FALL 1996~~~~~~~~~~~~~~Chemistry 160~~~~~~~~~~~~Instructor:

cMODULE 159-2(ESR)
J.V. Acrivos, SJSU

ESR.2:

I. Given:

(a) System of particles possessing angular momentum in units of .

(b)Spin Angular Momentum == S;

orbital angular momentum == L;

nuclear spin angular momentum == I_i,

(c) The objective of the lecture is to determine the shape of the esr absorption line versus externally applied magnetic field B_z.

II.

The background knowledge for the lecture is the classical Larmor precession frequency solved in freshman physics is then:

w_L = g_e B_z0, (1.6')

where B_z0 is the field at the center of the absorption line and the finite width due to the lifetime of states given by the Uncertainty Principle:

, (1.7)

leading to a half width at half height (HWHH) of:

, (1.8)

where T₂ is called the spin state lifetime; it is determined by the surrounding molecules as well as the chemical reactions that the species carrying an unpaired spin undergoes.
The intensity of the transition is determined by the area under the absorption curve, it depends on the magnitude of the moment square and on the difference in population between the states which in turn depends on the energy separation hv and the absolute temperature T:

N₊/N_- = exp(-hv/kBT), (1.9)

where the signal intensity is proportional to the fractional excess population:

(N₊ - N_- )/N_total = (if T >> hv/k_B) ~ hv/k_BT, (1.10)

The magnetization M = k M_z + j M_y + i M_x, in an external flux field

B = kB_z + iB_x

obeys the kinetics:

dM_z/dt = -(M_z-M₀)/T₁; dM_x/dt = -M_x/T₂; dM_y/dt = -M_y/T_2,(1.11)

where T₁, T₂ are called the spin - lattice and spin - spin relaxation times respectively. The solution obtains the Bloch Equations:

, (1.12)

, (1.12')

which gives the susceptibility:

At resonance the imaginary part of the susceptibility (called the absorption) is at a maximum and:

|M|/M₀ , (1.13)

The spin system absorbs power from the radiation field:

P = = dW/dt = <H_x dM_x/dt + H_y dM_y/dt> = w X" H₁². (1.14)

Tha absorption of power raises the spin temperature and equilibrium is reached when dW/dt is equal to the rate at which the spin system transfers energy to the lattice containing the spin system.

At exactly resonance w=w_L and for a = , the magnitude of component X"H₁of the magnetization in the equatorial plane (relative to z) is:

X" H₁/M₀ = - a H₁/(1 + (a H₁)² T₁/T₂), (1.15)

which passes through a maximum when |a H₁| = (T₂/T₁)^1/2. The absorbed power at resonance is:

P = dW/dt = M₀H/T₁ ((M₀-M_z)/M₀), (1.16)

where the quantity s == (M₀-M_z)/M₀ is called the saturation parameter

III.

ESR detection measures the derivative of the power P absorption with respect to the external field. For a Lorentzian line shape we obtain:

L' = dP_L/dB_z= d(X"H₁)/dB_z =N_total²S S(S+1) hv/k_BT a *

(a)²B₁ (B_z-B_z0)/[1 + (a (B_z-B_z0))²+(a)²T₁/T₂B₁²]² . (2.1)

It is convenient to define a== and x = (B_z-B_z0) then for T₁ = T₂ the normalized line shape is:

L' ==L'(B₁, x)/ {N_total² S S(S+1) hv/k_BT a} =

(aB₁) (a x)/ [1 + (a x)²+ (aB₁)²]² , (2.1')

and the fields of maximum/minimum slope then determine a, i.e.,

dL"/dx = 0 as (a B_1,0) 0, for (a x_ms) = 1/3^1/2 (1 + (a B₁)²)^1/2.

Here we note that the normalized relation relation (2.1') evaluated at x_ms depends on (a B₁).

L'(B₁, x_ms) == 16/3^3/2 a B₁/[1 + (a B₁)²]^3/2

In order to compare L' for different values of T₂, note that the line shape depends very strongly on the relaxation times, e.g., when T₁=T₂ doubles the last term in the denominator reduces the intensity of L' in the wings (Figure 1.1) but increases in absolute value more than twofold near x_ms. When the product (a B₁) increases, L'_ms== L'(x_ms) gives significant values only in the linear region of Figure 1.2.

Figure 1. 1: Typical Lorentzian ESR absorption derivative.

Figure 1. 2: Saturation of typical Lorentzian ESR absorption derivative.

The results in Figure 1.2 indicate that B₁ has to be reduced until the linear region in Figure 1.2 produces the best results. L' in Figure 1.3 shows the kind of problems that could be encountered when a B₁ increases above the linear region.

Figure 1. 3: Reduction of ESR absorption derivative by B₁ when T₁ = T₂=10^-6 s. Note that x_ms(saturated) ~ B₁/3^1/2allows for the callibration of B₁ directly.

A Gaussian ESR absorption line shape extends to the wings:

G' = dP_G/dB_z= Constant'/T * a/(T₂<>^1/2)

a²/(T₂²<>) B₁(B_z-B_z0) exp(-a²(B_z-B_z0)²/(2T₂²<>)), (2.2)

dn(G')/dB_z = [1 - a²/(T₂²<>)(B_z-B_z0))²]/(B_z-B_z0),

The value of T₂is determined from the line widths. <> is the second moment of the esr absorption line. The values for the spin-lattice relaxation times are determined from saturation experiments. The results depend on the line shapes. In the Curie-Weiss temperature regime the absorption can be measured relative to a standard which does not saturate easily, i.e.,

a B₁² T_1,reference/T2_,reference <<1.

An absorption ratio of sample to reference is defined as (a B_1,0)²T₁/T₂0 for boththe reference and the sample for Lorentzian line shapes:

1/r_{0 = =}(L'_ms x_ms²)_reference/(L'_ms x_ms²)_sample)=

(T_sample/T_reference)* (N_{total,
reference}/ N_{total, sample})*

((S_reference(S_reference+1)/ (S_sample (S_sample+1)), (2.3)

N_{total, sample} is the only unknown to be determined from r₀in (2.3) in the absence of saturation. As the amplitude B_1,0 increases above a limiting value, the ratio r₀ decreases because the sample may saturate, i.e.,