Examination Radiation Physics - 8N120, 2 November 2012 - 14.00-17.00

Four general remarks:

• This exam consists of 6 assignments on a total of 3 pages. There is a table on page 3 listingthe maximum number of points that can be achieved in each assignment.

• You are not allowed to use books, notes, lecture notes, notebooks, telephones, tablets etcduring the exam.

• It is allowed to use a ordinary calculator.

• You can use the appended list of constants and formulas during the exam.

• All answers should be formulated and motivated clearly.

1. (a) What is the energy in electron-volt of a photon with a wavelengths of 550 nm?

(b) How many photons are emitted each second by a 5 watt LED lamp, that emits onlyphotons of 550 nm? You may assume that all the energy used by the lamp is transferredto the photons.

2. Consider the Compton effect where an incoming photon with an energy of 1.0 MeV interactswith a free electron. The trajectory of the scattered photon has an angle θ = 180◦ withrespect to the trajectory of the incoming photon.

(a) Compute the wavelength of the incoming photon.

(b) Compute the wavelength and energy of the scattered photon.

Next we consider the Compton effect for an incoming photon with an energy of 10.0 MeV,that is again scattered by an angle θ = 180◦ with respect to the trajectory of the incomingphoton.

(c) Compute the wavelength of the incoming photon.

(d) Compute the wavelength and energy of the scattered photon.

The effect where an incoming photon is scattered over an angle of (approximately) 180◦ iscalled Compton backscattering.

(e) What is in the case of Compton backscattering the maximal energy of the scatteredphoton? Hint: Look at incoming photons with very high energy.

3. Suppose an electron is trapped in a region (”the hydrogen atom”) of width ∆x = 2a0, wherea0 = 0.0529 nm is the Bohr radius.

(a) What is the uncertainty of the velocity in the x direction of this electron?

(b) What is in the Bohr model the velocity of an electron in the ground state of hydrogen?Hint: For a circular orbit 1

2mv2 = −E , where E is the energy of the electron.

1

4. In this exercise we consider a one-dimensional particle with mass m that can only move alongthe positive x-axis (so 0 < x <∞), in a potential U(x) = − q

x , where q is a positive constant.This potential occurs for instance if the particle has an electric charge and is attracted byanother electric charge that is fixed in x = 0.

(a) Give the Schrodinger equation for this case.

Given is that a solution of the Schrodinger equation for this case is of the form Ψ(x) =Axe−ax.

(b) Compute the first and second derivative of this wave function.

(c) Substitute Ψ(x) and the second derivative of Ψ(x) in the Schrodinger equation. Byconsidering the coefficients of e−ax and x e−ax we obtain two equations, from which aand E can be found. Give these two equations.

(d) Solve these two equations and give the corresponding energy value E and the correspon-ding wave function Ψ(x).

(e) The constant A is not yet known. Which condition for the wave function Ψ(x) can beused to find A?

(f) Compute the value of A. You may use that∫∞

0 xne−bxdx = n!/bn+1, for integer n ≥ 0and b > 0.

5. Consider the α emissionAZX → A−4

Z−2 Y + α .

(a) Give the Q-value of this expression, expressed in nuclear masses. Use AZM for the nuclear

mass and AZM for the corresponding atomic mass.

(b) Show how this Q-value can be rewritten in terms of atomic masses.

(c) Assuming that the mother X has no kinetic energy before the reaction and that all ofthe decay energy is distributed between the α-particle and daughter nucleus Y , derivethe formulas that gives the distribution of Q between α-particle and daughter. Give theactual kinetic energies of the α-particle and of the daughter nucleus.

6. Consider the part of the nuclides chart in Figure 1.

(a) Give the three possible beta decay reaction from 198Au and compute the Q-values forthese three forms of beta decay.

(b) Which of these three beta decays are energetically possible?

(c) Following the information on the chart of nuclides, only one of these beta decay reactionsis observed experimentally. Which reaction is that?

(d) Explain which combinations of successive decay processes on the nuclides chart corres-ponds approximately with this beta decay reaction. (Note 1: use the right part of the198Au entry in the chart; the left part describes excited states of 198Au. Note 2: on thenuclides chart the energies of alpha and beta decay are indicted in MeV, the energies ofgamma decays are in keV.)

2

Figuur 1: part of the nuclides chart for assignment 6

Points: (total 60)

Assignment 1a: 3 points Assignment 2a: 2 points Assignment 3a: 3 pointsAssignment 1b: 3 points Assignment 2b: 2 points Assignment 3b: 3 points

Assignment 2c: 2 pointsAssignment 2d: 2 pointsAssignment 2e: 3 points

Assignment 4a: 1 point Assignment 5a: 3 points Assignment 6a: 4 pointsAssignment 4b: 3 points Assignment 5b: 3 points Assignment 6b: 1 pointAssignment 4c: 3 points Assignment 5c: 6 points Assignment 6c: 2 pointsAssignment 4d: 3 points Assignment 6d: 4 pointsAssignment 4e: 1 pointAssignment 4f: 3 points

3

4

Radiation Physics - 8N120Formulas and constants - 2012/2013

physical constants:

h = 6.626 · 10−34 Js (Planck’s constant)kB = 1.381 · 10−23 J/K (Boltzmann’s constant)c = 2.998 · 108 m/s (speed of light)e = 1.602 · 10−19 C (elementary charge)σ = 5.670399 · 10−8 W/(m2 K4) (Stefan-Boltzmann constant)Na = 6.022 · 1023mole−1 (Avogadro’s constant)Vm = 22.4 liter (at 1 atm, 0◦C) (Molair volumeR∞ = 1.097 · 107 m−1 (Rydberg constant)µB = 9.274 · 10−24 J/T (Bohr magneton)ε0 = 8.854 · 10−12 F/m (vacuum permittivity)a0 = 0.0529 · 10−9 m = 0.0529 nm (Bohr radius)hydrogen ionization energy= 13.6 eV

unit conversions:

1 eV = 1.602 · 10−19 J (electron-volt to Joule)1 J = 6.242 · 1018 eV (Joule to electron-volt)1 u= 1.66056 · 10−27 kg = 931.5 MeV/c2 (atomair mass unit to kg to energy)hc = 1240 eV nm (hc in electronVolt nanometer)

masses:

me = 9.109 · 10−31 kg =0.000549 u =511 keV/c2 (mass of electron)

mp = 1.672 · 10−27 kg =1.0072766 u (mass of proton)

mn = 1.675 · 10−27 kg =1.0086654 u (mass of neutron)

M(11H) = 1

1M = 1.007825 u (mass of hydrogen atom)

M(42He) = 4

2M = 4.00260325 u (mass of helium atom)

M(19880 Hg) = 198

80 M = 197.966769 u (mass of mercury isotope)

M(19879 Au) = 198

79 M = 197.968242 u (mass of gold isotope)

M(19878 Pt) = 198

78 M = 197.967893 u (mass of platinium isotope)

Photons and radiation laws:

λ ν = c (relation wave length frequency)

E = hν (energy of photon)

λmaxT = 2.898 · 10−3mK (Wien’s displacement law)

R(λ) =8πλ4kT

c

4(radiation law of Rayleigh Jeans)

R(λ) =c

48πλ4

hc

λ

1ehc/(λkBT )

(Planck’s radiation law)

I = σT 4 (law of Stefan Boltzmann)

1

Fotoelectric effect and Compton effect:

hf = eVs + φ (foto electric equation)

λ′ − λ =h

mec(1− cos(θ)) (Compton scattering)

Relativistic energy and impuls:

E2 = (mc2)2 + (p c)2

p =mv√

1− v2/c2

Atom model of Bohr:

f = cR∞

(1n2− 1m2

)(Rydberg formula)

rn = a0n2 =

4πh2

mee2(radius nth orbit)

En = − 1(4πε0)2

me4

2h2

1n2

= −13.6n2

eV (energy of nth orbit of hydrogen

En = −13.6Z2

n2eV (idem, for other atoms (ions) with one electron)

Wave mechanics:

λ =h

p(De Broglie wave length)

− h2

2md2Ψdx2

+ UΨ = EΨ (1-dim. Schrodinger equation)

∆x∆px ≈ h (uncertainty relationship)

∆E∆t ≈ h (uncertainty relationship)

Nuclear physics:

R = r0A1/3 (radius on nucleus)

Btot(A,Z) = ZMH + (A− Z)Mn − AZM (binding energy)

Q = (minitial −mfinal) c2 (Q-value)

Some codings on the Chart of Nuclids:

β+ : beta plus decay, ε : electron capture, β− : beta minus decay, α : alpha decay

2