What is Nuclear Fusion?

Nuclear fusion is a type of nuclear reaction. In nuclear fusion, two or more than two lighter atomic nuclei combine to form a heavier nucleus. During this process, an enormous amount of energy is released. This energy is called nuclear energy. Nuclear fusion is the energy source of the sun and stars.

Theory of Nuclear Fusion

We know that an atom is made up of protons, neutrons, and electrons. The nucleus of the atom consists of protons and neutrons. The neutrons and protons are collectively called nucleons. Nucleons are held together by two types of forces i.e., weak nuclear force and strong nuclear force.

In the nucleus, two or more protons are held together by a strong nuclear force. The protons are positively charged particles. From electrostatics, we know that like charges repel each other. Nuclear forces that hold protons should overcome the electrostatic repulsion. The protons are held by strong nuclear force. The proton-proton force is called strong nuclear force. Similarly, protons and neutrons in a nucleus are held by a weak nuclear force. Since the neutron is neutrally charged, the force required to hold the nucleons is relatively weaker than the strong nuclear force. Hence, this force is called weak nuclear force.

Nuclear force is a short-range force, which means the force acts only when the particles are very close to each other. Also, nuclear force is always attractive. Let us consider an example of the fusion of deuterium and tritium isotopes of hydrogen. The nuclear fusion of deuterium and tritium (D-T) produces helium and a neutron.

${}_1{}^{2}H+{}_1{}^{2}H\to {}_2{}^{4}He+{}_0{}^{1}n$

As the isotopes undergo fusion process, the nucleus of deuterium and tritium exert repulsion due to electrostatic field. The reactant nuclei has to overcome this electrostatic repulsion. This repulsive energy is called the coulomb barrier. The interaction energy between the reactant nuclei must be greater than the coulomb barrier energy. Hence, to initiate a nuclear fusion reaction, a large quantity of energy has to be supplied to overcome the coulomb barrier.

The coulomb barrier energy is small for lighter nuclei whereas, the coulomb barrier energy for heavier nuclei is very large. Nuclear energy is not strong enough between the heavier nuclei as the distance between the nuclei is large. Since nuclear force is a short-range force, the heavier nuclei cannot undergo nuclear fusion reactions. 

Hence, the light elements release energy when they undergo nuclear fusion. Heavier nuclei do not release energy during fusion, instead energy has to be supplied to fuse the atoms of heavier nuclei. Heavier nuclei undergo nuclear fission releasing an enormous amount of energy while, lighter nuclei do not undergo nuclear fission reactions.

Binding energy

Binding energy is defined as the minimum energy required to separate nuclei into their constituent nucleons. Binding energy is related to the stability of the nuclei. The nuclei with high binding energy are more stable. Binding energy is not used to compare different nuclei with a different number of nucleons. Hence, the binding energy per nucleon is used to compare different nuclei.

The binding energy per nucleon is given as,

$BEN=\frac{BE}{A}$

Where,

A is the atomic mass number.

BE is binding energy.

The binding energy of nuclei is obtained using,

$BE=\left[Z{m}_p+(A-Z)m_p-M\right]c^2$

Where,

Z is the atomic number of the nuclei.

A is the atomic mass of the nuclei.

M is the mass of the nuclei (in kg or amu).

$m_p$ is the mass of the proton (in kg or amu).

$m_n$is the mass of neutron (in kg or amu).

Binding energy per nucleon is a positive number for all the stable nuclei. When nuclei undergo nuclear reaction, binding energy is released as nuclear energy.

The energy released during Nuclear Fusion

In the example of the fusion of deuterium and tritium, the combined mass of the product is less than the combined mass of reactants. The mass difference is released as nuclear fusion energy. The released energy is calculated as follows.

We know that the atomic mass of deuterium is $m_d=2.014102 amu$

The atomic mass of tritium is $m_t=3.016050 amu$

Hence, the total mass of the reactant is $m_R=5.030152 amu$ 

Also, the atomic mass of helium is $m_{He}=4.002603 amu$

The mass of the neutron is $m_n=1.008665 amu$

Hence, the total mass of product is $m_P=5.011258 amu$

The quantity of mass lost is,

$∆m=(5.030152-5,011268) amu$

$∆m=0.018884 amu$

The mass difference can be given in kg as follows,

$\begin{array}{rcl}∆m& =& (0.018884amu)\frac{1.66056\times {10}^{-27}kg}{1amu}\\ ∆m& =& 3.136\times {10}^{-29}kg\end{array}$

During the fusion reaction, 0.018884 amu of mass is converted to energy. The amount of fusion energy can be calculated using Einstein’s mass-energy equation.

$E=∆m{c}^2$

Where c is the velocity of light.

The fusion energy released during the fusion reactions is,

$\begin{array}{rcl}E& =& \left(2.823\times {10}^{-12}J\right)\frac{6.242\times {10}^{12}MeV}{1 J}\\ E& =& 17.621 MeV\end{array}$

During the fusion of the deuterium-tritium nucleus, 17.621 MeV of energy is released.

Nuclear fusion in stars

Nuclear fusion is the energy source of stars. Stars release a huge amount of energy by the fusion of lighter elements like hydrogen to form helium. Sun-like stars undergo a process called a proton-proton chain reaction. In stars, hydrogen nuclei act as fusion fuel. In this process, four hydrogen nuclei undergo fusion to form a helium nucleus and two positrons and two neutrinos. Stars that are heavier than the sun undergo a process called the CNO cycle. In which helium is produced using carbon, nitrogen, and oxygen as catalysts.

Proton-Proton cycle

In stars, four protons (or positive hydrogen ions) combine to form helium nuclei. This reaction is called the proton-proton chain reaction. This reaction is observed in the stars of mass less than or equal to the mass of the sun. This process consists of three steps.

Step 1: Initially two protons combine to form deuterium nuclei, positron, and neutrino,

${}_1{}^{1}H+{}_1{}^{1}H\underset{}{\to }{}_1{}^{2}D+e{}_{}{}^{+}+v_e$

The reacting nuclei have to overcome the electrostatic repulsion between protons. Hence sufficient energy is required to fuse the protons. Stars have very high temperatures due to gravitational collapse. Hence this reaction is initiated as the reactants have enough energy to overcome the repulsion.

Step 2: In this step, the deuterium formed is fused with a proton to form a Helium-3 isotope.

${}_1{}^{2}D+{}_1{}^{1}H\underset{}{\to } {}_2{}^{3}He$

In this step, gamma rays are emitted along with helium-3 isotope.

Step 3: In final step two helium-3 isotopes combine to form a stable helium-4 nucleus also two protons are formed as byproduct.

${}_2{}^{3}He+{}_2{}^{3}He\stackrel{}{\to }{}_2{}^{4}He+2{}_1{}^{1}H$

Nuclear Fusion reactors

During the nuclear fusion reaction, an enormous amount of energy is released. The main disadvantage of the fusion reaction is that fusion power is not controllable. Also, to initiate nuclear fusion large amount of energy has to be supplied. Hence, it is challenging to construct a nuclear fusion power reactor. 

Scientists and researchers are learning to harness nuclear fusion energy. An international research project called ITER is founded by seven countries. ITER aims to create a feasible nuclear fusion reactor. ITER is the largest nuclear fusion power plant in the world.

Formulas

The fusion energy released during the nuclear fusion reactions is,

$E=∆m{c}^{2 }(in Joules)$

where, $∆m$ is the mass defect (in kg or amu).

c is the velocity of light (in m/s).

The binding energy of a nucleus can be calculated using,

$BE=\left[Z{m}_p+(A-Z)m_p-M\right]c^2$

Where,

Z is the atomic number of the nuclei.

A is the atomic mass of the nuclei.

M is the mass of the nuclei (in kg or amu).

$m_p$ is the mass of the proton (in kg or amu).

$m_n$ is the mass of the neutron (in kg or amu).

Context and Applications

This topic is significant in physics for both undergraduate and graduate courses, especially for Masters in physics, and Bachelors in physics.

Practice Problems

Question 1: Which of the following holds protons in the nucleus together?

(a) Weak nuclear force

(b) Strong nuclear force

(c) Electrostatic force

(d) None of the above

Answer: Option (b) is the correct answer.

Explanation: The protons are positively charged; the nuclear energy force that holds the protons are the strong nuclear force.

Question 2: The protons in the nuclei repel each other due to ___.

(a) Strong nuclear force

(b) Weak nuclear force

(c) Electrostatic force

(d) Magnetic force

Answer: Option (c) is the correct answer.

Explanation: Protons are positively charged; hence they repel each other with electrostatic force.

Question 3: Usually lighter nuclei undergo ___.

(a) Nuclear fusion

(b) Nuclear fission

(c) Both (a) and (b)

(d) None of the above

Answer: Option (a) is the correct answer.

Explanation: Lighter nuclei can easily undergo fusion since binding energy is less compared to that of heavier nuclei. They also have low coulomb barrier energy.

Question 4: Which of the following is the source of energy of stars?

(a) Combustion

(b) Chemical reaction

(c) Nuclear fission

(d) Nuclear fusion

Answer: Option (d) is the correct answer.

Explanation: Nuclear fusion is the energy source of stars. In stars, hydrogen nuclei undergo fusion to form helium nuclei and energy.

Question 5: The main disadvantage of nuclear fusion is ___.

(a) Large quantity of energy is produced

(b) Uncontrollable

(c) Availability of fuel

(d) Nuclear waste management

Answer: Option (b) is the correct answer.

Explanation: The main disadvantage of nuclear fusion is that it is uncontrollable. Since a huge quantity of energy is produced during the fusion reaction.