Quantum and nuclear physics

Question 1

What is a photon?

Answer 1

• A small quantized bundle of energy.

- The energy of a beam of light is a product of hf (the energy of one photon) and N (the number of photons)

Question 2

What is the photoelectric effect? Describe the apparatus used to test it.

Answer 2

• A phenomenon where electromagnetic radiation incident on a metallic surface causes electrons to be emitted from the surface.
• An evacuated tube: light passes through an opening and falls on the photo-surface, which emits electrons. These arrive at a collecting plate. Wires connecting the two allow a current to flow.

Question 3

How can you experimentally and calculatory work out the maximum kinetic energy of electrons emitted as part of the photoelectric effect?

Answer 3

Place a power supply in the circuit so that the collecting plate repels electrons. Only electrons with the most energy reach the plate. The voltage is increased, and the magnitude of the voltage at which the current becomes zero is called the stopping voltage, Vs.
Emax = eVs
since the work done is the change in kinetic energy.

Question 4

How does the intensity of light affect the current produced by the photoelectric effect?

Answer 4

• It does not affect the stopping voltage.

- Less intense light will produce a lower current.

Question 5

How does the frequency of light affect the current produced by the photoelectric effect?

Answer 5

• The higher the frequency, the higher the current.
• The higher the frequency, the higher the magnitude of the stopping voltage.
• The critical or threshold frequency fc is the frequency below which no electrons are emitted at all, even if the light is very intense.

Question 6

What happens when you plot maximum kinetic energy against frequency for different photo-surfaces?

Answer 6

• A straight line with no negative y (Ek) values.
• The x axis intercept is the critical frequency.
• Different photo-surfaces are the parallel lines.
• A y intercept at the work function.

Question 7

What are the four surprising observations of the photoelectric effect that cannot be understood in terms of light as a wave and why?

Answer 7

• Intensity of light does not affect the energy of emitted electrons: If light is a wave, a more intense beam should carry more energy and so electrons should have more energy.
• Electron energy depends on frequency of incident light: the formula for energy of light as a wave does not include frequency, so it should not affect it.
• There is a minimum frequency below which no electrons are emitted. “
• Electrons are emitted with no time delay: In the wave model, electrons would have to wait to accumulate enough energy from a low intensity beam to escape the metal, causing a delay.

Question 8

How does Einstein’s explanation of the photoelectric effect work?

Answer 8

• A single photon of frequency f is absorbed by a single electron in the photo-surface, increasing the energy of the electron by hf.

Question 9

What is the work function?

Answer 9

• Φ The minimum energy required for an electron to escape a photo-surface.
• The rest of the energy transferred by the photon is converted to kinetic energy:
  Ek = hf - Φ

Question 10

What is the de Broglie hypothesis?

Answer 10

• The concept that particles can have wave-like properties; ie. they obey basic phenomena such as diffraction and interference and have an assigned wavelength.
• Called the duality of matter
• For a particle of momentum p, λ = h/p

Question 11

How can you derive an equation for the wavelength of an electron accelerated through a potential difference V?

Answer 11

Ek =p²/2m
eV = p²/2m
p = √(2meV)
λ = h/√(2meV)
λ proportional to 1/√V

Question 12

Pair annihilation and pair production

Answer 12

Pages 490-491

Question 13

What is the Bohr condition for the quantization of angular momentum?

Answer 13

The angular momentum of an electron is quantised in integral values of h/2π :
angular momentum = mvr = nh/2π
where n is the quantum (orbital) number (integer)
r is the orbital radius

Question 14

What does the Bohr model of hydrogen predict about allowed orbits of electrons?

Answer 14

Allowed orbits are ones for which an integral number of electron wavelengths fit on the circumference of the orbit, so that they form a standing wave.

Question 15

Pages 492-4

Answer 15

a

Question 16

What is the wave function, Ψ?

Answer 16

The probability of finding an electron at a distance r from its source is
P(r) = IΨI²ΔV where V is the volume considered.
- Where Ψ² is the highest, you are most likely to find the electron.

Question 17

What is the Heisenberg uncertainty principle for position and momentum?

Answer 17

• It is not possible to measure simultaneously the position and momentum of a particle with indefinite precision. (a fundamental property of nature)
  ΔxΔp ≥ h/4π
  Δx and Δp are uncertainty in position and momentum
  h is Planck’s constant
  You can also replace the inequality with approximation.

Question 18

What is an equation for kinetic energy in terms of momentum?

Answer 18

Ek = p²/2m

Question 19

How can the uncertainty principle be applied to energy and time?

Answer 19

If the energy of a state is measured with an uncertainty ΔE, then the lifetime of the state is of order Δt such that:
ΔEΔt ≥ h/4π
Also applies to decaying particles where ΔE is the uncertainty in the measured value of energy released and Δt is the lifetime of the particle.
You can also replace the inequality with approximation.

Question 20

What is some evidence for the wave nature of electrons?

Answer 20

When electrons are fired one at a time through a double slit, it creates an interference pattern, which suggests that the electron is interfering with itself. This can only be explained if the electron is also a wave.

Question 21

Describe how the quantisation of angular momentum of the electron in a hydrogen atom led Neils Bohr to a successful understanding of the spectrum.

Answer 21

The Bohr model predicts that the electron in the hydrogen atom has discrete/quantised energy. This explains the absorption and emission spectra of hydrogen.

Question 22

What must be conserved during pair production and annihilation?

Answer 22

• Momentum before and after must be the same. This is why two photons are produced in opposite directions.
• The total mass and energy of the system at the start must be equivalent to the total mass and energy at the end.

Question 23

What is quantum tunneling?

Answer 23

Objects can ‘borrow’ energy, in order to pass barriers. This can be illustrated by the wave function, which may have a small (but non-zero) amplitude on the other side of the barrier, indicating a small chance of finding the object there.

Question 24

How can you calculate the total energy of an electron for a particular orbital number?

Answer 24

E = -13.6/n²
E = the total energy = Ek + Ep
n = the orbital number (starting at 1)

Question 25

What is quantum tunneling?

Answer 25

Objects can ‘borrow’ energy, in order to pass barriers.
Because the wave function must be continuous, there must be a small (but non-zero) amplitude on the other side of the barrier, indicating a small chance of finding the object there.

Question 26

What factors affect the likelihood that quantum tunneling will occur? [3]

Answer 26

• The thickness of the barrier. The further the particle has to tunnel, the smaller the amplitude of the wave function on the other side, therefore less likely.
• The amount of energy required
• The mass of the object: heavier objects often require more energy to overcome the barrier, and so are less likely to tunnel.

Question 27

How can Rutherford scattering be used to estimate the radius of the nucleus?

Answer 27

As an alpha particle fired directly at the nucleus gets closer, its kinetic energy is transferred to potential energy, until it stops, and is accelerates away.

Question 28

How can Rutherford scattering be used to estimate the radius of the nucleus?

Answer 28

• As an alpha particle fired directly at the nucleus gets closer, its kinetic energy is transferred to electric potential energy, until it stops, and is accelerates away.
  Ekmax = Epmax
  1/2mv = kQq/R where R is the separation of the alpha particle and the nucleus.
• Increasing the initial Ek of the alpha particles decreases R, called the method of closest approach.

Question 29

What is some evidence for the energy levels of a nucleus?

Answer 29

The energies of the alpha particles and gamma ray photons that are emitted by nuclei in alpha and gamma decays are discrete.

Question 30

How is the radius of a nucleus related to its nucleon number?

Answer 30

R = R₀A^1/3  DB
R = radius / m
R₀ = radius of one nucleon (Fermi) = 1.2 x 10^-15
A = nucleon number

Question 31

How can Beta decay be explained in regards to the discrete energy of the nucleus?

Answer 31

Unlike alpha particles, beta particles can be emitted with a range of kinetic energies, anywhere up to a maximum value.
A neutrino is also produced, and the total energy (shared between the electron, the neutrino and the nucleus) is discrete.

Question 32

How can you calculate the radius of the nucleus by measuring electron diffraction?

Answer 32

Making corrections for the relativistic speeds of the electron,
sinθ ≈ λ/D
θ = angle between first minimum and central max of diffraction pattern
λ = de Broglie wavelength
D = diameter of the nucleus

Question 33

How can you calculate the radius of the nucleus by measuring electron diffraction?

Answer 33

Making corrections for the relativistic speeds of the electron,
sinθ ≈ λ/D DB
θ = angle between first minimum and central max of diffraction pattern
λ = de Broglie wavelength
D = diameter of the nucleus

Question 34

What is the decay constant? How can it be used to calculate the activity?

Answer 34

λ is the probability that a nucleus will decay per unit time.
A = λN (Bq)
N is the number of nuclei in the sample.

Question 35

What is the decay constant? How can it be used to calculate the activity?

Answer 35

λ is the probability that a nucleus will decay per unit time.
A = λN (Bq)
N is the number of nuclei in the sample.

Question 36

How can you calculate the half life from the decay constant?

Answer 36

t1/2 = ln2/λ

Question 37

What is the decay constant? How can it be used to calculate the activity?

Answer 37

λ is the probability that a nucleus will decay per unit time.
A = λN (Bq) NDB
λ is the slope on a graph of A against N
N is the number of nuclei in the sample.

Question 38

How can you calculate the half life from the decay constant?

Answer 38

λ x t1/2 = ln2 NDB

Question 39

How can you calculate the number of nuclei left in a radioactive sample?

Answer 39

N = N₀e^-λt DB
N₀ = original number of nuclei
Note that as long as t is the same units as λ, the equation works

Question 40

How can you calculate the activity of a sample?

Answer 40

A = λN₀e^-λt DB
N₀ = original number of nuclei
Note that as long as t is the same units as λ, the equation works

Question 41

How can you measure the half life of a radioactive substance that has a short half life?

Answer 41

Measure the activity over a long time period, and use , A = A₀e^−λt to find λ. From there calculate the half life using
λ x t1/2 = ln2 NDB

Question 42

How can you measure the half life of a radioactive substance that has a very long half life?

Answer 42

Take a very fresh sample of the element in which only a negligible number of the nuclei have decayed. Measure the mass, and calculate the number of radioactive nuclei, using n = N/NA (Avogadro)
From there use A = λN (Bq) NDB