Landau Distribution and Straggling: Energy Loss in Silicon

The statistical nature of the ionising process during the passage of a fast charge particle through matter results in large fluctuations in the energy loss (Δ) in absorbers, whose thickness is thin compared to the particle range.

The number of electron-hole pairs (J) is related to Δ by the expression J=Δ/P, where P is a proportional factor, which for silicon equals 3.68 eV. Both Δ and J are stochastic quantities. The probability functions f(Δ) and φ(J) are usually called energy loss distribution or straggling functions.

They can be described schematically by the position of the maximum of distribution function (Δp) and the full width at half maximum (w), with Δp being located at lower value compared to the mean energy loss obtained from Bethe-Bloch ‹Δ› (Fig.1).

The energy-loss calculator computes the Bethe-Bloch mean loss and the electron-hole pair yield for your own particle and thickness.

The mean energy loss: Bethe-Bloch

The collisions are random, of course, but their number per macroscopic path length is generally large and this is the reason why average quantities are generally used. One of the most important quantities is the mean energy loss per unit length, often called stopping power. Many theories have been developed during the first half of the twentieth century in order to characterize this quantity. The correct quantum mechanical calculation was given by Hans Bethe (1930, with the relativistic form in 1932) and extended by Felix Bloch (1933), with later corrections by other authors. The formula reads:

with:

2π Na re2 mec2 = 0.1535 MeV cm2/g	β: v/c of incident particle
re: classical electron radius (2.817 × 10-13 cm)	ρ: density of absorbing material
me: electron mass	γ: 1/√(1 − β2)
Na: Avogadro’s number	δ: density correction
Z: atomic number of absorbing material	C: shell correction
A: atomic weight of absorbing material	I: mean excitation potential
z: charge of incident particle	Wmax: maximum energy transferable in a single collision

W_max can be calculated using the equation:

where: s=m_e/M and η=βγ.

The mean excitation energy I depends on the orbital frequencies and binding energies of the electrons in the absorber, and there is no precise formula to calculate it. Values of I for many materials have been deduced from measurements and are tabulated by the ICRU. The last two terms in the parentheses of the formula are the density and the shell corrections and they were added to the original Bethe-Bloch formulation to improve its agreement with experimental results in particular energy ranges. The density correction takes into account the effect of the electric field produced by incoming particles and is more evident at high velocity. Instead, the shell correction is noticeable when the velocity of incident particle is comparable to the orbital velocity of the bound electrons of the target material.

At this low energy some other complicated effects come into play and the Bethe-Bloch formula breaks down. When the velocity is comparable with the speed of orbital electrons of the target material the energy loss reaches a maximum that depends slightly on the sign of the charge (Barkas effect), and at lower energy it drops sharply. At higher energy (that means higher velocity) dE/dx is dominated by the 1/β² factor and decreases until βγ ≈ 3 to 4 (β ≈ 0.96), where a minimum is reached. A particle with this energy is called a minimum ionizing particle (MIP).

As the energy increases further, the losses rise only slowly due to the density effect (Fermi plateau) until the radiative components, such as the Cherenkov radiation and Bremsstrahlung, start to be relevant. The Cherenkov radiation arises when a charged particle in a medium moves faster than the speed of light in that same medium (βc>c/n, with n: index of refraction): in such case an electromagnetic shock wave is created, just as an aircraft that moves faster than sound.

Especially for light particles, such as electrons or positrons at very high energy, the Bremsstrahlung emission represents the main energy loss mechanism. The deflection and the deceleration of the particle due to the interaction with the nuclei of the target cause the emission of photons; this effect is much greater the lighter the particle (in fact the emission probability by Bremsstrahlung varies as the inverse square of the particle mass) and higher is the atomic number of target material. While ionization loss rates rise logarithmically with energy, Bremsstrahlung losses rise linearly and dominate at high energy (for electrons, above a few tens of MeV in most materials). Fig. 1 below shows the mean energy loss (the stopping power) for positive muons in copper, over nine orders of magnitude in muon momentum, from 0.1 MeV/c to 100 TeV/c.

Correction to Bethe-Bloch for electrons and positrons

Electrons and positrons need particular consideration. First, their small mass implies the possibility of a large deflection due to a single collision too; moreover the collisions are between identical particles, so that the calculation must take into account their indistinguishability. As result the maximum transferable energy in a single collision becomes:

with T_e: kinetic energy of the incident particle, and the Bethe-Bloch formula can be rearranged as:

with:

where suffix “+” means positrons and “–” means electrons.

From the mean to the distribution: Landau and Vavilov

Landau and Vavilov performed theoretical calculations on this distribution. Each of these solutions, however, has a different region of applicability and the distinguishing parameter in all these theories is the ratio k = ‹Δ› / E_max where E_max is the maximum transferable energy in a single collision.

Landau solved this problem for all situations where k ≤ 0.01, deriving the expected energy loss distribution by solving an integral transport equation:

Here f(x,Δ) represents the distribution probability that the incident particle will lose an amount Δ of energy on traversing a layer of thickness x. W(E)dE denotes the probability per unit path length of a collision transferring energy E to an electron in the material.

The function W(E)dE is not generally known, but Landau was able to derive an approximate solution by using the free electron (Rutherford) cross section:

with

keV μm-1 in silicon.

In the previous equation, z is the charge of the incident particle, Z and A are the atomic number and weight of the material, and ρ is the density. The Landau distribution is therefore given by [1]

with φ(λ) a universal function of the variable λ only

and

where C_E is the Euler constant equal to 0.5772. The Landau distribution, fL(Δ), is asymmetric with a tail extending to Emax with a maximum at λ ≈ −0.223 and a full width at half maximum w = 4.018ξ. This tail is mainly due to fast-emitted δ-rays. The energy loss corresponding to the maximum of the function fL(Δ) is the most probable energy loss [2]

Δ_p = ξ [ ln(2m_ec²β²γ²/I) + ln(ξ/I) + j − β² − δ ], with j = 0.200 [2]

where I is the mean excitation potential and δ is the density correction.

Subsequently, Vavilov [3] derived an improved solution that takes into account the spin of the incident particle and introduces the physical limits coming from Emax. For the collision cross-section, Vavilov used the form

Vavilov also demonstrated that his solution tends to the Landau function for k ≤ 0.01, region where ‹Δ› is approximated to ξ. For 300 µm of silicon, k falls below 0.01 for protons above roughly 1 GeV/c. For k > 10, the number of collisions is very large and, by the Central Limit Theorem, the Vavilov function coincides with Gaussian distribution, where the standard deviation is

Binding corrections: why the measured width is broader

Landau-Vavilov solutions have been derived under the assumption that scatterings occur on quasi-free electrons. Therefore, they neglect the electron-binding energies. This assumption is a valid approximation only for close collisions.

Further corrections to the theory, taking into account the fact that the electrons in the material are not free, have been attempted by Blunck and Leisegang [5], Shulek [6] and Bichsel [7].

In the case of solid media, these calculations are aimed at the experimental observation that, while the most probable energy loss agrees rather well with the prediction of the theory, the width of the distribution is broader than expected and cannot be accounted for by electronic noise or imperfect resolution. The effect is particularly noticeable for very thin absorbers [8], of the order of a few hundred μm or less. The modified energy loss distribution can be improved by using a modified cross section to take into account the electron binding energy and the atomic shell structure. The modified energy straggling function can be expressed as:

In other words, the experimentally observed energy spectrum can be calculated by convoluting the Landau distribution with a normal distribution of variance δ₂. The result of the convolution is a broader distribution with a peak value that is usually increased by a small amount compared to the Landau theory. From the experimental observations, δ₂ has been estimated as:

δ₂ = (8/3)(ξ/x) Σ_i f_i I_i ln(2m_ec²β²/I_i)

where f_i is the fraction of electrons in shell i and I_i its binding energy. In silicon, this gives

δ₂ ≈ 1.1 × 10³ keV² cm^-1 for β = 1 [6]. Bichsel’s more detailed calculation [13] implies a larger binding correction, equivalent to about 1.8 × 10³ keV² cm^-1.

The resulting improved energy loss distribution has an overall value of w, which is roughly given by:

w ≈ √[(4.018 ξ)² + 5.56 δ₂]

As the material thickness decreases, δ₂ becomes more and more the dominant term, which determines the overall w of the straggling function. Conversely, it is not expected to provide an additional broadening of the distribution at large thicknesses.

As an example, in a 300 μm thick silicon detector, ξ ≈ 5.34 keV and √δ₂ ≈ 5.7 keV, giving an overall w of about 25.3 keV (about 18% larger than the Landau width) for a relativistic (β ≈ 1, z = 1) particle. With Bichsel’s larger correction, √δ₂ ≈ 7.3 keV and w is about 27.6 keV, roughly 28% larger.

Prefer a bare calculator, or want the electron-hole pair yield per micrometre for your own case? Open the energy-loss tool on its own page →

References

1. L. Landau, On the Energy Loss of Fast Particles by Ionization, J. Phys. USSR 8 (1944) 201.
2. Particle Data Group, Passage of particles through matter, in S. Navas et al., Review of Particle Physics, Phys. Rev. D 110, 030001 (2024). Edition originally cited: W.-M. Yao et al., J. Phys. G 33, 1 (2006).
3. P.V.Vavilov, Ionization losses of high energy heavy particles, Soviet Physics JETP, 5:749, 1957.
4. S. Meroli et al., Energy loss measurement for charged particles in very thin silicon layers , JINST,  6 P06013 doi: 10.1088/1748-0221/6/06/P06013
5. O. Blunck and S. Leisegang, Zum Energieverlust schneller Elektronen in dünnen Schichten, Z. Physik 128 (1950) 500.
6. P. Shulek et al., Fluctuations of Ionization Loss, Sov. J. Nucl. Phys 4 (1967) 400.
7. H. Bichsel, Straggling of Heavy Charged Particles: Comparison of Born Hydrogenic-Wave-Function Approximation with Free-Electron Approximation, Phys. Rev. B1 (1970) 2854
8. H. Esbensen et al., Random and channeled energy loss in thin germanium and silicon crystals for positive and negative 2-15-GeV/c pions, kaons, and protons, Phys. Rev. B 18 (1978) 1039
9. William R. Leo, Techniques for Nuclear and Particle Physics Experiments. Berlin and Heidelberg: Springer, 1987.
10. Claude Leroy, Pier Giorgio Rancoita, Principles of Radiation Interaction in Matter and Detection. Singapore: World Scientific Publishing, 2004.
11. International Commission on Radiation Units and Measurements. [Online]. https://www.icru.org/
12. S. M. Sze, Kwok Kwok Ng, Physics of Semiconductor Devices. John Wiley & Sons, 2007.
13. H. Bichsel, Straggling in thin silicon detectors, Rev. Mod. Phys. 60 (1988) 663.

@online{scienceshot-the-straggling-function-energy-2024,
  author       = {Piro, Viros},
  title        = {{Landau Distribution and Straggling: Energy Loss in Silicon}},
  organization = {ScienceShot},
  date         = {2024-01-09},
  year         = {2024},
  url          = {https://scienceshot.com/post/the-straggling-function-energy-loss-distribution-of-charged-particles-in-silicon-layers}
}