PC301 Lab Manual

Terry Sturtevant

January 2, 2002

Contents

1 Determination of Planck’s Constant Using the PhotoelectricEffect 1-11.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

2 PN Junction Characteristics 2-12.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-12.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-1

3 Conductivity of N and P Type Germanium 3-13.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1

3.1.1 Room Temperature Conductivity . . . . . . . . . . . . 3-13.1.2 Conductivity as a Function of Temperature . . . . . . 3-13.1.3 The Hall Effect . . . . . . . . . . . . . . . . . . . . . . 3-1

3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-23.2.1 Conductivity . . . . . . . . . . . . . . . . . . . . . . . 3-23.2.2 Temperature Effects . . . . . . . . . . . . . . . . . . . 3-23.2.3 Extrinsic Range . . . . . . . . . . . . . . . . . . . . . . 3-33.2.4 Intrinsic Range . . . . . . . . . . . . . . . . . . . . . . 3-33.2.5 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 3-3

4 The Common Emitter Amplifier 4-14.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-1

5 The Common Source JFET Amplifier 5-15.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-1

6 Oscillator Circuits 6-16.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-1

iv CONTENTS

6.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-16.2.1 Transistor Oscillators . . . . . . . . . . . . . . . . . . . 6-16.2.2 Operational Amplifier Oscillators . . . . . . . . . . . . 6-46.2.3 Digital Oscillators . . . . . . . . . . . . . . . . . . . . . 6-8

7 Photon Counting and Spectroscopy in the Gamma–Ray En-ergy Range 7-17.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-17.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-1

7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 7-17.2.2 Detection . . . . . . . . . . . . . . . . . . . . . . . . . 7-27.2.3 Start–up . . . . . . . . . . . . . . . . . . . . . . . . . . 7-47.2.4 Pulse Counting . . . . . . . . . . . . . . . . . . . . . . 7-47.2.5 Counting Statistics . . . . . . . . . . . . . . . . . . . . 7-47.2.6 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . . 7-5

8 Determination of Planck’s Constant Using LEDs 8-18.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-1

List of Figures

3.1 Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4

6.1 Phase Shift Oscillator . . . . . . . . . . . . . . . . . . . . . . . 6-26.2 Crystal Equivalent Circuit . . . . . . . . . . . . . . . . . . . . 6-36.3 Crystal Oscillator Circuit . . . . . . . . . . . . . . . . . . . . . 6-46.4 Operational Amplifier Phase Shift Oscillator . . . . . . . . . . 6-56.5 Operational Amplifier Wein Bridge Oscillator . . . . . . . . . 6-66.6 Square Wave Oscillator . . . . . . . . . . . . . . . . . . . . . . 6-76.7 TTL Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 6-9

vi LIST OF FIGURES

List of Tables

viii LIST OF TABLES

Chapter 1

Determination of Planck’sConstant Using thePhotoelectric Effect

1.1 Objective

Determine Planck’s constant, h, and W , the work function of the photosen-sitive material using the photoelectric effect apparatus in the lab. Determinewhether the values you obtained are reasonable. Also, determine the cutofffrequency of the glass surrounding the photosensitive material.

Familiarize yourself with the apparatus and make sure you know how itworks before you begin.

You will have available a series of light sources, interference filters andcut-off filters. Because of the different transmission characteristics of theinterference filters and the cut-off filters, it will often be necessary to usethem together.

1-2Determination of Planck’s Constant Using the Photoelectric

Effect

Chapter 2

PN Junction Characteristics

2.1 Objective

Experimentally determine the forward biased current–voltage characteristicof the diode at room temperature. From your data and Equation 2.2 below,determine I0 and η. Explain your analysis.

Collect data for voltage as a function of temperature for constant cur-rent, and from this, determine the energy gap of the semiconductor (in eV).Explain your analysis in detail. Compare your result with the accepted valueof Eg for either silicon or germanium and suggest which material you thinkmakes up your diode. Show the derivation of Equation 2.5 below. Use yourexperimentally determined values of η and Eg to predict the sensitivity of acurrent biased diode to temperature shifts. Express your answer in mV/Kfor the case when V = 0.7 volts at room temperature and discuss whetheror not this is consistent with your data.

2.2 Background

The standard textbook treatment of the electrical behaviour of a p–n junctionresults in a current–voltage characteristic of a diode given by:

I = I0

(eeV/kT − 1

)(2.1)

where e is the electronic charge, k is Boltzmann’s constant, T is the absolutetemperature in Kelvins and I0 is the reverse saturation current. Real semi-conductor diodes deviate from Equation 2.1 because of carrier generation

2-2 PN Junction Characteristics

and recombination within the depletion region. By introducing an empiricalparameter η, agreement with experiment can be improved, and we have theequation

I = I0

(eeV/ηkT − 1

)(2.2)

The reverse saturation current I0 is given by:

I0 = BT 3e−Eg/ηkT (2.3)

where Eg is the energy gap of the semiconductor material and B is constant(independent of temperature).

From Equations 2.2 and 2.3, show that, for eV kT ,

V ≈ Ege− ηkT

eln

(BT 3

I

)(2.4)

Equation 2.4 indicates that, within the temperature range of 273K to 373K,the diode voltage V varies almost linearly with temperature, provided thebias current remains constant. The logarithmic term in Equation 2.4 variesslowly with temperature in such a situation. Assuming that Eg is indepen-dent of temperature, it can be shown from Equation 2.4 that:(

dV

dT

)I=const.

= −(

3ηk

e+EgeT− V

T

)(2.5)

Chapter 3

Conductivity of N and P TypeGermanium

3.1 Objective

3.1.1 Room Temperature Conductivity

Determine the room temperature resistivity ρ = 1/σ of the two samplesmarked A and B, and verify that the contacts to the semiconductor areohmic. Familiarize yourself with the apparatus before you begin. Make sureyou know current and voltage limits, etc. so you do not damage anything.

3.1.2 Conductivity as a Function of Temperature

Determine the variation of conductivity in a doped semiconductor over thetemperature range 0C to 125C, and relate these results to the energy gapof the semiconductor.

Compare your values for the energy gap of samples A and B with theaccepted value for Germanium.

Make sure you know current and voltage limits, etc. so you do not damageanything.

3.1.3 The Hall Effect

Use the Hall effect and previous results to determine the mobility and carrierconcentrations for both samples.

3-2 Conductivity of N and P Type Germanium

Familiarize yourself with the apparatus before you begin. Make sure youknow current and voltage limits, etc. so you do not damage anything.

3.2 Background

3.2.1 Conductivity

For a sample of n-type semiconductor, the electrical conductivity σ is givenby

σ = neeµe (3.1)

where ne is the density of carriers (electrons), e is the charge of an electron,and µe is the mobility of the electrons. For a p-type material, the conductivityis given by

σ = nheµh (3.2)

where nh is the density of carriers (holes) and µh is the hole mobility.

3.2.2 Temperature Effects

1. At low temperatures, most of the donors (for n-type) or acceptors (p-type) in a semiconductor are un-ionised. In this region any increase intemperature will ionise additional impurities, thus causing the densityof carriers to quickly increase.

2. Above some particular temperature, virtually all of the donor and/oracceptors will become ionised. A temperature range exists in which thecarrier density remains virtually constant, and is determined strictly bythe doping levels.

3. As the temperature increases still further, the mean thermal energy(kT ) becomes comparable in magnitude to the energy gap of the (in-trinsic) semiconductor material. Significant numbers of electron-holepairs are thus created.

In this experiment, the latter two regions will be explored. These twodomains are referred to as extrinsic and intrinsic, respectively.

Recall that conductivity is equal to the product of three factors: charge,carrier density, and mobility. The temperature dependence of the mobilitymust also be accounted for in σ(T ).

3.2 Background 3-3

3.2.3 Extrinsic Range

In this domain, impurity scattering of carriers can be shown to lead to arelationship of

µ ∝ T+3/2

Since the carrier density is essentially independent of temperature in thisrange, the conductivity will have a temperature dependence of:

σ ∝ T+3/2 (3.3)

3.2.4 Intrinsic Range

In this domain, lattice scattering is the dominant process affecting the carriermobility, with the result that

µ ∝ T−3/2

However, the carrier density has a temperature dependence of the form:

ni = pi ∝ T 3/2e−Eg/2kT

where Eg is the semiconductor energy gap, k is Boltzmann’s constant and Tis the absolute temperature in Kelvins. Combining these equations leads usto predict a temperature dependence of the conductivity given by

σ ∝ e−Eg/2kT (3.4)

3.2.5 Hall Effect

When a semiconductor is subjected to a magnetic field oriented along aperpendicular to the face of the sample holder and a bias current is appliedas in the previous experiments, the Hall voltage will be measured acrossthe sample; see Figure 3.1.

From the theory of the Hall Effect, it may be shown that the Hall voltage,VH , is given by:

VH =

− 1nee

BzIxt

for electrons

+ 1nhe

BzIxt

for holes

3-4 Conductivity of N and P Type Germanium

Figure 3.1: Hall Effect

where ne (nh) is the carrier density for electrons (holes), Ix is the bias cur-rent, t is the sample thickness along the magnetic field direction, e is theelectronic charge, and Bz is the applied magnetic field(in the z direction).Obviously the polarity of the Hall voltage is a direct indication of the type ofcarrier (electrons or holes) in an unknown semiconductor. In addition, theseequations imply that the carrier density can be determined if VH , Ix, Bz andt are known.

A quantum mechanical calculation of the Hall voltage for Germaniumintroduces a refinement in the above equations, namely:

VH =

−0.93nee

BzIxt

for electrons

+1.40nhe

BzIxt

for holes

Chapter 4

The Common EmitterAmplifier

4.1 Objective

Determine the characteristic curves of a 2N3904 bipolar junction transis-tor at the operating point suggested by the manufacturer, and then designa single stage amplifier using a bipolar junction transistor in the commonemitter configuration. Measure the amplifier parameters and see how theyfit with your predictions. Simulate the amplifier in PSPICE and see if thesimulated behaviour matches what is observed. (See page 98-112, 164-170 ofFundamentals of Linear Circuits by Floyd.)

4-2 The Common Emitter Amplifier

Chapter 5

The Common Source JFETAmplifier

5.1 Objective

Determine the characteristic curves of a junction field effect transistor atthe operating point suggested by the manufacturer, and then design a singlestage amplifier using a 2N3819 JFET in the common source configuration.Measure the amplifier parameters and see how they fit with your predictions.Simulate the amplifier in PSPICE and see if the simulated behaviour matcheswhat is observed. (See page 117-124, 127-129, 176-180 of Fundamentals ofLinear Circuits by Floyd.)

5-2 The Common Source JFET Amplifier

Chapter 6

Oscillator Circuits

6.1 Objective

Choose an oscillator circuit, analyze its behaviour (including performing asimulation), build the circuit and determine whether its performance fitswhat you expect.

6.2 Background

It can be shown that the voltage gain of a feedback amplifier is given by

A′ =A

1− βA(6.1)

where A is the open loop gain of the amplifier. If a circuit is designed usingpositive feedback, so that

Aβ = 1 (6.2)

the closed loop gain would be infinite. This condition gives rise to self-sustaining oscillations. Equation 6.2 is called the Barkhausen criterion,and is met when the overall phase shift of the feedback is 360.

6.2.1 Transistor Oscillators

Phase Shift Oscillator

Figure 6.1 shows the circuit for a phase shift oscillator, in which the feedbackcircuit employs three cascaded RC sections to shift the phase by 180. An

6-2 Oscillator Circuits

Figure 6.1: Phase Shift Oscillator

additional shift of 180 is obtained by taking the feedback from the collec-tor of the transistor. Ignoring loading effects, β can be calculated over thefeedback network, and is given by:

β =ViVo

=1

1− 5/(ωRC)2 + j(1/(ωRC)3 − 6/ωRC)(6.3)

For a phase shift of 180, the imaginary part is zero, which leads to

ω0 =1√

6RC(6.4)

Then

β = − 1

29

and the gain required by the Barkhausen criterion is

A =1

β= −29

6.2 Background 6-3

Figure 6.2: Crystal Equivalent Circuit

Crystal Oscillator

You can see that the previous circuit lacks precision. Another way to designan oscillator is to set up a resonant circuit in the feedback network. In orderto increase the precision, the quality factor, Q, of the resonant circuit mustbe large.

High Q factors can be achieved by using a piezo-electric crystal, usuallymade of quartz. Piezo-electric means that when the crystal is put undermechanical stress across its faces, a potential difference will be developedbetween the opposite faces. Conversely, if a potential difference is appliedbetween the faces, the crystal will distort. Most crystalline materials willvibrate at a natural resonant frequency. If we apply an AC voltage to thequartz crystal, it will resonate and produce an electrical resonance that canbe amplified. Figure 6.2 shows the symbol and the electrical equivalent of acrystal. Crystal losses, represented by R, are small and therefore Q is large— on the order of 20,000 and higher. Note that the circuit equivalent showsthat the crystal has series and parallel resonance.

6-4 Oscillator Circuits

!"

Figure 6.3: Crystal Oscillator Circuit

6.2.2 Operational Amplifier Oscillators

Since operational amplifiers have almost infinite gain and infinite input impedance,they are ideal for use in oscillator circuits. Since the open loop gain A is al-most infinite, Equation 6.1 reduces to

A′ = − 1

β(6.5)

Therefore, if we use two feedback networks, where A′ sets the gain to −1/β,the condition for oscillation is met.

Phase Shift Oscillator

Figure 6.4 shows the circuit for a phase shift oscillator, using an op ampinstead of a transistor. Note the similarity to the corresponding transistoroscillator.

6.2 Background 6-5

!"#

Figure 6.4: Operational Amplifier Phase Shift Oscillator

6-6 Oscillator Circuits

Figure 6.5: Operational Amplifier Wein Bridge Oscillator

6.2 Background 6-7

!"#

Figure 6.6: Square Wave Oscillator

The Wein Bridge Oscillator

With reference to Figure 6.5, at a frequency of

f0 =1

2πRC(6.6)

the feedback ratio is

β =ViVo

=1

3(6.7)

where Vi and Vo are the voltages at the input and output of the feedbacknetwork, respectively. In order to obtain a loop gain greater than unity, themagnitude of the gain A′ must be greater than 3.

6-8 Oscillator Circuits

The Square Wave Generator

A square wave generator is shown in Figure 6.6. The feedback factor associ-ated with the circuit is

β =R2

R1 +R2

(6.8)

The output saturation levels of the op-amp, V +0sat

and V −0sat, are given in

the op amp manufacturer’s data sheet. The duration t1 for which the outputremains at V +

0satis given by

t1 = RC ln

(V +

0sat− βV −0sat

V +0sat

(1− β)

)(6.9)

and the duration t2 for which the output remains at V −0satis

t2 = RC ln

(V −0sat

− βV +0sat

V −0sat(1− β)

)(6.10)

If C = 0.1µF and R = 10KΩ, calculate t1, t2 and the output frequency forβ = 0.5.

6.2.3 Digital Oscillators

Crystal Oscillator

Many computer systems use TTL oscillators to generate required clock pulses.The circuit shown in Figure 6.7 oscillates at a frequency determined by thecrystal, which can have a value between 4MHz and 20MHz.

6.2 Background 6-9

Figure 6.7: TTL Oscillator

6-10 Oscillator Circuits

Chapter 7

Photon Counting andSpectroscopy in theGamma–Ray Energy Range

7.1 Objective

Determine whether a radioactive source or set of sources obey Poisson statis-tics, and whether the radiation is isotropic.

7.2 Background

7.2.1 Introduction

A gamma–ray photon can be emitted in an energy–level transition in anatomic nucleus, a process which is usually referred to as radioactive decay.The energy–level separations are very large so the photon is very energetic,with a typical energy on the order of 1 MeV or so compared to a few eV forvisible photons. The radiation is usually isotropic, so that

I ∝ 1

r2

where I is the intensity of radiation and r is the distance from the source.The emission is a quantum–mechanical process governed by statistical

laws with which we can predict the behaviour of a large number of radioac-tive nuclei, but individual decays are randomly distributed in time. The

7-2Photon Counting and Spectroscopy in the Gamma–Ray Energy

Range

separation of nuclear energy levels are unique to each nucleus (similar to theatomic energy levels which electrons occupy) and so the energy of the photonor photons released can be used to identify the nucleus which emitted it.

7.2.2 Detection

Any photon detection system must rely on a physical interaction between thephoton and the detector. In the gamma–ray energy range, the probability ofinteraction per unit length is orders of magnitude lower than in the visiblerange, and the detection is done through three major processes: the pho-toelectric effect, the Compton effect and pair production. With a suitablematerial, the energy transferred to the electron involved in these processescan be measured. Our main interest is in the photoelectric effect since, inthis interaction, all of the energy of the photon is deposited in the detector(there is not a range of possible energies as with the Compton effect).

In our case, the detector is an insulating crystal of NaI (sodium iodide)doped with thallium atoms. This material will scintillate when one of theinteractions mentioned above occurs in it releasing an energetic electron. Theenergy of the electron is dispersed in collisional processes which result in theemission of visible photons (these are the scintillations). One of these photonscan be easily detected by a photomultiplier tube which is an evacuated tubewith a phosphor–coated window at one end which gives off an electron whenhit by a photon (a TV set in reverse). This electron is accelerated in alarge electric field through several stages, at each of which more electronsare released by collisions with metal accelerating plates (secondary electronemission at dynodes). At the final stage (anode) there should be enoughelectrons to be detectable as a current pulse, or as a voltage rise of a fewmicrovolts on a capacitor. This pulse is amplified and the pulses can then becounted.

Normally, the various scintillation photons from one gamma–ray photonarrive at the phosphor within such a short time that they appear as part ofone pulse. The size of the pulse (voltage) is determined by:

1. the energy of the gamma–ray photon

which determines

2. the energy of the electron released in the scintillator crystal

which determines

7.2 Background 7-3

3. the number of electrons in a cascade of electrons released by collisionswith the first one

which determines

4. the number of scintillation photons produced (some of which impingeon the phosphor of the photomultiplier tube)

which determines

5. the number of primary electrons that are accelerated into the photo-multiplier tube

which determines

6. the number of electrons which finally reach the anode of the photomul-tiplier tube

which determines

7. the size of the voltage pulse that is measured on a capacitor

From this point on, the pulse is amplified in a pre–amplifier, amplifiedagain (and re–shaped) in a main amplifier, electronically evaluated to mea-sure its maximum height in an Analog–to–Digital Converter (ADC) andcategorized and counted by a Multi–Channel– Analyser (MCA). Finally,the results of this process are displayed as a frequency distribution of pulseheights.

It should come as no surprise that there is a fairly large uncertainty inthe measurement of the energy of a gamma–ray photon. The uncertainty ismainly random, however, so we can improve the measurement by doing theexperiment several times. With the electronic system provided, only a shorttime is required to measure the energy of a few million photons. This will(at least for the photoelectric effect) produce a peak in the energy spectrumwhose centroid (the mean of the energy distribution) is the best estimateof the energy of the detected photons and whose width (usually called theenergy resolution of the system) indicates the size of the random variationsin the signal processing chain described above. The upper limit of the energydistribution of a photon which undergoes a Compton interaction is “smearedout” for the same reason, and the position of the Compton edge in the energyspectrum is halfway up to the ledge that forms the Compton plateau.

7-4Photon Counting and Spectroscopy in the Gamma–Ray Energy

Range

7.2.3 Start–up

Turn on the power switch on the right side of the nuclear instrumentationpower supply and module cage (known in technotalk as a NIM bin). Thedetailed setting on the modules will already have been made for you and onlya few changes will be required as you go along. You will have to turn on thepower to the high–voltage power supply separately, although it is located inthe NIM bin also. The nearby cylindrical object is a housing containing aNaI(Tl) scintillator crystal, a photomultiplier tube, and a preamplifier. Theconnections to this detector are a high–voltage line (use great care in handlingor, better yet, do not move it at all) and a signal output. The signal runsto an amplifier where it is processed and then to the ADC. It is then fedto the MCA card in the PC and accessed by the Windows–based programMCA.EXE in the directory S100. The sophistication of this system requiressome training, so be prepared to take notes.

7.2.4 Pulse Counting

The voltage pulses that are produced by the photomultiplier tube must bedischarged through a resistor to allow the next pulse to be differentiated fromthe first. A trade–off in time constant is made here between

1. the requirement for fast discharge so that pulses that are close togetherin time can be measured as separate pulses

and

2. the requirement for large voltage signals so that pulse detection is easierand the pulse height has a larger dynamic range.

7.2.5 Counting Statistics

Each measurement of radioactive decay counts is independent of all othersuch measurements because radioactive decay is a random process. How-ever, for a large number of individual measurements, the count rates followthe Poisson statistical distribution with a well–defined mean value and stan-dard deviation. This is a special case of the binomial distribution where asimplification is achieved because the probability of success (decay of oneradioactive nucleus) is very small. The mean value is found in the usual way,

7.2 Background 7-5

and the standard deviation is the square root of the mean.

mean = variance = µ

P (x, µ) =µxe−µ

x!

This expression gives the Poisson probability that x events (counts) willbe observed in a given time interval.

Hint: To compare the observed and expected values, you will have to lookup information on the standard error of the standard deviation in astatistics textbook.

7.2.6 Isotropy

The intensity of isotropic radiation decreases with the square of the distancefrom the source. The mathematical expression for the count–rate dependencefor an isotropic source is given by

I =I0

4πr2

where I is the intensity at a separation of r and I0 is the total count rate.(Note that this is strictly only valid when the background radiation can beneglected. Also, due to the geometry of the detector, the effective source–detector distance may be different than the measured source–detector dis-tance.)

7-6Photon Counting and Spectroscopy in the Gamma–Ray Energy

Range

Chapter 8

Determination of Planck’sConstant Using LEDs

8.1 Objective

Determine Planck’s constant, h, by finding the voltage at which LEDs ofdifferent colours turn on. (This is in essence the photoelectric effect in re-verse.) You will need to know the frequency of the various LEDs in orderto find Planck’s constant. Determine whether the values you obtained arereasonable.