Neutron Scattering from Magnetic Materials part 1 pps

[...]... interval dE1 is ρk1 1 (E1 ) = V0 V0 2 dk1 = k dk1 Ω 8π3 8π3 1 Using the known result of quantum mechanics dE1 = h2 k1 dk1 ¯ m we have ρk1 1 (E1 ) = V0 mk1 Ω 8π3 h2 ¯ (18 ) The incident neutron flux is equal to the normalized number density times the neutron velocity Φ= v hk0 ¯ = V0 V0 m (19 ) Substituting equations (17 )– (19 ) in (16 ), we get dσ dΩ k0 ,σ0 ,λ0 →k1 , 1 1 = m 1 k1 N k0 2πh2 ¯ 2 k1 1 1 |V |k0... |V |k0 σ0 λ0 1/ 2 ik0 ·r 1/ 2 ik1 ·r e , V0 e where |k0 , |k1 denote the plane waves V0 vation of energy 2 , (20) From the law of conser- E = E0 − E1 = E 1 − Eλ0 , ( 21) we get d 2σ dΩ dE = k0 ,σ0 ,λ0 →k1 , 1 1 m 1 k1 N k0 2πh2 ¯ 2 k1 1 1 |V |k0 σ0 λ0 2 δ(E + Eλ0 − E 1 ) (22) Magnetic neutron scattering 9 We now sum over all final states of the sample 1 and final polarization states 1 , average over... of the magnetic atoms in condensed matter The corresponding neutron scattering is called nuclear neutron scattering and magnetic neutron scattering, respectively In the present book we will mainly consider magnetic neutron scattering However, neutron scattering intensity from magnetic materials is a superposition of both types of scattering In order to be able to separate magnetic scattering from nuclear... η dσ dΩ (15 ) k0 ,σ0 →k1 , 1 4.2 The master equation The theory of neutron scattering has been treated in several textbooks [10 ,12 14 ,16 ] and articles [15 ,18 ] We recall here some basic results We consider scattering of neutrons by a sample consisting of condensed matter which undergoes a change from a state λ0 to a state 1 while the state of the neutron changes from (k0 , σ0 ) to (k1 , 1 ) The corresponding... differential scattering cross-section is given by dσ dΩ k0 ,σ0 ,λ0 →k1 , 1 1 = 1 N Φ dΩ Wk0 ,σ0 ,λ0 →k1 , 1 , 1 , (16 ) k1 where Wk0 ,σ0 ,λ0 →k1 , 1 , 1 is the number of transitions per second from the state k0 , σ0 , λ0 to the state k1 , σ0 , 1 and Φ is the flux of incident neutrons The summation is over all values of k1 that lie in the small solid angle dΩ in the direction θ, φ, the values k0 , λ0 and 1 remaining... 2.072k 2 = 81. 81 1 λ2 = 0.685 × 10 12 ω = 4 .13 6ν, 1 1 1 1 λ = 6.283 = 3.956 = 9.045 √ = 30. 81 √ , k v E T (7) (8) where λ is in Å, k is in Å 1 , v is in km s 1 , E is in meV, T is in K, ω is in s 1 and ν is in THz 3 Neutron source We already mentioned that there exist two main types of neutron sources at which the neutron scattering experiments can be done, namely the reactor and spallation neutron sources... golden rule, Wk0 ,σ0 ,λ0 →k1 , 1 , 1 = 2π k1 1 1 |V |k0 σ0 λ0 h ¯ 2 ρk1 1 (E1 ) , (17 ) where V is the interaction potential between the neutron and the sample and ρk1 1 (E1 ) is the density of the final scattering states per unit energy interval The golden rule is based 8 T Chatterji on the validity of first-order perturbation theory It is certainly valid for nuclear neutron scattering since the nuclear... between the sample and the neutron 5 Nuclear neutron scattering 5 .1 Neutron scattering from a single nucleus The scattering of neutrons by a nucleus is caused by the nuclear forces which have a range of about 10 12 10 13 cm The wavelength of the thermal neutron is of the order of 10 −8 cm which is much larger than the range of nuclear forces It is well known from the theory of scattering that the scattered... neutron scattering 5 Table 2 Values of some physical constants Physical constant Value Planck constant, h Boltzmann constant, kB Bohr magneton, µB Nuclear magneton, µN 6.6260755(40) × 10 −34 J s 1. 380658 (12 ) × 10 −23 J K 1 9.274 015 4( 31) × 10 −24 J T 1 5.0507866 (17 ) × 10 −27 J T 1 Taking values of the physical constants from Table 2 we get the following useful equations: E = 0.08 617 T = 5.227v 2 = 2.072k 2 = 81. 81. .. the scattering ˆ system is unique and if we define a unit vector η along this direction then the directional ˆ dependence of the magnetic scattering is given by the factor 1 − (Q · η)2 In fact, in favorable cases this factor enables us to determine the orientation of the magnetic moments in crystalline materials Magnetic neutron scattering 15 6.2 Scattering of neutrons from crystalline magnetic materials . 9 5.2.Coherentandincoherentscattering 11 6.Magneticneutronscattering 12 6 .1. Scattering of neutrons from unpaired electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.2. Scattering of neutrons from. =E 0 −E 1 =E λ 1 −E λ 0 , ( 21) we get  d 2 σ dΩ dE  k 0 ,σ 0 ,λ 0 →k 1 ,σ 1 λ 1 = 1 N k 1 k 0  m 2π ¯ h 2  2   k 1 σ 1 λ 1 |V |k 0 σ 0 λ 0    2 δ(E +E λ 0 −E λ 1 ). (22) Magnetic neutron scattering. scattered neutron in the energy interval dE 1 is ρ k 1 σ 1 (E 1 ) = V 0 8π 3 dk 1 = V 0 8π 3 k 2 1 dk 1 Ω. Using the known result of quantum mechanics dE 1 = ¯ h 2 k 1 dk 1 m we have ρ k 1 σ 1 (E 1 ) = V 0 8π 3 mk 1 ¯ h 2 Ω.