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Chapter: 4 Atomic and nuclear physics
    Section: 4.5 Absorption of particles and dosimetry
        SubSection: 4.5.2 Attenuation length of electrons in solids
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4.5.2 Attenuation length of electrons in solids   

The electron attenuation length (AL), λ, is defined as the thickness of material through which electrons may pass with a probability e−1 that they survive without inelastic scattering. Conventionally the inelastic scattering is considered to be significant only for energy losses 1 eV, phonon excitations being ignored. (The AL should not be confused with the total electron range which may be 10 to 100 times greater.) The value of λ depends on both the material and electron energy. For electrons emitted at an angle θ to the normal from a solid surface, the unscattered intensity from a source at a depth z below the surface follows the approximate relation

 

    I = I0 exp(−z/λ cos θ)

  

Thus, in Auger and X-ray photoelectron spectra from solid surfaces, the average information depth is characterized by λ cos θ.


      For elements, an empirical and approximate description of λ in nm is given by Seah and Dench (1979)

 

λ = 538aE−2 + 0.41a3/2E1/2

where E eV is the electron energy and a3 nm3 is the volume of one atom of the element in the solid state. This description fits the experimental data on average to a standard deviation of 30%.
     A selection of recent measurements of λ is given in the table, either as the values at certain energies E eV or, when sufficient data exists in the energy range 200–2000 eV, as the value at 1 keV and the power of the energy dependence where n is defined in the equation (Wagner et al., 1980)

 

   λ = λ(1 keV)(E/1000)n

More recent studies of the theoretical parameter, the inelastic mean free path (IMFP), show an energy dependence closer to that of Wagner et al. (1980). The IMFP, λi, is the average total distance that the electrons traverse between inelastic scattering whereas the AL reflects the average net distance travelled. The AL is some 20% less than the IMFP due to the elastic scatterings which deflect the electron trajectories. The general form of the IMFP relation (Tanuma et al., 1993) is

 
λi =  E
Ep2[β ln γE − C/E + D/E2]

where the various constants may be calculated from tables of values of basic materials constants.

References

M. P. Seah and W. A. Dench (1979) Surface and Interface Analysis, 1, 2–11.
S. Tanuma, C. J. Powell and D. R. Penn (1993) Surface and Interface Analysis, 20, 77–89.
C. D. Wagner, L. E. Davis and W. M. Riggs (1980) Surface and Interface Analysis, 2, 53–55.



Attenuation length of electrons
 

Element
 

λ/(nm)
at 1 keV

n
 

λ/(nm),
E/(eV)

Inorganic compound
 

λ/(nm)
at 1 keV

n

λ/(nm),
E/(eV)

 

 

 

 

 

 

 

 

  Ag   .   .   .   .   .   .

1.5

0.50

 

  Al2O3   .   .   .   .   .   .   .   .

1.4

0.54

 

  Al    .   .   .   .   .   .

1.9

0.74

 

  Cr2O3   .   .   .   .   .   .   .   .

1.9

0.52

 

  Au   .   .   .   .   .   .

1.7

0.50

 

  Fe3O4   .   .   .   .   .   .   .   .

 

 

2.3,   519

  Be   .   .   .   .   .   .

1.6

0.50

 

  GeO2    .   .   .   .   .   .   .   .

 

 

0.6,   250

  C (amorphous)

 

 

1.4, 1000

  KI .   .   .   .   .   .   .   .   .   .

6.3

0.59

 

  C (diamond)   .   .

 

 

2.0, 1000

  NaCl .   .   .   .   .   .   .   .   .

3.8

0.56

 

  C (graphite)    .   .

 

 

3.7, 1000

  NaF  .   .   .   .   .   .   .   .   .

4.2

0.56

 

  Co  .   .   .   .   .   .

 

 

1.2, 1194

  NiO  .   .   .   .   .   .   .   .   .

 

 

1.0,   511

  Cr  .   .   .   .   .   .

 

 

1.1, 1211

  SiO2     .   .   .   .   .   .   .   .

2.2

0.70

 

  Cs  .   .   .   .   .   .

 

 

2.6, 1260

  WO3    .   .   .   .   .   .   .   .

 

 

2.6, 1450

  Cu  .   .   .   .   .   .

1.4

0.75

 

 

 

 

 

  Fe  .   .   .   .   .   .

 

 

1.3, 1200

Organic compound

 

 

 

  Ge  .   .   .   .   .   .

2.3

0.60

 

  alpha-iodostearic acid   .   .

 

 

14.0,  867

  In   .   .   .   .   .   .

 

 

1.0,   408

  barium stearate     .   .   .   .

5.0

0.62

 

  K   .   .   .   .   .   .

 

 

2.1,   173

  butylamine    .   .   .   .   .   .

4.2

0.48

 

  Mo .   .   .   .   .   .

1.6

0.71

 

  cadmium arachidate  .   .   .

3.4

0.50

 

  Na  .   .   .   .   .   .

 

 

1.3,   173

  graphite fluoride    .   .   .   .

 

 

4.4,   969

  Ni   .   .   .   .   .   .

 

 

1.3, 1186

  polybromoparaxylylene .   .

 

 

1.9, 1065

  Pt   .   .   .   .   .   .

 

 

0.9, 1168

  polychloroparaxylylene  .   .

 

 

1.9, 1065

  Rb   .   .   .   .   .   .

 

 

2.0,   173

  polyethylene   .   .   .   .   .  .

 

 

8.4,  969

  Si    .   .   .   .   .   .

2.9

0.62

 

  polyethylene terephthalate

 

 

 6.8,  969

  Sn   .   .   .   .   .   .

 

 

1.8,   520

  polyparaxylene   .   .   .   .   .

1.5

2.0

 

  Ti    .   .   .   .   .   .

 

 

1.0,   422

  polystyrene    .   .   .   .   .   .

 

 

 7.4, 969

  V    .   .   .   .   .   .

 

 

1.1, 1216

  polytetrafluoroethylene   .   .

 

 

 7.2, 969

  W   .   .   .   .   .   .

1.3

0.36

 

  polyvinylchloride    .   .   .   .

 

 

 8.4, 969

  Zn   .   .   .   .   .   .

 

 

1.8, 1167

  polyvinylidene fluoride    .   .

1.4

1.4

 

 

 

 

 

 

 

 

 

M.P. Seah

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