An unknown magnetic field produces a set of lines for a transition from a state with l = 3 to one with l = 2, between which there is a maximum energy difference of 6.1x10-4 eV. How many lines are there? What is the magnitude of the field?
Consider a H-atom in a magnetic field. Sketch the possible transitions between the n = 3, l = 2 and n = 2, l = 1 levels. Which lines will not appear if we require that ml = 1,0 and -1 only? How many different frequencies will be seen in the spectrum? What will be the frequencies of the lines be if the magnetic field has a magnitude B = 4T ?
senýn gýbý malýn amkyum
The transitions Δml = -3 , 3 , -2 ,2 are not allowed
The allowed trasitions Δml = -1 ,0 ,1 only
number of transitions 9
Energy of the trasitions ΔE = Δml μB B
Here B magetic field , μB = bohr magnton = 9.264*10^-24 J/ T
frequencies f = hc / ΔE
Here h = 6.63*10^-34 J.s
c = 3*10^8 m/s
plug all values for corresponding Δml
To determine the number of lines and the magnitude of the magnetic field, we need to use the formula for the energy difference between different angular momentum states in a magnetic field.
The energy difference (ΔE) between two states with angular momentum quantum numbers l₁ and l₂ can be given by:
ΔE = gμB * B * Δm
where g is the Landé g-factor, μB is the Bohr magneton, B is the magnetic field strength, and Δm is the difference in the magnetic quantum number between the two states.
In this case, we have a transition from l = 3 to l = 2 with a maximum energy difference of 6.1x10^(-4) eV. Let's assume the maximum value of Δm is Δm_max.
To find the number of lines, we need to determine the possible values of Δm. The number of lines will be equal to the number of different possible Δm values.
The possible values for Δm are given by:
Δm = m₂ - m₁
where m₁ and m₂ are the magnetic quantum numbers for the initial and final states, respectively. In this case, since we are going from l = 3 to l = 2, the possible Δm values will be -3, -2, -1, 0, 1, 2, or 3.
Each Δm value corresponds to a specific line. So, the number of lines will be equal to the number of different Δm values, which is 7.
Now, let's calculate the magnitude of the magnetic field using the given maximum energy difference:
ΔE = 6.1x10^(-4) eV
To convert this to joules, we use the conversion factor:
1 eV = 1.602x10^(-19) J
ΔE = 6.1x10^(-4) * 1.602x10^(-19) J
Now, rearranging the equation for energy difference in terms of the magnetic field (B):
B = ΔE / (gμB * Δm)
Substituting the known values:
B = (6.1x10^(-4) * 1.602x10^(-19)) / (g * μB * Δm_max)
The Landé g-factor and the Bohr magneton are constants.
The Landé g-factor for electrons in an atom is approximately 2.
The Bohr magneton (μB) is given by:
μB = eħ / (2me)
where e is the elementary charge, ħ is the reduced Planck constant, and me is the electron mass. The values of these constants are:
e = 1.602x10^(-19) C
ħ = 1.055x10^(-34) J·s
me = 9.10938356x10^(-31) kg
Substituting these values and the given Δm_max value into the equation, we can calculate the magnitude of the magnetic field (B).