Transcript Interaction of Charged Particles with Matter

Sizes
Measuring Nuclear Masses
Used with ISOL target to measure exotic reaction product nuclei
New nuclides are
produced in high
charge (q) states
good mass resol.
TRIUMF/Canada
2
Lorentz force

F  q E v B
Combination Wien Filter - TOF
2 sets of magnetic & electric dipoles,
MCP-TOF system in focal plane

 Wien vel. Filter
mv 
q 
m

q
E 
 
q 
Nuclear Masses & BE
B B 
TOF :  v , m, K
Abs :
W. Udo Schröder, 2007
Future :
m
m
103
m
 105 | Rel : 108
m
Nuclear Masses & BE
3
TRIUMF-Dragon Recoil Mass Spectrometer
W. Udo Schröder, 2007
Basic Constituents of Atomic Nuclei
Helium
4
A
Z
X N  24 He2
A nucleons: protons (H+) plus neutrons
Z protons (equals the number of electrons in a neutral
atom)
N=A-Z neutrons
Charges
ep = +e = 1.602·10-19C (Coulomb)
en = 0
(e2=1.440MeVfm)
Nuclear Masses & BE
Masses
mp = 1.673·10-27kg = 938.279 MeV/c2 = 1.00728 u
mp = 7.289 MeV rel
12C
mn = 1.675·10-27kg = 939.573 MeV/c2 = 1.00867 u
mn = 8.071 MeV rel
12C
1u = m(12C)/12 = 1.6606·10-27kg = 931.502 MeV/c2 = standard nucleonic mass
Expect approximately:
W. Udo Schröder, 2007
m  ZA X N   Z  m p  N  mn  Z  me
Radiative Capture of Nucleons
Eg
E>0
Y (r)
0
r
5
E<0
Bound
States
Nuclear Masses & BE
Potential
Elastic NN scattering (E  0) does not lead
to a bound (E < 0) NN system (nucleus).
Need to “dissipate” extra energy 
radiate out some of the mass as g-ray.
Hypothetical nuclear “condensation”:
Eg radiated as g-rays lost from nucleus
 less energy than original free nucleons
Exothermal capture
E  0, E  Eg  0
Observe radiation emitted in capture of n, p,… eW. Udo Schröder, 2007
Nuclear Masses
mH=1.00782u≠ mp
mp= 1.00728u
mn= 1.00866u
Nuclear Masses & BE
6
Element
Deuterium
Helium 4
Lithium 7
Beryllium 9
Iron 56
Silver 107
Iodine 127
Lead 206
Polonium 210
Uranium 235
Uranium 238
D
4He
7Li
9Be
56Fe
107Ag
127I
206Pb
210Po
235U
238U
Binding energy
B   Z  mH  N  mn   m  ZA X N   0
Mass of
Nuclear Mass Binding Energy Binding Energy
Nucleons (u)
(u)
(MeV)
MeV/Nucleon
2.01594
2.01355
2.23
1.12
4.03188
4.00151
28.29
7.07
7.05649
7.01336
40.15
5.74
9.07243
9.00999
58.13
6.46
56.44913
55.92069
492.24
8.79
107.86187
106.87934
915.23
8.55
128.02684
126.87544
1072.53
8.45
207.67109
205.92952
1622.27
7.88
211.70297
209.93683
1645.16
7.83
236.90849
234.99351
1783.8
7.59
239.93448
238.00037
1801.63
7.57
Note: hydrogen atomic mass mH is used in tables, rather than mp.
W. Udo Schröder, 2007
Systematics of Experimental Binding Energies
For heavy nuclei, B/A  8MeV
Light nuclei:
odd-even staggering
Nuclear Masses & BE
7
gg:
uu:
ug:
gu:
Exceptions:
8Be unstable,
8Be
N
N
N
N
even, Z even
odd, Z odd
odd, Z even
even, Z odd
 2 4He
There is no stable A=5 nucleus.
W. Udo Schröder, 2007
gg nuclei are most
tightly bound
uu nuclei least bound
4
8
16
20
He
,
Be
,
O
,
2
4
8
10 Ne,...
Energetics of 8Be
8 free nucleons=:0
tabulated
8Be
Nuclear Masses & BE
8
-B/A 4n 4p
0
-1
B for
8Be
2
8
8m(12C)/12
Be
B
-2
m  4.942MeV
-3
4n  4 p
-4
m  4   8.071  7.289  MeV  61.44MeV  "0 " per def .
-5
2
-6
m  2  2.425MeV  4.850MeV
-7
8Be
W. Udo Schröder, 2007
2
8Be
 56.49 MeV
 56.59MeV
is bound but highly
unstable (E ≈100keV) against
disintegration into 2  particles T1 2  7  1017 s
The Semi-Empirical Mass Formula
m  Z, N    Z  mH  N  mn   B  Z, N  c 2 ; B  0 neglect e- binding
Original: Weizsäcker, Z. Physik 96,431(1935)
B  A, Z   aV A  aS A2 3  aC
9
aV  15.835 MeV
aS  18.33 MeV
aC  0.714 MeV
Z2
A1 3
 aa
 A  2Z 2
A
  A 1 2
Wapstra, Handb. Physik, Vol. XXXVIII
11.2 MeV for o  o nuclei

 
0 MeV for odd  A nuclei
 11.2 MeV for e  e nuclei

Nuclear Masses & BE
aa  23.20 MeV
Assumption: ”Leptodermous” structure less nucleus R A1/3
aV = volume, “condensation”  A
aS = surface, less binding
 A2/3
aC = Coulomb repulsion of protons  A-1/3
aa = asymmetry lessens binding for
like particles, compared to unlike
quantal
 = “pairing,”
effects
unpaired particles less tightly bound
W. Udo Schröder, 2007
Sharp surface (liquid drop)
Relative Contributions to Nuclear Mass
R  A1 3  Vnucleus  A
const. contribution from each
nucleon “saturated” force
10
fewer interactions on surface 
reduce contribution from each
23
surface nucleon S  A
different interactions between
like and unlike nucleons
(Fermion statistics, isospin)
depends on |N-Z|, reduces B
Nuclear Masses & BE
Coulomb self energy becomes
large for large Z, heavy nuclei,
makes nucleus unstable
reduces B
ECoul
e2 Z 2
(r )(r ) 3 e2 Z 2
3
3


 d rd r 
2
r  r
5 RC
W. Udo Schröder, 2007
hom. sharp sphere
Coulomb Radius
RC  1.24 A1 3fm
The Surface Symmetry Energy Component
m  Z, N    Z  mH  N  mn   B  Z, N  c 2 ; B  0 neglect e- binding
Parameterization: G. Audi, A.H. Wapstra, C. Thibault, Nucl. Phys. A729, 337 (2003)
Nuclear Masses & BE
11
B  N, Z   aV A  aS A2 3  aC
Refit data with 2 symmetry
energy terms:
aVs = volume symm. energy
aSs = surface symm. energy
Z
2
A1 3
N  Z 
2
 aVs
A
aV  15.7 MeV
aS  18.1 MeV
aVs  26.5 MeV
aSs  19.0 MeV
aC  0.71 MeV
N  Z 
2
 aSs
A4 3
  A 1 2
12.8 MeV for o  o nuclei

 
0 MeV for odd  A nuclei
 12.8 MeV for e  e nuclei

“Symmetry energy” results from quantal Pauli
Exclusion Principle  density  dependent
Surface has lower average density than volume 
symmetry energy lower in surface (note the
different signs !)
Fits: aVs and aSs are strongly correlated 
impossible to determine them independently
Surface symmetry energy has x2 uncertainty!
(N-Z)2 dependence of symmetry terms comes from
semi-classical expansion for large A. Better
approximation contains |N-Z| Wigner + Shell terms.
W. Udo Schröder, 2007
Diffuse surface
(unlike liquid drop)
Coulomb vs. Symmetry Energy
Light nuclei with few nucleons:
W. Udo Schröder, 2007
Medium-weight to heavy nuclei:
Binding energy BE(Z) is smooth function
of Z  average (“gross”) behavior of
nuclei most important
Z=A/2
Binding Energy (MeV/A)
Nuclear Masses & BE
12
Z=A/2
Few isotopes, N ≈ Z ≈ A/2
BE(Z)A=const is non-monotonic:
For any A, there is a maximum BE for a
characteristic Z(A) < A/2
NOT: smallest charge densities, Nuclei
with large N are also unstable
correlated values of Z and N (neutrons
and protons) Quantum Pauli correlations
But high Coulomb self-energy shifts
max binding energy/A to higher N.
Adapted from N.D. Cook, Models of the Atomic
Nucleus, Springer Verlag Heidelberg, 2006
The Odd-Even (Pairing) Effect of Binding Energies
Even-A  even-N + even-Z (e e) or odd-N + odd-Z (o o)
Effect not visible for odd-A nuclei
Nuclear Masses & BE
13
e-e nuclei have slightly, but systematically, stronger binding
than o-o neighbors
W. Udo Schröder, 2007
ee ee
ee ee
oo
oo
Structure Effects in the Binding Energies
14
Most of BE
systematic well
described by
spherical Liquid Drop
Model
Next approximation:
odd-even mass
differences, BE
largest for “paired”
nucleons
Average trend:
BLiquid Drop  Bexpt  (N, Z )
 n   p   A 1 2
Nuclear Masses & BE
Remaining structure and
fluctuations not random:
Effects due to intrinsic
nuclear structure (“shell
model”) and collective
deformation.
Weak indications of special structure at “magic” neutron and/or proton
numbers (N, Z = 8, 20, 28, 50, 82, 126,…): several isotopes/isotones
W. Udo Schröder, 2007
The Misaligned b-Stable Valley
Symmetry energy not strong enough to enforce N=Z
ZA 
A
2
Nuclear Masses & BE
15
Z
Nature:
170 even-Z, even-N
60 odd-A
4 odd z, odd N
ZA 
A
2
N
W. Udo Schröder, 2007
valley of b stability
 Z A B max

A
2  A2 3aC
2asym 
Nuclear Masses & BE
16
Quality of Droplet-Model Mass Fit
W. Udo Schröder, 2007
Total
binding
energy of
heavy
nuclei
~1600
MeV,
accuracy
of LDM fit:
±0.5 MeV
Droplet Model
W.D. Myers & W. J. Swiatecki, Ann. Phys. 55, 395(1969); 84, 186 (1974)
Extension of LDM  1 higher order in A-1/3, I=(N-Z)/A
17
finite compressibility, deformed shapes, n and p different surfaces.
Most accurate: Finite-Range Droplet Model (also: most parameters)
1
1


B(N, Z; shape)   a1  J 2  K  2  M 4  A 
2
2



9 J2 2  2 3
  A Bsurf
 a2 
4 Q


 a3 A1 3Bcurv  c1 Z 2 A1 3BC  c2 Z 2 A1 3Bred  c5 Z 2 Bw  c3 Z 2 A1  c4 Z 4 3 A1 3
Nuclear Masses & BE
 
I  (3c1 16Q)ZA2 3BV
1  (9 J 4Q)A1 3Bsurf
;
bulk symmetry
  2a2 A1 3Bsurf  L 2  c1Z 2 A 4 3BC  / K
deviation from average density
2
c1  3e2
5c
c4  1
4
5c1  b 
c12  1 18 
5r0  ; c2  336  J  K  ; c3 
  ;
2


 r0 
23
 3 
 2 


W. Udo Schröder, 2007
1 c12
; c5 
64 Q
Functions B describe contributions
of shape/spherical
Energetics of Transmutation
9
Non-monotonic behavior,
6
5
4
3
B/A  A
Energy released
by fission
7
Energy released
in fusion
Binding energy per nucleon (MeV)
18
8
Fe/Ni most strongly bound
2 different regions in A,
different energetically
preferred transmutations:
2
1
0
0
20
40
60
80
100
120 140
160 180 200 220
240
Nuclear Masses & BE
Mass number (A)
226Ra
(226.0254 MeV)
(222Rn + 4He)
(222.0176 + 4.00260) MeV
Mass defect = mRa- mRnHe = 0.0052 u
 exothermic,
energy released Q=+4.84 MeV
Transmutation energetically possible
W. Udo Schröder, 2007
Heavy nuclei are unstable,
exothermic spontaneous
emission of  particles (B
=28 MeV!)
Heavy nuclei can split
(“fission”) nuclear power.
2 light nuclei can fuse and
produce energy
(stars, nuclear power)
Mechanical Shape Instability: Nuclear Fissility



Axially symmetric shapes R( )  R0 1   b P (cos  )  ; V  const.
 2


R( )
Consider small quadrupole deformations: only b2 ≠ 0

z
19
Calculate balance repulsive Coulomb vs. cohesive surface E
2


Es (b2 )  Es (b2  0) 1  b22 
5


1 

ECoul (b2 )  ECoul (b2  0) 1  b22 
5 

1
2
Stability : ECoul (b2 )  ECoul (0) b22  Es (b2 )  Es (0) b22
5
5
Fission Instability
Bohr-Wheeler fissility parameter
Es (b2  0)  17.8 A2 3 MeV
x 
ECoul (0)
2Es (0)
Measure of stability
ECoul (b2  0)  0.71 Z 2 A1 3 MeV
 x  f (Z 2 A)
Spontaneous fission instability (x  1) :
W. Udo Schröder, 2005
Z 2 A  (Z 2 A)crit  50
Fissility of Nuclei
H
x
L
fission
fragments
Nuclear Masses & BE
20
Qf  B( AL , ZL )  B( AH , ZH )  B( A, Z )
 Bs  BCoul
Volume and symmetry terms cancel
 Balance of Coulomb vs. surface
energy
 fissility parameter x
x 
239U:
Es = 650MeV EC =950MeV
Fiss.frgm. : Es = 813MeV
Balance:
Es =-163MeV EC =343MeV
Qf = 180 MeV
W. Udo Schröder, 2007
EC =607MeV
EC0
2Es0
3e2 5r0  Z 2
Z2 A


2as
A
50
Nuclei with A  95 are fissile
Energetics of Multifragment Decay
21
Important for spallation
and applications: n
fragments at infinity r
Qn > 0??
n fragments
Binary fission exothermic for x>0.35
according to LD model
Threshold Fissility for Qn  0
Nuclear Masses & BE
2
EC
Z 2 3e 5r0
x 

2Es
A
2as
x ≈ 1, n ≈ 20 (many)
Hasse & Myers, Geometrical Relationships Of
Macroscopic Nuclear Physics,Springer, New York,1988
W. Udo Schröder, 2007
However: Needs to be pushed over nbody transition state,
competition with b decay for every
emission.
With increasing deformation (b2),
ECoul decreases, Esurf increases
Equilibrium point/saddle = transition state.
Beyond saddle Coulomb drives system
apart, accelerates fission fragments
Equilibrium/Saddle 
Ground State
22
LDM-Fission Equilibrium (“Saddle”) Shapes
LDM-Fission Equilibrium Shapes
Fission Instability
b2
Cohen & Swiatecki, 1974
W. Udo Schröder, 2005
Chemical Instability: Beta Decays of Odd-A Nuclei
m  A, Z     A   b  A  Z  g  A  Z 2  

 chemical potential

4as  mn  mp  me c 2  A
b

m  mmin : Z A 
 
bottom of valley
2g
2 4as  aC A2 3


Expand around ZA: Mass parabola m(Z)  ( A)     b  Z  Z A 
23
2
m
Beta decay and K-electron capture (EC)
Nuclear Masses & BE
odd-A
isobars
=0
b
b
b b
Z
A
W. Udo Schröder, 2007
Z
b
b
EC
b
to
K-hole
Bethge,
Kernphysik
b Decay of an Even-A Nucleus
A=108
e-e strongly bound
Nuclear Masses & BE
24
 11.2
 A MeV for o  o nuclei


0 MeV for odd  A nuclei
 11.2

MeV for e  e nuclei
A


stable “Traps”
Z108
W. Udo Schröder, 2007
o  o
even A  
e  e
o-o weakly bound
 for even A, 2 distinct
mass parabolas
separated by
2 = 22.4·A-2 MeV
Z108≈47 is relative
minimum for oo A=108
nuclei, but not for ee.
Energetics of b Decay
Beta decay and K-capture (EC): Energetics for entire atoms
 Z, A   Z  1, A  e   e | b 
25
m(Z, A)  m(Z  1, A)


 Z, A   Z  1, A   e   e | b 
m(Z, A)  m(Z  1, A)  2me
 Z, A  e   Z  1, A   e | EC

1 extra e+
1 extra e-
Nuclear Masses & BE
m(Z, A)  m(Z  1, A)
6

11
5



Example :  11
C

6
e


Be

6
e

e
  e  Qb
5

6



b
Qb>0
5
1
exotherm.
Mass balance:
m(11 Be)c2  mec2   mec 2  Qb
m(11 C )c2 



b


Qb  m(11 C )c2  m(11 Be)c 2  2mec 2
Decay Q-value smaller by 2mec2 for b+ decay than for b-
W. Udo Schröder, 2007
Energetics of “Proton Decay”
Isotone-Parabolas
Nuclear Masses & BE
26
Binding Energy (MeV)
odd-A
even-A
Decay by p emission
energetically possible if
Qp =-[B(Z,A) – B(Z-1,A-1)] >0
equivalent to
Qp  m(Z , A)c 2 
 m(Z  1, A  1)  mp  me  c 2  0
p
p
p
Daughter
Half Life
p
p
Z
S. Hofmann, Handbook of Nuclear Decay
Modes, CRC Press 1993
W. Udo Schröder, 2007
Proton Emitter Decay Q Value
Search for location of p-Drip Line
continues
Energetics of  Decay
One of the most frequent decays of heavy nucleus because B = 28.3
MeV is relatively large.
(A,Z)

Q  B( A  4, Z  2)  28.3MeV   B( A, Z )  0

Nuclear Masses & BE
27

W. Udo Schröder, 2007

No shell corrections
B
(A-4,Z-2)+B
 energetically allowed
(along b-stable valley) for
A > 150,
 rare earth region of
elements (Gd).
 decay from nuclear ground
state is mostly slower than b
decay, except for very heavy
nuclei > Pb, actinides, e.g.,
Po, Am, Cf,…
Alpha Q-Values
Lise calculation
28
U
Nuclear Masses & BE
Q>0
Bi
W. Udo Schröder, 2007