PDF Introduction to Nuclear Fusion

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I ntroduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

1

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What is nuclear fusion?

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3

Matter and Energy

• Chemical reactions (combustion)

heating fuels

reaction energy products

E_out

M_out E_in

M_in

Matter-Energy Transformation

reactor/

generator

eV O

H O

H 3

2 1

2 2

2

+ → +

K K J

eV J 11600

/ 10

38 . 1

10 6

.

1 1

₂₃

19

=



= 

₋⁻

eV

KT = 1

(4)

Matter and Energy

• Chemical reactions (combustion)

• Fission process

• Fusion process

• Total energy conservation including rest mass energy

• If Δm = M_out-M_in < 0, then we can get E_out > E_in

out out

in

M E M

E + → +

heating fuels

reaction energy products

E_out

M_out E_in

M_in

Matter-Energy Transformation

reactor/

generator

http://www.meteoweb.eu/2011/09/e- possibile-superare-la-velocita-della- luce-teoria-della-relativita-a-

rischio/88437/, Dec, 2014

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5

Mass Defect Energy of Nuclear Reaction

e d

b

a + → +

reactants products

*

after before

E E =

) (

)

( E

_k_,_a

+ m

_a

c

²

+ E

_k_,_b

+ m

_b

c

²

= E

_k_,_d

+ m

_d

c

²

+ E

_k_,_e

+ m

_e

c

²

Total energy

( m

a

c

²

+ m

b

c

²

) (  m

d

c

²

+ m

e

c

²

) + E

k_,d

+ E

k_,e

) (

)

(

_d _e _a _b

ab

m m m m

m = + − +



ab b

k a

k

E Q

E

_,

+

_,



For

( ) ( )

 

2

) (

m c

c m m

m m

Q

ab

e d

b a

ab



−

=

+

− +

=

2 2

,

2

1 2

1

e e d

d e

k d

k

ab

E E m v m v

Q  + = +

(6)

Mass Defect Energy of Nuclear Reaction

2 2

,

2

1 2

1

e e d

d e

k d

k

ab

E E m v m v

Q  + = +

e e d

d

v m v

m =

ab e

d d e

k ab

e d

k

Q

m m

E m m Q

m

E m  

 





 +

 

 





 +

_,

,

Momentum conservation for reactions with CM at rest

Ex) d-t fusion reaction

d + t → n +  MeV

6 .

= 17 Q

dt

kg m

m m

p

p n

27 27

10 674927

. 1

10 672621

. 1

−



=



=





 



 +

= +



 



 +

= +



 



 +

= +

2 2

2 2 2

2 2

2

2 2

2 1 2

1

2 1 2

1

2 1 2

1

d d e

d e e d e

d e

d d d

e d e

d e

d d e

e d

d

v m m v

m m m m

m m

v m m v

m m

v m m

m m

v m Derive! m

MeV 5

. 5 3

1 , MeV 5 14.1

4

,

,_n



_dt



_k



_dt



k

Q E Q

E

_

eV O

H O

H 3

2 1

2 2

2

+ → +

(7)

Fusion in Nature

• Nuclei are made up of protons and neutron, but the mass of a nucleus is always less than the sum of the individual masses of the protons and neutrons which constitute it.

• Binding energy: the amount of energy released when a particular nucleus is formed.

. . )

(A Z n X B E

Zp+ − →_Z^A +

2

)]

2

) (

[(

.

. E m Zm A Z m c mc

B  −

_X

−

_p

− −

_n

= − 

Δm < 0: released energy

(exothermic or exoergic)

http://www.daviddarling.info/encyclopedia/B/binding_energy.html 7

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Fusion in Nature

Sun producing 3.8x10²⁶ J/s equivalent to 4.3x10⁹ kg

Sun composed of proton (73%), He (25%), etc

How old is the sun?

How long is the sun’s lifetime?

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Fusion in Nature

Nobel prize in physics 1967

“for his contribution to the theory of nuclear reactions, especially his discoveries concerning the energy production in stars”

Hans Albrecht Bethe

(1906. 7. 2 – 2005. 3. 6)

•

Fusion reactions by which stars convert hydrogen to helium

- The PP (proton-proton) chain: in stars the mass of the Sun and less - The CNO cycle (Bethe-Weizsäcker-cycle): in more massive stars

http://www.nobelprize.org/nobel_prizes/physics/laureates/1967/bethe-bio.html 9

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Fusion in Nature

One of the most impressive discoveries was the origin of the stars, that makes them continue to burn. One of the men who

discovered this was out with his girl friend the night after he realized that nuclear reactions must be going on in the stars in order to make them shine. She said “Look at how pretty the stars shine!” He said “Yes, and right now I am the only man in the

world who knows why they shine.” She merely laughed at him.

She was not impressed with being out with the only man who, at that moment, knew why stars shine. Well, it is sad to be alone, but that is the way it is in this world.

- The Feynman Lectures on Physics I, p.3-7

•

Fusion reactions by which stars convert hydrogen to helium

- The PP (proton-proton) chain: in stars the mass of the Sun and less - The CNO cycle (Bethe-Weizsäcker-cycle): in more massive stars

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Fusion in Nature

• The PP (Proton-Proton) Chain

Step 1: Smash two protons together to make deuterium

MeV d

p

p + → + 

⁺

+  + 1 . 2

Positron: antiparticle of the electron-like an electron with charge of +1e

Neutrino: electrically neutral, weakly interacting elementary subatomic particle

http://burro.astr.cwru.edu/Academics/Astr221/StarPhys/ppchain.html

MeV n

p → + 

⁺

+  − 0 . 782

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Fusion in Nature

Gamma ray: electromagnetic radiation of an extremely high frequency (very high energy photon)

MeV He

d

p + →

³

+  + 5 . 5

Do Steps 1 and 2 again so to have two ³He nuclei.

Step 2: A proton crashes into a deuterium nucleus, making ³He

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13

Fusion in Nature

MeV p

He He

He

³ ⁴

2 12 . 9

3

+ → + +

Step 3: Mash two helium-3 nuclei together to make helium-4

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Fusion in Nature

MeV d

p

p + → + 

⁺

+  + 1 . 2 MeV He

d

p + →

³

+  + 5 . 5

MeV p

He He

He

³ ⁴

2 12 . 9

3

+ → + +

 +

→ + He Be He

⁴ ⁷

3



 → + +

⁻

Li

Be

⁷

7

nucleosynthesis

PPII Chain (14-23x10⁶ K)

He He

p

Li

⁴ ⁴

7

+ → +

PPIII Chain ( > 23x10⁶ K)

 +

→ + p B Be

⁸

7



 + +

→ Be

⁺

B

⁸

8

He He

Be

⁴ ⁴

8

→ +

• The PP (Proton-Proton) Chain

http://csep10.phys.utk.edu/astr162/lect/energy/ppchain.html

10-14x10⁶ K

CX → years

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Fusion in Nature

- Most of ⁴He nuclei being produced in the Sun are born in the PP chain (98.3%).

6

K 10 15 

T_Sun



http://www.nasa.gov/mission_pages/galex/20070815/f.html

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Fusion in Nature

MeV N

p

C

¹³

1 . 9

12

+ → +

MeV C

N

¹³

1 . 5

13

→ + 

⁺

+  +

MeV N

p

C

¹⁴

7 . 6

13

+ → +

MeV O

p

N

¹⁵

7 . 3

14

+ → +

MeV N

O

¹⁵

1 . 8

15

→ + 

⁺

+  +

MeV C

p

N

¹²

5 . 0

15

+ → +  +

MeV p 2 2 25 . 1

4 →  + 

⁺

+  +

• The CNO Cycle

K T_star 1310⁶

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17

Fusion in Nature

keV Be 92

8

−

→ + 



MeV C

Be

¹²

7 . 367

8

+  → +  +

• The triple alpha process

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Fusion in Nature

http://jcconwell.wordpress.com/2009/07/20/formation-of-the-elements/

http://eqseis.geosc.psu.edu/~cammon/HTML/Classes/IntroQuakes/Notes/earth_origin_lecture.html

Layers of Fusion in the final stage of a massive star

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19

Fusion reaction

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Physical Characterization of Fusion Reaction

r r q F

_c _a

q

^a₃^b

0

,

4

1

= 

a b

r r m G m

F

_g_,_a

= −

^a₃ ^b

• The electrostatic force caused by positively charged nuclei is very strong over long distances, but at short distances the nuclear force is stronger.

• As such, the main technical difficulty for fusion is getting the nuclei close enough to fuse.

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21

Physical Characterization of Fusion Reaction

(

^A ^A

)

^R

( )

^m

R R

p t R d

q R q

U

p b

a p

b a

15 3

/ 1 3

/ 1 0

0 0 0

10 7

. 1 3 . 1

,

, , for MeV 4

. 0 4 ~

) 1 (

 −

−

= +



= 

] 1 exp[

) Pr(

r b a

r v

q q tunneling  v −

Reflection and tunneling of an

electron wave packet directed at a potential barrier

A. B. Balantekin and N. Takigawa, 'Quantum tunneling in nuclear fusion', Rev. Mod. Phys. 70, 77 (1998).

V0

E

( )

e a

k k

a k

k T k



2 2

2 0 2

0

2 2 2 0 2 2 0

2 2 0

4

sinh )

/ 2 (

) / 2 (

 −









 +

+

= +

0

Attractive strong nuclear force

Sun: 1.4 keV

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Physical Characterization of Fusion Reaction

• By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunneling. After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunneling.

(Max Born, Nobel Prize in Physics 1954)

Max Born (1882-1970) George Gamow

(1904-1968)

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Fusion Reaction Cross Sections

- Fusion cross section for low energy E_CM < U(R₀) by quantum mechanical tunneling process:

E B

ab

e

E

E ) A

^/

( =

⁻



Gamow theory

(1938)

0 2

2 / 1 2

/

1

/

2

., B  m Z Z e h 

const

A = =

⁻ _r _a _b

2 28 2

24

10

1 barn =

⁻

cm =

⁻

m

(24)

Fusion Reaction Rate Parameter (Reactivity)

• Fusion reaction rate density

ab r b

a

fu

N N v

R =   

( )

_a _b _a _a _b _b _a _b

v v

b a ab

ab v v v v F v F v d v d v

v

a b

3

) 3

( )

( 



  − −

=



^

 

^

• σ-v parameter

r b a

fu

N N v

R  v

_r

v 

_a

v 

_b

−

=

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25

- Thermodynamic equilibrium - Both species at the same

temperatures

http://www.scienceall.com/jspJavaPopUp.do?classid=CS000140&articleid=611&bbsid=146&popissue=java

) ( )

(

_x _x _x

x

v M v

F →

(26)

• Fusion reaction rate density

ab r b

a

fu

N N v

R =   

fu ab b

a fu

fu

R Q N N v Q

P = =   

• Fusion power density

( )

_a _b _a _a _b _b _a _b

v v

b a ab

ab v v v v F v F v d v d v

v

a b

3

) 3

( )

( 



  − −

=



^

 

^

• σ-v parameter

r b a

fu

N N v

R  v

_r

v 

_a

v 

_b

−

=