.93 Home Work Se No. (Professor Sow-Hsin Chen Spring Term 5. Due March 7, 5. This problem concerns calculaions of analyical expressions for he self-inermediae scaering funcion (ISF of he es paricle in a liquid which capures boh he shor-ime and he long-ime behavior of he funcion correcly. In he classical limi he self-isf in he Gaussian approximaion is given by iq[ X (X (] F s (Q, e e iq X ( e Q ( X ( e Q W ( ( where he widh funcion W( is defined by W( ( X( d d V x ( V x ( d' ( ' V x (V x ('. ( (a Show ha Eq.( is valid when he velociy auo-correlaion funcion V x ( V x ( depends only on he ime difference -, such as in a seady sae. (b Define he normalized velociy auo-correlaion funcion ψ( V x (V x ( V x, where V x V k B T M (3 and is Fourier ransform (he densiy of saes funcion ψ(ω de iω ψ( π π. ( d cosωψ( Show ha (by using Eq.( and Eq.( he widh funcion can be wrien as W( V cosω dω ψ(ω ω. (5 (c Show ha if he velociy of he es paricle saisfies he Langevin equaion, hen he normalized velociy auo-correlaion funcion is given by ψ( e τ. (6
Calculae he widh funcion W( and give is shor-ime and long-ime limis. Relae he relaxaion ime τ o he fricion consan of he es paricle in he liquid. Give also he densiy of saes funcion in his case. (d By examining long ime behavior of he widh funcion, derive he general relaionships beween he diffusion consan D of he es paricle and he normalized velociy auo-correlaion funcion ψ(. Give also he relaion beween D and he densiy of saes funcion ψ(ω. (e The Brownian paricle model given above is a one-parameer model in which he parameer is he relaxaion ime τ. I gives he shor ime and he long ime limis of he self-isf correcly. One can formulae a more advanced model, call he relaxing cage model of Desai and Yip (Physical Review 66, 9 (968. This model is a wo-parameer model in which he densiy of saes funcion is modeled as ψ(ω ω /τ π ω ( ω ω /τ (. (7 Show ha he normalized velociy auo-correlaion funcion is given by ψ( dω cosωψ(ω e τ cosω τ Ω sinω (8 where Ω ω τ. (f Give he expression of he widh-funcion. Check is shor-ime and long-ime behavior. You shall find ha his model describes he moion of he es paricle as a vibraional one wih a characerisic frequency ω a shor ime, and a diffusional one wih a fricional coefficien ω τ a long ime. In erms of a ypical aom in a liquid, he picure corresponds o puing he aom in an exernal parabolic poenial well which relaxes in ime so ha evenually he aom diffuses away experiences only he fricional force. Thus i is appropriae o call i a relaxing cage model.
3. The objecive of his problem is o calculae he oal cross secion of scaering of cold neurons from hydrogen molecules in gas phase where iner-molecular correlaion effec can be negleced. (a If we denoe he neuron and nuclear spin operaors respecively by s ˆ and i ˆ, he scaering lengh operaor bˆ can be wrien as: bˆ A Bŝ î (9 where A and B are consans o be deermined. Now denoe he oal spin operaor (sum of he neuron and nuclear spins by ˆ ŝ î, and noing he operaor relaionship ( ˆ î ŝ ŝ î, ( which has an eigen value of ( i(i s(s [ ] ( i(i 3. By demanding ha he bˆ operaor has an eigen value b when i b when i, obain he following values for he wo consans:, and an eigen value A i i b i i b b coh and B ( i b b _. ( (b Now urn our aenion o scaering of cold neurons of wave lengh 5 A (energy ~ mev which is larger han he iner-nuclear separaion of.7 A in a hydrogen molecule. As a resul, he spaial inerference effec due o he wo hydrogen aoms in a molecule can be negleced in he calculaion of scaering cross secions. We can calculae he differenial scaering cross secion by he formula ' k' k ' b ˆ ξ exp(iq R ξ ξ ( where and ' denoe he iniial and he final saes of he molecule in he scaering process, and he index ξ runs over and hydrogen aoms in he molecule. Since we can neglec he special correlaion due o he finie separaion of he wo proons in a molecule, he exponenial facor in he marix elemen can be pu equal o uniy. In order o ge he measured differenial scaering cross secion, we furher sum over he final saes ' in Eq.( o obain 3
k' k ( ˆ b ˆ b ( ˆ b ˆ b (3 The relevan iniial sae of he molecule and he neuron is Î î î I, m I s, m s, where he firs facor represens he oal spin sae of he molecule,. By using he scaering lengh operaor given in Eq.(9 for he and proons, you can calculae he marix elemen in Eq.(3. Show ha ( bˆ bˆ (bˆ bˆ ( A Bŝ I A (AB B ŝ Î B Î ( (d In order o obain he resul of Eq.(, you need o show he operaor relaion: ( Î ŝ Î ŝ Î (5 (e Subsiue he resul of Eq.( ino Eq.(3 and calculae he marix elemen by assuming an unpolarized inciden neuron beam, show ha unpol k' k A B I(I (6 (f There are wo kinds of molecular hydrogen. When he wo nuclear spins of he wo proons are parallel, i is in he orho-sae (I. On he oher hand, when he wo spins are ani-parallel, i is in he para-sae (I. Show ha dσ dω orho k' k b (b b (7 dσ dω para k' k b (8 (g Inegraing he above expressions over he solid angle o ge he free molecular cross secions, like ha is done in he lecure noes, show ha
5 π σ orho b (b b 9 (9 σ para 9 π b ( Explain how does he facor /9 in he above expressions arise. 5