Magnetic Resonance: Principles and Applications 2 Hours

PHY422

PHY472

Data Provided: Formula sheet and physical constants

Data Provided: A formula sheet and table of physical constants is attached to this paper.

DEPARTMENT OF PHYSICS & Autumn Semester 2009-2010 ASTRONOMY

DEPARTMENT OF PHYSICS AND ASTRONOMY ADVANCED QUANTUM MECHANICS 2 hours Autumn 2014

AnswerMagnetic question ONE Resonance: (Compulsory) Principles and TWO other and questions, Applications one each from 2 hours section A and section B. Instructions: All questionsAnswer THREE are marked questions. out All of questions ten. The are breakdown marked out ofon twenty. the right-hand The breakdown side on of the the right-hand paper sideis meant of the paperas a guide is meant to as the a guide marks to thethat marks can thatbe obtained can be obtained from from each each part. part.

Please clearly indicate the question numbers on which you would like to be examined on the front

cover of your answer book. Cross through any work that you do not wish to be examined.

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1. (a) Give four examples of nuclear isotopes having zero spin. [2] (b) Give four examples of nuclear isotopes having non-zero spin. [2] (c) Give four examples of nuclear isotopes having non-zero spin and zero quadrupole moment. [2] (d) Give four examples of nuclear isotopes having non-zero quadrupole moment. [2] (e) Consider a nucleus having a spin of 5/2. The nucleus is placed in an external magnetic field. (i) Give an expression for the energy levels due to the interaction of the magnetic moment of such a nucleus with the applied magnetic field. [2] (ii) Using the expression obtained in (i), calculate the frequencies of the allowed spin transitions. [2] (iii) For the case considered in (ii), sketch the corresponding energy levels and indicate the calculated frequencies on your diagram. [2] (f) Consider a nucleus having a non-zero quadrupole moment. The nucleus is placed in an external magnetic field. A non-zero electric field gradient is also present. (i) Give an expression for the energy levels due to the electromagnetic moments of such a nucleus. [2] (ii) Using the expression obtained in (i), calculate the frequencies of the allowed spin transitions using as an example a spin-3/2 nucleus. [2] (iii) For the case considered in (ii), sketch the corresponding energy levels and indicate the calculated frequencies on your diagram. [2]

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2. (a) A mole of protons is placed in an external magnetic field of 8 T in a cryostat with a temperature of 1 milli-Kelvin. The proton gyromagnetic ratio is 26.75 107 1 1 × rad T − s − . (i) Calculate the average spin for a proton in equilibrium in this environment. [4] (ii) Calculate how much energy would be released if spins of all protons were taken out of equilibrium and orientated along the magnetic field. [4] (b) Calculate the average nuclear spin for indium-115 atoms placed in an external magnetic field of 8 T at an ambient temperature of 100 K. Use the gyromagnetic ratio 115 7 1 1 for In of 5.89 10 rad T − s − . Justify any assumptions you make. [4] × (c) Sketch the dependences of the average spin on the spin temperature for the follow- ing nuclei placed in an external magnetic field of 10 T: (i) proton, [4] (ii) indium-115. [4] Sketch both dependences on the same graph and add tick labels on both axes.

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3. Consider a system of spin-1/2 nuclei having a gyromagnetic ratio γ and very long T1 and T2 times. The spins are initially aligned along an external magnetic field B0 directed along the z axis. The external field is highly uniform, so that the dephasing of spins in the experimental conditions described below is negligible. (a) A rotating frame is introduced that is used to facilitate the description of the spin motion in the magnetic field. What would be the commonly used frequency and direction of rotation of this frame for the given set of nuclei? [2] (b) A ’π/2’ radio-frequency pulse is applied using a magnetic field oscillating along the

x axis in the laboratory frame and having an amplitude of B1. For the given spin system, provide a mathematical description of the spin evolution as a function of time in the rotating frame introduced in (a) (i) during the π/2-pulse, [3] (ii) after the π/2-pulse. [3] Describe any assumptions that you made. (c) For the experiment described in (b), provide a mathematical description of the spin evolution as a function of time in the laboratory frame (i) during the π/2-pulse, [4] (ii) after the π/2-pulse. [4] Describe any assumptions that you made. (d) A B-ﬁeld gradient is now applied. Find the value of the B-ﬁeld gradient so that two protons situated 0.1 mm away from each other with no initial phase, develop a phase difference of π after a time of 2 ms. Which time constant would be used

normally to describe this type of dephasing: T2 or T2∗? Justify your answer. [4]

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4. (a) Explain why it is not possible to measure the transverse relaxation time T2 in a free induction decay (FID) experiment. [2] (b) Describe the NMR experiment used for the measurement of the spin-spin relaxation

time T2. Include in your answer a description of the sequence of radio-frequency pulses used (indicating the time intervals between the pulses), the evolution of nuclear spins and the signal measured by a pick-up coil. Specify what frame, laboratory or rotating, is used in your description of nuclear spin evolution. [4] (c) A spin-echo experiment is carried out on a set of proton spins having the dephasing time T2=0.5 s distributed uniformly in a volume of 3 litre. The initial sample mag- 3 netisation along the external magnetic field is 10− A/m. A multiple-echo technique is used with π-pulses applied at 50 ms intervals. The NMR signal is measured with 3 a pick-up coil having a uniform magnetic field per unit current of 10− T/A over the sample. Find the amplitude of the voltage measured by the coil of the fourth spin-echo, if the experiment is carried out in magnetic field of 1 T. Use the proton 7 1 1 gyromagnetic ratio of 26.75 10 rad T − s − . [6] × (d) Explain the need for spin-echo techniques in NMR experiments. [2]

(e) Consider an ensemble of spin-1/2 nuclei having T2∗=1.5 ms and T2=500 ms. (i) An FID experiment is carried out and the NMR spectrum for this nuclear spin ensemble is measured. Find the full width at half maximum (FWHM) in this NMR spectrum. [3] (ii) A multiple spin-echo experiment is carried out. What would be the narrowest FWHM in the NMR spectrum measured in this experiment? [3]

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5. (a) Sketch a typical coil arrangements used in order to obtain (i) a highly uniform static magnetic field, [2] (ii) a gradient magnetic field. [2] For each case sketch the magnetic field distribution due to each coil and the total field for the pair. State the relationship between the coil radius and the distance between the coils. (b) Explain the principles of frequency encoding used in magnetic resonance imaging. [2] (c) Give the 1D-imaging equation and explain the meaning of the k-space. [3] (d) Explain the factors determining the spatial resolution in magnetic resonance imaging experiments. [3] (e) Spins with gyromagnetic ratio γ are uniformly distributed with linear spin density

ρ0 along the z axis from z0 to z0 in a one-dimensional imaging experiment. Sup- pose they are excited at t −= 0 by a radio-frequency pulse such that the demodulated NMR signal at that instant would be given by 2z0ρ0. A negative constant gradient field G is immediately applied at t = 0+ and flipped to the positive constant gradient− field +G at time t = T. Find an expression for the signal at t > T. Find the time, at which the gradient echo will be observed. [4] (f) Sketch a sequence of radio-frequency and B-field gradient pulses that could be applied for a 2D-imaging experiment. Mark on your sketch the periods of time when the signal is read. Sketch the k-space diagram corresponding to this pulse sequence and explain how the applied gradient pulses allow transversing through the k-space. [4]

END OF EXAMINATION PAPER

6 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE

Physical Constants electron charge e = 1.60×10−19 C −31 −2 electron mass me = 9.11×10 kg = 0.511 MeV c −27 −2 proton mass mp = 1.673×10 kg = 938.3 MeV c −27 −2 neutron mass mn = 1.675×10 kg = 939.6 MeV c Planck’s constant h = 6.63×10−34 J s Dirac’s constant (~ = h/2π) ~ = 1.05×10−34 J s −23 −1 −5 −1 Boltzmann’s constant kB = 1.38×10 JK = 8.62×10 eV K speed of light in free space c = 299 792 458 m s−1 ≈ 3.00×108 m s−1 −12 −1 permittivity of free space ε0 = 8.85×10 F m −7 −1 permeability of free space µ0 = 4π×10 H m 23 −1 Avogadro’s constant NA = 6.02×10 mol gas constant R = 8.314 J mol−1 K−1 −1 ideal gas volume (STP) V0 = 22.4 l mol gravitational constant G = 6.67×10−11 N m2 kg−2 7 −1 Rydberg constant R∞ = 1.10×10 m Rydberg energy of hydrogen RH = 13.6 eV −10 Bohr radius a0 = 0.529×10 m −24 −1 Bohr magneton µB = 9.27×10 JT ﬁne structure constant α ≈ 1/137 Wien displacement law constant b = 2.898×10−3 m K Stefan’s constant σ = 5.67×10−8 W m−2 K−4 radiation density constant a = 7.55×10−16 J m−3 K−4 30 mass of the Sun M = 1.99×10 kg 8 radius of the Sun R = 6.96×10 m 26 luminosity of the Sun L = 3.85×10 W 24 mass of the Earth M⊕ = 6.0×10 kg 6 radius of the Earth R⊕ = 6.4×10 m

Conversion Factors 1 u (atomic mass unit) = 1.66×10−27 kg = 931.5 MeV c−2 1 Å (angstrom) = 10−10 m 1 astronomical unit = 1.50×1011 m 1 g (gravity) = 9.81 m s−2 1 eV = 1.60×10−19 J 1 parsec = 3.08×1016 m 1 atmosphere = 1.01×105 Pa 1 year = 3.16×107 s Polar Coordinates

x = r cos θ y = r sin θ dA = r dr dθ

1 ∂ ∂ 1 ∂2 ∇2 = r + r ∂r ∂r r2 ∂θ2

Spherical Coordinates

x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dV = r2 sin θ dr dθ dφ

1 ∂ ∂ 1 ∂ ∂ 1 ∂2 ∇2 = r2 + sin θ + r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2

Calculus f(x) f 0(x) f(x) f 0(x) xn nxn−1 tan x sec2 x ex ex sin−1 x √ 1 a a2−x2 ln x = log x 1 cos−1 x − √ 1 e x a a2−x2 −1 x a sin x cos x tan a a2+x2 cos x − sin x sinh−1 x √ 1 a x2+a2 cosh x sinh x cosh−1 x √ 1 a x2−a2 −1 x a sinh x cosh x tanh a a2−x2 cosec x −cosec x cot x uv u0v + uv0 u0v−uv0 sec x sec x tan x u/v v2

Deﬁnite Integrals

Z ∞ n −ax n! x e dx = n+1 (n ≥ 0 and a > 0) 0 a

+∞ r Z 2 π e−ax dx = −∞ a Z +∞ r 2 −ax2 1 π x e dx = 3 −∞ 2 a Z b dv(x) b Z b du(x)

Integration by Parts: u(x) dx = u(x)v(x) − v(x) dx a dx a a dx Series Expansions

(x − a) (x − a)2 (x − a)3 Taylor series: f(x) = f(a) + f 0(a) + f 00(a) + f 000(a) + ··· 1! 2! 3! n X n n n! Binomial expansion: (x + y)n = xn−kyk and = k k (n − k)!k! k=0 n(n − 1) (1 + x)n = 1 + nx + x2 + ··· (|x| < 1) 2! x2 x3 x3 x5 x2 x4 ex = 1+x+ + +··· , sin x = x− + −· · · and cos x = 1− + −· · · 2! 3! 3! 5! 2! 4! x2 x3 ln(1 + x) = log (1 + x) = x − + − · · · (|x| < 1) e 2 3 n X 1 − rn+1 Geometric series: rk = 1 − r k=0

Stirling’s formula: loge N! = N loge N − N or ln N! = N ln N − N

Trigonometry

sin(a ± b) = sin a cos b ± cos a sin b

cos(a ± b) = cos a cos b ∓ sin a sin b tan a ± tan b tan(a ± b) = 1 ∓ tan a tan b sin 2a = 2 sin a cos a cos 2a = cos2 a − sin2 a = 2 cos2 a − 1 = 1 − 2 sin2 a 1 1 sin a + sin b = 2 sin 2 (a + b) cos 2 (a − b) 1 1 sin a − sin b = 2 cos 2 (a + b) sin 2 (a − b) 1 1 cos a + cos b = 2 cos 2 (a + b) cos 2 (a − b) 1 1 cos a − cos b = −2 sin 2 (a + b) sin 2 (a − b) eiθ = cos θ + i sin θ 1 1 cos θ = eiθ + e−iθ and sin θ = eiθ − e−iθ 2 2i 1 1 cosh θ = eθ + e−θ and sinh θ = eθ − e−θ 2 2 sin a sin b sin c Spherical geometry: = = and cos a = cos b cos c+sin b sin c cos A sin A sin B sin C Vector Calculus

A · B = AxBx + AyBy + AzBz = AjBj ˆ ˆ ˆ A×B = (AyBz − AzBy) i + (AzBx − AxBz) j + (AxBy − AyBx) k = ijkAjBk A×(B×C) = (A · C)B − (A · B)C A · (B×C) = B · (C×A) = C · (A×B) ∂φ ∂φ ∂φ grad φ = ∇φ = ∂ φ = ˆi + ˆj + kˆ j ∂x ∂y ∂z ∂A ∂A ∂A div A = ∇ · A = ∂ A = x + y + z j j ∂x ∂y ∂z ∂A ∂A ∂A ∂A ∂A ∂A curl A = ∇×A = ∂ A = z − y ˆi + x − z ˆj + y − x kˆ ijk j k ∂y ∂z ∂z ∂x ∂x ∂y ∂2φ ∂2φ ∂2φ ∇ · ∇φ = ∇2φ = + + ∂x2 ∂y2 ∂z2 ∇×(∇φ) = 0 and ∇ · (∇×A) = 0 ∇×(∇×A) = ∇(∇ · A) − ∇2A