CHEM 312 Lecture 9: Nuclear Reactions
• Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and
Radiochemistry, Chapter 4
• Notation
• Energetics of Nuclear Reactions
• Reaction Types and Mechanisms

Barriers

Scattering
• Nuclear Reaction Cross Sections
• Reaction Observables
• Scattering
• Direct Reactions
• Compound Nuclear Reactions
• Photonuclear Reactions
• Nucleosynthesis
9-1
•
•
•
•
Nuclear Reactions
Nucleus reactions with a range of particles

nucleus, subatomic particle, or photon to
produce other nuclei

Short time frame (picosecond)
First nuclear reaction from Rutherford

What reaction was this?
Number of terms conserved during nuclear
reactions

Number of nucleons
 except in reactions involving creation
or annihilation of antinucleons

charge

Energy
1 . 193 MeV

momentum
atom

angular momentum

parity
Q is the energy of the reaction
14
7
N  24He178 O 11H  Q
14
N ( , p ) O
17
Q=-1.193 MeV
x
6 . 02 E 23 atoms
mole
x
1 . 6 E  13 J
MeV
 1 . 15 E 11
J
mole
 positive Q corresponds to energy release
 negative Q to energy absorption
• Q terms given per nucleus transformed
9-2
Energetics
• Reaction Q values
 Not necessarily equal to kinetic energy of bombarding particles
for the reaction to occur
 Need more energy than Q value for reaction to occur
* Reaction products will have kinetic energy that needs to
come from reaction
• Conservation of momentum
 Some particles’ kinetic energy must be retained by products as
kinetic energy
• Amount retained as kinetic energy of products
 Based on projectile mass
 Retained kinetic energy becomes smaller with increasing target
mass
APr ojectile
Equation for kinetic energy (T):
T 
Q
• What does this mean about reaction
APr ojectile  AT arg et
 Heavier target or heavier projectile?
248Cm + 18O266Rf
18
T 
Q  0 . 068 Q
248
248  18
T 
Q  0 . 932 Q 248Cm Projectile
248  18
18O
Projectile
9-3
Energetics: Reaction Barrier
•
Need to consider comparison of laboratory and
center of mass frame

Laboratory frame
 conservation of momentum considers
angle of particles
Q  T x (1 

T cm
•
•
mx
mR
)  T p (1 
mp
mR
)
2
mR
( m p T p m x T x ) cos 
Q  Tx  T p  TR
Center of mass
 Total particle angular momentum is
zero
vpm p
2
( m p  m T ) v cm
v cm 

(m p  mT )
2
Kinetic energy carried by projectile (Tlab) is not
fully available for reaction

Tlab - Tcm = T0
For reaction to occur Q + T0 must be achieved

Basis for threshold reaction

Q + T0 > 0
T cm  Tlab (
mp
m p  mT
)
9-4
•
Reaction Barrier
Threshold energy (minimum energy for reaction)
Q  T lab  T CM  0 ; T cm  T lab (
T lab  T lab (
T lab (1  (
mp
m p  mT
mp
m p  mT
T lab 
(1  (
T  Q
mp
m p  mT
Solve of laboratory T
)
)  Q
))   Q
Q
mp
m p  mT
Q

))
APr ojectile  AT arg et
(
m p  mT
m p  mT
(
mp
m p  mT

))
Q
mT
m p  mT
A for mass
MeV
AT arg et
•
Fraction of bombarding particle’s kinetic energy retained as kinetic energy of
products becomes smaller with increasing mass of target

Heavier target or heavier projectile?
248Cm + 18O266Rf

9-5
Reaction Barrier: Threshold Energy
•
Consider the 14N(,p)17O reaction
APr ojectile  AT arg et

Find threshold energy
T  Q
MeV
 Q from mass excess
AT arg et
* Q=2.425 + 2.863 – 7.289 – (-0.809) = -1.19 MeV
T   (  )1 . 19
•
•
•
•
4  14
MeV  1 . 53 MeV
14
Reaction barrier also induced by Coulomb interaction

Need to have enough energy to react and overcome Coulomb barrier
 From charge repulse as particle approach each other
2
* R is radius
Z 1Z 2 e
1/3
Vc 
R  ro A
* ro =1.1 to 1.6 fm
R1  R 2
Equation can vary due to ro
Z 1Z 2
Vc can be above threshold energy
V c  0 . 96 1 / 3
MeV
1/3
A1  A2
2*7
V c  0 . 96 1 / 3
MeV  3 . 36 MeV
1/3
4  14
Center of mass, need to bring to laboratory frame

Consider kinetic energy carried by projectile

3.36x ((14+4)/14) = 4.32 MeV alpha needed for reaction
9-6
Cross Section Limits
• Reaction cross section of R2 is approximated at high
energies
 Wave nature of incident particle causes upper limit of
reaction cross section to include de Broglie wavelength
So cross section can be larger that area
 r   (R  )
2
• Collision between neutron and target nucleus characterized
by distance of closest approach
 B is impact parameter
9-7
Cross section
l is partial cross section
of given angular
momentum l
 l    [( l  1)  l ]    ( 2 l  1)
2
•
2
2
2
Quantum-mechanical treatment Tl is the
transmission coefficient for reaction of a
neutron with angular momentum l

Represents fraction of incident
particles with angular momentum l
that penetrate within range of
nuclear forces
 Provides summing term to
increase cross section
 Reason why cross section can be
larger than physical size of
nucleus
•

 r  
2
 2 l  1T
l0
l
General trends for neutron and charged particles

Charged particle cross section minimal at
low energy

Neutron capture cross section maximum
at low energy
9-8
Types of Experiments: Excitation Functions
• variation of reaction cross section with incident energy
• shape can be determined by exposing several target foils in same beam with
energy-degrading
• provide information about probabilities for emission of various kinds of
particles and combinations of particles in nuclear reactions
 formation of given product implies what particles were ejected from the
target nuclide
• Range of cross sections can be evaluated
9-9
•
Low-Energy Reactions with Light
Projectiles
Elastic scattering

kinetic energy conserved
• Slow-Neutron Reactions
 Purest example of compoundnucleus behavior
1/v law governs most
neutron cross sections in
region of thermal energies
 neutrons available only from
nuclear reactions
 Range of energies can be
obtained
• Deuteron Reactions
 Prevalence of one nucleon
stripping
large size and loose
binding of deuteron
Only proton and
neutron in deuteron
nucleus
* Proton charge carries
both nucleons
9-10
High Energy Reactions
•
•
Spallation Products
 products in immediate neighborhood of target element found in
highest yields
 within 10 to 20 mass numbers
 yields tend to form in two regions
  stability for medium-weight products
 neutron-deficient side of stability with increasing Z of products
 Used to produce beam of neutrons at spallation neutron source
 Heavy Z will produce 20-30 neutrons
 Basis of Spallation neutron source
(http://neutrons.ornl.gov/facilities/SNS/)
High-Energy Fission
 single broad peak in mass-yield curve instead of double hump seen
in thermal-neutron fission
 many neutron-deficient nuclides
 especially among heavy products
 originate from processes involving higher deposition energies
 lower kinetic energies
 do not appear to have partners of comparable mass
 arise from spallation-like or fragmentation reactions
9-11
High-Energy Reactions
•
•
Mass-Yield Curves
 at low energies, compound-nucleus picture dominates
 as energy increases importance of direct reactions and preequilibrium (pre-compound nucleus)
emission increase
 above 100 MeV, nuclear reactions proceed nearly completely by direct interactions
 products down to mass number 150 are spallation products
 those between mass numbers 60 and 140 are fission products
Cascade-Evaporation Model

Above 100 MeV reactions

energy of the incident proton larger than interaction energy between the nucleons in the nucleus

Wavelength less than average distance between nucleons
 proton will collide with one nucleon at a time within the nucleus
* high-energy proton makes only a few collisions in nucleus
* Produces nucleons with high energy
9-12
Heavy Ion Reactions
• Inelastic scattering
 scattering in which some of projectile’s kinetic energy
transformed into excitation of target nucleus
greatest importance at large impact parameters
 heavy ions valuable
can excite high-spin states in target nuclei because of
large angular momenta
• Can experience Coulomb excitation
 high charges
 below Coulomb barrier heights and excite nuclei by purely
electromagnetic interactions
• Transfer Reactions
 stripping and pickup reactions prevalent with heavy ions
take place at impact parameters just below those at
which interactions are purely Coulombic
 angular distributions show oscillatory, diffraction-like
pattern when transfer reaction to single, well-defined state
observed
9-13
Heavy Ion Reactions: Deep Inelastic Reactions
• Relatively large amounts of nuclear matter
transferred between target and projectile
 Show strongly forward-peaked angular
distributions
 “Grazing contact mechanism”
• Products with masses in vicinity of projectile mass
appear at angles other than classical grazing angle
 Relatively small kinetic energies
• Total kinetic energies of products strongly correlated
with amount of mass transfer
 Increasing mass difference of product and
projectile lowers kinetic energy
• Product will dissociate into two fragments
 Appreciable fraction of incident kinetic energy
dissipated and goes into internal excitation
9-14
Compound-Nucleus Reactions
• compound-nucleus formation can only
take place over a restricted range of
small impact parameters
 can define critical angular
momentum above which
complete fusion cannot occur
 cf/R decreases with increasing
bombarding energy
• Neutron deficient heavy ions produce
compound nuclei on neutron-deficient
side of  stability belt
• Heavy ion of energy above Coulomb
barrier brings enough excitation energy
to evaporate several nucleons
 5-10 MeV deexcitation for neutron
evaporation
• heavy-ion reactions needed for
reaching predicted island of stability
around Z=114 to Z=184
• U is excitation energy, MA and Ma
masses of target and projectile, Ta is
projectile kinetic energy, Sa is projectile
binding energy in compound nucleus
M
U 
M
A
Ta  S a
A
M
a
9-15
Photonuclear reactions
• Reactions between nuclei and lowand medium-energy photons
dominated by giant resonance

Excitation function for
photon absorption goes
through a broad maximum a
few MeV wide
 Due to excitation of
dipole vibrations of
protons against neutrons
in the nucleus
• Resonance peak varies smoothly
with A

24 MeV at 16O

13 MeV at 209Bi
• Peak cross sections are 100-300 mb
• (, p), (, n), (,) reactions
http://www.engin.umich.edu/research
/cuos/ResearchGroups/HFS/Research
/photonuclear_reactions.html
9-16
Origin of Elements
•
•
•
•
Gravitational coalescence of H and He into clouds
Increase in temperature to fusion
Proton reaction
1H + n → 2H + 

2H + 1H → 3He

2H + n → 3H

3H + 1H → 4He + 

3He + n → 4He + 

3H + 2H → 4He + n

2H + 2H → 4He + 

4He + 3H → 7Li + 

3He+4He → 7Be + 

 7Be short lived
 Initial nucleosynthesis lasted 30 minutes
* Consider neutron reaction and free neutron half life
Further nucleosynthesis in stars

No EC process in stars
9-17
•
•
Stellar
Nucleosynthesis
He burning
4He + 4He ↔ 8Be

+ γ - 91.78 keV
 Too short
lived

3 4He → 12C + γ +
7.367 MeV
12C + 4He →16O

16O + 4He →20Ne

Formation of 12C based
on Hoyle state

Excited nuclear
state
 Somewhat
different
from ground
state 12C

Around 7.6 MeV
above ground
state

0+
9-18
Stellar Nucleosynthesis
•
•
CNO cycle
12C + 1H →13N + 

13N →13C + e++ νe

13C + 1H →14N + γ

14N + 1H →15O + γ

15O →15N + e+ + νe

15N + 1H →12C +

4He

Net result is
conversion of 4
protons to alpha
particle
 4 1H → 4He
+2 e++ 2 νe +3
γ
Fusion up to Fe

Binding energy
curve
9-19
Formation of elements A>60
Neutron Capture; S-process

A>60
68Zn(n, γ) 69Zn, 69Zn → 69Ga +  n


mean times of neutron capture reactions longer than beta decay
half-life
 Isotope can beta decay before another capture

Up to Bi
9-20
Nucleosynthesis: R process
• Neutron capture time scale very much less than - decay lifetimes
• Neutron density 1028/m3

Extremely high flux

capture times of the order of fractions of a second

Unstable neutron rich nuclei
• rapidly decay to form stable neutron rich nuclei
• all A<209 and peaks at N=50,82, 126 (magic numbers)
9-21
•
•
•
•
•
•
P process
Formation of proton rich nuclei
Proton capture process
70<A<200
Photonuclear process, at higher Z (around 40)

(, p), (,), (, n)
190Pt and 168Yb from p process

Also associated with proton capture process (p,)
Variation on description in the literature
9-22
• Proton-rich nuclei
with Z = 7-26
• (p,) and + decays
that populate the prich nuclei
 Also associated
with rapid
proton capture
process
• Initiates as a side
chain of the CNO
cycle
 21Na and 19Ne
• Forms a small
number of nuclei
with A< 100
rp process (rapid proton
capture)
9-23
Review Notes
• Understand Reaction Notation
• Understand Energetics of Nuclear Reactions
 Q values and barriers
• Understand the Different Reaction Types and
Mechanisms
 Particles
 Energy
• Relate cross sections to energy
• Describe Photonuclear Reactions
• Routes and reactions in nucleosynthesis
• Influence of reaction rate and particles on
nucleosynthesis
9-24
Questions
• Describe the different types of nuclear reactions shown on 9-24.
• Provide notations for the following

Reaction of 16O with 208Pb to make stable Au

Formation of Pu from Th and a projectile
• Find the threshold energy for the reaction of 59Co and an alpha
that produces a neutron and a product nuclei
• What are the differences between low and high energy reactions?
• How does a charged particle reaction change with energy? A
neutron reaction?
• How are actinides made in nucleosynthesis?
• What is the s-process?
• What elements were produced in the big bang?
• Which isotopes are produced by photonuclear reactions?
• What is interesting about the production of 12C
9-25
Pop Quiz
• Provide the Q value, threshold energy, and
Coulomb barrier for the compound nucleus
reaction of 18O with 244Cm
• Provide comment in blog
• Bring to next class (31 October)
9-26
CHEM 312 Lecture 10: Chemical Speciation
• Use of constants to model chemical form
 Thermodynamic and kinetic
 Determine property of radioelement based
on speciation
Chemical species in system
• Review
 Equilibrium constants
 Activity
 Use of constants in equation
9-27
Reaction Constants
• For a reaction
 aA + bB <--> cC + dD
• At equilibrium ratio of product to reactants is a
constant
 Constant can change with conditions
Not particularly constant
 By convention, constants are expressed as products
over reactants
c
d
K 
[C ] [D ]
a
[A ] [ B]
b
• Conditions under which the constant is measured
should be listed
 Temperature, ionic strength
9-28
Complete picture: Activities
• Strictly speaking, activities, not concentrations should be used
 C [C ]  D [ D ]
c
K 
•
•
•
•
d
 A [ A ] a  B [ B ]b
Activities normalize concentration to amount of anions and cations in
solution
At low concentration, activities are assumed to be 1
constant can be evaluated at a number of ionic strengths and overall
activities fit to equations
Debye-Hückel (Physik Z., 24, 185 (1923))
2
0.5085 Z a 
 log  A 
1  0.3281 R A 
ZA = charge of species A
µ = molal ionic strength
RA = hydrated ionic radius in Å (from 3 to 11)
First estimation of activity
9-29
Activities
• Debye-Hückel term can be written as:
D 

0.5107
1  1.5 
• Specific ion interaction theory
 Uses and extends Debye-Hückel
long range Debye-Hückel
Short range ion interaction term
ij = specific ion interaction term log  i 
2
 Z D   ij 
2
log ß ( )  log ß( 0 )   Z i D    ij 
• Pitzer
 Binary (3) and Ternary (2) interaction
parameters
9-30
Experimental Data shows change in
stability constant with ionic strength
6.6
Ion Specific Interaction Theory
Cm-Humate at pH 6
6.5
lo gß
6.4
6.3
6.2
K+
6.1
6.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
sqrt Im
Ca2+
Al3+
Fe(CN)64-
9-31
Constants
• Constants can be listed by different names
 Equilibrium constants (K)
Reactions involving bond breaking
* 2 HX <--> 2H+ + X22 Stability constants (ß), Formation constants (K)
Metal-ligand complexation
* Pu4+ + CO32- <--> PuCO32+
* Ligand is written in deprotonated form
 Conditional Constants
An experimental condition is written into equation
* Pu4+ + H2CO3 <--> PuCO32+ +2H+
Constant can vary with concentration, pH
Must look at equation!
9-32
Using Equilibrium Constants
• Constants and balanced equation can be used to evaluate
concentrations at equilibrium
[H ] [ X ]
K

 2 HX <--> 2H+ + X22[ HX ]
 K=4E-15
 If you have one mole of HX initially, what are the
concentration of all species at equilibrium?
 Try to write species in terms of one unknown
Start with species of lowest concentration
2
2
[X22-]=x, [H+]=2x, [HX]=1-2x,
[ x ][ 2 x ]
[ x ][ 2 x ]
3
K



4
x
2
 Since K is small, x must be small
[1  2 x ]
1
Use the approximation 1-2x ≈ 1
3
Substitute x and rearrange K
4 E  15  4 x
 Solve for x
3
1 E  15  x
• [X22-]=1E-5, [H+]=2E-5
 2
2
2
2
x  1E  5
9-33
Realistic Case
• Metal ion of interest may be in complicated environment
 May different species to consider simultaneously
• Consider uranium in an aquifer
 Example is still a simplified case
• Species to consider in this example include
 free metal ion: UO22+
 hydroxides: (UO2)x(OH)y
 carbonates: UO2CO3
 humates: UO2HA(II), UO2OHHA(I)
• Need to get stability constants for all species
 Example: UO22+ + CO32- <--> UO2CO3
• Know or find conditions
 Total uranium, total carbonate, pH, total humic
concentration
9-34
Stability constants for selected uranium species at 0.1 M
ionic strength
Species
UO2 OH+
UO2(OH)2
UO2(OH)3UO2(OH)42(UO2)2OH3+
(UO2)2(OH)2+
UO2CO3
UO2(CO3)22UO2(CO3)34UO2HA(II)
UO2(OH)HA(I)
logß
8.5
17.3
22.6
23.1
11.0
22.0
8.87
16.07
21.60
6.16
14.7±0.5
Other species may need to be
considered. If total uranium
concentration is low enough,
binary or tertiary species can
be excluded.
Chemical thermodynamics of uranium: http://www.oecd-nea.org/dbtdb/pubs/uranium.pdf
9-35
Equations
• Write concentrations in terms of species
• Total uranium in solution, [U]tot, is the sum of all solution
phase uranium species
 [U]tot= UO22+free+U-carb+U-hydroxide+U-humate
 [CO32-]free=f(pH)
From Henry’s constant for CO2 and K1 and K2
from CO3H2
log[CO32-]free=logKHK1K2+log(pCO2)-2log[H+]
* With -log[H+]=pH
log[CO32-]free=logKHK1K2+log(pCO2)+2pH
 [OH-] = f(pH)
 [HA]tot = UO2HA + UO2OHHA+ HAfree
9-36
Uranium speciation equations
• Write the species in terms of metal, ligands, and constants
 Generalized equation, with free uranium, free ligand A and
free ligand B
 xab 
[( UO 2 ) x Aa B b ]
[UO
2
2
x
a
] [ A] [ B ]
b
[( UO 2 ) x Aa B b ]   xab [UO
2
2
x
a
] [ A] [ B ]
b

Provide free ligand and metal concentrations as pX value
 pX = -log[X]free
 pUO22+=-log[UO22+]
• Rearrange equation with pX values
 Include –logxab, treat as pX term
 [(UO2)xAaBb] = 10-(xpUO2+apA+bpB-logxab)
• Specific example for (UO2)2(OH)22+
 [(UO2)2(OH)22+]=10-(2pUO2+2pOH-22.0)
• Set up equations where total solution uranium concentration is
sum of all species and solve for known terms
9-37
Speciation calculations:
Excel spreadsheets
CHESS Program
9-38
U speciation with different CO2 partial
pressure
0% CO2
1.0
1% CO2
UO HA(II)
2
2
2
UO OHHA(I)
2
UO (OH)
2
M ole F ra c tion of U (V I) S p e c ies
UO
0.8
2
0.6
0.4
0.2
0.0
2.0
UO OHHA(I)
UO HA(II)
3
2
4.0
6.0
8.0
UO (CO
2+
2
2
4-
)
3 3
0.8
0.6
0.4
UO (CO
2
)
2-
3 2
0.2
0.0
10.0
2.0
4.0
pH
6.0
8.0
10.0
pH
1.0
UO HA(II)
UO
M ole F ra c tion of U (V I) S p e c ies
M ole F ra c tion of U (V I) S pe c ie s
UO
1.0
-
UO (OH)
2
2+
2+
2
UO OHHA(I)
2
UO (CO
2
2
)
4-
3 3
0.8
0.6
10% CO2
0.4
UO (CO
2
)
2-
3 2
0.2
0.0
2.0
4.0
6.0
pH
8.0
10.0
9-39
Comparison of measured and calculated
uranyl organic colloid
1.0
0.8
10%
1%
t o t al
[U(V I)]
[U-c ollo id]
100%
0.6
0.4
0%
0.035%
0.2
0.0
2.0
4.0
6.0
8.0
10.0
pH
9-40
Energy terms
R ln ß
• Constants can be used to
evaluate energetic of
reaction
 From Nernst equation
∆G=-RTlnK
 ∆G=∆H-T∆S
-RTlnK = ∆H-T∆S
RlnK= - ∆H/T + ∆S
* Plot RlnK vs 1/T
Temperature effect on Np-Humate stability
Temp (°C)
56
48
40
32
24
16
76
74
72
70
²H = -22.2 ± 2.8 kJ/mol
²G
=-21.7 kJ/mol
298
68
²S=1.2±1.4 J/molK
66
64
0.003
0.0031
0.0032
0.0033
0.0034
0.0035
1/T (K)
9-41
Solubility Products
• Equilibrium involving a solid phase


 AgCl(s) <--> Ag+ + Cl[Cl ][ Ag ]
K 
[ AgCl ]
 AgCl concentration is constant
Solid activity and concentration is
treated as constant
By convention, reaction goes from solid
to ionic phase in solution
 Can use Ksp for calculating concentrations in
solution

K sp  K [ AgCl ]  [Cl ][ Ag

]
9-42
Solubility calculations
• AgCl(s) at equilibrium with water at 25°C gives
1E-5 M silver ion in solution. What is the Ksp??
 AgCl(s) <--> Ag+ + Cl-: [Ag+] = [Cl-]
 Ksp = 1E-52 = 1E-10
• What is the [Mg2+] from Mg(OH)2 at pH 10?
 Ksp = 1.2E-11= [Mg2+] [OH]2
 [OH] = 10-(14-10)
[Mg
2
]
1.2 E  11
1E  8
 1.2 E  3
9-43
Solubility calculations
• Ksp of UO2 = 10-52. What is the expected U4+ concentration
at pH 6. Generalize equation for any pH
 Solubility reaction:
UO2 + 2 H2OU(OH)4  U4+ + 4 OH Ksp= [U4+][OH-]4
 [U4+]= Ksp /[OH-]4
pOH + pH =14
At pH 6, pOH = 8, [OH-]=10-8
 [U4+]= 10-52 /[10-8]4= 10-52 /10-32 = 10-20 M
 For any pH
[U4+]= 10-52 /[10-(14-pH)*4]
Log [U4+]= -52+((14-pH)*4)
9-44
Limitations of Ksp
• Solid phase formation limited by concentration
 below ≈1E-5/mL no visible precipitate forms
colloids
• formation of supersaturated solutions
 slow kinetics
• Competitive reactions may lower free ion concentration
• Large excess of ligand may form soluble species
 AgCl(s) + Cl- <--> AgCl2-(aq)
Ksp really best for slightly soluble salts
9-45