In
physics , Bargmann–Michel–Telegdi (BMT) equation describes the
spin
precession of an
electron in an external
electromagnetic field . It is named after
Valentine Bargmann ,
Louis Michel , and
Valentine Telegdi .
Background
A particle with
spin angular momentum s has a corresponding
magnetic moment μ , which will interact with an external
magnetic field B . The classical equation of motion for the spin of a particle subject follows from "torque equals rate of change of angular momentum" (rotation analogue of
Newton's second law )
d
s
d
t
=
μ
×
B
{\displaystyle {\frac {d\mathbf {s} }{dt}}={\boldsymbol {\mu }}\times \mathbf {B} }
and the spin and magnetic moment are proportional to each other
s
∝
μ
{\displaystyle \mathbf {s} \propto {\boldsymbol {\mu }}}
This applies to the rest frame of the particle. A covariant generalization to any frame is given by the BMT equation.
History
Equations
Tensor agebra
For an electron of
electric charge e ,
mass m ,
magnetic moment μ , the equation is
[1]
d
a
τ
d
s
=
e
m
u
τ
u
σ
F
σ
λ
a
λ
+
2
μ
(
F
τ
λ
−
u
τ
u
σ
F
σ
λ
)
a
λ
,
{\displaystyle {\frac {da^{\tau }}{ds}}={\frac {e}{m}}u^{\tau }u_{\sigma }F^{\sigma \lambda }a_{\lambda }+2\mu (F^{\tau \lambda }-u^{\tau }u_{\sigma }F^{\sigma \lambda })a_{\lambda },}
where aτ is the
polarization four vector , and uτ is
four velocity of electron. The
electromagnetic field tensor Fτσ is externally applied. We have the relations
a
τ
a
τ
=
−
u
τ
u
τ
=
−
1
{\displaystyle a^{\tau }a_{\tau }=-u^{\tau }u_{\tau }=-1}
u
τ
a
τ
=
0
{\displaystyle u^{\tau }a_{\tau }=0}
Using the
Lorentz force , the general equations of motion for any charged particle
m
d
u
τ
d
s
=
e
F
τ
σ
u
σ
,
{\displaystyle m{\frac {du^{\tau }}{ds}}=eF^{\tau \sigma }u_{\sigma },}
one can rewrite the first term on the right side of the BMT equation as
(
−
u
τ
w
λ
+
u
λ
w
τ
)
a
λ
{\displaystyle (-u^{\tau }w^{\lambda }+u^{\lambda }w^{\tau })a_{\lambda }}
, where
w
τ
=
d
u
τ
/
d
s
{\displaystyle w^{\tau }=du^{\tau }/ds}
is the
four acceleration of the electron. This term describes
Fermi–Walker transport and leads to
Thomas precession . The second term is associated with
Larmor precession .
When electromagnetic fields are uniform in space or when gradient forces like
∇
(
μ
⋅
B
)
{\displaystyle \nabla ({\boldsymbol {\mu }}\cdot {\boldsymbol {B}})}
can be neglected, the particle's translational motion is described by
d
u
α
d
τ
=
e
m
F
α
β
u
β
.
{\displaystyle {du^{\alpha } \over d\tau }={e \over m}F^{\alpha \beta }u_{\beta }\;.}
The BMT equation is then written as
[2]
d
S
α
d
τ
=
e
m
g
2
F
α
β
S
β
+
(
g
2
−
1
)
u
α
(
S
λ
F
λ
μ
U
μ
)
,
{\displaystyle {\;\,dS^{\alpha } \over d\tau }={e \over m}{\bigg [}{g \over 2}F^{\alpha \beta }S_{\beta }+\left({g \over 2}-1\right)u^{\alpha }\left(S_{\lambda }F^{\lambda \mu }U_{\mu }\right){\bigg ]}\;,}
The Beam-Optical version of the Thomas-BMT, from the Quantum Theory of Charged-Particle Beam Optics , applicable in accelerator optics
[3]
[4]
Geometric algebra
The BMT equation in geometric algebra is
[5]
d
s
d
t
=
e
m
F
⋅
s
+
(
g
−
2
)
e
2
m
(
F
⋅
s
)
∧
v
v
{\displaystyle {\frac {ds}{dt}}={\frac {e}{m}}F\cdot s+(g-2){\frac {e}{2m}}(F\cdot s)\wedge vv}
See also
References
^ V. Bargmann,
L. Michel , and V. L. Telegdi, Precession of the Polarization of Particles Moving in a Homogeneous Electromagnetic Field , Phys. Rev. Lett. 2, 435 (1959).
^ Jackson, J. D., Classical Electrodynamics , 3rd edition, Wiley, 1999, p. 563.
^ M. Conte,
R. Jagannathan ,
S. A. Khan and M. Pusterla, Beam optics of the Dirac particle with anomalous magnetic moment, Particle Accelerators, 56, 99-126 (1996); (Preprint: IMSc/96/03/07, INFN/AE-96/08).
^
Khan, S. A. (1997).
Quantum Theory of Charged-Particle Beam Optics , Ph.D Thesis ,
University of Madras ,
Chennai ,
India . (complete thesis available from
Dspace of IMSc Library ,
The Institute of Mathematical Sciences , where the doctoral research was done).
^ Doran, Lasenby (29 May 2003).
Geometric algebra for physicists . Cambridge University Press. p. 163–164.
ISBN
978-0-521-71595-9 .
External links