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26-S
ep-1
6 1
Nuc
lear
ene
rgy
prod
uctio
n
Aim
: eva
luat
e en
ergy
-gen
erat
ion
rate
per
uni
t mas
s
.
Sun:
(c
heck
L
/M
, hu
man
)
ener
gy-g
ener
atio
n ra
te p
rodu
ced
from
fusi
on o
f tw
o nu
clei
a +
A:
ener
gy re
leas
edpe
r rea
ctio
nre
actio
n ra
tepe
r uni
t vol
ume
(incl
udes
cro
ss s
ectio
nan
d ve
loci
ty d
istri
butio
n
)
-Nuc
lear
reac
tions
in s
un ta
ke p
lace
with
in in
ner ~
10%
of t
otal
sol
ar m
ass.
-Est
imat
ion
of e
nerg
y ge
nera
tion
by “h
ydro
gen
burn
ing”
reac
tions
Nuc
lear
ene
rgy
prod
uctio
n
-nuc
lear
tim
e sc
ale:
-mas
s lo
ss:
Nuc
lear
ene
rgy
prod
uctio
n
Ener
gy
by
each
reac
tion
-Mos
t sta
rs li
ve fr
om s
o-ca
lled
ther
mon
ucle
ar fu
sion
, whe
re d
ue to
ther
mal
mot
ion
light
er n
ucle
i fus
e to
form
hea
vier
ele
men
ts.
-Dur
ing
fusi
on p
roce
ss, s
ome
of th
e m
ass
(e.g
. 0.7
% fo
r hyd
roge
n bu
rnin
g)of
orig
inal
nuc
lei h
as b
een
conv
erte
d in
to e
nerg
y ac
cord
ing
to
.
-Mas
s lo
ss
o
rigin
s in
diff
eren
t bin
ding
ene
rgie
s
of
the
invo
lved
nuc
lei.
-is
ene
rgy
requ
ired
to s
epar
ate
nucl
eons
(pro
tons
& n
eutro
ns) a
gain
st th
eir
mut
ual a
ttrac
tion
of th
e st
rong
, sho
rt-ra
nge
forc
es, o
r the
gai
n if
nucl
eons
are
brou
ght t
oget
her(
with
in 1
0-12
cm) f
rom
infin
ity.
tota
l res
t mas
sof
neu
trons
tota
l res
t mas
sof
pro
tons
mas
s of
nucl
eus
Inte
ger a
tom
ic w
eigh
t:
Nuc
lear
ene
rgy
prod
uctio
n
Ener
gy
by
each
reac
tion
-for
com
parin
g nu
clei
, bet
ter
p
er n
ucle
on:
Inte
ger a
tom
ic w
eigh
t:
-for
A<5
6, f
, bec
ause
of s
hort-
rang
e, s
trong
-fo
rces
affe
ctin
g on
ly n
ucle
ons
in it
s im
med
iate
ne
ighb
ourh
ood
only
(geo
met
rical
effe
ct: s
urfa
ce
incr
ease
s w
ith r
slow
er th
an v
olum
e, i.
e. w
ith A
).
-for
A>5
6, f
, bec
ause
of r
epul
sive
Cou
lom
bfo
rces
bet
wee
n pr
oton
s, w
hich
are
far-r
each
ing.
-56 F
e m
ost t
ight
ly b
ound
nuc
lei (
smal
lest
m/n
ucle
i).
-any
reac
tion
brin
ging
resu
lting
nuc
lei c
lose
to f m
axw
ill be
exo
ther
mic
:
(a)
for A
<56
by fu
sion
(e.
g. s
tella
r cor
es)
(b)
for A
<56
by fi
ssio
n (e
.g. r
adio
activ
ity)
26-S
ep-1
6 2
Nuc
lear
ene
rgy
prod
uctio
n
Ener
gy
by
each
reac
tion
-Con
side
r gen
eral
reac
tion:
-The
ene
rgy
re
leas
ed in
this
reac
tion
is:
Shor
t not
atio
n:
-Sin
ce lo
ss o
f mas
s is
gen
eral
ly v
ery
smal
l, m
ore
conv
enie
nt to
use
m
ass
exce
ss
prot
ons
neut
rons
avai
labl
e in
form
of t
able
s(e
.g. C
layt
on 1
968)
.at
omic
mas
s ex
cess
es
Nuc
lear
ene
rgy
prod
uctio
nTh
e cr
oss
sect
ions
defin
ed a
s an
are
a, w
hich
des
crib
es th
e re
latio
n be
twee
n pr
obab
ility
pof
a re
actio
n an
d a
geom
etric
al a
rea.
- -rea
ctio
n be
twee
n nu
clei
is c
ause
d by
the
stro
ng fo
rces
, act
ing
betw
een
prot
ons
and
neut
rons
(rang
e lim
ited
to e
xten
t of n
ucle
us).
-for
a re
actio
n to
occ
ur, n
ucle
i mus
t be
brou
ght s
o cl
ose
toge
ther
that
they
touc
h ea
chot
her
re
quire
s to
ove
rcom
e C
oulo
mb
repu
lsio
nbe
twee
n th
em.
inte
ract
ion
radi
us
r 0=r
nucl
~A1/
3 10-
13cm
r~10
-8cm
E CB
-ene
rgy
to o
verc
ome
Cou
lom
b re
puls
ion,
i.e.
heig
ht o
f Cou
lom
b ba
rrier
-how
ever
, ave
rage
kin
etic
ene
rgy
of p
artic
les
nucl
ear (
stro
ng) f
orce
Cou
lom
bfo
rce
Nuc
lear
ene
rgy
prod
uctio
nTh
e cr
oss
sect
ions
Mea
sure
d cr
oss
sect
ion
for r
eact
ion
Stel
lar c
ore
cond
ition
s~2
5 ke
VEn
ergy
(MeV
)
Nuc
lear
ene
rgy
prod
uctio
nTh
e cr
oss
sect
ions
-rea
sona
ble
extra
pola
tion
of
to
low
er
c
an b
e ob
tain
ed b
y se
para
ting
stro
nges
t -
depe
nden
t con
tribu
tions
cau
sed
by (q
uant
um-m
echa
nica
l) pe
netra
tion
prob
abilit
y th
roug
h C
oulo
mb
barri
er.
-an
addi
tiona
l -
depe
nden
ce e
nter
s in
the
‘geo
met
rical
ext
ent’
of th
e nu
clei
:
… d
e Br
oglie
wav
elen
gth
geom
etric
al c
ross
sec
tion
-qua
ntum
-mec
hani
cal e
ffect
foun
d by
G. G
amov
(192
8) is
the
‘tunn
el-e
ffect
’, in
whi
ch n
ucle
i can
pen
etra
te (t
unne
l) th
e po
tent
ial b
arrie
r with
a p
roba
bilit
y
rela
tive
velo
city
bet
wee
n th
e tw
o nu
clei
: v~
E1/
2
,
26-S
ep-1
6 3
Nuc
lear
ene
rgy
prod
uctio
nTh
e cr
oss
sect
ions
-cro
ss-s
ectio
nfa
ctor
(ast
roph
ysic
al c
ross
sec
tion)
“ast
roph
ysic
al”
geom
etric
alfa
ctor
Cou
lom
b ba
rrier
-her
e
is
ener
gy o
f par
ticle
s in
the
cent
re-o
f-mas
s sy
stem
… re
duce
d at
omic
wei
ght
whe
re
Rea
ctio
ns a
mon
gst n
ucle
i of
low
cha
rges
are
fast
er a
nd a
re
poss
ible
at l
ow te
mpe
ratu
res.
Nuc
lear
ene
rgy
prod
uctio
nTh
e cr
oss
sect
ions
“cro
ss s
ectio
n fa
ctor
”
Nuc
lear
ene
rgy
prod
uctio
n
“ast
roph
ysic
al c
ross
sec
tion”
-dep
artu
re fr
om (n
ear)
cons
tant
val
ueat
low
E re
sults
from
‘scr
eeni
ng’f
rom
el
ectro
ns in
the
targ
et6 L
i or 7
Li a
tom
s,
whi
ch re
duce
s C
oulo
mb
pote
ntia
l ar
ound
nuc
leus
, and
hen
ce in
crea
ses
.
Nuc
lear
ene
rgy
prod
uctio
n
-we
need
rela
tion
betw
een
an
d to
tal r
ate
of re
actio
ns, b
etw
een
parti
cles
aan
d A
Prob
abilit
yth
at o
ne p
artic
le a
, the
pro
ject
ile, w
illha
ve a
col
lisio
n al
ong
dlis
i.e.
-spe
cial
cas
e A
=a:
The
reac
tion
rate
s
-par
ticle
flux
of p
roje
ctile
:
-rea
ctio
ns p
er u
nit t
ime
in w
hole
box
:
-rea
ctio
ns p
er u
nit t
ime
and
volu
me:
proj
ectil
es p
assi
ng
dAev
ery
seco
nd
26-S
ep-1
6 4
Nuc
lear
ene
rgy
prod
uctio
n
-In
real
ity A
is n
ot a
t res
t and
ther
e is
a d
istri
butio
n of
the
rela
tive
velo
citie
s v.
If
f(v)d
vis
the
fract
ion
of p
airs
of p
artic
les
with
rela
tive
spee
d be
twee
n v
& v+
dv
with
.
The
reac
tion
rate
s
-We
assu
me
that
bot
h nu
clei
hav
e M
-B d
istri
butio
n, th
en (s
ee C
layt
on 1
968)
the
dist
ribut
ion
of v
is a
lso
M-B
,
whe
re
… re
duce
d at
omic
wei
ght.
with
redu
ced
mas
s
intri
nsic
nucl
ear
reac
tions
Nuc
lear
ene
rgy
prod
uctio
n
and
we
obta
in
incl
udes
geo
met
ry o
f par
ticle
sM
-B fa
ctor
of
ther
mal
en
ergy
Cou
lom
b ba
rrier
(tunn
elin
g)
The
reac
tion
rate
s
Gam
ovpe
ak
com
petin
g ef
fect
s
resu
lting
in
max
imum
at
Nuc
lear
ene
rgy
prod
uctio
n
-ana
lytic
al a
ppro
xim
atio
n of
Gam
ovpe
ak:
-we
negl
ect v
aria
tion
of S
(E) o
ver
Gam
ovpe
ak [
S(E
) S
(E0)
], an
d ap
prox
imat
eth
e fu
nctio
n
by
a G
auss
ian
whe
re th
e w
idth
pro
vide
s th
e sa
me
curv
atur
e at
max
imum
.Af
ter s
ome
alge
bra
we
obta
in
,
with
For f
airly
larg
e
,
is
a d
ecre
asin
g fu
nctio
n of
:
for
.
The
reac
tion
rate
s
Nuc
lear
ene
rgy
prod
uctio
n
-in
orde
r to
‘see
’ the
T-d
epen
denc
em
ore
clea
rly, w
e fu
rther
sim
plify
the
expr
essi
onov
er a
lim
ited
Tra
nge
arou
nd T
=T0,
def
inin
g T 0
such
that
and
adop
t the
pow
er la
w
Beca
use
,
the
T-se
nsiti
vity
of
incr
ease
s st
rong
ly w
ith
:
e.g.
n=
4…
...5
fo
r H-b
urni
ng in
Sun
n=13
…23
f
or C
NO
-cyc
le .
The
reac
tion
rate
s
26-S
ep-1
6 5
Nuc
lear
ene
rgy
prod
uctio
n
We
may
now
writ
e th
e re
actio
n ra
tes
(reac
tions
/s /c
m3 )
as
The
ener
gy g
ener
atio
n ra
te p
er u
nit m
ass
is th
en
whe
re w
e us
ed
abun
danc
es
by m
ass
,
.
Rat
e of
cha
nge
in
d
ue to
reac
tions
with
nuc
lei o
f typ
e
The
reac
tion
rate
s
Nuc
lear
ene
rgy
prod
uctio
nEl
ectr
on s
cree
ning
-rep
ulsi
ve C
oulo
mb
forc
e is
impo
rtant
for e
stim
atin
g.
-fre
e e-
have
influ
ence
on C
oulo
mb
forc
e, i.
e. E
CB.
-app
roac
hing
par
ticle
will
“feel
” neu
tral c
ongl
omer
ate
of ta
rget
nuc
leus
&su
rroun
ding
e- -
clou
d.
-e- a
re a
ttrac
ted
from
nuc
leus
of c
harg
e +Z
ee-
have
slig
htly
n
ene
ar n
ucle
us,
and
ions
are
repe
lled,
i.e.
ni
clus
terin
g of
e- .
Nuc
lear
ene
rgy
prod
uctio
n
-par
ticle
den
sity
nw
ith c
harg
e ‘q
’ is
mod
ified
in th
e pr
esen
ce o
f an
elec
trost
atic
pot
entia
l
acco
rdin
g to
# de
nsity
for
= 0
… d
eriv
ed fr
omD
ebye
-Hüc
kelt
heor
y(s
ee e
.g. J
acks
on 1
975)
-Typ
ical
ly
a
ppro
xim
atio
n:
Whi
ch s
how
s di
rect
ly th
e de
crea
se o
f ion
den
sity
and
incr
ease
of e
- den
sity
.
Elec
tron
scr
eeni
ng
Nuc
lear
ene
rgy
prod
uctio
n
-tot
al (e
lect
rical
) cha
rge
dens
ity
for
all
type
s of
ions
(ni)
(neu
tral g
as)
i.e.: :
-Com
bine
last
two
term
s to
obt
ain
whe
rear
e
Elec
tron
scr
eeni
ng
26-S
ep-1
6 6
Nuc
lear
ene
rgy
prod
uctio
n
-an
dar
e al
so c
onne
cted
via
the
‘Poi
sson
equ
atio
n’
whi
ch h
as th
e so
lutio
n (fo
r a p
oint
cha
rge
Ze) a
nd fo
r sph
eric
al g
eom
etry
with
……
… b
eing
the
Deb
ye-ra
dius
(to s
ome
sens
e th
e ro
f e-cl
oud;
r D=1
0-8 …
10-9
cm.)
For ‘
norm
al s
tars
’
unsh
ield
edpo
tent
ial
clas
sica
l par
ticle
dis
tant
at
Gam
ov p
eak
Elec
tron
scr
eeni
ng
Nuc
lear
ene
rgy
prod
uctio
n
-Ac
cord
ingl
y th
e C
oulo
mb
barri
er E
CB
is re
duce
d by
Deb
ye e
nerg
y E
D
Deb
ye e
nerg
y E
D
Ther
eby
incr
easi
ng th
e pr
obab
ility
with
whi
ch p
artic
les
can
tunn
elth
roug
h th
e C
oulo
mb
barri
er, l
eadi
ng to
an
incr
ease
of
Cou
lom
bba
rrier
M-B
tail
of th
erm
alen
ergy
scre
enin
gfa
ctor
“f”
Incr
ease
by
“f”:
“ f”
Elec
tron
scr
eeni
ng
.
.
Nuc
lear
ene
rgy
prod
uctio
n
-stro
ng s
cree
ning
:
-wea
k sc
reen
ing:
For s
olar
cas
e:
Scre
enin
g fa
ctor
dep
ends
stro
ngly
on
& T
Elec
tron
scr
eeni
ng
(con
trolle
d by
wea
k in
tera
ctio
n)
2x
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(a) p
p ch
ain:
no 4
-bod
y re
actio
ns
sub
-reac
tions
T~
15 m
ill. K
mai
ntai
n ch
arge
bal
ance
(con
serv
e le
pton
num
ber)
26-S
ep-1
6 7
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(a) p
p ch
ain:
no 4
-bod
y re
actio
ns
sub
-reac
tions
2x
(con
trolle
d by
wea
k in
tera
ctio
n)
T~
15 m
ill. K
mai
ntai
n ch
arge
bal
ance
(con
serv
e le
pton
num
ber)
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(a) p
p ch
ain:
no 4
-bod
y re
actio
ns
sub
-reac
tions
2x
(con
trolle
d by
wea
k in
tera
ctio
n)
T~
15 m
ill. K
mai
ntai
n ch
arge
bal
ance
(con
serv
e le
pton
num
ber)
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(a) p
p ch
ain:
no 4
-bod
y re
actio
ns
sub
-reac
tions
shie
ldin
g fa
ctor
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(b) C
NO
cyc
le:
CN
O (I
)
CN
O(II
)[1
04tim
es
less
like
ly]
(C, N
& O
are
cat
alys
ts)
T~
20 m
ill. K
e e e
posi
tron
deca
y (p
n
)
26-S
ep-1
6 8
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(b) C
NO
cyc
le:
CN
O (I
)
CN
O(II
)[1
03tim
es
less
like
ly]
(C, N
& O
are
cat
alys
ts)
T~
20 m
ill. K
e e e
posi
tron
deca
y (p
n
)
Nuc
lear
ene
rgy
prod
uctio
nH
ydro
gen
burn
ing
(b) C
NO
cyc
le:
Nuc
lear
ene
rgy
prod
uctio
nH
eliu
m b
urni
ngTr
iple
re
actio
n:
perfo
rmed
in 2
ste
ps:
Also
:
(dec
ays
afte
r 10-
16s)
Onc
e su
ffici
ent
is p
rodu
ced,
add
ition
al to
trip
le
reac
tion,
suc
cess
ive
-cap
tion:
Nuc
lear
ene
rgy
prod
uctio
nC
arbo
n bu
rnin
gW
hen
He
is e
xhau
sted
, nex
t ele
men
t to
reac
t is
mos
t pro
babl
e
(end
othe
rmic
reac
tion)
with Scre
enin
g fa
ctor
may
bec
ome
impo
rtant
.
26-S
ep-1
6 9
Nuc
lear
ene
rgy
prod
uctio
nO
xyge
n bu
rnin
g
Cou
lom
b ba
rrier
alre
ady
so h
igh
that
tem
pera
ture
s ar
e re
quire
d.
mos
t fre
quen
t
2nd
mos
t fre
quen
t
Phot
odis
inte
grat
ion
and
subs
eque
nt
capt
ure:
(Tso
hig
h th
at p
hoto
ns in
MeV
rang
e; th
ese
may
be
abso
rbed
by
nucl
ei, c
ausi
ngth
em to
bre
ak u
p, ty
pica
lly in
an
deca
y), e
.g. n
eon
disi
nteg
ratio
n(b
urni
ng)
(end
othe
rmic
occu
rs b
efor
eox
ygen
bur
ning
)
net r
eact
ion:
pa
rticl
e us
ed in
He
burn
ing:
Silic
on b
urni
ng:
two
28Si
5
6 Fe
end
supe
rnov
a ex
plos
ion.
Nuc
lear
ene
rgy
prod
uctio
n
Nuc
lear
ene
rgy
prod
uctio
nN
eutr
inos
Hav
e ex
trem
ely
smal
l cro
ss s
ectio
n
Neu
trino
s in
MeV
rang
e ha
ve m
ean
free
path
(=1
)
Nor
mal
ste
llar m
atte
r :
:
colla
psin
g st
ella
r cor
e:re
abso
rbed
with
in s
tar.
Also
impo
rtant
for c
oolin
g co
res
of s
tars
.
Nuc
lear
ene
rgy
prod
uctio
nN
eutr
inos
Hyd
roge
n bu
rnin
g (e
nerg
y lo
ss d
ue to
esc
apin
g ne
utrin
os):
With
~ 4
x10-
5er
g/cy
cle
Sun
prod
uces
~ 2
x103
8(e
lect
ron)
neu
trino
s ev
ery
seco
nd!
Flux
of ~
101
1so
lar n
eutri
nos
per c
m2
and
seco
nd a
t Ear
th!
Urc
a pr
oces
s at
ext
rem
e de
nsiti
es (n
o nu
clea
r rea
ctio
ns n
eces
sary
):
e-ca
ptur
e:
Nuc
leus
with
cha
rge
Zan
d at
omic
wei
ght A
dege
nera
te
26-S
ep-1
6 10
Nuc
lear
ene
rgy
prod
uctio
nN
eutr
inos
: E-
EM w
ave:
dis
pers
ion
eq.:
coup
les
with
e-
othe
r pro
cess
es w
ithou
t nuc
lear
reac
tions
anni
hila
tion
once
in 1
019
times
of e
-
