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Theory of the magnetic resonance spectrum of spin-polarized hydrogen gas Zhou, Haosheng 1987

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T H E O R Y O F T H E M A G N E T I C R E S O N A N C E S P E C T R U M O F S P I N - P O L A R I Z E D H Y D R O G E N G A S by H A O S H E N G Z H O U B . S c , South Ch ina Institute of Technology, 1982 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Physics We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A November 1987 © Haosheng Zhou, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of P H Y U c i The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6(3/81) Theory of the Magnetic Resonance Spectrum of Spin-Polarized Hydrogen Gas Abstract The Green's function method is applied to investigate the magnetic spin res-onance spectra of three-dimensional and two-dimensional spin-polarized quantum gases. The Hartree-Fock approximation is employed to calculate the one-particle Green's function of the atoms, then this one-particle Green's function is used for the calculation of the vertex part of the Green's function. Such a combination yields a self-consistent result. The absorption spectra are obtained from the calculation of the susceptibility in terms of the two-particle Green's function (bubble diagram). Some general expressions for the dispersion relation, for the effective mass of a spin wave, and for the dipolar frequency shift are given in the calculation. In order to estimate the shift of the electron-spin-resonance (ESR) frequency, the effective dipole-dipole interactions among the hydrogen atoms are included in the calculation. These effective interactions are deduced from the ladder approximation, and hence are characterized by the scattering amplitude. The scattering amplitude is calculated numerically. The result shows that the theoretical value of the shift is smaller than the experimentally observed value by about 35%. ii Table of Contents Page Abst rac t i i Lis t of Figures v Acknowledgement v i 1. Introduction 1 2. One-Particle Green's Function 2.1 Equat ion of M o t i o n of F ie ld Operator 20 2.2 Equat ion of Mot ion of One-Particle Green's Function 22 2.3 The Hartree-Fock Approximat ion 23 2.4 Quasi-Part icle Energy Spectrum 24 3. Der iving the Susceptibility from the Vertex Part of Green's Functions 3.1 The Relat ion between La^a{p,to) and Susceptibility 30 3.2 The Green's Function Equat ion 33 3.3 Dispersion Relations 36 4. The Calculat ion of L^]_a 4.1 The Bubble Diagram 37 4.2 Real Par t of LL°J.(P,W) 39 4.3 Spin Wave Dispersion Relat ion 41 4.4 Summary of the Results 45 5. Es t imat ion of Frequency Shift 5.1 Scattering Ampl i tude 46 5.2 Calcula t ion of the Scattering Ampl i tude ; 47 6. Conclusions 56 Bibliography 59 i i i Appendix A A . 1 B - S (Bethe-Salpeter) Equat ion 63 A . 2 Gal i tski i ' s Integral Equat ion 65 A . 3 Solution of Gali tski i 's Integral Equat ion 67 Appendix B 70 Appendix C 73 Appendix D 76 Appendix E 80 iv List of Figures Figure Page 1.1 The Principle of Operation 2 1.2 The Hyperfine Energies in A Magnetic F ie ld 3 1.3 The Microwave Absorption x" v s - Magnetic F ie ld 12 1.4 The Singlet and Triplet Potentials 15 1.5 P lo t of A w vs. Density n . 1 9 2.1 Interaction between Two Particles 20 2.2 Hartree-Fock Approximat ion 23 3.1 Vertex Funct ion 33 3.2 Bubble Diagram wi th Vertex Correction 34 4.1 Bubble Diagram 37 5.1 P lo t of the Phase Shift 6 vs. Ek 51 5.2 Probabi l i ty Density p vs. Phase Shift 6 53 5.3 Experimental Da ta of Au> vs. n 54 6.1 P lo t of A w vs. n 58 v Acknowlegment I am sincerely grateful to D r . A . J . Berl insky for his supervision and support throughout the course of this work. In addit ion, I would like to express my gratitude to D r . W . N . Hardy and M . W . Reynolds for their providing me wi th their detailed results of E S R experiments. v i Chapter 1 Introduction Atomic hydrogen can be stabilized against recombination into H 2 molecules by spin-polarization at low temperature and high magnetic fields. Such hydrogen is called spin-polarized hydrogen. The reaction H + H —> H2 is suppressed because the triplet potential between two H atoms is repulsive. O n the other hand, the temperature of the system should be low so that the rate of spin flips is low. Such flipping of spins is caused by collisions among the atoms. The principle of operation is shown in F i g . 1.1. To make the discussion more concrete, we take the nuclear spin into account. The four hyperfme states for atomic hydrogen in its orbital ground state are [la] |a) = ( U t ) - e | T ± » / ( l + e 2 ) 1 / 2 l*> = I i i) \c) = (\n) + e\l}))/(l + e2y/2 \d) = ' | T t > where indicates the component of the electron spin, and ^ indicates the component of the proton spin, e is a small parameter in a strong field (~2x 10 - 3 at 10 T field.) The hyperfine energies for atomic hydrogen in a magnetic field are i l lustrated in F i g . 1.2. States \a) and \b) are essentially electron spin "down" states, whereas states |c) and \d) are spin "up". Atoms in states \a) and |6) are attracted into a high magnetic field; atoms in |c) and \d) are repelled. A t low temperature, all hydrogen atoms wi l l be in their Is electronic ground state, therefore the potential between two atoms depends only on the distance be-tween two centers of the atoms. Keeping this in mind , we wi l l not mention any effects due to orbital internal degrees of freedom in the following parts of this thesis. Note that the state |a) is a mixed state,containing both electronic spin states. In 1 6.7 K Potential Energy B o 1 Hhermal Energy at 0.3K -6.7 K 10 Tesla • J I I J I J J M Spin Parallel to B Spin Antiparallel - 0 . 4 1 -F i g . 1.2 The Hyperfine Energies in A Magnetic F ie ld 3 this state, the atoms recombine more easily into molecules. Under some condition, the mixed state density na decreases quickly, and finally the system consists only of hydrogen atoms in the pure state |6), in which the nuclear spin is also polarized ("Xj bottleneck" effect [lb].) We wi l l assume that the spin-polarized hydrogen gas is in the state |6) in the following discussion. In a nuclear-magnetic-resonance ( N M R ) experiment, transitions occur between the states \a) and |6), while in an electron-spin-resonance (ESR) experiment, tran-sitions occur between the states \b) and \c). The transition |6) —>• \d) is forbidden because it requires that both the spins of the electron and the proton flip at the same time. Therefore we ignore the transition \b) —• \d) in the E S R experiment. Moreover, the resonant frequencies of E S R and N M R are different by several orders (ESR ' s is in the region of G H z while N M R ' s is in that of M H z . ) Therefore the two frequencies w i l l not interfere wi th each other either in an E S R experiment or in a N M R experiment. Consequently, we can treat atomic hydrogen as a fictitious spin | system where the "spin" may be either the electron or proton depending on the experimental situation. The theory in this thesis applies to both cases. It not only works for E S R , but also is suitable for N M R . In the remainder of this chapter, we wi l l review the semi-classical theory of collective spin-wave oscillations in spin-polarized quantum systems, and give the connection between the resonant frequency and the dispersion relation of the spin waves. This theory applies to a gas of spin | parti-cles w i th spin-independent interparticle interactions. The latter restriction wi l l be removed later in this thesis. The collective spin-wave oscillations of spin-polarized quantum systems have been studied for over two decades. The polarized spin may be either an elec-tron spin or a nuclear spin, depending on the system we investigate. In all of these systems, the temperature is so low that the thermal de Broglie wavelength A 4 (= (2nh2/mksT)1/ 2) of the particles is significantly greater than the atomic d i -mensions ro- Such systems are called quantum fluids which may be liquids (e.g., l iquid 3 H e ) or gases (e.g., spin-polarized atomic hydrogen gas). The quantum ef-fects generated by identical-particle symmetrization are known to play a crucial role in the propagation of spin waves. Unlike localized spin systems described by the Heisenberg model, the interactions on which the propagations of spin waves depend may be spin-independent for spin-polarized quantum fluids. The development of the theory of spin waves in spin-polarized quantum fluids can be traced back to the 1950s. In 1957, Si l in derived a Fermi-liquid-type kinetic equation in the semi-classical approximation [Id]. He predicted the occurrence of spin oscillations in a Fermi l iquid placed in a magnetic field. In 1966, P la tzman and Wolff used this Fermi-l iquid theory to investigate the properties of spin waves in th in slabs of metals [2]. Thei r predictions were immediately verified experimentally by Schultz and Dunifer [3]. In 1968, Leggett and Rice used Silin's theory to pre-dict effects occurring in spin echo experiments (the Leggett-Rice effect [4]). These predictions have been confirmed in experiments wi th l iquid 3 H e at very low temper-atures and used to measure the Fermi l iquid interaction parameter A for pure 3 H e and 3 H e - 4 H e mixtures [5]. The earliest works focused on strongly interacting, degenerate Fermi systems such as electrons in metals and l iquid 3 H e , whereas recent work has turned to d i -lute, weakly interacting, spin-polarized systems such as spin-polarized H and 3 H e gases. In 1981, Bashkin predicted the propagation of weakly damped spin waves in spin-polarized gaseous H and 3 H e [6]. In 1982, Lhuil l ier and Laloe constructed macroscopic equations for the spin dynamics of a polarized quantum gas. These equations also imply the existence of spin oscillations [7]. In 1984, Johnson et al . carried out the N M R experiments which were the first to reveal collective spin modes 5 in spin-polarized H [8]. A t about the same time, Laloe's group observed nuclear spin waves in spin-polarized 3 H e [9]. Levy and Ruckenstein [10] used a quasipar-ticle approach to derive a quantitative interpretation of the experimental data of [8]. Bashkin has also derived a detailed theory of collective quantum phenomena in Maxwel l ian spin-polarized gases [11]. The theory of N M R in two-dimensional H adsorbed on a surface of l iquid 4 H e has also been investigated by several authors [12-14]. In the following several paragraphs, we w i l l give a brief review of Silin 's semi-classical theory for spin-polarized systems. B y solving the Fermi-liquid-type kinetic equation (see (1.1)), we derive an expression for the magnetic susceptibility x+ depending on the wave vector k and frequency u> (see (1.21)). The dispersion relation is determined by finding the frequency w(k) for a given k, at which x+ is peaked. For the case of weakly damped spin waves, the condition determining the dispersion relation is given by (1.25), and the result is (1.31). Let us consider a spin-polarized quantum system of spin | Bosons. A n example of such a spin | Bose gas is spin-polarized atomic hydrogen. Al though electron spin resonance (ESR) probes its electrons spin (with spin | ) , the hydrogen atom itself is a composite Boson. In order to polarize the electron spin, a strong, uniform external magnetic field B o is applied to the system. For convenience, we choose the z-axis along the direction of B o . Moreover, assuming that the condition ft1/3 » r 0 is fulfilled, where n is the density of the hydrogen gas, and ro is the atomic dimensions, we can approximately treat the hydrogen atoms as point particles and ignore the effect of the orbital wave function (it spreads out an electron only in the space wi th dimension ro.) The idea for semi-classical treatment is that we treat the coordinate r and mo-mentum p of a quasi-particle as c-numbers while retaining al l other operators as 6 2x2 matrices in spin space in order to take the quantum effects into account, where a quasi-particle means a state for electrons, not for spin wave. We shall use the phrase "a spin wave" to refer to a state of the collective spin-wave oscillations of the system. By expanding the equation of motion in powers of h, and keeping only the lowest order terms in the equation, Silin derived a Fermi-liquid-type kinetic equation or so called semi-classical Boltzmann equation [id]: dn i. 1 / de dn dn de\ 1 /de dn dn de\ - . - ^ - ^ [ e , n j + - ^ — • — + — • — J - - 1 ^ — • — + — • — J = J (1.1) where n = n(p, r, t) is a 2 x 2 matrix, the generalization of the Wigner distribution function to the case of spin e = e(p,r) is the Hamiltonian for a quasi-particle, [e,n] is a commutator in spin space, and J is the collision integral. In the low temperature and low density limit, J is negligible, (see Appendix D) Therefore we ignored the collision integral J in the following calculation. Introducing a distribution function / for the particle density in phase space and a vector function s for the distribution of the spin density: /(p,r,0 = Tr(n) (1.2) s(p,r,i) = Tr(an) where Tr(...) denotes the trace over the spin space, and noting that e(p,r) can be written in the form e(p,r) = e x(p,r)+c7-e 2(p,r) (1.3) then one obtains df dex df dei df de 2 ds de2 ds 1 • • 1 • — • — — = 0 (1.4a) dt dp dr dr dp dpj drj drj dpj and ds dt \ dp dr (del d\ fdex d\ 2, . (df d \ (df d\ , t . where double indices are summed over. In the absence of external perturbations, one may evaluate ej and e2 by the so called Fermi renormalization method [11]*: e , = M B B 0 J + 9 / ^ e < ° V ) (1.6) where 9 = 1-7 m and a is the scattering length. The integral in (1.6) acts as an internal field generated by the so called identical spin rotation effect, which wi l l be derived in chapter 2. The physical meaning of this effect is discussed by Bouchaud and Lhui l l ier [15]. Notice that in the absence of external perturbation, / ( ° ) and s(°) do not depend on r. In the presence of a perturbation Hi(r, t) oc e x p ( z ^ — iu)i) , the linear response of the system to this perturbation is Sf = f - / ( 0 ) = 0 6s = s - s(°) = A (1.8) where In this case k • r A oc exp(i — iuit) (1.9) p1 f d3p' ( 0 ) 2m J (27m)d /d3p' ( 2 7 ^ S ^ P ' , r ' ^ (1.10) * In fact, this is the self-energy calculated in Hartree-Fock approximation. Renor-malizat ion means that the interactions are replaced by a (^-function interaction V(r) = (47ih2a/ m)6(r) which is also spin-independent in this case. 8 Substituting (1.8) and (1.10) into (1.4) and linearizing it in the small quantities A and H i , one has d6f dti 36f dF(A,Hi) ds{0)_ (1.11a) dt \m dr dt dp dr dtj dpj A + | ( F ( A . H , ) x S<°>) f | { . B B o l + . / ^ .<•>&,•)} x A -where d/ ( 0 ) a \ , F(A, H , ) = w , H , (r, t) + s | A(p', r, () In the long wave-length limit, dF(A,H!) (1.116) (1.12) dry |k|0(A,Hi) -^0 Therefore (1.11a) is negligible. That is why one can set 6f = 0 beforehand. Using the relations s<°) = (n+ ~ n J ) z / < ° ) = n+ + n_ a/ ( 0 ) p / d n + an. and assuming that dp m \ dti de\ (H!) 2 = 0 (1.13) (1.14) (1.15) (1.16) one has, in terms of the circular variables A± = \ x ± i\y and H± = Hix ± iHiy, _ ~ 2 / / B j B o " 2 9 i ( 2 ^ p ( n + - n - } J x + + (1.17) = 0 Therefore A + = ^ - ^ - 2^B5O - 2 g f [n+ - » _ ) (1.18) Integrating (1.18), one finds the magnetic moment 6M+ = SMX + i6My = J A +(p',r,0 (1.19) induced by the perturbation H+ 6M+ = x+H+ (1.20) Consequently, where HB(R(u,k) + Q(u,,k)) x + 1+ k) + Q(w,k)) K • ] ( 2 ^ ) 3 - - J ( L 2 2 ) ^ k ) - - / ^ ( l 7 + l 7 ) ^ ^ <"»> Qo>,k = / . — ± - — 1.24 v ' J [2Trh)3 huj - k -p/m v ' Bashkin has used a different approach to obtain an expression similar to (1.21) [11]. For a given k, the value of u> where the magnetic susceptibility (1.21) is peaked, corresponds to a spin wave energy. This is determined by 1 + gRe{R{u, k) + Q(w, k)) = 0 (1.25) In the long-wavelength region, |k| —> 0, the dominant contribution to (1.25) comes from Q. In the case when the distribution functions n+ and n_ are Maxwellian, the function Q can be expressed as [16] ^ ,v 2(7V+ - N-) /mp\2 1 , 1 . , ReQ(u,,k)«-^-±_ -.(1 + 2 ^ ) (1,26) where * s t ( - t J ( 1 - 2 7 )  N± = I t B r n ± { p ' r ) ( 1 ' 2 8 ) 10 Substi tuting (1.26) into (1.25), one obtains (1.29) where XQ = 2g(7v_ - N+) fm(3 (1.30) Therefore from (1.27) and (1.22), k2 (1.31) 2m* where 1 1 (1.32) m* g{N_ - N+)m(3 E q . ( l . 3 l ) is the spectrum of spin waves in a non-degenerate spin-polarized Bose gas. The treatment shown above is the semi-classical theory. We have made the discussion as simple as possible, and not included the possibility of spin dependent interactions and, in particular, of interactions which wi l l cause a frequency shift of the spin wave spectrum. In 1984, Reynolds et al . observed unexpected sidebands in the E S R spectra of spin-polarized hydrogen in the presence of l iquid 4 He-coated walls (F ig . 1.3). Reynolds et al. [17] proposed that these sidebands were the spectra of H atoms on the walls of the cavity and that the shift was caused by dipole-dipole interactions among the atoms. In this thesis, we are going to examine this proposition by means of the method of Green's functions. Like other many-body theories, the Green's function method also involves ap-proximation. One way of controlling the approximation is to assure that they are consistent wi th the symmetries of the problem. For example, if the atom-atom interaction Hj(i) commutes w i th the total spin operator: [HI(t),Stot(t)}=0 (1.33) 11 I I I I I I 0 1 2 3 4 AB (Gauss) F i g . 1.3 The Microwave Absorpt ion x" vs. Magnetic F i e ld 12 there will be no shift in spin wave spectra, where Hi(t) is the interaction operator, and Stot[t) is the total spin operator. In fact, for a general operator in the form 0{t) = a{Stot{t))x + b{Stot{t))y + c{Stot{t))z (1.34) where the indices x, y, z denote the coordinate components of St0t(<)? a n d b,c are c-numbers, the equation of motion of 0(t) may be written as ^-O(t) = ^[H0(t) + HI(t),O(t)} (1.35) where Ho(t) is the Hamiltonian without interaction. If (1.33) is satisfied, then the equation above becomes ~0(t)=l-[HQ(t),0(t)} (1-36) So the interaction does not affect the motion of 0(t). The spin wave creation oper-ator for p = 0 has the form of (1.34): S+{t) = (Stot(O)* + HStot{t))y (1.37) Therefore the atom-atom interactions satisfying (1.33) will not shift the p = 0 spin wave frequency. It is also easy to see that an interaction which is invariant under uniform rotation of the spin satisfies (1.33). In general, the dispersion relation of long wavelength spin waves can be written in the form (see (4.35), for p+ = M-) hu = 2fiBB0 + fiAw + —f— (1.38) 2m* where p is the wave-number of spin waves, and Au; is the shift of the ESR frequency. What is usually measured in ESR experiments is the frequency given by (1.38) at p = 0, hu0 = 2nBB0 + ft Aw (1.39) 13 Now, we are going to discuss the interactions between hydrogen atoms more in detail . The combination of the orbital wave functions of the electrons and the wave functions of the spins in a pair of hydrogen atoms provides one singlet state and three triplet states 3 £ + [lc]. The potential between two hydrogen atoms depends on the states J E + and 3 £ + , and the distance r between the two centers of the atoms (the two protons.) The curves of the potentials are shown in F i g . 1.4. In terms of the spin parameters, these potentials can be wri t ten as V(r) = \vD{r) + WE(r)ax-o2 where o\ and o2 are the Paul i matrices for two electrons of the atoms respectively. Vo(r) and V]s(r) are known as the direct and exchange potentials respectively. The direct and exchange interactions Vr> and VE between the H atoms wi l l not cause a frequency shift for p = 0 because they are rotationally invariant, i.e., they are of the form V{r) = f{r) + g{r)a1-o2, (1.40) where f(r) and g(r) are only functions of r — |r|, and r is the relative position vector between two particles. O n the contrary, the dipole-dipole interaction between two electrons is not in-variant under rotations in spin space since it has the form Vdd(r) = (<7i • a2 - 3(<7, • h)(a2 • n)) (1.41) where o\ and o2 are the Pau l i matrices, r is the relative position vector between two particles, and ft is the unit vector r/ r . In a spin-polarized system, Vdd(r) depends on the relations between the direction of r and the direction of the polarization axis. In three dimensions, r is equally likely to point in any direction and hence the average dipolar shift is zero. However in two dimensions, r is constrained to lie in the plane 14 of the surface, and therefore is no longer invariant under uniform rotation of the spins. Consequently, if we take only Vp and Vp into account in our calculation, there will be no frequency shift in the dispersion relation. On the other hand, the dipole-dipole interaction is certainly a candidate for causing the frequency shift. Further evidence comes from direct calculation: the line which is shifted to low field comes from atoms on the side wall of the cylindrical cavity, which is parallel to the magnetic field Bo, and the line which is shifted to high field comes from atoms on the top and bottom surfaces of the cavity, which are normal to the magnetic field Bo- The ratio of these shifts, with respect to the resonant peak of the three-dimensional (3-D) hydrogen gas, is —1 : 2(see Section 4.4). This is in agreement with the experiment data. The theoretical result also agrees with the.experimental data in that the shift is proportional to the density of gas. A self consistent calculation of the frequency shifts in the interacting gas involves two main steps: 1) calculating the self-energy of a single quasi-particle in the Hartree-Fock approx-imation. This gives the one-particle Green's function for quasi-particles: (see (2.40) and (2.41)) Gafi{p,zv) = *!f (1.42) zv - h Ea[p) where Ea(p) is the energy for a quasi-particle, zv is an imaginary frequency, and a,f3 (— ±) are the spin indexes. 2) using (1.42) in the calculation of the vertex correction, we obtain an expression for the "bubble diagram", L (-(p,w) (see Section 3.2.) 16 The relation between Z/_ + (p,u;), the Fourier transform of Lap(l,2) with a = — and /3 = + , and the susceptibility x- + (p,w) is (see Section 3.1, (3.18)) X-+(p,w) = nBh~lL- + (v,u + ie) (1.43) where [iB is the Bohr magneton, and e is an infinitesimal positive number. A calculation similar to that of (1.21) to (1.31), gives the dispersion relation (1.38) h2P2 hoj = 2p,BBo + hAu H 2m' where Aw depends only on the parameters that characterize the dipole-dipole inter-action. The H-H interaction used in this calculation is a spin-dependent potential in-cluding V D , VE and the dipole-dipole interaction. Unfortunately, the equation for the vertex part is usually very difficult to solve, and the singularity of the dipole-dipole interaction precludes the usual approximation of 6-function interactions. To avoid such difficulties, we consider only the low density limit in which the interac-tion potential may be replaced by an effective potential deduced from the ladder approximation. Such a replacement makes the vertex equations solvable because 1) the effective potential is a 6-function in the coordinate representation, which turns the integral equations into algebraic equations in the momentum repre-sentation. 2) the effective potential is not singular like the dipole-dipole interaction. In the ladder approximation, the effective interaction V(q, q',P) is related to the scattering amplitude /(q,q',P) by the formula (see (5.3)) h2 F(q,q',P)«-/(q,q',P) I 1- 4 4) m where V is written in the frame of the center of mass, P is the momentum of the center of mass, q and q' are the relative momenta. Thus the problem reduces to 17 one of calculation of the scattering amplitude. The numerical estimation of the scattering amplitude yields the result shown in F i g . 1.5. Note that because of the "sweep field" technique used in the experiment, a negative frequency shift results in a positive field shift and vice versa. Moreover, some more general discussions about the effective mass of spin waves are presented in Section 4.3. The calculation shows that in the low temperature l imi t (T —» 0), the effective mass approaches the mass of a free particle while in the high temperature l imit (T —> oo), the effective mass is given by an expression similar to (1.32). Final ly, the organization of this thesis is as follows. In chapter 2, the one-particle Green's function is derived in the Hartree-Fock approximation, and the quasi-particle energy spectrum is given. Also in this chapter, a discussion of the identical spin rotation effect is given. In chapter 3, the vertex part of the Green's function is calculated, and the complete two-particle Green's function is given. In chapter 4, the zero-order r ing diagram is calculated, and spin-wave spectra for 2-D and 3-D are derived. Also in this chapter, the E S R shift is inferred from the results. In chapter 5, the effective potential for dipole-dipole interactions in the presence of a hard core is estimated numerically for the 2-D case, and a numerical estimate of the frequency shift is given. Final ly , a summary and discussion is given in chapter 6. 1 8 Fig. 1.5 Plot of Aw vs. Density n 19 Chapter 2 One-Part ic le Green's Funct ion 2.1 Equa t i on of M o t i o n of F ie ld Opera to r The Hamiltonian of the atomic hydrogen system can be written as H = Hk + Hz + Hj (2.1) where Hk is kinetic energy, Hz is Zeeman energy, and Hj is interaction energy. For convenience, the z-axis is chosen along the external magnetic field. Let tpx(r,t),\ = ± be the field operator. Then Hk, Hz, and Hi can be expressed as H * = E / d d r ^ v ^ t ] ' ( 2 2 ) H2 = HBBoY,X f ddrtpl{rit)rpx{T,t) (2.3) A J where the applied magnetic field is taken to be uniform, and H ' = \ E /^^Vl .(r ,O^(r # .OV'vA ,»,(|r--r ' | )0 ,(r ' f*)^(r ,O (2.4) where d is the dimensionality of the system (d = 2 or 3 ), /JB is Bohr magneton, Bo is the external magnetic field, and VA'Ai7'i?(lr —R'|) k the spin-dependent interaction potential between two H atoms: Fig. 2.1 Interaction between Two Particles The field operator satisfies the following commutation relations: [VA(M),^(r ' , t)] = 0 (2.5) 20 and \Mr,t)^l(T\t)] = 6XriS(T-T') (2.6) One easily finds that \i>a(r,t),Hk(t)} = - ^ V 2 i ( M ) (2.7) Zm \rpa{T,t),Hz{t)} = anBBorpa{T,t) (2.8) and [ 0 a ( r , O , # / ( * ) ] = E / ^ > v ( r ' O ^ A a r , ( | r ' - r | ) ^ A ( r ' , f ) ^ ( r , 0 (2.9) A'Arj where the symmetry fvA„<„( | r ' - r | ) = V ^ ^ A d r ' - r |) (2.10) has been used. From (2.7)-(2.9), one has the equation of motion of xjja(r,t): ih—Mr,t) = \ipa(r,t),H(t)] h2 = - — V 2 4 ( r , t ) + anBB0il>a(T,t) + zm A'Ar? or ih— + — V 2 )^a{r,t) = afiBB0ipa{r,t) + at 2m + E / ^ ' ^ ( r ' . O V A ' A ^ d r ' - r D ^ ^ O ^ C r , * ) + E / d V ^ ' ( r ' ' O ^ A ' A a , (|r' - r |) V A ( r ' s t)^(r, t) (2.11) 21 2.2 E q u a t i o n o f M o t i o n o f O n e - P a r t i c l e G r e e n ' s F u n c t i o n The definition of the one-particle Green's function is G a j 9 ( l , 2 ) = i ( T ( ^ a ( l ) 0 j ( 2 ) ) > (2.12) where T represents the Wick time-ordering operation, (• • •) is an equil ibr ium thermal average, and the indexes 1 and 2 represent r ^ i and r2t2 respectively. B y means of the relation ^ < r ( i M l ) V £ ( 2 ) ) > - <T( J - i M l ) ^ ( 2 ) ) > = 6aP6(l - 2) (2.13) where 6{l-2) = 6{r1-r2)6{tl-t2) (2.14) one has the equation of motion of Gap(l,2): ( d h2 \ in— + — V 2 )Gap[l,2) = h6aP6(l - 2) + aiiBBQGap{l,2) + at 1 Lm ) / d d f V V A a „ ( | r - r 1 | ) G r ? A / 3 V ( l l , 2 l + ) A'Ar? ^ where G a / 9 a ^ ( l , 2 ; l ' , 2 ' ) = ( r ( ^ a ( l ) ^ ( 2 ) ^ ( 2 ' ) A ( l ' ) ) > ( 2 - 1 6 ) is a two-particle Green's function, and the notation 1 + is intended to serve as a reminder that the time argument of ip^(l) must be chosen to be infinitesimally larger than the time arguments of the t/>'s. (2.15) 22 2.3 The Hartree-Fock App rox ima t i on From (2.15), it is clear that the two-particle Green's function is required. The equation of motion of the two-particle Green's function will in turn involve three-particle Green's functions, and so on. These Green's functions form a family or a hierarchy. In practice one can not find solutions for such a hierarchy involving an infinite number of equations. Therefore a truncation is needed which involves an approximation. ' The simplest way to do this is to express two-particle Green's function approx-imately in terms of the products of one-particle Green's functions, which is the well-known Hartree-Fock approximation, G a / W ( l , 2 ; l ' , 2 ' ) » G Q Q , (1 ,1 ' )G^(2,2 ' ) + G>(l,2')G>«.(2,l') (2.17) '—>—2 2' 2' *-Fig. 2.2 Hartree-Fock Approximation Substituting (2.17) into (2.15), one has I*ffVvAa,(|r-r1|)[Gf^(l,2)GAv(I,'l+)+ ( 2 " 1 8 ) + G,v( l , i + )G^( I ,2 ) ] Because of translational invariance, it is useful to introduce Gaf,(h2) = J 70-dGae(p,h -« 2)e-W'«-*-) (2.19) 23 where p is a wavevector. Now in this momentum representation, one has i f t — - JG a / 0 (p , i i - t2) = h6apS(ti - t2) + apBB0Gap(p,ti - t2) + y\r,J [ ' ^ I fa)* ddp' X'Xri + i X / ^ r^v^'^r,{p' -p)Gvy{p',0')Gx/3{p,t1-t2) (2.20) where •„(p) = J ddrVyWr,(T)e^r (2.21) 2.4 Q u a s i - P a r t i c l e E n e r g y S p e c t r u m Replacing Gap(p,0~) by —i8ap{nPja) in (2.20), where ( n p > a ) is the average density of particles wi th wavenumber p and spin a , namely ( » P , « > = UP) = e / n B a { p ) - r a ) _ t (2-22) then one has {^hd~T ~ ~ 2 ~ m ~ ) G a / ^ P , f l ~~ ^  = hS<xpb{ti - t2) + ansBoGapip^! - t2) + + *'E /.fj^rp + ^ a > ( p ' - p ) ] /A(p , )G ' r , / ? (p , t i - t2) (2.23) or, in a matr ix form, (lih-^--E>JG[p,t1-t2) = h6ap6(U - t2) (2.24) where the elements of G are given by {G)al3 = GClp{p,t1-t2) (2.25) and those of E are given by [E)a0 = + ap,BB0^h6ap + t £ J [V A A a /j(0) + F A / 3 q A ( P ' - p)]/ A(p') A (2.26) 24 Notice that in general E is not necessarily a diagonal matrix. Therefore the quasi-particle energy spectrum will be generally given by the eigenvalues of E. If E does have non-zero off-diagonal elements (d-d interaction will cause such terms), then the eigenstate of a quasi-particle is neither | ] ) nor | J. ). It will be a mixed state, namely, a linear combination of two "pure" states | j ) and | | ). For simplicity, we shall ignore the effects of the mixed states. It is shown later that for the d-d interaction, it is reasonable to neglect the off-diagonal elements of E. Further calculation requires more information about the potential Va>apip. The interaction between a pair of hydrogen atoms consists of two parts. The first part of the potential is described by the so-called K-W potential [18]. At low temperature and low density, the form of the potential is unimportant. Therefore in practice we use a (^-function potential. The coefficient of the <5-function is derived from the exact s-wave phase shift in the low energy (scattering length) limit. The second part of the potential is the dipole-dipole interaction between the spins of two electrons. The total potential hence can be written as HI = 6(r)(^-VD+l-VEol-o2^ + 2 MB — 3 — ~ - r ) ( ^ - r ) (2.27) where o\ and 02 are Pauli matrices,and Vp and VE are given by 2 V D + 8 V E - ^ n ~ 2 8 m where and fs are the scattering amplitudes. For three-dimensional hydrogen gas, the values of fx and fs are [19] ^ = 0.72 A 47T — = 0.12 A 2 5 Our numerical calculations show that ay increases for T > 0 w i th increasing T. Therefore VD and VE are temperature dependent. B y means of the relations oz\a) = a\a) ox\cc) = \-a) } (2.28) oy\a) = ia\ — a) one may write the matr ix element of the potential as Va.ap.p{r) = ((3'a'\Hi\aP) = VD 2 3fi 6(r)6a>aSp>p + ^ E ^ ^ + -^r) {8a-p8a'af>P'P + -^Sa'aSp'p)-r 3fi 5 [{x - afiy )6a>-a6p>_p + afiz 6a>a6p,p}--[i{a + fi)xy6a>-a6p>_p + xz(0Sa>^a6pip + a6a>a6pi_p) + + ia(3yz(6a>-a6p>p + 6a'af>p'-p)} (2.29) The Fourier components of (2.29) has the form Va'aP'pip) = Y ^ ' a ^ ' / 3 + + 2Vr(p)^ (da-pdaia6p>p + -^-8a'a6p>p) -- [(Wxs(p) - a/Wyi,(p))6a._afy#_0 + a/?W«(p)tfa'«fy'/?]-- [i(a + f3)Wxy{p)6a>-a6p'-p + Wxz{j>)(f36a>-a6p>p + a6a>ot6pl-p) + + ia/?Wj,2(p)(6a'_Q6/3'/3 + 6aia8pi_p)\ (2.30) where Vr = l*B d d r e - ^ - } f xx ( 2 3 1 ) WXtX](p) = 3n2B fr-LJ-e-**" xx,x,=x,y,z) In order to neglect the off-diagonal elements of J51, one should neglect the cross terms such as Wxy(p). Fortunately, the diagonal elements of E are relatively large in absolute value because of the very strong uniform field Bo- A t low temperature, the 2 6 most important region is |p| —> 0, and in this l imit the cross terms vanish. Therefore the cross terms w i l l be neglected in further calculations. Looking at (2.31) closely, one finds that Vr and the W ' s are singular at |p | = 0. The singularity results from contributions around r = 0. In the real system, the hard core repulsion of the hydrogen atoms prohibits any two atoms from being less than a hard core radius apart. Therefore this repulsion removes the singularity. This means that one should replace the d -d potential by an effective potential which includes the effect of the hard core. Now assume an appropriate effective potential has been found so that one may write the total potential as Va'a/?'/?(p) = -y-<5a'a<5/9'/? + + 2 ^ {^oc-^a'af>p'/3 + -~^a'a^j3>p)~ - \{WXX - apwyy)6a,_a6f3'-p + ct/3Wzz6a>a6p>p} where Vr and W ' s are independent of p . Substituting (2.32) into (2.23), one has (*h^ ~ ~2m~)GaP(p,tl ~*2) = h6*P6(tl ~ * 2 ) + + {anBB0 + £#> + /4 2 ) ) G a / ? ( p , ' i - t2) (2.33) which implies that the energy for a quasi-particle is Ea(p) = + [OCUBBO + + E ^ ) (2.34) where 4 1 } ( P ) = ( ^ + ^ + ^ ) ( 2 n a + n _ a ) (2.35) # i 2 ) ( p ) = ~(Wzx + Wyy)n-a - Wzz{2na - n _ a ) (2.36) Here we have deliberately separated the energy correction into two parts for conve-nience. 2 7 To derive an expression for Gap, let us expand Gap{p,ti — t2) in a Fourier series which automatically takes the quasi-periodic boundary condition into account [20]: 0 < it i < ph G*p(j>M -t2) = — r e - ^ ( i l - i ! l g ( p ) 2 , ) for (2.37) - l P h u 0<it2</3h Therefore (2.19) now becomes G a " ( 1 ' 2 ) = / 7 ^ p ^ ( p ^ , ) e - i p ( r i - r 2 ) - ^ ( t l - t 2 ) (2-38) where zv — V h~1pa v — even integer (2.39) Substituting (2.38) into (2.33), one has G«/9(P,*„) = ^ r — (2.40) 'ot where Ea is given by (2.34). Now consider so called identical spin rotation effect. For simplicity, let VT = WXX = Wyy = WZZ = 0 then (2.34) becomes E a = \m~ + a ( i B B ° + ( i f + i f ) { 2 n a + n ~ a ) { 2 A 1 ) which can be rewritten as h2p2 3 fVD VE\. 1 (VD VE . , , +apB Therefore, the effective field that quasi-particle experiences is (2.42) b - « = * + ^ ( t + tK- B-> (2'43) 2 8 Thus exchange effects give rise to an effective field which is proportional to the magnetization. Note that this effective field does not require spin-dependent inter-actions. It is non-zero even if VE = 0. Eq.(2.43) should be compared to (1.6). It is important to note that the E S R frequency is not affected by such an internal field because the E S R frequency is determined by where the susceptibility is peaked, not by h~ {E+ - E-). In fact, we have shown this in chapter 1 (see the derivation of (1.31)). Moreover, it is easy to see that VR acts like VE (see (2.27) and (2.30)). Therefore, if there is any frequency shift, it should depend on W ' s only. 2 9 C h a p t e r 3 D e r i v i n g the Susceptibility f r o m the Ver t e x P a r t of Green's Functions 3 .1 T h e R e l a t i o n between La-a(p,uj) and Suscep t i b i l i t y Consider how the system is perturbed by an external field. The Hamiltonian for this perturbation is Hex(t) = J A s ^ i j - H ^ i ) (3.1) where S (r, t) = ( V l (r, t) AI (r,t)) o ( *+_ £ J J ) (3.2) and H i ( r , t ) is a c-number external field with ( H x ) z = 0 (3.3) Introduce H±{r,t) =Hlx{r,t)±iHly(r,t) (3.4) S±(r,0 =^(M)<MM) (3-5) then iP*(*) = MB]T / <*dr (5+(r,t)^-(r , 0 + S_(r,t)H+(r,t)) (3.6) A ^ From (2.5) and (2.6), one can easily show that [ S + ( r , * ) , S + ( r\ 0 ] = O (3-7) [5_(r , 0 , 5+(r ',0] = 26(r - r')(^+(r',t)xp+(r,t) - ^ i ( r , i ) V - ( r ' , i ) ) (3.8) = 2S*(r,<)Mr-r') 3 0 therefore, following ref. [22], t 0<S+(r,*)> = l- j dt'({H™(t'),S+(r,t)}) -oo t h -oo + 00 J dt' j ( f V([5 +(r , i),5_(r ' , 0]}^ +(r ' , 0 (3.9) = ^ j dt' j ddr'DR{r,t;r'b,t,)H+{r',t') —00 where DR is a retarded correlation function iDR(r,t-y,t') = (\S+(r,t),S„(r',t')})6(t-t') (3.10) and 1, for x > 0; ^ ~ 1 0, for x < 0. To calculate DR, we introduce a Green's function that depends on the imaginary-time variable: L_+(l,2) = -(T[5+( r i,f a)5_(r 2t 2)]) (3.11) 1 or, more generally, L/3a = ^{TlSaiTutJSfiiTiiti)]) (3.12) Eqs. (3.10) and (3.11) have the usual Fourier representations ddq du D R ( r u t i ; r 2 , t 2 ) = j ^i^ ei«.-(n- 3)--(*.-* a)^( q > u,) (3.13) -i/3h where WK Z„ • h 1(y^_ — v — even integer. i/3h The usefulness of the imaginary time Green's function Lap results from the fact that DR(q, oo) is equal to the analytic continuation of L (_ from zv to u + ie [23]: DR{q,u) = lim . Z,_ + (q,z„) (3.15) 3 1 Now introduce the Fourier representation of H+(r,t): H + { r > t ] = / « i q- r- i w t^ +(q,«) (3-i6) Substi tut ing (3.16) and (3.13) into (3.9), one obtains 6(S+{q,u>)) = ^Bh-1DR{^u)H+{q,oj) (3.17) Consequently, 6(S+(q,uj)} H+{q,oj) = (^Bh-1DR(q,u;) (3-18) = ^Bh~l L_ +(q,w + ier) 3 2 3.2 The Green's Funct ion Equat ion In order to calculate L^Q(q,u> + is), we start by considering the vertex function L^ a(2,1,3). The equation of motion for this function is 1,^(2,1,3) = • G o a ( 3 , l ) G / v ( 2 , 3 ) + + Yi J ^ 2 G a i Q ( l , l ) G ^ l ( 2 , 2 ) t c 7 a 3 a i ^ ^ j 2 - l ) L ^ 2 a 2 ( 2 , l , 3 ) + Oiftasft o (3.19) + Y, f <^'GPia{Z\l)GM{2tZ')iUMia3aiP 0 1 0 1 * 2 0 2 (J where Ua.*f,.f,{l - 2) = V a ^ f l r , - T2\)6{h - t2) (3.20) Fig. 3.1 Vertex Function and the G's are given by (2.38). From (2.32), one may write in coordinate representation (3.21) Substituting (3.21) and (3.20) into (3.19), integrating over 2 and 3, and summing 33 over some spin indices, one has where L^(2,l,3) = Lj2(2,l,3) + + E / ^i{2(2,l,I)V f c o W lL A f c(l, l,3)+ Afc { (3-22) -%ph +E / ^4°2( 1' 1' i)W0 1^ 10 2(i ' i>3) 0i0a o 4°2(2,1,3) = » G Q O ( 3 , l ) G ^ ( 2 , 3 ) (3.23) Now let 2 = 1 and 3 —• 2. By doing so one has an equation for the "bubble diagram" with vertex corrections: Lpa(l,2) = Fig. 3.2 Bubble Diagram with Vertex Correction The time and space Fourier transform of Lpa(l,2) is clearly the function Lpa(<i,u + iE) which determines X0a (Qiu) . The approximation used to obtain (3.22) is the so-called "self-consistent Hartree Fock approximation" including vertex cor-rections. One can write ^(1,2) = 42(1,2)+ -ipn + Y, J < f i i j 2 ( l . I ) I W + W f t | i f t A ( U ) 0103 O (3.24) 34 Because G Q Q ( 2 , l ) , a = ± is only a function of 2 — 1, then Lpa(\,\,2) is also only a function of 2 — 1. Therefore one may write L ^ ( l - 2 ) = 4 ° 2 ( l - 2 ) + o In general L / j a ( l — 2) satisfies the boundary condition (3.25) t2=o W 1 - 2) therefore one may write L ^ ) ( 1 " 2 ) = ^ ^ y (2^p = e ^ " - ^ ! ^ ! - 2 ) (3.26) t!=-i/3h 1 [ d p i p . ( r i _ r 2)_i 2 and where 7TZ/ 2|y = i(3h v = even integer (3.28) (3.29) W i t h (3.27) and (3.28), one has, in the momentum representation LPa{p,zu) = L{pl(p,zu) + L{pl{p,zu)^(VfcapPl +V/3ap2p1)Lplp2{p,zl/) (3.30) Pi Pi Substi tuting (2.32) into (3.30) and taking the transformation (3 —• a , a —> — a, one has L a , - a ( P , 3 | / ) = £ i ° ! - a ( P > * i / ) + La^-a(P> - (VT X X + V K y y - W „ ) ^ ^ ( p . z , , ) -^ 2 L i ° ) _ Q ( p , 2 , ) ( W I I - W y y ) I , _ Q ! a ( p , 2 t / ) The continuation from zv to the frequency OJ yields (3.13) L a , _ Q ( P , c ) = 4°L a ( P , w ) + 4"L(p^) 2/ a , -e.(p,w) - ( w z x + W y y - Wzz) - 2L{°la{p,u){Wxx - Wyy)L^a{p,u) 3 5 (3.32) This is a matrix equation of the form 6_L +_ + a _ L _ + = (3.33) where aa = l-l£°Ltt(p,w) 6 a = -2L^ )_ a(p,u;)(^ I-W ! / y) xx i ^^yy wzz) (3.34) 3.3 Dispersion Relations The spin wave dispersion relations are determined by the poles of x !-(<!»<*>). From (3.18) and (3.33), one obtains an equation det a+ b+ 6_ a_ = 0 For small damping, the imaginary part of X- + (q>w) is small, then the dispersion relation is determined by [24] Re det 6_ = Re{a +a_ — b+b_} = 0 (3.35) (3.36) Because Ws are small, one may neglect and then the above equations become Re(a+) = 0 or Re(a_) = 0 Consider the specific case a = —, then the dispersion relation is determined by Re(a_) = 0 or 1 -( ^ T + X + V r ) " ( ^ I Z + ^ ~ W z z ) Re{LL°|(p,w)}=0 (3.37) In the next chapter, L^(p,w) will be calculated. 3 6 Chap te r 4 The Ca lcu la t ion of L^ia 4.1 The Bubb l e D i a g r am The function X^°2a is a so-called "bubble diagram". In position space, one may write 4-a(l>2) = t G Q Q ( 2 , l ) G _ a _ a ( l , 2 ) . (4.1) Fig. 4.1 Bubble Diagram Substituting (2.38) into (4.1), ddp ddp' r«» / d p d p ' 1 Q ~ a [ ' j " V Wd (2*) ' z„» - h-*E.a(P) x I e - * ' ( * v " - * „ ' ) ( * i - * 3 ) + t ( p - p ' ) - ( r i - » a ) «„. -fe _ 1£; Q(p') (4.2) where * — l zv> = —— + n H-zv« = —r=r + ft M ^ J \ i/" = even integers (4.3) Multiplying (4.2) by e t ' n " ( < 1 - t 2 ) - , ' k ( r > _ r 2 > and integrating over tx through 0 to -ifih and over ri yields the Fourier coefficient of L^]_a (4.4) where Uu =—— + h~1(n-a - Ha), v = even integer (4.5) —ipn 37 Now making use of the frequency summation formula [25] where ' M s > • « = ( 4 - 7 ) and C is the contour that encircles all the poles of h(z) in the positive sense, one has ddp' f dz 1 1 + k) z - J T ^ f p ' ) /• rfV fa(Ea(p')) - /q(£-«(p' + k) - fen.) J (27r)d hn1/ + Ea(p')-E_a(p' + ^) (4.8) Recall that where n„ = uv + h (/z_a - ^ Q ) (4-9) 0j„ = (4.10) Then one may write r(o) r i r 1 - ft / rfV /«(g a(pQ) - /-«(S-«(p' + k) - uy) where fa is given by (4.7). Finally, the continuation from uy to u; yields r(o) f n f ^ - f t / /a(gq(q ) ) - / - a(g-a(q+P)) M l 2 > Q ~ a l P ' J y ( 2 7 r ) ^ W + M _ Q - / , Q + J E ; a ( q ) - J E _ Q ( q + p) l " ' where the relation e = 1 (4.13) has been used. 3 8 4.2 Real Part of L (^(p,w) Now consider the case a = —. For convenience, introduce the notation ; i ,j y (27r)d + -/*_+ £_(q)-£+(q + k) (4.14) where P represents the principle value. The total number density of H atoms with spin a is given by »« = / 7 ^ / « ( £ a ( p ) ) (4-15) For n_ » n +, one may neglect /+ in (4.14). By means of the expansion oo UEa) = Yt*-nP{Ea-'i-) (4-16) n=l one has ~ f d d a e-n/9(£;_(q)-M-) J2(p,w) = fiY P/ y ^ T 7 - T ; r (4.17) Substituting (2.34) into (4.17), one has d d q n = l (2TT)< •X X e x p { - n / ? [ ^ - W 5 o + ^  + - M_] } !g Mne ^ y m (4.18) where M = exp{-/?[-/xBB0 + E{_1] + E{2) - M_]} (4.19) A = feu; + (i+ — (l.- — 2HBB0 + E(y + E(y - - E { 2 ) (4.20) Using cylindrical polar coordinates q = q^ + g^p where p is the polar axis, one may 3 9 write O O + ° ° j , + ° ° riph2 ( . » \ : — oo — oo m — oo m where ; 2 \ h m Introducing the notation 2m mA hp\ ( (3 ph 2 ) \ 2 m one has C O + ° ° Z—' 27Tftp / X + « = 1 ™ — oo ^ — d+l °° 27^ ftp n = 1 where — oo (4.21) A = - Z T - (4-22) y (4.24) (4.25) + 0 0 2 F{t) = - \ p f d y ^ - (4.26) When p —> 0, and ui —» yu_ — / n + -f 2 p B B o , one expects that x —> oo. The following calculation wi l l focus on this case. The asymptotic behavior of F(t) is Using (4.27), one may write 40 On the other hand, n_ = ddp J (27T (2ir)d e/?(s_(P)-M-) _ i oo n-1 then where Xn^mfl rii + i 1 2n?hp\x Vd 2xs oo n=l 4.3 Sp in Wave Dispers ion Re la t ion The condition (3.37) may now be written as x = + Y + Vr) ~ (Wxx + Wyy -Wzz) Xri-m ( ^i + i 1 271-2% \ ri& 2x 2 An approximate solution for (4.32) is where XQ x « x 0 1 + z-o TJd 2X Q + ^ + Vr) - (W« + W y y - W„) From (4.24), (4.22), and (4.20), one has fcw(p) = /*_ - //+ + 2fiBB0 + E{A + E *(2) + n_ 1 + Ara_m 27T2 fep ^ + i 1 ^ 4 2x2 (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) 2„2 + 2^ 2 2m* 2m (4.35) where ftAw = E{2) - E(J> - ( W x x + Wyy - Wzz)n_ , ( 2 ) 2{2WZZ - Wxx -Wyy)n. 1 m* 1 m 1 + rid + i ^ (^D + | ^ + y r - W X I - W y ! / + ^ ) n _ / 3 . (4.36) (4.37) 41 here A w represents a shift of the spectrum, and m* represents the effective mass of a spin wave. Now consider the shift A w and the effective mass m* in two cases: 2-d and 3-d. a) 3-d case. In 3-dimensional system, d — 3, and Vr = W where W is a constant. Therefore, (4.38) A w = 0 (4.39) In the high temperature limit, M —> 0* then (see appendix B, (B.2)) Vi 25 where M is given by (4.19). Then 1 _ 1 m* m 1 + ( 1 * M ) (4.40) 2§ ' {\VD + \VE)n-P_ In the low temperature limit, consider first the temperature T « Tc (T > T c), where Tc is the critical temperature for Bose-Einstein condensation. By introducing the notation a = /3\-fiBB0 + E{_1] + Ei2) - //-(/?)] (4.41a) then from (4.19) and (4.29), oo n- = \~dY2n~i-e~a (4.416) n=0 When T « Tc, a 0+ (4.41c) * It is shown in appendix C that in general, M —• 0 when T —> +oo, and a —> 0+ when T -> Tc (3-D) or when T -» 0 (2-D). (for a, see (4.41)) 4 2 where we have noted that /z_ depends on /?.** From (5.18) in appendix B , *?<»4-i r ( - V ( - ^ a^0 3 + 1 - l l j n 2 J r l + - 7 3 f ] (4.42) *S r ( | ) f ( | ) ^ f( |) where T(i/) is the G a m m a function, $(v) is the Riemann's Zeta function, and 71 = 0 then 1 1 m* m , , 5 f ! ) ( » - » ) ( 4 . 4 3 ) r ( ^ c ( - ) a ^ 3 (4.44) r ( I M f ) L f ( | ) J ( i v b + i ^ ) » - / ? j b) 2 -D case. In the 2-dimensional system, d = 2. From (E.5) in the Appendix E , the general angular dependence of A w is ftAw = 2W (3 s i n 2 6 - 2) (4.45) where 6 is the angle between the normal of the plane h and the external field B o -Depending on the relation between the normal to the plane of the system and the direction of the field B o , there are two different l imit ing cases, i) the normal coincides wi th the direction of B o -Let 6 = 0 in (4.45), one has A w = -4h~lWn_ (4.46) Also , Vr = \w (4.47) where W is a constant to be determined. In the high temperature l imi t , M —> 0 as before. Then from (B.2) in appendix B , T)d 4 ** For non-equilibrium system, we do not know where, when and how a w i l l ap-proach 0 + . B u t for equi l ibr ium system, we do have some expressions for 3 - D and 2 - D cases respectively, (see appendix C) 4 3 one has m m 1 + (1-IM)TT, (4.48) 4 ' {\VD + \VE-iW)n-(}\ In low temperature limit (T —• 0), one also has a —»• 0+.* From (-B.9) in appendix B, »7i 2 .f(2) In a 6 In a then 1 1 m ii) the normal direction of system is perpendicular to the direction of Bo-Let 6 = 7r/2 in (4.45), one has (4.49) Au = 2h~1Wn- (4.50) Similarly, Vr = -W 3 (4.51) where W is a constant to be determined. Because of symmetry, Ws are the same in both (4.46) and (4.50). Similarly, in high temperature limit, 1 m* 1 m 1 + ( 1 - - M ) -4 ' {\VD + \VE + lW)n-p whereas in low temperature limit (T —* 0), 1 1 m ( " ) -V 6 l n a ' {\VD + \VE + § W ) n _ / ? J (4.52) (4.53) * For non-equilibrium system, there is no expression available to show how a appoaches 0 +. However, for equilibrium system, there is an expression to show this, (see Appendix C) 44 4.4 S u m m a r y o f the R e s u l t s Eq.(4.35) shows the spin-wave dispersion relation in the l imit p —> 0. However, the A w ' s are different for different cases. From (4.39), (4.46), and (4.50), one may conclude that the ratio of the shifts w i t h respect to the 3 -D resonant peak ( A w = 0 ) is — 1 : 2 for fields parallel and perpendicular to the surface, and al l A w ' s are proportional to the density n _ . These results are in agreement with the experiment data (F ig . 1.1-2). 4 5 Chapter 5 Es t imat ion of Frequency Shi f t 5 . 1 Scat ter ing Amp l i t ude In this chapter, attention is focused on the calculation of the constant W appearing in (4.46) and (4.50). We consider only the low density and low energy limit. For spin-independent interactions, a well-known effective potential which is deduced from the ladder approximation, is [26] V ( q , q ' , P ) = r ( q , q ' , P ) « — (5.1) m where V is written in the frame of the center of mass, P is the momentum of the center of mass, q and q' are the relative momenta, and a is the scattering length. The conditions for validity of this approximation are that [27] |q|a <C 1, |q'|a<l and na3 <C 1 where n is the density of the particles. For spin-dependent interactions, one may derive a general result (see Appendix A) h2 Va'a/?'/?(q,q',P) » — /a'a/?'j8(q,q',P) (5.2) where faia/3'p is the scattering amplitude. Eq.(5.2) may be written in the form of tensor: K(q,q',P) * -/(q,q',P) . (5.3) m where now V and / are tensors. In the first order of approximation, / does not depend on q, q', and P, therefore V has the same expression in both the frame of the center of mass and the laboratory frame. 4 6 5.2 Calculation of the Scattering Amplitude To calculate the scattering amplitude, one must solve the Schrodinger equation ( ~ | l V 2 + H I + »BBO{O1Z®UE + UE®O22) + ^-S12SJIP= Erp (5.4) where Hj is the singlet-triplet potential, UE is a 2 x 2 unit matr ix, <8> denotes the tensor product of two matrices. The dipole-dipole interaction has been factorized as H dd Si, (5.5) in which 5 1 2 = ox • o2 - S(o1 • h)(a2 • h) (5.6) where o is Pau l i matr ix , and h is the unit vector r/|r|. To simplify the calculation, one may neglect the off-diagonal elements because in a high field, the diagonal elements of Hj +pBBo(oizlS)UE-\-UE®a2z) + HBSI2/ r 3 are large in magnitude (~ HBBQ). The result is a set of uncoupled equations. Solving one of the equations, one can estimate the scattering amplitude. Let us take the z-axis as the quantization axis, and consider the simplest 2 -D case where the gas is in the x - y plane. Using the basis W = ^(l + - > - | - + » ll,l> = l + +> (5.7) H,o>^(| + -> |l,-l> = |--> + ) ) one has S12 — 0 (5.8) 0 0 o A 1 0 -3e _ 2 i^ 0 0 - 2 0 VO -3e 2 t^ 0 1 / where <p is the angle between r and the x-ax is . Neglecting the off-diagonal terms, /0 0 0 0 0 1 0 0 0 0 - 2 0 V0 0 0 1 (5.9) 4 7 Compar ing (5.4) wi th the equation ^ - | ^ V 2 + Hj + nBBo{olz ®UE + Ue® o2g)Sjipo = Eipo (5.10) one easily sees that Hdd w i l l cause some corrections to the scattering amplitudes or equivalently to the phase shifts. This reflects the realistic influence of the d -d interaction on the scattering process. Therefore, these corrections w i l l be used to estimate the effect of the d-d interaction. Notice that in the basis (5.7), Hj is diagonal. Taking one of the uncoupled equations, for example, that for the particles in the state | }, one has ( - | ^ V 2 -f VT — 2nBB0 + ^jrp= EXJJ (5.11) where Vj- is the triplet potential. F rom (5.10), one also has ( - ^ V 2 + VT - I U B B ^ Q = ExPo (5.12) Now there are two radial equations: 1 d ( dRo\ (12 TJ m2 r * v r - * ; n * D r ° ( 5 1 3 ) and ;*('S + ( * ' - ^ - f £ - £ ) * = 0 < 5»> where 2a h2 and k2 = [E + 2nBBQ)'-^2 (5.15) 2fi ' f t 2 Solving (5.14) and (5.15) numerically, one can calculate the phase shift easily by comparing the positions of the nodes. In the low energy l imi t , only s-wave scattering (m = 0) is important. 4 8 Silvera has found [28] a useful approximate analytic expression for the triplet potential, VT(r) = exp(0.09678 - 1.10173r - 0.03945r2)-- {ri(r - 10.0378) + T?(10.0378 - r) exp[-(l0.0378/r - l) 2]}x (5.16) x (6.5/r6 + 124/r8 + 3285/r10) where ( 1, for x > 0; r,(x) = (5.17) 0, for x < 0. Now let us derive the relation between the scattering amplitude and the phase shift 6o. The definition of / as derived in the Appendix A, is: / s , ^ ( p ' k ) = (dfc)i / ^ ^ ^ ( p ^ i p ' ^ ^ ^ p ' ) (5-18) Comparing this definition with the usual definition given by [29] (Eqs. (3.10)-(3.12)), then one has 7 = - 4 V / f / / ( 5 - 1 9 ) where /' is the amplitude defined in [29]. From (2.13) of [29], one has, in the low energy limit, /' « yj^So (5.20) Therefore, /«-4<5 0 (5.21) According to (5.3), the effective interaction is Ah2 V e f f(p) = So (5.22) m This is also the Fourier transform of Veff(r). By writing ^ = 4> ( x 2 + 2 / 2 ) (5-23) y*3 y+ 5 49 one may identify VM = \(WXX + Wyy) = 2-W (5.24) Consequently, from (4.46), one has hAuj = -AWn_ = -6V e f f(p)n_ = 24h280n_ (5-25) m or by wri t ing (4.35) in the form h2v2 fcw(p) = n - - n+2pB(B0 + AS) + — ~ 2m* we obtain the expression HAOJ 12h260n_ A 5 = = — gauss (5.26) 2/J,b mfxB The Eqs. (5.11) and (5.12) have been solved numerically by the R u n g e - K u t t a method [30]. The results of the phase shift 6(Ek) have been fitted in the low energy region by the expression (Fig. 5.1) 6(Ek) 0.15466 1.0974 3.0413 3.1037 (In £ f c + 0 . 8 3 8 ) 2 (In EK +0 .838) 3 (In EK +0 .838) 4 (In EK +0 .838) 5 ' for 0 < Ek < 10~2 m e V ; -5.3912 x 10~ 4 + 4.1268 x 10~ 4 \nEk + 4.9663 x 10~ 5 ( ln Ek)2 for 1 0 " 2 m e V < Ek < 0.5 m e V . (5.27) The reason for fitting the data in this form is that the phase shift of the two-dimensional low-energy scattering is [12] 6 { k ) = 2h7ka[l + ° i l / l n k a ) ] ( 5 , 2 8 ) for k -» 0, where a is the scattering length. For T < 0.1 °K(= 8.6 x 1 0 ~ 3 m e V ) , the high energy behaviour of 6(Ek) is less important. For convenience, we extend (5.27) to Ek < 3.15136 m e V without further correction. A t Ek = 3.15136 m e V , 6 = 0. A n d for Ek > 3.15136 meV, we simply let 6 = 0. 5 0 -«(xlO-») 1.4 1.2 1.0 0.8 0.6 • 0.4 0.2 0 5 x 10 - s io~a 4 5 6 7 («) 5x 10 Ek(xlO-3meV) 8 9 10 5 x 10-1 Fig. 5.1 Plot of the Phase Shift 6 vs. Ek 51 Now let us find out what is the probability density p(6). The function p(6) satisfies v m a x / P{6) dS = 1 where <5max i 1.3964 x 10" (5.29) (5.30) For the distribution function of a non-degenerate gas, + 0 O k 1 f E - E k / k s T d E k = 1 B-L J (5.31) By changing the integral variable to 6, one obtains ^ J |exp(- — ) kRT -5 max dEbl(6) d6 + e X p ( - ^ } dEb2{6) dS | d6 = 1 (5.32) where Eb\ and Eb2 are the two branches of Ek{6). From the last equation, 1 / En(S) P(6) = TT~^ \ e x P ( — I T ^ r ) kBT kRT dEbl{6) d6 dEb2(6) d6 } (5.33) The numerical result is shown in Fig. 5.2. From (C.16) in appendix C, the condition for the validity of the high temperature approximation is - I n 2nh2 n m*kBT » 1 or n m*kB ,o , - <C f - = 2.4 x 10 1 3 c m _ 2 K - 1 (5.34) For most experimental data (Fig. 5.3), 2.4 x 10 1 1 cm 2 < n < 5.4 x 10 1 1 cm 2. Consequently, one has the condition T > 0.010 ~ 0.023 °K (5.35) The lowest experimental temperature (T « 0.066 °K) roughly satisfies this condi-tion. 5 2 53 54 From the experimental data (Fig. 5.3), one obtains AJ5 = -1.84 x 10 _ 1 2n_ (5.36) therefore from (5.26), Sp = -2.14 x 1CT3 (5.37) where the index p denotes the most probable value of 6, or where the function p(6) peaks. The numerical calculation shows that at T = 0.066 °K, 8P « -6max « -1.40 x 10~3 (5.38) This value is about 35% less than that expected. The result above is in disagreement with the experimental data. In fact, the layer of Hj atoms on the 4He surface is about 10A thick. The 3-D effect on the effective d-d interactions [34] will be significant. According to the theory, this effect will reduce the value of 6P because the average over the bond state (denoted by (f>(z) in [34]) usually reduces the value of Bp. Therefore this effect would reduce the theoretical result even further and make the agreement with the experimental data worse. 5 5 C h a p t e r 6 C o n c l u s i o n s Details of the calculations carried out for this thesis are presented in the pre-ceeding chapters. In summary, a self-consistent Green's function method has been developed to derive the spectra of spin-waves in spin-polarized system wi th spin-dependent interactions including dipole-dipole interactions. The main conclusions resulting from the calculations are listed below. 1) the spin-wave spectra generally has the form h2p2 hoj(p) = + h A w + 2/J.BBQ In the 3 -D case, A w = 0 whereas i n the 2 - D case, the d-d interaction causes a non-zero shift. 2) the many-body method allowed us able to calculate the effective mass in the very low temperature region. For 3 -D case [T > Tc but T ~ T c ) , the result is 1 m* 1 m r ( f ) f ( § ) [ 1 + f ( § ) J ( j i ^ + ivk )«_ /? . where a = (3\-uBBo - E{y + E{2) - /*_(/?)] ~ 0+ fl i r ( ^ - l ) ( ^ - 2 ) - - - ( ^ - n )  K = E |n + l - i / | n ! n=0 ' ' m is the mass of a H atom. For 2 -D cases (T ~ 0), a) when the normal direction of system is coincident wi th the direction of B o , 1 m" 1 m ^ 6 l n a ; ( i ^ + ! ^ - f W ) n _ / ? J b) when the normal direction of system is perpendicular to the direction of B o , 2 m* m ( " ) V 6 l n o ; (\VD + \VE + § W ) n _ / ? . 5 6 On the other hand, in the high temperature region, the effective mass for 3-D case is given by 1 m* 1 m 1 1 + (1 M ) 2§ ' {\VD + \VE)n-p_ which is consistent with (1.32), the quasi-classical result. While for 2-D cases, a) when the normal direction of system is coincident with the direction of B o , 1 m* 1 m 1 + ( 1 - 7 ^ ) 7 T 4 >(±VD + \VE-lW)n_(3\ b) when the normal direction of system is perpendicular to the direction of B o , 1 m* 1 m 4 >{\VD + \VE + lW)n_P 3) the general angular dependence of the shift Aw is given by hAw = 2VF(3sin20 - 2) where W is a constant and 6 is the angle between B o and the normal of the plane of the 2-D system. The simple estimation of scattering amplitude by approximately solving one of the Schrodinger equation has led to the result AB 1.20 x 10 _ 1 2n_ gauss where AB is defined by AB = hA w 2fi B here Aw is the frequency shift. The comparison of this result with the experi-mental data is shown in Fig. 6.1. This result is smaller than the experimentally observed value by about 35%. We are unable to account for the discrepancy between theory and experiment. In fact we would expect that a more realis-tic calculation, including the effect of the spatial extent of the wave function perpendicular to the surface, would further reduce 6P. 5 7 58 Bibliography [la] Eric D. Siggia and Andrei E. Ruckenstein, Phys. Rev. Lett.44(21), 1423 (1980). [lb] Richard W. Cline, Thomas J. Greytak, and Daniel Kleppner, Phys. Rev. Lett.47(17), 1195 (1981). [lc] Donald Rapp, Quantum Mechanics, Holt, Rinehart and Winston, Inc., p367-373. [Id] V. P. Silin, JETP 6 (33), 945 (1958). [2] P. M. Platzman and P. A. Wolff, Phys. Rev. Lett.18 (8), 280 (1967). [3] Sheldon Schultz and Gerald Dunifer, Phys. Rev. Lett.18 (8), 283 (1967). [4] A. J. Leggett and M. J. Rice, Phys. Rev. Lett.20, 586 (1968). [5] L. R. Corruciui, D. D. Osheroff, D. M. Lee, and R. C. Richardson, Phys. Rev. Lett.27, 650 (1971). [6] E. P. Bashkin, JETP Lett.33, 8 (1981). [7] C. Lhuillier and F. Laloe, J. Phys. (Paris)43, 197, 225, 833 (1982). [8] B. R. Johnson, J. S. Denker, N. Bigelow, L. P. Levy, J. 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Statt, Phys. R e v . B 3 1 , 7503 (1985). [18] W . Kolos and L . Wolniewicz, J . Chem. Phys. 43 , 2429 (1965); J . Chem. Phys. 24, 457 (1974). [19] D . G . Friend and R . D . Etters, J . Low Temp. Phys. 39, 409 (1980). [20] Leo P . Kadanoff and Gordon B a y m , Q u a n t u m S t a t i s t i c a l M e c h a n i c s , The Benjamin/Cammings Publ ishing Company, Inc., 1962, p l 8 - 1 9 . [21] Leo P . Kadanoff and Gordon B a y m , Q u a n t u m S t a t i s t i c a l M e c h a n i c s , The Benjamin/Cammings Publ ishing Company, Inc., 1962, p91. [22] Fetter &; Walecka, Q u a n t u m T h e o r y o f M a n y - P a r t i c l e S y s t e m , M c G r a w -H i l l Book Co . , 1971, p299. [23] Fetter & Walecka, Q u a n t u m T h e o r y o f M a n y - P a r t i c l e S y s t e m , M c G r a w -H i l l Book Co . , 1971, p301. [24] Fetter & Walecka, Q u a n t u m T h e o r y o f M a n y - P a r t i c l e S y s t e m , M c G r a w -H i l l Book Co . , 1971, p l 8 1 . [25] Leo P. Kadanoff and Gordon B a y m , Q u a n t u m S t a t i s t i c a l M e c h a n i c s , The Benjamin/Cammings Publ ishing Company, Inc., 1962, p l 9 6 . [26] Fetter & Walecka, Q u a n t u m T h e o r y o f M a n y - P a r t i c l e S y s t e m , M c G r a w -H i l l Book Co . , 1971, p l 4 3 , 219. [27] Fetter & Walecka, Q u a n t u m T h e o r y o f M a n y - P a r t i c l e S y s t e m , M c G r a w -H i l l Book Co . , 1971, p219 and p221. [28] see Ref. [10] in [19]. [29] Sadhan K . Adh ika r i , A m . J . Phys .54 (4) , 362 (1986). [30] M i l n e , N u m e r i c a l S o l u t i o n o f D i f f e r e n t i a l E q u a t i o n , Appl ied Mathematics Series, Edi ted by I. S. Sokolnikoff, 1953. [31] S. T . Beliaev, J E T P 34 (7) , 299 (1958). [32] Roger G . Newton, S c a t t e r i n g T h e o r y o f W a v e s a n d P a r t i c l e s , M c G r a w - H i l l Book Company, 1966, Chapter 10. [33] Thomas J . Greytak and Daniel Kleppner , L e c t u r e s o n S p i n - P o l a r i z e d H y -d r o g e n , unpublished, 1982, p35-36. 60 [34] B. W. Statt, Ph.D thesis. [35] F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Book Company, 1965, p525. 6 1 Append i x A It is shown in this appendix that the relation between the effective interaction deduced from ladder approximation and the scattering amplitude is eff h* ~ m where fsiuisu is expressed in centre-mass frame as Av< s,( P,k) = — J ^ ^ t ( 2 M ) ( p | ^ | p ' ) ^ ( + ) ( k ^ ; p ' ) (A.2) in which fj, is the reduced mass, xl> 1S ^ n e wavefunction in spin space, is the state of scattered particle, and finally, (p\V|p') = j ddrddr' exp[t(p' • r' - p • r)] (r|#j|r') (A.3) Usually (T\HJ\T') = (r\Hj(r)\r') = Hj(r)6(r — r'), where Hj(r) is the interaction potential. 6 2 A . l B-S (Bethe-Salpeter) Equation In the low density case, the ladder diagrams are the most important. The B-S equation in the ladder approximation is r a'a/3'/9(Pl ' P 2 ' ^ l ( 2 l / 2 < ; P i P 2^i / 1 2 I / 2 ) = Va>apip(pV - Pi) + + p ^ E E / ^ ^ G i ° x , ( P i . ^ ) G < ° ) ( p 5 , 2 l , 1 ) F < > , A , , „ ( p „ - p I ) x x(27T)3(-Z/?)(5(pi + P 2 -P!# - p 2 ' ) < 5 / x J + ^ j M l , + ^ , r A a r / / 3 ( P l P 2 ^ i ^ ; P l P 2 ^ 1 2 I , 2 ) (AA) where one has assumed that The following formulas will be developed in the 3-D case for convenience. However, it is easy to modify the results for the 2-D case. Carrying out the sum over v2 and the integral over p2 in (A.4), one has ra'Q/3'/3(pi'P2'2l/1,^i/2,;PlP2^12i/2) = Va>apip{pi' - P i ) + Ar/ vq ' xV/Q/A^r7(q)rAar,/3(Pi' - Q ,P2 ' + q .^i/ , , ~ z v ^ z v % , + z V q \ p i p 2 z U l z v „ ) (A.6) where vq is all even integers. For convenience, one may work in mass-centre frame in which P = P l + P 2 = Pi' + P 2 ' [A.7) p' = ^(Pi' - P 2 O (A.8) P = ^ ( P i - p 2 ) U-9) and 2 Zjyt — Zv1 -\- Zi>^ — ~\~ ZiSr,t 1 . ZV> = -(21/,, - 2^,) (A.10) Zy — ^{^fi ^ 1 / 2 ) 63 where (Pz„t) are the total wave vector and frequency, and (p'2^') and (j>zu) are the relative wave vectors and frequencies. Moreover, one denotes that P 2 ' zu1i zv2i j P l P 2 2 i / j zi/2 ) — ^ a'aP'P\P Pzu'zu\* zvt then (A.6) becomes T a< ap< p(p'pzu' zv,*P zvt) = VW/?'/?(p' - p) + AT? i', v ' ) (^12) Notice that T can be expressed by V(p' — p), therefore T is actually independent of the relative frequencies [31]: r a' a/?'/3(p'p^'Z 1 /;P2 I,J =Ta>api/3(p'p;'Pzl/t) (A.13) Keeping this in mind one can sum over vq, and using the sum rule (see (4.6)-(4.7)) one has the B-S equation ro'Q/?'0(p'p;P2i/t) = Va>ap>p{p' - p)+ +tT J ^ z»< - + q) - - q) + «^ , ( P , q ) e w ( q P ' " J (A15) where e a is the energy of a quasi-particle with spin a, £ is a positive infinitesimal number, and JVA„(P,q) = 1 + / M ^ P + q) - MA) + / M ^ P - q) - A*„) (A16) in which 64 A.2 Galitskii's Integral Equation From (A.16) one knows that iV A r / = 1 when considering a system containing only two particles. Replacing JVAt7 by 1 in (A.15) yields r2i/?'/?(p'p;p^t) = V « ' « / 9 ' / 9 ( P ' - P)+ Va'Xl3'r,{p' - q) ^ J (2TT)3 ^ _ C A ( I P + q ) _ 6 r ; ( l p _ q) + j e * W (A.18) notice that in this case T has been replaced by T ^ 0 ^ . Now one is going to express T in terms of r(°). From (A.15) and (A.18), one has 0 r r a ' Q /?^(p'p;P^ t ) -r£ ) a / 3 , / ? (p'p;P^ t ) = J2 J 7^yi^'A/?'r , (p' -q)x X iVA„(P,q) [zUt - £ A(IP + q) - £„(IP - q) + tW A„(P, qje 1 - e A ( i p + q) - c , ( l p - q ) + , - . IE x r A Q 7 ) ( 8 ( q p ; P z I / J + + T f — rAar,/3(qp;P^ t ) - i j 0 ^ ( q p; Pz„ t) z ^ - o d P + q J - c d P - q J + te va'A/9'i,(p' - q) (A.19) For convenience one may introduce an operator DJ so that (A.18) can be written as a r S ^ p ' p j P ^ j = v a, a / 9^( P ' - P) (A.20) with this convention now (A.19) can be written as » { r « . a ^ ( p ' p ; P ^ , ) - r £ ? 0 - . - ( p ' p ; P ^ , ) } = V / I W „ ( P ' - q) X X ^ ( P , q ) zVt - e A ( | P + q) - e„(±P - q) + tJV A l,(P ,q)e 1 z„t - e A ( i P + q) - e„(±P - q) + te x T A a r 7 ( g ( q p ; P z 6 5 (A.21) Replacing V by using (A.20) and mult ip lying 5R 1 from the left-hand side in (A.21), one finally has (2*y X - € A ( i P - r q ) - 6 „ ( i P - q ) - ( - t e . x r A « u / 9 ( q p ; P 2 i / t ) (A22) This is Gal i tski i ' s integral equation. For low density, N\n —• 1, and one expects that (A.22) w i l l converge quickly when iterated. Part icularly, one has an approximation r«'a/9'/9(p'p;Pz«<.J - r i ° L / ? ' / 3 ( p ' p ; p ^ t ) (A.23) Further calculations require the switching of the spin representation to hyperfine representation, i.e., using singlet and triplet states as basis. Let s be the total spin of a pair of particles, and v be the z-component of the total spin, i.e., f 0 v = 0; 5 \ 1 i / = 1 ,0 , -1 then (A.18) and (A.23) become r i t i S > ' p ; P ^ J = ^ v , s , ( p ' - p) + (A.24) + and V f ^ , s ' V " ( p ' - q ) r (o ) f a D p , \ r v ^ ( p ' p ; P ^ « ) ^ r i ? i , i S J / ( p ' P ; p ^ t ) The relation between element Aa<i>',sv a n d Aaiap>p is given by (A.26) 1 1 (A.27) "2 2 2'2' 66 A . 3 S o l u t i o n o f G a l i t s k i i ' s I n t e g r a l E q u a t i o n To connect the function T to the scattering amplitudes, we now investigate the relations between the scattering amplitudes fs'visv and the interaction elements V < i ' s'v'sv In the following calculations, Newton's convention [32] is used. In coordinate representation, one has the expression for a plane wave i M k w / . r ) = M l x ' e x p ( i k - r ) (A.28) (2TT)2 which satisfies J dsr 4(k^,r)^ 0(kVi/',r) = 6{E - E')8n{k k')6Ba.6vv, (A.29) where h2k2 E = ^ - (A.30) 2/x and 6n(k,K') is defined by J dnr6n{f,f')f{r) = / ( f ) (A.31) and fx is reduced mass. Also ipo satisfies (complete set) oo ] T fdE I dflk ipo{ksis, r)^J (k5 i / , r') = 6{r - r') (A.32) The Green's function is which satisfies ( ^ V 2 + £)G ±(£;r,r') = 6(r - r') (A.34) Then the solution of the Schrodinger equation is V> ( + )(ksi/;r) = 0 o (k5^;r) + ^ d 3 r ' d 3 r " G + ( j E;;r,r")(r " | F|r , ) ^ ( + ) ( k s i/;r') (A.35) 6 7 Switching to the momentum representation, one has G±{E;p,p') = ^ = and ^+>(k«,;p) = ( ^ £ x t ' V * W ( k - P) + 4^ 4^ ] (^ -36) fc2 — p 2 + IE where dzp' y t 2 M i (uk)i is the scattering amplitude, and /,'„'.„(p,k) = - ± - r j ^ X S ; > ^ ( P \ V \ P ' ^ + H ^ - , P ' ) (AM) {p\V\p') = j d3rd3r' exp[t(p' • r' - p • r)](r|#j|r') [A.38) Eliminate from (A.S7) and (A38), one has an integral equation for /: /,'„'S„(p,k) = Xstl^(p\V\k)xi+ f d3p> X*'1' %{p\V\p')xi'»fs»u"av{p'*) + sum g»„» / — — — 7 :  2u f d3p' MVa>u'8"v"{p-p')fa"if"au[p',^) = F v^.„ (p - k ) + £ y ^ (A39) where V W * „ ( p - k) = xi.\p\V\k)xl (AAO) To compare (A.39) with (A.25), one may multiply (A.25) by |§, and transform p' to p and p to k respectively, then ^ r S ' , s J p k ; p ^ J = £F*Vi/',«/(p-k)+ d V tt^ v',s»,»(p-p')(^ )ri?,U^ (p'k;P2,() ^ r d3P' 2u r ft2 fc2 - p'2 + »e 6 8 2 M K - 6 A ( | P + P ' ) - e „ ( J P - P ' ) ) + p r^ (, s i /(p'k;P^) (A.41) Introduce an operator S so that (A.39) can be writ ten as 3 / a V « i / ( p , k ) = 7?V , i„ ' f l„(p - k) (A.42) then (A.41) can be writ ten as » [ p r i ? i l | W ( p k ; p ^ ) ] = p t w , „ , ( P - k ) + 7 3 „ / v-> f d*p' 2u 2u{z„t - e A ( | P + p') - e„(|P - p')) + is k2 - p'2 + ie j — r( 0 ) fo'k-Pz ) ^2 1 s"i/"-,siA" K ' rzvt) (A.43) Replace ^Vs>v>av by (A.42) and mult iply S 1 from the left-hand side, one has + ( ^ F / S , ^ ( P ' P 2»(zUt - c A ( i P + p') - £„(±P - p')) + ^ 1 — r ( 0 ) ro'k-Pz ) fc2 - p'2 + ie Therefore one has the expression connecting T^ 0) to the scattering amplitude In the first order, one has (A.44) h2 r ^ ' , ^ ( p K ; P ^ J - ^ W « / ( p , k ) (A.45) W i t h (A.26) and (A.45), one obtains a practical approximation: h2 = / s V s i / ( P , k ) m (A.46) 6 9 Appendix B Consider the series oo e n a n n=l For a > 0, this series converges. In the limit a —• +00, one has S^a) * e~* + • = e - ( l + Now let us consider the most interesting case o —> 0 +. Since + 0 0 + 0 0 0 0 n = 1 0 0 + ° °  n = 1 ~ , + 0 0 °° -na r n = l ^ { 00 — na one has n=l + 0 0 a + °° „_w _ 2 ^ - l (B.2) b\u,a) = ——- / dx 0 + 0 0 = ^ 7 ~ T / " " dx (B.3) r (u )J e*-l v ; e z - 1 a By expanding (1 - f in a series 0 - f ) " = f ( - i ) - ' ' ' - 1 » y - n ? - ' y - » » ( f ) -one obtains + 0 0 K ' 7 r i / ^ l ; rz! 7 e 1 - 1 v ; x ' n=0 7 0 One of the special cases is u = k = 1,2,3, • • •. From (5.4), )+oo / ^ r i > for k = 1; a k-l +oo r f e E ( - ! )"(*;>"/ for*>l. V n=0 a f + O O for A; = 1; k-2 +oo ffe E ( - i r ( * ; V f ( * - n) + ( - l ) ^ 1 ^ / - r r j n=0 V ' rv n=0 where f (x) is the Riemann's Zeta function, oo f (x) = £V* n = l Since a l i r * a ~ P / - r - T d x = *' f o r ^ > ° ' a—o+ y e 1 — ! /? and in the limit a —> 0 +, lim *-»o+ In + oo i a J ex — 1 dx = —1, for k > 1. (5-5) (5.6) (5.7) (5.8) — In a, for A: = 1; S{k,a) « <[ ?(2) + alna, for A: = 2; OfcirjlfM " 7 > ? 2 ) ! ? ( * " 2), for AT > 2. For v /integer, we can- write (5.9) where P(u,a) S{u,a) = P{v,a) + Q{v,a), fl/l— 1 +oo 1 . , n ( i / - l ) ( i/-2 ) - - - ( i / -n) B 1 (5.10) V ( _ i r ^ - ^ - ^ - ^ - n j ftn r ^ z r i d x { B 1 1 ) w (t / - l ) ( i/-2)---( i / -n) n / - 1 ct + oo /xu-n—1 ^ , ( 5 . 1 2 ) 7 1 and \u\ is the greatest integer in v. Because each integral in P(u,a) converges when a -» 0+, [i/] —1 . . . . ^. . 1 , \v — 1)\v — 2) • • • \v — n) „ , P M = f f ^ E J La^(u-n)-' ' " f ' ( - 1 ) > 2 » - ) t t . / ^ < f a ( B . 1 S ) re-u Q Combination of (5.13) with (5.7) yields, at the limit a -> 0+, P(t/, a) = 0 for 0 < i/ < 1 (5.14) P , , , Q ) J ^ M - ™ - FOTL<"<2; (JU5) In addition to (5.7) and (5.8), we also have oo a f X~P 1 lim ap / dx = -, for /3 > 0 (5.16) a then similarly ^^fSE^r '^ 1 ^ 2 ' ^ " ' " 0 < , < o o <*17) n = [ i / ] Therefore from (5.14), (5.15), and (5.17), a) r » <(») i » I r( .) r » for 0 < < 1; for 1 < v < 2; (5.18) where n = 0 ' 1 (5.19) 72 Appendix C In this appendix we investigate some properties of the 3-D and 2-D gas systems. Let us consider a 3-D system first. For simplicity, assume that the system is a one-component Boson system. The density of the particles is - 1 / t/.3fc  U ~ (2lr]3 J c/»(«00-M) - 1 [ C A ) where \i is the chemical potential, and E{k) = Ek + V0n (C.2) here Ek is the kinetic energy of a Boson, h2k2 VQU is the self-energy correction, and m* is the effective mass. By changing the integral variable, 3 +°° 1 (m*kgT):2 f x?dx n = •n2hZy/2 o /x^dx . ^ . m - 1 < c ' 4> where a(T)=P{V0n-ii). (C.5) Noting that n is a constant, then when T —> Tc, (this means T is small enough,) one must have a(T) —> 0 +. In this limit, from (-B.11) one obtains, + oo j / c ^ m . 1 " 4 ) - a k [ T ) i i ( c ' 6 ) 0 where 0± is given by (B.19). Combination of (C.4) and (C.6) yields 7 3 (C.7) where A = 2irh2 m*k,BT (C.8) Then, - ' ^ - i r l ^ - r i i j (C.9) where 2TT/^ n m * * B V ? ( i ) r ( i ) y ' When T < Tc, one has a = 0 but n is no longer a constant. Therefore (CIO) T c\ 2 n - n0(T) n = 0 or n \TC (C.ll) where no(T) is the density of the condensate fraction. This mean-field result is valid only when T ~ Tc and T 3 3 , (C.12) r , 3, _. , i - » C ( - ) r ( j ) a „ 5 where a is the scattering length. [33] The case T —» +oo is quite simple. Because n is a constant, then from (C.4), one knows that when T —» +oo, it must be a —• +oo. Consequently, + oo n (m*kBT)$c_a e - a ( T ) (T) A 3 then a(T) « -ln(A 3n) 74 (C.13) Now let us consider a 2 - D system. The density of the particles is d2k or ! T T ) 2 / ( 2 T T ) 2 J e ^ O O - M ) - l 2nh2 / e * + « ( T ) - 1 o For T —* 0, a (T ) goes to zero as before, then from (B.9), +oo (C.14) m*kBT f dx n « — -m*kBT 2nh2 \na(T) 2nh2n For T —>• +oo, also one has o:(T) —> +oo, then similarly + c /  oo m*fc B T m o 27rft2 ~ A 2 therefore, a(T) w - l n ( A 2 n ) (C.16) 7 5 Append i x D To show the effect of the collision integral, let us consider the simplest situation: a spin 0 system under the influence of an external field F . In this case, the Boltzmann equation can be written as [35] at or m ox where Q(r,v,t) is the collision integral: Q(r,v,t) = J J(f'f[ - ff^VodQ'd^ (D.2) and /'s are the density of the particles: / = /(r,v,t), / i = / i ( r , V l , 0 /' = /'(r,v',*), f[ = f[(Tyitt) where V is the magnitude of the relative velocities (J9.3) V = |v- v i | = |v'- v'J, (DA) a is the differential scattering cross section, o = o(y'-\'1) (D.5a) and dffl is the solid-angle about the vector v' — v'j with respect to v - Vj. In the low temperature limit, we can replace a by a constant, namely, a = a2 (D.5b) where a is the scattering length. Consequently, Q(r, v,*) = a J J (/'/{ - ffjVdtl'tPv! (D.6) 7 6 Assuming that the system is not far removed from equilibrium, one can write / in the form / = /(°)(l + $), with $ < 1 [D.7) where /(°) is the distribution function in equilibrium. Neglecting the small quadratic term one has / / i = / ( 0 )/ 1 (° )(l + * + *i) (D.8a) Similarly, /7i = / ( 0 ) / i ( 0 ) ( i + *' + *i) where we have used the fact that f(0)'f{0)' _ j(0) j(0) Substituting (D.8) into (D.6), Q(r,v,t) =o j j d3vl(m' /(°»/fVA where A $ = $' + $ 1 - $ - $ ! To estimate the order of the magnitude of Q, we may write Q[T,V,t) = = where the relaxation time is defined by * = -a j j d3vxdW f[0)V (D&b) (D.9) (D.10) (D.11) (D.12) (D.13) and we have used the relations 6f = / - /(°) = The upper limit of 1/r is given by the following inequality -t<° j j dzvxdtf f[0)V (D.U) 7 7 Since / dfi' |A$/ $| has an order of 1, this upper limit can be estimated by a J J d3vl(in' f[0)V tm ( |v-v i | ) (D.15) where (...) denotes the thermal average, and n = J dvf{0) (D.16) For v RS 0 (long wave-lengths), <|v-v,|) M where m is the mass of a particle. Therefore in low temperature limit, we roughly have 1 fkBT — ~ c (7J.18) 2 / kBT = a n 1m Now, referring back to Chapter 1, the self-energy correction in (1.6) will enter linearly in the equation of motion. For simplicity, we assume that in Chapter 1, the temperature and the density of the particles are so low that the condition i « J ^ l l (27.20) T n is satisfied. Under such conditions, the effect of the collision integral is small com-pared to that of the correction term |Aei|/ft. Therefore the collision integral J is negligible. From (1.7), we have , assuming 100% polarization case for simplicity, |Aei| 4-nah -n (D.21) h m where n is the density of the particles. With (D.18), the condition (D.20) may be written as a2n - E — < n (D.22) \ m ) m 7 8 This inequality implies that a < A (7J.23) where A = (2nh2 j m A ^ T ) 2 " is the thermal de Broglie wavelength. Note that the density does not appear in this inequality and that therefore the assumption of low temperature is compatible wi th the situation of a non-degenerate gas. 7 9 Append i x E To see the angular dependence of Aw, let us consider the 2-D system on a plane with normal n. The angle between h and the external field B o is 0. Obviously, Aw depends on 0 only because the symmetry. Let us denote the original frame with z-axis along the external field B o as F,. and the frame on the plane with z'-axis along h as F'. For convenience, we choose the y-axis and y'-axis along the direction z x n, the x-axis along the direction y x z , and the x'-axis along the direction y' x z'. Under such a choise, the relation of the coordinates between two frames is x = x' cos 0 + z' sin 0 y = y' (E.i) z = —x's'md + z' cosO Now we are going to calculate Ws. From (2.31), f x 2 WXiX. « 3 M B / d2r' for p - 0 (E.2) where xt = x,y,z, d2r' is the surface element of the plane, and r' is the position vector on the plane. In the frame F', we have z' = 0, therefore from (E.I), x = x cos 0 y = y' (£.3) z = — x sin 6 Substituting (E.3) into (E.2), we have Wxx = W cos2 0 Wyy = W (EAa) Wzz = W sin 2 0 80 where , / 2 W = 3fBj d V ^ = 3 , 4 J dV (£ .46) Substi tut ing (E.4) into (4.36), finally we have kAu = 2 W ( 3 s i n 2 0 - 2) (E.5) 81 

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