Pulsed EPR

23 1.7.1.2 The spin Hamiltonian concept

All the aforementioned interactions can be described within the spin Hamiltonian concept. In this concept all interactions can be given by one Hamiltonian ℋ₀ as given in eq. 13.

ℋ̂₀= ℋ̂_𝐸𝑍+ ℋ̂_𝑁𝑍+ ℋ̂_𝐻𝐹+ ℋ̂_𝐷𝐷 with

eq. 13

ℋ̂_𝐸𝑍= 𝒈𝜇_𝐵𝑆̂𝐵_𝑜, ^{eq. 14}

ℋ̂_𝑁𝑍 = 𝒈𝑵𝜇𝐵𝐼̂𝐵𝑜, ^{eq. 15}

ℋ̂_𝐻𝐹= 𝑆̂𝑨𝐼̂, and

eq. 16

ℋ̂_𝐷𝐷 = 𝑆̂1𝑫𝑆̂2 eq. 17

The terms given in eq. 13 to eq. 17 describe the electron Zeeman interaction ℋ̂_𝐸𝑍, the nuclear Zeeman interaction ℋ̂_𝑁𝑍, the hyperfine coupling interaction ℋ̂_𝐻𝐹, and the dipole-dipole coupling between two electron spins ℋ̂_𝐷𝐷. This concept is the basis for the programs used to simulate the EPR spectra.

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1.7.2.1 The Hahn echo experiment

The Hahn echo experiment corresponds to a simple ^𝜋

2− 𝜏 − 𝜋 − 𝜏 − 𝑒𝑐ℎ𝑜 pulse sequence (Figure 16A). The ^𝜋

2 pulse rotates the net magnetization Mz into the +y-direction (Figure 16B and C). This shifts the magnetization out of its thermal equilibrium and induces phase coherence. The spin packets start to dephase during the time 𝜏 because of the distribution of their Lamor frequencies. A following 𝜋 pulse flips the spin packets into the –y-direction refocusing their phase maximally after the same time interval 𝜏 after the 𝜋 pulse. The magnetization built up in –y-direction can be detected and is referred to as Hahn echo (Figure 16D).

1.7.2.2 Relaxation

After the pulse excitation, the spins will start to relax back to thermal equilibrium. This is driven by two main relaxation processes i.e., the longitudinal (T1) and the transversal (T2) relaxation.

T1 is governed by processes that restore the thermal equilibrium as given by the Boltzmann distribution. The spins dissipate their energy into the surrounding, the lattice, by returning to the z-axis. Accordingly, this process is characterized by an energy transfer between the spin system and the lattice. T1 dictates the repetition rate of a pulsed EPR experiment that allows the spins to return to equilibrium between two cycles of a pulse sequence. The repetition rate is set to find an optimal signal-to-noise ratio of a pulsed EPR experiment in time. The inverse of T1 is the relaxation rate constant for the decay process and is temperature dependent. Experimentally, T1 is assessed through the inversion recovery experiment. This three-pulse experiment consists of a 𝜋 pulse

Figure 16. (A) Hahn echo pulse sequence. (B) Illustration of the net spin magnetization in z-direction. (C) Illustration of the dephasing of the spins after applying a ^𝜋

2 pulse. (D) Illustration of the refocussing of the spins after applying a 𝜋 pulse. The resulting magnetization is detected as Hahn echo.

25 applied prior to the variable delay time t followed by the aforementioned ^𝜋

2− 𝜏 − 𝜋 − 𝜏 − 𝑒𝑐ℎ𝑜 sequence. The echo intensity, which recovers towards thermal equilibrium, is measured as function of the time t. The exponential recovery curve of the echo is fitted, yielding T1.

T2 is assigned to relaxation processes in the plane perpendicular to the applied magnetic field, the x,y-plane. The transverse relaxation process is characterized by spin exchange mechanism i.e., spin flip-flops, which destroy the spin coherence. The driving force for this mechanism is the gain in entropy. The relaxation time T2 can be measured via a two pulse electron spin echo envelope modulation (2PESEEM) experiment (Figure 17). The integrated echo, which is generated by applying a ^𝜋

2− 𝜏 − 𝜋 − 𝜏 − 𝑒𝑐ℎ𝑜 sequence, is measured as function of the interpulse delay 𝜏. The resulting Hahn echo decay curve is fitted assuming an exponential decay yielding the phase memory time TM. In terms of more complicated processes wherein different components are involved, a stretched bi-exponential decay as given in eq. 18 can be used to analyze the components of the phase memory time¹⁵⁸

Since the relaxation process occurs over both periods of 𝜏 in the Hahn echo sequence, eq. 18 contains 2 𝜏 in the exponent. TM1 and TM2 are the phase memory times, and x1

and x2 are the corresponding stretch parameters. A1 is the pre-exponential weighting factor of the first relaxation component of the Hahn echo decay curve, and the contribution of the second component is given by A2, using the restriction A1+A2=1.¹⁵⁸ The stretch parameters provide insight into the underlying relaxation process that contributes to the respective component of TM. If x is close to 1, TM is attributed to instantaneous diffusion. This process occurs when the second pulse of the Hahn echo sequence flips another electron spin in the vicinity of the observed electron spin. In this way, the observerd electron spin cannot fully refocus because it is moved out of resonance due to dipolar coupling between both spins. The extent of the contribution of the instantaneous diffusion is influenced by the pulse lengths and the local spin

𝑦(2𝜏) = 𝐴1× 𝑒⁽⁻

2𝜏 𝑇𝑀1)

𝑥1

+ 𝐴2× 𝑒⁽⁻

2𝜏 𝑇𝑀2)

𝑥2

.

eq. 18

Figure 17. 2PESEEM sequence. The integrated echo is measured as function between the two pulses.

The observed decay is analyzed as function of TM.

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concentration.150,151,158

Another main contributor to TM is nuclear spin diffusion. If x is 2 - 2.5, TM is dominated by this process. Nuclear spin diffusion includes all processes where nuclear spin flips modulate the electron-nuclear dipolar coupling. The electron spins lose their coherence in the x, y - plane and thereby dephases. The larger the magnetic moment of the dipolar coupled nuclear spin is, the greater the impact. Thus, the smaller magnetic moment of

2H compared to ¹H decreases the impact of the nuclear spin flip on the electron and lengthen TM. Hence, using deuterated buffers or placing the spin label into an aprotic environment can lead to a longer TM.151,159–161 Not only solvent protons, but also those of the biomolecule151,159,160,162–164 or the spin label himself influence TM.151,165,166 Thus, TM

can be used to characterize the spin label environment, especially regarding polar and non-polar biomolecule surfaces.151,159,160 It further indicates the influence of the gem-substituents of the spin label and thus the design strategy of a spin label. Classical rotation of gem-dimethyl groups is responsible for a shortened TM because the averaged inequivalent coupling of nuclear spins influences the electron spin dephasing.150,151,165,166

1.7.2.3 The pulsed electron-electron double resonance experiment In the work reported here, pulsed electron-electron double resonance (PELDOR), also called double electron-electron resonance (DEER), is employed to determine distances between two nitroxide spin labels on RNA strands. The four-pulse PELDOR sequence used is depicted in Figure 18A and consists of four microwave pulses applied at two different microwave frequencies, the detection frequency (𝜈det) and the pump frequency (𝜈_pump). The three microwave pulses applied at the detection frequency 𝜈_det excite only those spins in resonance with the detection frequency, referred to as the A spins.

Likewise, the microwave pulse generated at the pump frequency 𝜈pump excites only those spins in resonance with the pump frequency, designated as the B spins. The first two detection pulses separated by the delay 𝜏₁ create a Hahn echo of the A spins after the time 𝜏₁ after the first π - pulse. The second π - pulse applied at a time 𝜏₂ after the Hahn echo refocuses the spin echo after another time delay 𝜏₂ after this pulse. The introduction of the π-pump pulse, applied between the second and third detection pulse, flips the magnetization of the B spins. Due to dipole-dipole coupling between A and B spins, the local magnetic field at the A spins changes so that the Lamor frequencies of the A spins are shifted. The echo intensity of the A spins recorded as function of the time t yields the PELDOR time trace (Figure 18B), where the echo intensity oscillates with cos (𝜔_𝐴𝐵𝑡). To obtain reliable data, at least 1.5 full oscillation periods have to be recorded.

27 Besides the oscillation frequency, the PELDOR time trace is characterized by the modulation depth Δ, which depends on the efficiency of the pump pulse excitation and on the fraction of coupled spins (Figure 18B). For two coupled nitroxide spin labels under the usual Q-band measurement conditions the expected modulation depth is approximately 35%. The dipolar spectrum, the Pake pattern, is obtained by Fourier transformation of the PELDOR time trace.

Chapter 1.7.1.1 showed that the dipole-dipole interaction provides in addition to the interspin distance the possibility to obtain the relative orientation (𝜃) of two spin labels providing the spin labels have a fixed orientation to 𝐵₀. This is the case for rigid spin labels. The non-random detection of 𝜃 leads to an incomplete Pake pattern. Orientation-selective measurements can be analyzed to determine the spatial orientation of the spin labels e.g., with PeldorFit,¹⁶⁷ or by summing up a set of orientation selective time traces recorded at different pump and probe frequencies.^168–170

Figure 18. (A) Four-pulse PELDOR sequence. (B) PELDOR time trace. The modulation depth Δ and a fully oscillation are indicated.

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