Bloch equation

Bloch equation

http://scai.fraunhofer.de/HuPSON#SCAIVPH_00000129

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Term information

PMID

20832224

definition

In physics and chemistry, specifically in NMR (nuclear magnetic resonance) or MRI (magnetic resonance imaging), or ESR (electron spin resonance) the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M = (Mx, My, Mz) as a function of time when relaxation times T1 and T2 are present. These are phenomenological equations that were introduced by Felix Bloch in 1946. [1] Sometimes they are called the equations of motion of nuclear magnetization. ... Let M(t) = (Mx(t), My(t), Mz(t)) be the nuclear magnetization. Then the Bloch equations read: \frac {d M_x(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _x - \frac {M_x(t)} {T_2} \frac {d M_y(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _y - \frac {M_y(t)} {T_2} \frac {d M_z(t)} {d t} = \gamma ( \bold {M} (t) \times \bold {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1} where is the gyromagnetic ratio and B(t) = (Bx(t), By(t), B0 + Bz(t)) is the magnetic field experienced by the nuclei. The z component of the magnetic field B is sometimes composed of two terms: * one, B0, is constant in time, * the other one, Bz(t), may be time dependent. It is present in magnetic resonance imaging and helps with the spatial decoding of the NMR signal. M(t) B(t) is the cross product of these two vectors. M0 is the steady state nuclear magnetization (that is, for example, when t ; it is in the z direction. source: http://en.wikipedia.org/wiki/Bloch_equations

exact synonym

equation of motion of nuclear magnetization

has enhanced presentationMathML

has latex math

\frac {d M_x(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _x - \frac {M_x(t)} {T_2}

\frac {d M_z(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1}

\frac {d M_y(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _y - \frac {M_y(t)} {T_2}

has raw presentationMathML

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <msub> <mi>M</mi> <mi>y</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>γ</mi> <msub> <mfenced close=")" open="("> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mo>×</mo> <mi>B</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </mfenced> <mi>y</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>M</mi> <mi>y</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <msub> <mi>T</mi> <mn>2</mn> </msub> </mfrac> </mrow> <annotation encoding="SnuggleTeX">\[ \frac {d M_y(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _y - \frac {M_y(t)} {T_2} \]</annotation> </semantics> </math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <msub> <mi>M</mi> <mi>z</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>γ</mi> <msub> <mfenced close=")" open="("> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mo>×</mo> <mi>B</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </mfenced> <mi>z</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>M</mi> <mi>z</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mo>-</mo> <msub> <mi>M</mi> <mn>0</mn> </msub> </mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> </mfrac> </mrow> <annotation encoding="SnuggleTeX">\[ \frac {d M_z(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _z - \frac {M_z(t) - M_0} {T_1} \]</annotation> </semantics> </math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mi>d</mi> <msub> <mi>M</mi> <mi>x</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>γ</mi> <msub> <mfenced close=")" open="("> <mrow> <mi>M</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> <mo>×</mo> <mi>B</mi> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> </mfenced> <mi>x</mi> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mfenced close=")" open="("> <mi>t</mi> </mfenced> </mrow> <msub> <mi>T</mi> <mn>2</mn> </msub> </mfrac> </mrow> <annotation encoding="SnuggleTeX">\[ \frac {d M_x(t)} {d t} = \gamma ( {M} (t) \times {B} (t) ) _x - \frac {M_x(t)} {T_2} \]</annotation> </semantics> </math>

Term relations

Subclass of:

kinetics equation
has_variable some (
variable and
is_about some proportionality to)
has_variable some (
variable and
is_about some magnetism)