General theory of light propagation and triplet generation for studies of spin dynamics and triplet dynamic nuclear polarisation

A general theory is presented describing the photo-excitation dynamics of transient, paramagnetic triplet states of aromatic molecules including their absorption properties in a molecular host crystal when irradiated by a pulsed laser beam. It is applied to the system of a pentacene doped naphthalene single crystal, where the photo-excitation of pentacene results in a highly non-equilibrium population of intermediate electronic triplet states. These paramagnetic states can be addressed to align the proton spins of the naphthalene host at moderate experimental conditions using dynamic nuclear polarisation (DNP). The photo-excitation profile and the resulting triplet state distribution, which is determined by the material and the laser parameters, leads to a corresponding proton polarisation profile. Measurements of the resulting polarisation inhomogeneity by spatially resolved neutron transmission confirm the theory. The theoretical model then allows the extraction of crucial parameters that enable quantitative investigations of the dynamics of spin systems. For applications of triplet DNP with pentacene-doped naphthalene crystals, polarisation maximum, homogeneity and build-up rate are key factors, and the theory allows the identification of optimal experimental parameters depending on the priorities placed on these factors. GRAPHICAL ABSTRACT


Introduction
Photo-excited triplet states are a powerful alternative for ground state electron spins to serve as polarising agents in dynamic nuclear polarisation (DNP) [1,2]. Using ground state electron spins, DNP can only be efficient when the electron spin is significantly polarised by thermal contact to its environment via spin-lattice relaxation which requires a temperature of 1 K or below and a magnetic field of several tesla [3,4]. Photo-excited triplet states, however, are selectively populated by spin-orbit coupling, a mechanism that allows aromatic molecules with conjugated π-systems to attain a transient high electron spin polarisation and DNP does neither need very low temperatures nor high magnetic fields [5]. This reduces the instrumental complexity significantly. In practice, temperatures around or below 100 K for cooling the system against heating from microwave and laser irradiation and magnetic fields of about 0.3 T are typically used, providing robustness to operate the system reliably [6]. Another key aspect of this system manifests itself in the transient nature of the triplet states: since the electron spin decays to a singlet ground state after photo-excitation, the paramagnetic contribution to the spin-lattice relaxation of the nuclear spin environment is quenched, significantly extending the lifetime of the nuclear polarisation for proton spins in the system naphthalene/pentacene typically to 50 h at 80 K and 0.5 T and enabling the transport of polarised samples over long distances [7]. This has the potential of greatly simplifying applications of DNP, for instance to enhance resolution in magnetic resonance imaging (MRI) [8,9] and increasing sensitivity of nuclear magnetic resonance (NMR) [10].
Photo-excited triplet states also have a strong potential to provide new and complementary insights in the dynamics of electron and nuclear spins. While unpaired ground state electron spins e.g. from stable radicals or transition metals [11,12], are permanently present, triplet electron spins only exist a short time after photoexcitation. While the dynamics of unpaired ground state electron spins is determined by their mutual interactions and electron spin-lattice relaxation, the dynamics of photo-excited triplet states is determined by their creation and annihilation. While for unpaired ground state electron spins their interaction with the nuclear spins is permanently present, for triplet electron spins this interaction is only present during their existence, which can be controlled by varying the repetition rate of the laser and its pulse shape. Thus studies using photoexcited triplet states can provide complementary and otherwise difficult to obtain data on the dynamics of spin systems.
Recently this potential has been exploited to solve a decades old problem regarding the relaxation of nuclear dipolar energy. At low temperature in strong magnetic fields the relaxation time of nuclear dipolar energy has been observed to be up to four orders of magnitude shorter than the relaxation time of nuclear Zeeman energy-for an overview by Abragam and Goldman see [13], Chapter 6. While these authors were able to relate this large ratio to parameters of the spin system, no plausible explanation for this fast dipolar relaxation had been provided until recently. A study, in which photo-excited triplet states instead of ground state electron spins cause dipolar relaxation provided the solution to the problem [14]. Extremely fast nuclear dipolar relaxation was found to be due to a new mechanism involving flipflop transitions between nuclear spins near the electron spins, and a quantitative analysis of the dipolar relaxation time could be obtained. This required an extensive study of the dynamics of the triplet spin system yielding a complete and quantitative knowledge of all parameters involved. These include not just the concentration of molecules that can be photo-excited, but also the fraction that is actually photo-excited into the triplet state in each laser pulse, the occupation of the triplet sub-levels after photo-excitation, their decay to the singlet ground state and possible gradients of these parameters along the path of the laser beam. It should be noted that the optimisation of DNP using photo-excited triplet states is also greatly facilitated by a good understanding of the photo-excitation process. Therefore we set out to study all these parameters in detail, and this article presents the results. Notice that some early results were already reported in [14].
This article concentrates on the system of naphthalene doped with pentacene, one of the best studied triplet spin systems. After an initial investigation of its triplet state in 1980 [15], these studies were driven by the development of DNP using the photo-excited triplet state of pentacene as a polarising agent [5,[16][17][18][19][20]. The system has been successfully used to build reliably working spin filters for neutrons [7,[21][22][23][24]. Recently pioneering experiments demonstrated the feasibility of using this system to improve signal sensitivity in liquid-state NMR. Eichhorn et al. [25] reported that the hyperpolarisation of naphthalene crystals when dissolved can be transferred to different target molecules via inter-molecular cross relaxation in the liquid state at room temperature and moderate magnetic fields. We expect that the understanding of triplet production in this system may be also useful for the study of radical-chromophore hybrid systems, where transfer of non-thermal electron polarisation of the triplet to a bound radical has recently been demonstrated. These findings are of great interest for applications in molecular electronics [26] and for magic-angle spinning (MAS) DNP, which may even be performed without microwave irradiation [27].
We start out presenting a general theory describing the photo-excitation of aromatic molecules and the subsequent population of triplet paramagnetic states by intersystem crossing (ISC) using rate equations. This theory we then apply to naphthalene single crystal doped with pentacene. For comparison with experiment we use the triplet state of pentacene to polarise the proton spins in naphthalene by means of DNP and subsequently determine the proton polarisation as a function of the penetration depth of the laser beam by means of neutron transmission. From this polarisation we calculate the distribution of triplet states in the sample and compare it with the predictions of our model. We will show that good agreement can be obtained, also on an absolute scale, which allows us, not only to provide optimum conditions for DNP using photo-excited triplet states, but moreover to extract crucial parameters necessary for studies of the dynamics of the spin systems.

Theory of photo-excitation
In this section we present a general theory that describes the photo-excitation of aromatic molecules doped in a host single crystal and the subsequent population of paramagnetic triplet state that can be used for DNP.

Rate equations
Our aim is to determine the density of the aromatic molecules that are photo-excited into the lowest triplet state shortly after each laser pulse. In particular we wish to determine this density as a function of the penetration depth z of the laser light. For this purpose we construct a set of rate equations, from which we can solve the probability ρ T (z, t) that an aromatic molecule is in its lowest triplet state as a function of time t and light penetration depth z. Table 1 summarises the notations used for the theoretical development. Figure 1 presents the level scheme showing the transitions and the corresponding rates leading to the creation and decay of these lowest triplet states, where we have made some assumptions as listed below. The laser light excites aromatic molecules from the S 0 singlet ground state to a vibronic excited singlet state -which we denote as S v 1 -with a rate c n σ N ph , which is proportional to the photon density N ph . Next, the molecules relax to the S 0 1 zero phonon first excited singlet state with the internal conversion rate τ −1 IC . Subsequently, the molecules decay partly to the T 1 lowest triplet state with the intersystem crossing rate ISC τ −1 F and partly back to the S 0 singlet ground state with a rate (1 − ISC )τ −1 F . Here τ F is the Figure 1. Simplified Jablonski diagram for the optical excitation of an aromatic molecule, with the transition rates. The electrons can be excited from the ground S 0 level to a vibrational S v 1 , which then decay to the excited state S 0 1 via internal conversion with a lifetime τ IC about 20 ps. In addition to direct fluorescence, the excited pentacene molecule has a probability ISC of going into a triplet state via intersystem crossing (ISC). The fluorescence time τ F is typically a few tens of ns, while the triplet has a much longer lifetime τ T of the order of 50 μs. Note that only the lowest triplet level T 1 is drawn and the splitting of the triplet level is not shown for simplicity. Additionally, in the diagram c denotes the speed of light in vacuum, n the refractive index of the material, σ the light absorption cross section and N ph the photon number density as a function of the light penetration depth z and time t. singlet decay time. Finally the molecules that ended up in the T 1 lowest triplet state decay back to the singlet ground state with the triplet life time τ T . The splitting of the triplet level is not considered here and the sum of the transition rates is given. Of course, the transition rate to each triplet sub-level is different, resulting in a nonequilibrium population used for triplet-DNP, also the decay rates of the sub-levels differ from each other. In this description we have made the following assumptions: • Only the local excitation of the aromatic molecules are involved in the light absorption process and the host is completely transparent. • The repetition rate of the laser is slow enough so that all photo-excited aromatic molecules completely decay back to the ground state before the next laser pulse. • Each vibronic state is represented by a discrete energy, but at all temperature ranges, due to line-broadening, these individual states are not resolved and overlap to form a continuous absorption cross section profile which is essentially flat for the laser line width. In addition the molecules of the host material also couple to the states of the photo-excited molecules via phonons, thus the absorption and fluorescence of the sample depends on the host material. • In principle, the molecules decay to a S v 0 vibronic state, which further decays back to the S 0 ground state by internal conversion. The latter process is extremely fast (∼ 20 ps, see below) so we can simplify the model to a direct decay to the S 0 singlet ground state.
• In the intersystem crossing, the molecules first go to a higher triplet state and then decay to the lowest T 1 triplet state. Since this decay between triplet states is fast, we simplify the model and consider only one triplet state T 1 . • We further neglect triplet-triplet absorption, since the population in the triplet state is expected to be extremely small (see Figure 4).
The rates at which these processes occur vary widely in order of magnitude. The system of a pentacene-doped naphthalene single crystal can serve as an example, to which we will apply the theory below. Details on the crystal system are given in the supplementary material. Setting the index of refraction for an unpolarised beam perpendicular to the Xb-plane of the naphthalene crystal at n = 1.834 [28] and the vacuum velocity of light at c = 3 × 10 8 ms −1 , the transit time of photons through a sample of 5 mm is only 31 ps. The lifetime τ IC of the vibronic excited singlet state S v 1 was found to be of the same order of magnitude (20 ps) in [29]. On the other hand, we measured the singlet decay time to be τ F = 18.5 ± 1.5 ns in pentacene-d14:naphthalene-h8 at 100 K [30]. 1 Furthermore the width τ p of the laser pulses is 225 ns, so both are three orders of magnitude longer. Finally, the triplet lifetime τ T is of the order of 50 μs and another two to three orders of magnitude longer.
We wish to determine the probability that aromatic molecules are in their triplet state just after the optical pulse is finished and these molecules have reached a temporary equilibrium through singlet decay and intersystem crossing. On the short time scale that the optical pulse is applied and these two processes create such a local equilibrium, triplet decay is negligible and we will omit this process in what follows.
Next we consider the most general case where the crystal is birefringent, thus the optical beam needs to be split into two rays -the fast and slow rays with the photon number densities N f ph and N s ph and the refractive indices n f and n s , respectively. Then, we set up a set of non-linear equations for the processes, neglecting the decay from the T 1 lowest triplet state to the S 0 ground state: Here N 0 (z, t) and N v 1 (z, t) are the number densities of the aromatic molecule in the S 0 singlet ground state and S v 1 vibronic excited singlet states. N 0 1 (z, t) and N T (z, t) are the pentacene number densities in the S 0 1 zero phonon excited singlet state and the T 1 excited triplet state. Furthermore, σ v f and σ v s are the light absorption cross sections to the S v 1 vibronic excited singlet state when the light is polarised along the fast and slow axes.
With these equations and the knowledge of the photon density N ph (z, t) (see (10)) we can solve the desired triplet number density = N T and probability ρ T (= N T /N) that the aromatic molecules will end up in the triplet state after application of each laser pulse.

Short internal conversion rate
We estimate the maximum value of the rate (c/n)σ N ph (z, t) for the excitation of the molecules by one of the rays from the S 0 singlet ground state into one of the S v 1 vibronic excited singlet states. The beam of the laser enters the sample with a maximum power P max and a spot size A giving the maximum energy intensity I max = P max /A. Then the energy maximum density in the sample is equal to (n/c) I max , in which c is the velocity of light in vacuum and n the index of refraction. As a result the maximum photon density in the sample is equal to λ is the energy of the photons in the laser beam. As a result We see that the refractive index cancels in the final result.
We evaluate these expressions for our 515 nm laser setup as used in the experiments (see 3.1). The maximum intensity of the laser beam is We set the velocity of light in vacuum at 3 × 10 8 ms −1 . Then at λ = 515 nm the photon energy is equal to ω = 2π c/λ = 0.386 × 10 −18 J. Then, using a typical value of σ ≤ 4 × 10 −17 cm 2 , we obtain c n σ N ph (z, t) ≤ 1 × 10 6 s −1 . Thus we find that the rate, at which one of the S v 1 vibronic excited single state is populated by optical excitation, is five orders of magnitude slower than the rate τ −1 IC at which this state decays to the S 0 1 zero phonon excited singlet state and still two orders of magnitude slower than the rate at which the latter state decays further to the S 0 singlet ground state and the T 1 lowest triplet state. This suggest that in our observation scale, there is hardly any population in the vibronic excited states, . This observation allows for a simplification of the rate equations. Furthermore, we eliminate the internal conversion process by adding (2), and (3) for all the v, and approximate to where we define As a result, we obtain the set of rate equations for the evolution of the number densities N 0 (z, t), N 0 1 (z, t) and N T (z, t) that aromatic molecules are in the S 0 , S 0 1 and T 1 states. Note that there is no further need for a rate equation for the probability N v 1 (z, t) that an aromatic molecule is in the S v 1 vibronic excited singlet state. Notice that addition of the three rate equations yields which simply reflects the conservation of the total number of aromatic molecules.

Fast photon transit
Now the only question left is to obtain the information of photon density as a function of time and depth. Taking into account the short internal conversion rate, we expect it to have the shape where ξ = f or s. Furthermore, we consider the transit time of a photon over z = 1 mm and n = 1.5, t = z n/c = 5 ps, which is more than three orders of magnitude faster than all processes in the set of rate Equations (6) to (8). This suggests that the time derivative term in (10) is much smaller than the space derivative term, and thus can be neglected, so the photon density is quasi-stationary.
We then solve these equations by integration over the interval 0 ≤ z ≤ z: Upon insertion of these expressions in (6) to (8), we obtain a closed set of three rate equations from which we can determine the number densities N 0 (z, t), N 0 1 (z, t) and N T (z, t) of the aromatic molecules in the S 0 singlet ground state, the S 0 1 zero phonon excited singlet state and the T 1 lowest triplet state or the corresponding probabilities ρ 0 (z, t), ρ 0 1 (z, t) and ρ T (z, t) that an aromatic molecule is in the corresponding states.

Experimental
In order to verify the model introduced in the previous section, we performed triplet-DNP experiments in naphthalene single crystals doped with aromatic pentacene molecules using an X-band DNP system operating at 9.4 GHz [31]. The initial growth rate of the proton polarisation is proportional to the density of photo-excited triplet states. Measuring this initial growth rate as a function of position in the sample using neutron transmission allows us to verify the theoretically predicted density of triplet states as a function of the penetration depth of the laser beam. Furthermore, the comparison of polarisation build-ups using different laser sources with simulations provides additional evidence for the viability of the model.
In order to make quantitative predictions, the model requires numerical values for the pentacene light absorption cross sections as input parameters. A formal description of the light propagation in the biaxial, anisotropic, absorptive medium has been developed and is described in the supplementary material. It allows to extract the cross sections from simple light transmission measurements, which were performed in the wavelength range of interest (500 nm ≤ λ ≤650 nm) between room temperature and 8 K. Here the absorption cross sections near the vibronic transitions of 515, 556 and 600 nm are of special interest, since theses are expected to yield an optimum triplet excitation, prerequisite to efficiently perform DNP.

Materials and methods
The crystals investigated here were cut out of a large single crystal grown with a self-seeding vertical Bridgman technique from zone-refined naphthalene doped with typically a few 10 −5 mol/mol pentacene-d 14 . Two samples were prepared including a highly doped crystal for the purpose of creating a strong polarisation gradient (see 3.2) and one with a lower doping to compare the DNP efficiency of two different laser sources (see 3.3). Their pentacene concentrations have been determined by light absorption measurements to be N 1 = 12.9 × 10 23 m −3 (= 24.0 × 10 −5 mol/mol) and N 2 = 4.45 × 10 23 m −3 (= 8.3 × 10 −5 mol/mol) with 10 % relative errors.
The samples were cut into cuboids with a length along the b-axis of 8 mm and 5 mm, respectively, so that their bottom/top lies in the ac-plane (details of the crystal structure can be found in the supplementary material). With the crystal b-axis vertical the samples were clamped to a Kel-F (chlorotrifluoroethylene) rod that is attached to a sample stick fitting into an insert that is mounted in a helium flow cryostat sitting between the pole pieces of an electro-magnet. The sample stick can be rotated in order to align the pentacene X-axis parallel to the horizontal magnetic holding field B 0 , which gives a high electron polarisation [30]. This orientation of X B 0 is used throughout all experiments. A fine alignment can be done by rotating the magnet. The vertical position of the sample can be precisely adjusted to within ±0.1 mm with the help of a digital calliper. When performing DNP, the home built 4 He flow cryostat, which is especially suited for neutron experiments, was operated at 25 K.
Two laser systems have been used for photo-excitation. The light of both systems can be coupled with high efficiency into a high numerical aperture multi-mode fibre that transports the light over long distances (up to 20 m), allowing to conveniently separate the laser system from the rest of the apparatus. The fibre is attached at the other end to an optical stage at the bottom of the cryostat, which collimates the unpolarised light in vertical direction onto the sample along the crystal b-axis. In addition, the fluorescence spectrum emitted by the sample is monitored with a photo spectrometer (Avantes AvaSpec-2048) with a wavelength long pass filter to block the transmitted laser light. The line broadening of the fluorescence spectrum by crystal phonons allows a determination of the actual sample temperature during laser irradiation. In the first laser system to be shortly denoted as the 515 nm laser, the output of a diode pumped Yb:YAG disk laser (Jenlas disk IR50) is doubled in a lithium triborate (LBO) non-linear crystal. It operates at a fixed wavelength of 515 nm and is used for photo-excitation into a vibronic excited singlet state. The second system to be shortly denoted as the 556 nm laser consists of a diode pumped Nd:YAG laser (CNI HPL-556-Q 50) using the 1112 nm line that is then frequency doubled with an LBO. It operates at a fixed wavelength of 556 nm and is used for photo-excitation into a lower vibronic excited singlet state than the former. The pulse shapes of these laser systems operating at 1 kHz were measured using a diode detector (Alphalas UPD-300-UP, rise time < 300 ps) and shown in Figure 2. The pulses are scaled according to the integrated pulse energy illuminating the sample as measured by a power meter. For the 515 nm laser the pulse energy was 0.9 mJ lasting for about 0.4 μs with an estimated beam spot at the sample position of A = 36 mm 2 . The pulse energy of the 556 nm laser was determined to be about 3.2 mJ over roughly 3.2 μ s with a slightly larger beam waist and a corresponding estimated spot size of A = 60 mm 2 .
The static magnetic field was set to about 0.36 T and DNP was performed with the Integrated Solid Effect (ISE) [16][17][18]32] on the so-called high field transition with the pulsed X-band ESR system operating at 9.4 GHz and synchronised to the laser. The spectrometer also enables us to observe the ESR signal of the triplet states, which we use to precisely align the crystal samples as described in the supplementary information and to optimise the timing of the pulse DNP sequence. A saddle coil wound on a Teflon frame outside the dielectric ring resonator but inside the cavity produces the fast magnetic field sweep over a range of ±40 mT to perform ISE. To build up the proton polarisation, 10 μs microwave pulses were applied during the field sweep of ∼ 0.3 mT/μs synchronised to the laser pulse at 1 kHz repetition rate. For this coherent transfer of electron polarisation to the surrounding nuclei, an efficiency was achieved that was close to the theoretical maximum of E ISE = 0.686 [17,18,32]. A pulsed NMR system was used to monitor the proton polarisation build-up and to optimise the parameters for the ISE process. Typical ESR and NMR signals are presented in the supporting information and details on the pulsed ESR/DNP apparatus can be found in [31].

Homogeneity and absolute polarisation measurements
As noted above, the initial growth rate of the proton spin polarisation is proportional to the density of photoexcited triplet states created along the light penetration axis. This opens a way to put our model to test by performing a polarisation gradient measurement in a crystal using the method of neutron transmission. The neutron proton scattering cross section is strongly spin dependent and well determined for naphthalene [33], so that the proton polarisation can be determined with high precision measuring the transmission of polarised neutrons [24]. With a well collimated thin neutron beam the proton polarisation can thus be determined as a function of position in the sample and on an absolute scale.
To perform this measurement, the DNP system was set up at the BOA beamline [34] at the SINQ spallation neutron source at PSI. We intentionally chose the highly doped sample (N 1 = 12.9 × 10 23 m −3 ), which we expected to exhibit a strong polarisation gradient after a short period of DNP build-up. The sample was polarised by ISE for 30 min using the 515 nm disk laser at 1 kHz repetition rate and the polarisation was then measured via transmission of a thin neutron beam perpendicular to the vertical light penetration axis as a function of the penetration depth by moving the crystal vertically in equidistant steps. The neutron beam was well collimated and a small slit of 0.5 mm in height was placed in front of the crystal on the cryostat. The measured polarisation gradient after 30 min ISE is plotted in Figure 3.

Comparison between experiment and simulation
We now want to compare the measurement of the polarisation gradient with the prediction of the theory. We numerically solve the differential Equations (6) to (8) in steps of time and depth using Matlab. The solution at one time step gives the initial values for the next time step. This has the advantage that instead of an analytical expression, only a numerical form of the photon number density of the laser pulse is required. Other parameter values used for the simulations are: ISC = 0.3 [35], τ F = 19.5 ns, c = 3 × 10 8 ms −1 , n α = 1.525 and n γ = 1.945. During ISE, the cryostat is stabilised at 25K. Taking into account the laser heating estimated from fluorescence spectra, we use the light absorption coefficients values at 40 K for simulation, which are σ α = 0.79 × 10 −18 cm 2 and σ γ = 12.72 × 10 −18 cm 2 (see supplementary material).
For these experimental DNP conditions the simulation yields a triplet probability ρ T after each laser pulse. From the triplet production, we can further predict the proton polarisation build-up speed dP dt on an absolute scale. Theoretically, the build-up rate at P = 0 can be described with the formula where E ISE is the efficiency of the ISE polarisation transfer, which has a theoretical maximum value of 0.686 [17]. We set the value to 0.68 since we can optimise our ISE system very close to the optimum. Further, N I = 4.3 × 10 28 m −3 is the number density of protons (taking into account that each naphthalene molecule has 8 protons) and R the ISE repetition rate of 1 kHz. N T (= N ρ T ) denotes the triplet state number density after each laser pulse andN T the average triplet number density available during ISE due to the triplet decay. Similarly,P S denotes the average triplet electron polarisation available during ISE, due to the different decay rates of each triplet level. After photo-excitation, the initial population and decay parameters are N T 0 = 0.91 N T , N T + = N T − = 0.045 N T , τ T 0 = 61 ± 1 µs and τ T + = τ T − = 290 ± 70 µs [30]. In our case of d-pentacene in naphthalene, no electron spinlattice relaxation has been observed under our experimental conditions. Hence, and Neglecting relaxation, which is reasonable for triplet-DNP as the spin-lattice relaxation time under laser irradiation is estimated to be in the order of 40 h, the polarisation build-up should follow the differential equation dP/dt = (P S − P)/τ b , which has as solution an exponential rise P =P S (1 − e t/τ b ). Here, τ b is the rise time and P S is the electron polarisation, which is also the theoretical maximum proton polarisation that can be reached. A comparison with (12) yields which is a function of penetration depth z. The simulated polarisation gradient after 30 min ISE is plotted together with the experimental measurement in Figure 3. Considering an estimated relative error of the simulation to be at least 25 % due to the uncertainties of the pentacene concentration, the absorption coefficient, the laser power, the DNP efficiency, the ISC efficiency, the spin-lattice relaxation and non-linearities in the build-up, which will be discussed in more details later, the theoretical prediction is in good agreement with the experimental result, not only in the form but also on the absolute polarisation scale.

Triplet production and resulting DNP performance: comparison between the two laser systems
We now proceed and investigate how well our theory can describe the polarisation build-up achieved with the two available laser systems. Already the optical measurements show that the 556 nm laser excites the pentacene molecules in our sample more efficiently (see Table 2), not mentioning its much higher power. Hence its DNP performance should be much better compared to the 515 nm laser. To quantify this, the second 5 mm long (b-axis) crystal with a pentacene concentration of N 2 = 4.45 × 10 23 m −3 was polarised with both laser systems while keeping all the other parameters the same. The concentration and the sample length were chosen so that the whole crystal can be highly polarised with both laser systems. To compare the experimental results with the theory, the triplet number density N T (z) needs to be evaluated. For illustration, we plot here the probability ρ T (= N T /N) that the aromatic molecule enters the triplet state after application of a laser pulse, for our experimental conditions (Figure 4). We immediately see that the percentage of pentacene molecules ending up in the triplet state is relatively low even for the 556 nm laser that is about 3 times more efficient than the 515 nm laser. To further improve the triplet production, a more powerful laser source would be required. Note that the ability to predict the triplet number density for given experimental parameters is crucial for optimisation of DNP, but even more for fundamental studies of spin dynamics as in [14].

Comparison between experiment and simulation
As shown in the previous section, the triplet number density is reflected in the nuclear polarisation build-up, which is measured by means of NMR that determines the average polarisation of the whole crystal (16) where τ b is given by (15). Figure 5 compares the experimental values of the polarisation build-up fitted with a simple exponential rise function (left) with the simulation results using (16) (right). For the 515 nm laser, we use the same parameters as in the previous section (light absorption cross sections at 40 K due to heating, laser beam size of A = 36 mm 2 , E ISE = 0.68 and ISC = 0.3). For the polarisation build-up with the 556 nm laser, we use the same ISE and ISC efficiencies and a laser beam size of A = 60 mm 2 , but take into account that the laser generates more heating, resulting in a temperature of about 60 K during DNP with corresponding light absorption cross section values of σ α = 2.86 × 10 −18 cm 2 and σ γ = 17.05 × 10 −18 cm 2 (see supplementary material).
The simulated build-up curves are also fitted to a simple exponential rise function with the rise time shown in the figure. For both lasers we find an excellent agreement with the experimental values. Both, the simulation and the experimental measurements show that the 556 nm laser polarises the crystals about 2 to 3 times better than the 515 nm laser in terms of DNP performance. This again confirms that our model is able to simulate the triplet production well for any combination of sample and excitation light properties.

Discussion of the results
Besides the error bar of the light absorption coefficients and the pentacene concentration, many more parameters in the simulation have uncertainties. Especially the experimental parameters set in the DNP experiment are difficult to precisely quantify. These include the laser beam spot size A, the actual ISE efficiency E ISE given by the parameters of the DNP process [17,18,32], or the electron polarisationP S that depends on the sample cut and orientation.
In addition, some physical processes are not included in the simulation. Most important is spin-lattice relaxation that can have an impact on the polarisation buildup. However, the relaxation during ISE, T ISE 1 , depends strongly on the triplet density in the system, hence different laser light intensities and different pentacene concentrations will cause different relaxation times. Even within one sample, different parts of the sample will have different relaxation times. This makes it extremely hard to simulate the effect. From experience we estimate that the T ISE 1 at 1 kHz laser repetition rate is in the order of 40 h. This is much longer than the typical rise time so the effect is small. Furthermore, our model assumes that spin diffusion is fast enough to polarise the nuclei between the pentacene molecules, but too slow to change the homogeneity of polarisation over the entire crystal length. In our crystals the pentacene number density is typically in the order of N = 4 × 10 23 m −3 so the distance between the pentacene molecules is roughly l = 3 √ 1/N ∼ 13 nm. The spin diffusion time is given by t = r 2 4D , where r is the distance and D is the spin diffusion constant, which in the order of 8 × 10 −12 cm 2 s −1 [36]. Spin diffusion covers a distance of l/2 = 6.5 nm in about 1.3 ms, which is comparable to our typical pulse repetition time of 1 ms. Thus the spin diffusion is fast enough to spread the polarisation between the pentacene molecules, while it is not able to homogenise polarisation on a macroscopic scale, e.g. it takes about 1000 h for spin diffusion to cover r = 0.1 mm. This justifies our assumption.
Despite these uncertainties, the model is a comprehensive tool for describing the physical processes in our crystal system, including light absorption, triplet production and build-up of nuclear polarisation. It also gives us considerable insight into and understanding of these processes, which we can use to further investigate the spin dynamics and optimise our experiments.

Conclusion
We have introduced a theoretical model describing the photo-excitation of aromatic molecules into their triplet state by a laser beam propagating in a biaxial anisotropic single crystal. We applied this model to the case of a naphthalene single crystal doped with pentacene, in which photo-excitation creates strongly spin polarised triplet states that can then be used for DNP of the proton spins on naphthalene. To make quantitative predictions the model requires as input parameters the optical absorption cross sections of pentacene, which have been extracted from optical absorption measurements. The model furthermore requires the parameters of the two laser systems used for photo-excitation. To validate the model, after DNP neutron transmission is used to determine the distribution of the proton polarisation in the sample, and from this the density of triplet states. Next this density is compared to the results of simulations using our model. Good agreement was found. For instance, the model perfectly predicts the observed 2 to 3 fold performance increase by using the 556 nm instead of the 515 nm laser.
For applications of triplet DNP with pentacene-doped naphthalene crystals, the theory allows the identification of optimal experimental parameters depending on the priorities placed on e.g. polarisation maximum, homogeneity or build-up rate. Thus, on the one hand, a homogeneous proton polarisation of 80% can be achieved in cubically shaped crystals with volumes of > 100 mm 3 in several hours or 40% polarisation within 30 min in a highly doped 25 mm 3 crystal, as is described in detail in the supplementary material. These simulations can also be used to optimise the choice of the parameters of the laser used for photo-excitation. Moreover, the model presented above and the parameters extracted from the measurements are now available for studies of the dynamics of spin systems involving photo-excited triplet states and nuclear spins. Above we already noticed that such studies allowed for the discovery of a new mechanism explaining the fast relaxation of nuclear dipolar energy, a process that was thus far not understood. Thanks to the results reported here, it could be shown that this new mechanism not only explains this nuclear dipolar relaxation qualitatively, but also quantitatively. Future work will concentrate on further studies of these dynamics, such as the relaxation of nuclear Zeeman energy induced by photo-excited triplet states. As these studies are again complementary to those involving ground state electron spins, we may expect new insights in the processes.
It should be noted that the model is sufficiently general, that it can also be applied to other systems of aromatic molecules with a short triplet lifetime. For a given set of experimental parameters, e.g. the concentration of aromatic molecules, laser wavelength, laser power, sample temperature, etc., it can make predictions about the expected triplet density and its spatial distribution. Thus it can provide valuable guidelines for optimising a triplet DNP experiment on the one hand, and allows for the choice of the most suitable triplet systems for studies of the dynamics of spin systems on the other hand.