the direction of the magnetic field is qualitatively investigated.

Fig.1: Experimental set-up for the Zeeman effect.

Set-up

-The electromagnet is put on the rotating table for heavy loads and mounted with the two pole-shoes with holes in

such a way that a gap large enough for the Cd-lamp (9-11 mm) remains for the Cd-lamp. The pole-shoes have to be

well tightened in such a way that they cannot move later on when the magnetic flux is established. The Cd-lamp is

inserted into the gap without touching the pole-shoes and connected to the power supply for spectral lamps.


-The coils of the electromagnet are connected in parallel and via an ammeter connected to the variable power

supply of up to 20 VDC, 12 A. A capacitor of 22000 mF is in parallel to the power output to smoothen the DC-

voltage.

The optical bench for investigation of the line splitting carries the following elements (their approximate position in

cm is given in brackets):

The iris diaphragm is eliminated for initial adjustment and for the observation of the longitudinal Zeeman effect.

During observation of the transverse Zeeman effect the iris diaphragm is illuminated by the Cd-lamp and such it acts

as the light source. The lens L1 and a lens of f = 100 mm, incorporated in the etalon, create a nearly parallel light

beam which the Fabry-Perot etalon needs for a proper interference pattern.

The etalon contains a removable colour filter that lets the red cadmium line at 643.8 nm pass. The lens L2 produces an

interference pattern of rings which can be observed through L3. The ring diameters can be measured using the CCD-

camera and the software supplied with it. In the classical version the interference pattern is produced within

the plane of the screen with a scale mounted on a slide mount which can laterally be displaced with a precision of

1/100th of a millimeter. The measurement here can be done for instance, by systematic displacement of the slash

representing the „0― of the scale.

Fig.1b: Set-up for the classical version of the experiment.

-The initial adjustment is done in the following way: The rotating table with electromagnet, pole-shoes and Cd lamp

already mounted is adjusted so that the canter of the holes in the pole-shoes lies about 28 cm above the table. The

optical bench with all elements (except iris diaphragm and CCD-camera) mounted, is then moved closer to the

electromagnet in such a way that one of the outlet holes of the poleshoes coincides with the previous position of the

iris diaphragm. L1 is then adjusted so that the outlet hole is within the focal plane of it. All other optical elements of

Fig. 2. are subsequently readjusted with respect to their height correspondingly. The current of the coils is set for some

time to 8 A (increase in light intensity of the Cd-lamp !) and the ring interference pattern in axial direction is observed

through L3 by the eye. The pattern must be cantered and sharp which is eventually achieved by a last, slight

movement of the etalon (to the right or to the left) and by displacement of L2 (vertically and horizontally).


Fig. 2: Arrangement of the optical components.

Finally the CCD-camera with the 8 mm lens attached is mounted to the optical bench and adjusted in horizontal and

vertical position as well as in tilt and focus until a clear picture of the ring pattern is visible on the computer screen.

For installation and use of the camera and software please refer to the manual supplied with the camera.

In the classical version the screen with scale is shifted in a way that the slash representing the „0― of the scale is

clearly seen coinciding, for instance, with the centre of the fairly bright inner ring. The scale itself must be able to

move horizontally along the diameter of the ring pattern. (set-up see Fig. 1b)

Hint: best results are achieved when the experiment is carried out in a darkened room.

-The electromagnet is now turned by 90°, the iris diaphragm is inserted and the analyzer turned until the π-line

(explanation follows) disappears completely and the two ϭ-lines appear clearly visible.

Remark: For later evaluations the calibration curve of the magnetic flux density versus the coil current has to

be traced previously. This can be done if a teslameter is available. Otherwise the results of Fig. 3 must be used.

The curve of Fig. 3 was traced by measuring the flux density in the canter of the gap in the absence of the Cd-lamp.

For the evaluations these centre-values were increased by 3.5% to account for the non-uniform flux distribution within

the gap.

Measurement and Evaluation

Provided the ring pattern has been properly established as explained in the section ―set-up‖ above, the radii of the

rings have to be measured at different magnetic flux densities. Then it is possible by using equation (10) to determine

the corresponding difference in wave numbers Δν. We proceed in two steps: first we take pictures of the ring patterns

at different coil currents/magnetic field intensities. Then in a second step the ring diameters in these pictures are

measured.

The above procedure is repeated using different magnetic fields for instance, with coil currents of 5 A (or maybe less),

6 A, 8 A and 10 A. Once these pictures have been collected,

The slash of the scale „0― is shifted horizontally along a diameter through the ring pattern until it coincides, for

instance, with the fourth ring to the left. A magnetic field corresponding to a coil current of lets say 4 A is established

and the splitting of the rings observed. The analyzer is put into the vertical position so that only the two ϭ-lines appear.

The „0― slash is now adjusted to coincide perfectly with the outer ring of the two rings, into which the fourth ring has

split. The first reading on the socket of the sliding mount is taken. The „0― slash is then moved from left to right

through all the rings. The last reading is taken when the „0― slash coincides with the outer ring of the fourth ring to the

right.

The etalon spacing is t = 3 ・ 10-3 [m].


Equation (10) was used to calculate the difference in wave numbers of the two ϭ-lines as a function of the magnetic

flux density and the coil current respectively. The following table summarizes the results:

how to collect data: 1. Before applying the magnetic field, take a record as the table below:

Ϭ1 Ϭ2 Δ= Ϭ2- Ϭ1

Original shells prier applying

magnetic field-sub. In eq.10

2. Take also records while magnetic field is applying, as the table below:

I amp B(mT) d1 d2 δ= d2- d1 𝛥𝑣 (m-1) 1 After applying magnetic fields Eq.10

2

3

4

Some experimental results, taken for the used set up.


Fig. 3: Magnetic flux density B in the centre of gap without the Cd-lamp (gap width: 9 mm) as a function of twice the

coil current.

Theory

As early as 1862, Faraday investigated whether the spectrum of coloured flames changes under the influence of a

magnetic field, but without success. It was not until 1885 that Fievez from Belgium was able to demonstrate an effect,

but it was forgotten and only rediscovered 11 years later by the Dutchman Zeeman, who studied it together with

Lorentz.

This experiment, which was of importance to the development of the theory of the atomic shell, can now be carried

out with modern equipment in the students‘ experiment laboratory. The splitting of the Cd-spectral line λ = 643.8 nm

into three lines, the so-called Lorentz triplets, occurs since the Cd-atom represents a singlet system of total spin S

= 0. In the absence of a magnetic field there is only one possible D → P transition of 643.8 nm, as indicated by Fig. 4.

In the presence of a magnetic field the associated energy levels split into 2 L + 1 components. Radiating transitions

between these components are possible, provided that the selection rules

are taken into account. In this case, therefore, there are a total of nine permitted transitions. These nine transitions can

be grouped into three groups of three transitions each, where all transitions in a group have the same energy and hence

the same wavelength. Therefore, only three lines will be visible.


Fig. 4: Splitting up of the components in the magnetic field and permitted transitions.

The first group where ΔML = –1 gives a ϭ-line the light of which is polarized vertically to the magnetic field. The

middle group ΔML = 0 gives a π-line. This light is polarized parallel to the direction of the field. The last group where

ΔML = +1 gives a ϭ-line the light of which is again polarized vertically to the magnetic field.

In the absence of the analyser all three lines can be seen simultaneously. Each ring which was observed in the absence

of a magnetic field is split into three rings when a magnetic field is applied. Inserting the analyser the two ϭ-lines can

be observed exclusively if the analyser is in the vertical position, while only the π-line appears if the analyser is turned

into its horizontal position (transverse Zeeman effect). Turning the electromagnet by 90° the light coming from the

spectral lamp parallel to the direction of the field can also be studied since the pole-shoes have been drilled. It can be

shown that this light is circular polarized light. Whatever the position of the analyser may be, each of the rings seen

without a magnetic field is now permanently split into two rings in the presence of a magnetic field (longitudinal

Zeeman effect). Fig. 5 summarizes the facts.

Fig. 5: Longitudinal and transverse Zeeman effect.

Turning the electromagnet back for the observation of the two ϭ-lines of the transverse Zeeman effect it is easy to see

that the size of the splitting increases with increasing magnetic field strength. For a quantitative measurement of this

splitting in terms of number of wavelengths, a Fabry-Perot interferometer is used, the functioning of which may


briefly be explained. The Fabry-Perot etalon has a resolution of approximately 300000. That means that a wavelength

change of approximately 0.002 nm can still be detected.

Fig. 6: Reflected and transmitted rays at the parallel

surface (1) and (2) of the etalon. The etalon spacing is t.

The etalon consists of two parallel flat glass plates coated on the inner surface with a partially reflecting layer. Let us

consider the two partially transmitting surfaces (1) and (2) in Fig. 6 separated by a distance t. An incoming ray

forming an angle 6 with the normal to the plates will be split into the rays AB, CD, EF, etc. the path difference

between the wave fronts of two adjacent rays (for example, AB and CD) is

where, obviously, BK is normal to CD. With

we obtain

and for a constructive interference to occur one must demand:

where n is an integer. If the refractive index of the medium between the plates is μ#1, the equation still has to be

modified in the following way:

Equation (1) is the basic interferometer equation. Let the parallel rays B, D, F, etc. be brought to a focus by the use of

a lens of focal length ‗f‘ as shown in Fig. 7.

Light entering the etalon at an angle θ is focused onto a ring of radius r = ƒ θ where ƒ is the focal length of the lens.

Then, when θ fulfils equation (1), bright rings will appear in

the focal plane, their radius being given by

1

2

Fig. 7: Focusing of the light emerging from a Fabry-Perot etalon.


For small values θn, e.g. rays nearly parallel to the optical axis.

We can obtain:

If θn is to correspond to a bright fringe, n must be an integer. However, n0, which gives the interference at the centre

(cos θ = 1 or θ = 0 in equation [1] ), is in general not an integer. If n1 is the interference order of the first ring, clearly

n1 < n0 since n1 = n0 cos θnt. We then let

where n1 is the closest integer to n0 (smaller than N0). Thus, we have in general for the p-th ring of the pattern, as

measured from the centre out,

Combining equation (4) with equations (2) and (3), we obtain for the radii of the rings, substituting rp for rnp,

we note that the difference between the squares of the radii of adjacent rings is a constant:

ε can be determined graphically plotting r

2 p versus p and extrapolating to r

2 p = 0. Now, if there are two components of

a spectral line (splitting of one central line into two components) with wavelengths λa and λb, which are very close to

one another, they will have fractional orders at the centre εa and εb:

where n1,a, n1,b is the interference order of the first ring. Hence, if the rings do not overlap by a whole order n1,a = n1,b

and the difference in wave numbers between the two components is

4

5

6


Furthermore, using equations (5) and (6), we get

Applying equation (8) to the components a and b, yields

By substituting these fractional orders into equation (7), we get for the difference of the wave numbers:

From equation (6) it is clear that the difference between the squares of the radii of component a,

is equal to (within a very small part) the same difference for component b

Hence,

whatever the value of p may be. Similarly, all values

must be equal, regardless of p and their average can be taken as may be done for the different Δ-values. With δ and Δ

as average values we get for the difference of the wave numbers of the components a and b, anticipating µ = 1,

Equation (10)

* gives evidence of the fact that Δ𝑣 > does not depend on the dimensions used in measuring the radii of

the ring system nor on the amplification of the interference pattern.

7

8

9

10

Documents

Experiments in Modern Physics Lab. · Elementary charge and Millikan experiment 10. Zeeman Effect . 3 Experiments in modern physics | Physics Dept. Experiment 1 Microwave Wavelength