1. Chapter 9 – COHERENT AND INCOHERENT NEUTRON SCATTERING 9:1. COHERENT AND INCOHERENT CROSS SECTIONS 9:2. SPIN INCOHERENCE 9:3. COHERENT SCATTERING CROSS SECTION 9:7. CONTRAST FACTORS 9:11. TABULATED SCATTERING LENGTHS AND CROSS SECTIONS 9:4. ISOTOPIC INCOHERENT SCATTERING CROSS SECTIONS 9:12. NEUTRON TRANSMISSION 20:5. ESTIMATION OF INCOHERENT SCATTERING

2. 9:1. COHERENT AND INCOHERENT CROSS SECTIONS Microscopic cross section (in barns): Macroscopic cross section (in cm -1 ): coherent part incoherent part Q-dependentQ-independent Standard deviation:

3. 9:2. SPIN INCOHERENCE spin operators 2(I-1/2)+1=2I 2(I+1/2)+1=2I+2 I + 1/2 eigenvalues number of degenerate states I - 1/2 incident neutron of spin 1/2 scattering nucleus of spin I

4. DERIVATIONS Coherent scattering length: Incoherent scattering length: Average over spins:

5. 9:3. COHERENT SCATTERING CROSS SECTION 9:7. CONTRAST FACTORS Contrast factor: Coherent cross section: With the fractions: Macroscopic (cm -1 ) and microscopic (barns) cross sections:

6. 9:4. ISOTOPIC INCOHERENT SCATTERING CROSS SECTIONS Spin incoherence cross section: Isotopic incoherence cross section:

7. 9:11. TABULATED SCATTERING LENGTHS AND CROSS SECTIONS Elementb Fermi  c Barn  i Barn  s Barn  a Barn H-3.7391.75780.3082.0570.333 D6.6715.5922.057.6420.000 C6.6465.5500.0015.5510.003 N9.3611.010.5011.511.90 05.8034.2320.0004.2320.000 F5.6544.2320.0014.2330.000 Na3.631.661.623.280.530 Si4.1492.1630.0042.1670.171 P5.133.3070.0053.3120.172 S2.8471.0170.0071.0240.53 Cl9.57711.5265.316.82633.5

8. 9:12. NEUTRON TRANSMISSION Total cross section: scatteringabsorption coherent incoherent  t : total cross section d: sample thickness Neutron transmission:

9. 20:5. ESTIMATION OF INCOHERENT SCATTERING Incoherent scattering cross section: Including multiple scattering: Resuming the series: When incoherent scattering dominates: Way to estimate the incoherent cross section:

10. V.F. Sears, “Neutron Scattering Lengths and Cross Sections”, Neutron News 3, 26-37 (1992). A scattering length density calculator can be found online at: http://www.ncnr.nist.gov/resources/sldcalc.html REFERENCES

11. COMMENTS -- Coherent scattering probes the sample structure (Q-dependent). Incoherent scattering does not contain information about the structure (Q-independent). -- Scattering length densities and contrast factors can be estimated using the online calculator.