# A Quantum Ruler for Magnetic Deflectometry

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## Abstract

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## 1. Quantum Interference of Organic Molecules

^{5}${\lambda}_{dB}$, and hundreds or thousand times the size of the particle. At the same time a single, complex molecule can be composed of hundreds or even a thousand atoms, and each atom itself is composed of dozens of nuclei and electrons. This physical picture is complemented by acknowledging the presence of hundreds of vibrational modes and excited rotational states. At molecular temperatures around 500 K most of these modes are excited, leading to molecular rotation frequencies around $\mathsf{\Omega}\simeq 2\pi \times {10}^{9}$ rad/s and structural or conformational changes on the sub-nanosecond time scale. The molecules can thus be prepared in superpositions of position and momentum even though we can assign classical attributes such as internal temperatures, polarizabilities, dipole moments, magnetic susceptibilities and so forth to them. This philosophical aspect of macromolecular interferometry has very practical applications in metrology for the measurement of electronic, optical, and even magnetic molecular properties. Earlier work has shown that such parameters can be readily measured both in classical beam experiments [4,5] and in Talbot-Lau deflectometry [6,7,8]. Here, we propose that the de Broglie interference can also be a promising tool for photochemistry. The optically induced change of molecular geometry is often well-understood in solution, but little explored in the gas phase. We are interested in how such atomic rearrangements influence magnetic properties and study this for the example of 7-dehydrocholesterol (7-DHC) where isomerization causes a ring opening and a change of the particle’s magnetic susceptibility. We describe successful matter-wave interference with 7-DHC and a thought experiment that exploits the fact that modifications of the magnetic susceptibility will be seen as a relative shift of the de Broglie interference fringe pattern in an external magnetic field.

## 2. The Quantum Wave Nature of 7-Dehydrocholesterol

_{x}. Each slit is nominally 110 nm wide and the confinement of the molecular wavefunction in any slit suffices to expand its coherence function by several orders of magnitude further downstream. The center-of-mass wave function of wavelength ${\lambda}_{dB}$ diffracted at each slit of width $s$ thus obtains a spatial coherence ${W}_{c}\simeq 2{\lambda}_{dB}L/s$, which grows with distance to the source, such that the center-of-mass coherence function spreads over the extension of at least two slits when arriving at the second grating. The standing light wave G2 is obtained by retro-reflecting a 532 nm laser beam at a plane mirror. In the antinodes of the grating the light shifts the phase of the transmitted matter-wave mainly by the optical dipole potential, but the full quantum model includes absorption of photons as well [17]. At the center of the Gaussian laser beam, this phase depends on the power P and the vertical beam waist ${w}_{x}$ of the laser, as well as on the molecular optical polarizability $\alpha $ (532 nm), and the forward velocity of the molecule ${v}_{z}$. The coherent evolution of the molecules in phase-space leads to the formation of a molecular density pattern of period d = 266 nm, which can be sampled by the mechanical mask G3. This pattern forms periodically along the beam line and consecutive patterns are separated by the Talbot length ${L}_{T}={d}^{2}/{\lambda}_{dB}$ [18]. Tracing the number of transmitted molecules as a function of the position of G3, one finds a nearly sinusoidal fringe pattern, as shown in Figure 2a with a visibility V = (S

_{max}− S

_{min})/(S

_{max}+ S

_{min}), where S

_{max}and S

_{min}are the maximal and minimal count rates.

## 3. Photo-Switching

^{2}to be also a good approximation for molecules in the gas phase at T = 450 K. Recent experiments with photo-cleavable peptides showed that ultraviolet (UV) absorption cross sections of molecules in this complexity range can be comparable in the gas phase and in solution [31]. When a $v=100$ m/s fast 7-DHC molecule traverses a gaussian laser beam of power P and waist ${w}_{0}=0.3$ mm it will absorb on average $n=\frac{2P}{\pi {w}_{0}}\frac{\lambda}{hc}\frac{{\sigma}_{abs}}{v}$ photons. The average $n=1$ is reached for $\lambda =$ 266 nm and $P=$ 40 W. Single pass frequency doubling of a green solid state lasers can reliably generate ultraviolet light of P = 1 W and a power enhancement of 50–80 is conceivable in low finesse UV cavities, even in high vacuum where UV optics often suffer from outgassing [32]. Also, commercial high-power nanosecond lasers can produce up to 30 W average power at 266 nm and even 200 W at 355 nm, with repetition rates of 100 kHz. This is sufficient to ensure that all molecules interact with the laser beam. Positioned before the first grating, one or even two photoisomerization processes can be completed before the molecules enter the interferometer region. The following considerations focus on the feasibility of detecting such state changes via an interferometric monitoring of a change in molecular magnetism.

## 4. Magnetic Manifestations of Molecular Photoisomerization in the Gas Phase

**J**. The magnetic moment ${\mathit{\mu}}_{J}$ interacts with the flux density

**B**and experiences an orientation-dependent force $\mathit{F}=-\text{}\nabla \text{}\left({\mathit{\mu}}_{\mathit{J}}\text{}\mathit{B}\right)$, which will pull an aligned magnetic dipole towards the field maximum and push the anti-aligned particle away. A thermal beam of molecules with random orientations of their figure and rotation axes will therefore be broadened, when exposed to a B-field gradient. In matter-wave interferometry, this broadening will reduce the interference fringe contrast. This resembles the observations for electric dipole moments in electric fields [8,42,43]. In the gas phase first order paramagnetism will always dominate over all other magnetic effects, unless the magnetic dipole moment vanishes. In the following we focus on those molecules, with

**J**= 0 in the ground state.

^{13}C with an abundance of 1.1%. In natural fullerene for instance, 48% of all C

_{60}molecules hold at least one nuclear spin and 10% even exactly two. In 7-DHC, still 26% of all molecules hold at least one nuclear spin. Since

^{13}C has a nuclear spin of ½ and a nuclear magnetic moment of ${\mu}_{C13\text{}}=$ +0.7${\mu}_{N}$ the nuclear response will be about two thousand times weaker than that of a single unpaired electron, but nuclear paramagnetism can actually be comparable to electron diamagnetism or van Vleck paramagnetism and must not be ignored for

**J**= 0.

^{12}C

_{60}fullerenes whose magnetic response represents a lower limit to most of the interesting aromatic molecules. The molar magnetic susceptibility of C

_{60}has been measured to be ${\chi}_{C60}=-1.08\times {10}^{-9\text{}}{\mathrm{m}}^{3}\xb7{\mathrm{mol}}^{-1}$ [44]. This translates into a molecular magnetic polarizability of ${\beta}_{C60}=1.4\times {10}^{-27}{\text{}\mathrm{Am}}^{4}\xb7{\mathrm{V}}^{-1}\xb7{\mathrm{s}}^{-1}$. For L = 0.2 m, ${L}_{1}=\text{}0.04$ m, ${L}_{2}=$ 0.04 m, $v=100$ m/s, m = 720 amu, and K = 0.003, the interference fringe can be shifted by about 25 nm for $(\mathit{B}\nabla ){B}_{x}$ = 70 T

^{2}·m

^{−1}. If a field of that order of magnitude can be prepared, the fringe shift can still be resolved, the interferometer can still be sensitive to ${\chi}_{C60}$. The case of fullerene C

_{60}gives a conservative limit, since the deflection depends on the magnetic polarizability-to-mass ratio $\beta /m$. For example, the fully aromatic molecule benzene C

_{6}H

_{6}exhibits five times greater $\beta /m$.

## 5. Design of the Required Magnetic Structures

^{2}/m that is homogeneous within 2% of its peak value across an area of 1000 × 200 µm

^{2}; i.e., across the full molecular beam profile inside a KDTL interferometer.

## 6. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of Kapitza–Dirac–Talbot–Lau (KDTLI) interferometry, including the proposed extensions for magnetic deflectometry. The molecules evaporate to form a molecular beam in high vacuum. The molecular beam velocity is selected by its free-fall parabola using three horizontal slits. The molecular v-distribution can be recorded by chopping the beam in a pseudorandom sequence and measuring its arrival time at the quadrupole mass detector. The KDTLI comprises two nanofabricated absorptive masks, G1 and G3, and one optical phase grating G2. A tailored magnetic field (Halbach magnet) can exert a homogeneous force onto the molecules and deflect the molecular beam in proportion to the particles’ magnetic susceptibility. If the molecules exhibit a permanent magnetic dipole moment, the interference fringes will broaden, and contrast will be reduced.

**Figure 2.**(

**a**) Matter-wave interference of 7-dehydrocholestorol with a molecular beam velocity v

_{mean}= 212 ± 78 m/s (FWHM). The dots show the molecular count rate at the respective position of the third grating, and the continuous line is a sinusoidal fit to the data exhibiting a fringe contrast of 23.1 ± 1.5%. The grey shaded area indicates the dark counts of the detector; (

**b**) The interference contrast varies with the laser power in the diffraction grating G2, following the line shape of the quantum model. We compared the achieved fringe contrast to the theoretical maximum by calibration measurements with the well characterized fullerene C

_{60}and found a reduction of 10%, which we attribute to grating misalignment. This is still well compatible with fringe-assisted molecule metrology.

**Figure 3.**The photoisomerization (1) from 7-dehydrocholesterol (7-DHC, molecular weight MW = 384 amu) to previtamin D3 is well understood in solution, but little studied in the gas phase. This is also true for the spontaneous isomerization (2) from previtamin D3 to vitamin D3 (cholecalciferol).

**Figure 4.**Finite element simulation of the modified Halbach cylinder. (

**a**) Magnetic flux: the arrows show the direction of magnetization of the individual segments; (

**b**) magnetic force field: the deflection of a molecular beam is proportional to $(\mathit{B}\nabla ){B}_{x}$. The diameter of the magnet is 55 mm with an inner bore of 16 mm. The white rectangle indicates the location of the molecular beam, where the force is constant within 2%.

**Figure 5.**Spiropyran can isomerize to merocyanine upon absorption of a UV photon. This opens one ring which we expect to significantly change the magnetic susceptibility. In contrast to the case of 7-DHC, the process changes the electric dipole moment here by a large factor, from 7 Debye for spiropyran [46] to between 20–50 Debye for merocyanine [26]. Such huge changes will be easily detectable in interferometric electric deflectometry [8].

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**MDPI and ACS Style**

Mairhofer, L.; Eibenberger, S.; Shayeghi, A.; Arndt, M. A Quantum Ruler for Magnetic Deflectometry. *Entropy* **2018**, *20*, 516.
https://doi.org/10.3390/e20070516

**AMA Style**

Mairhofer L, Eibenberger S, Shayeghi A, Arndt M. A Quantum Ruler for Magnetic Deflectometry. *Entropy*. 2018; 20(7):516.
https://doi.org/10.3390/e20070516

**Chicago/Turabian Style**

Mairhofer, Lukas, Sandra Eibenberger, Armin Shayeghi, and Markus Arndt. 2018. "A Quantum Ruler for Magnetic Deflectometry" *Entropy* 20, no. 7: 516.
https://doi.org/10.3390/e20070516