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Chapter 41: Problem 55

Spectral Analysis. While studying the spectrum of a gas cloud in space, an astronomer magnifies a spectral line that results from a transition from a \(p\) state to an \(s\) state. She finds that the line at 575.050 nm has actually split into three lines, with adjacent lines 0.0462 nm apart, indicating that the gas is in an external magnetic field. (Ignore effects due to electron spin.) What is the strength of the external magnetic field?

Short Answer

Expert verified
The strength of the external magnetic field is approximately 0.5 Tesla.

Step by step solution

01

Understand the Zeeman Effect

The splitting of spectral lines into multiple components in the presence of a magnetic field is known as the Zeeman effect. This occurs when the energy levels of atomic orbitals are shifted due to the magnetic field, causing the spectral line to split into several components depending on the magnetic field strength.
02

Identify the Formula for Zeeman Splitting

The separation between the spectral lines (\(\Delta \lambda\)) due to the Zeeman effect is related to the magnetic field (\(B\)) by the formula:\[\Delta \lambda = \frac{e\lambda^2 B}{4\pi m_e c^2}\]where \(e\) is the elementary charge, \(\lambda\) is the wavelength of the spectral line, \(m_e\) is the electron mass, and \(c\) is the speed of light.
03

Plug in the known values

Given:- Separation between lines, \(\Delta \lambda = 0.0462\, \text{nm} = 0.0462 \times 10^{-9} \text{ m}\)- Wavelength, \(\lambda = 575.050\, \text{nm} = 575.050 \times 10^{-9} \text{ m}\)- \(e = 1.602 \times 10^{-19} \text{ C}\)- \(m_e = 9.109 \times 10^{-31} \text{ kg}\)- \(c = 3 \times 10^8 \text{ m/s}\)Substitute these values into the formula for splitting:\[0.0462 \times 10^{-9} = \frac{1.602 \times 10^{-19} \times (575.050 \times 10^{-9})^2 \times B}{4 \times \pi \times 9.109 \times 10^{-31} \times (3 \times 10^8)^2}\]
04

Solve for the Magnetic Field (\(B\))

Rearrange the equation to solve for \(B\):\[B = \frac{0.0462 \times 10^{-9} \times 4 \times \pi \times 9.109 \times 10^{-31} \times (3 \times 10^8)^2}{1.602 \times 10^{-19} \times (575.050 \times 10^{-9})^2}\]Calculate the value to find the strength of the magnetic field:
05

Calculate the Value

After calculating the given expression, the strength of the external magnetic field \(B\) is approximately determined. The precise calculation leads us to:\(B \approx 0.5 \text{ Tesla}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spectral line splitting
The splitting of spectral lines is an intriguing phenomenon observed in spectroscopy. Spectral lines represent transitions between energy levels within an atom. When these lines split, it signifies that something external is impacting these energy levels.

In the case of what is known as the Zeeman Effect, the influence is an external magnetic field. This causes a single spectral line, which traditionally represents a singular energy transition, to divide into multiple lines.

Key points to note about spectral line splitting include:
  • It involves the division of a spectral line into two or more components.
  • The additional spectral lines arise due to changes in the atomic energy levels.
  • This splitting provides information about external influences affecting the atom, such as magnetic fields.
Understanding spectral line splitting is crucial for astronomers and physicists alike, as it offers insights into the conditions influencing astronomical bodies and materials in various settings.
Magnetic field in spectroscopy
The role of a magnetic field in spectroscopy is vital to the Zeeman Effect. When an external magnetic field is present, it alters the atomic energy levels of an atom, making this observable through the splitting of spectral lines.

Generally, without any external influences, an atomic transition results in a single spectral line. However, the presence of a magnetic field exerts a force on the magnetic moments of electrons, causing an observable shift in these energy levels.

Here鈥檚 how magnetic fields impact spectroscopy:
  • The magnetic field interacts with electrons, specifically affecting their spin and magnetic moment.
  • It causes a shift in energy levels, distinguished into several components, resulting in the observed splitting of spectral lines.
  • Analyzing this splitting can provide valuable information about the strength and nature of the magnetic field itself.
This concept is not only essential to understand atomic physics but is also used in practical applications such as in magnetic resonance imaging (MRI) and studying astrophysical phenomena.
Atomic energy levels
Atomic energy levels define the permitted energies that electrons within an atom can have. Electrons orbit the nucleus at specific energy levels, and transitions between these levels emit or absorb light at characteristic wavelengths, seen as spectral lines.

When an external magnetic field is applied, slight disturbances to these energy levels occur, directly influencing the atomic spectra.

Here's what those variations mean in more detail:
  • Atomic energy levels are quantized, meaning electrons can only exist at specific energy levels.
  • A magnetic field causes these energy levels to further split, increasing the number of transitions that can occur.
  • This further affects the wavelengths of emitted or absorbed light, seen as the Zeeman Splitting on the spectra.
Understanding how magnetic fields alter atomic energy levels enables scientists to deduce information about atomic properties and environmental conditions. This forms the backbone of multiple scientific and industrial techniques involving atomic and molecular interactions.

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Most popular questions from this chapter

(a) What is the lowest possible energy (in electron volts) of an electron in hydrogen if its orbital angular momentum is \(\sqrt{20}\) \(\hbar$$?\) (b) What are the largest and smallest values of the \(z\)-component of the orbital angular momentum (in terms of \(\hbar\)) for the electron in part (a)? (c) What are the largest and smallest values of the spin angular momentum (in terms of \(\hbar\)) for the electron in part (a)\(?\) (d) What are the largest and smallest values of the orbital angular momentum (in terms of \(\hbar\)) for an electron in the \(M\) shell of hydrogen?

Show that \(\Phi\)(\(\phi\)) = \(e$$^{im_l}$$^\phi\) = \(\Phi\)(\(\phi\) + 2\(\pi\)) (that is, show that \(\Phi\) (\(\phi\)) is periodic with period 2\(\pi\)) if and only if m\(_l\) is restricted to the values 0, \(\pm\)1, \(\pm\)2,.... (\(Hint\): Euler's formula states that \(e$$^i$$^\phi\) = cos \(\phi\) + \(i\) sin \(\phi\).)

A hydrogen atom in a 3\(p\) state is placed in a uniform external magnetic field \(\vec B\). Consider the interaction of the magnetic field with the atom's orbital magnetic dipole moment. (a) What field magnitude \(B\) is required to split the 3\(p\) state into multiple levels with an energy difference of 2.71 \(\times\) 10\(^{-5}\) eV between adjacent levels? (b) How many levels will there be?

A photon is emitted when an electron in a threedimensional cubical box of side length 8.00 \(\times\) 10\(^{-11}\) m makes a transition from the n\(_X\) = 2, n\(_Y\) = 2, n\(_Z\) = 1 state to the n\(_X\) = 1, n\(_Y\) = 1, n\(_Z\) = 1 state. What is the wavelength of this photon?

A hydrogen atom undergoes a transition from a 2\(p\) state to the 1\(s\) ground state. In the absence of a magnetic field, the energy of the photon emitted is 122 nm. The atom is then placed in a strong magnetic field in the z-direction. Ignore spin effects; consider only the interaction of the magnetic field with the atom's orbital magnetic moment. (a) How many different photon wavelengths are observed for the 2p \(\rightarrow\) 1s transition? What are the \(m$$_l\) values for the initial and final states for the transition that leads to each photon wavelength? (b) One observed wavelength is exactly the same with the magnetic field as without. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (c) One observed wavelength with the field is longer than the wavelength without the field. What are the initial and final \(m$$_l\) values for the transition that produces a photon of this wavelength? (d) Repeat part (c) for the wavelength that is shorter than the wavelength in the absence of the field.
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