Sizes
N
=
Z
Measuring Nuclear Masses
Used wit h I SOL t arget t o m easure exot ic react ion product nuclei
Be a
m
DRAGON
2
(TRIUMF/Canada)
Combination
Wien Filter - TOF
new nuclides are
produced in high
charge states
good mass resolution
Lorent z force
F
2 sets of magnetic and electric dipoles,
MCP TOF system in focal plane
B
E
mv
q
E
q
TOF :
Abs :
W. Udo Schröder, 2004
B
Wien vel . Filt er
B
Nuclear Masses
q v
m
Abs
:
Princ. accuracy
m
10
m
q
v , m, K
m
m
5
10
3
| Rel : 10
8
Nuclear Masses
3
TRI UMF- Dragon Recoil Mass Spect rom et er
W. Udo Schröder, 2004
Basic Const it uent s of At om ic Nuclei
Helium
XN
4
2
He2
4
A
Z
A nucleons: protons (H+) plus neutrons
Z pr ot ons (equals t he number of elect r ons in a neut r al
atom)
N=A- Z neutrons
Charges
ep = +e = 1.602· 10 - 19C (Coulomb)
en = 0
( e2 =1.440MeVfm)
Nuclear Masses
Masses
mp = 1.673·10 -27kg = 938.279 MeV/ c2 = 1.00728 u (mass unit s)
mn = 1.675·10 -27kg = 939.573 MeV/ c2 = 1.00867 u
1u = m( 12C)/ 12 = 1.6606·10 -27kg = 931.502 MeV/ c2
Expect approximately
W. Udo Schröder, 2004
m
A
Z
XN
Z mp
N mn
Z me ??
Radiat ive Capt ure of Nucleons
E> 0
(r)
0
5
r
E< 0
Bound
States
Nuclear Masses
Potential
Elastic NN scattering (E 0) does not
lead t o a bound ( E < 0) NN syst em
(nucleus).
N e e d t o dissipa t e e x t r a e n e r gy
r a dia t e ou t som e of t h e m a ss.
Hypot het ical nuclear collapse: -r a ys, E
radiated, lost from nucleus
less energy
than original free nucleons
Exot herm
ENucleus 0
Observe radiat ion em it t ed in capt ure of e - , n,
W. Udo Schröder, 2004
Nuclear Masses
m H=1.00782u
m p = 1.00728u
m n = 1.00866u
Nuclear Masses
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
A
Z
XN
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 m p .
W. Udo Schröder, 2004
Syst em at ics of Experim ent al Binding Energies
For heavy nuclei, B/ A
8MeV
7
For light nuclei, oddeven staggering
gg nuclei most tightly
bound
gg: N even, Z even
uu: N odd, Z odd
4
8
16
20
He
,
Be
,
O
,
2
4
8
10 Ne, ...
Nuclear Masses
exceptions: 8 Be unstable
8 Be
2 4 He
W. Udo Schröder, 2004
there is no A=5 nucleus
Energet ics of 8 Be
8 free nucleons=:0
tabulated
8 Be
- B/A 4n 4p
0
2
8
Nuclear Masses
-1
B for
8 Be
8
8 m(12C)/12
Be
-2
m
-3
4n
4p
-4
m
4
B
4.942 MeV
8.071
56.49 MeV
7.289 MeV
61.44 MeV
" 0 " per def .
2
-5
m
-6
-7
8 Be
W. Udo Schröder, 2004
2
2 2.425 MeV
8 Be
4.850 MeV
is st rongly bound but highly
unstable ( E 100keV) against
disintegration into 2 particles
56.59 MeV
Energet ics of Transm ut at ion
9
Non- monotonic behavior,
6
5
4
BE
Ene r gy r e le a se d
by fission
7
Ene r gy r e le a se d
in fusion
Bin din g e n e r gy pe r n u cle on ( M e V )
9
8
Fe/ Ni m ost st rongly bound
3
2
1
0
0
20
40
60
80 100 120 140 160 180 200 220
240
Nuclear Masses
Mass number (A)
226Ra
(226.0254MeV)
( 222Rn + 4 He)
(222.0176 + 4.00260)MeV
Mass defect = m Ra- m RnHe = 0.0052 u
exothermic, energy released Q=+4.84 MeV
transmutation energetically possible
W. Udo Schröder, 2004
A
2 different regions in A,
different energetically
preferred transmutations:
Heavy nuclei are particle
unst able, exot herm ic
spontaneous emission of
particles (B =28 MeV!) or
fission ( nuclear power) .
2 light nuclei can fuse and
produce energy ( st ars,
nuclear power)
The Sem i- Em pirical Mass Form ula
m Z, N
Z mH
N mn
B Z, N
c2 ;
0 neglect e- binding
B
Original: Weizsacker, Z. Physik 96,431(1935)
Nuclear Masses
10
B Z, N
aV A
23
aS A
aC
Z
2
A1 3
aa
A
2Z
A
2
A
12
Wapstra, Handb. Physik, XXXVIII
aV
15.835 MeV
aS
18.33 MeV
11.2 MeV for o
aC
0.714 MeV
aa
23.20 Me V
0 MeV for odd Anuclei
11.2 MeV for e e nuclei
Implicit assumption: Leptodermous nucleus
aV = volume, condensation
aS = surface, less binding
a C = Coulomb repulsion of protons
a a = asymmetry lessens binding for
like particles, compared to unlike
= unpaired particles are less
tightly bound
W. Udo Schröder, 2004
quantal
effects
o nuclei
Relat ive Cont ribut ions t o Nuclear Mass
R
A1 3
Vnucleus
A
const. contribution from each
nucleon
saturated force
11
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
Coulomb self energy becomes
large for large Z, heavy nuclei,
makes nucleus unstable
reduces B
ECoul
e2 Z 2
d 3r d 3r
2
W. Udo Schröder, 2004
(r ) (r )
r r
3 e2 Z 2
5 RC
hom. sharp sphere
Coulomb Radius
RC
1.24 A1 3 fm
Pairing Energies
odd- even mass
differences, B
largest for paired
nucleons
12
Average trend:
Nuclear Masses
n
W. Udo Schröder, 2004
p
A
12
Remaining structure
and fluctuations:
shell and deformation
effects.
The - St able Valley
N
=
Z
ZA
A
2
Nuclear Masses
13
Z
Nature:
170 even- Z, even- N
60 odd- A
4 odd z, odd N
ZA
A
2
N
ZA
W. Udo Schröder, 2004
A
2
A2 3 aC
2 asym
Nuclear Masses
14
Qualit y of Droplet - Model Mass Fit
W. Udo Schröder, 2004
Total
binding
energy of
heavy
nuclei
~1600
MeV,
accuracy
of LDM fit:
±0.5 MeV
Bet a Decays of Odd- A Nuclei
m A, Z
m
A
m min : Z A
A Z
4 as
2
A Z2
mn
2 4 as
me c2 A
mp
23
aC A
( A)
Z
ZA
2
15
Expand around ZA: Mass parabola m( Z )
bottom of valley
m
Beta decay and K- elect ron capt ure ( EC)
Nuclear Masses
odd- A
isobars
=0
Z
ZA
W. Udo Schröder, 2004
EC
to
K- hole
Bethge,
Kernphysik
Energet ics of
Decay
Beta decay and K- capture
16
Z, A
Z
1, A
e
m ( Z , A)
m( Z
1, A)
Z, A
e
m ( Z , A)
Nuclear Masses
Exam ple :
Z
m( Z
11
6 C
1, A
e
|
e
1, A
m( Z
e
m ( 11 C ) c 2
|
2 me
1 extra e+
1 extra e-
1, A)
6e
e
1, A)
| EC
11
5 Be
6e
5
e
e
Q >0
exotherm.
Q
1
m ( 11 Be) c 2
me c2
m ( 11 Be) c 2
Decay Q- value smaller by 2mec2 for
W. Udo Schröder, 2004
Z
m ( Z , A)
Mass balance:
m ( 11 C ) c 2
Q
Z, A
me c2
Q
2 mec2
+
decay than for
-
Decay of an Even- A Nucleus
o o
e e
o-o weakly bound
even A
A=108
e- e strongley bound
oo
17
11.2 MeV for o o nuclei
0 MeV for odd Anuclei
Nuclear Masses
ee
11.2 MeV for e
st able Traps
Z 108
W. Udo Schröder, 2004
e nuclei
for even A, 2 distinct
mass parabolas
separated by
2 = 22.4.A- 2 MeV
Z 108 47 is relat ive
minimum for oo A=108
nuclei, but not for ee.
Energet ics of Prot on Decay
Isotone- Parabolas
Nuclear Masses
Bin din g Ene r gy ( M18e V)
odd- A
even- A
p decay energet ically possible if
Qp = - [B(Z,A)
equivalent to
Qp
m( Z , A) c2
m( Z
p
B(Z- 1,A- 1)] >0
1, A 1)
Proton Emitter Decay Q Value
mp
Daughter
me c2
Half Life
p
p
p
p
Z
S. Hofmann, Handbook of Nuclear Decay
W. Udo Schröder, 2004
Modes, CRC Press 1993
Search for location of p- Drip Line
continues
0
Energet ics of
Decay
One of t he m ost frequent decays of heavy nucleus because B = 28.3
MeV is relatively large.
(A,Z)
Q
B( A
4, Z
2)
28.3 MeV
B( A, Z )
0
Nuclear Masses
19
(A- 4,Z- 2)+B
W. Udo Schröder, 2004
No shell corrections
B
energetically allowed
(along - stable valley) for A
> 150, e.g., in the rare
eart h region of elements
( Gd).
decay from nuclear
ground st at e is mostly
slower than decay, except
for very heavy nuclei > Pb,
act inides, e.g., Po, Am , Cf,
Alpha Q- Values
Lise calculation
20
U
Nuclear Masses
Q >0
Bi
W. Udo Schröder, 2004
Fissilit y of Nuclei
H
L
B( AL, ZL)
21
Qf
Bs
B( AH , ZH )
fission
fragments
B( A, Z)
BCoul
Nuclear Masses
Volume and symmetry terms
cancel
Balance of Coulomb
vs. surface energy
fissility parameter x
x
239U:
W. Udo Schröder, 2004
2 Es0
3 e 2 5 r0 Z 2
2 as
A
Es = 650MeV EC =950MeV
Fiss.frgm. : Es = 813MeV EC =607MeV
Balance:
EC0
Es = - 163MeV EC =343MeV
Qf =
180 MeV
Z2 A
50
Energet ics of Mult ifragm ent Decay
22
Important for spallation
and applications: n
fragments at infinity
Qn > 0??
n fragments
Nuclear Masses
Threshold Fissilit y for Qn
x
EC
2 Es
x
1, n
2
Z 2 3 e 5 r0
A
2 as
20 ( m any)
However: com pet it ion wit h
decay for every emission.
Hasse & Myers, Geometrical Relationships Of
Macroscopic Nuclear Physics,Springer, New York,1988
W. Udo Schröder, 2004
0
Droplet Model
W.D. Myers & W. J. Swiatecki, Ann. Phys. 55, 395(1969); 84, 186 (1974)
1 higher order in A- 1/3, I=(N- Z)/A
Extension of LDM
finit e com pressibilit y, deform ed shapes, n and
Most accurate: Finite- Ra n ge D r ople t M ode l
different surfaces.
23
p
B( N , Z ; shape)
a3 A1 3 Bcurv
Nuclear Masses
I
a1
J
c1 Z 2 A
13
(3c1 16Q) ZA
23
1 (9 J 4Q) A
13
bulk asymmetry
c1
3e2
c4
5 c1 3
4 2
5 r0 ;
W. Udo Schröder, 2004
Bsurf
;
c2
c12 1
336 J
; c5
1 c12
64 Q
23
2
1
M
2
c2 Z 2 A1 3 Bred
BC
BV
1
K
2
2
2a2 A
4
A
a2
c5 Z 2 Bw
13
9 J2
4 Q
c3 Z 2 A
Bsurf
L
2
2
1
A2 3 Bsurf
c4 Z 4 3 A
c1 Z 2 A
43
13
BC / K
deviation from aver density
18
;
K
c3
5 c1 b
2 r0
2
;
Functions B describe contributions
of shape/spherical