Magnetic Resonance Imaging of Granular Materials Ralf Stannarius Institute of Experimental Physics, Otto-von-Guericke-University, Universit¨atsplatz 2, D-39106 Magdeburg, Germany (Dated: March 6, 2017) Magnetic Resonance Imaging (MRI) has become one of the most important tools to screen humans in medicine, virtually every modern hospital is equipped with an NMR tomograph. The potential of NMR in 3D imaging tasks is by far greater, but there is only ’a handful’ of MRI studies of particulate matter. The method is expensive, time-consuming, and requires a deep understanding of pulse sequences, signal acquisition and processing. We give a short introduction into the physical principles of this imaging technique, describe its advantages and limitations for the screening of granular matter and present a number of examples of different application purposes, from the exploration of granular packing, via the detection of flow and particle diffusion, to real dynamic measurements. Probably, X-ray computed tomography is preferable in most applications, but fast imaging of single slices with modern MRI techniques is unmatched, and the additional opportunity to retrieve spatially resolved flow and diffusion profiles without particle tracking is a unique feature. I. INTRODUCTION peting very successful imaging method. Hospitals soon became interested to spend large sums for medical imag- ing, where X-ray Computed Tomography (X-ray CT) and Nuclear Magnetic Resonance (NMR) imaging is a ma- MRI became strong competitors. ture but still rapidly developing technique that has an ample range of clinical applications today. It is impres- Today, contemporary MRI equipment achieves re- sive to recall the history of this imaging method during markable benchmarks. Structural MRI nowadays has the the past 80 years: The first observation of the absorption sensitivity to detect microliter volumes. Special tech- of electromagnetic waves by nuclear spins in a molec- niques can also encode local velocities in the recorded ular beam was reported in 1937 by Isidor Isaac Rabi signals and map local fluid flow. Diffusion tensor imag- (1898-1988), who modified the classical Stern-Gerlach ing (DTI) can quantify diffusion in fluids and thereby experiment by adding an oscillating magnetic field [1]. motion on a scale much below the size of a voxel (volume In 1944, he received the Nobel Prize for this discov- element of the tomogram). Spatially resolved diffusion ery. The method soon proved to be very useful for tensors can be reconstructed. Functional MRI (fMRI), as the study of magnetic and mechanic properties of nu- another example, allows to examine brain activities from clei. The precise measurement of resonance frequencies blood flow in the cortex, this is commonly achieved with of nuclei laid the foundation for countless applications, blood-oxygen-level dependent (BOLD) contrast measure- from atomic and molecular beam experiments to Masers ments. and contemporary NMR imaging. Edward Mills Purcell Many of these highly sophisticated methods have the (1912-1997, Nobel Prize 1952) and Felix Bloch (1905- potential to explore ensembles of granular particles as 1983, Nobel Prize 1952) independently discovered the well. Among the interesting problems for structural magnetic resonance phenomenon in condensed matter in NMR imaging is the revelation of packing structures, seg- 1946 [2, 3]. Progress in the field grew steadily, Herman regation, orientational order of grains and the composi- Y. Carr (1924-2008) was the first to apply defined mag- tion of granular mixtures. Flow measurements with rheo- netic field gradients to achieve a (one-dimensional) spa- NMR [8] can be useful in slurries or fluids inside porous tial mapping of nuclear magnetic spins. This discovery structures. Furthermore, grain displacement caused by would probably not have had the giant impact we state agitation can be mapped. arXiv:1703.01211v1 [cond-mat.soft] 3 Mar 2017 today, had not Raymond Damadian (born 1936) demon- Before we describe applications in particle imaging, strated that tissues and tumors can be distinguished by it will be instructive to introduce the basic physics of the relaxation times of their nuclear spins [5]. Medical NMR and principles of MR Imaging. The fundamentals application boosted the development of the method. Paul of NMR are found in quantum mechanics. However, for Lauterbur (1929-2007, Nobel Prize 2003) and Sir Peter most of the techniques described in this overview, only Mansfield (born 1933, Nobel Prize 2003) substantially macroscopic physical quantities of large spin ensembles contributed to the development of Magnetic Resonance are relevant. We can therefore resort to classical, phe- Imaging (MRI), the former presented the first MR im- nomenological models and get around the quantum me- ages of a living mouse [6], the latter achieved immense chanical description. Focus is laid on information needed progress in image reconstruction procedures, multipulse for applications in grainy matter and particle imaging. NMR, and the mathematical analysis of the radio sig- The introduction shall help the reader to decide whether nals. Efforts to establish MRI for clinical applications MRI may be useful to solve a certain particle imaging were further triggered by Hounsfield’s introduction of X- problem, and which technique may give optimal results. ray based computed tomography [7] in 1973 as a com- We will then discuss typical applications in granular mat- 2 ter research, and finally compare advantages and draw- principle choose other nuclei, but tomographs are usually backs of MRI to competitive techniques. tuned to the proton resonance frequency and other iso- topes represent nonstandard applications. Even though they may be used in principle, all published MRI studies II. BASICS OF NMR AND MRI in granular matter physics (except a few applications of hyperpolarized Noble gases mentioned below) have em- 1 A. Nuclear spins, nuclei suitable for NMR ployed H resonance. The origin of the electromagnetic signal in NMR are nuclear spins, which are related to a very small mag- netic moment. The spin of a nucleus is composed of the spins of the individual nucleons. The simplest atom, B. Sensitivity 1H, with only one proton, has the spin quantum num- 1 ber I = 1/2. The H nucleus has two energy levels, The major problem of NMR signal detection is the very − ± E = γIzB = ¯hγB/2, with the gyromagnetic ratio low population difference between the Zeeman levels, be- γ. Transitions between these two levels are related to cause of their low energy difference, e. g. ∆E ≈ 7·10−6kT emission/absorption at the Larmor frequency at room temperature for protons at fields of the order of ω γ one Tesla. Thus the difference in the occupation of the ν = L = B. (1) L 2π 2π Zeeman levels is only a handful of spins in one million. We mention in passing that hyperpolarization of Noble For the proton, γ =2π · 42.58 MHz/T. In nuclei consist- gases like 129Xe or 3He by optical pumping (see, e. g. ing of multiple nucleons, the individual spins may pair- Ref. [9]) can serve as a sophisticated but effective means wise compensate each other, so that not all isotopes pos- to achieve huge excess populations (up to 70%) of one sess a magnetic moment. By far most frequently used 1 level, which can increase the NMR signal by several or- for NMR experiments is the hydrogen H nucleus. Ta- ders of magnitude. Such techniques have been success- ble I lists some of the most common isotopes for NMR fully applied for dynamic flow measurements in granular spectroscopy. Nuclei with both even proton numbers and matter [10]. neutron numbers (like the naturally highly abundant 12C 4 ~ or He) have spin zero and are NMR-’invisible’. In the following, we will focus on the magnetization M, which in equilibrium is the sum of the nuclear magnetic moments ~µ = γIz~ez of the nuclei. The equilibrium value Isotope Spin Relative NMR frequency [MHz] M0 in a magnetic field B0 is abundance [%] (at 2.349 T) 1 2 H 1/2 99.98 100.000 Nγ2¯h I(I + 1) − M0 = B0. (2) 2H 1 1.5·10 2 15.35 3kT 3H 1/2 0 106.67 − 3He 3/2 1.3·10 4 76.18 Here, k is the Boltzmann constant, T the temperature of 13 the spin system, N the number of spins. Higher mag- C 1/2 1.108 25.15 netic fields provide higher sensitivities because of the 14N 1 99.63 7.227 increased population density difference (proportional to 23 B0) and higher energy of the radiation (linear with B0). Na 3/2 100 25.13 1/2 27 The noise level also increases (approximately with B0 ) Al 5/2 100 26.06 because of the larger bandwidth in signal detection, thus 29Si 1/2 4.7 19.87 the signal to noise ratio (SNR) improves only with a lower 2 31 power than B0 . Often, the SNR is assumed proportional P 1/2 100 40.49 3/2 to B0 . The conclusion is that for sensitivity reasons, 129Xe 1/2 26.4 27.81 it is generally advantageous to use high B0 fields. There are a few exceptions. For example, in grainy materials with large inhomogeneities of the magnetic susceptibility TABLE I: Table of natural abundances and Larmor frequen- it can be useful to reduce the involved statistical field cies of a selection of typical isotopes. Because most of these gradients by choosing lower B0. Today, typical NMR isotopes have lower resonance frequencies than the 1H isotope, magnets provide fields from about 1 T to 23.5 T (the 1 H is almost exclusively the nucleus of choice in MR imaging. latter corresponding to a Larmor frequency of ≈ 1 GHz). Contemporary human MR scanners work at B0 fields up Medical MRI almost exclusively uses the 1H isotope. to 9.4 T (400 MHz), and devices with up to 11.7 T are un- It is abundant in living bodies, mostly in water and fat der installation.