1

dE/dx

Let’s next turn our attention to how charged particles lose energy in matter

To start with we’ll consider only heavy charged particles like muons, pions, protons, alphas, heavy ions, … Effectively all charged particles except

electronsThe mean energy loss of a charged

particle through matter is described by the Bethe-Bloch equation

2

dE/dx

You’ll see

ρdxx

g

MeVcm

dx

dE

g

MeVcm

ρdx

dE

cm

MeV

dx

dE

thicknessmass theis where

of units with

of units with

of units with

2

2

3

dE/dx

In particle physics, we call dE/dx the energy loss

In radiation and other branches of physics, dE/dx is called the stopping power or linear energy transfer (LET) and dE/dx is called the mass stopping power

4

dE/dx

Assume Electrons are free and initially at rest p is small so the trajectory of the heavy

particle is unaffected Recoiling electron does not move appreciably

We’ll calculate the impulse (change in momentum) of the electron and use this to give the energy lost by the heavy charged particle

5

dE/dx

22

422

2

2

2

2then

2

42 says law sGauss'

direction e transversin the is forcenet hesymmetry tby

bvm

ez

m

IbE

bv

zeI

b

zedxE

zebdxE

v

dxEedtEeFdtI

ee

T

T

TT

6

dE/dx

min

max2

24

2

42

ln4

4-

2-

number atomic theis and atoms/V # theis

electrons 2 are there

and between sa thicknes inNow

b

b

vm

ZNze

dx

dE-

b

db

vm

NZez

dx

dE

bdbbENZdx

dE

ZN

πbdbdxNZ

dbbbdx

e

e

7

dE/dxNote

We only consider collisions with atomic electrons and can neglect collisions with atomic nuclei because

Except for ions on high Z targets at low energy

nucleielectrons

pe

dx

dE

dx

dE

Am

Z

m

Z

8

dE/dx

bmin (short distance collisions) In an elastic collision between a heavy

particle and an electron

2

2

min

2min

2

4222

min

22max

2max

max

22

2

2

2

vm

zeb

bvm

ezvmbE

vmT

vmT

vmp

e

ee

e

e

e

9

Classical dE/dx

bmax (long distance collisions)

We can invoke the adiabatic principle of QM

There will be no change if the interaction time is longer than the orbital period

states allover averagedfrequency mean a is

)(frequency

1

(velocity)

max

vv

vb

v

b

10

dE/dx

Substituting we have

This is very close to Bohr’s 1915 result Actually Bohr calculated the energy transfer

to a harmonically bound electron and found

vze

vmNZ

vm

ez

dx

dE

b

bNZ

vm

ez

dx

dE

e

e

e

2

32

2

42

min

max2

42

ln4

ln4

2

2

2

32

2

42

2

123.1ln

4

c

v

vze

vmNZ

vm

ez

dx

dE e

e

11

dE/dx

Our approximation is not too bad

12

dE/dx

Notes on the essential ingredients

Energy loss depends only on the velocity of the particle, not its mass At low velocity, dE/dx decreases as 1/2

Reaches a minimum at =0.96 or =3 At high velocities, dE/dx increases as ln2

Called the relativistic rise

Energy loss depends on the square of the charge of the incident particle

Energy loss depends on Z of the material

222

21 ln

cZzc

dx

dE

13

Bethe-Bloch dE/dx

=p/E=E/m

14

Quantum Effects

Real energy transfers are discrete QM energy > classical energy but the

transfer occurs in a few collisions Bethe calculated probabilities that the energy

transferred would cause excitation or ionization

bmin must be consistent with the uncertainty principle One needs to use the larger of

Bethe also included spin effects

vmvm

Ze

ee

and 2

2

15

Density EffectSo far we calculated the energy loss to

one electron of one atom and then performed an incoherent sum

For large , bmax > atomic dimensions The atoms in between will be affected by

the fields and these atoms themselves can produce perturbing fields at bmax

Atoms along the field will become polarized thus shielding electrons at bmax from the full electric field of the incident particle

Density effect = induced polarization will be greater in denser mediums

16

Density Effect

Calculation Fermi (1940) was first Sternheimer Phys Rev 88 (1952) 851 gives

additional gory details Jackson contains a calculation as well

The net effect is to reduce the logarithm by a factor of

Instead of a relativistic rise we observe a less rapid rise called the Fermi plateau And the remaining slow rise is due to large

energy transfers to a few electrons

17

Density Effect

18

Density Effect

The density effect is usually estimated using Sternheimer’s parameterization

See tables on next slide

19

20

Bethe-Bloch Equation

K=0.307075 MeVcm2/g I = mean excitation energy Tmax is the maximum kinetic energy that can be

imparted to a free electron

Accurate to about 1% for pion momenta between 40 MeV/c and 6 GeV/c

At lower energies additional corrections such as the shell correction must be made

21

Bethe-Bloch dE/dx

=p/E=E/m

22

Tmax

Tmax is the maximum energy that can be imparted to electrons Note it is in the logarithm and is also

responsible for part of the dE/dx increase as the energy increases

Tmax is given by

Sometimes a low energy approximation is used

2

222

max//21

2

MmMm

cmT

ee

e

222max 2 cmT e

23

Tmax

Alpha particles from 252Cf fission

24

Mean Excitation Potential I

Approximately I/Z = 12 eV for Z < 13 I/Z = 10 eV for Z > 13

Constants exist for most elements and should be used if more accuracy is needed

25

Other EffectsShell effect

At low energies (when v~orbital velocity of bound electrons), the atomic binding energy must be accounted for

At velocities comparable to shell velocities, the dE/dx loss is reduced

Shell corrections go as -C/Z where C=f() Relatively small effect (1%) at =0.3 but it

can be as large as 10% in the range 1-100 MeV for protons

Bremsstrahlung For heavy charged particles, this is

important only at high energies (several hundred GeV muons in iron)

26

Low Energy

One large effect at low energies is that the incident particle will capture an electron for some of the time thus neutralizing itself Thus the ionization losses will

decrease Energy losses from elastic scattering

with nuclei also become important (and may dominate for heavy ions)

27

Low Energy

dE/dx~(z/)2Z

28

dE/dx ValuesFor low energies (< 1000 MeV) tables of

stopping power are available from NIST http://physics.nist.gov/PhysRefData

For high energies, one can use dE/dxmin as a good estimate http://pdg.lbl.gov

29

dE/dx Values

How much energy does a cosmic ray muon (E>1 GeV) deposit in a plastic scintillator 1 cm thick?

MeVxdx

dE2103.196.1