Neutron Scattering

aQ . r 3 àQ . N . QR . aQ .t k
_MLQ) = fMS (N)e- d r =le-'Sik=je - -t1e -' 'S ik (3 .53)
ik i k
To calculate the scattering cross section, we now have to determine the expectation value of
this operator for the quantum mechanical state of the sample averaged over the
thermodynamic ensemble . This leads to
M(Q) =- 2 PB - fm(Q) -7,eiQ .Rt - Si (3 .54)
The single differential cross section for elastic scattering is thus given by :
2
_du = eiQRi ~ (Tnro)2 f.LQ)L Si1 (3 .55)
M
i
Here, fm(Q) denotes the form factor, which is connected wich the spin density of the atour via
a Fourier transform :
.fm (Q)= fps(?")eiQ .rd3r (3 .56)
Atom
With the form (3 .55), we have expressed the cross section in simple atomic quantities, such as
the expectation values of the spin moment at the various atoms . The distribution of the spin
density within an atom is reflected in the magnetic form factor (3 .56) .
For ions with spin and orbital angular momentum, the cross section takes a significantly more
complicated foira [4, 5] . Under the assumption that spin- and orbital- angular momentum of
each atour couple to the total angular momentum J (L/S-coupling) and for rather small
momentum transfers (the reciprocal magnitude of the scattering vector has to be small
compared to the size of the electron orbits), we can give a simple expression for this cross
section in the so-called dipole approximation :
_ 2 J 1 Q' ,I2
dS2-(Ynro) ' .Îm(QJ~Ji1ea-R .
i
(3 .57)
3-27