Polder tensor

The Polder tensor is a <a href="/facts/Tensor/pNjFo5tV">tensor</a> introduced by <a href="/facts/Dirk_Polder/AcLFvrS5">Dirk Polder</a> for the description of magnetic <a href="/facts/Permeability_(electromagnetism)/GA7CcRLU">permeability</a> of <a href="/facts/Ferrite_(magnet)/kBTM7Dqk">ferrites</a>. The tensor notation needs to be used because ferrimagnetic material becomes <a href="/facts/Anisotropy/YMqTQWVB">anisotropic</a> in the presence of a magnetizing field.
The tensor is described mathematically as:

B
        =
        
          
            [
            
              
                
                  μ
                
                
                  j
                  κ
                
                
                  0
                
              
              
                
                  −
                  j
                  κ
                
                
                  μ
                
                
                  0
                
              
              
                
                  0
                
                
                  0
                
                
                  
                    μ
                    
                      0
                    
                  
                
              
            
            ]
          
        
        H
      
    
    {\displaystyle B={\begin{bmatrix}\mu &j\kappa &0\\-j\kappa &\mu &0\\0&0&\mu _{0}\end{bmatrix}}H}

Neglecting the effects of damping, the components of the tensor are given by

μ
        =
        
          μ
          
            0
          
        
        
          (
          
            1
            +
            
              
                
                  
                    ω
                    
                      0
                    
                  
                  
                    ω
                    
                      m
                    
                  
                
                
                  
                    ω
                    
                      0
                    
                    
                      2
                    
                  
                  −
                  
                    ω
                    
                      2
                    
                  
                
              
            
          
          )
        
      
    
    {\displaystyle \mu =\mu _{0}\left(1+{\frac {\omega _{0}\omega _{m}}{\omega _{0}^{2}-\omega ^{2}}}\right)}

κ
        =
        
          μ
          
            0
          
        
        
          
            
              ω
              
                ω
                
                  m
                
              
            
            
              
                
                  
                    ω
                    
                      0
                    
                  
                
                
                  2
                
              
              −
              
                ω
                
                  2
                
              
            
          
        
      
    
    {\displaystyle \kappa =\mu _{0}{\frac {\omega \omega _{m}}{{\omega _{0}}^{2}-\omega ^{2}}}}

where

ω
          
            0
          
        
        =
        γ
        
          μ
          
            0
          
        
        
          H
          
            0
          
        
         
      
    
    {\displaystyle \omega _{0}=\gamma \mu _{0}H_{0}\ }

ω
          
            m
          
        
        =
        γ
        
          μ
          
            0
          
        
        M
         
      
    
    {\displaystyle \omega _{m}=\gamma \mu _{0}M\ }

ω
        =
        2
        π
        f
      
    
    {\displaystyle \omega =2\pi f}

γ
 =
 1.11
 ×
 
 10
 
 5
 
 
 ⋅
 g
 
 
 
 
 {\displaystyle \gamma =1.11\times 10^{5}\cdot g\,\,}
 
 (rad / s) / (A / m) is the effective <a href="/facts/Gyromagnetic_ratio/Afg6mY6M">gyromagnetic ratio</a> and 
 
 
 
 g
 
 
 {\displaystyle g}
 
, the so-called effective <a href="/facts/G-factor_(physics)/kmDsyF5g">g-factor (physics)</a>, is a ferrite material constant typically in the range of 1.5 - 2.6, depending on the particular ferrite material. 
 
 
 
 f
 
 
 {\displaystyle f}
 
 is the frequency of the RF/microwave signal propagating through the ferrite, 
 
 
 
 
 H
 
 0
 
 
 
 
 {\displaystyle H_{0}}
 
 is the internal magnetic bias field, 
 
 
 
 M
 
 
 {\displaystyle M}
 
 is the <a href="/facts/Magnetization/zxrtXzTN">magnetization</a> of the ferrite material and 
 
 
 
 
 μ
 
 0
 
 
 
 
 {\displaystyle \mu _{0}}
 
 is the magnetic <a href="/facts/Permeability_of_free_space/nKxRk6J1">permeability of free space</a>.
To simplify computations, the radian frequencies of 
 
 
 
 
 ω
 
 0
 
 
 ,
 
 
 ω
 
 m
 
 
 ,
 
 
 
 {\displaystyle \omega _{0},\,\omega _{m},\,}
 
 and 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 can be replaced with frequencies (Hz) in the equations for 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
 and 
 
 
 
 κ
 
 
 {\displaystyle \kappa }
 
 because the 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
 factor cancels. In this case, 
 
 
 
 γ
 =
 1.76
 ×
 
 10
 
 4
 
 
 ⋅
 g
 
 
 
 
 {\displaystyle \gamma =1.76\times 10^{4}\cdot g\,\,}
 
 Hz / (A / m) 
 
 
 
 =
 1.40
 ⋅
 g
 
 
 
 
 {\displaystyle =1.40\cdot g\,\,}
 
 MHz / Oe. If <a href="/facts/CGS_units/qbHxv78q">CGS units</a> are used, computations can be further simplified because the 
 
 
 
 
 μ
 
 0
 
 
 
 
 {\displaystyle \mu _{0}}
 
 factor can be dropped.

Polder tensor open-in-new

Polder tensor