Transcript Interaction of Charged Particles with Matter

Sizes
Z
N
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
Lorentz force
2

F  q E v B

 Wien velocity
Combination Wien Filter - TOF
2 sets of magnetic & electric dipoles,
MCP-TOF system in focal plane
Filter
mv 
q 
m

q
K 
E  
q 
TOF :  v , m, K
Nuclear Masses & BE
B B 
Abs :
Future :
W. Udo Schröder, 2009
m
m
103
m
 105 | Rel : 108
m
Nuclear Masses & BE
3
TRIUMF-Dragon Recoil Mass Spectrometer
W. Udo Schröder, 2009
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)
A
A
4
N=A-Z neutrons Z X N  X (ex : He)
{Briefly at E* » 0 also (virtual) p, D,..}
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: m  ZA X N   Z  m p  N  mn  Z  me
Find: m  ZA X N   Z  m p  N  mn  Z  me
W. Udo Schröder, 2009
m  c 2  B
Mass Defect
Radiative Capture of Nucleons
Eg
E
Unbound States
E>0
Y (r)
5
0
E<0
Need to “dissipate” extra energy 
radiate out some of the mass as g-ray.
r
Bound
States
Potential
Nuclear Masses & BE
Elastic NN scattering (E  0) does not lead
to a bound (E < 0) NN system (nucleus).
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, 2009
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
mc 2   mi c 2 Binding energy B
i
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, 2009
Systematics of Experimental Binding Energies
For heavy nuclei, B/A  8MeV
Light nuclei:
odd-even staggering
Nuclear Masses & BE
7
gg:
uu:
ug:
gu:
N
N
N
N
even, Z even
odd, Z odd
odd, Z even
even, Z odd
gg nuclei are most
tightly bound
uu nuclei least bound
4
8
16
20
He
,(
Be
),
O
,
2
4
8
10 Ne,...
Exceptions:
8Be unstable,
8Be
 2 4He
There is no stable A=5 nucleus.
W. Udo Schröder, 2009
Energetics of the A=8 System
Ref: 8 free nucleons=:0
8m(12C)/12
Systems with less BE
can transform
(decay) into more
strongly bound
systems
B for
8Be
8
-B/A 4n 4p
0
-1
m  c 2
61.44MeV
8Be
2
mc 212
4n  4 p m  4  8.071  7.289  MeV
-2
C
B
 61.44MeV
B  " 0" per def .
Nuclear Masses & BE
-3
8
-4
Be m  4.942MeV (comp 8  m12
C
/ 12)
-5
 7.06 A MeV
2  m  2  2.425MeV  4.850MeV
-6
-7
 56.49 MeV
 56.59MeV
 7.07 A MeV
8Be
W. Udo Schröder, 2009
2
8Be
is bound but unstable (E ≈100keV)
against decay  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
A  2Z 

Z2
23
B  A,Z   aV A  aS A  aC 1 3  aa
  A1 2
A
A
aV  15.835 MeV Wapstra, Handb. Physik, Vol. XXXVIII
Original: Weizsäcker,
Z. Physik 96,431(1935)
9
2
aS  18.33 MeV
aC  0.714 MeV
Nuclear Masses & BE
aa  23.20 MeV
11.2 MeV for o  o nuclei

   0 MeV for odd  Anuclei
 11.2 MeV for e  e nuclei

Implicit assumption: ”Leptodermous” structure less nucleus
aV = volume, “condensation”
aS = surface, less binding
aC = Coulomb repulsion of protons
aa = asymmetry lessens binding for
like particles, compared to unlike
 = “pairing,”
unpaired particles less tightly bound
W. Udo Schröder, 2009
quantal
effects
Thin 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
surface nucleon S  A2 3
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
e2Z 2 3 3 (r )(r ) 3 e2Z 2


 d rd r 

2
r r
5 RC
W. Udo Schröder, 2009
Coulomb Radius
RC  1.24 A1 3fm
hom. sharp sphere
Energetics of Transmutation
9
Non-monotonic behavior,
6
5
4
3
BE  A
Energy released
by fission
7
Energy released
in fusion
Binding energy per nucleon (MeV)
11
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, 2009
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)
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
12
B  N, Z   aV A  aS A2 3  aC
Refit data with 2 symmetry
energy terms:
aVs = volume symm. energy
aSs = surface symm. energy
Z2
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
 difficult to determine them independently
Surface symmetry energy has x2 uncertainty!
(N-Z)2 dependence of symmetry terms comes from semiclassical expansion for large A.
Better approximation: Wigner (|N-Z|) + Shell corrections.
W. Udo Schröder, 2009
Diffuse surface
(unlike liquid drop)
Stability: Coulomb vs. Symmetry Energy
Light nuclei with few nucleons:
W. Udo Schröder, 2009
Medium-weight to heavy nuclei:
Binding energy BE(Z) is smooth function
of Z  average (“gross”) behavior of
nuclei most important
BE(Z)A=const is non-monotonic:
For any A, there is a maximum BE for a
characteristic Z(A) < A/2
Z=A/2
NOT: smallest charge densities, Nuclei
with large N are also unstable
Z=A/2
Binding Energy (MeV/A)
Nuclear Masses & BE
13
Z=A/2
Few isotopes, N ≈ Z ≈ A/2
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
14
e-e nuclei have slightly, but systematically, stronger binding
than o-o neighbors
W. Udo Schröder, 2009
ee ee
ee ee
oo
oo
Structure Effects in the Pairing Energies
odd-even mass differences
15
B largest for “paired”
nucleons
Average trend:
BLD  Bexpt  (N, Z )
Nuclear Masses & BE
 n   p   A 1 2
Remaining structure and
fluctuations: Effect 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, 2009
The b-Stable Valley
Nature:
170 even-Z, even-N
60 odd-A
4 odd z, odd N
ZA 
A
2
Nuclear Masses & BE
16
Z
ZA 
A
2
The b-Stable Valley
N
W. Udo Schröder, 2009
ZA 
A
2  A2 3 aC
2asym 
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
5c1  b 
c12  1 18 
5r0  ; c2  336  J  K  ; c3 
  ;
2


 r0 
23
5c  3 
c4  1 
4  2 
W. Udo Schröder, 2009
1 c12
; c5 
64 Q
Functions B describe contributions
of shape/spherical
Quality of Droplet-Model Mass Fit
Nuclear Masses & BE
18
W.D. Myers & W. J. Swiatecki, Ann. Phys. 55, 395(1969); 84, 186 (1974)
W. Udo Schröder, 2009
Total
binding
energy of
heavy
nuclei
~1600
MeV,
accuracy
of LDM fit:
±0.5 MeV