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 describe the applications of esr instrumentation.

II.

The background knowledge for the lecture is found in mod159-1 mod159-2. 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"B₁)/dB_z =

N_total ² S(S+1) hv/k_BT a *

(a)² B₁ (B_z-B_z0)/[1 + (a (B_z-B_z0))² +(a)²T₁ /T₂ B₁²]² . (2.1)

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

The effect of the relaxation times on the esr spectra are shown by the Nd³⁺, ⁴F_9/2 esr spectra in Nd(Nd_.05Ba_.95)₂Cu₃O₇ (Acrivos et al., 1994, 1995). The esr spectrum indicates that the free electron is associated with Nd, identified by the large hyperfine structure in two isotopes ¹⁴³Nd and ¹⁴⁵Nd both with spin I=7/2.

The real spectrum is complicated by the fact that the two isotopes have different moments and there is a hint of ^63,65Cu (I=3/2) splitting. The ideal spectrum is shown in Figure 3.1 (a to c) for different relaxation times. This is the best way to get familiar with esr.

Organic conductors also show esr absorption. There is the triplet esr spectrum at half field which characterizes it (Vivó Acrivos et al., 1995, 1996).

Here the measurement of the saturation of the esr versus temperature allows for the measurement of T₁. This can be used to identify the superconducting transition temperature T_c.

Figure 3.1: Calculated ideal esr absorption derivative vs static field B_z for ¹⁴⁵Nd³⁺, ⁴F_9/2 (in the neighborhood of a ^63,65Cu nucleus) in Nd(Nd_0.05Ba_0.95)₂Cu₃O₇ for different values of T₁=T₂ from 10^-6 to10^-8.5s. As the line width increases the microwave amplitude can be increased from H₁=0.001 to say 50 G, also the modulation amplitude can be increased to an enhance the reduced signal to noise observed as T₂ decreases. Note that care must be taken to ascertain what are the causes of line width. The relaxation time is a natural property of the sample while H₁ must be lowered until the sample properties can be determined.

III.

The applications can be found in several publications. The students are to write term papers on publications later than 1990. The suggested subjects ( which can be searched in the chemical journals, e.g., JACS) are:

* Bloch Equations

* ESR Instrumentation

* ESR Software

* Applications to Biology

* Applications to Organic Chemistrty

* Applications to Superconductivity, when x = (B_z-B_z0) then for T₁ = T₂ the normalized line shape is:

L' ==L'(B₁, x)/ {N_total ² S(S+1) hv/k_BT a} =

(a B₁) (a x)/ [1 + (a x)² + (a B₁)²]² , (2.1')

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

L"= =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

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) ~ B₁/3^1/2 allows for the calibration of B₁ directly. The derivative amplitudes are equal even though the B₁ has been increased by a factor of ~252.

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/T₂ 0 for both the 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 r₀ in (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.,

r = =r₀(a B₁) /r₀(a B_1,0)/(B₁/B_1,0) =

(1+(a B_1,0) ²T₁/T₂)^3/2/(1+(a B₁)²T₁/T₂)^3/2, (2.3')

so that as (a B_1,0)²T₁/T₂ approaches zero, the value of T₁T₂ is obtained with good accuracy for:

1/r = (1+(a B₁)²T₁ /T₂)^3/2

= 10 to 1.33

with errors r/r 5% and B₁/B_1,0 to better than 0.1% obtain:

(T₁ /T₂)^1/2 = (1/r^2/3-1)^1/2/(aB₁)

with an absolute error of

(T₁ T₂)^1/2(r/3r)/(1 - r^2/3)

(T₁ T₂)^1/2depends on the chemical dynamics.

IV.

Gorter related T₁ to the heat capacity and heat capacity and heat transfer coefficient at constant field H, C_H and_H:

T₁ = C_H/_H. (2.4)

The sample temperature is T_S and the lattice's is T₀ at field H and T_ST0 at the rate:

dT_S/dt = (T₀-T_S)/T₁, (2.5)

when |T₀-T_S|/T₀<<1.

Phase transitions at a critical temperature, say T_c, may be identified by the changes in both C and and consequently T₁:

T₁(T_c)/T₁ C_H/C_H -_H/_H.

T₁ NMR measurements by Slichter et al., use it for nuclei in hyperfine contact with the Bose condensate. For cuprates:

C_H(T_c)/C_H

and it appears that:

_H(T<T_c)/ _H < C_H(T_c)/C_H

So that by esr T₁ relaxation measurements one can find T_c of phase transitions.

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