Presentazione sul tema: "Magnetochimica AA 2012-2013 Marco Ruzzi Marina Brustolon 1. The coupling of Angular Momenta 2. EPR in a nutshell 3. The exchange spin Hamiltonian 4. The."— Transcript della presentazione:
Magnetochimica AA 2012-2013 Marco Ruzzi Marina Brustolon 1. The coupling of Angular Momenta 2. EPR in a nutshell 3. The exchange spin Hamiltonian 4. The Zero Field Splitting spin Hamiltonian 5. Radicals with delocalized electron spin density
Magnetochimica AA 2011-2012 Marco Ruzzi Marina Brustolon The coupling of Angular Momenta
The coupling of angular momenta 1 with eigenvalues or, considering the two particles together The wavefunctions of the two particles can be referred to these quantum numbers, therefore: with eigenvalues Two non interacting particles, each with a constant angular momentum, are characterized each by its own eigenvalues of the operators magnitude of the vectors and component along z.
There are The coupling of angular momenta 2 states For example, if the two momenta are two spin ½, there are 2x2 =4 states: We can use a shorter notation, as J 1 and J 2 are always 1/2:
The coupling of angular momenta 3 For the orbital momenta of two p electrons, J 1 = J 2 =1, therefore states The states are eigenstates of J z1, J z2 and also of J z, as these three operators commute. Each of these states is an eigenstate of J z1 and J z2. Moreover, they are eigenstates of with eigenvalues
The coupling of angular momenta 4 M tot = m 1 +m 2 2 1 1 0 0 0 -2 These states are eigenfunctions of the operators: with quantum numbers but they are not eigenfunctions of as does not commute with
The coupling of angular momenta 5 So, we have two choices: either use the basis set of eigenfunctions of: (and ) : State M =m 1 +m 2 2 1 1 0 0 0 -2 or find a basis set of eigenfunctions of: Uncoupled basis Coupled basis 1. The dimensions of the basis sets are the same. 2. The values of J vary between j 1 +j 2, j 1 +j 2 -1,…, | j 1 +j 2 | 3. Each function of the coupled basis with a value M k is a linear combination of the functions of the uncoupled basis with the same M k value.
The coupling of angular momenta 6 -2 0 0 0 1 1 2 M=m1+m2M=m1+m2 State Therefore if j 1 =1 and j 2 =1, the possible J values are:J = 2,1, 0 For each J value there are 2J+1 states, with M J = J, J-1,…,-J For J = 2 we have five functions:: The two functions in orange can give two independent linear combinations: one of the two is this coupled function. The two functions in blue can give two independent linear combinations: one of the two is this coupled function.
The coupling of angular momenta 7 0 0 0 1 1 M=m1+m2M=m1+m2 State J = 2,1, 0 For J = 1 there are 3 states, with M J = 1, 0, -1 For J = 1 we have three functions:: The other linear combination is this coupled function.
The coupling of angular momenta 8 0 0 0 M=m1+m2M=m1+m2 State J = 2,1, 0 For J = 0 there is 1 state, with M J = 0 For J = 0 we have one function: These three functions can give three independent linear combinations: this one, and the others indicated in the previous two slides.
The coupling of angular momenta 9 The coefficients of the linear combinations of the uncoupled basis to give the coupled one are the Clebsch-Gordan coefficients: with M =m 1 +m 2 The C-G coefficients can be obtained with recursion formulae, or can be found in tables.
Valore di J tot M Base disaccoppiata Coefficiente della combinazione lineare Base disaccoppiata Coefficienti del tripletto Coefficienti del singoletto Tables of Clebsch-Gordan coefficients For two spins =1/2 J tot =
Two spin =1/2, uncoupled basis: Coupled basis, following the Clebsch-Gordan table: Triplet, J=1, M=1 Singlet, J=0,M=0 etc.
La somma delle loro proiezioni sullasse z è sempre definita Due momenti angolari accoppiati I due momenti sono disaccoppiati, cioè ciascuno può essere sul suo cono di precessione in qualunque posizione indipendentemente dallaltro. I due momenti sono accoppiati, e la loro somma vettoriale dà il momento totale J. Ciò significa che non sono in una posizione qualsiasi uno rispetto allaltro, ma sono accoppiati in modo da dare sempre come somma vettoriale J. Si noti inoltre che negli stati nei quali è definito J, restano definiti j 1 e j 2, ma non sono più definiti m 1 e m 2, ma solo la loro somma M. Due momenti angolari disaccoppiati
s1s1 s2s2 S=1 M S =+1 M S =-1 S=1 s1s1 s2s2 s2s2 s1s1 M S =0 Rappresentazione vettoriale dello stato di tripletto M S =0 s2s2 s1s1 S=0 Rappresentazione vettoriale dello stato di singoletto
Using the raising and lowering spin operators 1 We know that the eigenfunctions of angular momentum operators J 2 and J z are characterized by quantum numbers J and M. For each J value we have a family of functions with different values of M: with The effect of the so called raising and lowering operators: is to transform a function with M J respectively to the one with M J +1 and M J -1
The effects of raising and lowering operators on a function characterized by J and M J are the following (see note*) : Using the raising and lowering spin operators 2 For example let us consider the simple pair of spin functions and : *We use here different symbols: I instead of J, m I instead of M J Therefore: I - = I + = 0 I - = 0 I + =
Exercise: Obtain the spin functions of the coupled basis from an uncoupled basis for two electron spins (or any other angular momentum with J=1/2), by using the raising and lowering operators.
So, which basis of eigenfunctions for two or more angular momenta should be used? Coupled or uncoupled? The answer stays in the type of spin Hamiltonian, as we will see.
Nel sito WEB della Stanford University con questo indirizzo trovate una utile serie di slides sui momenti angolari: http://www.google.it/url?sa=t&rct=j&q=&esrc=s& source=web&cd=2&ved=0CCwQFjAB&url=http %3A%2F%2Fwww.stanford.edu%2Fgroup%2Ff ayer%2Flectures%2FChapter15- 08.ppt&ei=zK5yULLyEaLg4QS2iID4Cw&usg=AF QjCNEqloSuvqmXYnMWOidt3G-_Wwj4Ag