Question

Not only particles such as electrons and protons but also entire atoms can be classified as fermions or bosons, depending on whether their overall spin quantum numbers are, respectively, half-integral or integral. Consider the helium isotopes ${}^{\mathbf{3}}\mathbf{He}$and ${}^{\mathbf{4}}\mathbf{He}$. Which of the following statements is correct? (a) Both are fermions. (b) Both are bosons. (c)${}^{\mathbf{4}}\mathbf{He}$is a fermion, and ${}^{\mathbf{3}}\mathbf{He}$is a boson. (d)${}^{\mathbf{3}}\mathbf{He}$is a fermion, and ${}^{\mathbf{4}}\mathbf{He}$is a boson. (The two helium electrons form a closed shell and play no role in thisdetermination.)

Short answer

(a) Not correct

(b) Not correct

(c) Not correct

(d) Correct

Step-by-step solution

Identifying the data given in the question

${}^3\text{He}$and ${}^4\text{He}$are two isotopes of helium.

Concept used to solve the question

Fermion

Any particle with an odd half-integer spin, such as 1/2 , 3/2 and so on, is referred to as a fermion.

Bosons

Bosons are particles with integer spins 0,1,2.

Step 3(a) : Checking statement (a)

Fermions have a half-integer spin and bosons have an integer spin.

${}^4\text{He}$has two protons and two neutrons.

In localid="1663128163011" ${}^4\text{He}$ total spin is zero since its neutron and proton spins are positioned in opposition to one another Therefore it is a boson.

${}^3\text{He}$has two protons and one neutron The proton spins in ${}^3\text{He}$are oppositely aligned so they have total spin 0, whereas the last neutron has spin 1/2. ${}^3\text{He}$has a 1/2 spin as a result it is a fermion.

Since both are not fermions therefore this statement is not true.

Step 4(b) : Checking statement (b) 

Fermions have a half-integer spin, whereas bosons have an integer spin.

${}^4\text{He}$has two protons and two neutrons.

${}^4\text{He}$has total spin is zero since its neutron and proton spins are positioned in opposition to one another. Therefore it is a boson.

${}^3\text{He}$has two protons and one neutron The proton spins in ${}^3\text{He}$are oppositely aligned (total spin 0 ), whereas the last neutron has spin 1/2 . ${}^3\text{He}$has a spin 1/2 as a result it is a fermion.

Since both are not boson therefore this statement is not true.

Step 5(c) : Checking statement (c) 

Fermions have a half-integer spin and bosons have an integer spin.

${}^4\text{He}$has two protons and two neutrons.

${}^4\text{He}$has total spin zero since its neutron and proton spins are positioned in opposition to one another Therefore it is a boson.

${}^3\text{He}$has two protons and one neutron The proton spins in ${}^3\text{He}$are oppositely aligned (total spin 0), whereas the last neutron has spin 1/2 . ${}^3\text{He}$has a 1/2 spin as a result it is a fermion.

Since ${}^3\text{He}$not is a boson and ${}^4\text{He}$is not a fermion therefore this statement is not true.

Step 6(d) : Checking statement (d) 

Fermions have a half-integer spin and bosons have an integer spin.

${}^4\text{He}$has two protons and two neutrons.

${}^4\text{He}$has total spin zero since its neutron and proton spins are positioned in opposition to one another Therefore it is a boson.

${}^3\text{He}$has two protons and one neutron The proton spins in ${}^3\text{He}$are oppositely aligned (total spin 0 ), whereas the last neutron has spin 1/2 . ${}^3\text{He}$has a 1/2 spin as a result it is a fermion.

Since is a fermion and ${}^4\text{He}$is a boson therefore this statement is true