Nuclear Reactions
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Transcript Nuclear Reactions
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 24He178 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 + 18O266Rf
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 + 18O266Rf
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 1T
l0
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 –logxab, 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 H2OU(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