AP Phys BTest Review

Modern Physics

5/9/2008

Overview

Basics Photoelectric Effect Bohr Model of the atom

• Energy Transitions

Nuclear Physics

Basics

Quantization: the idea that light and matter come in discreet, indivisible packets• Wave-particle duality in light and matter

• Matter behaves both as a wave and as a particle.

Energy of a photon

Blackbody radiation• Ultraviolet catastrophe

• Planck came up with the idea that light is emitted by certain discreet resonators that emit energy packets called photons

• This energy is given by:

E h

Photoelectric Effect Schematic

When light strikes E, photoelectrons are emitted

Electrons collected at C and passing through the ammeter are a current in the circuit

C is maintained at a positive potential by the power supply

Photoelectric Current/Voltage Graph

The current increases with intensity, but reaches a saturation level for large ΔV’s

No current flows for voltages less than or equal to –ΔVs, the stopping potential

• The stopping potential is independent of the radiation intensity

Features Not Explained by Classical Physics/Wave Theory

No electrons are emitted if the incident light frequency is below some cutoff frequency that is characteristic of the material being illuminated

The maximum kinetic energy of the photoelectrons is independent of the light intensity

The maximum kinetic energy of the photoelectrons increases with increasing light frequency

Electrons are emitted from the surface almost instantaneously, even at low intensities

Einstein’s Explanation A tiny packet of light energy, called a photon, would be

emitted when a quantized oscillator jumped from one energy level to the next lower one

• Extended Planck’s idea of quantization to electromagnetic radiation

The photon’s energy would be E = hƒ Each photon can give all its energy to an electron in the

metal The maximum kinetic energy of the liberated

photoelectron is

KE = hƒ – Φ

Explanation of Classical “Problems” The effect is not observed below a certain cutoff

frequency since the photon energy must be greater than or equal to the work function

• Without this, electrons are not emitted, regardless of the intensity of the light

The maximum KE depends only on the frequency and the work function, not on the intensity

The maximum KE increases with increasing frequency

The effect is instantaneous since there is a one-to-one interaction between the photon and the electron

Verification of Einstein’s Theory

Experimental observations of a linear relationship between KE and frequency confirm Einstein’s theory

The x-intercept is the cutoff frequency

cf h

27.4 X-Rays

Electromagnetic radiation with short wavelengths• Wavelengths less than for ultraviolet

• Wavelengths are typically about 0.1 nm

• X-rays have the ability to penetrate most materials with relative ease

Discovered and named by Roentgen in 1895

Production of X-rays

X-rays are produced when high-speed electrons are suddenly slowed down• Can be caused by the

electron striking a metal target

A current in the filament causes electrons to be emitted

Production of X-rays

An electron passes near a target nucleus

The electron is deflected from its path by its attraction to the nucleus

It will emit electromagnetic radiation when it is accelerated

27.8 Photons and Electromagnetic Waves

Light has a dual nature. It exhibits both wave and particle characteristics• Applies to all electromagnetic radiation

The photoelectric effect and Compton scattering offer evidence for the particle nature of light• When light and matter interact, light behaves as if it

were composed of particles Interference and diffraction offer evidence of

the wave nature of light

28.9 Wave Properties of Particles

In 1924, Louis de Broglie postulated that because photons have wave and particle characteristics, perhaps all forms of matter have both properties

The de Broglie wavelength of a particle is

The frequency of matter waves is

mv

h

h

Eƒ

The Davisson-Germer Experiment

They scattered low-energy electrons from a nickel target

The wavelength of the electrons calculated from the diffraction data agreed with the expected de Broglie wavelength

This confirmed the wave nature of electrons Other experimenters have confirmed the wave

nature of other particles

27.10 The Wave Function

In 1926 Schrödinger proposed a wave equation that describes the manner in which matter waves change in space and time

Schrödinger’s wave equation is a key element in quantum mechanics

Schrödinger’s wave equation is generally solved for the wave function, Ψ

i Ht

The Wave Function

The wave function depends on the particle’s position and the time

The value of |Ψ|2 at some location at a given time is proportional to the probability of finding the particle at that location at that time

27.11 The Uncertainty Principle

When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements• Classical mechanics offers no fundamental

barrier to ultimate refinements in measurements

• Classical mechanics would allow for measurements with arbitrarily small uncertainties

The Uncertainty Principle Quantum mechanics predicts that a barrier to

measurements with ultimately small uncertainties does exist

In 1927 Heisenberg introduced the uncertainty principle• If a measurement of position of a particle is made

with precision Δx and a simultaneous measurement of linear momentum is made with precision Δp, then the product of the two uncertainties can never be smaller than h/4

The Uncertainty Principle

Mathematically,

It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a particle

Another form of the principle deals with energy and time:

4

hpx x

4

htE

Early Models of the Atom

Rutherford’s model

• Planetary model

• Based on results of thin foil experiments

• Positive charge is concentrated in the center of the atom, called the nucleus

• Electrons orbit the nucleus like planets orbit the sun

Experimental tests

Expect:

1. Mostly small angle scattering

2. No backward scattering events

Results:

1. Mostly small scattering events

2. Several backward scatterings!!!

Difficulties with the Rutherford Model Atoms emit certain discrete characteristic

frequencies of electromagnetic radiation

• The Rutherford model is unable to explain this phenomena

Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency

• The radius should steadily decrease as this radiation is given off

• The electron should eventually spiral into the nucleus

28.2 Emission Spectra

A gas at low pressure has a voltage applied to it

When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed• Each line has a different

wavelength and color

Emission Spectrum of Hydrogen The wavelengths of hydrogen’s spectral lines can be

found from

• RH is the Rydberg constant

• RH = 1.0973732 x 107 m-1

• n is an integer, n = 1, 2, 3, …

• The spectral lines correspond to different values of n

A.k.a. Balmer series

22H n

1

2

1R

1

Absorption Spectra

An element can also absorb light at specific wavelengths An absorption spectrum can be obtained by passing a

continuous radiation spectrum through a vapor of the gas

The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum• The dark lines of the absorption spectrum coincide with the bright

lines of the emission spectrum

28.3 The Bohr Theory of Hydrogen

In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory

His model includes both classical and non-classical ideas His model included an attempt to explain why the atom was

stable

Bohr’s Assumptions for Hydrogen The electron moves in circular orbits

around the proton under the influence of the Coulomb force of attraction

Only certain electron orbits are stable

• These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation

• Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion

Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state

• The “jump” cannot be treated classically

i fE E hf

Bohr’s Assumptions

More on the electron’s “jump”:• The frequency emitted in the “jump” is related to the

change in the atom’s energy

• It is generally not the same as the frequency of the electron’s orbital motion

The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum

i fE E hf , 1, 2,3,...2e

hm vr n n

Results The total energy of the atom

•

Newton’s law

This can be used to rewrite kinetic energy as

Thus, the energy can also be expressed as

221

2 e e

eE KE PE m v k

r

r2

ekE

2e

2 2

2e e e

e vF m a or k m

r r

2 2

2 2emv e

KE kr

Bohr Radius

The radii of the Bohr orbits are quantized

• This shows that the electron can only exist in certain allowed orbits determined by the integer n

• When n = 1, the orbit has the smallest radius, called the Bohr radius, ao

• ao = 0.0529 nm

,3,2,1nekm

nr

2ee

22

n 2h

Radii and Energy of Orbits

A general expression for the radius of any orbit in a hydrogen atom is

• rn = n2 ao

The energy of any orbit is

• En = - 13.6 eV/ n2

The lowest energy state is called the ground state

• This corresponds to n = 1

• Energy is –13.6 eV The next energy level has an energy of –

3.40 eV The ionization energy is the energy

needed to completely remove the electron from the atom

Energy Level Diagram The value of RH from Bohr’s analysis is in

excellent agreement with the experimental value

A more generalized equation can be used to find the wavelengths of any spectral lines

• For the Balmer series, nf = 2

• For the Lyman series, nf = 1 Whenever a transition occurs between a

state, ni and another state, nf (where ni > nf), a photon is emitted

2i

2f

H n

1

n

1R

1

Quantum Number Summary

The values of n can increase from 1 in integer steps The values of ℓ can range from 0 to n-1 in integer steps The values of m ℓ can range from -ℓ to ℓ in integer steps

Atomic Transitions – Energy Levels An atom may have

many possible energy levels

At ordinary temperatures, most of the atoms in a sample are in the ground state

Only photons with energies corresponding to differences between energy levels can be absorbed

Atomic Transitions – Stimulated Absorption

The blue dots represent electrons

When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level

• These higher levels are called excited states

• ΔE = hƒ = E2 – E1

Atomic Transitions – Spontaneous Emission

Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon

This process is called spontaneous emission

Atomic Transitions – Stimulated Emission An atom is in an excited

stated and a photon is incident on it

The incoming photon increases the probability that the excited atom will return to the ground state

There are two emitted photons, the incident one and the emitted one

29.1 Some Properties of Nuclei

All nuclei are composed of protons and neutrons• Exception is ordinary hydrogen with just a proton

The atomic number, Z, equals the number of protons in the nucleus

The neutron number, N, is the number of neutrons in the nucleus

The mass number, A, is the number of nucleons in the nucleus• A = Z + N• Nucleon is a generic term used to refer to either a proton or

a neutron• The mass number is not the same as the mass

Charge and mass

Charge: The electron has a single negative charge, -e (e = 1.60217733 x 10-19

C) The proton has a single positive charge, +e

• Thus, charge of a nucleus is equal to Ze The neutron has no charge

• Makes it difficult to detect

Mass: It is convenient to use atomic mass units, u, to express masses

• 1 u = 1.660559 x 10-27 kg Mass can also be expressed in MeV/c2

• 1 u = 931.494 MeV/c2

The Size of the Nucleus

First investigated by Rutherford in scattering experiments

The KE of the particle must be completely converted to PE

2

2

4 ek Zedmv

2 1 221

2 e e

e Zeq qmv k k

r d or

Size of Nucleus

Since the time of Rutherford, many other experiments have concluded the following• Most nuclei are

approximately spherical

31

oArr

Density of Nuclei The volume of the nucleus (assumed to be

spherical) is directly proportional to the total number of nucleons

This suggests that all nuclei have nearly the same density

Nucleons combine to form a nucleus as though they were tightly packed spheres

Nuclear Stability

There are very large repulsive electrostatic forces between protons• These forces should cause the nucleus to fly apart

The nuclei are stable because of the presence of another, short-range force, called the nuclear (or strong) force• This is an attractive force that acts between all nuclear particles

• The nuclear attractive force is stronger than the Coulomb repulsive force at the short ranges within the nucleus

Nuclear Stability chart Light nuclei are most

stable if N = Z Heavy nuclei are most

stable when N > Z• As the number of protons

increase, the Coulomb force increases and so more nucleons are needed to keep the nucleus stable

No nuclei are stable when Z > 83

Isotopes The nuclei of all atoms of a particular element must contain

the same number of protons They may contain varying numbers of neutrons

• Isotopes of an element have the same Z but differing N and A values

C116 C14

6C136C12

6

29.2 Binding Energy The total energy of the

bound system (the nucleus) is less than the combined energy of the separated nucleons• This difference in energy is

called the binding energy of the nucleus• It can be thought of as the

amount of energy you need to add to the nucleus to break it apart into separated protons and neutrons

Binding Energy per NucleonBinding Energy per Nucleon

Binding Energy Notes

Except for light nuclei, the binding energy is about 8 MeV per nucleon

The curve peaks in the vicinity of A = 60• Nuclei with mass numbers greater than or less than 60

are not as strongly bound as those near the middle of the periodic table

The curve is slowly varying at A > 40 • This suggests that the nuclear force saturates

• A particular nucleon can interact with only a limited number of other nucleons

29.3 Radioactivity

Radioactivity is the spontaneous emission of radiation Experiments suggested that radioactivity was the result of

the decay, or disintegration, of unstable nuclei Three types of radiation can be emitted

• Alpha particles• The particles are 4He nuclei

• Beta particles• The particles are either electrons or positrons

• A positron is the antiparticle of the electron

• It is similar to the electron except its charge is +e

• Gamma rays• The “rays” are high energy photons

Distinguishing Types of Radiation

The gamma particles carry no charge

The alpha particles are deflected upward

The beta particles are deflected downward• A positron would be

deflected upward

Penetrating Ability of Particles

Alpha particles• Barely penetrate a piece of paper

Beta particles• Can penetrate a few mm of aluminum

Gamma rays• Can penetrate several cm of lead

The Decay Constant

The number of particles that decay in a given time is proportional to the total number of particles in a radioactive sample

• λ is called the decay constant and determines the rate at which the material will decay

The decay rate or activity, R, of a sample is defined as the number of decays per second

NR N

t

N N t

Decay Curve

The decay curve follows the equation

The half-life is also a useful parameter

693.02lnT 21

0tN N e

Units

The unit of activity, R, is the Curie, Ci• 1 Ci = 3.7 x 1010 decays/second

The SI unit of activity is the Becquerel, Bq• 1 Bq = 1 decay / second

• Therefore, 1 Ci = 3.7 x 1010 Bq

The most commonly used units of activity are the mCi and the µCi

Alpha Decay

When a nucleus emits an alpha particle it loses two protons and two neutrons• N decreases by 2

• Z decreases by 2

• A decreases by 4HeYX 4

24A2Z

AZ

Beta Decay

During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is one less

In addition, an electron (positron) was observed The emission of the electron is from the nucleus

• The nucleus contains protons and neutrons

• The process occurs when a neutron is transformed into a proton and an electron

• Energy must be conserved

Beta Decay – Electron Energy The energy released in the decay process

should almost all go to kinetic energy of the electron

Experiments showed that few electrons had this amount of kinetic energy

To account for this “missing” energy, in 1930 Pauli proposed the existence of another particle

Enrico Fermi later named this particle the neutrino

Properties of the neutrino

• Zero electrical charge

• Mass much smaller than the electron, probably not zero

• Spin of ½

• Very weak interaction with matter

Gamma Decay

Gamma rays are given off when an excited nucleus “falls” to a lower energy state

• Similar to the process of electron “jumps” to lower energy states and giving off photons

The excited nuclear states result from “jumps” made by a proton or neutron The excited nuclear states may be the result of violent collision or more likely of

an alpha or beta emission Example of a decay sequence

• The first decay is a beta emission

• The second step is a gamma emission

• The C* indicates the Carbon nucleus is in an excited state

• Gamma emission doesn’t change either A or Z

C*C

e*CB126

126

126

125