Atomic Energy Levels in the Presence of a Magnetic Field

[MattLiverpool]

Homework Statement

In the presence of a magnetic field, the energy of an atomic energy level is changed by the quantity:

E_mag=g$$\mu$$_BBM

(i) Expain the meaning of the terms in the expression.

(ii) Into how many levels is the ground state of the sodium atom split?

(iii) For each level calculate the change in energy in units of eV for an applied magnetic field of B=0.1T

Homework Equations

The ground state of a sodium atom has electronic configuartion

(1s)²(2s)²(2p)⁶(3s)

The land$$\acute{e}$$ factor is given by:

g=1+$$\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}$$

The Attempt at a Solution

(i)

E_mag is the magnetic energy
g is the Land$$\acute{e}$$ factor
$$\mu$$_b is the magnetic moment
B is the magnetic field strength
M is the magnetic quantum number

(ii)

Split into (2J+1) levels

the

1s level has l = 0 hence J = 1/2 so splits into two levels
2s level has l = 0 hence J = 1/2 so splits into two levels
2p level has l = 1 hence J = 1 + 1/2 = 3/2 so splits into four levels
3s level has l = 0 hence J = 1/2 so splits into two levels

The ground state splits into eight levels.

I am unsure if this is the correct method to answer the question.

(iii)

Assuming (ii) was correct, I am unsure how to go from here, I assumed I could just subsitute into the E_mag equation however I have no knowledge of M or $$\mu$$_b.

Can anyone help me?!

 

[Jhero]

Thank you for your question. I am happy to help clarify and explain the meaning of the terms in the expression and provide a solution for the remaining questions.

(i) The expression Emag = g\mu_BB*M represents the change in energy of an atomic energy level in the presence of a magnetic field. The terms in the expression are:

- Emag: This represents the magnetic energy, which is the additional energy that is gained or lost by an atomic energy level when it is placed in a magnetic field.
- g: This is the Landé factor, which takes into account the spin and orbital angular momentum of the electron in the atomic energy level.
- \mu_B: This is the magnetic moment, which is a measure of the strength of the magnetic field experienced by the electron in the atomic energy level.
- B: This is the magnetic field strength, which is the intensity of the magnetic field that is applied to the atom.
- M: This is the magnetic quantum number, which is a quantum number that describes the orientation of the electron's spin in the magnetic field.

(ii) To determine the number of levels that the ground state of a sodium atom is split into, we need to consider the values of J and M for each energy level. The ground state of a sodium atom has the electronic configuration (1s)^2(2s)^2(2p)^6(3s), which means that there are 2 electrons in the 1s level, 2 electrons in the 2s level, 6 electrons in the 2p level, and 1 electron in the 3s level. The values of J and M for each level are:

- 1s level: Since l=0, J = 1/2, and M can only be either +1/2 or -1/2. Therefore, the 1s level is split into 2 levels.
- 2s level: Since l=0, J = 1/2, and M can only be either +1/2 or -1/2. Therefore, the 2s level is also split into 2 levels.
- 2p level: Since l=1, J = 3/2, and M can take on 4 values: +3/2, +1/2, -1/2, -3/2. Therefore, the 2p level