Download Introduction to Subatomic Physics

Introduction to Subatomic Physics
"Revolution in the science can be done only by excellent, perfectly simple theory and on the base
of very carefully performed and unambiguously interpreted experimental measurement."
Vladimír Wagner
Nuclear Physics Institute of ASCR, Řež
E_mail:[email protected]
1) Three levels of submicroscopic domain
2) Tools for microscopic domain description
3) Relativistic properties
4) Quantum properties
5) Measurement in submicroscopic domain
6) Structure of matter
7) Collision kinematics
8) Cross section quantity
WWW: http://hp.ujf.cas.cz/~wagner
9) Basic properties of nucleus and nuclear forces
10) Models of atomic nuclei
11) Radioactivity
12) Basic properties of nuclear reactions
Recommended textbooks:
1) W.S.C. Williams : Nuclear and Particle Physics, Oxford Science Publications, 2001
2) Ashok Das, Thomas Ferbel: Introduction to Nuclear and Particle Physics, John Wiley & Sons, 1994
3) B. Povh, K. Rith, Ch. Scholz, F. Zetsche: Particles and Nuclei. An Introduction to the Physical Concepts,
Springer 2004
4) A. Beiser: Concepts of Modern Physics, McGraw-Hill Companies Date Published, 1995
5) Glen F. Knoll: Radiation Detection and Measurement, John Wiley & Sons, Inc., 2000
Three size levels of submicroscopic domain studies
Variety of our ordinary surroundings (macroscopic world) is consisted of atoms and molecules
originating in chemical bonding of atoms
Scale in
Description of the submicroscopic domain:
Scale in
Atom
Atomic physics – physics of an electron cloud of an
atom, chemical bonding of atoms to molecules, only
electromagnetic interaction
Nucleus
Nuclear physics – physics of atomic nuclei and
interactions inside, interaction of a nucleus and
electron shells, interaction of nucleus and elementary
particles, physics of nuclear matter, strong, weak and
electromagnetic interactions
Proton
Quark
Electron
Subnuclear physics (elementary particle physics or also high energy physics) – physics of
elementary particles, and interactions - strong, weak and electromagnetic
Scale
Size
[m]
Atomic
~10-10
Nuclear
~10-14
Subnuclear ~10-15
1)
2)
Energy 1)
[MeV]
~ 0,00001
~ 8
~ 200
Momentum 2)
[MeV/c]
elmg (molecul.)
 0,002
strong (nuclear)
 20
strong
 200
Interaction
Bonding energy of an electron in a atom or energy of molecular bonding, nucleon bonding energy,
energy needed for elementary particles creation (comparable with their rest energies – masses)
Calculated using characteristics size and Heisenberg uncertainty principle Δp·Δx ~ ħ
Characteristic rest masses:
matom  mnucleus = 938   260 000 MeV/c2
(mp = 1836 me ;
mp = 938.27 MeV/c2 = 1,67262·10-27kg)
mparticles = 0,511 MeV/c2 (electron)  91 187 MeV/c2 (Z0 boson)
Characteristic times:
1/c=3,3·10-9 s /m, transit through nucleus ~ 4·10-23 s; interactions – strong ~ 10-23 s;
weak ~ 10-10 - 10-6 s and electromagnetic. ~ 10-16 - 10-6 s .
Science is searching for a objective description of our world
19th and 20th centuries - new tools for description (applied within the range of extreme
values of physical quantities):
Microscopic domain - special theory of relativity - high velocities, transferred energies
- quantum physics – very small values of masses, particle distances,
transferred action
Megaworld - special theory of relativity – high velocities, transferred energies
- general theory of relativity – very big intensities of the gravitation field
Uniformity of Universe description on whole time and space scale enables the possibility
of extrapolation  on the base of present state → past or future state are predicted
Tools for microscopic domain studies
Special theory of relativity:
Differences between classical Newtonian mechanics
and Einstein´s special theory of relativity
(Galileo and Lorentz transformations) are
significant only for velocities v of the body
against reference frame near to the
velocity of the light c (3·108 m/s).
Motion of relativistic particles in the accelerator.
Dependence of special theory of relativity
manifestation on velocity.
Quantum physics:
Show itself during processes
with action transfer in the range:
h = 6.626·10-34 Js = 4.14·10-21 MeV s
Influence of measurement alone on
the measured object.
Fundamental uncertainty of measurement:
px  ħ
Et  ħ
Stochastic character.
Comparison of de Broglie wave length λ for
objects with different mass m (me – electron
mass, mj nucleus mass, rj – nucleus radius)
Relativistic properties
Relation between whole energy and mass: E = mc2
For rest energy of system in the rest:
E0 = m0c2
For kinetic energy then:
EKIN = E-E0 = mc2 - m0c2
For relativistic systems possibility to determine of energy changes by measurement of mass
change and vice versa
Nonrelativistic objects (EKIN  m0c2)  changes of mass are not measurable.
(further p and v are magnitudes of momentum and particle
velocity.)
Relation between energy E and momentum p and
kinetic energy EKIN=f(p):
E2 = p2c2 + (m0c2)2  EKIN =  (p2c2 + (m0c2)2) - m0c2
Nonrelativistic aproximation (p  m0c) – correspondence
principle:
EKIN = m0c2  (p2/(m0c)2 +1) - m0c2  p2/2m0 = (m0v2)/2
(for square root were take first members of binomial
development – it is valid (1x)n  1nx pro x1)
Ultrarelativistic approximation (p  m0c): EKIN  E  pc
Invariant quantity: m0c2 =  (E2 - p2c2)
It is valid for velocity v: v = pc2/E  pc2/EKIN
for p  m0c or m0 = 0: v = c
Quantum properties
The smaller masses and distances of particles – the more intensive manifestation of quantum
properties.
Quantitative limits for transformation of classical mechanics to the quantum one is given by
Heisenberg uncertainty principle ΔpΔx ≥ћ. Correspondence principle is valid - for ΔpΔx >> ћ
occurs transformation of quantum mechanics to the classical mechanics.
Exhibition of both wave and particle properties.
Discrete character of energy spectrum and other quantities for quantum objects (spectra of atoms,
nuclei, particles, their spins …).
Quantum physics is on principle statistic theory. This is difference from the classical statistic
theory, which assumes principal possibility to describe trajectory of every particle (the big
number of particles is problem in reality)
It is done only probability
distribution of times of
unstable nuclei or particles
decay
Objects act as particles as well as waves
Quantum mechanics (as well as classical) must include:
1) Description of state of the physical system in the given time.
2) The equations of motion describing changes of this state with the time.
3) Relations between quantities describing system state and measured physical quantities.
Classical theory
Quantum theory
Particle state is described by six numbers
x, y, z, px, py, pz in given time
Particle state is fully described by complex function
 (x,y,z) given in whole space.
Time development of state is described by
Hamilton equations:
dr/dt = ∂H/∂p
dp/dt = - ∂H/∂r
where H is Hamilton function
Time development of state is described by
Schrödinger equation:
iħ∂Ψ/∂t = ĤΨ
where Ĥ is Hamilton operator.
Quantities x and p describing state are
directly measurable.
Function  is not directly measurable quantity.
Classical mechanics is dynamical theory.
Quantum mechanics has stochastic character. Value
of |(r)|2 gives probability of particle presence in
the point r. Physical quantities are their mean
values.
Measurements in the microscopic domain
Man is macroscopic object – all information about microscopic domain are mediated.
The smaller studied object → the smaller wave length of studying radiation → the higher E and p
of quanta (particles) of this radiation. High values of E and p → affection even destruction of
studied object – decay and creation of new particles.
Main method of study – bombardment of different targets by different particles and detection of
produced particles at macroscopic distances from collision place by macroscopic detection systems.
Important are quantities describing collision (cross sections, transferred momentums, angular
distributions …).
Stochastic character of quantum processes in the microscopic domain → statistical character
of measurements.
Relation between wave length  and kinetic energy
EKIN for different particles. (rA – size of atom, rj –
radius of nucleus, rL – present size limit)
Relation between accuracy of mean value
determination of stochastic quantity on number
of measurement N (assumption of independent
measurement – Gaussian distribution).
Structure of matter:
Standard model of matter and interactions
Hadrons – baryons (proton, neutron …)
- mesons (π, η, ρ …)
Composed from quarks, strong interaction
Bosons - integral spin
Fermions - half integral spin
Interactions and their character
Interaction – term describing possibility of energy and momentum exchange or possibility of creation
and anihilation of particles
The known interactions: 1) Gravitation 2) Electromagnetic 3) Strong 4) Weak
Description by field – scalar or vector variable, which is function of space-time coordinates, it
describes behavior and properties of particles and forces acting between them.
Quantum character of interaction – energy and momentum transfer through v discrete quanta
Exchange character of interactions – caused by particle exchange
Real particle – particle for which it is valid
E   m02c4  p 2c2
Virtual particle – temporarily existed particle, it is not valid relation (they exist thanks Heisenberg
uncertainty principle):
E   m02c4  p 2c2
Four known interactions:
Interaction
intermediate boson
interaction constant
range
Gravitation
graviton
2·10-39
Weak
W+ W- Z0
7·10-14 *)
10-18 m
Electromagnetic
γ
7·10-3
infinite
Strong
8 gluons
1
10-15 m
infinite
*) Effective value given by large masses of W+, W- a Z0 bosons
Interaction intensity α
Interaction intensity given by interaction coupling constant – its magnitude changes with increasing
of transfer momentum (energy). Variously for different interactions → equalizing of coupling
constant for high transferred momenta (energies)
Exchange character of interactions:
Mediate particle – intermedial bosons
strong SU(3)
unified
SU(5)
weak SU(2)
electromagnetic U(1)
Range of interaction depends on mediate
particle mass
Magnitude of coupling constant on their
properties (also mass)
Energy E [GeV]
Leveling of coupling constants for high
transferred momentum (high energies)
Example of graphical representation of exchange
interaction nature during inelastic electron scattering on
proton with charm creation using Feynman diagram
Introduction to collision kinematic
Study of collisions and decays of nuclei and elementary particles – main method of microscopic
properties investigation.
Elastic scattering – intrinsic state of motion of participated particles is not changed  during
scattering particles are not excited or deexcited and their rest masses are not changed.
Inelastic scattering – intrinsic state of motion of particles changes (are excited), but particle
transmutation is missing.
Deep inelastic scattering – very strong particle excitation happens  big transformation of the
kinetic energy to excitation one.
Nuclear reactions (reactions of elementary particles) – nuclear transmutation induced by external
action. Change of structure of participated nuclei (particles) and also change of state of motion.
Nuclear reactions are also scatterings. Nuclear reactions are possible to divide according to
different criteria:
According to history ( fission nuclear reactions, fusion reactions, nuclear transfer reactions …)
According to collision participants (photonuclear reactions, heavy ion reactions, proton induced
reactions, neutron production reactions …)
According to reaction energy (exothermic, endothermic reactions)
According to energy of impinging particles (low energy, high energy, relativistic collision,
ultrarelativistic …)
Nuclear decay (radioactivity) – spontaneous (not always – induced decay) nuclear transmutation
connected with particle production.
Elementary particle decay - the same for elementary particles
Set of masses, energies and moments of objects participating in the reaction or decay is named as
process kinematics . Not all kinematics quantities are independent. Relations are determined by
conservation laws. Energy conservation law and momentum conservation law are the most
important for kinematics.
Transformation between different coordinate systems and quantities, which are conserved during
transformation (invariant variables) are important for kinematics quantities determination.
Rutherford scattering
Target: thin foil from heavy nuclei (for example gold)
Beam: collimated low energy α particles with
velocity v = v0 << c, after scattering v = vα << c
The interaction character and object structure are not
introduced



Momentum conservation law: m v0  m v  mt v t .. (1.1)
and so:
square:


m 
v 0  v  t v t
m

m   m
v02   v2  2 t v  v t   t

m
 m

Energy conservation law:
….. (1.1a)
2
 2 
 v t
 
….. (1.1b)
mt 2
2
2
1
1
1
2
2
2
v

v

vt
(1.2a)
and
so:
0

m  v 0  m  v  m t v t
m
2
2
2
 m
Using comparison of equations (1.1b) and (1.2b) we obtain: v 2t 1  t
 m
For scalar product of two vectors it holds:
.. (1.2b)

 
  2v  v t ………… (1.3)

  
a  b  a b cos  so that we obtain:
Reminder of equation (1.3)
If mt<<mα:
 m
v 2t 1  t
 m

 
 
  2v  v t  2  v  v t  cos

Left side of equation (1.3) is positive → from right side results, that target and α particle are moving
to the original direction after scattering → only small deviation of α particle
If mt>>mα:
Left side of equation (1.3) is negative → large angle between α particle and reflected target nucleus
results from right side → large scattering angle of α particle
Concrete example of scattering on gold atom:
mα  3.7·103 MeV/c2 , me  0.51 MeV/c2 a mAu  1.8·105 MeV/c2
1) If mt =me , then mt/mα  1.4·10-4:
We obtain from equation (1.3): ve = vt = 2vαcos ≤ 2vα
We obtain from equation (1.2b): vα  v0
Reminder of equation (1.2b):
v 02  v2 
mt 2
vt
m
Then for magnitude of momentum it holds: meve = m(me/m) ve ≤ m·1.4·10-4·2vα  2.8·10-4mv0
Maximal momentum transferred to the electron is ≤ 2.8·10-4 of original momentum and
momentum of α particle decreases only for adequate (so negligible) part of momentum .
Maximal angular deflection α of α particle arise, if whole change of electron and α momenta are
to the vertical direction. Then (α  0):
α rad  tan α = meve/mv0 ≤ 2.8·10-4  α ≤ 0.016o
Reminder of equation (1.3)
 m
v 2t 1  t
 m

 
 
  2v  v t  2  v  v t  cos

Reminder of equation (1.2b): v 02  v2 
2) If mt =mAu , then mAu/mα  49
mt 2
vt
m
We obtain from equation (1.3): vAu = vt = 2(mα/mt)vα cos  2(mαvα)/mt
We introduce this maximal possible velocity vt in (1.2b) and we obtain: vα  v0
2
mt 2
m t  2m α v α 
4m 2
2
2
2

  v2 
v

v

v

v

v  v2
because: 0

t

m
m  m t 
mt
Then for momentum is valid: mAuvAu ≤ 2mvα  2mv0
Maximal momentum transferred on Au nucleus is double of original momentum and α particle can
be backscattered with original magnitude of momentum (velocity).
Maximal angular deflection α of α particle will be up to 180o.
Full agreement with Rutherford experiment and atomic model:
1) weakly scattered  - scattering on electrons
2)  scattered to large angles – scattering on massive nucleus
Attention remember!!: we assumed that objects are point like and we didn't involve force character.
Inclusion of force character – central repulsive electric field:
Thomson atomic model
Electrons
Thomson model – positive charged cloud with radius of atom RA :
Electric field intensity outside:
Electric field intensity inside:
Positive charged cloud
Rutherford atomic model
Electrons
Q
40 r 2
E(r  R A ) 
1
1
Qr
40 R 3A
The strongest field is on cloud surface and
2eQ
force acting on  particle (Q = 2e) is: FMAX  2eE(r  R A ) 
40 R 2A
This force decreases quickly with distance and it acts along trajectory
L  2RA  t = L/ v0  2RA/ v0 . Resulting change of particle 
momentum = given transversal impulse:
4eQ
p  FMAX t 
40 R A v 0
Maximal angle is:
Positive charged nucleus
E(r  R A ) 
tan   p /p  
4eQ
40 R A m v 02
Substituting RA 10-10m, v0  107 m/s, Q = 79e (Thomson model):
rad  tan   2.7·10-4 →   0.015o only very small angles.
Estimation for Rutherford model:
Substituting RA = RJ  10-14m (only quantitative estimation):
tan    2.7 →    70o  also very large scattering angles.
Possibility of achievement of large deflections by multiple scattering
Foil at experiment has 104 atomic layers. Let assume:
1) Thomson model (scattering on electrons or on positive charged cloud)
2) One scattering on every atomic layer
3) Mean value of one deflection magnitude   0.01o. Either on electron or on positive
charged nucleus
Mean value of whole magnitude of deflection after N scatterings is (deflections are to all directions,
therefore we must use squares):
2
N
 N 
   i   i2  N 2
i 1
 i1 
2

  N 
…..…. (1)
We deduce equation (1). Scattering takes place in space, but for simplicity we will show
problem using two dimensional case:
Deflections i are distributed both in positive and negative
directions statistically around Gaussian normal distribution
for studied case. So that mean value of particle deflection from
original direction is equal zero:
N
N
i 1
i 1
  i  i  0
Multiple particle scattering
i  
the same type of scattering on each atomic layer:
i2   2
Then we can derive given relation (1):
2
N 1 N
N 1 N
N
 N 2
 N   N 2
  i     i  2  i j    i  2  i j   i2  N 2
i 1 ji 1
i 1 j i 1
i 1
 i1   i1
 i 1
Because it is valid for two inter-independent random quantities a and b with Gaussian distribution:
1 N
1 M
1 N M
1 NM
abk  ab
a  b  ai  b j 
 ai  b j  N  M 
N i 1 M j1
N  M i 1 j1
k 1
And already showed relation is valid:
  N
We substitute N by mentioned 104 and mean value of one deflection  = 0.01o. Mean value of
deflection magnitude after multiple scattering in Geiger and Marsden experiment is around  1o.
This value is near to the real measured experimental value.
Certain very small ratio of particles was deflected more then 90o during experiment (one particle
from every 8000 particles). We determine probability P(), that deflection larger then  originates
from multiple scattering.
If all deflections will be in the same direction and will have mean value, final angle will be ~100 o (we
accent assumption  each scattering has deflection value equal to the mean value). Probability of
this is P = (1/2)N =(1/2)10000 = 10-3010. Proper calculation will give:
P    e   
2
We substitute:


P  90o  e 90
o
1o
2  e 8100  10 3500
Clear contradiction with experiment – Thomson model must be rejected
Derivation of Rutherford equation for scattering:
Assumptions:
1)  particle and atomic nucleus are point like masses and charges.
2) Particle and nucleus experience only electric repulsion force – dynamics is included.
3) Nucleus is very massive comparable to the particle and it is not moving.

Acting force: Charged particle with the charge Ze produces a Coulomb potential: Ur   4 r
0

Two charged particles with the charges Ze and Z‘e and the distance r  r
experience a Coulomb force giving rice to a potential energy :
Vr  
1
Ze
ZZe 2
40 r
1
Coulomb force is:
 
 
1) Conservative force – force is gradient of potential energy: Fr   Vr 


2) Central force: Vr   V r   Vr 
Magnitude of Coulomb force is
ZZe 2
Fr  
and force acts in the direction of particle join.
40 r 2
1
Electrostatic force is thus proportional to 1/r2  trajectory of  particle is a hyperbola with
nucleus in its external focus.
We define:
Impact parameter b – minimal distance on which  particle comes near to the nucleus in the case
without force acting.
Scattering angle  - angle between asymptotic directions of  particle arrival and departure.
First we find relation between b and  :

Nucleus gives to the  particle impulse  Fdt  particle
momentum changes from original value p0 to final
value p:

 

p  p  p 0   Fdt
…………. (1)
Using assumption about target fixation we obtain that
kinetic energy and magnitude of  particle momentum
before, during and after scattering are the same:
p0 = p = mv0=mv
Geometry of Rutheford scattering.
We see from figure:
1
 
 
p  m v 0  sin    p  2m v 0  sin  
2
2
2
……….. (2)
Because impulse is in the same direction as the change of
momentum, it is valid:
…………… (3)
 F  cos   dt
Momenta in Rutheford scattering:


where  is running angle between F and p along particle trajectory.

We substitute (2) and (3) to (1):
 
2m v0  sin     F  cos  dt
2 0
 
2m v 0  sin   
2
We change integration variable from t to :
………….....................……(4)
 1 2   
-1 2

  
F  cos  

dt
d
d
…. (5)
where ddt is angular velocity of  particle motion around nucleus. Electrostatic action of nucleus

on particle is in direction of the join vector  r  F  0  force momentum do not act 
angular momentum is not changing (its original value is mv0b) and it is connected with angular
velocity = d/dt  mr2 = const = mr2 (d/dt) = mv0b
2
then:
dt
r

d v 0 b
 1 2   
 
2m v b  sin   
Fr 2 cos   d

 2  1 2   
2
0
we substitute dt/d at (5):
We substitute electrostatic force F (Z=2):
We obtain:
 1 2   
2
2Ze 2
Fr cos   d 

40
1 2   
 1 2   
because it is valid:
1 2
 1 2   
1 2
2Ze 2
F
4 0 r 2
1
Ze 2
  d 
cos



  d sin   
cos

 1 2   
 1 2   

................................… (6)
0
 
cos 
2
  
 
 2 sin     2 cos 
2 2
2
2
   Ze
 
cos 
We substitute to the relation (6): 2m v b  sin   
 2  0
2
2
Scattering angle  is connected with
   20 m v0 b 40 E KIN
cotg   

b
collision parameter b by relation:
Ze 2
Ze 2
2
2
0
The smaller impact parameter b the larger scattering angle .
… (7)
Energy and momentum conservation law
Just these conservation laws are very important. They determine relations between kinematic
quantities. It is valid for isolated system:
nf
nf
E  E
Conservation law of whole energy:
k 1
 m c
ni
k
j
j1
k 1
 m c    E    m c    E 
nf
k 1
nf
2
0
k
k 1
nj
ni
KIN k
j1
j
j1
 E KIN
   m c
nj
k
j1
0
2
 E KIN
M f0 c 2  E fKIN  M i0 c 2  E iKIN
2
0
0
2
KIN j
Nonrelativistic approximation (m0c2 >> EKIN): EKIN = p2/(2m0)
M f0  M i0
M f0 c 2  M i0 c 2
Together it is valid for elastic scattering: E
f
KIN
E
ni
 p2 
 p2 

   


k 1  2m 0  k
j1  2m 0  j
nf
i
KIN
Ultrarelativistic approximation (m0c2 << EKIN): E ≈ EKIN ≈ pc
E E
f
i
E
f
KIN
E
i
KIN
nf
ni
k 1
j1
 p k c   p jc
nf
Conservation law of whole momentum:
ni


p

 k pj
k 1
j1
nf
ni
k 1
j1
 pk   p j

j
We obtain for elastic scattering:
Using momentum conservation law:
0  p1 sin   p2 sin 
and p1  p1 cos  p2 cos
We obtain using cosine theorem:
p22  p12  p12  2p1p1 cos 
Nonrelativistic approximation:
Using energy conservation law:
p12
p12
p22


2m1 2m1 2m 2
We can eliminated two variables using these equations. The energy of reflected target particle
E‘KIN 2 and reflection angle ψ are usually not measured. We obtain relation between remaining
kinematic variables using given equations:
 m 
 m 
m
p12 1  1   p12 1  1   2 1 p1p1 cos   0
m2
 m2 
 m2 
 m 
 m 
m
EKIN1 1  1   E KIN1 1  1   2 1 E KIN1EKIN1 cos   0
m2
 m2 
 m2 
Ultrarelativistic approximation:
Using energy conservation law:
p1  p1  p2  p2  p1  p1  2p1p1
2
We obtain using this relation and momentum conservation law:
2
2
cos   1 and therefore:   0
Conception of cross section
1) Base conceptions – differential, integral cross section, total cross section, geometric
interpretation of cross section
2) Macroscopic cross section, mean free path.
3) Typical values of cross sections for different processes
Chamber for measurement of cross sections of
different astrophysical reactions at NPI of ASCR
Ultrarelativistic heavy ion collision
on RHIC accelerator at Brookhaven
Introduction of cross section.
Formerly we obtained relation between Rutherford scattering angle and impact parameter
( particles are scattered):
   40 E KIN
cotg   
b
Ze 2
2
………………… (1)
The smaller impact parameter b, the bigger scattering angle .
Impact parameter is not directly measurable and new directly measurable quantity must be define.
We introduce scattering cross section  for quantitative description of scattering processes:
Derivation of Rutherford relation for scattering:
Relation between impact parameter b and scattering angle   particle with impact parameter
smaller or the same as b (aiming to the area b2) is scattered to a angle larger than value b given
by relation (1) for appropriate value of b. Then applies:
(b) = b2 ……………….……....………. (2)
(then dimension of  is m2, barn = 10-28 m2)
We assume thin foil (cross sections of neighboring nuclei are not overlapping and multiple
scattering does not take place) with thickness L with nj atoms in volume unit. Beam with
number NS of  particles are going to the area SS. (Number of beam particles per time and area
units – luminosity – is for present accelerators up to 1038 m-2s-1).
The number of target nuclei on which  particles are impinging is: Nj = njLSS. Sum of cross sections
 for scattering to angle b and more is:
(b) = njLSS.
Fraction f(b) of incident  particles scattered to angle larger then b is:
Reminder of equation (2)
(b) = b2
Reminder of equation (1)
N (  b )  n j LS S 
f(  b )  


 n j L  b 2
NS
SS
SS
   40 E KIN
cotg   
b
2
2
Ze
 
2
We substitute of b from equation (1):
Sketch of the Rutherford experiment
 Ze 2 

 cot g 2   ………………… (3)
f (  b )    n jL
2
 40 E KIN 
Angular distribution of scattered particles
Reminder of pictures
Reminder of equation (3)
2
 Ze 2 

 cot g 2  
f (  b )    n jL
2
 40 E KIN 
During real experiment detector measures  particles scattered to angles from  up to +d.
Fraction of incident  particles scattered to such angular range is:
2
 Ze 2 


 cotg   sin 2  d
df    n j L
2
2
 40 E KIN 
We can write for detector area in distance r from the target:
   
dS  (2  r sin  )( rd )  2  r 2 sin   d  4  r 2 sin   cos d
2 2
Number N() of  particles going to the detector per area unit is:
2
 Ze 2 


 cotg   sin 2  d
N S  n j L
dN  N S df
2
2
 40 E KIN 


dS
dS
   
4  r 2 sin   cos d
2 2
Such relation is known as Rutherford equation for scattering.
dN 

dS
N S n jLZ 2 e 4
80 2 r 2 E 2KIN sin 4   
2
… (4)
Different types of differential cross sections:
angular
d
( ,  )
d
d
( )
d
d
(E)
dE
spectral
spectral angular
d
( E,  ,  )
dEd 
double or triple differential cross section
Integral cross sections: through energy, angle
Values of cross section:
Very strong dependence of cross sections on energy of beam particles and interaction character.
Values are within very broad range:  10-47 m2 ÷  10-24 m2 →  10-19 barn ÷  104 barn
Strong interaction (interaction of nucleons and other hadrons):
 10-30 m2 ÷  10-24 m2 →  0.01 barn ÷  104 barn
Electromagnetic interaction (reaction of charged leptons or photons):
 10-35 m2 ÷  10-30 m2 →  0.1 μbarn ÷  10 mbarn
Weak interaction (neutrino reactions):
 10-47 m2 = 10-19 barn
Cross section of different neutron
reactions with gold nucleus
Macroscopic quantities:
Particle passage through matter: interacted particles disappear from beam (N0 – number of
incident particles):
dN

 n j  dx
N
N
x
dN
N N  n j 0 dx
0
ln N – ln N0 = – njσx
N  N0 e
 n j  x
Number of touched particles N decrease exponential with thickness x:
Number of interacting particles: N0  N  N0 (1  e
For x→0 : e
 n j  x
 n j  x
)
 1  n j  x
N0 – N  N0 – N0(1-njx)  N0njx
and then:
dN N 0  N

 n j  x
N
N0
Absorption coefficient  = nj
Mean free path l = is mean distance which particle travels in a matter before interaction.
l
1
n j
Quantum physics  all measured macroscopic quantities , l are mean values (l is statistical
quantity also in classical physics).
Phenomenological properties of nuclei
1) Introduction - nucleon structure of nucleus
6) Magnetic and electric moments
2) Sizes of nuclei
7) Stability and instability of nuclei
3) Masses and bounding energies of nuclei
8) Nature of nuclear forces
4) Energy states of nuclei
5) Spins
Introduction – nucleon structure of nuclei.
Atomic nucleus consists of nucleons (protons and neutrons).
Number of protons (atomic number) – Z. Total number nucleons (nucleon number) – A.
Number of neutrons – N = A-Z.
A
Z
Pr N
Different nuclei with the same number of protons – isotopes.
Different nuclei – nuclides.
Different nuclei with the same number of neutrons – isotones. Nuclei with N1 = Z2 and
N2 = Z1 – mirror nuclei
Different nuclei with the same number of nucleons – isobars.
Neutral atoms have the same number of electrons inside atomic electron shell as protons inside
nucleus.
Proton number gives also charge of nucleus: Qj = Z·e
(Direct confirmation of charge value in scattering experiments – from Rutherford equation
for scattering (dσ/dΩ) = f(Z2))
Atomic nucleus can be relatively stable in ground state or in excited state to higher
energy – isomers (τ > 10-9s).
Stable nuclei have A and Z which fulfill approximately empirical equation: Z 
A
1.98  0.0155A 2/3
Reliably known nuclei are up to Z=112 in present time (discovery of nuclei with Z=114, 116
(Dubna) needs confirmation).
Nuclei up to Z=83 (Bi) have at least one stable isotope. Po (Z=84) has not stable isotope.
Th , U a Pu have T1/2 comparable with age of Earth.
Maximal number of stable isotopes is for Sn (Z=50) - 10 (A =112, 114, 115, 116, 117, 118, 119, 120,
122, 124).
Total number of known isotopes of one element is till 38.
Number of known nuclides: > 2800.
Sizes of nuclei
Distribution of mass or charge in nucleus are determined.
We use mainly scattering of charged or neutral particles on nuclei.
Density ρ of matter and charge is constant inside nucleus and we observe fast density decrease on
the nucleus boundary. The density distribution can be described very well for spherical nuclei by
0
relation (Woods-Saxon):
 (r ) 
 ( r R )
1 e
where α is diffusion coefficient. Nucleus radius R is distance from the center, where density is half of
maximal value. Approximate relation R = f(A) can be derived from measurements: R = r0A1/3
where we obtained from measurement r0 = 1,2(1) 10-15 m = 1,2(2) fm (α = 1,8 fm-1). This shows on
permanency of nuclear density. Using Avogardo constant
or using proton mass:

Am p
4   R3
3

mp
4   r03
3

1.67 1027 kg
4   1.2 10
3
15
m
3
High energy electron scattering (charge distribution)  smaller r0.
Neutron scattering (mass distribution)  larger r0.
Larger volume of neutron matter is done by larger number
of neutrons at nuclei (in the other case the volume of protons
should be larger because Coulomb repulsion).
Distribution of mass density connected with charge ρ = f(r)
measured by electron scattering with energy 1 GeV
we obtain   1017 kg/m3.
Deformed nuclei – all nuclei are not spherical, together with smaller values of deformation of some
nuclei in ground state the superdeformation (2:1  3:1) was observed for highly excited states. They
are determined by measurements of electric quadruple moments and electromagnetic transitions
between excited states of nuclei.
Neutron and proton halo – light nuclei with relatively large excess of neutrons or protons
→ weakly bounded neutrons and protons form halo around central part of nucleus.
Experimental determination of nuclei sizes:
1) Scattering of different particles on nuclei: Sufficient energy of incident particles is necessary
for studies of sizes r = 10-15m. De Broglie wave length λ = h/p < r:
Neutrons: mnc2 >> EKIN →   h/ 2mE KIN → EKIN > 20 MeV
Electrons: mec2 << EKIN → λ = hc/EKIN → EKIN > 200 MeV
2) Measurement of roentgen spectra of mion atoms: They have mions in place of electrons
(mμ = 207 me): μ,e – interact with nucleus only by electromagnetic interaction. Mions are ~200
nearer to nucleus → „feel“ size of nucleus (K-shell radius is for mion at Pb 3 fm ~ size of nucleus)
3) Isotopic shift of spectral lines: The splitting of spectral lines is observed in hyperfine structure of
spectra of atoms with different isotopes – depends on charge distribution – nuclear radius.
4) Study of α decay: The nuclear radius can be determined using relation between probability of
α particle production and its kinetic energy.
Masses of nuclei
Nucleus has Z protons and N=A-Z neutrons. Naive conception of nuclear masses:
M(A,Z) = Zmp+(A-Z)mn
where mp is proton mass (mp  938.27 MeV/c2) and mn is neutron mass (mn  939.56 MeV/c2)
where MeV/c2 = 1.78210-30 kg, we use also mass unit: mu = u = 931.49 MeV/c2 = 1.66010-27 kg.
Then mass of nucleus is given by relative atomic mass Ar=M(A,Z)/mu.
Real masses are smaller – nucleus is stable against decay because of energy conservation law.
Mass defect ΔM:
ΔM(A,Z) = M(A,Z) - Zm + (A-Z)m
p
n
It is equivalent to energy released by connection of single nucleons to nucleus - binding energy
B(A,Z) = - ΔM(A,Z) c2
Binding energy per one nucleon B/A:
Maximal is for nucleus 56Fe (Z=26, B/A=8.79 MeV).
Possible energy sources:
1) Fusion of light nuclei
2) Fission of heavy nuclei
8.79 MeV/nucleon  1.4·10-13 J/1.66·10-27 kg = 8.7·1013 J/kg
(gasoline burning: 4.7·107 J/kg)
Binding energy per one nucleon for stable nuclei
Measurement of masses and binding energies:
Mass spectroscopy:
Mass spectrographs and spectrometers use particle motion in electric and magnetic fields:
Mass m=p2/2EKIN can be determined by comparison of momentum and kinetic energy. We use
passage of ions with charge Q through “energy filter” and “momentum filter”, which are realized
using electric and magnetic fields:



 
 
FE  QE
FB  Qv  B for Bv is FB = QvB
and then F = QE
The study of Audi and Wapstra from 1993 (systematic review of nuclear masses) names 2650
different isotopes. Mass is determined for 1825 of them.
Frequency of revolution in magnetic field of ion storage ring is used. Momenta are equilibrated by
electron cooling → for different masses → different velocity and frequency.
Comparison of frequencies (masses)
of ground and isomer states of
52Mn. Measured at GSI Darmstadt
Electron cooling of
storage ring ESR
at GSI Darmstadt
Excited energy states
Nucleus can be both in ground state and in state with higher energy – excited state
Every excited state – corresponding energy→ energy level
Quantum physics → discrete values of possible energies
Scheme of energy levels:
Deexcitation of excited nucleus from higher level to lower one by photon irradiation (gamma ray)
or direct transfer of energy to electron from electron cloud of atom – irradiation of conversion
electron. Nucleus is not changed. Or by decay (particle emission). Nucleus is changed.
Three types of nuclear excited states:
1) Particle – nucleons at excited state EPART
2) Vibrational – vibration of nuclei EVIB
3) Rotational – rotation of deformed nuclei EROT
(quantum spherical object can not have rotational energy)
it is valid: EPART >> EVIB >> EROT
Energy level structure of 66Cu nucleus
(measured at GANIL – France,
experiment E243)
Spins of nuclei
Protons and neutrons have spin 1/2. Vector sum of spins and orbital angular momenta is
total angular momentum of nucleus I which is named as spin of nucleus
Orbital angular momenta of nucleons have integral values → nuclei with even A – integral spin
nuclei with odd A – half-integral spin
  
l  r  p . At quantum physic by appropriate operator,
Classically angular momentum is define
ˆ ˆas ˆ
which fulfill commutating relations: l  l  i l
There are valid such rules:

̂
1) Eigenvalues I 2 are ˆI 2  I(I  1) 2 , where number I = 0, 1/2, 1, 3/2, 2, 5/2 …
angular momentum magnitude is |I| = ħ [I(I-1)]1/2
2) From commutation
̂ 2 relations it results, that vector components can not be observed individually.
Simultaneously I and only one component – for example Iz can be observed .
3) Components (spin projections) can take values Iz = Iħ, (I-1)ħ, (I-2)ħ, … -(I-1)ħ, -Iħ together 2I+1
values.
4) Angular momentum is given by number I = max(Iz). Spin corresponding to orbital angular
momentum of nucleons is only integral: I ≡ l = 0, 1, 2, 3, 4, 5, … (s, p, d, f, g, h, …),
intrinsic spin of nucleon is I ≡ s = 1/2.
ˆ ˆ 
5) Superposition for single nucleon j  l  ŝ leads to j = l  1/2. Superposition for system of more
particles is diverse. Extreme cases:
 
   
LS-coupling, where ˆI  Lˆ  Sˆ , Lˆ   ˆli , Sˆ   ŝi
i
i
jj-coupling, where
ˆ
ˆ
I   ji
i
Magnetic and electric momenta
Magnetic dipole moment μ is connected to existence of spin I and charge Ze. It is given by relation:


  g jI
  g j I
where g is gyromagnetic ratio and μj is nuclear magneton:
Bohr magneton:
j 
B 
e
 3.15 1014 MeVT 1
2mp c
e
 5.79 10 11 MeVT 1
2m e c
For point like particle g = 2 (for electron agreement μe = 1.0011596 μB). For nucleons μp = 2.79 μj
and μn = -1.91 μj – anomalous magnetic moments show complicated structure of these particles.
Magnetic moments of nuclei are only μ = -3 μj  10 μj, even-even nuclei μ = I = 0 → confirmation
of small spins, strong pairing and absence of electrons at nuclei.
Electric momenta:
Electric dipole momentum: is connected with charge polarization of system. Assumption: nuclear
charge in the ground state is distributed uniformly → electric dipole momentum is zero.
Agree with experiment.
Electric quadruple moment Q: gives difference of charge distribution from spherical. Assumption:
Nucleus is rotational ellipsoid with uniformly distributed charge Ze:
2
Q

Z(c 2  a 2 )
(c,a are main split axles of ellipsoid) deformation δ = (c-a)/R = ΔR/R
5
Results of measurements:
1) Most of nuclei have Q = 10-29 10-30 m2 → δ ≤ 0.1
2) In the region A ~ 150  180 and A ≥ 250 large values are measured: Q ~ 10-27 m2. They are larger
than nucleus area. → δ ~ 0.2  0.3 → deformed nuclei.
Stability and instability of nuclei
Stable nuclei: for small A (<40) is valid Z = N, for heavier nuclei N  1,7 Z. This dependence can be
express more accurately by empirical relation:
A
Z
1.98  0.0155A 2/3
For stable heavy nuclei excess of neutrons → charge density and destabilizing influence of Coulomb
repulsion is smaller for larger number of neutrons.
N
Z
number of stable nuclei
Even-even nuclei are more stable → existence of pairing even
even
156
even
odd
48
odd
even
50
odd
odd
5
Magic numbers – observed values of N and Z with increased stability.
At 1896 H. Becquerel observed first sign of instability of nuclei – radioactivity. Instable
nuclei irradiate:
Alpha decay → nucleus transformation by 4He irradiation
Beta decay → nucleus transformation by e-, e+ irradiation or capture of electron from atomic cloud
Gamma decay → nucleus is not changed, only deexcitation by photon or converse electron irradiation
Spontaneous fission → fission of very heavy nuclei to two nuclei
Proton emission → nucleus transformation by proton emission
Nuclei with livetime in the ns region are studied in present time. They are bordered by:
proton stability border during leave from stability curve to proton excess (separative energy
of proton decreases to 0) and neutron stability border – the same for neutrons. Energy level
width Γ of excited nuclear state and its decay time τ are connected together by relation τΓ ≈ h.
Boundery for decay time Γ < ΔE (ΔE – distance of levels) ΔE~ 1 MeV→ τ >> 6·10-22s.
Nature of nuclear forces
The forces inside nuclei are electromagnetic interaction (Coulomb repulsion), weak (beta decay)
but mainly strong nuclear interaction (it bonds nucleus together).
For Coulomb interaction binding energy is B  Z (Z-1)  B/Z  Z for large Z  non saturated
forces with long range.
For nuclear force binding energy is B/A  const – done by short range and saturation
of nuclear forces. Maximal range ~1.7 fm
Nuclear forces are attractive (bond nucleus together), for very short distances (~0.4 fm) they
are repulsive (nucleus does not collapse). More accurate form of nuclear force potential can be
obtained by scattering of nucleons on nucleons or nuclei.
Charge independency – cross sections of nucleon scattering
are not dependent on their electric charge. → For nuclear
forces neutron and proton are two different states of single
particle - nucleon. New quantity isospin T is define for
their description. Nucleon has than isospin T = 1/2 with
two possible orientation TZ = +1/2 (proton) and TZ = -1/2
(neutron). Formally we work with isospin as with spin.
Spin dependence – explains existence of stable deuteron (it exists only at triplet state – s = 1 and no
at singlet - s = 0) and absence of di-neutron. This property is studied by scattering experiments
using oriented beams and targets.
Tensor character – interaction between two nucleons depends on angle between spin directions and
direction of join of particles.
Expect strong interaction electric force influences also. Nucleus has positive charge and for
positive charged particle this force produces Coulomb barrier (range of electric force is larger
then this of strong force). Appropriate potential has form V(r) ~ Q/r.
In the case of scattering centrifugal barier given by angular momentum of incident particle acts in
addition.
Models of atomic nuclei
1) Introduction
2) Drop model of nucleus
3) Shell model of nucleus
Octupole vibrations of nucleus.
(taken from H-J. Wolesheima,
GSI Darmstadt)
Extreme superdeformed states were
predicted on the base of models
Introduction
Nucleus is quantum system of many nucleons interacting mainly by strong nuclear interaction.
Theory of atomic nuclei must describe:
1) Structure of nucleus (distribution and properties of nuclear levels)
2) Mechanism of nuclear reactions (dynamical properties of nuclei)
Development of theory of nucleus needs overcome of three main problems:
1) We do not know accurate form of forces acting between nucleons at nucleus.
2) Equations describing motion of nucleons at nucleus are very complicated – problem
of mathematical description.
3) Nucleus has at the same time so many nucleons (description of motion of every its particle is not
possible) and so little (statistical description as macroscopic continuous matter is not possible).
Real theory of atomic nuclei is missing  only models exist.
Models replace nucleus by model reduced physical system.  reliable and sufficiently
simple description of some properties of nuclei.
Models of atomic nuclei can be divided:
A) According to magnitude of nucleon interaction:
Collective models (models with strong coupling) – description of properties of nucleus given by
collective motion of nucleons
Singleparticle models (models of independent particles) – describe properties of nucleus given by
individual nucleon motion in potential field created by all nucleons at nucleus.
Unified (collective) models – collective and singleparticle properties of nuclei together are reflected.
B) According to, how they describe interaction between nucleons:
Phenomenological models – mean potential of nucleus is used, its parameters are determined
from experiment.
Microscopic models – proceed from nucleon potential (phenomenological or
microscopic) and calculate interaction between nucleons at nucleus.
Semimicroscopic models – interaction between nucleons is separated to two parts: mean potential
of nucleus and residual nucleon interaction.
Liquid drop model of atomic nuclei
Let us analyze of properties similar to liquid. Think of nucleus as drop of incompressible liquid
bonded together by saturated short range forces
Description of binding energy: B = B(A,Z)
We sum different contributions: B = B1 + B2 + B3 + B4 + B5
1) Volume (condensing) energy: released by fixed and saturated nucleon at nuclei:
B1 = aVA
2) Surface energy: nucleons on surface → smaller number of partners → addition of negative
member proportional to surface S = 4πR2 = 4πA2/3:
B2 = -aSA2/3
3) Coulomb energy: repulsive force between protons decreases binding energy. Coulomb energy for
uniformly charged sphere is E  Q2/R. For nucleus Q2 = Z2e2 a R = r0A1/3: B3 = -aCZ2A-1/3
4) Energy of asymmetry: neutron excess decreases binding energy
5) Pair energy: binding energy for paired nucleons increases:
+  for even-even nuclei
B5 = 0 for nuclei with odd A
where   aPA-1/2
-  for odd-odd nuclei
We sum single contributions and substitute to relation for mass:
TZ2
(Z  A 2) 2
B 4  a A
 a A
A
A
Volume energy
Surface energy
Coulomb energy
Asymmetry energy
M(A,Z) = Zmp+(A-Z)mn –
M(A,Z) = Zmp+(A-Z)mn–aVA+aSA2/3 + aCZ2A-1/3 + aA(Z-A/2)2A-1±δ
B(A,Z)/c2
Binding energy
Weizsäcker semiempirical mass formula. Parameters are fitted
using measured masses of nuclei.
(aV = 15.85, aA = 92.9, aS = 18.34, aP = 11.5, aC = 0.71 all in MeV/c2)
Shell model of nucleus
Assumption: primary interaction of single nucleon with force field created by all nucleons.
Nucleons are fermions  one in every state (filled gradually from the lowest energy).
Experimental evidence:
1) Nuclei with value Z or N equal to 2, 8, 20, 28, 50, 82, 126 (magic numbers) are more stable
(isotope occurrence, number of stable isotopes, behavior of separation energy magnitude).
2) Nuclei with magic Z and N have zero quadrupol electric moments  zero deformation.
3) The biggest number of stable isotopes is for even-even combination (156), the smallest for
odd-odd (5).
4) Shell model explains spin of nuclei. Even-even nucleus  protons and neutrons are paired. Spin
and orbital angular momenta for pair are zeroed. Either proton or neutron is left over in odd
nuclei. Half-integral spin of this nucleon is summed with integral angular momentum of rest of
nucleus  half-integral spin of nucleus. Proton and neutron are left over at odd-odd nuclei 
integral spin of nucleus.
Shell model:
1) All nucleons create potential, which we describe by 50 MeV deep square potential well with
rounded edges, potential of harmonic oscillator or even more realistic Woods-Saxon potential.
2) We solve Schrödinger equation for nucleon in this potential well. We obtain stationary states
characterized by quantum numbers n, l, ml. Group of states with near energies creates shell.
Transition between shells  high energy. Transition inside shell  law energy.
3) Coulomb interaction must be included  difference between proton and neutron states.
4) Spin-orbital interaction must be included. Weak LS coupling for the lightest nuclei. Strong
jj coupling for heavy nuclei. Spin is oriented same as orbital angular momentum → nucleon is
attracted to nucleus stronger. Strong split of level with orbital angular momentum l.
without spin-orbital
coupling
with spin-orbital
coupling
Number
Number
per shell
Energy
per level
Sequence of energy levels of nucleons given by
shell model (not real scale) – source A. Beiser
Total
number
Radioactivity
1) Introduction
2) Decay law
3) Alpha decay
4) Beta decay
5) Gamma decay
6) Fission
7) Decay series
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.
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.
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
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.
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
After supplial of energy – induced fission – energy supplied by photon (photofission), by neutron, …
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
Nuclear reactions
1) Introduction
2) Nuclear reaction yield
3) Conservation laws
4) Nuclear reaction mechanism
5) Compound nucleus reactions
6) Direct reactions
Fission of 252Cf nucleus
(taken from WWW pages of
group studying fission at LBL)
Introduction
Incident particle a collides with a target nucleus A → different processes:
1) Elastic scattering – (n,n), (p,p), …
2) Inelastic scattering – (n,n‘), (p,p‘), …
3) Nuclear reactions:
a) creation of new nucleus and
particle - A(a,b)B
b) creation of new nucleus and
more particles - A(a,b1b2b3…)B
c) nuclear fission – (n,f)
d) nuclear spallation
from point of view of used projectile:
e) photonuclear reactions - (γ,n), (γ,α), …
f) radiative capture – (n, γ), (p, γ), …
g) reactions with neutrons – (n,p), (n, α) …
h) reactions with protons – (p,α), …
i) reactions with deuterons – (d,t), (d,p), (d,n) …
j) reactions with alpha particles – (α,n), (α,p), …
k) heavy ion reactions
Reaction can be described in the form A(a,b)B, for example:
27Al(n,α)24Na
or 27Al + n → 24Na + α
input channel - particles (nuclei) enter into reaction and their characteristics (energies,
momenta, spins, …)
output channel – particles (nuclei) get off reaction and their characteristics
Cross section σ depends on energies, momenta, spins, charges … of involved particles
Dependency of cross section on energy σ (E) – excitation function.
Threshold reactions – occur only for energy higher than some value.
Reaction yield – number of reactions divided by number of incident particles.
Thin target – does not changed intensity and energy of beam particles
Thick target – intensity and energy of beam particles are changed
Nuclear reaction yield
Reaction yield – number of reactions ΔN divided by number of incident particles N0: w = ΔN /N0
Depends on specific target
Thin target – does not changed intensity and energy of beam particles → reaction yield:
w = ΔN /N0 = σnx
where n – number of target nuclei in volume unit, x is target thickness → nx is surface target
density.
Thick target – intensity and energy of beam particles are changed. Process depends on type of
particles:
1) Reactions with charged particles – energy losses by ionization and excitation of target atoms.
Reactions occur for different energies of incident particles. Number of particle is changed by
nuclear reactions (can be neglected for some cases). Thick target (thickness d > range R):
dN = N(x)nσ(x)dx ≈ N0nσ(x)dx
(reaction with nuclei are neglected N(x) ≈ N0)
R
Reaction yield is (d > R):
ΔN
w
 n   (x)dx  n
N0
0
E KINa

0
 (E KIN )
dE
 KIN
dx
dE KIN
Higher energies of incident particle and smaller ionization losses → higher range and yield
w=w(EKIN) – excitation function
R
Mean cross section:
1
    (x)dx
R0
→
w  n  R
2) Neutron reactions – no interaction with atomic shell, only scattering and absorption on nuclei.
Number of neutrons is decreasing but their energy is not changed significantly. Beam of
monoenergy neutrons with yield intensity N0. Number of reactions dN in a target layer dx for
dN = -N(x)nσdx
deepness x is:
where N(x) is intensity of neutron yield in place x and σ is total cross section σ = σpr + σnepr + σabs + …
We integrate equation: N(x) = N0e-nσx
for
0≤x≤d
Number of interacting neutrons from N0 in target with thickness d is: ΔN = N0(1 – e-nσd)
Reaction yield is:
w
N  R  R


(1  e  n d )
N0 

σ – total cross section
σR – cross section of given reaction
3) Photon reactions – photons interact with nuclei and electrons → scattering and absorption →
decreasing of photon yield intensity:
I(x) = I0e-μx
where μ is linear attenuation coefficient (μ = μan, where μa is atomic attenuation coefficient and n is
number of target atoms in volume unit).
For thin target (attenuation can be neglected) reaction yield is:
where ΔI is total number of reactions and from this
reactions.
We obtain for thick target with thickness d:
w
I

a
w
I 

 n  d
I0 a
is number of studied photonuclear
I 



(1  e  a nd )
I0 a a
Conservation laws
Energy conservation law and momenta conservation law:
Described in the part about kinematics. Directions of fly out and possible energies of reaction
products can be determined by these laws.
Type of interaction must be known for determination of angular distribution.
Angular momentum conservation law – orbital angular momentum given by relative motion of
two particles can have only discrete values l = 0, 1, 2, 3, … [ħ]. → For low energies and short range
of forces → reaction possible only for limited small number l. Semiclasical (orbital angular
momentum is product of momentum and impact parameter):
pb = lħ → l ≤ pbmax/ħ = 2πR/ λ
where λ is de Broglie wave length of particle and R is interaction range. Accurate quantum
mechanic analysis → reaction is possible also for higher orbital momentum l, but cross section
rapidly decreases. Total cross section can be split:    l
l
Charge conservation law – sum of electric charges before reaction and after it are conserved.
Baryon number conservation law – for low energy (E < mnc2) → nucleon number conservation
law
Mechanisms of nuclear reactions
Different reaction mechanism:
1) Direct reactions (also elastic and inelastic scattering) - reactions insistent very briefly τ ≈ 10-22s →
wide levels, slow changes of σ with projectile energy
2) Reactions through compound nucleus – nucleus with lifetime τ ≈ 10-16s is created → narrow levels
→ sharp changes of σ with projectile energy (resonance character), decay to different channels
Reaction through compound nucleus
Reactions during which projectile energy is distributed to more nucleons of target nucleus → excited
compound nucleus is created → energy cumulating → single or more nucleons fly out.
Compound nucleus decay 10-16s.
Two independent processes: Compound nucleus creation
Compound nucleus decay
Cross section σab reaction from incident channel a and final b through compound nucleus C:
σab = σaCPb
where σaC is cross section
for compound nucleus creation and Pb is probability of compound nucleus decay to channel b.
Direct reactions
Direct reactions (also elastic and inelastic scattering) - reactions continuing very short 10-22s
Stripping reactions – target nucleus takes away one or more nucleons from projectile, rest of
projectile flies further without significant change of momentum - (d,p) reactions.
Pickup reactions – extracting of nucleons from nucleus by projectile
Transfer reactions – generally transfer of nucleons between target and projectile.
Diferences in comparison with reactions through compound nucleus:
a) Angular distribution is asymmetric – strong increasing of intensity in impact direction
b) Excitation function has not resonance character
c) Larger ratio of flying out particles with higher energy
d) Relative ratios of cross sections of different processes do not agree with compound nucleus model