Ap Physics B Review - Modern Physics

AP Phys B Test 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 = hν

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-toone 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

 fc  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

1895

and named by Roentgen in

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

•



The photoelectric effect and Compton scattering offer evidence for the particle nature of light

•



Applies to all electromagnetic radiation

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 h λ= mv



The de Broglie wavelength of a particle is



The frequency of matter waves is

E ƒ= h

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

 i  H t 

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

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/4π

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: ∆E∆t ≥ h

h ∆x∆p x ≥ 4π

4π

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: • •

Mostly small angle scattering No backward scattering events

Results: • •

Mostly small scattering events 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 1 1  1 = RH  2 − 2  λ n  2

• • • 

RH is the Rydberg constant

•R

H

= 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

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 nonclassical 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

Ei  E f  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

Ei  E f  hf 

 h  me vr  n  , n  1, 2,3,...  2 

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

Results  The

•

total energy of the atom

2 1 e E  KE  PE  me v 2  ke 2 r

F  me a or  Newton’s

 This

law

can be

e2 v2 ke 2  me r r

mv 2 e2 KE to rewrite  ke used 2 2r

kinetic energy as k ee2 E=− 2r

Bohr Radius  The

radii of the Bohr orbits are quantized

n2 2 rn = m ek e e 2

n = 1, 2, 3, 

h h

2

• 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

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

 1 1 1 = RH  2 − 2  λ  n f ni 

• •



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

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

• • •

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

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 

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

2e   Ze   1 2 q1q2 mv  ke  ke 2 r d

or

4ke Ze 2 d mv 2

Size of Nucleus 

Since the time of Rutherford, many other experiments have concluded the following

•

Most nuclei are approximately spherical

r = ro A

1 3

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 11 6

C

12 6

C

13 6

C

14 6

C

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

Binding 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

• Beta particles

He nuclei

4

• The particles are either electrons or positrons

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 N R  N N   N  t  t

• 

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

Decay Curve 

The decay curve follows the equation N  N 0 e  t



The half-life is also a useful parameter T1 2 =

ln 2 0.693 = λ λ

Units  The

unit of activity, R, is the Curie, Ci

• 1 Ci = 3.7 x 10

 The

Bq

10

decays/second

SI unit of activity is the Becquerel,

• 1 Bq = 1 decay / second

• Therefore, 1 Ci = 3.7 x 10

 The

10

Bq

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 4

A Z

X→

A −4 Z −2

Y + He 4 2

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

• •

−

12 5

B→ C * + e + ν

12 6

C*→126 C + γ

12 6

The C* indicates the Carbon nucleus is in an excited state Gamma emission doesn’t change either A or Z