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 collision on RHIC accelerator at Brookhaven

Introduction of cross section. Formerly we obtained relation between Rutherford scattering angle and impact parameter (  particles are scattered): ………………… (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  b 2 ) is scattered to a angle larger than value b given by relation (1) for appropriate value of b. Then applies:  (  b ) =  b 2 ……………….……....………. (2) (then dimension of  is m 2, barn = m 2 ) We assume thin foil (cross sections of neighboring nuclei are not overlapping and multiple scattering does not take place) with thickness L with n j atoms in volume unit. Beam with number N S of  particles are going to the area S S. (Number of beam particles per time and area units – luminosity – is for present accelerators up to m -2 s -1 ).

Fraction f(  b ) of incident  particles scattered to angle larger then b is: We substitute of b from equation (1): Sketch of the Rutherford experiment Angular distribution of scattered particles The number of target nuclei on which  particles are impinging is: N j = n j LS S. Sum of cross sections  for scattering to angle b and more is:  (  b ) = n j LS S . Reminder of equation (1) Reminder of equation (2)  (  b ) =  b 2 ………………… (3)

We can write for detector area in distance r from the target: Number N( ) of  particles going to the detector per area unit is: Such relation is known as Rutherford equation for scattering. 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 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 N S = 1 induces on one target nucleus n j L = 1 scattering to the angle to unit solid angle: Because we obtain Rutherford equation for scattering in the form: Let us define total cross section: 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: because we can write: Reminder of equation (4)

Different types of differential cross sections: angular spectralspectral angular 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 m 2  . In the centre-of-mass frame, obtained results are same also without this condition. Energy E KIN will be kinetic energy of particle relative motion E KIN = (1/2)  v 1 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: We obtain for elastic scattering (already derived relation is used: where  = m 1 /m 2 ) We make derivation with respect to and we obtain: Then we obtain for transformation of differential cross sections: Because:

Geometrical interpretation of cross section: Let us obtain differential cross section for elastic scattering on stiff sphere with radius R. We obtain: because In our case relation between angles are: 2  + =    =  /2 - /2  sin  = cos ( /2) Impact parameter: b=R  sin  = R  cos( /2)  (db/d ) = (R/2)sin( /2) Then we can write ( sin = 2sin( /2)cos( /2) ): 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  ~ m 2 = 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 (N 0 – number of incident particles): ln N – ln N0 = – n j σx Number of touched particles N decrease exponential with thickness x: Number of interacting particles: For x→0 : N 0 – N  N 0 – N 0 (1-n j  x)  N 0 n j  x and then: Absorption coefficient  = n j  Mean free path l = is mean distance which particle travels in a matter before interaction. 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:  m 2 ÷  m 2 →  barn ÷  10 4 barn Strong interaction (interaction of nucleons and other hadrons):  m 2 ÷  m 2 →  0.01 barn ÷  10 4 barn Electromagnetic interaction (reaction of charged leptons or photons):  m 2 ÷  m 2 →  0.1 μbarn ÷  10 mbarn Weak interaction (neutrino reactions):  m 2 = barn Cross section of different neutron reactions with gold nucleus