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# BASIC PHYSICS I Physics 3

UCI

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This 28 page Class Notes was uploaded by Carlotta Dare DVM on Saturday September 12, 2015. The Class Notes belongs to Physics 3 at University of California - Irvine taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/201926/physics-3-university-of-california-irvine in Physics 2 at University of California - Irvine.

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Physics 3A Using Newton39s Laws 39 Getting back to the quotreal worldquot FRICTION 0 What is friction Shoup 7 89 Physics 3A Using Newton39s Laws 39 Getting back to the quotreal worldquot FRICTION 0 What is friction 39 Is a force which opposes impending or actual motion 0 We depend on friction every day walking running etc Friction is also a bane quotsucks energy out of mechanical systemsquot 0 Properties of Friction Two Cases 39 Case 1 No motion between surfaces in contact g0 H l l r ll r 39 Consider mg Shoupr 89 Physics 3A Using Newton s Laws 39 Case 1 No motion between surfaces in contact continue 0 acts between 2 surfaces in contact 0 increases as applied force parallel to surfaces increases up to a maximum level 0 independent of quotarea of contactquot 0 maximum value is proportional to normal force so u ECOEfflClEHt of static friction f m 5 a depends on material in contact mg Shoup r 90 Physics 3A Using Newton s Laws 39 Case 2 Motion between surfaces in contact F with motion lt F no motion f fmax 0 Again Ff is proportional to normal force k 2coefficie nt of kinetic friction I1 Usually uk lt us MJ 1 7 7 I r a 7 iquot l l Molinn l l J 1 H Hl F n mg Shoup 7 91 Physics 3A Using Newton s Laws Example coefficients of static and kinetic friction l TABLE mman l39l39ulimlquot Note The direction of fs is parallel to surfaces in contact and opposite to quotapplied forcequot Show Physics 3A Using Newton s Laws What causes friction 39 attraction between atoms electromagnetic only parts that are in actual contact o Ffmax is force needed to break quotbondsquot mg which are formed by contact a Ff kinetic lt Ffmax because in motion bonds can39t form over as large an area as when no motion 0 friction force is independent of total area in contact Shoup r 93 Physics 3A Using Newton s Laws 39What causes friction Shoup r 93 Physics 3A Using Newton s Laws Another type of force that opposes motion of an object is a resistive force 0 Situations involving resistive forces 39 A swimmer swimming through water 39 A skydiver falling from a perfectly good plane 0 A marble dropped in molasses 39 Consider no air resistance i What if air reistance is included Shoup r 94 Physics 3A Using Newton s Laws Another type of force that opposes motion of an object is a resistive force 39 Situations involving resistive forces A swimmer swimming through water A skydiver falling from a perfectly good plane 39 A ball dropped in molasses Consider A no air resistance Wlth all I resistance I Shoup r 95 Physics 3A Using Newton s Laws Write Newton39s 2 1 for this case 2 Fy mg Rmg bvmami 55 39This is a differential equation in v Solving it yields bt v1 e m VT1 e T 56 Shoup r 97 Physics 3A Using Newton s Laws Resistive force is always opposite the direction of the motion Rea1life resistive forces can be very complicated functions we consider only two quotmodelsquot of these 0 Model 1 Resistive force is proportional to object39s velocity where v is the velocity and b is a constant that depends on the medium and the shape and dimensions of the object Example Shoup r 96 Physics 3A Using Newton s Laws 0 Where In VT g and 1 b b avT is called the terminal speed its the speed where the weight mg is balanced by the resistive force R thus a 0 1 is called the quottime constantquot time it takes for speed to reach 0635vT 1 e 10635 bt v1 e m VT1 e T V Shoup r 98 Physics 3A Using Newton39s Laws I Model 2 Resistive force is proportional to object s speed squared R DpAV2 an Sky surfer I Example can be modeled by Shoup 7 99 Physics 3A Using Newton39s Laws I Model 2 Resistive force is proportional to object s speed squared R DpAV2 an I The resistive force on large objects moving through air can be modeled approximated by 57 where I p is the density of air I A is the crosssectional area of the object I v is the object s speed I D is the quotdrag coefficientquot I usually measured for each object I 05 for spherical bodies can be up to 2 for irregular bodies I Questions I Is it better to y in low density or high density air Low I Is it better to y slow or fast Shoup 7 100 Physics 3A Using Newton39s Laws I Model 2 Resistive force is proportional to object s speed squared DpA an I The resistive force on large objects moving through air can be modeled approximated by 57 where I p is the density of air I A is the crosssectional area of the object I v is the object s speed I D is the quotdrag coefficientquot I usually measured for each object I 05 for spherical bodies can be up to 2 for irregular bodies I Questions I Is it better to y in low density or high density air Shoup 7100 Physics 3A Using Newton39s Laws I Model 2 Resistive force is proportional to object s speed squared z pr an I The resistive force on large objects moving through air can be modeled approximated by 57 where I p is the density of air I A is the crosssectional area of the object I v is the object s speed I D is the quotdrag coefficientquot I usually measured for each object I 05 for spherical bodies can be up to 2 for irregular bodies I Questions I Is it better to y in low density or high density air Low I Is it better to y slow or fast Slow I Is it better to y a square plane or a quotroundquot plane Shoup 7100 Physics 3A Using Newton s Laws 39 Model 2 Resistive force is proportional to object39s speed squared DpAV2 57 0 The resistive force on large objects moving through air can be modeled approximated by 57 Where I p is the density of air 39 A is the cross sectional area of the object 39 v is the object39s speed 3 D is the quotdrag coefficientquot 39 usually measured for each object I 05 for spherical bodies can be up to 2 for irregular bodies 0 Questions I Is it better to y in low density or high density air Low 39 Is it better to y slow or fast Slow 39 Is it better to y a square plane or a quotroundquot plane Round Shoupr 100 Physics 3A Using Newton s Laws 0 We can compute the terminal velocity vT in this case also by settinga0 f DpA D A ag L 0 g 2m 2m Z V TABLE 52 1 llHllIl1l AlkllH munh l WW 7 Hmmuh n l mm W V39r u i i i i H i i quotg Shoup 7 102 Physics 3A Using Newton s Laws Can analyze this situation using Newton39s 2 1 Law Consider a ball in free fall in air R ZFma mgRma v1 mg lD pAv2ma 58 mg D A ag p V2 59 2m dV dv D A but a so 0 V2 dt dt 2m g another differential equation Shouprlol Physics 3A Using Newton s Laws 9 Apply this to a skydiver Fnet C 3 a 1000 N 100 kg 5 51000 N a 100 mm2 3 down Shoup r 103 Physics 3A Using Newton s Laws What does Newton39s Laws say about Uniform Circular Motion 39 recall J 3 5 39 Newton39s 2m1 law say if a is not zero there must be a force 39 What39s its magnitude and direction Shoup 104 Physics 3A Using Newton s Laws Consider ball on a string top View What happens if string breaks Shoup r 105 Physics 3A Using Newton s Laws What does Newton39s Laws say about Uniform Circular Motion 0 recall 4 a5 39 Newton39s 2m1 law say if a is not zero there must be a force 39 What39s its magnitude and direction in toward center of circle with magnitude 2 139 Shoup r 104 Physics 3A Using Newton s Laws Consider ball on a string Shoup r 105 Physics 3A Using Newton39s Laws 39 In uniform circular motion the speed is constant Lets generalize this to noniconstant speeds 39 From chapter 3 we had aaa n where ar is the quotchange in directionquot of the velocity per unit time a and al is the change in magnitude of the velocity per unit time 39 As before Newton39s 2 1 says there must be a force responsible for this a 2Frmar 2am changes direction changes magnitude 39ty of velocity of veloc1 Physics 3A Using Newton39s Laws 39 Forces The Real Story I Fundamentally what are contact forces quotactually they are due to the electromagnetic forcequot 0 Up until about 1967 physicists thought there were 4 fundamental forces a Gravitational force weakest force of nature Electromagnetic force 2 1 strongest force of nature Strong force also called nuclear force Strongest force 39 Weak force 2 1 weakest force of nature 39 In 1967 it was predicted that the Electromagnetic force and the weak force were really parts of a force called the Electroweak force This was con rmed in 1984 with the discovery of a particle call the W particle Lets look at each of these in more detail Shoupr l 7 Physics 3A Using Newton39s Laws 39 Forces The Real Story I Fundamentally what are contact forces Shoup 7 m7 Physics 3A Using Newton39s Laws aGravitational Force 39 In trying to understand why the planets moved the way they did Newton deduced a Universal Law of Gravitation It states quotthat every particle in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them Shoup 7 103 Lecture 1 Motivation for course The title of this course is condensed matter physics77 which includes solids and liquids and occasionally gases There are also intermediate forms of matter eg glasses rubber polymers and some biophysical systems Basically this is the branch of physics that covers the things we see and touch in everyday life ie real stu r77 Most of the materials we meet in every day life are amorphous but since we understand crystalline materials so much better that is what we will spend most of our time talking about Why should we study condensed matter physics H Because its there77 N3 Real life physics 03 Frontier of complexity 7 more is di cerent77 Think of a spin a multitude gives all sorts of magnetism due to interactions Hgt Analogies with elementary particle physics eg Higgs mechanism topological winding numbers broken symmetry etc 5quot Practical applications eg transistors Drude Theory of Metals a Phenomenology of metals 7 high electrical conductivity shiny re ecting ductile malleable high thermal conductivity etc Found generally in columns 1A and 2A of the periodic table among heavier lll Vl column elements and in transition metals and rare earths In general they have 1 2 extra electrons above a closed shell Typically pmml N few uQ cm versus pmwlam N 1017 Q cm for insulators like polystyrene b Basic concepts The extra electrons are called conduction electrons and they are free to move within the volume Core electrons stay home The number of conduction electrons N 3 n5 ZnAmgadm N 1022 7 1023 electronscm 11 Where Z is the chemical valence see table 11 of AM Electronic density is often de ned in terms of T5 radius of sphere Whose volume is equal to the volume per conduction electron V 1 mg 3 13 i n N n 3 7 7 47m Typically rs N 1 7 3121 Natural unit is Bohr radius 10 7377162 0529 X 10 8 cm 5276 lt10 For comparison7 note that a typical atomic ionic radius is N 03 7 2121 So conduction electrons occupy a larger sphere than ions c Electrical Conductivity resistivity 1 0 ja Epj Let A cross sectional area of Wire7 L length7 I I ZandVEL VpLipZLiIR i R 2 or p R Longer Wires have more resistance Larger A means more manuverability for electrons and less resistance As we said before7 p3000K N 1uQ cm At not too low T7 pT N T phonon scattering As T a 07 pT a po residual resistivity due to scattering of impurities This yields Matthiessen7s Rule p0po T l Assumptions of the Drude model i Electrons move independently under the in uence of local electric eld between col lisions ii Collisions are instantaneous7 With some unspeci ed but energy nonconserving mech anism iii Collisions are random7 With probability dtT per unit time no history dependence iv Electrons totally thermalized to local temperature by inelastic collisions DC Conductivity 07 E const Electric current j 7 7265 12 The minus sign is due to the negative charge of the electrons There are two contributions d to m total dj 7 if dj dt total collision field Field a da 1 a a Force Fip m ieE dt dt d d r 2E i ij i i nev 726 ll field dt m Collisions Collisions knock electrons out of the current ow So we expect lt 0 call degrade current prob of collision N fraction of particles a ected 739 is relaxation time CW dt coll T j I ll 739 So 13 dt 7262 E j 7 m total In a steady state with E 00723757 j must be constant 6LT 0 dttotal 2 n T j mE 00E When E E t7 lt7 0a7 ie7 the conductivity has frequency dependence From experimental values of lt70 and 717 we can work out 739 see AM7 table 12 Typ ically7 739 N 10 14 7 10 15 sec at room temperature 7 1 N T At low T7 739 g 10 9 sec and is limited by impurity scattering Matthiessen7s rule 7 1 N 751 7 1 T We can de ne a mean free path E N 17739 How do we estimate 17 Drude used kinetic theory of gases and said va ng 17W N107 at T 300K i E N 1 7 10121 N lattice spacing or distance between ions But this is misleading Should use UF N 108 cms Conduction in a Magnetic Field In the presence of a magnetic eld B an additional Lorentz force acts on the electrons ruler 1376am XB This leads to the Hall E ect Consider a metal bar with current owing in it carried by electrons with average velocity 17 Z 6 e e e gt Vd 0 9 6 gt Y 0 0 6 0 ml ML X Now suppose we apply a magnetic eld in the i direction This initially causes a down ward de ection of the moving electrons O O are E 6 e are Negative charge builds up at the bottom positive charge at the top The transverse electric eld E counters the magnetic force so that the electrons again ow in the 717 direction e e e A gt e e E E 0 6 B Notice that if the charge carriers had been positively charged7 t would point in the opposite direction f is in the same direction as before Thus if we measure the voltage di erence between top and bottom7 the sign should tell us the sign of the carriers We expect negative7 but sometimes its positive More on this later It is easy to determine the magnitude of E by balancing the electric force with the magnetic force in the Z direction Let7s use q rather than 76 th qBB Et 3B c c We know j nq jnq 1 Et JB RHJB nqc 15 where g is called the Hall coe icient For q 76 RH if Experimentally7 R H Note that because E cancels the e ect of the magnetic eld7 we still have jy TOEy 1 erent coor st an ou can c ec t is y 00 mg at i Xperimenta y7 d dhAMY hkh blk E ll t t I this isnt always true Drude model is too simple 0 a AC Conductivity 3 07505 0w Consider an electric eld that is varying in time Et E0 cos wt Re E0 67 The response of the electrons as well as the current Will also vary in time This leads to a frequency dependent conductivity t Re 0671 Where awE0 In general 0a Will be complex7 indicating that j is out of phase With Calculate 0a Start With dj 7 n62Et i dt total m Plug in t joe mt and Email to get d 62ED J0 WJO m 739 1 a nez 72w70 0 739 m a n62 1 a 7 7 E J0 m lt71wigt 0 00 n627 1 i 00 i m 1721417 1721417 a a E If E E0 coswf7 then f 00 0 1 W272 coswt 7 6 Where tan6 441739 We can relate a to the frequency dependent dielectric constant 5a Consider a piece of metal that is free standing Suppose we irradiate it With electromagnetic radiation There Will be no free current ff but there Will be a polarization current because the electrons slosh back and forth a 813 7 1 0E 20E J i P 7 7 7 at 7244 7244 w D8EE4WP i 5 47r i 14mlt7 E w 47r2390w a Plasma Frequency 40739 gtgt 1 At high frequencies 441739 gtgt 1 0390 100 0a V 2 7 1 7 MT 441739 4mm 47139 n62 002 g 1 7 1 7 1 7 l 8a 402739 402 m 2 2 Where m This is called the plasma frequency What does this mean physically top is the characteristic frequency for the electrons to slosh back and forth These are called plasma oscillations7 or plasmons AM give a simple model of this lmagine displacing the entire electron gas7 as a Whole7 through a distance d With respect to the xed positive background of ions surface charge ned N Z ions ml QIlllll N electrons surface charge ned Plasmon longitudinal excitation without E ext The resulting surface charge gives rise to an electric eld of magnitude 47w7 Where 0 is the charge per unit area recall Gauss7 Law The electron gas obeys the equation of motion F iNeE Eat 0 This E is internally generated de iNeE 7N6l47wl 7Ne47mde de 747m62Nd F mi lm i 2 N total number of electrons to A 472 i N n V There is yet another way to derive the plasma frequency go back to 8f 7 at 7 nez f think of sloshing electrons m 739 as producing a polarization current At high frequencies 441739 gtgt l7 to gtgt we can neglect the last term This leaves 8f 7 7262 a 57m 18 Recall the continuity equation 75 7V I So 8f 7262 a 82 7262 a 47m62 V 77E 777VE7 at m l 8752 m m p 2p 2 2 47m62 or W iwpp Where again WP 7 mi 71 form Transverse EM Waves If we shine EM radiation on a rnetal7 it Will not penetrate very far and in fact7 it Will be re ected for low frequencies because the electrons respond quickly enough to screen it At high frequencies7 however7 Ad gtgt WP the electrons aren7t fast enough to respond to EOE and the radiation gets through Thus the metals becorne transparent to ultraViolet light To see this rnathernatically7 go back to Maxwell7s eqns and derive the wave equation p0 vE0 v 30 setu51 VXB47W 305 VXEaaif VXVXEWVXB WV E 7v2E 1341a l87E lof3 CONTENTS 39 Preface g Help 1 About the author 1 Introduction NMRI or MRI Opportunities in MRI Tomographic Imaging Microscopic property responsible for MRI 2 The Mathematics of NMR Exponential Functions Trigonometric Functions Differentials and Integrals Vectors Matrices Coordinate Transformations Convolutions Imaginary Numbers The Fourier Transform 000000000 3 Spin Physics 0 Spin Properties of Spin Nuclei with Spin Energy Levels NMR Transitions Energy Level Diagrams Continuous Wave NMR Experiment Boltzmann Statistics Spin Packets T1 Processes Precession T2 Processes Rotating Frame of Reference Pulsed Magnetic Fields Spin Relaxation Bloch Equations 000000000000000 4 NMR Spectroscopy 0 Time Domain NMR Signal Frequency Convention 90 FID SpinEcho Inversion Recovery Chemical Shift 00000 5 Fourier Transforms 05292004 0619 PM 000000000 Introduction The and Frequency Problem The Fourier Transform Phase Correction Fourier Pairs The Convolution Theorem The Digital FT Sampling Error The TwoDimensional FT 6 Imaging Princigles 00000 Introduction Magnetic Field Gradient Frequency Encoding Back Projection Imaging Slice Selection 7 Fourier Transform Imaging Principles 0 0000 Introduction Phase Encoding Gradient FT Tomographic Imaging Signal Processing Image Resolution 8 Basic Imaging Techniques 0 0000000 Introduction Multislice imaging Oblique Imaging SpinEcho Imaging Inversion Recovery Imaging Gradient Recalled Echo Imaging Image Contrast Signal Averaging 9 Imaging Hardware 0000000 Hardware Overview Magnet Gradient Coils RF Coils Qadrature Detector Safety Phantoms 10 Image Presentation 0000 Image Histogram Image Processing Imaging Coordinates Imaging Planes l 1 Image Artifacts 0 0 0 0 20f3 Introduction RF Quadrature B o Inhomo geneity Gradient 05292004 0619 PM RF Inhomogeneity Motion Flow Chemical Shift Partial Volume Wrap Around Gibbs Ringing 0000000 12 Advanced Imaging Technigues Introduction Volume Imaging 3D Imaging Flow Imaging MRI Angiography Diffusion Imaging Fractional NeX amp Echo Imaging Fast SpinEcho Imaging Chemical Shift Imaging Fat Suppression Echo Planar Imaging Functional MRI Spatially Localized Spectroscopy Chemical Contrast Agents Magnetization Transfer Contrast Variable Bandwidth Imaging T1 T2 amp Spin Density Images Tissue Classi cation Hyperpolarized Noble Gas Imaging Parallel Imaging Magnetic Resonance Elastography Electron Spin Resonance 0 00000 000000000000 13 Your MRI Exam Introduction Screening The Imager The Exam Your Results 0 0000 14 Clinical Images Angiography HeadampNeck Spine Extremities Glossa List of Symbols References Usage Statistics Software License Go to the cover Copyright 1996 2004 JP Hornak All Rights Reserved 3 of 3 05292004 0619 PM Contents Help Glossarx Symbols 1 of 1 05292004 0619 PM The Basics of MRI Chapter 8 BASIC IMAGING TECHNIQUES Introduction Multislice imaging Obligue Imaging SpinEcho Imaging Inversion Recovery Imaging Gradient Recalled Echo Imaging Image Contrast S39gnal Averaging Problems Introduction In the previous chapter you learned the principles of Fourier transform magnetic resonance imaging The examples presented were for a simpli ed 90FID imaging sequence Although the principles were correct some aspects were simpli ed to make the presentation easier to understand Some of these principles will be presented in a little more depth in this section The 90FID imaging sequence will be presented as a gradient recalled echo sequence in this section The principles of multislice imaging and oblique imaging will be introduced Two new imaging sequences called the spinecho sequence and inversion recovery sequence will be introduced Multislice Imaging An imaging sequence based on a 90FID was introduced in Chapter 7 Based on this presentation the time to acquire an image is equal to the product of the TR value and the number of phase encoding steps If TR were one second and there were 256 phase encoding gradient steps the total imaging time required to produce the image would be 4 minutes and 16 seconds If we wanted to take 20 images across a region of interest the imaging time would be approximately 15 hours This will obviously not do if we are searching for pathology Looking at the timing diagram for the imaging sequence with a one second TR it is clear that most of the sequence time is unused E This unused time could be made use of by exciting other slices in the object The only restriction is that the excitation used for one slice must not affect those from another slice This can be accomplished by applying one magnitude slice selection gradient and changing the RF frequency of the 900 pulses E Note that the three frequency bands from the pulses do not overlap In this animation there are three RF pulses applied in the TR period Each has a different center frequency 1 2 and 3 As a consequence the pulses affect different l of 8 05292004 0619 PM slices in the imaged object E Oblique Imaging Orthogonal imaging planes along the X Y or Z axes are easily produced with the imaging sequence presented in the Chapter 7 However what ifthe anatomy of interest does not lie along one of the three orthogonal imaging planes This is where the concept of oblique imaging comes in Oblique imaging is the production of images which lie between the conventional X Y and Z axes Oblique imaging is performed by applying linear combinations of the X Y and Z magnetic field gradients so as to produce a slice selection gradient which is perpendicular to the imaged plane a phase encoding gradient which is along one edge of the imaged plane and a frequency encoding gradient which is along the remaining edge of the image For example if we wanted to image a slice lying along the X axis but passing between the Z and Y axes such that it made an angle of 30 with respect to the Y axis and 60 with the Z axis E the following combination of gradients would he need E Slice Selection Gradient GZ GS Sin 600 G G Cos 60 y s E Phase Encoding Gradient GZ G Sin 300 G G Cos 30 y E Frequency Encoding Gradient Gx Gf The frequency and phase encoding gradients are interchangeable The timing diagram for the sequence looks as follows E SpinEcho Imaging In Chapter 4 we saw that signal could be produced by a spinecho sequence An advantage ofusing a spinecho sequence is that it introduces T2 dependence to the signal Since some tissues and pathologies have similar T1 values but different T2 values it is advantageous to have an imaging sequence which produces images with a T2 dependence The spinecho imaging sequence will be presented in the form of a timing diagram only since the evolution of the magnetization vectors from the application of slice selection phase encoding and frequency encoding gradients are similar to that presented in Chapter 7 The timing diagram for a spinecho imaging sequence has entries for the RF pulses the gradients in the magnetic field and the signal E A slice selective 90 RF pulse is applied in conjunction with a slice selection gradient E A period oftime equal to TE2 elapses and a 180 slice selective 180 pulse is applied in conjunction with the slice selection gradient A phase encoding gradient is applied between the 90 and 180 pulses E As in the previous imaging sequences the phase encoding gradient is varied in 128 or 256 steps between G4m and G m E The phase encoding gradient could be applied after the 1800 pulse however if we want to minimize the TE period the pulse is applied between the 90 2 of 8 05292004 0619 PM and 180 RF pulses E The frequency encoding gradient is applied after the 180 pulse during the time that echo is collected E The recorded signal is the echo The FID which is found after every 90 pulse is not used One additional gradient is applied between the 90 and 180 pulses E This gradient is along the same direction as the frequency encoding gradient It dephases the spins so that they will rephase by the center of the echo This gradient in effect prepares the signal to be at the edge ofkspace by the start ofthe acquisition of the echo The entire sequence is repeated every TR seconds until all the phase encoding steps have been recorded Inversion Recovery Imaging In Chapter 4 we saw that a magnetic resonance signal could be produced by an inversion recovery sequence An advantage of using an inversion recovery sequence is that it allows nulling of the signal from one component due to its T1 Recall from Chapter 4 that the signal intensity is zero when TI T1 ln2 Once again this sequence will be presented in the form ofa timing diagram only since the evolution of the magnetization vectors from the application of slice selection phase encoding and frequency encoding gradients are similar to that presented in Chapter 7 An inversion recovery sequence which uses a spinecho sequence to detect the magnetization will be presented The RF pulses are 18090180 An inversion recovery sequence which uses a 90FID signal detection is similar with the exception that a 90FID is substituted for the spinecho part of the sequence The timing diagram for an inversion recovery imaging sequence has entries for the RF pulses the gradients in the magnetic field and the signal E A slice selective 180 RF pulse is applied in conjunction with a slice selection gradient EA period of time equal to T1 elapses and a spinecho sequence is applied The remainder of the sequence is equivalent to a spinecho sequence This spinecho part recorded the magnetization present at a time T1 after the first 1800 pulse A 90FID sequence could be used instead of the spinecho All the RF pulses in the spinecho sequence are slice selective The RF pulses are applied in conjunction with the slice selection gradients Between the 90 and 180 pulses a phase encoding gradient is applied The phase encoding gradient is varied in 128 or 256 steps between Gdam and G m The phase encoding gradient could not be applied after the first 180 pulse because there is no transverse magnetization to phase encode at this point The frequency encoding gradient is applied after the second 180 pulse during the time that echo is collected The recorded signal is the echo The FID after the 90 pulse is not used The dephasing gradient between the 90 and 180 pulses to position the start of the signal acquisition at the edge of kspace as was described in the section on spinecho imaging The entire sequence is repeated every TR seconds Gradient Recalled Echo Imaging 3 of 8 05292004 0619 PM The 1magmg sequences mentioned thus far have one major d1sadvantage For maX1mum signal they all require the transverse magnetization to recover to its equilibrium position along the Z aXis before the sequence is repeated When the T1 is long this can significantly lengthen the imaging sequence If the magnetization does not fully recover to equilibrium the signal is less than if full recovery occurs E If the magnetization is rotated by an angle less than 90 its MZ component will recover to equilibrium much more rapidly but there will be less signal since the signal will be proportional to the Sin B So we trade off signal for imaging time In some instances several images can be collected and averaged together and make up for the lost signal The gradient recalled echo imaging sequence is the application of these principles Here is its timing diagram E In the gradient recalled echo imaging sequence a slice selective RF pulse is applied to the imaged object El This RF pulse typically produces a rotation angle ofbetween 10 and 90 A slice selection gradientis applied with the RF pulse E A phase encoding gradient is applied next E The phase encoding gradient is varied between G m and G m in 128 or 256 equal steps as was done in all the other sequences A dephasing frequency encoding gradient is applied at the same time as the phase encoding gradient so as to cause the spins to be in phase at the center of the acquisition period EThis gradient is negative in sign from that of the frequency encoding gradient turned on during the acquisition of the signal An echo is produced when the frequency encoding gradient is turned on because this gradient refocuses the dephasing which occurred from the dephasing gradient E A period called the echo time TE is defined as the time between the start of the RF pulse and the maximum in the signal E The sequence is repeated every TR seconds The TR period could be as short as tens of milliseconds Image Contrast In order for pathology or any tissue for that matter to be visible in a magnetic resonance image there must be contrast or a difference in signal intensity between it and the adjacent tissue The signal intensity S is determined by the signal equation for the specific pulse sequence used Some ofthe intrinsic variables are the 2 The spin density is the concentration of signal bearing spins The instrumental variables are the 4 of 8 05292004 0619 PM Repetition Time TR Echo Time TE Inversion Time TI Rotation Angle T2 T2 falls on both lists because it contains a component dependent on the homogeneity of the magnetic eld and the the molecular motions The signal equations for the pulse sequences presented thus far are SpinEcho S k leXpTPUT1 eXpTET2 Inversion Recovery 18090 S k l2eXpTITleXpTPUT1 Inversion Recovery 18090180 S k l2eXpTIT1eXpTPUT1 eXpTET2 Gradient Recalled Echo S k leXpTPUT1 Sin eXpTET2 l Cos eXpTIUT1 In each of these equations S represents the amplitude of the signal in the frequency domain spectrum The quantity k is a proportionality constant which depends on the sensitivity of the signal detection circuitry on the imager The values of T1 T2 and are speci c to a tissue or pathology The following table lists the range of T1 T2 and values at 15T for tissues found in a magnetic resonance image of the human head E Tissue T1 S T2 Ins CSF 08 20 110 2000 70230 White 076 108 61100 7090 Gray 109 215 61 109 85 125 Meninges 05 22 50 165 5 44 Muscle 095 182 20 67 45 90 Adipose 02 075 53 94 50 100 Based on 1 11 for 2mM aqueous Ni l The contrast C between two tissues A and B will be equal to the difference between the signal for tissue A SA and that for tissue B SB csA sB 5 of 8 05292004 0619 PM SA and SB are dete1m1ned by the s1gnal equatlons g1ven above For any two t1ssues there will be a set of instrumental paramenters which yield a maximum contrast For example in a spinecho sequence the contrast between two tissues as a function of TR is graphically presented in the accompanying curve E A contrast curve for tissues A and B as a function of TE is presented in the accompanying curve To assure that signals from all phase encoding steps possess the same signal properties a few equilibrating cycles through the sequence are added to the beginning of every image acquisition The necessity of this can be seen by examining the MZ and MXY components as a function of time in a 90FID type sequence E Note that the amount of transverse magnetization from a 900 pulse reaches an equilibrium value after a few TR cycles This practice lengthens the imaging time by a few TR periods The magnetic resonance community has adopted nomenclature to signify the predominant contrast mechanism in an image Images whose contrast is predominantly caused by differences in T1 of the tissues is called a Tlweighted image Similarly for T2 and the images are called T2weighted and spin density weighted images The following table contains the set of conditions necessary to produce weighted images 1 2 1 2 1 2 It is impressive to see how the choice of the instrumental parameters TR TE TI and affect the contrast between the various tissues of the brain In the accompanying set of graphics you can select an imaging sequence and the imaging parameters the resultant image will be displayed in the graphics window The spinecho images are actual magnetic resonance images of the human brain The remaining images are calculated images based on the signal equations above and a set of measured overall T1 T2 and images ofthe human brain The two bright circles to the bottom right and left sides of each calculated image are spin density standards or phantoms placed next to the head SpinEcho Images 6 of 8 05292004 0619 PM Inversion Recovery Images 18090 Gradient Recalled Echo Images T E 5 ms Signal Averaging The signaltonoise ratio SNR of a tissue in an image is the ratio of the average signal for the tissue to the standard deviation of the noise in the background of the image E The signaltonoise ratio may be improved by performing signal averaging Signal averaging is the collection and averaging together of several images The signals are present in each of the averaged images so their contribution to the resultant image add Noise is random so it does not add but begins to cancel as the number of spectra averaged increases The signaltonoise improvement from signal averaging is proportional to the square root of the number ofimages averaged NeX NeX is more commonly referred to as the number of excitations SNR olt NeX12 Compare the results of averaging together the following number ofimages of a bottle of water 7 of 8 05292004 0619 PM Physics 3A Torque amp Equilibrium 39Lets talk about what causes rotational motion We know forces cause translational motion 39 Another vector quantity derived from forces cause rotational motion 39 This is Torque 39 Consider the figure below We define the torque as 1018 9 We can understand this by considering how one opens 0 a a door r Linc of 3 You push perpendicular to d 7 action the door the door rotates Q 0 You push parallel to the door the door does not rotate Shoupr 111 Physics 3A Torque amp Equilibrium 39Consider the gure to the right 6 What is the sum or the net torque TnetT1T2Fl d1F2 d2 WARNING torque is NOT a force force cause torque 3 torque depend on the force but also on the point of application of the force 39Torque is a vector quantity so we need to define its direction Consider when I push on the bicycle wheel 39 When wheel is rotating counterclockwise we said its angular velocity is up righthand rule 0 Seems natural to say a torque which produces counterclockwise rotation should also be quotupquot Shouprm Physics 3A Torque amp Equilibrium Pushing parallel is the F cos 1 part gt doesn39t produce rotation Pushing perpendicular is the F sin I part TErFsin 4 Fsin g0 Another way to look at it is A quot TEFrsin 1 0 where r sin I is called the 1 iquot Lil of moment arm of the force d zli aclion 0 Its the perpendicular distance O from the rotation axis to the line I of action The larger the moment arm the larger the torque 39We also define the quotsignquot of the torque by quotquot if its resulting rotation would be counterclockwise and by quot quot if its resulting rotation would be clockwise Shouprllz Physics 3A Torque amp Equilibrium lf we choose this then we can define the torque mathematically as 1010 This is the quotvector productquot or quotcross productquot of the vectors r and F D The vector product of two vectors is a vector thus has a magnitude and a direction 39 Magnitude of C A X B czl lalAllslsine 1021 A 39 The direction is given by the quotright hand rulequot 0 point fingers of right hand in direction of A first vector 39 curl fingers into B second vector 39 Thumb points in direction of C Shoup r 1 14

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