This presentation is about magnetic traps for neutral particles, first achieved for cold neutrons and nowadays widely used in cold-atom physics. The level is that of undergraduate electrodynamics and tensor calculus courses. Tackling this topic within a computer algebra worksheet as shown below illustrates well the kind of advanced computations that can be done today with the Physics package. A new feature minimizetensorcomponents and related functionality is used along the presentation, that requires the updated Physics library distributed at the Maplesoft R&D Physics webpage.
 

 

Magnetic traps in cold-atom physics

 

Pascal Szriftgiser1 and Edgardo S. Cheb-Terrab2 

(1) Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France

(2) Maplesoft

 

We consider a device constructed with a set of electrical wires fed with constant electrical currents. Those wires can have an arbitrary complex shape. The device is operated in a regime such that, in some region of interest, the moving particles experience a magnetic field that varies slowly compared to the Larmor spin precession frequency. In this region, the effective potential is proportional to the modulus of the field: LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z)), this potential has a minimum and, close to this minimum, the device behaves as a magnetic trap.

 

 

 

Figure 1: Schematic representation of a Ioffe-Pritchard magnetic trap. It is made of four infinite rods and two coils.

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Following [1], we show that:

 

  

a) For a time-independent magnetic field  `#mover(mi("B"),mo("→"))`(x, y, z) in vacuum, up to order two in the relative coordinates X__i = [x, y, z] around some point of interest, the coefficients of orders 1 and 2 in this expansion, `v__i,j` and `c__i,j,k` , respectively the gradient and curvature, contain only 5 and 7 independent components.

  

b) All stationary points of LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z))^2 (nonzero minima and saddle points) are confined to a curved surface defined by det(`∂`[j](B[i])) = 0.

  

c) The effective potential, proportional to LinearAlgebra[Norm](`#mover(mi("B"),mo("→"))`(x, y, z)), has no maximum, only a minimum.

 

Finally, we draw the stationary condition surface for the case of the widely used Ioffe-Pritchard magnetic trap.

  

 

  

Reference

  

[1] R. Gerritsma and R. J. C. Spreeuw, Topological constraints on magnetostatic traps,  Phys. Rev. A 74, 043405 (2006)

  

 

The independent components of `v__i,j` and `c__i,j,k` entering B[i] = u[i]+v[i, j]*X[j]+(1/2)*c[i, j, k]*X[j]*X[k]

   

The stationary points are within the surface det(`∂`[j](B[i])) = 0

   

U = LinearAlgebra[Norm](`#mover(mi("B",fontweight = "bold"),mo("→",fontweight = "bold"))`)^2 has only minima, no maxima

   

Drawing the Ioffe-Pritchard Magnetic Trap

   


 

Download MagneticTraps.mw  or in pdf format with the sections open: MagneticTraps.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft


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