1. Oct 2006, Lectures 6&7 Nuclear Physics Lectures, Dr. Armin Reichold 1 Lectures 6 & 7 Cross Sections

2. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 2 6.0 Overview 6.1 Definition of Cross Section Why concept is important Experimental Definition Reaction Rates 6.2 Breit-Wigner Line Shape or resonance 6.3 How to calculate  Fermi’s Golden Rule Breit Wigner Crossection 6.4 QM calculation of Rutherford Scattering

3. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 3 6.1 Definition of Crossection What do we want to describe? The collisions of quantum mechanical objects (nuclei)  There is no event by event certainty about the outcome We want to know how “likely” a certain scattering event is We need a statistical property of the collision that tells us about any distributions of variables that may be present in the final states, i.e. scattering angles, momentum transfers, etc. How do we want to describe this? We use the concept of an average, effective area associated with the collision (total crossection) averages are taken over all possible collisions We ask how this area changes when we vary a) the properties of the initial state (i.e. centre of mass energy) b) the properties of the final state (i.e. scattering angle)  differential cross section

4. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 4 #of a trying per unit time interaction probability 6.1 Definition of Cross Section Consider reactions between two particles: a+b  x x can be any final state (also just a+b again) Beam of particles, type: a thickness dx flux: N a N a (x=0) particles of type a per unit time hit a target made from particles of type b Target has n b particles of type b per unit volume Number of b particles / unit area = n b dx We DEFINE: Probability that an a interacts with any b when traversing a target of thickness dx : P(a,b) dx =  n b dx We call  the total cross section of this reaction target made of b x How many reactions dN a would we get per dx and per dt ? dN a =-N a n b dx  (integrate)   N a (x)=N a (0) exp(-x/ ) ; =1/(n b  )

5. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 5 6.1 Cross sections and Reaction Rates n a beam particles/unit volume at speed v Areal flux density (particles per unit area and unit time): F= n a v Reaction rate per target b atom: R=F  For a thin target of thickness x<< Total rate per area in the thin target R tot =n b F  x The above is the total cross section. We can also define a differential cross sections, as a function of final state energy, transverse momentum, angle etc. but for a given set of final state particles where  c is the angle that particle c makes with the direction of particle a

6. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 6 6.2 Breit-Wigner Line Shape We want to use non relativistic QM to compute the distribution of energy in the decay of “resonance” (a nearly bound state) We assume that we know the Hamiltonian H that perturbs our resonance and lets it decay We assume we are dealing with a spin less resonance Book for the algebra of this section: Cottingham & Greenwood, Appendix D

7. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 7 6.2 Breit-Wigner Line Shape i) Start with non. rel., time dependent Schrödinger equation: Where  n is any complete ortho-normal set of stationary wave functions that describes the unperturbed system Insert this Ansatz into the TDSE multiply above by  m * and integrate over all space using orthonormality of the set: H mm =E m  cancel the m th term of sum on RHS with second term on LHS

8. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 8 6.2 Breit-Wigner Line Shape ii) … and divide by the exponential factor on the LHS Now specify the initial resonance state further Let  (t=0)=  l, i.e. a l (t=0)=1 and a m  l (0)=0 and consider only first order transitions, i.e. over a short time and only from  l directly to any other state  m then only H lm remains in the above sum and we have: If further H nm has any non diagonal elements and … … if it is not explicitly time dependent then … … the initial state  l will decay exponentially with time  Ansatz: where  is the energetic width (uncertainty) of our initial resonance

9. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 9 6.2 Breit-Wigner Line Shape iii) Inserting Ansatz: into:

10. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 10 6.2 Breit-Wigner Line Shape iii) Look at the probabilities of  l (E l ) going to a final state  m (E m ) We call P(E’) the normalised Breit-Wigner line shape

11. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 11 6.2 Breit-Wigner Line Shape iv) If we started in our initial state  l and if  l was short lived with life time  then its energy was uncertain to a level of  =h/2  due to the uncertainty principle What is probability of going from  l at any energy inside its width into a final state  m of energy E m ? We have to consider any energy value that the initial state might have been in due to its uncertainty =1

12. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 12 6.2 Breit-Wigner Line Shape v) We will see the Breit-Wigner shape many times in atomic, nuclear and particle physics  =FWHM We determine lifetimes of states from their energetic width as measured via many decays

13. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 13 6.3 Fermi Golden Rule i) Want to be able to calculate reaction rates in terms of matrix elements of H. Note: We will use this many times to calculate  but derivation not required for exams, given here for completeness. Define a decay channel: Range of kinematic variables (i.e. Energy) of final states  f over which the matrix element H fi does not vary significantly The final state is in the energy continum The density of final states is assumed flat across the channel Note: The Breit-Wigner resonance was assumed narrower than a channel

14. 14 6.3 Fermi Golden Rule ii) Decays to a channel denoted by f has range of states with energies from E 1 to E 2 Density of states n f (E ) is flat in this region Assume an initial state which was a narrow resonance with mean E 0 (narrow compared to the range E 1 to E 2 ) Look for probability of going to this channel f E1E1 E2E2 n f (E ) E0E0 since n f (E ) ≈const and P(E-E 0 ) normalised P(E-E 0 ) define partial width into channel f,  f and total and partial decay rate R tot and R f via: Fermis Golden Rule

15. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 15 6.3 The Breit-Wigner cross section Q: What is the rate of transitions from an initial state i (particles a+b, energy  i ) via an intermediate resonance X around E x to a final channel f (particles c+d, energy  f ) Split this into four parts A: What is the probability that X is formed from i B: What is the probability that X decays to f C: from A: and B: form the rate of i going to f D: and convert that rate into a cross section

16. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 16 6.3 The Breit-Wigner cross section A: probability that X is formed from i use in reverse we can do this because H is hermitian and |H mn | 2 =|H nm | 2 we get: as probability that initial state ends up forming X using FGR:

17. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 17 6.3 The Breit-Wigner cross section A: replace the |H xi | 2 with the  x  i via FGR

18. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 18 6.3 The Breit-Wigner cross section B: What is the rate with which X decays to f It is the partial decay rate of X to f which was defined FGR: C: So the rate for R(i  x  f) is given via R i  f =P(i  x)*R x  f which is: D: How do we get a crossection from a Rate?

19. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 19 6.3 Cross Sections from rate C: Relation between rate R, cross section  and flux F: Let’s calculate F for a free particle (initial state i ): normalised to 1 particle per volume V if the normalised flux where is the particle velocity is the density of states of free particles in k space normalised to 1 particle per volume V and calculate n i (k) for our initial state:

20. Oct 2006, Lectures 6&7 20 6.3 Cross Sections from rate Now we can compute the cross section:

21. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 21 6.3 Breit-Wigner Cross Section n + 16 O  17 O

22. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 22 6.4 Rutherford Scattering i) What do we want to describe: Scattering between two spin less nuclei due to Coulomb interactions Non relativistic scattering energies ( E cm << smallest of the two nuclear masses) Use the Born approximation plane waves going into and coming out of scattering no disturbance of wave functions during the scattering acceleration happens at one instance in time nuclei stay what they were (no break-up or emission of other particles etc.) First nucleus, denoted by i1 and f1 is light compared to the first one to guarantee no recoil has charge Z 1 Second nucleus denoted by i2 and f2 is heavy  no recoil is stationary in the lab frame before collision has charge Z 2 Do this quantum mechanically and not classically. You would get the right result but by accident! Good book for this is: “Basic Ideas and Concepts in Nuclear Physics”, K. Heyde, page 51-54

23. Oct 2006, Lectures 6&7 23 result of d  integration 6.4 Rutherford Scattering (computing H fi ) The scattering potential in natural units: The wavefunction of the incoming and outgoing first nucleus: The matrix elements of the Coulomb interaction Hamiltonian: chance variables: Choosing z -axis parallel to q :

24. Oct 2006, Lectures 6&7 24 6.4 Rutherford Scattering (computing H fi ) Substitute:

25. Oct 2006, Lectures 6&7 25 6.4 Rutherford Scattering (computing d  /d  ) Fermi Golden Rule: Final state consists of two free non relativistic particles  density of states dependent on two variables: E tot,f : total energy in the final state P cm,f1 : CM momentum of one of the final state particles inserted into gives: and inserting gives:

26. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 26 6.4 Rutherford Scattering (computing d  /d  ) We now want to see the depedence on the scattering angle  : p i1 p f1  inserted into:

27. Oct 2006, Lectures 6&7 27 6.4 Rutherford Scattering (low energy experiment) Compare with experimental data at low energy Q: what changes at high energy ? Scattering of  on Au & Ag  agree with calculation assuming point nucleus sin 4 (  /2) dN/dcos  ~ ds/dcos 

28. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 28 6.4 Rutherford Scattering (high energy experiment) Deviation from Rutherford scattering at higher energy  determine charge distribution in the nucleus. Similarity to diffraction pattern in optics Form factor is F.T. of charge distribution Electron – Gold scattering a’la Rutherford looks like a diffraction pattern on top of a falling line

29. Oct 2006, Lectures 6&7Nuclear Physics Lectures, Dr. Armin Reichold 29 6.4 Thinking about the last two lectures What would happen to d  rutherford /d  if: V coulomb  -V coulomb V coulomb were ~ 1/r 2 if  tot =sum(  partial ), is the width of the energy distribution for decays into a single decay channel only a fraction of the total width