DETECTING BODY CAVITY BOMBS WITH NUCLEAR QUADRUPOLE RESONANCE

Transcription

1 DETECTING BODY CAVITY BOMBS WITH NUCLEAR QUADRUPOLE RESONANCE AThesisPresented By Michael London Collins to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Master of Science in the field of Electrical Engineering Northeastern University Boston, Massachusetts August, 2014

2 1 ABSTRACT 1 Abstract Nuclear Quadrupole Resonance (NQR) is a technology with great potential for detecting hidden explosives. Past NQR research has studied the detection of land mines and bombs concealed within luggage and packages. This thesis focuses on an NQR application that has received less attention and little or no publicly available research: detecting body cavity bombs (BCBs). BCBs include explosives that have been ingested, inserted into orifices, or surgically implanted. BCBs present a threat to aviation and secure facilities. They are extremely di cult to detect with the technology currently employed at security checkpoints. To evaluate whether or not NQR can be used to detect BCBs, a computational model is developed to assess how the dielectric properties of biological tissue a ect the radio frequency magnetic field employed in NQR (0.5-5MHz). The relative permittivity of some biological tissue is very high (over 1,000 at 1MHz), making it conceivable that there is a significant e ect on the electromagnetic field. To study this e ect, the low-frequency approximation known as the Darwin model is employed. First, the electromagnetic field of a coil is calculated in free space. Second, a dielectric object or set of objects is introduced, and the free-space electric field is modified to accommodate the dielectric object ensuring that the relevant boundary conditions are obeyed. Finally, the magnetic field associated with the corrected electric field is calculated. This corrected magnetic field is evaluated with an NQR simulation to estimate the impact of dielectric tissue on NQR measurements. The e ect of dielectric tissue is shown to be small, thus obviating a potential barrier to BCB detection. The NQR model presented may assist those designing excitation and detection coils for NQR. Some general coil design considerations and strategies are discussed. ii

3 CONTENTS Contents 1 Abstract ii 2 Introduction 1 3 Basic NQR Theory Purpose Quadrupole Moment Energy Splitting Exciting and Detecting Transitions Principle Echoes and Relaxation Times NQR as a Detection Tool Introduction The Potential of NQR NQR Challenges and Current Research Free Space NQR Model Shinohara s Model Powdered Samples Calculating the NQR Signal An improved method for calculating NQR signal Programming the Model Validating the Model Experimental Hardware - NQR Spectrometer The Experiment Results and Discussion Utility of the Free Space Model Sensitivity Map Signal from example targets Coil Design Considerations Desirable qualities Constraints Methodology of Design NQR in Biological Tissue - Background 35 iii

4 LIST OF FIGURES 8 Correcting for Dielectric Media The Darwin Model The Algorithm Summary Calculating Electric Field - Jacobi Method Calculating the E ect of Dielectric Media with Axial Symmetry Calculating the initial electromagnetic field Defining the Dielectrics Finding the correction charges and current density Calculating the Correction to Magnetic Field Results and Discussion Generalization to Arbitrary Geometry Concept Free Space Fields Defining Objects Correction Charge and Current Correction Magnetic Field The e ect of frequency Other Coil Designs Nonuniform Grid Conclusion Implications for BCB detection by NQR A Code for calculating correction electric field 80 B Code for calculating correction magnetic field 85 List of Figures 1 Nuclear Orientation Quadrupole Visualization Energy Level Diagram NQR Spectra of Various Explosives Nuclear Magnetization - Bessel Function Spectrometer Front End iv

5 LIST OF FIGURES 7 Solenoid and NaNO Nutation Curves Plane of Interest Gradiometer Sensitivity Maps for a Gradiometer Diagram of Cylindrical Target in Two Possible Orientations Nutation Curves for Two Cylinder Orientations Permittivity of Biological Tissue Correcting Electric Field - Part Correcting Electric Field - Part Magnetic Flux Density of a Circular Coil Electric Field of a Circular Coil Magnetic Flux Density of a Circular Coil - Planar View Sample Object Edge Detection Correction Current Density Simulated Liver Tissue Correction Magnetic Field Relative Correction Field Magnetization Response to Changes in Nutation Angle Magnetic Flux Density - Cartesian Defining Dielectric Objects Current Density - Cartesian Correction Magnetic Field - Cartesian E ect of Frequency on Correction Magnetic Field Logarithmic Spiral Coil Gradiometer Made of Compressed Linear Spirals Nonuniform Grid Construction v

6 2 INTRODUCTION 2 Introduction At airports, o cials are responsible for detecting contraband such as knives, firearms, explosives, and illegal drugs. They employ an array of technologies in this e ort, each with advantages and drawbacks concerning cost, speed, and the kind of contraband which each is capable of detecting. Metal detectors are good at detecting large metal objects such as conventional knives and firearms but are incapable of finding ceramic or plastic weapons, explosives, or drugs. More recently, whole body imaging (WBI) technology has been implemented. WBI systems use X-rays, mm waves, or terahertz waves to penetrate clothing and form an image of the person being screened. Computer algorithms or trained operators then examine these images to identify anomalies which indicate the presence of concealed objects. WBI systems are capable of detecting and localizing threats of all kinds that are concealed beneath clothing. WBI systems are the focus of current research aiming to improve their e cacy and lower their cost. However, the electromagnetic radiation used in WBI is not capable of penetrating skin deeply, and so any contraband concealed inside of a person will remain invisible to these methods despite ongoing improvements. The final technology in widespread use is trace detection. Trace detectors are sensitive to the tiny amounts of explosive that are likely to be present near bombs or the humans who handled them. While the technology has a lot of promise, it may be ine ective against hermetically sealed explosives that are handled with high care. Currently, the largest gap in personal screening systems is the ability to detect nonmetallic objects that have been swallowed, surgically implanted, or inserted into orifices. While illegal drugs have been smuggled in this manner for a long time, the problem of detecting objects concealed inside the body has taken on new urgency with the rise of terrorism. Terrorists may use Body Cavity Bombs (BCBs) to attack individuals or airplanes [9,10]. For example, a Europol report concluded that the attempted assassination of Muhammad bin Nayef in 2009 may have used a BCB. While some known search methods could be e ective at detecting BCBs, they are inappropriate for use at checkpoint security. For example, transmission X-ray systems produce too much radiation, and cavity searches are a violation of privacy. Nuclear quadrupole resonance (NQR) may be capable of detecting BCBs safely and non-invasively. This thesis begins by introducing NQR from a theoretical standpoint. It then considers NQR from a practical point of view and examines its potential for detecting BCBs. Finally, it analyzes how body tissue may a ect NQR measurements. 1

7 3 BASIC NQR THEORY 3 Basic NQR Theory 3.1 Purpose Nuclear quadrupole resonance is a quantum phenomenon in which a nucleus transitions between discrete energy levels, absorbing and emitting photons in the process. While a complete description of NQR at the level of quantum mechanics is beyond the scope of this thesis, this section introduces the main concepts that are necessary to use NQR for material detection and identification. Examining the nucleus and its electric environment using classical means goes a long way toward describing NQR. We will begin with the classical description to develop an intuitive understanding of NQR and then declare the conclusions of the full quantum mechanical treatment when necessary. Further discussion of NQR theory can be found in the excellent works of [15, 46, 88, 90, 98]. 3.2 Quadrupole Moment This discussion of NQR begins with an understanding of the quadrupole which is at the heart of the phenomenon. Consider a nucleus to be a distribution of positive charge with density, inspacewhichhaselectricfieldpotential. is external to and independent of the nucleus. Classically, the electrostatic energy, V, is given by integrating over the volume of the nucleus: Z V= d (1) The field potential varies slowly over the scale of a nucleus, so it may be well represented using a Taylor s series expansion in Cartesian coordinates: V= 0 Z d + i=1 i 0 r i d X 3X i=1 j 0 Z r i r j d +... (2) Here, (x 1,x 2,x 3 )=(x, y, z), (r 1,r 2,r 3 )arethedistancesfromtheoriginalong(x, y, z), and the symbol 0 indicates evaluation at the origin. The origin is chosen to be center of the nucleus. The three terms of Eq. 2 represent the electric monopole, dipole, and quadrupole moments contributions to the nucleus energy. Each will be considered separately. Experiments and quantum theory have shown the dipole contribution to be equal to zero [41, 90]. Intuitively, this can be explained through a concept known as definite parity wherein protons are just as likely to appear on one side of the nucleus as the opposite side, canceling any potential dipole terms [18]. Definite parity also precludes the existence of 2

8 3 BASIC NQR THEORY any higher multipole terms with 2 n -poles where n is odd. In general, other high multipole terms do exist, but are so weak, that they do not need to be considered in simple NQR experiments [98]. For the nucleus which will be the focus of this thesis, 14 N, there are no multipole moments higher than quadrupole [41]. Accordingly, we drop these terms from Eq. 2. The quadrupole moment term of Eq. 2 can be simplified by choosing a coordinate system in which the cross terms, i 6= j, vanish. Onesuchcoordinatesystemisdefinedby the electric field gradient (EFG) and is called the EFG principal axis system, an idea which will be returned to shortly. The second coordinate system, which we shall use presently, is defined by the nucleus orientation. This nuclear coordinate system is referred to here as (x 0,y 0,z 0 ), where the nucleus is symmetric about the z 0 axis. This symmetry causes the cross terms to cancel [41]. Adopting these coordinates, Eq. 2 becomes: V= 0 Z d Z rx 0 d 02 0 Z ry 0 d 02 0 r 2 z 0 d (3) Since potential varies slowly over space, the monopole term of Eq. 3 is dominant. Energy due to the monopole term is minimized when the nucleus occupies a position where potential,,islowest. Anucleuswithinasolidormoleculewillcometorestatanequilibriumposition where the electric field is zero [88]. Since the monopole term is constant for any one location, it is dropped in order to focus on the quadrupole term: In Eq. V= Z Z rx 0 d 02 0 Z ry 0 d 02 0 r 2 z 0 d 4, it is clear that energy depends on the second spatial derivative of potential. Since potential is the integral of electric field, its second derivative is simply the electric field gradient (EFG). To interpret the EFG s e ect on the nucleus, consider the case of a spherical nucleus. In that case, the integrals of Eq. 4 would equal one another: V= Z (4) rz 2 0 d (5) Since the EFG is established by fields external to the nucleus, and the nuclear charge distribution is very small, LaPlace s equation can be used. Note that in special cases, Poisson s equation is used instead [88], but here we apply LaPlace s equation: r 2 =0 (6) 3

9 3 BASIC NQR THEORY Then Eq. 5 reduces to zero and the nucleus energy is independent of the EFG. However, some nuclei are not spherically symmetric, but instead possess an oblate or prolate distribution of charge. The energy of these nuclei depends upon their orientation to the EFG. For a simplified view, consider Figure 1 in which a prolate nucleus is located at a point where the electric field is zero but the gradient is nonzero. Note how the nucleus energy depends upon its orientation. Figure 1: A cigar-shaped charge distribution represents a nucleus in the presence of positive and negative charges +q and q. If the nucleus is positive, orientation b has lower energy than orientation a even though the externally defined electric field and nucleus position is the same in both cases. Figure from Slichter [88]. It is also helpful to visualize the decomposition of a nuclear charge distribution into its monopole and quadrupole terms as shown in Figure 2. Figure 2: An oblate nucleus (left) is visually decomposed into its monopole term (right) and quadrupole term (center). Figure from Miller and Barrall [58]. The nucleus quadrupole moment can be classified by a single scalar parameter. To do so, we again rely on nuclear symmetry about the z 0 axis, which gives: Z r 2 x 0 d = Z ry 2 0 d (7) so Eq. 4 becomes: V= r 2 z 0 d Z r 2 x 0 d (8) 4

10 3 BASIC NQR THEORY and via LaPlace s equation: V= V= r 2 z Z r 2 x 0 d (rz 2 0 r2 x0) d (9) To arrive at the standard result, it is necessary to convert distance along the x 0 -axis, r x 0, into the total distance from the origin, r 0 : r 0 = r 2 x 0 + r2 y 0 + r2 z 0 =2r2 x 0 + r2 z 0 (10) V= Z (3r 2 z 0 r02 ) d (11) Normalizing by the charge of a proton, e, anddroppingtheefgterm,wearriveatthe scalar nuclear quadrupole moment, Q, a property of the nucleus which measures how oblate (squashed) or prolate (stretched) the nucleus is about its axis of symmetry: Q = 1 Z (3rz 2 e 0 r02 ) d (12) 3.3 Energy Splitting EFG Principal Axis System (PAS): So far, we have been using axes defined by the nucleus. Another coordinate system which is useful is defined by the EFG and is called the EFG principal axis system (PAS), or sometimes the molecular or crystal coordinate system. The EFG s physical origin is the collection of charges that are external to the nucleus. This includes the electrons immediately surrounding the nucleus, other nuclei and electrons within the molecule, and even adjacent molecules. As such, the EFG is highly specific not only to the chemical bonds of an atom but to the overall crystalline structure. The EFG is therefore a ected by factors such as temperature and crystalline inhomogeneities. The EFG and its interaction with the quadrupole moment are partially explainable through both valence bond theory and molecular orbital theory [41]. To define the EFG PAS quantitatively, we once again examine the components of the EFG, evaluated at a particular nuclear site. In general, for a cartesian coordinate system 5

11 3 BASIC NQR THEORY (x 1,x 2,x 3 ), i x j 6=0 i 2{1, 2, 3},j 2{1, 2, 3} (13) For i 6= j, aparticularcoordinatesystemcanalwaysbefoundinwhich ij =0[46,64]. This is is the PAS, and its axes are designated (x, y, z). By convention, (x, y, z) arenamedsuch that zz yy xx. zz is given the special name eq. The asymmetry of the PAS is defined by [108]: = xx yy eq (14) Including Quantum E ects: So far, our description has been entirely classical and is useful for understanding how the nucleus orientation to the EFG a ects its energy. However, the description is incomplete because quantum e ects dictate that only certain orientations are permitted. This leads to discrete states with quantized energy. To identify these states, it is necessary to formulate the system s Hamiltonian, from which the discrete energy levels, or eigenvalues, can be taken. When the EFG is axially symmetric ( = 0), the Hamiltonian is given by substituting the spin angular momentum operator into the classical expression for energy [41]. In this case, the xx and yy terms can be ignored. This leads to energy eigenvalues: E m = eq zz 0 4I(2I 1) 3m2 I(I +1) = e 2 qq 4I(2I 1) 3m2 I(I +1) (15) where eq is given by Eq. 12, I is the spin quantum number of the nucleus, and (m = I, I +1,...,I 1,I). Spin quantum number is a property of the nucleus. The above equations are only valid for spin quantum numbers I>1/2 sincethosewithi apple 1/2 do not possess quadrupole moments directly observable by NQR (they may, however, possess intrinsic quadrupole moments observable by other experiments [46]). It is interesting that even after taking quantum mechanics into consideration, the result is expressible in terms of classically derived scalars eq, eq, and [88]. Transition Frequencies: NQR involves inducing and observing transitions between eigenstates. These transitions are associated with discrete amounts of energy, E = E m1 E m2. Transition energies are related to frequency via Planck s relation, E = hv, wheree is energy, h is the Planck constant, and v is frequency. These are the NQR frequencies for a particular nucleus in a particular EFG. The next section will discuss the mechanisms of excitation and detection, but first, we take a closer look at transition frequencies. 6

12 3 BASIC NQR THEORY For the remainder of this thesis, we will focus on one nucleus, 14 N, since it is ubiquitous in conventional explosives such as TNT, RDX, and PETN, and has been the subject of much research. 14 NhasI =1,andsoweexpectthreeeigenstates(m = 1, 0, 1) as well as three transition frequencies associated with the di erences between states. Calculating the eigenstates with Eq. 15, we see that the m = 1andm =1statesaredegenerate,leaving only a single transition frequency associated with the transition from m = 0toeitherm = 1 or m = 1andviceversa.Thistransitionfrequencyiscalled Q : Q = 3e 2 qq 4I(2I 1)h (16) Eq. 16 is only valid when =0and Q depends only on zz as a result (recall that this assumption was made when obtaining Eq. 15). To account for the asymmetrical nature of EFGs encountered in real life, a quantum mechanical derivation must be used. Closed form solutions are known only for I =1,andI =3/2. While derivations are beyond the scope of this thesis, their result is satisfyingly simple and can be written in terms of the parameters already attained. We look at the I = 1 solution since it applies to 14 N. For I =1,EFGasymmetrybreaks the degeneracy of the m =1andm = 1levels. Thisgivesthreestatesandthreetransition frequencies: 0 = 2 3 Q, ± =(1± 3 ) Q (17) Figure 3a depicts the three energy levels for 14 Nandassociatedtransitionfrequencies. Note that transition frequencies are sometimes expressed in angular form, i =2! i. 3.4 Exciting and Detecting Transitions Principle Coupling to the nucleus: In NQR, energy is applied to a nucleus in order to excite a transition to a higher energy state. In principle, the applied energy could take the form of an artificially generated EFG that interacts with the quadrupole moment directly. The large EFG that would be required makes this an impractical approach [17]. Some work has been done generating an EFG indirectly with acoustic energy, a technique known as NAR [98]. In NQR, however, the energy is applied as a radio frequency (RF) magnetic field. Nuclei spin about their axis of symmetry, z 0. This essentially forms a current loop since the nucleus is made of positively charged particles. A magnetic dipole moment therefore 7

13 3 BASIC NQR THEORY Figure 3: For an I =1nucleus in an EFG with =0, there are only two energy states and a single associated transition frequency, as shown in b. When asymmetry is introduced, the degeneracy is broken, and three energy states become possible. The angular transition frequencies! x,! y, and! z are associated with the di erence between energy levels. Figure from Lee [46]. arises along z 0. Nuclei with spin quantum number greater than 0 exhibit this property. When an external magnetic field is applied, the nucleus experiences torque through its dipole moment. To excite nuclei to higher energy states, magnetic fields can be applied at the proper transition frequencies which couple to the dipole moment. In a simple implementation, an RF magnetic field is applied by a transmitting coil as ashortpulseoflength which excites nuclei to a higher energy state. After the pulse has turned o, nuclei reorient themselves to their ground state. During this reorientation, the moving magnetic dipole moment generates a field which may be detected by a receiving coil (typically the same one used for excitation). For an axially symmetric EFG, Bloom et al. provide a geometric description of the nucleus motion (precession) following excitation by a magnetic pulse [6]. However, for the general case, no simple visualization exists that can be explained in classical terms [15]. Nevertheless, there are some simple geometric principles critical to a basic understanding of NQR detection systems. Direction of applied RF field and nuclear magnetization: To predict the e ect of an RF pulse on a nucleus, physicists add a Hamiltonian term (due to the RF field) to the original Hamiltonian due to the quadrupole term [15, 46]. A result of this analysis is that the direction of the applied RF pulse is critical to its e ect on the nucleus. The applied field may be broken down into its components along the PAS axes (x, y, z). Each component has the potential to excite a single one of the three possible transitions discussed above. If we call the applied field B, B z may excite 0,B y may excite,andb x may excite + [46, 82]. 8

14 3 BASIC NQR THEORY Excitation will only occur if the applied field frequency su ciently matches its associated transition. After a resonant pulse has ceased, the nucleus behaves as a magnetic dipole oscillating linearly along this same direction. So for example, if a magnetic field pulse is applied at + along the x direction of the PAS and then ceases, the nucleus will temporarily behave as a dipole oscillating along the x axis. If the original pulse were applied at the same frequency yet had both an x and y component, only the x component would a ect the nucleus. Nuclear magnetization in a single crystal: The total oscillating magnetic dipole may be quantified as an ensemble of the dipoles due to each nucleus being excited. Take the example of a single crystal composed of many identical molecules aligned in the same direction. Let us select a particular nuclear site within the molecule (a specific 14 Nnucleus,for example). The PAS of that nucleus will align with the PAS of the same nuclear site within every other molecule in the crystal. In this case, the nuclei s net magnetization, M s,canbe easily written as the sum of each nucleus magnetic dipole moment [86]. Assuming that a magnetic field of amplitude B is applied directly along an axis of the PAS at the associated resonant frequency: where M s =Csin( B ) (18) is the gyromagnetic ratio of the nuclear spin (intrinsic property of the nucleus), is the length of time that the pulse is applied, and C is given by: C=N h 3 where N is the number of nuclei in the sample, k is the Boltzmann factor, and T is the absolute temperature. The described magnetization oscillates linearly following the pulse, allowing a receiving coil to detect its signal. h kt thermal equilibrium and the net dipole moment decays. (19) As time progresses, the system relaxes to Echoes and Relaxation Times Much of the NQR research that has been done attempts to identify the best methods to excite nuclei and detect their return signal for detection applications. Consider a simple example in which an investigator is attempting to determine whether a single, specific explosive or other material of interest (MOI) is present within an unknown volume. First, in a laboratory setting, an MOI s NQR frequencies are characterized along with their depen- 9

15 3 BASIC NQR THEORY dence upon environmental factors such as temperature [98]. Armed with this knowledge, an investigator attempts to excite one of the MOI s NQR frequencies in an unknown sample. If the MOI is present within the sample, a return signal will be observed. In the simplest case, the investigator would apply a single pulse to the sample with a transceiver coil, then detect the oscillating magnetic dipole produced as a result. The return signal often decays very quickly (S / e t/t 2 )wheret 2 is known as the e ective relaxation time. The signal s exponential decay immediately following an excitation pulse is known as Free Induction Decay (FID). This presents a problem to the investigator since there is only a a very short time window in which to detect the signal. Furthermore, the transceiver coil is typically part of a tuned circuit and is likely to experience ringing even after the applied pulse has been turned o. It can therefore be di cult to distinguish between an FID signal and residual ringing in the coil. The signal decay during FID is due to individual spins losing coherence [24,36,58]. It is possible to refocus the spins with an additional pulse so that the signal re-emerges at a later time. This means that the signal emerges after ringing e ects have subsided. Refocusing pulses can be linked consecutively in a train so that the signal returns many times. This is important because the signal observed in real-life detection applications is often very low. By increasing the number of measurements made, N, the signal to noise ratio (SNR) improves (SNR / p N). The strength of each successive echo decreases exponentially according to the spin-lattice relaxation time, T 1. T 1 is the time constant associated with the return of the system to thermal equilibrium. It establishes the wait time necessary between successive excitations. In addition to the pulsed (transient) methods described above, work has been done with continuous wave methods where the sample is exposed to a continual AC magnetic field that is either fixed at a single frequency or slowly sweeps across frequencies. While common in early work and spectroscopy [50], it tends to provide a smaller signal than is ideal for real-world detection problems [16]. That said, modern variations on non-pulsed methods (such as adiabatic half passage) show potential for field detectors [56]. 10

16 4 NQR AS A DETECTION TOOL 4 NQR as a Detection Tool 4.1 Introduction An ideal explosives detector would be capable of instantly scanning a given region and then report the presence, identity, location, and quantity of explosives found. It would be sensitive to all known explosives, work in real-life conditions, have no adverse e ects on its surroundings, and be impossible to thwart. It would always find any explosive present (high probability of detection) and not give false alarms (low false positives). In addition all this, it must be a ordable and accepted by societal standards. While no single technique is capable of meeting all criteria of this wish list, they are helpful to keep in mind while evaluating a given technology. This section briefly considers NQR in light of these criteria. We will first discuss the capabilities that NQR has to o er. We will then look at the challenges that researchers have encountered and the work which has been done to overcome these di culties. 4.2 The Potential of NQR Pure NQR was first observed experimentally in 1946 by Nierenberg et al. [61, 62]. It was described theoretically in 1950 by Pound, whose work coincided with experiments by Dehmelt and Kruger [15, 70]. Through his work with the British military, Pound became interested in using NQR to detect land mines, making the idea of using NQR as an explosives detector almost as old as the technology itself [58]. In the twentieth century, NQR was never deployed widely for explosives detection, but it did become an important laboratory tool for chemists who use it to understand molecular structure [4, 91]. Recent advances in theory, electronics, signal processing, and computational power have once again made NQR s utility as a detection tool an active research topic. As an explosives detector, NQR has some prominent advantages. Nitrogen is highly ubiquitous in solid explosives including TNT, RDX, PETN, and HMX. In principle, NQR can be used to detect any of these since the 14 Nisotopeis99.63%naturallyabundant[7]. In general, there are three NQR frequencies for each Nitrogen nucleus found within a molecule (TNT, for example has the chemical formula C 7 H 5 N 3 O 6 ; thus, three nitrogens and nine total NQR frequencies). Even nitrogen atoms with similar valence bonds may have distinct spectra since the EFG is a function not only of chemical bonding, but also overall molecular and crystalline structure. For example, TNT possesses two NO 2 groups, with distinct NQR frequencies for the nitrogen contained in each. The spectra for various compounds is spread 11

17 4 NQR AS A DETECTION TOOL over a wide range (0 to 6MHz) and the line width of each line is typically just 3kHz [8], making the spectra of each substance very unlikely to match any other substance. This specificity enables NQR to uniquely identify particular explosives. Furthermore, by observing resonance lines unique to a particular substance, it makes very high detection rates and low false positive rates possible [16, 24, 29, 38, 39, 101]. High specificity also qualifies NQR for use detecting counterfeit pharmaceuticals, pharmaceutical quality control, and detecting concealed narcotics [3, 4, 19, 23, 37, 85]. Figure 4: Each solid containing nitrogen has a unique NQR spectra. This allows unique identification of many explosives. Figure from Miller and Barrall [58]. As shown in Figure 4, NQR frequencies for the common explosives RDX, TNT, PETN, and tetryl are below 6MHz. The electromagnetic radiation that must be generated to excite nuclei is therefore very low energy and non-ionizing. At su ciently low power levels, there is no danger presented by the AC magnetic fields typically employed in NQR. Because these frequencies overlap with those used in AM, maritime, and amateur radio communications, organizations such as the FCC have compiled ample data on their safety [13]. NQR frequencies of 0.5-5MHz correspond to wavelengths of m in air. These fields pass easily through most dielectric media. This gives NQR the ability to see through materials such as earth, luggage, or tissue in order to detect deeply concealed MOI. This thesis will give special attention to how the fields pass through biological tissue for use detecting explosives concealed within people. 4.3 NQR Challenges and Current Research While having potential for high detection rates, low false positives, and unique identification, NQR has been hampered by long detection times that are the consequence of peren- 12

18 4 NQR AS A DETECTION TOOL nially low signal to noise ratio. As noted in the previous section, nuclei act as oscillating magnetic dipoles during NQR. Unfortunately, their signal is very weak. In field applications, it is common for the NQR signal to be about the same intensity as the transceiver coil s thermal noise, making the signal to noise ratio (SNR) very low [16,27,38]. Since SNR scales with the square root of the number of measurements taken, (SNR / p N), a decisive measurement may often be possible, yet take too long for field applications as a method for primary screening. For this reason, it is often suggested that NQR is used as a secondary screening technology to confirm or clear detections made by a faster, yet less discriminate technology [5, 48]. Improving the SNR and finding other ways to lower detection time have been active fields of NQR research. The previous section introduced the concept of a pulse sequence that produces spin echoes. Di erent patterns of pulse sequences have been studied to identify those which enable most e cient detection for various MOI depending upon their T 1, T 2, and NQR frequencies [53, 55, 76, 79, 80, 84]. Another excitation method is to apply the magnetic field at a frequency slightly o -resonance [53]. Another active area of research is to improve the SNR through advanced signal processing techniques [37 39, 95, 101]. NQR frequencies are generally temperature dependent, adding a complication to detection scenarios where the region being scanned has an unknown temperature. Temperature dependence can be so strong that detailed characterization even allows NQR to act as a sensitive thermometer [105]. In detection applications, methods to deal with temperature dependent spectra shift may be as simple as estimating the target s temperature or selecting a specra line with low temperature dependence. A di erent approach is to exploit temperature dependence as an identifying MOI characteristic. A recent algorithm claims to improve detection this way, even when sample temperature is unknown [38]. The frequencies used in NQR often overlap with those used in A.M., shortwave, and maritime radio communications. While this is a good thing in terms of the fields safety near organisms, it adds the issue of radio frequency interference (RFI). One way to avoid RFI is to operate the NQR detection in a shielded environment. This strategy is more feasible in controlled applications such as checkpoint screening than in landmine detection. When unavoidable, a strategy for compensating for RFI is to add reference antennas designed to measure RFI so that it can be subtracted from the detection coil s signal [5,48,49,100,102]. Alternatively, a gradiometer can be used as the transceiver coil. Gradiometers are insensitive to large-scale ambient signals [35, 96]. At first, it could seem as though using a gradiometer could reduce the fraction or strength of the resonating nuclei which the coil is 13

19 4 NQR AS A DETECTION TOOL sensitive to. However, if used for both transmission and detection, reciprocity guarantees that all coil designs (including gradiometers) are inherently optimal for receiving any signal from the nuclei which they are responsible for exciting. The limiting condition then becomes that the coil be capable of generating a field of su ciently uniform magnitude. This is discussed in more detail in sections 5.2 and In addition to minimizing RFI, coils are designed to maximize sensitivity to small signals and to generate as little thermal noise as possible. Coil designs which have been studied include simple loops, uniform and non-uniform spiral coils, birdcage coils, and planar and axial gradiometers [26, 47, 54, 60, 86, 96]. In addition to conventional coils, researchers have investigated Superconducting Quantum Interference Devices (SQUIDs) [1, 34], high temperature superconductors [109], and atomic magnetometers [45, 83]. For field applications, conventional coils seem to perform best due to their simplicity and cost. Cross-polarization techniques o er particular promise for enhancing NQR signals [42, 59, 63, 71, 78]. In these methods, a pulsed, static magnetic field is applied before NQR measurements are taken. This pre-polarizes the sample, raising the energy of the system by exciting protons (hydrogen nuclei) before 14 Nnucleiareexcited. Energyistransferred from the protons to 14 Nviathecouplingofmagneticdipoles. WorkbyKimandRudakov shows that the resulting increase in NQR signal can be an order of magnitude [42,78]. This technique may not require magnetic fields as powerful or well-characterized as those that are necessary in Nuclear Magnetic Resonance (NMR) experiments, making field use feasible. If technology were to succeed at boosting the SNR to levels adequate for field use, NQR would still have a few drawbacks. In the standard detection schemes, MOI are searched for one-at-a-time. While this specificity has the advantages described earlier, it introduces its own challenges. It requires setting the hardware s excitation and detection frequencies for each spectra line that is searched for. This may be accomplished using orthogonal coils capable of detecting multiple frequency lines at the same time or switching hardware that changes the characteristic frequency of coils between searching for one MOI and the next. However, it may still take an appreciable amount of time to attempt to resonate the spectra lines of many di erent explosives which may be hidden within an area. Another issue has to do with characterization. Much of NQR research to-date has focused on detecting explosives within land mines in which the MOI is either known beforehand, or is one of just a few di erent kinds of well-characterized military explosives. In applications where the threat is unknown, a larger number of MOI must be searched for. Especially when searching for terrorist s bombs, one would also have to characterize home- 14

20 4 NQR AS A DETECTION TOOL made explosives and all of the variability that their non-commercial manufacture involves. For example, impurities are known to e ect NQR characteristics such as signal intensity, T2, and line-width [8,40,68,89], and small impurities in TNT are known to reduce the signal by a factor of ten [27]. Even the physical form of a material a ects NQR measurements (such as the particle size in a powdered sample) [8, 89]. While most of these e ects are not large enough to interfere with a detector s operation, some are important to consider. Adding NQR characteristics to explosives databases would help this e ort. The other major disadvantage of NQR is its inability to detect liquids. In liquids, the NQR signal disappears due to motional averaging [46]. This prevents NQR from being able to detect either liquid explosives or liquid precursors to improvised bombs. With its ability to detect liquid explosives, NMR is a complementary technology to NQR [2,20,21]. Much of the hardware used in NMR is also useful in NQR, so it may be convenient to package the two technologies together so that they share use of electromagnetic shielding from background noise, detection coils, and even polarizing coils (allowing use of NQR with cross polarization techniques). Finally, it is worth considering the e ect of metal on NQR measurements. An MOI shielded in metal may become invisible to an NQR detector. Experiments have shown NMR (which operates at similar frequencies) to be e ective even through shielded containers such as aluminum cans [52]. This is because the skin depth of low frequency fields is so large. Therefore, it is possible that NQR could confront the problem of light shielding directly. However, it may be more robust to plan on using NQR in conjunction with a metal detector to detect shielding; the NQR hardware itself could even be used as a metal detector. Another issue with metal is that it may introduce spurious signals that complicate NQR measurements (such as paper clips within a piece of luggage) [72]. These ringing e ects are mostly transient, however, so can be dealt with by using a pulse sequence that makes the NQR signal observable after transients from ringing metal have subsided. 15

21 5 FREE SPACE NQR MODEL 5 Free Space NQR Model Before examining how biological tissue a ects NQR measurements, we begin with a model of NQR in free space. Some analytical and computational models have been developed by physicists to explain how complicated interactions between nuclei a ect NQR in systems with more than a single atom. For example, there is coupling between the magnetic dipole moments of various nuclei undergoing quadrupole resonance that has been accounted for [11]. These models are useful for constructing optimal pulse sequences and in processing the NQR signal [38]. However, such models do not account for the overall geometry of the problem or the inhomogeneities in the magnetic field that are expected in real life. An analytic model has been developed by Shinohara et al. that is capable of predicting the e cacy of various coil designs at detecting targets in various positions via NQR [86]. The group that developed it demonstrated a basic and successful implementation of the model in Microsoft Excel VBA. This section introduces the model and then details our implementation and evaluation of the algorithm. The model provides a solid understanding of how geometry a ects NQR measurements in free space. The next section examines how magnetic fields are altered by biological tissue and how this model can be used to evaluate the significance of that e ect. 5.1 Shinohara s Model Powdered Samples In section 3.4.1, we noted that following a pulse delivered exactly on resonance at frequency!, a single crystal of material will act as an oscillating magnetic dipole with magnetization M s given by Eq. 18. Note that this equation assumes that the applied magnetic field is identically aligned with the single axis of the principle axes system (PAS) which corresponds to the NQR frequency being excited. If the applied field were orthogonal to this axis, zero magnetization would be induced. In real life, explosives and other MOI are typically found as powders or in other poly-crystalline forms composed of many tiny crystals with random orientation. Within such an arrangement, only a tiny fraction will be aligned such that they are maximally sensitive to the applied field while a similar fraction will be almost entirely invisible. We can account for the arbitrary alignment of a given crystal s PAS with respect to the applied field. When the applied field and relevant axis of the PAS form an angle, we simply substitute the projection of the magnetic field into Eq. 18. M s continues to point 16

22 5 FREE SPACE NQR MODEL along the relevant axis of the PAS: M s =Csin( Bcos ) (20) As an example, consider the case where the 0 transition in a single crystal is excited. This transition is associated with the z 0 axis of the EFG PAS. A magnetic field is applied along the laboratory frame axis z. z and z 0 form an angle. ThenM s points along z 0,and may contain components in each of (x, y, z). To account for the myriad crystals within a sample, one could sum the contributions M s due to each. However, it is much easier to integrate M s due to a single crystal over all positions that it is capable of attaining. In doing so, we are assuming that a given small volume of material can be expressed as a single magnetization M T.Thisintegrationiseasiest to visualize if we refer to each crystal s z 0 axis by its angular displacement from lab frame z in polar coordinates 0 and 0. If a magnetic field is applied along z, thez component of M s is then: M sz =Csin( 0 )sin( Bcos 0 ) (21) We then integrate this expression over all possible values of 0 and 0 to obtain M Tz : Z 2 Z M Tz =C d 0 sin( 0 )sin( Bcos 0 )d 0 (22) 0 0 Using the same procedure to integrate the x and y components leads to their selfcancellation, so M TZ order 3/2, which is a well-known result: = M T. The expression becomes dominated by a Bessel function of r M T =C 2 B J 3/2( B ) (23) Eq. 23 means that a small volume of powdered material will act as a magnetic dipole during NQR. M T points in the same direction as the applied field. This result has been experimentally verified [65]. The maximum signal that can be expected from a powdered sample is 43.6% of that which is attainable from a single crystal with the same number of nuclei. Although smaller, this signal is independent of coil orientation, unlike the case of a single crystal in which certain orientations create blind spots. The product B is referred to as the nutation angle [56]. The nutation angle which maximizes M T is 119,butpulses which meet this nutation angle are often referred to as 90 pulses. A normalized plot of Eq. 23 is shown in Figure 5. 17

23 5 FREE SPACE NQR MODEL Figure 5: Nuclear magnetization as a function of nutation angle is dominated by the 3/2 order Bessel Function. The nutation angle is directly proportional to the applied magnetic field s magnitude as well as pulse length. The /2 pulse occurs at 2.08 rad while the pulse occurs at 4.49 rad Calculating the NQR Signal In [86], the authors discretize an NQR target (such as a block of explosive) into voxels. Each voxel is treated as a magnetic dipole, with M T calculated using Eq. 23. This requires knowing the magnetic field at that point. The magnetic field of a coil can be calculated using the assumption that because NQR frequencies are so low, the problem is essentially magnetostatic. The Biot-Savart law can then be employed: B = µ Z 0 Idl r (24) 4 C r 3 where µ 0 is the permeability of free space, dl is the vector describing the length and direction of a segment of current-carrying wire, I is the current, and r is the vector pointing from dl to the observation point. The coil is numerically discretized into many small elements dl to employ Eq. 24. Once Eq. 23 has been used to obtain the magnetization at various points within the target, the task turns to calculating the signal induced in the coil. The authors do so by employing Faraday s law: I E dl = ds (25) 18

24 5 FREE SPACE NQR MODEL where H E dl is the voltage induced in the coil and ds is the time derivative of magnetic flux over the coil s surface. To use Faraday s law directly, one must calculate the magnetic flux over the coil. Treating each voxel of the target as a magnetic dipole, the magnetic field due to each can be found over the surface of the coil. Summing these contributions, the total magnetic flux if obtained. At some position r, awayfromamagnetic dipole, the dipole s magnetic field, B d,isgivenby[12]: and the flux due to a dipole over the coil s surface is: Z S B d = r µ 0 M T r (26) 4 r 3 B d ds = µ Z 0 r M T r ds (27) 4 S r 3 Since the nuclei resonate sinusoidally, phasor notation can be used. Multiplying flux by j! then gives its time derivative and the NQR signal s peak voltage due to each dipole. Summing the contributions due to each dipole gives the total signal. Considering N total dipoles, each with magnetization M Ti, the total signal received is V: V= j! µ 0 4 NX Z i=1 S r M Ti r r 3 ds (28) 5.2 An improved method for calculating NQR signal more di While conceptually simple, complicated coil designs may bound surfaces which are much cult to define than in the case of a single loop coil. When the coil used to excite the nuclei is the same as that used for detection, reciprocity can be used to calculate the received signal much faster. This method was introduced to us by Dr. Peter Volegov of Los Alamos National Laboratory who uses it for modeling NMR problems. For a vector field, A, thestokes theoremstates: I l Z A dl = S r A ds (29) where l is a contour enclosing the surface S with infinitesimal normal elements ds. Applying Stokes theorem to Eq. 27, we obtain: Z S B d ds = µ I 0 MT r dl (30) 4 r 3 19

25 5 FREE SPACE NQR MODEL then using the scalar triple product, (B C) A = B (C A), this expression becomes: Z Z S S B d ds = µ I 0 4 B d ds = M T µ0 4 M T r dl r 3 I dl r r 3 (31) Recall that we are using the same coil as both a transmitter and receiver. In this case, part of the part of the right hand side of Eq. 31 is identical to the Biot-Savart law (Eq. 24) when current equals unity. The expression then reduces to: Z S B d ds = M T B u (32) where B u is the magnetic field at the location of the magnetic dipole due to the transceiver coil when unit current is running through it. Earlier, we discussed how the magnetic field of the coil and induced magnetization are perfectly aligned in the case of powdered samples. The dot product then reduces to multiplication. Again treating the dipole as time harmonic, the signal amplitude (V d )inducedinatransceivercoilbyasingle dipole can be derived from Eq. 32 by applying Faraday s Law along the contour defined by the coil wires, and V d = j!m T B u (33) Since the computational algorithm originally begins by calculating the magnetic field due to the coil, B u is already known (or attainable by a scaling factor). Finally, the total signal (V) is the sum due to the N dipoles which the target is being modeled as: V= NX j!m T B u (34) i=1 This method is both simple and unambiguous. It is also a good demonstration of how the principle of reciprocity applies to NQR. A coil that establishes nuclear magnetization within a sample is ideally situated to detect that magnetization. 5.3 Programming the Model We worked in conjunction with a group at Los Alamos National Laboratory that works principally on Nuclear Magnetic Resonance (NMR). In NMR, energy level splitting in nuclei 20

26 5 FREE SPACE NQR MODEL occurs as a result of their magnetic dipoles interaction with a static magnetic field, as opposed to the quadrupole interaction in NQR s case. Like in NQR, NMR uses an RF magnetic field to excite nuclei and establish magnetization. From the point of view of electromagnetic modeling, NMR and NQR are quite similar. The Los Alamos group had previously developed a toolbox in MATLAB to 1) calculate the magnetic field produced by a coil in the magnetostatic approximation, 2) calculate the magnetization formed by excited nuclei due to NMR in response to the field, and 3) calculate the return signal due to this magnetization. We wrote additional functions for this toolbox that employ Eq. 23 to calculate magnetization due to NQR. This function replaces step 2 of the described sequence, allowing NQR problems to be modeled. The code that we used and extended at Los Alamos cannot be released to the general public and is not included here. Our own implementation of the free space model in MATLAB is conceptually identical to that employed by [86] except for own method of final signal calculation described in 5.2. Our contributions to the basic model include computation of nutation curves which were compared against experimental data (section 5.4) and the use of 2D sensitivity maps to visualize the data (section 5.5.1). 5.4 Validating the Model Before using the model in any critical work, it is important to check it against experimental data. Shinohara et al. programmed the model in Microsoft Excel VBA and used it to generate a one dimensional sensitivity profile [86]. That is, they used the model to predict how the NQR signal generated by a small cube of HMT would change as its position was translated within a plane parallel to the coil. They did this with both a circular loop coil and a planar gradiometer. They compared their program s results against experiment and the data agree well. To compare our implementation of the model against experiment, we used data taken previously by Los Alamos researchers. This section describes the physical experiment then compares its result to the simulation s prediction Experimental Hardware - NQR Spectrometer Dr. Young Jin Kim and Dr. Todor Karaulanov of Los Alamos built an NQR spectrometer, then tested it on the 4.64 MHz line of sodium nitrite NaNO 2.Thespectrometerisbased on the design described by Hiblot et al. [30]. Grandinetti provides an excellent description of spectrometry hardware in his NMR book [25], while other NQR-specific spectrometer design considerations can be found elsewhere [24,75,99]. This subsection introduces how the spectrometer used in our experiment functions. 21

27 5 FREE SPACE NQR MODEL The RF pulse or pulse sequence is programmed into a PC. The PC is connected to a Tecmag Apollo console which generates the specified pulse train. Its output passes through an amplifier capable of boosting the signal to approximately 350W and 2.75A peak current. The amplifier s output is connected to the front end of the spectrometer. The front end includes the transceiver coil and matching and tuning components. After the pulse has passed through the coil, the coil must be used in receive mode. Therefore, the front end is also connected to a preamplifier. The signal from the preamp passes through a secondary amplification stage (again, using the Apollo console) and back to the PC in order to record the NQR signal. The front end serves multiple functions and is generally home-built whereas the other components are available as commercial, o -the-shelf technology. It consists of all the components between the power amplifier (transmit mode) and preamplifier (receive mode). Figure 6showsthefrontendDr.KaraulanovandDr.Kimchosetouseintheirspectrometer[30]. Using the nomenclature of this figure, L is the transceiver coil, while C M and C T are the matching and tuning capacitors respectively. Taking into account the transceiver coil s inductance L, the values of C M,andC T are carefully selected to form a circuit which is resonant at the particular NQR frequency chosen. That is the primary purpose of C T. Meanwhile, the primary purpose of C M is to match the coil s impedance to that of the preamplifier in order to maximize the amount of power received by the preamp. The spectrometer used in our experiment uses vacuum variable capacitors for C M and C T to allow for fine tuning (and to switch to a di erent NQR frequency for another experiment). Care must be taken to ensure that power from the amplifier flows into the coil, yet does not damage the pre-amp during transmission. However, the power of the NQR signal must be maximally transmitted to the preamp during receive mode. The remaining components accomplish this. The components in the blue box (L 0, C 1, C 2 )arechosentoforma /4 circuit at the NQR frequency being used. The /4 circuitallowsoptimaltransmissionto the preamplifier from the coil during receive mode. During transmit mode, the pair of cross diodes near the pre-amplifier allows the transmit pulse to see a path to ground - making the /4 circuit behave as an open circuit. This blocks the transmit pulse from reaching (and possibly damaging) the preamplifier. Since the signals used in NQR are very small (on the order of nv), it is important to limit sources of noise. A second pair of cross diodes next to the amplifier blocks small signals present in the amplifier from entering the rest of the circuit during receive mode. Asolenoidalcoilisusedbecauseitcreatesastrong,fairlyuniformfieldthroughthe sample. It is made of 40 turns of 21AWG wire wrapped around a machined form. It is 3.5cm in diameter and 7cm long. 22

28 5 FREE SPACE NQR MODEL Figure 6: The front end of the NQR spectrometer used is shown between the power amplifier and preamplifier. The elements in the blue box form a quarter wavelength circuit. The elements in the red box provide tuning and matching. Figure from [30] 23

29 5 FREE SPACE NQR MODEL The Experiment Measurements: A 100g sample of powdered sodium nitrite, NaNO 2, was placed inside of a cylindrical container and into the solenoidal transceiver coil of the spectrometer. A sinusoidal pulse with frequency 4.64MHz was applied to the sample for a time. The free induction decay (FID) signal was measured 250µs afterapplyingthepulsetoallowringinge ectsin the coil to subside prior to measurment. ranged from 5 to 240µs. For all measurements, the spectrometer s power amplifier was set to 250W, which we estimate creates 2.22A peak current through the coil. The experiment was conducted at room temperature (295K). Simulation: We simulated this experiment using the NQR software described above. The coil was modeled as a solenoid discretized into 1280 sections. The sodium nitrite was modeled as a cylinder centered within the solenoid. We used the estimate of 2.22A for peak current flowing through the coil. The characteristics of sodium nitrite (density, molar mass, NQR frequency) were programmed along with the gyromagentic ratio of 14 Nandthesample s temperature. The maximum FID signal was calculated using Eqs. 24, 23, and 31. Pulse lengths covered the same range as the measured data and were incremented 1µs atatime. Figure 7: A 100g cylinder of sodium nitrite (green) is simulated inside of a solenoidal coil of 40 turns of wire (blue) to replicate the physical experiment. 24

30 5 FREE SPACE NQR MODEL Results and Discussion The measured and simulated results are plotted alongside one another in Figure 8. This nutation curve describes signal intensity as a function of pulse length. For both the experiment and simulation, we plot the absolute value of the received signal. Figure 8: The measured and simulated nutation curves for a 100g sample of sodium nitrite. The received NQR signal is plotted as a function of pulse length while applied power is held constant. The simulated and measured data agree fairly well. There are two notable di erences between the data sets. First, the simulated data predicts that the length of the 90 and 180 pulses are slightly shorter than those which were measured (the plot of measured data appears to be shifted ahead of the simulated data). For example, the 90 pulse (that which maximizes signal) was predicted by simulation to be 80µs, whereas experimentally, it was observed to be closer to 90µs. The second notable discrepancy is that the measured signal did not return all the way to zero for the 180 pulse (that which brings the signal to zero). There are many possible explanations for these discrepancies: Both discrepancies could be partially attributed to variations in the sample s NQR characteristics across its volume. These variations can be created by subtle factors such as temperature gradients, particle size, and the powder s physical packing as discussed in 4.3. If these variations were present within the sample, ine cient excitation could result - thereby requiring longer pulse times than predicted. The variations could also cause the net signal to not average to zero during the 180 pulse since frequency shifts. These e ects would take 25

31 5 FREE SPACE NQR MODEL place due to C (Eq. 19) of the magnetization expression (Eq. 23) acting as a weighting operator during summation (Eq. 31). The fact that a non-zero NQR signal was recorded for the 180 pulse could be due to the noise floor of the experimental apparatus. Measured NQR signal intensity is no more than the FFT of the received signal. So it is very possible that the lowest values that were recorded (0.08 in the normalized plot) are simply the values of noise in the system. Unfortunately, we do not have any baseline measurements available with which to check whether this were actually the case. Another unknown is the time which is required for the applied pulse to reach its maximum amplitude. The simulation assumes that the pulse reaches its maximum value instantaneously and the antenna has no ringing. In real life, there is both a ring-up and ring-down time. This could cause a discrepancy between commanded and e ective pulse lengths. We do not have measurements of the ring up and ring down times for this particular experiment, but the same spectrometer has displayed ringing in the past. The peak to peak current of the applied pulse was estimated. If it was estimated to be too high, that would explain the apparent shift of the measured data ahead of simulated in the plot. A di erent error source could be the complex interactions between nuclei such as dipole-dipole interactions. These e ects are not considered in the simple model. In spite of these potential sources for error, the simulation and measured data agree closely enough to demonstrate the simulation s basic e ectiveness. Before being used in critical work, it should be compared more rigorously against experiment to better understand why and how it deviates from reality. For now, we consider our results and those taken by Shinohara et al. to have demonstrated that the simulation is e ective in free space [86]. 5.5 Utility of the Free Space Model Sensitivity Map The free space model can be used as a tool to optimize coil design. One way to evaluate coils is with sensitivity maps. A sensitivity map is a plot showing how sensitive an NQR coil is to material located at di erent points in space. To the best of our knowledge, this is the first time that 2D sensitivity maps have been created for NQR. To generate a sensitivity map, a plane is chosen a certain distance away from the coil. It is colorized according to the NQR signal that would result from each voxel within that plane if a particular MOI were present there. Sensitivity maps can be used to examine how changes in coil design, MOI, pulse strength, and pulse length a ect a detector s sensitivity. 26

32 5 FREE SPACE NQR MODEL Figure 9: Plane of interest 10cm away from a planar gradiometer coil. Detector sensitivity to material within this plane is mapped. For our example, we design a planar gradiometer coil comprised of two circular coils. Each circular coil has five loops of wire, spaced 5mm apart from one another. The inner and outer diameters of each circular coil are 10cm and 14cm as shown in Figure 10. Figure 10: The planar gradiometer used in our example. ID= 10cm, OD=14cm The coil is set to detect the 3.5MHz line of RDX. Its peak current is set to 10A. Two planes of interest are defined; one is 3.5cm from the coil and the other 7.0cm. In Figure 11a, the magnetic field magnitude is plotted for these two planes. We choose magnitude as opposed to any particular component of the magnetic field since magnitude is what 27

33 5 FREE SPACE NQR MODEL determines the degree of excitation. We then choose choose =285µs, and generate sensitivity maps for the two planes, as shown in Figure 11b. We observe that the 285µs pulseresultsinahighreturnsignalfrom RDX that is located 3.5cm from the coil and within its central region. However, the signal at 7cm is about four times weaker. Next, we generate sensitivity maps for =600µs as shown in Figure 11c. In this case, sensitivity in the central region at 3.5cm has plummeted, and may not be usable at all. However, the peripheral region at 3.5cm returns a much stronger signal than was observed for = 285µs. Likewise, detection in the central region at 7.0cm has improved significantly from the case of the shorter pulse. The return signal for this region is over twice as strong as was observed for 285µs. It is important to observe that the =600µs createsmagnetizationwithintherdx at 7.0cm that is roughly the same as the magnetization created by =285µs at3.5cm. However, since the RDX is further from the coil at 7.0cm, the received signal is weaker despite magnetization being the same. We return to this example in the next section, 6 - Coil Design Considerations. First, consider another way to use the model to evaluate a coil design Signal from example targets Sensitivity maps are best suited for qualitatively evaluating which regions a coil is most sensitive to. It is also possible to study sensitivity to particular objects, as we did to create the nutation curve in section This section reviews how total signal from an example object is calculated and illustrates how that capability may be useful in designing a coil. First, an object s material and resonant frequency are defined as with the making of sensitivity maps. The object is then given a particular size, shape, and position near the coil. The voxels which the object occupies are identified. Let these N voxels be indexed by the index i. The simulation is then run, calculating the sensitivity of each voxel over a large volume just as was done to generate sensitivity maps. Once again, sensitivity is the peak voltage that would be returned from that voxel if an MOI was present there. By summing the sensitivities for each of the N voxels that comprise the test object, the total signal received from that object is obtained. That is, s t = P N i=1 s i.thetotalsignal,s t,istheinitialamplitudeofthe FID signal received from the object. To demonstrate this, we use the same gradiometer coil as in the previous example with 28

34 5 FREE SPACE NQR MODEL (a) Magnetic field (b) Sensitivity Maps for = 285µs (c) Sensitivity Maps for = 600µs Figure 11: The left column shows results for the plane 3.5cm from the coil. The right column shows results at 7.0cm. In part a, the magnetic field magnitude is shown. Parts b and c show sensitivity maps to RDX for pulse lengths of 285µs and 600µs respectively when the peak transmit current through the coil is 10A. The shorter pulse length is superior at detecting objects where the magnetic field is greatest. The longer pulse length is superior at detection objects where the magnetic field is weaker. 29

35 5 FREE SPACE NQR MODEL peak current of 10A. We create at cylindrical target to simulate 60g of RDX. Its height is 5.55cm and diameter is 2.75cm. We o set the cylinder from the coil s center line and place it so that its nearest point is 2.0cm from the coil as shown in Figure 12a. We calculate s t for various pulse lengths from 100µs to 360µs to generate a nutation curve. We then rotate the cylinder so that its end now faces the coil as shown in Figure 12b. Again its closest point is 2cm away from the coil, but its orientation causes its center of mass to be moved further from the coil. We repeat the experiment. Nutation curves for the cylinder s two orientations are shown in Figure 13. The end-on orientation resulted in a weaker signal that required a slightly longer pulse for maximum excitation. This modeling capability may be very useful when designing a coil. An array of test objects can be created representing the spectrum of possible threats that a coil is designed to detect. Any given coil can then be quickly tested against these test objects. (a) cylinder oriented with its length parallel to coil (b) cylinder oriented with its end facing the coil Figure 12: A 60g cylinder of RDX is simulated in two di erent orientations with respect to the transceiver coil. 30

36 5 FREE SPACE NQR MODEL Figure 13: Normalized nutation curves for the two simulated orientations of the cylinder. The cylinder with its end facing the coil has a lower signal than the cylinder parallel to the coil. 31

37 6 COIL DESIGN CONSIDERATIONS 6 Coil Design Considerations The previous examples suggest that the computational model can be a useful tool for optimizing coil design and pulse length. For illustration, let us return to the example of section where it was shown that a much longer pulse length is required for optimal stimulation 7cm from the coil than at 3.5cm. Suppose an operator is interested in detecting potential objects at both those ranges. The operator could choose to use the short pulse ( =285µs) to excite both ranges if a sensitivity of about Volts/Voxel is enough to overcome the noise present in the detector. Whether this sensitivity is high enough depends on the type, quality, and bandwidth of the coil and circuits used. It also depends on how many detections the operator has time to average in order to su ciently improve the SNR (recall, SNR / p N, where N is the number of measurements). Time constraints (such as minimum required throughput for an airport detector) may limit the number of measurements. This is especially the case for materials with long T1 such as TNT since this increases the rest time necessary between successive measurements [24,27,36]. If =285µs provides insu cient sensitivity at 7.0cm, it may be worth finding a compromise value of pulse length between 285µs and600µs that provides su cient sensitivity at all ranges. If no such compromise value exists, the operator may be forced to conduct two separate searches - one at the short pulse length and one at the long. However, it may be possible to eliminate this issue altogether by a better coil design. This section introduces desirable qualities in a coil as well as constraints that a designer will encounter Desirable qualities Uniform Magnetic Field Magnitude: To overcome the di culty of two or more optimal pulse lengths for various detection regions, a coil should generate a magnetic field of uniform magnitude. In the case of a field with perfectly uniform magnitude, a single pulse length would optimally excite the MOI in any position. For this reason, coils should be designed to project a field of uniform magnitude over the region being investigated. While it may be obvious, it is important to note that uniform magnitude is not the same as uniform field. Dependence on magnitude rather than field is a result of reciprocity, as discussed in section 4.3. This is the reason why planar gradiometers may be suitable coils despite creating a very inhomogenous field. Sensitivity: Coils that maximize the amount of magnetic flux received from the excited sample maximize the received signal. This is accomplished by increasing the coil s surface area through increase in radius or the number of turns. Of course, doing so increases the 32

38 6 COIL DESIGN CONSIDERATIONS Q-factor of the coil, Q. This relationship between sensitivity and Q is mentioned in [24]: SNR / Q 1/2. If these were the only considerations, one could simply construct large coils made of thousands of turns of wire. Unfortunately, there are both operational constraints and those due to physical e ects that limit the length of wire within a coil and its overall size Constraints Size Constrained by Application: Certain applications may require small coils incapable of projecting a uniform field for very far. The field of surface coils falls o rapidly after about half a radius from the coil [24]. Coil inductance: For the simple circuitry described in 5.4.1, there is a maximum acceptable value for coil inductance. Exceeding this value will cause unacceptably long ring times. Furthermore, the bandwidth of the coil may become too narrow to cover the range of frequencies being examined or to accommodate unknown shifts in NQR frequency associated with unknown temperature or impurities. More advanced circuits could be used to facilitate use of higher Q coils [109]. Johnson-Nyquist noise: Another constraint is the Johnson-Nyquist noise generated by the coil. The RMS thermal noise, v n,ofalengthofwirewithacresistancer is given by: v n = p 4k B TR f (35) where, as before, k B is the Boltzmann constant and T is temperature. f is the bandwidth in hertz over which the noise is being measured - in our case, we would use the bandwidth of the coil. Since noise scales with resistance (which in turn scales with wire length), larger coils produce more thermal noise than smaller ones. When a coil is su ciently large, its thermal noise may dominate over that of the pre-amp and become the limiting factor of the SNR. Thermal noise may be exacerbated by the skin e ect and proximity e ect. These e ects can be mitigated by using litz wire [83] and hollow wire (such as the outer shield of a coaxial cable) [35]. Thermal noise in NMR coils (which are very similar to NQR coils) has also been studied [33]. 33

39 6 COIL DESIGN CONSIDERATIONS Methodology of Design If designing a coil from the beginning, physical size and coil inductance are the first parameters to be limited. From there, obtaining an optimal design is a matter of balancing uniformity of magnetization, sensitivity to various regions, and thermal noise. If thermal noise of the coil is determined to be a significant problem for a given application, it may be worth the e ort to include its e ect in the NQR model. For wire with well-characterized noise properties, approximating thermal noise is not hard. The larger di culty would lie in combining this noise approximation with received NQR signal in order to estimate SNR. Up to this point, we have been calculating the peak value of the FID signal. To translate peak FID into a good estimate of SNR, one would have to account for the decay of the FID signal according to the e ective relaxation time, T2. More broadly, one would also have to consider other relaxation times and the specific pulse sequence that is being employed. Since a lot of NQR research has focused on signal processing, it may be worthwhile to combine that knowledge with an electromagnetic-based NQR model such as the ones discussed in this thesis. 34

40 7 NQR IN BIOLOGICAL TISSUE - BACKGROUND 7 NQR in Biological Tissue - Background To our knowledge, no one has studied how biological tissue a ects the electromagnetic fields used in NQR. The remainder of this thesis discusses a model that we developed to address this question. There are three principle ways in which biological tissue may e ect the fields used in NQR: 1) field interaction with tissue s magnetic permeability, 2) field interaction with tissue s electric permittivity, and 3) losses due to conduction. One can estimate that e ects due to tissue s magnetic permeability are negligible since the relative permeability of biological tissue is one [87]. The problems of electric permittivity and conduction losses deserve more attention. The e ect of losses due to conductive media is well demonstrated both theoretically and experimentally by Suits et al. [96]; additional mention of lossy media includes [26,29,31,33, 76]. Lossy media reduces the e ective Q factor of the coil by introducing ohmic losses. The practical result of this is that better SNR may be obtained by keeping the coil at a certain lifto distance away from the lossy media. Suits et al. characterize these ohmic losses and identified the optimal lifto distance for media with conductivity equal to and greater than that found in human tissue and fluid [22, 96]. We therefore consider the problem of lossy media for NQR to be solved. It should be noted that the eddy currents established in lossy media are considered too weak to otherwise impact the magnetic field. The final unknown for doing NQR near biological tissue is the e ect of dielectrics. The relative permittivity of biological tissue is very high at NQR frequencies. For NQR frequencies, permittivity typically increases as frequency decreases. Some examples for the upper and lower end of the NQR range are given in Figure 14. As far as we know, the e ect of dielectrics has never been studied for NQR although Suits et al. do mention it as a potential problem. The question has been discussed for Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI) which is based on NMR. Since MRI is used on human patients, any e ects observed due to the dielectric properties of the body help inform whether NQR can be successfully used to search for explosives concealed within tissue. NMR is very similar to NQR except that the energy splitting of the nucleus occurs due to an applied static magnetic field rather than the EFG established by atomic structure. Like in NQR, an RF magnetic pulse is applied to excite nuclei and the nuclei s subsequent radiation is detected. NMR is the phenomenon at the core of Magnetic Resonance Imaging (MRI). While studying dielectric e ects, the crucial di erence between NMR and NQR is that NMR frequencies are typically much higher. For protons (hydrogen nuclei), NMR 35

41 7 NQR IN BIOLOGICAL TISSUE - BACKGROUND Figure 14: Approximate values of relative permittivity for various biological tissue. Data taken from the graphs compiled by Gabriel and Gabriel [22] frequencies typically range from MHz depending on the strength of the static field applied. High field and low field experiments extend this range. One very recent NMR paper by Harimoto et al. examined the e ect of dielectrics such as saline on NMR measurements and the resulting MRI image [28]. They found that at 63.5MHz, the dielectric properties of tissue have a significant e ect on MRI images. They further used finite element modeling to show that this was because of distortion of the magnetic field. Other research has shown both analytically and experimentally that MRI measurements can be a ected by dielectrics [32,33,43,103,104,106,107]. However, the NMR research also suggests that at low frequencies, dielectric losses may be insignificant. For example, an analysis by Redpath and Hutchison modeled dielectric losses as stray capacitance [73]. They found dielectrics to be insignificant at the studied frequencies (less than 16MHz). While their experiment was specific to NMR rather than NQR, the frequencies used are similar. Our goal is to determine the significance of dielectric e ects on NQR measurements. To do so, we develop a computational algorithm appropriate for low frequencies. Rather than analyzing dielectric losses as stray capacitance, we study how the magnetic field itself is a ected. Following the timeline of actual development, we first present an algorithm designed to work in two dimensions by exploiting cylindrical symmetry. We then present a generalization to completely arbitrary coils and targets in three dimensions. 36

42 8 CORRECTING FOR DIELECTRIC MEDIA 8 Correcting for Dielectric Media 8.1 The Darwin Model In the free space model described previously, the electric field can be ignored since the magnetic field is determined solely by the distribution of current through the coil. However, when dielectrics are introduced, the electric field will reconfigure itself in response to them, changing the magnetic field in the process. To account for dielectrics, we must adopt a model that recognizes the coupling of electric and magnetic fields. Maxwell s equations are di cult to directly implement computationally for this problem. Techniques that our lab specializes in such as Finite Di erence Frequency Domain (FDFD) and Finite Di erence Time Domain (FDTD) could possibly be used but are better suited for higher frequencies. Furthermore, an FDFD or FDTD solution in three dimensions is very computationally intensive, and it can be di methods. cult to specify arbitrary coils using these The distinguishing feature of NQR systems is their low frequency, so quasistatic models can be used to simplify the problem. The electroquasistatic model (EQS) calculates the electric and magnetic fields directly from sources (the original charge distribution and current). This means that it fails to capture the link between electric and magnetic fields and is therefore unsuitable for our purpose. The magnetoquasistatic model (MQS) does a better job since it includes dependence of the electric field upon the time derivative of the magnetic field. However, it does not provide a reciprocal means of calculating the magnetic field from the electric. This is because it is only capable of handling current distributions with zero divergence (r J = 0, where J is current density). Although, the current distribution through the coil is divergenceless, it becomes necessary to describe the correction field due to dielectrics in terms of a diverging current distribution. E ectively, the requirement that r J = 0 prevents dielectrics from a ecting the magnetic field in our application. An excellent overview of the quasistatic models discussed here is given by Larsson [44]. The Darwin model is very similar to MQS except that it uses the continuity equation for charge density ( ) + r J =0 (36) As will be seen, this relationship is crucial to our method of calculating the e ect of dielectrics on the magnetic field. The other expressions that make up the Darwin model can be expressed in integral form. Magnetic field is calculated with the Biot-Savart law as 37

43 8 CORRECTING FOR DIELECTRIC MEDIA expressed in terms of time-dependent current density, B(r,t)= µ Z Z Z 0 J(r 0,t) R d 0 (37) 4 R 3 where r is field position, r 0 is source position, R is the vector from the source to field position, R is that vector s magnitude, and 0 is the region which includes all sources. The electric field is expressed in two separate parts: The Coulomb electric field, E C,comesfromchargedensity: E = E F + E C (38) E C (r,t)= 1 Z Z Z (r 0,t)R d 0 (39) 4 0 R 3 The Faraday electric field, E F,comesfromthemagneticfield: I Z E F dl = ( ) ds E F (r,t)= 1 Z Z 0,t) R R 3 d 0 (41) 8.2 The Algorithm In the case of NQR detection, there are no free charges, and so E C is initially zero. However, the Faraday portion of the electric field can be calculated either using Faraday s law (Eq. 40) or Eq. 41. Doing so, it will be found that E F is continuous. However, there is a boundary condition on the electric field at the interface of any two dielectric media. The perpendicular component of electric flux density across any boundary is equal to the surface charge density present on that boundary: E?1 1 E?2 2 = (42) where E?1 and E?2 are the components of E perpendicular to the interface, directly above and below the boundary. 1 and 2 represent respective dielectric constants. is the surface charge density that exists on the boundary between media and is equal to zero in the case of NQR problems. The Faraday portion of the electric field, E F,isafunctionof the magnetic field and is determined without any consideration of dielectric media. Since 38

44 8 CORRECTING FOR DIELECTRIC MEDIA =0,E F does not satisfy the boundary condition in general. Therefore, we must introduce acorrectiontermwhichallowsthetotalelectricfield,e, tosatisfytheboundarycondition. We can decompose the boundary condition into its Faraday and Coulomb components. The charge density is set to zero to reflect physical truth: E F?1 1 E F?2 2 +E C?1 1 E C?2 2 =0 (43) Next, we explicitly identify the amount by which E F fails to meet the boundary condition. This term can be thought of as a fictitious correction charge; we call it C for its connection to the Coulomb component of the electric field (E C ). It is attained by rearranging Eq. 43: E F?1 1 E F?2 2 = E C?1 1 +E C?2 2 E F?1 1 E F?2 2 = C (44) E C?1 1 +E C?2 2 = C (45) Noting the similarity in form of Eq. 44 and Eq. 45 to the boundary condition, we can treat C as a fictitious surface charge density. Its value is calculated from Eq. 44. Then, via Eq. 45, C creates E C. That is, E C is the correction term to E F. When E C is calculated this way and summed with E F,theoriginalboundarycondition(Eq.43)issatisfied. At this point, we have identified the proper electric field in the presence of dielectric media. The task then turns to finding the associated magnetic field. At first, this may seem like a di cult task since (in our model) the magnetic field is dependent only upon current density. It is possible because we have altered the current density from its original value by introducing the correction charge, C. Rather than computing the magnetic field directly from the electric, we can convert C into a current density then apply Eq. 37. Integrating both sides of Eq. 36 and employing the divergence theorem, we obtain: Z d 0 = { J s ds (46) where the right hand side integrates over that surface which encloses the volume over which the left hand side is integrated. We are interested in finding the current density due to the correction charges. We call this J C to distinguish it from the current density of the coil, J I.Weselectasmallportionoftheboundarybetweentwodielectricmediawitharea, A. The charge on that boundary is then q C = C A. By selecting A to be su ciently small, we can treat Q C as a point charge. Then, employing Eq. 46 with spherical symmetry, we 39

45 8 CORRECTING FOR DIELECTRIC MEDIA obtain an expression for current density. We assume sinusoidal excitation, and obtain J C = j!q C 4 r 2 ˆr (47) where r is the distance from the charge to the point at which J C is being evaluated and ˆr is the unit vector pointing from the charge to the point of evaluation. 8.3 Summary The initial current density of the coil is used with the Biot-Savart Law to calculate the initial magnetic field. The electric field associated with the initial magnetic field is called E F and fails to take account of the presence of dielectrics. The fictitious correction charge surface density C is calculated from E F at each interface where permittivity changes. The continuity relation is used to calculate the correction charge density, J C from C. The correction charge density is used with the Biot-Savart Law to calculate the correction to the magnetic field that exists as a result of nonuniform permittivity. Finally, the NQR model of section 5.1 can be applied with the corrected magnetic field. 40

46 9 CALCULATING ELECTRIC FIELD - JACOBI METHOD 9 Calculating Electric Field - Jacobi Method Before employing the described correction algorithm in an NQR application, we wish to test that it does a suitable job of correcting the electric field. We start with a two dimensional test. The code is given in appendix A. First, a square domain is defined in Cartesian coordinates that is 1.2m on each side. The domain is discretized into pixels 0.25mm on each side. There are approximately 16 million total pixels. A rectangular dielectric object is then defined in the middle of the grid. It is small compared to the grid, only 8cm x 20cm to keep it far from the domain s edges. It is assigned relative permittivity of 8. The edges of the dielectric object are a gradient with permittivity decreasing to unity over the span of 3.025mm. This is done to facilitate convergence of the solution. Next, an electric field is defined to exist with a uniform strength of one V/m, pointing in the x direction over the entire domain. This field represents the free space field created by a coil. It is physically unrealizable with the dielectric present. To correct the electric field, correction charges are identified on the edge of the object (in the gradient zone where permittivity decreases from 8 to 1). These charges were calculated with equation 44, which is the discontinuity in the normal component of electric flux density across a boundary. For our algorithm, the correction charges are then treated as free charges and the dielectric object is temporarily ignored. The electric potential is related to free charge density, f,bypoisson sequation: r 2 = f / (48) where is the permittivity of the medium. This equation can be solved using the Jacobi method [81]. We use the Jacobi method rather than a relaxation technique in part for its conceptual simplicity. With the Jacobi method, it is easy to exploit the vectorized method of calculation that MATLAB excels at. The potential took about nine hours to compute on a MacBook with 2.6GHz Intel i7 processor and 16GB of 1600MHz DDR3 RAM. Finally, the electric field was calculated from potential. This field is the correction field which must be subtracted from the original field in order to find the corrected field. Plots of the final, corrected electric field and electric flux density were examined and found to satisfy physical boundary conditions. That is, the component of electric field tangential to the boundary was continuous while the component of electric flux density normal to the boundary was continuous. Thus, the procedure of introducing correction 41

47 9 CALCULATING ELECTRIC FIELD - JACOBI METHOD charges and subtracting the field that they produce corrects for the presence of dielectric objects. The various steps of the procedure are shown in figures 15 and

48 9 CALCULATING ELECTRIC FIELD - JACOBI METHOD Figure 15: 1) A dielectric block is defined (shown in red). 2) An expanded view of the block. 3) The initial electric flux density appears identical to the block. This is because the electric field is initially uniform everywhere. 4) The correction charges are identified. Since the electric field points downward along the vertical direction of the plot, they appear along the top and bottom edge of the dielectric. In all figures of fields and flux density, only the x component is shown. The x component points down vertically in the figure. All units are SI. 43

49 9 CALCULATING ELECTRIC FIELD - JACOBI METHOD Figure 16: 1) The correction electric field is calculated from the correction charge density. It strengthens the electric field within the dielectric and weakens it in free space. 2) The total electric field is calculated by adding the correction field to the original. Note that it is appropriately continuous for the boundary parallel to its direction (downward vertical). 3) The total electric flux density now obeys the boundary condition that its normal component be continuous across boundaries. Compare this figure with the initial flux density in 3 of Figure 15. In all figures of fields and flux density, only the x component is shown. The x component points down vertically in the figure. All units are SI. 44

50 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY 10 Calculating the E ect of Dielectric Media with Axial Symmetry 10.1 Calculating the initial electromagnetic field For simplicity, the first simulation to employ the algorithm of section 9 exploited cylindrical symmetry to calculate the initial electromagnetic field established by the coil. This section describes that implementation. A coil made of a single, circular loop is studied. Cylindrical coordinates are used where points away from the coil s center, points in the direction of the coil s curve, and the z axis passes through the coil s center. The coil is centered about the z axis at z =3.25cm. The loop has a radius of 10cm and is discretized into three three hundred sections. A current of 1A is specified and the Biot-Savart Law is used to calculate the magnetic flux density. The field values were calculated at points along a plane orthogonal to the plane occupied by the coil (the plane s axes point in the z and directions). Magnetic flux density is shown in Figure 17. The plane covers a region of 15cm along the z axis and 20cm along the axis. Sampled every 0.5mm, this creates a grid of 401x301 points. Via symmetry, B =0. For a coil of radius a, the analytic expression for the axial component of magnetic field at =0is: B z = µ 0 2 a 2 I (49) 4 (z 2 + a 2 ) 3/2 The simulation and analytic expression agree very well for all points along the z axis (percent error less than 0.8%). We conclude that discretizing the coil into three hundred points is su cient. The circumferential component of E F can be calculated using Faraday s Law, while the other components remain unknown. This is a satisfactory approximation for our purpose since the circumferential component dominates for a circular loop. Obtaining the electric field requires that we introduce time variation to the problem. A frequency of 5MHz was chosen. To employ Faraday s law, we compute the magnetic flux passing through planes parallel to that occupied the coil. The result is plotted in Figure

51 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 17: The magnetic flux density generated by a coil with 10cm radius and 1A of current is plotted. This view shows a cross-section of the coil. As a result, the coil is only visible at two points within the plot (located at = ± 10cm, z = 3.25cm). The region that coil is centered over is the region used in NQR detection. This region is strongly dominated by the z component of the field. 46

52 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 18: The circumferential component of the electric field, E, generated by the circular coil. 47

53 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY 10.2 Defining the Dielectrics The boundary condition (Eq. 43) pertains to the component of the electric field that is normal to the boundary between dielectric media. In this example, the initial electric field has only a circumferential component. Therefore, to study the e ect of dielectric objects, it is necessary to simulate objects with discontinuities along the direction. This requires breaking the axial symmetry that has been used so far. To extend the volume being studied to three dimensions, we introduce the dimension. The 2-D solutions for initial electric and magnetic field are replicated across to attain the free-space 3-D solution. With this new 3-D representation, it is possible to generate plots of the electric and magnetic field in planes defined parallel to the coil as we did in section An example is Figure 19, which shows the z component of the magnetic field. Some care must be taken in handling the data now that it is stored in Cartesian coordinates. The data is initially stored as a cylindrical 3-D array which must be converted into Cartesian coordinates. The resulting array is not monotonic as is the usual case when acartesiangridismadefromthestandardmeshgrid.m and ndgrid.m functions. Figure 19: The z component of the magnetic flux density is plotted for a plane defined 1cm away from a coil. The coil has a radius of 10cm and 1A of current. 48

54 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY With the new 3-D coordinates, it is possible to define a geometry array, herein referred as the GT array. This is an array which contains information about the position of physical objects and their electric properties. For this example, we choose to make a very simple object. Our array containing information about the volume is discretized into voxels that have a chockstone shape. Voxels are defined to lie between radii 1 and 2,angles 1 and 2, andbetweenaxialpositionsz 1 and z 2.TheGTarraycontainsinformationaboutwhat substance each voxel in the volume is made of. Objects are constructed by defining the material of a certain set of voxels. For simplicity, we choose to construct objects that have the same basic chockstone shape as an individual voxel. To visualize the constructed object, several MATLAB functions initially appear suitable such as isosurface.m and isonormals.m. However, these functions rely on the underlying coordinate system being Cartesian and monotonic. While we have converted our coordinate system to Cartesian coordinates from cylindrical, it is still not monotonic. Therefore, we must write our own function to visualize objects defined in cylindrical coordinates. It is powered by the low-level patch.m function. This function takes the eight corner points of each voxel and uses them to define its six faces. Each face is broken into two triangles which patch.m can be used to plot. In reality, the two voxel faces that have normal components along the radial direction are curved. However, since each voxel is small, we can approximate them as being planar for visualization purposes. A sample object is shown in Figure 20. Figure 20: Simple chockstone shaped objects are created by defining adjacent voxels to be made of the same material. 49

55 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY 10.3 Finding the correction charges and current density With the dielectric objects defined, the correction charge can be calculated. First, the electric flux density, D F is calculated by multiplying E F at each voxel by the dielectric constant of the material occupied by that voxel (as defined by the GT array). Next, it is necessary to identify discontinuities in D F.Theboundarycondition(Eq.43) specifies that only the component of D F normal to the boundary between media must be considered. In this case, only the component of D F is known. Therefore, the boundaries that matter are those for which permittivity changes in the direction. Discontinuities in D F along the radial and axial directions can be ignored. To calculate discontinuities in the circumferential direction, we shift D F by one index position along, taking care that the last value stored in the array (at =(2 /N)(N 1)) properly wraps back to the first index position (at =0). TheshiftedD F array is subtracted from the original D F array in order to reveal discontinuities; the resulting array contains information about the correction charge density ( C ) present at each point in the volume. This shift-and-subtract operation is also useful for identifying locations where permittivity changes. For example, the operation can be applied to the GT array so that visualization shows only the boundaries between media in the direction. This operation is performed in Figure 21. Since the computer simulation is discretized, C calculated by the shift-and-subtract method will generally be nonzero between all voxels, even including those for which permittivity is constant and D F is continuous in reality. With su ciently fine discretization, C will approach zero for the points where D F is continuous. To make the simulation more e cient, we manually set C =0wherepermittivityisknowntobeconstant. Thisisdone by doing a shift-and-subtract operation on the GT array to identify boundaries. C is set equal to zero for all locations which do not coincide with a boundary. It is important to note that that the indices of the new C array do not align with the coordinate arrays. Instead, C exists shifted 1/2 an index position along the circumferential direction. We calculate Q C by multiplying C by the area of the boundary between voxels. Q C can then be treated as a point source. The correction current density due to each correction charge is calculated using Eq. 47. First, the current density at all points in the volume due to a single correction charge is calculated in a single vector operation. The current density due to each subsequent charge is calculated the same way and summed withf the prior contributions. Current density is calculated this way since attempting to fully vectorize the calculation (the contributions due to each point being calculated simultaneously) would overwhelm the computer s RAM. The 50

56 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY (a) Objects Defined (b) Edges Detected Figure 21: Top: Two objects are shown. Bottom: The shift-and-subtract operation is performed on the objects in the top of the figure. The result reveals the faces of the objects which have surfaces normal to. These are the faces for which C will be nonzero. 51

57 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY current density is best visualized with a quiver plot as shown in Figure 22. As expected, current density points from the boundary where the correction charge is positive to where it is negative. Current density is strongest close to the boundary Calculating the Correction to Magnetic Field Finally, the corrected magnetic field can be calculated from current density using Eq. 37. Using the low-frequency approximation, the time dependence of Eq. 37 can be ignored. Eq. 37 calls for current density to be integrated over the source volume. We accomplish this by multiplying each voxel s current density by the source voxel s volume. The computational procedure is similar to that used to find the current density: the magnetic field at all voxels due to a single voxel is calculated in a single operation and then summed with the contributions due to all other voxels. The magnetic field is output in Cartesian coordinates. Since this function must take N 2 cross products where N is the number of voxels in the volume, it can take a long time to run. For a volume discretized into a 25x50x25 grid, the function takes about eight minutes to run on a Windows PC with a 3.0GHz dual core intel processor and 4GB RAM. To perform our first assessment of biological tissue on magnetic fields, we choose to study liver tissue. With a dielectric constant of about 425 at 5MHz, it has one of the highest permittivities of any tissue found in the human body. It is also a fairly large organ, weighing about 1.5kg. We thus use the liver as an example of the worst case field distortion that the body may create upon the magnetic fields used in NQR. Achockstoneshapedobjectwithpermittivityof425iscreatedtosimulateapieceof liver tissue. Its length in the z dimension is 5.5cm and it has a total volume of 231cm 3.Itis positioned with its nearest face 0.8cm from the plane of the coil. The simulated tissue and its position are visualized in Figure 23. The domain of the simulation is discretized into a 68x70x39 grid for 185,640 total voxels. The correction magnetic field due to the simulated liver tissue is calculated. The calculation takes 1.9 hours on a MacBook with 2.6GHz Intel i7 processor and 16GB of 1600MHz DDR3 RAM Results and Discussion The change in magnetic field: The z component of the magnetic field is studied since it is strongly dominant for a circular coil, as is shown by Figure 17. The e ect of the simulated tissue on the magnetic field is shown in Figure 24. The left column shows the initial magnetic field created by the coil. This is the field that would exist in free space. The right column 52

58 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY (a) Edges Detected (b) Edges Detected Figure 22: Top left: An object (red) is shown in a plane in front of the coil. Top right: The correction current density created in response to the object. Bottom: An enlarged view of the current density plot. Note how the current density originates at the boundaries of the object normal to electric field. 53

59 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 23: Left: the simulated liver tissue viewed in a 2D plane parallel to the coil. The plane is located 2cm from the coil. Right: a 3D view of the simulated liver from an angle. shows the correction to the original magnetic field. The three rows show the field at planes 2cm, 4cm, and 6cm from the coil. Qualitatively, the e ect of the dielectric object is as expected. The correction field is strongest within and just outside of the dielectric object. However, much more important than the shape of the correction field is its strength. It is far weaker than the free space field. Note that the color axis used for the initial field is three orders of magnitude larger than that used to depict the corrected plots. To quantify the di erence in strength between the initial and correction field, we divide the correction field by initial field. The result is plotted for the plane 2cm from the coil (Figure 25). The value of the correction field is never greater than 4/10,000 of the maximum initial field. The e ect on nuclear magnetization: To estimate whether the correction field would have any e ect on an NQR measurement, we recall Equation 23 which determines nuclear magnetization as a function of field strength and pulse length. It is also helpful to review Figure 5. The nutation angle which maximizes nuclear magnetization is B = 2.08 rad and this is referred to as the /2 pulse. This pulse is used to achieve maximum FID signal. Intuitively, it is apparent from the equation and Figure 5 that a correction field that is 4/10,000 the strength of the original will not have a significant e ect on magnetization for the /2 pulse. ThisisbecausethederivativeofEq. 23iszeroatthefunction smaximum. Afractionalchangeof4/10,000willresultinafractionalchangeinmagnetizationthatis 54

60 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 24: The left column shows the initial magnetic flux created by the coil. This is the flux that would exist in free space. The right column shows the correction to the original magnetic flux due to simulated liver tissue. The three rows show the field at planes 2cm, 4cm, and 6cm from the coil from top to bottom. 55

61 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 25: The correction flux is divided by , the highest flux of the initial pattern for the plane 2cm from the coil (see the left column of Figure 24). The correction flux is small compared to initial flux. 56

62 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY even smaller. The e ect of departing from this optimal value is examined in Figure 26. It shows that the percent change in magnetization is less than the percent change in nutation angle for small changes. For example, a nutation angle 5% less than is optimum results in magnetization that is 99.71% of the maximum value. Even with a 15% decrease in nutation angle, magnetization remains over 97% its maximum value. Nutation angle is directly proportional to the applied magnetic field. If pulse length is held constant, the x-axis of Figure 26 could simply be relabeled percent change in magnetic field. Therefore, the correction field calculated in our example for liver tissue would have an unnoticeable e ect on maximum magnetization. The correction field calculated for liver tissue was 4/10,000 the strength of the free space field. Liver has one of the highest electrical permittivities in the body, so this is a worstcase simulation. For argument, suppose that the correction field that exists in real life is fifty times stronger than calculated. It would then be 2% of the free space field. Referring to Figure 26, the nuclear magnetization would still be 99.95% of its maximum value. We therefore expect the correction field to have a negligible e ect on NQR measurements for any field application that uses 90 pulses. However, many modern pulse sequences also make use of pulses to refocus the spins and create echoes. The pulse is that which creates the first zero of nuclear magnetization and occurs at 4.49 rad. Here, the derivative of magnetization is not zero, and so changes in field strength will have larger e ect, plotted in Figure 26. A five percent change from the pulse creates nuclear magnetization that is 10% of its maximum value when it should be zero. For a pulse sequence using both /2 and pulses, inhomogeneities in the magnetic field will have larger impact on the pulse s e ect. Although the derivative of the magnetization expression is not zero for the pulse, nor is it particularly large. For example, a 1% departure from the pulse would create magnetization that is 2% of its maximum value. The correction field s total e ect on a series of echoes may be larger than the e ect of an individual pulse on the FID signal since there is the chance for error to accumulate during each refocusing pulse. Nevertheless, a correction field that is 4/10,000 the strength of the original field is very small. This change is much smaller than the di erence in initial field strength that is likely to exist between two points of a given target. Therefore, even in the case of refocusing pulses, dielectrics are unlikely to have a noticeable e ect on the NQR signal. 57

63 10 CALCULATING THE EFFECT OF DIELECTRIC MEDIA WITH AXIAL SYMMETRY Figure 26: Top: If the nutation angle experience a small change from its /2 value, nuclear magnetization falls o very slowly from its maximum. Magnetization is plotted as a function of the percent change in nutation angle from the /2 pulse. Bottom: small deviations from the pulse have larger consequences. A 5% change from the pulse results in nuclear magnetization decreasing from 0 to -10% of its maximum value. 58

64 11 GENERALIZATION TO ARBITRARY GEOMETRY 11 Generalization to Arbitrary Geometry 11.1 Concept The simulation of the previous section exploited axial symmetry. This method was conceptually simple and allowed Faraday s Law to be employed directly to calculate the initial (free space) electric field. The drawbacks of this approach are that only circular coils can be studied, and that the radial and axial components of the electric field are ignored. This section describes a more general approach that allows arbitrary coil geometries. The code is given in Appendix B. Without being able to employ the integral form of Faraday s Law, we need a new way to compute the electric field. Equation 41 from the Darwin model provides this means. First, the magnetic field is calculated over a large region surrounding the coil. Equation 41 is then applied to calculate the Faraday component of electric field. Due to the close similarity in form between this equation and the Biot-Savart law, the problem is computationally identical to calculating the magnetic field from correction current density. Therefore, the same semi-vectorized approach discussed in the last section is applied. Once the initial (free space) electric field is known, the algorithm can proceed in a manner identical to the previous section. The correction charges due to dielectrics and the associated current density are identified. The correction magnetic field is then calculated Free Space Fields Cartesian coordinates are chosen for the generalized program. A circular coil is defined with a radius of 10cm. We simulate 5MHz frequency and 1A peak current. The z component of the magnetic field is shown in Figure 27. The electric field is calculated with equation 41 as discussed in section The grid used in this example is made of 101x101x25 points spanning 0.3 x 0.3 x 0.072m in 3mm increments. There are N total points and the time to compute the free space electric field from magnetic field is three hours on a MacBook with 2.6GHz Intel i7 processor and 16GB of 1600MHz DDR3 RAM. Since the computation of correction magnetic field from correction current density uses the same function, it also takes three hours for the N point grid. 59

65 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 27: The z component of magnetic field is shown for a plane located 1.0cm from the coil. 60

66 11 GENERALIZATION TO ARBITRARY GEOMETRY 11.3 Defining Objects As in the previous section, objects are created by choosing the permittivity of individual voxels. In principle, any shape of object can be created this way. In this example, we construct two di erent rectangular prisms of dielectric material as shown in Figure 28. They are each assigned relative permittivity of 500 to simulate kidney tissue, which has one of the highest permittivities of biological tissue. They are positioned with their nearest face 1cm from the plane of coil. Figure 28: A pair of objects with the same permittivity as kidney tissue are defined in cartesian coordinates. The coil is shown in blue Correction Charge and Current In the case of axial symmetry, only the faces of dielectric objects that were normal to the circumferential direction were considered. With the generalized program, all sides of the object are considered. The shift-and-subtract scheme of section 10.3 is applied for all three dimensions to find the correction charges. In this scheme, the x component of electric field is used to find charges that exist where media changes along the x direction. Likewise for the y and z directions. The resulting current density from all charges is then calculated. Figure 29 shows the current density created by the pair of dielectric objects. The result is as we expect. As was the case for axial symmetry (Figure 22), a net current density is established within the object to compensate for the object s presence. An interesting e ect is that the current density curls about some of the sharp corners 61

67 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 29: Top left: The dielectric prisms as viewed in a plane 2cm from the coil. The coil is shown in black. Top Right: The current density 2cm from the coil is plotted. The coil s position is shown in black. Bottom: A larger view of the current density formed by the lower left object. 62

68 11 GENERALIZATION TO ARBITRARY GEOMETRY of the object. These create half-loops of current that are expected to create strong magnetic field in their centers. This e ect may play a dominant role wherever dielectric objects present sharp corners Correction Magnetic Field The correction magnetic field is calculated from the current density using the Biot- Savart Law. The calculation takes three hours since each voxel of the domain contributes to the magnetic field at every other voxel. The results are shown in Figure 30. The magnetic field is strongest at the corners of the objects. At these points, the correction current either curls to form half loops of current or diverges strongly. As was the case for axial symmetry, the correction field is much smaller than the original field and is unlikely to have an e ect on NQR measurements. The correction field is on the order of Wb compared with 10 6 Wb for the initial field The e ect of frequency Permittivity increases as frequency decreases, so it is worth investigating the e ect of frequency on the correction field. As shown in Figure 14, the permittivity of biological tissue at 1MHz is generally two to four times the permittivity at 5MHz. However, the electric field is directly proportional to frequency, so the decrease from 5MHz to 1MHz results in a field one fifth as strong. This is expected to create proportionally weaker correction charges. Therefore, we expect the correction magnetic field to be smaller at 1MHz than at 5MHz. Meanwhile, the magnetic field has constant strength regardless of frequency. Therefore, the correction field is expected to be smaller in both absolute terms and relative to the initial field at 1MHz than at 5MHz. We test this hypothesis with the same geometry as described in the previous section. The permittivity of the objects is increased from 500 to The resulting correction magnetic field is shown in Figure 31. As expected, the field is very similar to the 5MHz case, but weaker all around Other Coil Designs Many coils popular for NQR use are not simply circular loops. The generalized program is capable of handling arbitrary coil shapes. Coils are discretized into a set of discrete sections. Each section is defined by its position in space, direction, and length of the wire which it represents. This can be thought of as a series of vectors lined up head to tail, 63

69 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 30: Top: The z component of the correction magnetic field 2cm from the coil. Bottom: The z component of the initial field. In both top and bottom, units are Wb. 64

70 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 31: The correction magnetic field at 1MHz. The color axis is kept the same as was used in Figure 30 for 5MHz. The basic shape is the same of the field is the same as in the 5MHz case, but it is weaker. connecting consecutive positions sampled along the coil. Using this very general formulation, many coil shapes are possible and there is no requirement of symmetry. Mozzhukhin discusses using a coil with logarithmically spaced windings [60]. Such a coil is shown in Figure 32. As another example, a gradiometer is made with linear spiral windings and then compressed towards its center axis. It is shown in Figure Nonuniform Grid The preceding calculations were performed with a single Cartesian grid with regular spacing between voxels. It may be useful to use a nonuniform grid in some cases. For example, in the region very close to the coil, the magnetic field changes much more rapidly than it does far away from it. In order to extract as much information as possible from the magnetic field in this region, fine discretization is necessary. However, fine discretization over the entire volume of interest would lead to excessively long computation times. A nonuniform grid may be used to achieve a high sampling rate near the coil while using a lower sampling rate far away to keep computation time low. We chose to employ a very simple nonuniform grid in which several uniform grids with 65

71 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 32: Top: A spiral coil with logarithmically spaced windings. Bottom left: The z component of magnetic field 1cm from the coil. Bottom right: The magnitude of magnetic field 1cm from the coil. 66

72 11 GENERALIZATION TO ARBITRARY GEOMETRY Figure 33: Top: A gradiometer coil with linearly spaced spiral windings. Bottom left: The z component of magnetic field 1cm from the coil. Bottom right: The magnitude of magnetic field 1cm from the coil. 67

73 11 GENERALIZATION TO ARBITRARY GEOMETRY di erent voxel size are placed adjacent to one another. The method is analogous to stacking slabs where each slab has di erent spatial resolution. For example, a narrow box is defined just large enough to contain the coil. This region is discretized with high resolution. Next, grids are defined on either side of the original box. These secondary grids have lower resolution than the first. A third grid is then added in the same manner. This style of grid construction is visualized in Figure 34. When a nonuniform grid is used, information in one grid section must be used to compute fields in another. This capability has been programmed. Vector fields such as current density or magnetic field in one grid can be used to determine magnetic field or electric field in a di erent grid. Figure 34: The blocks of this figure represent the di erent individual grids that contribute to the total grid. Each prism (color) has di erent spatial resolution. In this visualization, the bright blue slab would contain the coil and be discretized with fine resolution. 68