1. Home
  2. University physics volume 3
  3. Unit 2. modern physics
  4. Atomic structure
  5. Electron spin

8.3 Electron spin

<< Chapter < Page
  University physics volume 3     Page 1 / 9
Chapter >> Page >
By the end of this section, you will be able to:
  • Express the state of an electron in a hydrogen atom in terms of five quantum numbers
  • Use quantum numbers to calculate the magnitude and direction of the spin and magnetic moment of an electron
  • Explain the fine and hyperfine structure of the hydrogen spectrum in terms of magnetic interactions inside the hydrogen atom

In this section, we consider the effects of electron spin. Spin introduces two additional quantum numbers to our model of the hydrogen atom. Both were discovered by looking at the fine structure of atomic spectra. Spin is a fundamental characteristic of all particles, not just electrons, and is analogous to the intrinsic spin of extended bodies about their own axes, such as the daily rotation of Earth.

Spin is quantized in the same manner as orbital angular momentum. It has been found that the magnitude of the intrinsic spin angular momentum S of an electron is given by

S = s ( s + 1 ) ℏ ,

where s is defined to be the spin quantum number . This is similar to the quantization of L given in [link] , except that the only value allowed for s for an electron is s = 1 / 2 . The electron is said to be a “spin-half particle.” The spin projection quantum number m s is associated with the z -components of spin, expressed by

S z = m s ℏ .

In general, the allowed quantum numbers are

m s = − s , − s + 1 , … , 0 , … , + s − 1 , s .

For the special case of an electron ( s = 1 / 2 ),

m s = − 1 2 , 1 2 .

Directions of intrinsic spin are quantized, just as they were for orbital angular momentum. The m s = −1 / 2 state is called the “spin-down” state and has a z -component of spin, s z = −1 / 2 ; the m s = + 1 / 2 state is called the “spin-up” state and has a z -component of spin, s z = + 1 / 2 . These states are shown in [link] .

The two possible spin states of the electron are illustrated as vectors of equal length, one pointing up and right, representing vector S spin up, and the other pointing down and right, representing spin down. The two vectors are at the same angle to the horizontal. Spin up has a z component of plus h bar over two, and spin down has a z component of minus h bar over 2.
The two possible states of electron spin.

The intrinsic magnetic dipole moment of an electron μ e can also be expressed in terms of the spin quantum number. In analogy to the orbital angular momentum, the magnitude of the electron magnetic moment is

μ s = ( e 2 m e ) S .

According to the special theory of relativity, this value is low by a factor of 2. Thus, in vector form, the spin magnetic moment is

μ → = ( e m e ) S → .

The z -component of the magnetic moment is

μ z = − ( e m e ) S z = − ( e m e ) m s ℏ .

The spin projection quantum number has just two values ( m s = ± 1 / 2 ) , so the z- component of the magnetic moment also has just two values:

μ z = ± ( e 2 m e ) = ± μ B ℏ ,

where μ B is one Bohr magneton. An electron is magnetic, so we expect the electron to interact with other magnetic fields. We consider two special cases: the interaction of a free electron with an external (nonuniform) magnetic field, and an electron in a hydrogen atom with a magnetic field produced by the orbital angular momentum of the electron.

Electron spin and radiation

A hydrogen atom in the ground state is placed in an external uniform magnetic field ( B = 1.5 T ). Determine the frequency of radiation produced in a transition between the spin-up and spin-down states of the electron.

Strategy

The spin projection quantum number is m s = ± 1 / 2 , so the z- component of the magnetic moment is

μ z = ± ( e 2 m e ) = ± μ B ℏ .

The potential energy associated with the interaction between the electron magnetic moment and the external magnetic field is

U = − μ z B = ∓ μ B B .

The frequency of light emitted is proportional to the energy ( Δ E ) difference between these two states.

Solution

The energy difference between these states is Δ E = 2 μ B B , so the frequency of radiation produced is

f = Δ E h = 2 μ B B h = 2 ( 5.79 × 10 −5 eV T ) ( 1.5 T ) 4.136 × 10 −15 eV · s = 4.2 × 10 10 cycles s .

Significance

The electron magnetic moment couples with the external magnetic field. The energy of this system is different whether the electron is aligned or not with the proton. The frequency of radiation produced by a transition between these states is proportional to the energy difference. If we double the strength of the magnetic field, holding all other things constant, the frequency of the radiation doubles and its wavelength is cut in half.

Got questions? Get instant answers now!

Questions & Answers

what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
What are the treatment for autism?
Magret Reply
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective
Jharna
results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.
Jharna
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 6

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 3' conversation and receive update notifications?

Ask