Electronphonon interactions and Resonance Raman scattering in onedimensional. carbon nanotubes

 Tyrone Oliver
 11 months ago
 Views:
Transcription
1 Electronphonon interactions and Resonance Raman scattering in onedimensional systems: application to carbon nanotubes José Menéndez Giovanni Bussi Elisa Molinari Acknowledgements: M. Canonico, C. Poweleit, J. B. Page, G. B. Adams Supported by the National Science Foundation
2 Infrared absorption and Raman scattering Frequency (cm 1 ) Raman shift (cm 1 )
3 Energy units for optical spectroscopy hw = hck = 2phc l k wave vector; 1/l = wavenumber 1/l µ energy 1 ev = cm cm 1 = 37.2 mev
4 Ï Ô Ì Ô Ó Conservation rules for a Raman process hw = hw + h L S W phonon hk = hk + hq L S phonon BACKSCATTERING W (cm 1 ) 300 Raman K S q phonon L L L K L [ wl  ws]( q) = w( KL)  w( q KL) = ck c( q K ) = 2cK cq 4p 500 nm Brillouin p 0.5 nm q phonon
5 Phonons in crystals Ramanactive
6 What determines Raman intensities? R È S = 1 R Í R Í M «() t 4pcR 2 2 R Í R Î E ( t) = E coswt Ê ˆ M t P e iwt P ei t a() = 1 Á  * w ag + ag E Á 2Â Ë Ê ˆ P P w, u, u= c( f) d P g ab ab g g Á ab Ë P = P ( w, 0) + ab ab Â d d f Â f f f f g
7 Why a frequency shift? M () t = P(, tt ) E( t ) dt Ú M( w) = P( w, w ) E( w ) dw Ú If P(t,t ) = P(tt ) fi P(w,w ) = P(w) d(ww ) fim(w) = P(w) E(w) If a phonon is present: Ptt (, ) Pt ( + Tt, + T) Ê PÊw, w ˆ P( n) w d w w n = ( ) n T p Á  2 Ë Â Á Ë fi w = w nw phonon ˆ In QM this is just energy conservation!
8 The carbon nanotube family (n,n) Armchair (n,0) Zigzag (n,m) Chiral C = na + ma h 1 2
9 The electronic structure of graphene V ppp V ppp = 3.1 ev
10 Nanotubes: folding the graphene band structure Ê Á Á K Em( k) = E k 2 Á + mk, ( m = 0,..., N 1); g2dá 1 K Á Ë 2 ˆ (11,8) (15,0) (10,10)
11 Nanotubes: electronic density of states (11,8) (15,0) (10,10)
12 Vibrational modes in nanotubes
13 The radial breathing mode in carbon nanotubes w = 1170 cm R. (armchair) J. Kürti et al. Phys. Rev. B 58, R8869 (1998) w = 1170 cm 1 R (zigzag)
14 Why 1/r dependence? Let us consider an atom of mass M, The radial force is F r = 2F t cos q = F t a CC /r 0. But F t = K s Da CC. On the other hand, Da CC = 2u r cos q = u r a CC /r 0. Hence F r K a a = Ê s Ë Á ˆ Ê Á r Ë r Therefore CC CC 0 0 ˆ ur = K u eff r u r u r F t a CC u r q K = = M K M Ê a Á Ë r eff s CC w RBM 0 ˆ r 0 r 0 r0
15 Raman intensities in carbon nanotubes A.M. Rao et al., Science 275, 187 (1997)
16 Resonance Raman excitation profiles hw L Raman shift (cm 1 ) Raman shift (cm 1 ) Raman shift (cm 1 ) Raman shift (cm 1 ) Raman shift (cm 1 ) Intensity (arb. units) Laser Photon Energy (ev)
17 REP for 200 cm 1 RBM E 11 = ev Intensity h = 60 mev Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, (2002)
18 Resonance Raman scattering Light Phonon Light 0 j i 0 Light j j phononi i Light I µ Â 0 0 Ê ˆ Ê ij Áhw hw E hw E Á Ë Á Ë L phonon j L i ˆ 2
19 Raman cross section: quantum theory I The scattering cross section is defined as ds dw = ( radiated power into ) dw incident power per unit area The incident power per unit area is CMP ( ) IP N c Lhw L = n L Other RP IP (R.1) (R.2) where N L = number of incident photons per unit volume and c/n L is the speed of light. The radiated power is RP dw = number of incident photons hw S ( dw ) dt dw = ( NV L ) hw S ( dw ), dt (R.3)
20 Raman cross section: quantum theory II where dw/dt (dw) is the transition probability per unit time from a state with a photon (w L, k L,l L ) to a state with a photon with polarization l S and a wave vector k S within the solid angle dw plus a phonon of frequency W ph and wave vector q. For an arbitrary scattering wave vector k S the transition probability is given, according to Fermi s golden rule, by dw dt CMP Other d hw hw h 2p 2 W fi l S, l L L S Â = ( )   h k S ( W ) ph (R.4) where W fi is the matrix element of the transition operator. Combining R.4 with R.13, we obtain ds wl nv L = Ê dw Ë Á ˆ ÊË ˆ w c S 2 p Â Wfi S L d wl w ( l, l ) 2  S  h k ( dw ) S ( h h hw ) ph (R.5)
21 Raman cross section: quantum theory III The sum over k S can be transformed into an integral: k S 3 Â Ú 2 S 3 kdk S S 3 ( dw ) 2p 2p h c Ú CMP Inserting this into R.5, we finally obtain V V n = = Ê w d ( ) ( ) Ë ˆ ( hw ) Other 2 S S (R.6) ds nl dw = Ê Ë Á w L ˆ w Sn ÊË 4 c S ˆ V Wfi ws S w ÊË 2 ˆ 2 L L p (, l ;, l ) (R.7) 2 h The key quantity that contains resonance effects is the transition matrix element W fi
22 The matrix element W fi I We will use perturbation theory to compute W fi. We assume that the unperturbed hamiltonian H 0 can be written as H 0 = H el + H R + H L, with: A oneelectron (band structure) hamiltonian of the form A free radiation term (photons) of the form A harmonic vibrational hamiltonian (phonons): + Hel = E nsc nsc ns CMP k, n,s Other 1 ( 2 ) + HR = a a + H Â k k k Âhw k l k l k l k, l ( 1 ) 2 + = b b + L ÂhW q m q m q m q, m (R.8) (R.9) (R.10)
23 The matrix element W fi II Therefore the quantum states of the system are characterized by three sets of occupation numbers: (R.11) We will be able to simplify this notation considerably because only a few of these occupation numbers change in the scattering process. For example, the initial state consists of the electronic and phonon system in their ground CMPstates and one photon in state Other k L l L. The final state consists of the electronic system in the ground state, one photon in state k S l S, and one phonon in state qm, so that we can express these states as i f m { N } { N } { N },, l nk qm k = 0010,,, ; E = hw L S i L = 010,,, 1; E = hw + hw qm L S f S qm (R.12)
24 The matrix element W fi III The transition between these states is caused by the interaction hamiltonian H int. We will take H int = H el + H er, where H el is the electronphonon interaction and H er the electronphoton interaction. The electronphonon interaction can be written as nn + H = M b + b c + c el Â kqm qm qm k q, ns kn s kqnn ms CMP ( ) Other (R.13) Explicit expressions for M kqm will be discussed below. The electronradiation hamiltonian arises from the A p coupling term in the interaction of an electronsystem with an electromagnetic wave. It has a form similar to R.13: H er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w K K 12 Â kknn ls ˆ nn ( + + e p a + a ) c , c l kk Kl Kl k K ns kn s (R.14) where p kk is the matrix element of the momentum operator between the periodic parts of the Bloch wave functions for states nk and n (kk)
25 The matrix element W fi IV When applied to the specific problem at hand, the notation in these expressions can be simplified. We first limit the summations over electronic bands n, n to a single empty conduction band and a singly fully occupied valence band. We assume parabolic bands. CMP E t k Other 2me + h2 2 E t k  h2 2 2m h
26 The matrix element W fi V We keep the notation c, c + for annihilation/creation operators of electrons in the conduction band, and we introduce v, v + as annihilation/creation operators for electrons in the valence band. We also limit ourselves to the case when a photon of frequency w L is annihilated, a phonon is created, and a photon of frequency w S is created. We then rewrite the electronphonon hamiltonian as CMP Other ( ) Â el qm kqm k q, ns kn s kqm k q, ns kn s kqms + cc vv H = b M c + c + M v + v (R.15) Notice that we have limited ourselves to intraband terms of the form c + c and v + v. We have not included interband terms of the form c + v, etc. We will see later that the interband contribution is negligible.
27 The matrix element W fi VI For the electronradiation hamiltonian we further assume that the incident and scattered light polarizations are parallel to the same axis z. In nanotubes z must be the axis of the tube. Other directions give zero. We can now write with: H H e CMP Other cv = Ê 12 ˆ Ê 2p ˆ pz a c + v v + c Ë S  S  S m Á Â + 2,,, Ë Vn w + + er kk K k K s ks k K s ks S S ks 12  er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w L L H = H + H +  er er er, ( ) ˆ Â p vc z a c + v v +, c L + L, + + L, ks ( ) s s s s kk K k K k k K k Here we have neglected intraband terms. This is because intraband matrix elements of the momentum operator are very small for small wave vector transfers (by virtue of Green s theorem for periodic functions, see AshcroftMermin, Appendix I). (R.16) (R.17) (R.18)
28 The matrix element W fi VII Since each application of H er can change one photon state (the interaction is linear in the photon creation and annihilation operators) and each application of H el can only change one phonon state, we can only transition from the initial state to the final state by two applications of H er and one application of H el. In other words, we need to go to at least third order in H int : W fi fh H H i = Â int m m int n n int ( E  E ) mn m i ( En  Ei ) CMP Other fh H H i = Â int m m int n n int mn E  hw E hw ( ) ( ) m L n  L (R.19) Notice that the initial state i> contains a filled valence band and an empty conduction band. It is then clear that only the c + v terms in R.17 and R.18 can contribute. (This is of course a manifestation of Pauli s exclusion principle.) Of these, only R.18 can lead to a resonant enhancement by annihilating the incoming laser photon and creating an electronic excitation of equal energy, so that the rightmost denominator in R.19 becomes small.
29 The matrix element W fi VIII Maximum resonant enhancement is then obtained if the energy E m is comparable to the energy E n. When the phonon energy is small compared to the separation between the conduction and valence bands, this condition is met when the middle operator causes an intraband transtion. Thus the most resonant contribution to the Raman cross section arises from W fi +  fher m m HeL n n HeR i = CMP Â Other E  E E E mn ( ) ( ) m i n  i (R.20) By inserting R.15, R17, and R18 in R.20 we obtain sums over four indices k,k, k and q. (And also over the spin index s, but this gives trivial factors of 2). There are two types of terms that yield nonvanishing contributions:
30 The matrix element W fi IX CONDUCTION BAND TERMS These terms contain matrix elements of the form + + 0v c c c c + v 0 n ( W ) + 1 S q k  K k k  k k + K k qm where 0> refers to the electronic ground state, and n(w) is the phonon occupation number at temperature T. We assume that there is one incident photon and one scattered photon, so the application of the photon creation/annihilation CMP operators gives just 1. Other These elements are zero except when k = k + K L, k = k + K S, and q = K L  K S. When these conditions are satisfied the matrix element is equal to +1.The last condition is clearly the manifestation of crystal momentum conservation. VALENCE BAND TERMS These terms contain matrix elements of the form + + 0v c v v c + v 0 n ( W ) + 1 S q k  K k k  k k + K k qm These elements are zero except when k = k + K L  K S, k = k + K L, and q = K L  K S. When these conditions are met the matrix element is equal to 1. (Using commutation properties of Fermion creation/annihilation operators.) L L (R.21) (R.22)
31 The matrix element W fi X We thus end up with a single summation over the index k.we now take into account the fact that the relevant light wave vectors are negligible relative to a reciprocal lattice vector in a typical crystal. Thus we can set K L = K S = 0 in all matrix elements and energies. We thus obtain W fi e = Ê ˆ Ë m cv vc Ê 2ph ˆ p p M cc M vv z z m m Á Ê 1 ˆ  Á Â, k, k k0 k0 Ë VnLnS CMP Ë wlws ks ( Ek  hw Other S ) Ek  hws 2 12 where E k is the energy of the intermediate state given by E k We now recognize that the states closest to to k = 0 will make the dominant contribution because they are closest to the singularity in the density of states. We can then replace the momentum and phonon matrix elements for their values at k = 0 and take them out of the sum. We then obtain ( ) ( ) hk hk hk = Et + + = E + ( m ) = m + m 2m 2m 2 m ; with * * e h t e h (R.23) (R.24)
32 W fi Â ks where we have defined The matrix element W fi XI e = Ê ˆ Ë m Ê 2ph ˆ Á Ê 1 Á Ë Vn n Ë w w 2 12 L S L S 1 ( Ek  hw E CMP S ) k  ( hw ) cv P p p z = ( ) vc z, 0 z, 0 * S ˆ 2 Pz M cc vv m  Mm ( ) Other. The last factor can be converted into an integral: 1 r EdE (  ) Ú k E hw E  hw ( E  hw + ih) E  hw + ih Â s k ( ) = ( ) ( ) S k S S L (R.25) (R.26)
33 The matrix element W fi XII where we have added a phenomenological broadening parameter that represents the lifetime of the excited states. For a onedimensional solid: Other where the first factor of 2 takes into account the spin degeneracy. Using R.24 de Hence, using R.27 we obtain h = kdk = m L rkdk = 2 Ê d k (R.27) Ë 2 p ˆ h m 2 2 * * ( ) * È2m E  E Í Î 2 h L m r( E) = Ê * ˆ È Ë ph Í Î2 E  ( ) E t t dk (R.28) (R.29)
34 The matrix element W fi XIII We thus need to calculate the following integral Ú = 12 ( E  Et ) de ( E  hw + ih) ( E  hw + ih) = L ip È Í 1  Other 1 hw ph Î( hwl  Et ih) hws  Et ih These integrals are discussed in the review article by Martin and Falicov. Combining R.30 with R.29, R.25, and R.26, we finally obtain S ( ) e cc vv L m W = ( ) ( )( )( ) * p P  M n fi È Í Î m V n n w w z m L S L S ph hw  E ih hw  E ih ( ) L t S t m (R.30) ( )( )( ) ( W ) + 1 ( ) hw 2 12 (R.31) R. M. Martin and L. M. Falicov,"Resonant Raman Scattering", in Light Scattering in Solids, edited by M. Cardona (Springer Verlag, Berlin, 1975), Vol. 8, p. 79. m
35 Raman cross section: quantum theory IV so that the cross section finally becomes ds 1 e = Ê ˆ dw 2 Ë mc h 2 F( w ) L Ê ws S Á Ë w L ˆ Ê n Á Ë n L ˆ L Ê Á Ë W m ˆ 4 P M cc z m M vv 2 *  m m n ( ) ( Wm ) + [ 1] (R.32) with F( w ) = i L Ê Á Ë hw  E ih hw  E ih ( ) ( ) L t S t ˆ (R.33)
36 Raman cross section: quantum theory V The measured photon count rate at the detector is proportional to the cross section, but the proportionality factor is usually left undetermined, since the measurement of an absolute Raman cross section requires a careful calibration with a known standard. [Notice that we defined the cross section in terms of power. We can define a photon number cross section which is equal to R.1 times (w L / w S )] If no absolute cross section is measured, only the function F(w L ) is needed to analyze the experimental results, assuming that the approximation of a single conduction and a single Other valence band is valid. This is the case for armchair tubes. From fits of F(w L ) to the experimental data it is possible to determine the transition energy E t and the lifetime of the excited state. In order to study the properties of this function we define Fx ( ) = i Ê Á Ë x 1 ied x  D 1 ied ( ) ( ) ˆ (R.34)
37 Raman cross section: quantum theory VI There are two singularities associated with the incoming laser photon or the scattered photon having the same energy as the transition energy. Notice that the imaginary part has the expected asymmetric shape of a onedimensional density of states, but the real part has the opposite shape, leading to symmetric intensity profiles. Intensity Imaginary part D = 0.1 e = 0.01 Other Real part Real F, Imag F Normalized laser frequency x L 1.15
38 Raman cross section: quantum theory VII A more realistic value for D in carbon nanotubes is D = As the broadening of the electronic states increases, the intensity profile evolves to a single peak Intensity D = 0.01 e = 0.10 Other Intensity D = 0.01 e = Normalized laser frequency x L Intensity D = 0.01 e = Normalized laser frequency x L Normalized laser frequency x L
39 REP for 200 cm 1 RBM E 11 = ev Intensity h = 60 mev Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, (2002)
40 Trigonal warping in carbon nanotubes I The figure shows the 1D density of states for several metallic nanotubes of approximately the same diameter, showing the splitting of singularities (increasing from zero in armchair tubes to a maximum value in zigzag tubes) due to trigonal warping in the band structure of graphene. The associated optical transitions are very close to each other and therefore one has to include two conduction bands and two valence bands in the calculation of Raman cross sections. Other From R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B 61, 2981 (2000)
41 Trigonal warping in carbon nanotubes II Because the two transitions can interfere, it is critical to know the sign and magnitude of the prefactor cc P M  M vv m z ( )( ) 2 * 12 m m in the expression for the transition matrix element. The effective mass can be obtained by fitting a parabolic band to the calculated energy bands. The matrix element of p z can be easily obtained from pseudopotential calculations of the band structure. Other For the electronphonon matrix elements, we can derive a simple expression for the particular case m = RBM, the radial breathing mode. If we limit ourselves to this mode, the electron phonon interaction is H el H = d where d RBM is the normal coordinate for the RBM and H is the electronic hamiltonian. The derivative means that the atomic positions are displaced by along the phonon mode eigenvector. RBM d RBM (R.36) (R.37)
42 Trigonal warping in carbon nanotubes III The normal coordinate can be written in terms of the mode eigenvectors c and the atomic displacements u as d Â * f c lka f M k u lka = ( ) ( ) ( ) lka where l is the cell index, k the basis index, a a cartesian index, and M(k) the mass of atom k within the unit cell. For a periodic solid, Other 1 c( lka qm) = ka NM el ( q me ) c k iqr ( lk ) (R.38) (R.39) where N c is the number of unit cells and e is a unit vector normalized to unity: Â e * ( lka qm ) e ( lka q m ) = d qq d mm ka (R.40)
43 Trigonal warping in carbon nanotubes IV If we give all atoms a displacement Du( lka ) = DR e( ka RBM ), (R.41) where R is the nanotube radius, and we assume q RBM = 0, the corresponding change in the normal coordinate is Dd = N M D Other R RBMc where we have used R.38, 39, 40 and the fact that the atomic mass is the same for all atoms The electronphonon hamiltonian thus becomes H el = 1 NM c H R d i h RBM = 2NMw c RBM H ( R b  b + ) RBMRBM (R.42) (R.43)
44 Trigonal warping in carbon nanotubes V We therefore need matrix elements of the form where we have invoked the HellmanFeynman theorem. By comparing R37 with R.15, and using R.41, we can write cc vv M  M = i RBM RBM = i H n R n nk k k = ER h 2NMw c h 2NMw c RBM RBM Ê Ec Ë R Et R Other Ev  R Thus the relevant quantity for the calculation of Raman crosssections is the derivative of the transition energy relative to the radius of the nanotube. ˆ (R.44) (R.45)
45 Trigonal warping in carbon nanotubes VII E f 2.0 Energy (ev) ev 1.99 ev Other E f Energy (ev) G Z G Z 3.0
46 Trigonal warping in carbon nanotubes VIII Autovalori a G Transizioni a G Energia (ev) ev 1.99 ev 1.99 ev Other 1.62 ev Energia (ev) Raggio del nanotubo Raggio del nanotubo (A)
47 Trigonal warping in carbon nanotubes IX Squared momentum matrix element (1/Å) Square root of effective mass Other ev 1.99 ev ev 1.99 ev
48 Trigonal warping in carbon nanotubes X We can use this to compute the corresponding Raman cross section. When we reverse the sign of the radial derivative for one of the transitions, we obtain a different profile, indicating that interference effects cannot be neglected. Intensity E 2 E 1 = 0.37 ev h = 0.07 ev Intensity Other E 2 E 1 = 0.37 ev h = 0.07 ev de 2 /dr reversed Laser photon energy (ev) Laser photon energy (ev)
49 REP for 188 cm 1 RBM Intensity (12,6) (15,3); (16,1) (11,8) Photon energy (ev)
50 CONCLUSIONS Interference effects are important for the understanding of the Raman cross section in carbon nanotubes. The shape of the Raman cross section as a function of the excitation energy may provide an optical tool to identify the (n,m) values. It is unclear to what extend the inclusion of excitonic effects may affect the simple calculations presented here.
QUANTUM WELLS, WIRES AND DOTS
QUANTUM WELLS, WIRES AND DOTS Theoretical and Computational Physics of Semiconductor Nanostructures Second Edition Paul Harrison The University of Leeds, UK /Cf}\WILEY~ ^INTERSCIENCE JOHN WILEY & SONS,
More information5 Problems Chapter 5: Electrons Subject to a Periodic Potential Band Theory of Solids
E n = :75, so E cont = E E n = :75 = :479. Using E =!, :479 = m e k z =! j e j m e k z! k z = r :479 je j m e = :55 9 (44) (v g ) z = @! @k z = m e k z = m e :55 9 = :95 5 m/s. 4.. A ev electron is to
More informationRefering to Fig. 1 the lattice vectors can be written as: ~a 2 = a 0. We start with the following Ansatz for the wavefunction:
1 INTRODUCTION 1 Bandstructure of Graphene and Carbon Nanotubes: An Exercise in Condensed Matter Physics developed by Christian Schönenberger, April 1 Introduction This is an example for the application
More informationC. Show your answer in part B agrees with your answer in part A in the limit that the constant c 0.
Problem #1 A. A projectile of mass m is shot vertically in the gravitational field. Its initial velocity is v o. Assuming there is no air resistance, how high does m go? B. Now assume the projectile is
More informationELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES
ELECTRONIC ENERGY DISPERSION AND STRUCTURAL PROPERTIES ON GRAPHENE AND CARBON NANOTUBES D. RACOLTA, C. ANDRONACHE, D. TODORAN, R. TODORAN Technical University of Cluj Napoca, North University Center of
More informationElectrons in Crystals. Chris J. Pickard
Electrons in Crystals Chris J. Pickard Electrons in Crystals The electrons in a crystal experience a potential with the periodicity of the Bravais lattice: U(r + R) = U(r) The scale of the periodicity
More informationGraphene and Carbon Nanotubes
Graphene and Carbon Nanotubes 1 atom thick films of graphite atomic chicken wire Novoselov et al  Science 306, 666 (004) 100μm Geim s group at Manchester Novoselov et al  Nature 438, 197 (005) KimStormer
More informationchiral m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Three major categories of nanotube structures can be identified based on the values of m and n
zigzag armchair Three major categories of nanotube structures can be identified based on the values of m and n m = n Armchair m = 0 or n = 0 Zigzag m n Chiral Nature 391, 59, (1998) chiral J. Tersoff,
More informationOptical Lattices. Chapter Polarization
Chapter Optical Lattices Abstract In this chapter we give details of the atomic physics that underlies the Bose Hubbard model used to describe ultracold atoms in optical lattices. We show how the ACStark
More informationNeutron and xray spectroscopy
Neutron and xray spectroscopy B. Keimer MaxPlanckInstitute for Solid State Research outline 1. selfcontained introduction neutron scattering and spectroscopy xray scattering and spectroscopy 2. application
More informationA. F. J. Levi 1 EE539: Engineering Quantum Mechanics. Fall 2017.
A. F. J. Levi 1 Engineering Quantum Mechanics. Fall 2017. TTh 9.00 a.m. 10.50 a.m., VHE 210. Web site: http://alevi.usc.edu Web site: http://classes.usc.edu/term20173/classes/ee EE539: Abstract and Prerequisites
More informationMatterRadiation Interaction
MatterRadiation Interaction The purpose: 1) To give a description of the process of interaction in terms of the electronic structure of the system (atoms, molecules, solids, liquid or amorphous samples).
More informationMolecular spectroscopy
Molecular spectroscopy Origin of spectral lines = absorption, emission and scattering of a photon when the energy of a molecule changes: rad( ) M M * rad( ' ) ' v' 0 0 absorption( ) emission ( ) scattering
More informationLecture 5. HartreeFock Theory. WS2010/11: Introduction to Nuclear and Particle Physics
Lecture 5 HartreeFock Theory WS2010/11: Introduction to Nuclear and Particle Physics Particlenumber representation: General formalism The simplest starting point for a manybody state is a system of
More informationvan Quantum tot Molecuul
10 HC10: Molecular and vibrational spectroscopy van Quantum tot Molecuul Dr Juan Rojo VU Amsterdam and Nikhef Theory Group http://www.juanrojo.com/ j.rojo@vu.nl Molecular and Vibrational Spectroscopy Based
More informationElectron energy loss spectroscopy (EELS)
Electron energy loss spectroscopy (EELS) Phil Hasnip Condensed Matter Dynamics Group Department of Physics, University of York, U.K. http://wwwusers.york.ac.uk/~pjh503 Many slides courtesy of Jonathan
More informationS. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study
S. Bellucci, A. Sindona, D. Mencarelli, L. Pierantoni Electrical conductivity of graphene: a timedependent density functional theory study INFN Laboratori Nazionali Frascati (LNF), Italy Univ. Calabria,
More informationMOLECULAR SPECTROSCOPY
MOLECULAR SPECTROSCOPY First Edition Jeanne L. McHale University of Idaho PRENTICE HALL, Upper Saddle River, New Jersey 07458 CONTENTS PREFACE xiii 1 INTRODUCTION AND REVIEW 1 1.1 Historical Perspective
More informationC2: Band structure. CarlOlof Almbladh, Rikard Nelander, and Jonas Nyvold Pedersen Department of Physics, Lund University.
C2: Band structure CarlOlof Almbladh, Rikard Nelander, and Jonas Nyvold Pedersen Department of Physics, Lund University December 2005 1 Introduction When you buy a diamond for your girl/boy friend, what
More informationOBE solutions in the rotating frame
OBE solutions in the rotating frame The light interaction with the 2level system is VV iiiiii = μμ EE, where μμ is the dipole moment μμ 11 = 0 and μμ 22 = 0 because of parity. Therefore, light does not
More information12 Carbon Nanotubes  script
12 Carbon Nanotubes  script Carola Meyer Peter Günberg Insitute (PGI6), and JARAFIT Jülich Aachen Research Alliance  Future Information Technology Research Center Jülich, 52425 Jülich, Germany 12.2
More informationPhysics of Nanotubes, Graphite and Graphene Mildred Dresselhaus
Quantum Transport and Dynamics in Nanostructures The 4 th Windsor Summer School on Condensed Matter Theory 618 August 2007, Great Park Windsor (UK) Physics of Nanotubes, Graphite and Graphene Mildred
More informationOneStep Theory of Photoemission: Band Structure Approach
OneStep Theory of Photoemission: Band Structure Approach E. KRASOVSKII ChristianAlbrechts University Kiel Dresden, 19 April 2007 CONTENTS OneStep Theory Theory of Band Mapping Valence band photoemission
More informationLecture 0. NC State University
Chemistry 736 Lecture 0 Overview NC State University Overview of Spectroscopy Electronic states and energies Transitions between states Absorption and emission Electronic spectroscopy Instrumentation Concepts
More information( ) /, so that we can ignore all
Physics 531: Atomic Physics Problem Set #5 Due Wednesday, November 2, 2011 Problem 1: The acstark effect Suppose an atom is perturbed by a monochromatic electric field oscillating at frequency ω L E(t)
More informationPHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids
PHYSICS 4750 Physics of Modern Materials Chapter 5: The Band Theory of Solids 1. Introduction We have seen that when the electrons in two hydrogen atoms interact, their energy levels will split, i.e.,
More informationPH575 Spring Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5
PH575 Spring 2009 Lecture #26 & 27 Phonons: Kittel Ch. 4 & 5 PH575 Spring 2009 POP QUIZ Phonons are: A. Fermions B. Bosons C. Lattice vibrations D. Light/matter interactions PH575 Spring 2009 POP QUIZ
More informationUltravioletVisible and Infrared Spectrophotometry
UltravioletVisible and Infrared Spectrophotometry Ahmad Aqel Ifseisi Assistant Professor of Analytical Chemistry College of Science, Department of Chemistry King Saud University P.O. Box 2455 Riyadh 11451
More informationUltrafast study of Dirac fermions in out of equilibrium Topological Insulators
Ultrafast study of Dirac fermions in out of equilibrium Topological Insulators Marino Marsi Laboratoire de Physique des Solides CNRS Univ. ParisSud  Université ParisSaclay IMPACT, Cargèse, August 26
More informationQuantum field theory and Green s function
Quantum field theory and Green s function Condensed matter physics studies systems with large numbers of identical particles (e.g. electrons, phonons, photons) at finite temperature. Quantum field theory
More information2D Raman band of singlelayer and bilayer graphene
Journal of Physics: Conference Series PAPER OPEN ACCESS 2D Raman band of singlelayer and bilayer graphene To cite this article: V N Popov 2016 J. Phys.: Conf. Ser. 682 012013 View the article online for
More informationFermi surfaces which produce large transverse magnetoresistance. Abstract
Fermi surfaces which produce large transverse magnetoresistance Stephen Hicks University of Florida, Department of Physics (Dated: August 1, ) Abstract The Boltzmann equation is used with elastic swave
More informationIntermediate valence in Yb Intermetallic compounds
Intermediate valence in Yb Intermetallic compounds Jon Lawrence University of California, Irvine This talk concerns rare earth intermediate valence (IV) metals, with a primary focus on certain Ybbased
More informationAtomic cross sections
Chapter 12 Atomic cross sections The probability that an absorber (atom of a given species in a given excitation state and ionziation level) will interact with an incident photon of wavelength λ is quantified
More informationThe Bose Einstein quantum statistics
Page 1 The Bose Einstein quantum statistics 1. Introduction Quantized lattice vibrations Thermal lattice vibrations in a solid are sorted in classical mechanics in normal modes, special oscillation patterns
More informationExperimental Determination of Crystal Structure
Experimental Determination of Crystal Structure Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware, U.S.A. PHYS 624: Introduction to Solid State Physics http://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html
More informationSpins and spinorbit coupling in semiconductors, metals, and nanostructures
B. Halperin Spin lecture 1 Spins and spinorbit coupling in semiconductors, metals, and nanostructures Behavior of nonequilibrium spin populations. Spin relaxation and spin transport. How does one produce
More informationLinear dynamic polarizability and absorption spectrum of an exciton in a quantum ring in a magnetic field
Linear dynamic polarizability and absorption spectrum of an exciton in a quantum ring in a magnetic field A V Ghazaryan 1, A P Djotyan 1, K Moulopoulos, A A Kirakosyan 1 1 Department of Physics, Yerevan
More informationProblem 1: Spin 1 2. particles (10 points)
Problem 1: Spin 1 particles 1 points 1 Consider a system made up of spin 1/ particles. If one measures the spin of the particles, one can only measure spin up or spin down. The general spin state of a
More informationResonance Raman scattering in photonic bandgap materials
PHYSICAL REVIEW A, VOLUME 63, 013814 Resonance Raman scattering in photonic bandgap materials Mesfin Woldeyohannes, 1 Sajeev John, 1 and Valery I. Rupasov 1,2 1 Department of Physics, University of Toronto,
More informationν=0 Quantum Hall state in Bilayer graphene: collective modes
ν= Quantum Hall state in Bilayer graphene: collective modes Bilayer graphene: Band structure Quantum Hall effect ν= state: Phase diagram Timedependent HartreeFock approximation Neutral collective excitations
More informationBand Structure of Isolated and Bundled Nanotubes
Chapter 5 Band Structure of Isolated and Bundled Nanotubes The electronic structure of carbon nanotubes is characterized by a series of bands (sub or minibands) arising from the confinement around the
More informationCoherent states, beam splitters and photons
Coherent states, beam splitters and photons S.J. van Enk 1. Each mode of the electromagnetic (radiation) field with frequency ω is described mathematically by a 1D harmonic oscillator with frequency ω.
More informationSoft Carrier Multiplication by Hot Electrons in Graphene
Soft Carrier Multiplication by Hot Electrons in Graphene Anuj Girdhar 1,3 and J.P. Leburton 1,2,3 1) Department of Physics 2) Department of Electrical and Computer Engineering, and 3) Beckman Institute
More informationSOLID STATE PHYSICS. Second Edition. John Wiley & Sons. J. R. Hook H. E. Hall. Department of Physics, University of Manchester
SOLID STATE PHYSICS Second Edition J. R. Hook H. E. Hall Department of Physics, University of Manchester John Wiley & Sons CHICHESTER NEW YORK BRISBANE TORONTO SINGAPORE Contents Flow diagram Inside front
More information535.37(075.8) : /, ISBN (075.8) ISBN , 2008, 2008
.. 2008 535.37(075.8) 22.34573 66.. 66 : /... : , 2008. 131. ISBN 5982983128 ,, ,,, , ., 200203 «».,,, . 535.37(075.8) 22.34573 ,.. ISBN 5982983128.., 2008, 2008., 2008 2 , . :,,  ;,, ;
More informationSection 10 Metals: Electron Dynamics and Fermi Surfaces
Electron dynamics Section 10 Metals: Electron Dynamics and Fermi Surfaces The next important subject we address is electron dynamics in metals. Our consideration will be based on a semiclassical model.
More informationOptical Spectroscopy of SingleWalled Carbon Nanotubes
Optical Spectroscopy of SingleWalled Carbon Nanotubes Louis Brus Chemistry Department, Columbia University Groups: Heinz, O Brien, Hone, Turro, Friesner, Brus 1. SWNT Luminescence dynamics psec pumpprobe
More informationFREE ELECTRON LASER THEORY USING TWO TIMES GREEN FUNCTION FORMALISM HIROSHI TAKAHASHI. Brookhaven Natioanal Laboratory Upton New York, 11973
FREE ELECTRON LASER THEORY USING TWO TIMES GREEN FUNCTION FORMALISM HIROSHI TAKAHASHI Brookhaven Natioanal Laboratory Upton New York, 11973 In this paper, we present a quatum theory for free electron laser
More informationElectron spins in nonmagnetic semiconductors
Electron spins in nonmagnetic semiconductors Yuichiro K. Kato Institute of Engineering Innovation, The University of Tokyo Physics of noninteracting spins Optical spin injection and detection Spin manipulation
More informationThe many forms of carbon
The many forms of carbon Carbon is not only the basis of life, it also provides an enormous variety of structures for nanotechnology. This versatility is connected to the ability of carbon to form two
More informationElectronphonon scattering (Finish Lundstrom Chapter 2)
Electronphonon scattering (Finish Lundstrom Chapter ) Deformation potentials The mechanism of electronphonon coupling is treated as a perturbation of the band energies due to the lattice vibration. Equilibrium
More informationSecondorder harmonic and combination modes in graphite, single wall carbon nanotube bundles, and isolated single wall carbon nanotubes
Secondorder harmonic and combination modes in graphite, single wall carbon nanotube bundles, and isolated single wall carbon nanotubes V. W. Brar a, Ge. G. Samsonidze b, M. S. Dresselhaus a,b, G. Dresselhaus
More informationFermi Surfaces and their Geometries
Fermi Surfaces and their Geometries Didier Ndengeyintwali Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA (Dated: May 17, 2010) 1. Introduction The Pauli exclusion principle
More informationQuantum Mechanics: Foundations and Applications
Arno Böhm Quantum Mechanics: Foundations and Applications Third Edition, Revised and Enlarged Prepared with Mark Loewe With 96 Illustrations SpringerVerlag New York Berlin Heidelberg London Paris Tokyo
More informationLuz e Átomos. como ferramentas para Informação. Quântica. Quântica Ótica. Marcelo Martinelli. Lab. de Manipulação Coerente de Átomos e Luz
Luz e Átomos como ferramentas para Informação Quântica Ótica Quântica Inst. de Física Marcelo Martinelli Lab. de Manipulação Coerente de Átomos e Luz Question: Dividing the incident beam in two equal parts,
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kxwt) wave (w,k)
More informationPhysics of atoms and molecules
Physics of atoms and molecules 2nd edition B.H. Bransden and C.J. Joachain Prentice Hall An imprint of Pearson Education Harlow, England London New York Boston San Francisco Toronto Sydney Singapore Hong
More informationFermi polaronpolaritons in MoSe 2
Fermi polaronpolaritons in MoSe 2 Meinrad Sidler, Patrick Back, Ovidiu Cotlet, Ajit Srivastava, Thomas Fink, Martin Kroner, Eugene Demler, Atac Imamoglu Quantum impurity problem Nonperturbative interaction
More informationThe interacting boson model
The interacting boson model P. Van Isacker, GANIL, France Dynamical symmetries of the IBM Neutrons, protons and Fspin (IBM2) T=0 and T=1 bosons: IBM3 and IBM4 The interacting boson model Nuclear collective
More informationOptics and Response Functions
Theory seminar: Electronic and optical properties of graphene Optics and Response Functions Matthias Droth, 04.07.2013 Outline: Light absorption by Dirac fermions Intro: response functions The optics of
More informationSolid State Physics. Lecture 10 Band Theory. Professor Stephen Sweeney
Solid State Physics Lecture 10 Band Theory Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK s.sweeney@surrey.ac.uk Recap from
More informationPC Laboratory Raman Spectroscopy
PC Laboratory Raman Spectroscopy Schedule: Week of September 59: Student presentations Week of September 1923:Student experiments Learning goals: (1) Handson experience with setting up a spectrometer.
More informationOptical properties of wurtzite and zincblende GaNÕAlN quantum dots
Optical properties of wurtzite and zincblende GaNÕAlN quantum dots Vladimir A. Fonoberov a) and Alexander A. Balandin b) NanoDevice Laboratory, Department of Electrical Engineering, University of California
More informationAn Extended Hückel Theory based Atomistic Model for Graphene Nanoelectronics
Journal of Computational Electronics X: YYYZZZ,? 6 Springer Science Business Media, Inc. Manufactured in The Netherlands An Extended Hückel Theory based Atomistic Model for Graphene Nanoelectronics HASSAN
More informationPractical Quantum Mechanics
Siegfried Flügge Practical Quantum Mechanics With 78 Figures SpringerVerlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Contents Volume I I. General Concepts 1. Law of probability
More informationChapter 9: Electrons in Atoms
General Chemistry Principles and Modern Applications Petrucci Harwood Herring 8 th Edition Chapter 9: Electrons in Atoms Philip Dutton University of Windsor, Canada N9B 3P4 PrenticeHall 2002 PrenticeHall
More informationParticle Physics. Michaelmas Term 2011 Prof. Mark Thomson. Handout 2 : The Dirac Equation. NonRelativistic QM (Revision)
Particle Physics Michaelmas Term 2011 Prof. Mark Thomson + e  e +  + e  e +  + e  e +  + e  e +  Handout 2 : The Dirac Equation Prof. M.A. Thomson Michaelmas 2011 45 NonRelativistic QM (Revision)
More informationOptical Properties of Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India
Optical Properties of Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013 Light Matter Interaction Response to external electric
More informationAtomic Physics 3 rd year B1
Atomic Physics 3 rd year B1 P. Ewart Lecture notes Lecture slides Problem sets All available on Physics web site: http:www.physics.ox.ac.uk/users/ewart/index.htm Atomic Physics: Astrophysics Plasma Physics
More informationLecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2
Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1 The Born approximation for the differential cross section is valid if the interaction
More informationElectron levels in a periodic potential: general remarks. Daniele Toffoli January 11, / 46
Electron levels in a periodic potential: general remarks Daniele Toffoli January 11, 2017 1 / 46 Outline 1 Mathematical tools 2 The periodic potential 3 Bloch s theorem and Bornvon Karman boundary conditions
More informationRb, which had been compressed to a density of 1013
Modern Physics Study Questions for the Spring 2018 Departmental Exam December 3, 2017 1. An electron is initially at rest in a uniform electric field E in the negative y direction and a uniform magnetic
More informationLecture 21: Lasers, Schrödinger s Cat, Atoms, Molecules, Solids, etc. Review and Examples. Lecture 21, p 1
Lecture 21: Lasers, Schrödinger s Cat, Atoms, Molecules, Solids, etc. Review and Examples Lecture 21, p 1 Act 1 The Pauli exclusion principle applies to all fermions in all situations (not just to electrons
More informationOptomechanically induced transparency of xrays via optical control: Supplementary Information
Optomechanically induced transparency of xrays via optical control: Supplementary Information WenTe Liao 1, and Adriana Pálffy 1 1 MaxPlanckInstitut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
More informationconventions and notation
Ph95a lecture notes, //0 The Bloch Equations A quick review of spin conventions and notation The quantum state of a spin particle is represented by a vector in a twodimensional complex Hilbert space
More informationSolid State Device Fundamentals
4. lectrons and Holes Solid State Device Fundamentals NS 45 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 4N101b 1 4. lectrons and Holes Free electrons and holes
More informationMESOSCOPIC QUANTUM OPTICS
MESOSCOPIC QUANTUM OPTICS by Yoshihisa Yamamoto Ata Imamoglu A WileyInterscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore Preface xi 1 Basic Concepts
More informationSUPPLEMENTARY INFORMATION
SUPPLEMENTARY INFORMATION I. Experimental Thermal Conductivity Data Extraction Mechanically exfoliated graphene flakes come in different shape and sizes. In order to measure thermal conductivity of the
More information5. GrossPitaevskii theory
5. GrossPitaevskii theory Outline N noninteracting bosons N interacting bosons, manybody Hamiltonien Meanfield approximation, order parameter GrossPitaevskii equation Collapse for attractive interaction
More informationSupplementary Information for: Exciton Radiative Lifetimes in. Layered Transition Metal Dichalcogenides
Supplementary Information for: Exciton Radiative Lifetimes in Layered Transition Metal Dichalcogenides Maurizia Palummo,, Marco Bernardi,, and Jeffrey C. Grossman, Dipartimento di Fisica, Università di
More informationThis manuscript was submitted first in a reputed journal on Apri1 16 th Stanene: Atomically Thick Freestanding Layer of 2D Hexagonal Tin
This manuscript was submitted first in a reputed journal on Apri1 16 th 2015 Stanene: Atomically Thick Freestanding Layer of 2D Hexagonal Tin Sumit Saxena 1, Raghvendra Pratap Choudhary, and Shobha Shukla
More informationarxiv:quantph/ v5 10 Feb 2003
Quantum entanglement of identical particles Yu Shi Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom and Theory of
More informationCrystals. Peter Košovan. Dept. of Physical and Macromolecular Chemistry
Crystals Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Lecture 1, Statistical Thermodynamics, MC26P15, 5.1.216 If you find a mistake, kindly report it to the
More informationExcited states and tp propagator for fermions
Excited states and tp propagator for fermions So far sp propagator gave access to E 0 Groundstate energy and all expectation values of 1body operators E +1 Energies in +1 relative to ground state n E0
More informationXRay Magnetic Circular Dichroism: basic concepts and theory for 4f rare earth ions and 3d metals. Stefania PIZZINI Laboratoire Louis Néel  Grenoble
XRay Magnetic Circular Dichroism: basic concepts and theory for 4f rare earth ions and 3d metals Stefania PIZZINI Laboratoire Louis Néel  Grenoble I)  History and basic concepts of XAS  XMCD at M 4,5
More informationDecays, resonances and scattering
Structure of matter and energy scales Subatomic physics deals with objects of the size of the atomic nucleus and smaller. We cannot see subatomic particles directly, but we may obtain knowledge of their
More informationAn Investigation of Benzene Using Ultrafast Laser Spectroscopy. Ryan Barnett. The Ohio State University
An Investigation of Benzene Using Ultrafast Laser Spectroscopy Ryan Barnett The Ohio State University NSF/REU/OSU Advisor: Linn Van Woerkom Introduction Molecular spectroscopy has been used throughout
More informationQuantum Field Theory
Quantum Field Theory PHYSP 621 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory 1 Attempts at relativistic QM based on S1 A proper description of particle physics
More informationStudying Metal to Insulator Transitions in Solids using Synchrotron Radiationbased Spectroscopies.
PY482 Lecture. February 28 th, 2013 Studying Metal to Insulator Transitions in Solids using Synchrotron Radiationbased Spectroscopies. Kevin E. Smith Department of Physics Department of Chemistry Division
More informationTopological insulator with timereversal symmetry
Phys620.nb 101 7 Topological insulator with timereversal symmetry Q: Can we get a topological insulator that preserves the timereversal symmetry? A: Yes, with the help of the spin degree of freedom.
More informationLecture 6:Feynman diagrams and QED
Lecture 6:Feynman diagrams and QED 0 Introduction to current particle physics 1 The Yukawa potential and transition amplitudes 2 Scattering processes and phase space 3 Feynman diagrams and QED 4 The weak
More informationOther Devices from pn junctions
Memory (5/7  Glenn Alers) Other Devices from pn junctions Electron to Photon conversion devices LEDs and SSL (5/5) Lasers (5/5) Solid State Lighting (5/5) Photon to electron conversion devices Photodectors
More informationOUTLINE. CHARGED LEPTONIC WEAK INTERACTION  Decay of the Muon  Decay of the Neutron  Decay of the Pion
Weak Interactions OUTLINE CHARGED LEPTONIC WEAK INTERACTION  Decay of the Muon  Decay of the Neutron  Decay of the Pion CHARGED WEAK INTERACTIONS OF QUARKS  CabibboGIM Mechanism  CabibboKobayashiMaskawa
More informationInteraction between Singlewalled Carbon Nanotubes and Water Molecules
Workshop on Molecular Thermal Engineering Univ. of Tokyo 2013. 07. 05 Interaction between Singlewalled Carbon Nanotubes and Water Molecules Shohei Chiashi Dept. of Mech. Eng., The Univ. of Tokyo, Japan
More informationThe electronic structure of materials 2  DFT
Quantum mechanics 2  Lecture 9 December 19, 2012 1 Density functional theory (DFT) 2 Literature Contents 1 Density functional theory (DFT) 2 Literature Historical background The beginnings: L. de Broglie
More informationQuantized Electrical Conductance of Carbon nanotubes(cnts)
Quantized Electrical Conductance of Carbon nanotubes(cnts) By Boxiao Chen PH 464: Applied Optics Instructor: Andres L arosa Abstract One of the main factors that impacts the efficiency of solar cells is
More informationPAPER No. : 8 (PHYSICAL SPECTROSCOPY) MODULE No. : 5 (TRANSITION PROBABILITIES AND TRANSITION DIPOLE MOMENT. OVERVIEW OF SELECTION RULES)
Subject Chemistry Paper No and Title Module No and Title Module Tag 8 and Physical Spectroscopy 5 and Transition probabilities and transition dipole moment, Overview of selection rules CHE_P8_M5 TABLE
More informationFEYNMAN DIAGRAM TECHNIQUES IN CONDENSED MATTER PHYSICS
FEYNMAN DIAGRAM TECHNIQUES IN CONDENSED MATTER PHYSICS A concise introduction to Feynman diagram techniques, this book shows how they can be applied to the analysis of complex manyparticle systems, and
More informationClassical and Quantum Dynamics in a Black Hole Background. Chris Doran
Classical and Quantum Dynamics in a Black Hole Background Chris Doran Thanks etc. Work in collaboration with Anthony Lasenby Steve Gull Jonathan Pritchard Alejandro Caceres Anthony Challinor Ian Hinder
More information