Nuclear Stability
Some nuclides are stable,
existing indefinitely, e.g. 126C
Others are unstable and are thus radioactive
14 C
14 N +0 e
6
7
-1
These nuclides decay spontaneously to more stable
nuclides.
Nuclear Stability
2
Nuclear Stability
!"#$%&'()*+,-./*,0.123& 45&146& $7*8+$9!#5/5&146& $7*8?
;<!4=1(>+3(;!= nuclides %&'146& $?,@$$5A*>0,B,5&! C/4!=;<!29!
;,*-;!=,0.123& 4 (size of the nucleus.)
!=2H$"(!I;!=,0.123& 4 80*$J**(!>$*4/.,;!= n/p
L$=,0.123& $H L3"83==*, M-1+,&' .
Nuclear stability
;,*-;!=,0.123& 4 (size of the nucleus.)
Nuclear Stability
Look at the model of nucleus.
There are no stable nuclei heavier than
209
83
Bi
Shell model
!"#$%!
!&'&()&*+,-)&$'./$)
8I./* 5&)*,.,,0.23&!!,%&'! C/?,,0.123& 4%&'5&2.*5146& $4C=
%&'1$& (./* Magic numbers are nuclei %&'5&2/*1%/*(I 2, 8, 20, 28, 50,
L3" 82 4)*+$I protons +$9! 2, 8, 20, 28, 50, 82, L3" 126 4)*+$I
neutrons 23<* (I(*$-1$& =!0132>$!,.=,!(%&'5&2.*5146& $4C=
?,!">!5;!=(V*W1X9'! ( 2, 8, 18 , 32)
=8I./* ,0.123& 4%&'5&)*,., protons +$9! neutrons 1Z,13;2C/ 5&
2.*5146& $5*((./* ,0.#23-H%&'5&)*,.,,0.23&!!,1Z,13;2&' 1A/,
4 He 146& $(./* 3 He
2
2
7
• magic numbers 0 12 &&3)(/)4$&$'.
• !&%54!)& 6&0 7*&) %8
!!&%54!)&/ $)!&91
1Z,LII)*3!=;!=,0.123& 4%&'$.5L,.2.*520;!= liquid drop model 1;<*(I shell model
9
!>$*4/.,;!=
n/p =1:1
n/p
*(($*^ $"+./*= n/p ;!=
)*,.,,0.#23-H%B=+5- 2000
,0.#23-H 8I./*5&,0.#23-H%&'
146& $ 18& = 279 ,0.#23-H.
?,@*>a+,(!>$*4/.,;!= n/p >1
%)*?+<14<,2.*5146& $1(0-(*$
1I&' =1I,*(14<, n/p %&' = 1
Patterns of Nuclear Stability
.&# NeutronNeutron-toto-Proton Ratio
• 7*&) *+3:12*&$3! $54&4!/&#(!&!4 '7*&)8&;!1<==>&$64!4&3&
• &12'8'5'7*&)/6%54&4!<'% &!4 &&9
• 3:)&12<4*&$3<==>*+9*&$! $?5!'8'5'<%'%
&&,4'!7*&)
• /@)36!7*&)!8&;!!(4!8
• @)36!8)%!&)&!.02/6%?&:.7'!&'&
;!&$647*&)
15
Mass too high
α decay.
N/Z too high
β- decay.
N/Z too low
β+ decay
or electron
capture.
Patterns of Nuclear Stability
Nuclear Stability
NeutronNeutron-toto-Proton Ratio
E.g. Here is how the 238U
decay sequence looks on our
zone of stability graph.
• @*>a%&'5&2/* Z > 83 1Z,,0.
#23-H%&'#5/146& $%a(@*>a c4* >.?+<$=4&L!3^d* 189'!3)*,.,c$>!, 2 +,/. L3"
,0.>$!, 2 +,/. ?,(*$
43* >.2$B=1-& .L3"43* >.
?+<$=4&A,0-!9',e189'!3!>$*4/.,;!= n/p ?+<1;<*?(3<
1:1 5*(%&'4a19
Patterns of Nuclear Stability
Patterns of Nuclear Stability
NeutronNeutron-toto-Proton Ratio
NeutronNeutron-toto-Proton Ratio
• <'9 12)64)2!4
%7%?&
)/6% & β+ 12*+
3:$*&$314
)& )4*&$3*+!
81/6%7*&)'1
64 $)&.28 1
64 6&0!'!!&
)&,/3' %<*
&/ &!4
“electron captureJ
• <'9 12)645!4
%7%?&)
/6% & β 12*+ )&
&(5 81/6%
7*&).28 1 64 $
)&' 1 64
21
!" Nuclear Equations
• In nuclear equations to ensure conservation of
nucleons we write all particles with their atomic
and mass numbers:
Nucleons /!'!&)<'%':
1
0n
→ 11p+ + 0-1e- (β
β-emission)
0
-1e
-+0
1
1
0
p+
1
p+
1e
+
→ 0n +
+
0
-1
238 U
92
→ 200γ (positron annihilation)
1
e-
0
+
1e
(positron or
22
4
2α
→ 23490Th + 42α
represent for nucleus 42He (α
α-radiation)
β+-emission)
• In nuclear equations, the total number
of nucleons is conserved:
1
→ 0n (electron capture)
)*,.,,0.23&!!,%B=4!=;<*=1%/*(,
• Energy is conserved
23
24
#
" Nuclear force
#
"
#$%&
1Z,L$=$" "4B,1(0-;MB,?,I$01.J13f(e?,,0.123& 4
1Z,L$=%&'5&;,*-5*((./*L$=g3(;!=$"a#^^h*
1Z,L$=%&'5&;,*-5+*i*3 1(0-*(*$8$/!=;!=5.3
(mass defect) L3<.13&' ,#1Z,83==*,
The Einstein Equation
'"
$&#$
!&*2*. (∆
∆ E) /!&$!&!'
!&&)!*+
1/6%!'!&..02/,%/!&8'62
211p + 210n
&#<'%!!&*2*126<* (∆
∆ m; mass
defect )&$64;):#W9$&))%7.@9!)!&
4 He (E = -∆
B ∆E)
2
∆E = ∆mc2
m = c = &(/5X!Y (3x108m/sec)
27
Calculating “ Mass Defects”
28
6& n+p
%)*#-<c- ?A<+*5.3$.5;!= p+n L3<.
,)*5*+*g3>/*=c- 3I-<. 5.3$0=;!=,0.#23-H,B,
= (3 × mass of p) + (4×
× mass of n)
mass of proton = 1.00728 amu
mass of neutron = 1.00866 amu
)4
!&6 ∆m 73Li 12 7.016004 amu
+*)*,.,,0.>$!,*( n = A-Z
7 Li 5& neutrons = 7-3 = 4
3
6;)4!&<'9
∆m = (mass of p + n) - (atomic mass of nucleus)
29
30
What is the mass defect in formation
of a 42 He nucleus?
<'%
mass of p + n = 3(1.00728) + 4(1.00866)
= 3.02184+4.03464
= 7.05648 amu
(Mass of He nucleus (measured) = 4.00153 amu)
,4#/6%&9'%!!!
∆m = m[42He] - 2m[11H]-2m[10n]
= -0.03035 amu
mass defect of nucleus = 7.05648 - 7.016004
∆m = 0.040476 amu
31
Binding Energy (∆E)
32
The Electron Volt ( eV )
∆E = ∆mc2 7' m 64*+ kg $ &( c '
64*+ ms-1
4 He
211p + 210n
2
∆E = -28.2956 MeV
6&0 ∆E = -4.53*10-12 J
= -2.73*1012 J/mol
$<'%./64 5 (J) (1J = 1 kgms-1)
6&0/,% 1 amu = 931.5 MeV 12$'!$!!4
Note 1 MeV = 1.602177*10-13 J
34
Sample Problem
Energy Changes
/!&!' 42He
!&*2 *.
-28.2956 MeV 6&0 -2.73*1012 J/mol
o82!!4.12<'%!*p!&12<*?8 %14 ( ~107 14) &&98*+&12'
6Y
35
Calculate the binding energy per nucleon
for the isotopic species 3517Cl (mass 34.96885 amu.)
• Step 1 - write balanced nuclear equation.
• Step 2 - determine mass defect ...m.
• Step 3 - convert ∆m to ∆E
(1 amu = 931.494 MeV)
• Step 4 - convert ∆E to Eb ( sign change ) and
divide by the number of nucleons in the nucleus.
36
Cl-35 nuclear binding energy
12'/6X48
?&8&6&0???
= 4.900 x 10-11 J/atom
6&0 = 2.951 x 1013 J/mol
.&#!!' ! 4.8'62 .402'
/6X48.8'62!(2!8 !?
There are no stable nuclei heavier than
209
83
Bi
Binding Energy Per Nucleon
39
Some of the binding energies per nucleon
for some common elements
Element
Deuterium
Helium
Lithium
Beryllium
Iron
Silver
Iodine
Lead
Polonium
Uranium
Uranium
Neucleon
number
4
7
9
56
107
127
206
210
235
238
Mass of
nucleons
2.01594
4.03188
7.05649
9.07243
56.44913
107.86187
128.02684
207.67109
211.70297
236.90849
239.93448
Nuclear
mass
2.01355
4.00151
7.01336
9.00999
55.92069
106.8793
126.8754
205.9295
209.9368
234.9935
238.0004
Eb
(MeV)
2.23
28.29
40.15
58.13
492.24
915.23
1072.53
1622.27
1645.16
1783.8
1801.63
To compare nuclear stabilities, need to look at binding
energy per nucleon.
4 He
2
Eb = 28.2956 MeV
Eb/4 = 7.0739 MeV
35 Cl
17
Eb = 298.2085 MeV
Eb/17 = 17.5468 MeV
This figure gives a direct measure of the stability of the
He nucleus.
40
Trends of Binding Energy in Nuclear Stability
Eb per nucleon
(MeV)
1.12
7.07
5.74
6.46
8.79
8.55
8.45
7.88
7.83
7.59
7.57
41
42
• Binding energy per nucleon initially
increases with atomic mass.
• Maximum stability is reached in the vicinty
of 56Fe, which has Eb/A = 8.8 MeV.
• Stability decreases slightly for successively
heavier elements.
43