Conception of cross section

1) Base conceptions – differential, integral cross section, total cross section, geometric interpretation of cross section

2) Macroscopic cross section, mean free path.

3) Typical values of cross sections for different processes

Chamber for measurement of cross sections of different astrophysical reactions at NPI of ASCR

Ultrarelativistic heavy ion collisionon RHIC accelerator at Brookhaven

Introduction of cross section.

Formerly we obtained relation between Rutherford scattering angle and impact parameter ( particles are scattered):

bZe

E4

2cotg

2KIN0

………………… (1)

The smaller impact parameter b, the bigger scattering angle .

Impact parameter is not directly measurable and new directly measurable quantity must be define. We introduce scattering cross section for quantitative description of scattering processes:

Derivation of Rutherford relation for scattering:

Relation between impact parameter b and scattering angle particle with impact parameter smaller or the same as b (aiming to the area b2) is scattered to a angle larger than value b given by relation (1) for appropriate value of b. Then applies:

(b) = b2 ……………….……....………. (2)

(then dimension of is m2, barn = 10-28 m2)

We assume thin foil (cross sections of neighboring nuclei are not overlapping and multiple scattering does not take place) with thickness L with nj atoms in volume unit. Beam with number NS of particles are going to the area SS. (Number of beam particles per time and area units – luminosity – is for present accelerators up to 1038 m-2s-1).

2j

jbb bLn

S

LSn

SN

)(N)f(

S

S

SS

Fraction f(b) of incident particles scattered to angle larger then b is:

We substitute of b from equation (1):

2gcot

E4

ZeLn)(f 2

2

KIN0

2

jb

Sketch of the Rutherford experiment Angular distribution of scattered particles

The number of target nuclei on which particles are impinging is: Nj = njLSS. Sum of cross sections for scattering to angle b and more is:

(b) = njLSS.

Reminder of equation (1)

bZe

E4

2cotg

2KIN0

Reminder of equation (2)(b) = b2

………………… (3)

We can write for detector area in distance r from the target:

d2

cos2

sinr4dsinr2)rd)(sinr2(dS 22

d2

cos2

sinr4

d2

sin2

cotgE4

ZeLnN

dS

dfN

dS

dN

2

2

2

KIN0

2

jS

S

Number N() of particles going to the detector per area unit is:

2sinEr8

eLZnN

dS

dN

42KIN

220

42j

S

Such relation is known as Rutherford equation for scattering.

d2

sin2

cotgE4

ZeLndf 2

2

KIN0

2

j

During real experiment detector measures particles scattered to angles from up to +d. Fraction of incident particles scattered to such angular range is:

Reminder of pictures

2gcot

E4

ZeLn)(f 2

2

KIN0

2

jb

Reminder of equation (3)

… (4)

Diferencial and total cross section:

It is useful to know number of particles scattered to given angle independently on detector distance from the target. We determine number of particles going to the unit solid angle Ω instead unit area S. We define differential cross section, which gives probability, that one incident particle NS= 1

induces on one target nucleus njL = 1 scattering to the angle to unit solid angle:

dS

)(dN

LnN

1

d

d

j

Because we obtain Rutherford equation for scattering in the form: drdS 2

2

sin

1

E8

eZ

d

d

42KIN

20

42

Let us define total cross section:

d

d

dT

For axially symmetric cases particle will be scattered to given angle with the same probability for all azimuthal angles . Then we can take all particles scattered to the angle range to +d. Appropriate cross section is:

d

dsin2

d

d

because we can write:

dsin2d

dddsin

d

d

d

dd

d

d 2

0

2

0

d

2sinEr8

eLZnN

dS

dN

42KIN

220

42j

S

Reminder of equation (4)

Different types of differential cross sections:

angular ),(d

d

)(d

d spectral )E(

dE

d spectral angular ),,E(dEd

d

double or triple differential cross section

Integral cross sections: through energy, angle

Transformation of cross section from centre-of-mass frame to laboratory frame:

Rutherford equation for scattering we derived using assumption that target mass m2 . In the centre-of-mass frame, obtained results are same also without this condition. Energy EKIN will be kinetic energy of particle relative motion EKIN = (1/2)v1

2.

Obtained differential cross sections then must be transformed to laboratory frame:

We compare numbers of particles to corresponding elements of solid angle in the both coordinate frames:

~d~

d~

sin~d

~dddsin

d

d~d~

d

~dd

d

d~~

We obtain for elastic scattering (already derived relation is used:

where = m1/m2 )

212 )~

cos21(

~cos

cos

~We make derivation with respect to and we obtain:

2/322/322/12 ~cos21

~cos1~

sin~cos21

~cos

~sin2

2

1~

cos21

~sin

~d

dsin~

d

d

d

)d(cos~

d

)d(cos

Then we obtain for transformation of differential cross sections:

~

2/32

~d

~d~

cos1

~cos1

d

d

~dd Because:

Geometrical interpretation of cross section:

Let us obtain differential cross section for elastic scattering on stiff sphere with radius R. We obtain:

d

db

sin

b

d

d

d

dbb2

d

db)( 2

b

because

d

dsin2

d

d

In our case relation between angles are: 2 + = = /2 - /2 sin = cos (/2)

Impact parameter: b=Rsin = Rcos(/2) (db/d) = (R/2)sin(/2)

Then we can write ( sin = 2sin(/2)cos(/2) ):

4

R

2sin

2

R

2cos

2sin2

2cosR

d

d 2

22

T Rd4

R Total cross section is:

It conforms to visual idea, that total cross section is effective area (geometrical cross section) of sphere on which scattering proceeds.

Cross section – area affected by incident particles → probability of reaction increases with σ.

Value of total cross section for reactions with nuclei will be more or less equal to geometrical cross section of nucleus – that means ~ 10-28 m2 = 1 barn (assumption of closeness to geometrical cross section).

In the reality σ depends on interaction properties and beam energy → can be not equal to the geometrical cross section.

Macroscopic quantities:

Particle passage through matter: interacted particles disappear from beam (N0 – number of incident particles):

dxnN

dNj

x

0

j

N

N

dxnN

dN

0

ln N – ln N0 = – njσx xn0

jeNN

Number of touched particles N decrease exponential with thickness x:

Number of interacting particles: )e(1NNN xn00

j

For x→0 : xn1e jxn j

N0 – N N0 – N0(1-njx) N0njx

and then: xnN

NN

N

dNj

0

0

Absorption coefficient = nj

Mean free path l = is mean distance which particle travels in a matter before interaction.

jn1

l

Quantum physics all measured macroscopic quantities , l are mean values (l is statistical quantity also in classical physics).

Values of cross section:

Very strong dependence of cross sections on energy of beam particles and interaction character. Values are within very broad range: 10-47 m2 ÷ 10-24 m2 → 10-19 barn ÷ 104 barn

Strong interaction (interaction of nucleons and other hadrons): 10-30 m2 ÷ 10-24 m2 → 0.01 barn ÷ 104 barn

Electromagnetic interaction (reaction of charged leptons or photons): 10-35 m2 ÷ 10-30 m2 → 0.1 μbarn ÷ 10 mbarn

Weak interaction (neutrino reactions): 10-47 m2 = 10-19 barn

Cross section of different neutron reactions with gold nucleus