Multipole magnet

Multipole magnets are magnets built from multiple individual magnets, typically used to control beams of charged particles. Each type of magnet serves a particular purpose.

• Dipole magnets are used to bend the trajectory of particles
• Quadrupole magnets are used to focus particle beams
• Sextupole magnets are used to correct for chromaticity introduced by quadrupole magnets

The magnetic field of an ideal multipole magnet in an accelerator is typically modeled as having no (or a constant) component parallel to the nominal beam direction (

    z
  

{\displaystyle z}

direction) and the transverse components can be written as complex numbers:

B

        y
      
    
    +
    i
    
      B
      
        x
      
    
    =
    
      C
      
        n
      
    
    ⋅
    (
    x
    −
    i
    y
    
      )
      
        n
        −
        1
      
    
  

{\displaystyle B_{y}+iB_{x}=C_{n}\cdot (x-iy)^{n-1}}

where

    x
  

{\displaystyle x}

and

    y
  

{\displaystyle y}

are the coordinates in the plane transverse to the nominal beam direction.

      C
      
        n
      
    
  

{\displaystyle C_{n}}

is a complex number specifying the orientation and strength of the magnetic field.

      B
      
        x
      
    
  

{\displaystyle B_{x}}

and

      B
      
        y
      
    
  

{\displaystyle B_{y}}

are the components of the magnetic field in the corresponding directions. Fields with a real

      C
      
        n
      
    
  

{\displaystyle C_{n}}

are called 'normal' while fields with

      C
      
        n
      
    
  

{\displaystyle C_{n}}

purely imaginary are called 'skewed'.

For an electromagnet with a cylindrical bore, producing a pure multipole field of order

    n
  

{\displaystyle n}

, the stored magnetic energy is:

U

        n
      
    
    =
    
      
        
          n
          
            !
            
              2
            
          
        
        
          2
          n
        
      
    
    π
    
      μ
      
        0
      
    
    ℓ
    
      N
      
        2
      
    
    
      I
      
        2
      
    
    .
  

{\displaystyle U_{n}={\frac {n!^{2}}{2n}}\pi \mu _{0}\ell N^{2}I^{2}.}

Here,

      μ
      
        0
      
    
  

{\displaystyle \mu _{0}}

is the permeability of free space,

    ℓ
  

{\displaystyle \ell }

is the effective length of the magnet (the length of the magnet, including the fringing fields),

    N
  

{\displaystyle N}

is the number of turns in one of the coils (such that the entire device has

    2
    n
    N
  

{\displaystyle 2nN}

turns), and

    I
  

{\displaystyle I}

is the current flowing in the coils. Formulating the energy in terms of

    N
    I
  

{\displaystyle NI}

can be useful, since the magnitude of the field and the bore radius do not need to be measured.

Note that for a non-electromagnet, this equation still holds if the magnetic excitation can be expressed in Amperes.

The equation for stored energy in an arbitrary magnetic field is:

U =

        1
        2
      
    
    ∫
    
      (
      
        
          
            B
            
              2
            
          
          
            μ
            
              0
            
          
        
      
      )
    
    
    d
    τ
    .
  

{\displaystyle U={\frac {1}{2}}\int \left({\frac {B^{2}}{\mu _{0}}}\right)\,d\tau .}

Here,

      μ
      
        0
      
    
  

{\displaystyle \mu _{0}}

is the permeability of free space,

    B
  

{\displaystyle B}

is the magnitude of the field, and

    d
    τ
  

{\displaystyle d\tau }

is an infinitesimal element of volume. Now for an electromagnet with a cylindrical bore of radius

    R
  

{\displaystyle R}

, producing a pure multipole field of order

    n
  

{\displaystyle n}

, this integral becomes:

U

        n
      
    
    =
    
      
        1
        
          2
          
            μ
            
              0
            
          
        
      
    
    
      ∫
      
        ℓ
      
    
    
      ∫
      
        0
      
      
        R
      
    
    
      ∫
      
        0
      
      
        2
        π
      
    
    
      B
      
        2
      
    
    
    d
    τ
    .
  

{\displaystyle U_{n}={\frac {1}{2\mu _{0}}}\int ^{\ell }\int _{0}^{R}\int _{0}^{2\pi }B^{2}\,d\tau .}

Ampere's Law for multipole electromagnets gives the field within the bore as:

B ( r ) =

          n
          !
          
            μ
            
              0
            
          
          N
          I
        
        
          R
          
            n
          
        
      
    
    
      r
      
        n
        −
        1
      
    
    .
  

{\displaystyle B(r)={\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}.}

Here,

    r
  

{\displaystyle r}

is the radial coordinate. It can be seen that along

    r
  

{\displaystyle r}

the field of a dipole is constant, the field of a quadrupole magnet is linearly increasing (i.e. has a constant gradient), and the field of a sextupole magnet is parabolically increasing (i.e. has a constant second derivative). Substituting this equation into the previous equation for

      U
      
        n
      
    
  

{\displaystyle U_{n}}

gives:

U

        n
      
    
    =
    
      
        1
        
          2
          
            μ
            
              0
            
          
        
      
    
    
      ∫
      
        ℓ
      
    
    
      ∫
      
        0
      
      
        R
      
    
    
      ∫
      
        0
      
      
        2
        π
      
    
    
      
        (
        
          
            
              
                n
                !
                
                  μ
                  
                    0
                  
                
                N
                I
              
              
                R
                
                  n
                
              
            
          
          
            r
            
              n
              −
              1
            
          
        
        )
      
      
        2
      
    
    
    d
    τ
    ,
  

{\displaystyle U_{n}={\frac {1}{2\mu _{0}}}\int ^{\ell }\int _{0}^{R}\int _{0}^{2\pi }\left({\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}\right)^{2}\,d\tau ,}

U

        n
      
    
    =
    
      
        1
        
          2
          
            μ
            
              0
            
          
        
      
    
    
      ∫
      
        0
      
      
        R
      
    
    
      
        (
        
          
            
              
                n
                !
                
                  μ
                  
                    0
                  
                
                N
                I
              
              
                R
                
                  n
                
              
            
          
          
            r
            
              n
              −
              1
            
          
        
        )
      
      
        2
      
    
    (
    2
    π
    ℓ
    r
    
    d
    r
    )
    ,
  

{\displaystyle U_{n}={\frac {1}{2\mu _{0}}}\int _{0}^{R}\left({\frac {n!\mu _{0}NI}{R^{n}}}r^{n-1}\right)^{2}(2\pi \ell r\,dr),}

U

        n
      
    
    =
    
      
        
          π
          
            μ
            
              0
            
          
          ℓ
          n
          
            !
            
              2
            
          
          
            N
            
              2
            
          
          
            I
            
              2
            
          
        
        
          R
          
            2
            n
          
        
      
    
    
      ∫
      
        0
      
      
        R
      
    
    
      r
      
        2
        n
        −
        1
      
    
    
    d
    r
    ,
  

{\displaystyle U_{n}={\frac {\pi \mu _{0}\ell n!^{2}N^{2}I^{2}}{R^{2n}}}\int _{0}^{R}r^{2n-1}\,dr,}

U

        n
      
    
    =
    
      
        
          π
          
            μ
            
              0
            
          
          ℓ
          n
          
            !
            
              2
            
          
          
            N
            
              2
            
          
          
            I
            
              2
            
          
        
        
          R
          
            2
            n
          
        
      
    
    
      (
      
        
          
            R
            
              2
              n
            
          
          
            2
            n
          
        
      
      )
    
    ,
  

{\displaystyle U_{n}={\frac {\pi \mu _{0}\ell n!^{2}N^{2}I^{2}}{R^{2n}}}\left({\frac {R^{2n}}{2n}}\right),}

U

        n
      
    
    =
    
      
        
          n
          
            !
            
              2
            
          
        
        
          2
          n
        
      
    
    π
    
      μ
      
        0
      
    
    ℓ
    
      N
      
        2
      
    
    
      I
      
        2
      
    
    .
  

{\displaystyle U_{n}={\frac {n!^{2}}{2n}}\pi \mu _{0}\ell N^{2}I^{2}.}