Bloch equations

This article explains the behavior of nuclear and electron magnetization; For the interactions of a two-level system with an oscillating field ( optical Bloch equations) see Maxwell-Bloch equations .

The Bloch equations (after Felix Bloch , who published them for the first time in 1946) are a system of equations of motion for two-level systems . They enable a physical interpretation of the paramagnetic resonance effect in nuclear magnetic resonance and in paramagnetic electron resonance .

formulation

The Bloch equations apply to liquids , but only to a limited extent for solids . They represent equations of motion for the entire nuclear and electron magnetization of the sample under the influence of external magnetic fields and are written in vector : ${\displaystyle {\vec {M}}}$ ${\displaystyle H}$

${\displaystyle {d{\vec {M}} \over dt}=\gamma {\vec {M}}\times {\vec {H}}_{a}-{\vec {e}}_{x}{M_{x} \over T_{2}}-{\vec {e}}_{y}{M_{y} \over T_{2}}-{\vec {e}}_{z}{M_{z}-M_{0} \over T_{1}}}$

Describe therein:

• ${\displaystyle \gamma {\vec {M}}\times {\vec {H}}_{a}}$ the movement of magnetization in the magnetic field
  • ${\displaystyle \gamma }$the gyromagnetic ratio of the atomic nuclei or the electrons
• the last three summands on the right-hand side represent paramagnetic relaxation , which tends towards an equilibrium value through the interaction of the particles with each other and with their molecular environment.
  • ${\displaystyle {\vec {e}}_{x}}$, and the unit vectors in -, - and -direction ${\displaystyle {\vec {e}}_{y}}$ ${\displaystyle {\vec {e}}_{z}}$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$
  • ${\displaystyle T_{2}}$the transverse relaxation time (spin-spin relaxation)
  • ${\displaystyle T_{1}}$the spin-lattice relaxation time
  • the external magnetic field consists of two parts:
    • a strong constant magnetic field in the direction ${\displaystyle z}$
    • a high-frequency magnetic field applied perpendicular to it in the direction. ${\displaystyle x}$

Application to non-spin 1/2 systems

It was later shown that these equations of motion, originally designed for spin 1/2 systems, also describe every other two-level system. For this purpose, parts of the general "pseudo-spin-1/2-system" are associated with spin components and the interaction with external fields is treated like magnetic interactions.

In semiclassical radiation theory, the spin components correspond to the ground or excited state of a two-level atom, and the axes of the Bloch sphere provide information about the quantum mechanical coherence ( -, -axis) or the population difference ( -axis) of the system. The equations adapted for this purpose are called optical Bloch equations . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

literature

• Horst Stöcker: Pocket book of physics. 4th edition, Verlag Harry Deutsch, Frankfurt am Main 2000, ISBN 3-8171-1628-4

Individual evidence

1. ^ F. Bloch: Nuclear Induction. In: Phys. Rev. 70, 1946, pp. 460-474. doi : 10.1103 / PhysRev.70.460