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Radioactivity
1) Introduction
2) Decay law
3) Alpha decay
4) Beta decay
5) Gamma decay
6) Fission
7) Decay series
8) Exotic forms of decay
Detection system GAMASPHERE for study rays
from gamma decay
Introduction
Transmutation of nuclei accompanied by radiation emissions was observed - radioactivity. Discovery
of radioactivity was made by H. Becquerel (1896).
Three basic types of radioactivity and nuclear decay:
1) Alpha decay
2) Beta decay
3) Gamma decay
and nuclear fission (spontaneous or induced) and further, more exotic types of decay
Transmutation of one nucleus to another proceed during decay (nucleus is not changed during
gamma decay – only energy of excited nucleus is decreased).
Mother nucleus – decaying nucleus
Daughter nucleus – nucleus incurred by decay
Sequence of follow up decays – decay series.
Decay of nucleus is not dependent on chemical and physical properties of nucleus neighboring
(exception is for example gamma decay through conversion electrons which is influenced by
chemical binding).
Electrostatic apparatus of P. Curie for
radioactivity measurement (left) and
present complex for measurement of
conversion electrons (right)
Radioactivity decay law
Activity (radioactivity) A:
A
dN
dt
where N is number of nuclei at given time in sample [Bq = s-1, Ci =3.7·1010Bq].
Constant probability λ of decay of each nucleus per time unit is assumed.
Number dN of nuclei decayed per time dt:
dN
   dt
dN = -Nλdt
N
N
t
dN
Both sides are integrated:  N    dt
N0
0
N  N 0 e  t
ln N – ln N0 = -λt
Then for radioactivity we obtain:
A
dN
 N 0 e    t  A 0 e    t
dt
where A0 ≡ -λN0
Probability of decay λ is named decay constant. Time of decreasing from N to N/2 is decay
ln 2
N0
 T
half-life T1/2. We introduce N = N0/2:
T 
Mean lifetime τ:

1
2

For t = τ activity decreases to 1/e = 0,36788.
Heisenberg uncertainty principle: ΔE·Δt ≈ ħ → Γ · τ ≈ ħ
where Γ is decay width of unstable state: Γ = ħ /τ = ħ λ
 N 0e
12
12

Total probability λ for more different alternative possibilities with decay constants λ1,λ2,λ3 … λM:
M
M
   k
   k
k 1
k 1
Sequence of decay we have for decay series λ1N1 → λ2N2 → λ3N3 → … → λiNi → … → λMNM
Time change of Ni for isotope i in series: dN /dt = λ N - λ N
i
We solve system of differential equations
and assume:
i-1
i-1
i
i
N1  C11e  1t
N 2  C21e 1t  C22e 2 t
…
N M  CM1e 1t  ...  CMM e M 2 t
i1
i   j
Coefficients with i = j can be obtained from boundary
conditions in time t = 0:
Ni(0) = Ci1 + Ci2 + Ci3 + … + Cii
For coefficients Cij it is valid: i ≠ j
Cij  Ci 1, j
Special case for τ1 >> τ2,τ3 … τM : each following member has the same
number of decays per time unit as first. Number of existed nuclei is
inversely dependent on its λ. → decay series is in radioactive equilibrium.
Creation of radioactive nuclei with constant velocity – irradiation
using reactor or accelerator. Velocity of radioactive nuclei creation is P:
dN/dt = - λN + P
Solution of equation (N0 = 0): λN(t) = A(t) = P(1 – e-λt)
It is efficient to irradiate number of half-lives but not so long time –
saturation starts.
Development of activity during
homogenous irradiation
Alpha decay
High value of alpha particle binding energy → EKIN sufficient for escape from nucleus →
Relation between decay energy and kinetic energy of alpha particles:
A
Z
X AZ42Y 42 He
Decay energy: Q = (mi – mf –mα)c2
Kinetic energies of nuclei after decay (nonrelativistic approximation):
EKIN f = (1/2)mfvf2
EKIN α = (1/2) mαvα2
From momentum conservation law: mfvf = mαvα
→
From energy conservation law: EKIN f + EKIN α = Q
We modify equation and
we introduce:
vf 
m
v
mf
( mf >> mα → vf << vα)
(1/2) mαvα2 + (1/2)mfvf2 = Q
2
 1
m

m  mf
1  m
1
m f 
v   m v2  m v2    1  E KIN 
Q
2  mf
2
mf
 2
 mf

Kinetic energy of alpha particle:
E KIN 
mf
A4
Q
Q
m  m f
A
Typical value of kinetic energy is 5 MeV. For example for 222Rn: Q = 5.587 MeV and
EKIN α= 5.486 MeV.
Barrier penetration:
Particle (Z,A) impacts on nucleus (Z,A) – necessity of potential barrier overcoming.
For Coulomb barier is the highest point in the place,
where nuclear forces start to act:
Z Ze 2
1 Z Ze 2
VC 

40 r0 (A1α 3  A1 3 ) 40 R
1
Barrier height is VCB ≈ 25 MeV for nuclei with A=200.
Problem of penetration of α particle from nucleus through
potential barrier → it is possible only because of quantum
physics.
Assumptions of theory of α particle penetration:
1) Alpha particle can exist at nuclei separately
2) Particle is constantly moving and it is bonded at nucleus
by potential barrier
3) It exists some (relatively very small) probability of
barrier penetration.
Probability of decay λ per time unit:
bound state
quasistationary
state
λ = νP
where ν is number of impacts on barrier per time unit and P probability of barrier penetration.
We assumed, that α particle
is oscillating along diameter of nucleus:
E KIN
E KIN c 2
v



 1021
2
2
2R
2m R
2E 0 R
Probability P = f(EKINα/VCB). Quantum physics is needed for its derivation.
Centrifugal barrier depends on angular momentum of emitted or incident particle:
classical:
L2
Vl
F  m r 


m r 3
r
2
quantum: L2 → l(l+1)ħ2 →
l(l  1) 2
Vl 
2m r 2
Vl (r) 
L2
2 m r 2
Beta decay
Nuclei emits electrons:
1) Continuous distribution of electron energy (discrete was assumed – discrete values of energy
(masses) difference between mother and daughter nuclei). Maximal EEKIN = (Mi – Mf – me)c2.
→ postulation of new particle existence – neutrino.
mn > mp + mν → spontaneous process
n  p   e  
neutron decay τ ≈ 900 s (strong ≈ 10-23 s,
elmg ≈ 10-16 s) → decay is made by weak interaction
Relative electron intensity
2) Angular momentum – spins of mother and daughter nuclei differ mostly by 0 or by 1. Spin
of electron is but 1/2 → half-integral change
inverse process proceeds spontaneously only inside
nucleus
Process of beta decay – creation of electron
(positron) or capture of electron from atomic shell
accompanied by antineutrino (neutrino) creation
inside nucleus. Z is changed by one. A is not
changed.
Electron energy
Schematic dependence Ne = f(Ee) at beta decay
According to mass of atom with charge Z we obtain three cases:
1) Mass is larger than mass of atom with charge Z+1 → electron decay – decay energy
A
A

is split between electron a antineutrino, neutron is transformed to proton:
ZYZ1Y  e  
2) Mass is smaller than mass of atom with charge Z+1 but it is larger than mZ+1 – 2mec2 →
electron capture – energy is split between neutrino energy and electron binding energy. Proton
is transformed to neutron: A Y  e- A Y 
Z1
Z
3) Mass is smaller than mZ+1 – 2mec2 → positron decay – part of decay energy higher than
2mec2 is split between kinetic energies of neutrino and positron. Proton changes to neutron:
Discrete lines on continuous spectrum :
YAZY  e  
A
Z1
1) Auger electrons – vacancy after electron capture is filled by electron from outer electron shell
and obtained energy is realized through Röntgen photon. Its energy is only a few keV → it is
very easy absorbed → complicated detection
2) Conversion electrons – direct transfer of energy of excited nucleus to electron from atom
Beta decay can goes on different levels of daughter nucleus, not only on ground but also on excited.
Excited daughter nucleus then realized energy by gamma decay.
Some mother nuclei can decay by two different ways either by electron decay or electron capture
to two different nuclei.
During studies of beta decay discovery of parity conservation violation in the processes connected
with weak interaction was done.
Neutrino – particle interacted only by weak interaction, very small cross-section. Detection by inverse
beta decay:
  p  n  e
  n  p  e
Determination of neutrino mass from form of end of electron spectra
We can express function connected to dependency of number of electrons on their energy:
NE e 
 konst  (E MAX  E e )
F Z, E e 

where N(Ee) – number of electrons, F*(Z,Ee) – Fermi function containing of correction on Coulomb
field of nucleus and electron cloud. In the case on nonzero neutrino mass : EMAX=Q - mνc2. (Q –
decay energy). The graph of dependency is named as:
Relative electron intensity
Fermi graph – possibility of accurate determination of maximal energy (decay energy) – eventually
neutrino mass. The neutrino mass determined by tritium decay is 2 eV at present time.
Tests and transport of main vacuum
chamber of KATRIN spectrometer
Fermi graph for decay of tritium 3H
Electron energy
Gamma decay
Excited nucleus unloads energy by photon irradiation
After alpha or beta decay → daughter nuclei at excited state → emission of gamma quantum →
gamma decay
Multipole expansion and simple selective rules:
Different transition multipolarities:
Electric
Magnetic
EJ → spin I = min J, parity π = (-1)I
MJ → spin I = min J, parity π = (-1)I+1
Transition between levels with spins Ii and If and parities πi and πf :
I = |Ii – If| pro Ii ≠ If
I = 1 for Ii = If > 0
π = (-1)I+K = πi·πf
K=0 for E and K=1 pro M
Electromagnetic transition with photon emission between states Ii = 0 and If = 0 does not exist
Energy of emitted gamma quantum: Eγ = hν = Ei - Ef
More accurate (inclusion of recoil energy):
Momenta conservation law → hν/c = Mjv
1
1  h 
2
Energy conservation law → E i  E f  h  2 M j v  h  2M  c 

j 
2 2
h
E   h  E i  E f 
 E i  E f  E R
2M jc 2
where ΔER is recoil energy.
2
Level width Γ is connected with its live time by Heisenberg uncertainty principle: Γτ  ħ
And then Γ  ħ/τ ~ uncertainty of (Ei – Ef).
Nucleus can be excited by the same energy Eγ which can emit. Nucleus recoil must be included
(recoil is created also during absorption):
Γ ≥ 2∙ ΔER
for possibility of resonance absorption. This is right for free atom.
Transition Eγ = 14 keV for isotope 57Fe:
For level τ ~ 10-7 s → Γ ~ 10-8 eV and ΔER ~ 10-3 eV. → Γ << ΔER
Atom bounded in crystal lattice → momentum is transferred to whole crystal lattice → little energy
transfer → possibility of resonance absorption – Mössbauer phenomena.
Mössbauer phenomena makes possible very accurate measurements of level energy and width.
We have:
1) Source of gamma quanta
2) Moving absorber
3) Detector of gamma rays
Doppler phenomena changes with absorber velocity energy of gamma quantum by ΔE =
E∙v/c, we can measure intensity of absorption → form of Mössbauer lines is visible.
Mean lifetimes of levels are mostly very short ( < 10-7s – electromagnetic interaction is much stronger
than weak) → life time of previous beta or alpha decays are longer → time behavior of gamma
decay reproduces time behavior of previous decay.
They exist longer and even very long life times of excited levels - isomer states.
Probability (intensity) of transition between energy levels depends on spins and parities of
initial and end states. Usually transitions, for which change of spin is smaller, are more intensive.
System of excited states, transitions between them and their characteristics are shown by decay
schema.
Example of part of gamma ray spectra from
source 169Yb → 169Tm:
Decay schema of 169Yb → 169Tm:
Internal conversion
Direct transfer of energy of excited nucleus to electron from atomic electron shell (Coulomb
interaction between nucleus and electrons):
Ee = E γ – Be
Energy of emitted electron:
where Eγ is excitation energy of nucleus, Be is binding energy of electron
Alternative process to gamma emission. Total transition probability λ is:
λ = λγ + λe
The conversion coefficients α are introduced: It is valid: dNe/dt = λeN and
dNγ/dt = λγN
and then: Ne/Nγ = λe/λγ and λ = λγ (1 + α) where α = Ne/Nγ
We label αK, αL, αM, αN, … conversion coefficients of corresponding electron shell
K, L, M, N, …:
α = αK + αL + αM + αN + …
The conversion coefficients decrease with Eγ and increase with Z of nucleus.
Transitions Ii = 0 → If = 0: only internal conversion not gamma transition
The place freed by electron emitted during internal conversion is filled by other electron
and Röntgen ray is emitted with energy: Eγ = Bef - Bei
characteristic Röntgen rays of correspondent shell.
Energy released by filling of free place by electron can be again transferred directly to another
electron and the Auger electron is emitted instead of Röntgen photon.
Pair internal conversion – Eγ > 2mec2 → pair electron and positron pair can be created → it is not
connected to electron shell → probability increases with Eγ.
Nuclear fission
The dependency of binding energy on nucleon number shows possibility of heavy nuclei fission to two
nuclei (fragments) with masses in the range of half mass of mother nucleus.
A
Z
XAZ11Y1  AZ22Y2
Penetration of Coulomb barrier is for fragments more difficult than for alpha particles
(Z1, Z2 > Zα = 2) → the lightest nucleus with spontaneous fission is 232Th. Example of fission of 236U:
Energy released by fission Ef ≥ VC → spontaneous fission
We assume A1=A2=A/2 a Z1=Z2=Z/2. Then magnitude of
Coulomb potential barrier is:
2 2
VC 
Z 2  e
40 2r0 A 21 3
1
C
Z2
A1 3
For fission energy is valid: Ef/c2 = m(Z,A) – 2m(Z/2,A/2)
After substitution from Weizsäker formula:
Ef/ c2 = aSA2/3(1-21/3)+aCZ2A-1/3(1-2-2/3)
Ef = (1-21/3) c2aSA2/3+(1-2-2/3) c2aCZ2A-1/3 = aS‘A2/3+aC‘Z2A-1/3 = A2/3 (aS‘+aC‘Z2/A)
From this:
Ef > 0
Ef ≥ VC
→
Z2/A > -aS‘/aC‘~ 18
a S
Z2

 51
A C  a C
Ratio Z2/A (fission parameter) is critical for stability against spontaneous fission.
→
After supplial of energy – induced fission – energy supplied by photon (photofission), by neutron, …
Induced fission can be described by comparison of surface and Coulomb energy of symmetrical
sphere and deformed ellipsoid with
4
4
VKOULE    R 3    ab 2  VELIPSOID
half-axe a and b with the same volume V:
3
3
The same volume → the same volume energy for sphere and ellipsoid. We express a = R(1- ε)
and b = R(1-ε)-1/2
where ε is ellipsoid eccentricity. Ellipsoid surface is:
2
SELIPSOID  4  R 2 (1   2  ...)
5
2
E S  a S A 2 3 (1   2  ...)
5
Surface energy in Weizsäker formula then is:
Coulomb energy of charged ellipsoid:
3 1 Z2e 2
1
EC  
(1   2  ...)
5 40 R
5
Then from Weizsäker formula: E C  a C Z 2 A 1 3 (1  1  2  ...)
5
Deformation energy ED (ΔES and ΔEC differences between energies of ellipsoid and sphere ε = 0):
2
1
E D  E S  E C  a S A 2 3  2  a C Z 2 A 1 3   2  ...
5
5
After substitution of constants from Weizsäker formula:
ED = ε2(7.34∙A2/3 –0.14∙Z2A-1/3) = ε2∙A2/3 (7.34–0.14∙ Z2/A) [MeV]
Z2/A ≥ 51 → ED ≤ 0 → spontaneous fission
Energy Ea needed for overcoming of potential barrier – activation energy – for heavy nuclei is
small ( ~ MeV) → energy released by neutron capture is enough (high for nuclei with odd N).
Certain number of neutrons is released after neutron capture during each fission (nuclei with
average A have relatively smaller neutron excess than nucley with large A) → further fissions are
induced → chain reaction. 235
U + n → 236U → fission → Y1 + Y2 + ν∙n
Average number η of neutrons emitted during one act of fission is important - 236U (ν = 2.47) or per
one neutron capture for 235U (η = 2.08) (only 85% of 236U makes fission, 15% makes gamma decay).
How many of created neutrons produced further fission depends on arrangement of setup with
fission material
Ratio between neutron numbers in n and n+1 generations
of fission is named as multiplication factor k:
Its magnitude is split to three cases:
k < 1 – subcritical – without external neutron source
reactions stop → accelerator driven transmutors
– external neutron source
k = 1 – critical – can take place controlled chain reaction
→ nuclear reactors
k > 1 – supercritical – uncontrolled (runaway) chain
reaction → nuclear bombs
Fission products of uranium 235U. Dependency of their production on
mass number A: (taken from A. Beiser: Concepts of modern physics)
Decay series
Different radioactive isotope were produced during elements synthesis (before 5 – 7 miliards years).
Some survive: 40K, 87Rb, 144Nd, 147Hf
Beta decay: A is not changed
Summary of decay series:
A
The heaviest from them: 232Th, 235U a 238U
Alpha decay: A → A - 4
Series
Mother
nucleus
T 1/2 [years]
4n
Thorium
232Th
1.39·1010
4n + 1
Neptunium
237Np
2.14·106
4n + 2
Uranium
238U
4.51·109
4n + 3
Actinium
235U
7.1·108
Decay life time of mother nucleus of neptunium series is shorter than time of Earth existence.
Also all furthers → must be produced artificially → with lower A by neutron bombarding, with
higher A by heavy ion bombarding.
Some isotopes in decay series must decay by alpha as well as beta decays → branching
Possibilities of radioactive element usage:
1) Dating (archeology, geology)
2) Medicine application (diagnostic –
radioisotope, cancer irradiation)
3) Measurement of tracer element contents
(activation analysis)
4) Defectology, Röntgen devices
Exotic forms of decay
Proton emission – protons must penetrate Coulomb barrier → life time (also in μs and ms
range) is longer than characteristic nuclear time (time of nucleon transit through nucleus – 1021s) → existence of proton radioactivity. It is possible only for exotic light nuclei with large
excess of protons (for example 9B) – decay has sufficiently short decay time and hence it is not
suppressed by competitive positron beta decay.
Emission of couple of protons – made by coupling (maybe also in form of 2He) - year 2000 at Oak
Ridge laboratory for nucleus 18Ne
Delayed proton emission – emission of protons following after proton decay → nuclei
with large excess of protons → created nucleus at highly excited state emits proton
Neutron emission – life time of nuclei with big neutron excess, if neutron decay is energy possible,
is comparable with characteristic nuclear time – it can not be named as neutron radioactivity
Delayed neutron emission following after beta decay. Nucleus with big neutron excess → beta decay
with longer decay time → consecutive quick neutron emission during time comparable with
characteristic nuclear time.
Emission of hevier nuclei – 12C, 16O … → fragmentation of highly excited nuclei
Double beta decay (ββ2ν) – is possible if double decay is allowed by energy conservation and
single beta decay is not allowed.
A
A

Z X Z  2Y  2e  2 e
We have potentially 35 (ββ 2ν) – emitters. Nine was observed up to now (48Ca,
100Mo, …). Very long decay time T
19
24
1/2 = 10 – 10 years.
76Ge, 82Se,
Studied using underground experiments (main problem is background). For example new device
NEMO-3 (10 kg of 100Mo, Qββ = 3.038 MeV). Next possibility – geochemical measurements.
Examples of nuclei decayed by double beta decay
Neutrinoless double beta decay (ββ0ν) – possible only in the case of nonzero neutrino rest mass
and if neutrino is Majorana particle type (antiparticle is identical with particle – difference of
lepton number is still there). In this case two neutrons can change neutrino and antineutrino in
the process violated lepton number conservation and only pair of electrons is emitted. This decay
was not observed up to now . Limit is in the range of 1025 years measured with 76Ge → limit on
mass ~ 0.45 eV.
Device NEMO-3 for double beta studies
Device NEMO-3 in underground tunnel at Alps