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The number of nuclear spins aligned in the lower energy state is slightly greater than those in the higher energy state. In the presence of an external magnetic field, as the spins precess at the Larmor frequency, the excess population results in a net magnetization oriented along the z axis. When a pulse or a short burst of radio waves at the Larmor frequency is applied along the x axis, the coupling of frequencies causes resonance and flips the nuclear spins of the excess population from the...
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In the absence of an external magnetic field, nuclear spin states are degenerate and randomly oriented. When a magnetic field is applied, the spins begin to precess and orient themselves along (lower energy) or against (higher energy) the direction of the field. At equilibrium, a slight excess population of spins exists in the lower energy state. Because the direction of the magnetic field is fixed as the z-axis,  the precessing magnetic moments are randomly oriented around the z-axis.
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Magnetic Fields

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A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on any other moving charge or current that is present in the field. Like an electric field, the magnetic field is also a vector field. At any position, the direction of the magnetic field is defined as the direction in which the north pole of a compass needle points.
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All atomic nuclei are positively charged. When they have a nonzero spin, they behave like rotating charges. As a consequence of their charge and spin, these nuclei generate a magnetic field (B). This, in turn, gives rise to a magnetic moment (μ), which is randomly oriented in the absence of an external magnetic field. When an external magnetic field (B0) is applied, the magnetic moment vectors can align with the field or against it in 2 + 1 orientations. A hydrogen nucleus, which is just a...
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NMR-active nuclei have energy levels called 'spin states' that are associated with the orientations of their nuclear magnetic moments. In the absence of a magnetic field, the nuclear magnetic moments are randomly oriented, and the spin states are degenerate. When an external magnetic field is applied, the spin states have only 2 + 1 orientations available to them. A proton with = ½ has two available orientations. Similarly, for a quadrupolar nucleus with a nuclear spin value of one, the...
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The earth's gravitational field produces a 'twisting force' perpendicular to the angular momentum of a spinning mass (such as a spinning top) that causes the mass to 'wobble' around the gravitational field axis in a phenomenon called precession. Similarly, the magnetic moment (μ) of a spinning nucleus precesses due to an external magnetic field directed along the z-axis. The precession of the magnetic moment vector about the magnetic field is called Larmor precession,...
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Advanced Experimental Methods for Low-temperature Magnetotransport Measurement of Novel Materials
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Zero-field magnetic response functions in Landau levels.

Yang Gao1, Qian Niu2,3

  • 1Department of Physics, The University of Texas at Austin, Austin, TX 78712; ygaophysics@gmail.com.

Proceedings of the National Academy of Sciences of the United States of America
|June 29, 2017
PubMed
Summary

We introduce a new method to refine the Landau level quantization rule by incorporating zero-field magnetic properties. This approach corrects Onsager

Keywords:
Berry phaseHofstadter butterflyLandau levelmagnetic susceptibilitytopological insulator

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Solid-State Physics

Background:

  • The Landau level quantization rule is fundamental in understanding electron behavior in magnetic fields.
  • Onsager's rule provides a foundational understanding of Landau quantization but requires refinement for higher-order effects.
  • Experimental observations show deviations from the linear relationship predicted by basic theories.

Purpose of the Study:

  • To develop a novel theoretical framework for correcting and refining the Landau level quantization rule.
  • To provide a universal explanation for higher-order corrections beyond the standard Onsager's rule.
  • To offer a method for extracting fundamental physical parameters from experimental data.

Main Methods:

  • Successively including zero-field magnetic response functions (magnetization, susceptibility) at zero temperature.
  • Reinterpreting the corrected rule as a quantization of semiclassical electron density in solids.
  • Developing a theoretical approach to calculate Landau levels with enhanced accuracy.

Main Results:

  • Reproduces Onsager's rule at zeroth order and includes Berry phase and magnetic moment corrections at first order.
  • Universally explains the nature of higher-order corrections to Landau quantization.
  • Predicts the curvature in the Landau level index versus inverse magnetic field relationship observed experimentally.

Conclusions:

  • The proposed theory offers a fresh perspective on Landau level quantization, unifying lower and higher-order corrections.
  • Provides a practical method to extract zero-temperature Berry phase and magnetic susceptibility from experimental fan diagrams.
  • Enables accurate theoretical calculations of Landau levels for realistic solid-state models.