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On the inhomogeneous magnetised electron gas
Kettenis, M.M.
Publication date
2001
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Citation for published version (APA):
Kettenis, M. M. (2001). On the inhomogeneous magnetised electron gas. Ridderprint
offsetdrukkerij b.v.
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Chapterr 1
Introduction n
Inn this thesis, we will investigate an inhomogeneous gas of charged particles in the pres-encee of a hard wall. From the point of view of physics one would like to study a "real" plasma,, taking into account the inter-particle (Coulomb) interactions. Unfortunately, thiss would be a very difficult task. Since the interaction-less case could serve as a valu-ablee reference system, and various aspects of it have not been studied before, a study of aa gas of charged non-interacting particles in a magnetic field close to a hard wall seems appropriate.. As we will see in this thesis, this study turns out to yield some surprising resultss that describe a richer structure than one would expect for such a rather simple model. .
1.11 Diamagnetism
Sincee we do not consider the inter-particle interactions in the gas, our study essentially becomess a study of the inhomogeneous magnetised free-electron gas. This subject is closelyy related to the study of diamagnetism in electron theory, which has an interesting history. .
Inn the early days of (classical) electron theory, various attempts at explaining the mag-neticc properties of materials were made. One of the questions raised was, whether free electronss in such materials could cause diamagnetism. J J . Thomson [49] reasoned that pathss of electrons in a magnetic field are curved and that this curvature of the path will producee a magnetic field in the direction opposite to the external field. A piece of metal containingg free electrons will, therefore, act as a diamagnetic body. This view seemed too be generally accepted at the time. Unfortunately, this view is not correct. It is not entirelyy clear who was the first to point this out. In his Master's thesis [9], Bohr gives 1 1
aa qualitative argument, which he clarifies later in his Ph.D. thesis [10], In her Ph.D. thesis,, van Leeuwen refers to lectures given by Lorentz in 1910—1911 in which Lorentz gavee two proofs for the absence of diamagnetism in a free-electron gas. Both Bohr and vann Leeuwen argue that the presence of the magnetic field does not change the statisti-call distribution of the particles in the gas, and that therefore there will be no magnetic effectt whatsoever.
Inn modern-day language we can see this by calculating the free energy F of the system andd then use the thermodynamic relation
MM = - g (1.1) too calculate the magnetisation. In the canonical ensemble, the free energy is given by
d3p d3r e -p H ( p>r )) (1.2) PP Ni
FF = — — In
wheree H ( p , r ) is the Hamiltonian. In the presence of a (constant) magnetic field the Hamiltoniann is given by
H ( p , r )) = - ^ - ( p - ^ A )2 (1.3)
ln\,ln\, c
wheree A is the vector potential related to the magnetic field B = V x A. If we substitute 7tt = p — | A as the integration variable in (1.2), we immediately see that F becomes independentt of A and therefore independent of the strength B of the magnetic field. Thereforee the magnetisation must be zero. An essential problem with this reasoning is thatt it does not take into account the influence of the boundary on the system.
Therefore,, it was considered to be essential to look at the issue from the viewpoint off electron orbits, and include the influence of the wall on those orbits. Bohr [9, 10] andd van Leeuwen [37, 38] did just that. Under the assumption that electrons are re-flectedflected like elastic spheres from the walls of their container, they were able to prove thatt the magnetic effects of electrons moving in so-called skipping orbits close to the wall,, exactly cancel the contribution of the bulk electrons moving in closed orbits in thee interior. Figure 1.1 (taken from Bohr's Ph.D. thesis) shows a few electron orbits withh a fixed radius, and therefore the same tangential velocity, that move in the plane depictedd in the figure. The line ab represents the boundary of the container, its inte-riorr being on the right-hand side of the line. Bohr reasons that in the bulk, far from thee wall, there cannot be an anisotropy in the motion of electrons. At every point in thee region where the motion of electrons is restricted by the magnetic field such that theyy do not encounter the wall, their velocities are distributed equally in all directions.
1.1.1.1. Diamagnetism 3 3
Figuree 1.1: Electron orbits in the neighbourhood of a wall.
Fromm the picture it becomes clear that the paths of electrons reflecting from the wall formm continuations of the orbits they would follow if the wall were not present. One immediatelyy concludes that there cannot be an anisotropy in the electron motion close too the wall either. Now, since at any moment the magnetic effect of the motions of thee electrons in curved orbits is the same as if they were moving in straight lines with thee same velocity, it becomes clear that there is no diamagnetic response: the magnetic effectt of an electron is exactly cancelled by an electron moving at the same speed in thee opposite direction. Bohr then goes on by considering the paths of individual elec-trons.. The motion of a single bulk-electron in a circular orbit does produce a magnetic fieldfield in the opposite direction of the external magnetic field. However, if one follows thee motion of an electron that comes in contact with the wall, one sees that such an electronn "creeps" along the wall and describes an orbit in the opposite direction of the bulk-electrons.. This produces a magnetic field in the same direction as the external field andd the two contributions cancel exactly. In her Ph.D. thesis van Leeuwen gives a proof forr this cancellation using geometrical arguments.
Soo much for the classical case. How does quantum mechanics affect this picture? The electronn orbits become quantised. Landau [36] has shown that this leads to a diamag-neticc response in the free-electron gas. Landau's derivation involves calculating the free energyy from the (quantum-statistical) partition sum and employing ( 1 . 1 ) - much in the samee way as the reasoning given above for the classical case. Landau calculates the par-titionn sum by looking at the energy spectrum. If one only considers the motion of the
electronss in a plane perpendicular to the magnetic field, one finds that the eigenvalues aree given by
Enn = (n + J)5^51, 11 = 0,1,2,... (1.4)
mc c
Inn an infinite plane there are infinitely many states belonging to each of these energy eigenvalues,, but per unit area the degeneracy is
(1.5) )
27irtc c
Nowadayss we refer to these states as Landau levels.
Thee motion in the direction parallel to the magnetic field adds a simple kinetic term to En.. Landau then directly calculates the free energy by means of Poissons summation
formula,, which is a good approximation when U.B |Bj <C (3_1 (where JIB = ^mc is the Bohrr magneton). But he does not provide an explicit expression for the partition sum itself.. For Maxwell-Boltzmann statistics the partition function per unit volume would read d
|e|BB I m 1_ 47iricc V 2n$h2 sinh(uB 0B]
Whenn |1B(3B is small, which is the limit considered by Landau, one can expand the sinhh in the denominator to arrive at the often quoted result of
zz = -EZ:J.22^,:. »,- (1-6)
- ^ uB2( 3 BB (1.7)
forr the magnetisation per electron.
Thee partition function (1.6) is calculated without taking into account a particular boundaryy condition. This suggests that the magnetisation is independent of, or at least insensitivee to, the boundary conditions. At the time when Landau published his results, manyy people doubted this insensitivity of the magnetisation to the exact nature of the boundaryy condition. Especially since in the classical case it seemed that the boundary wass responsible for the total absence of a magnetic response. Therefore it took quite a bitt of time for Landau's result to become generally accepted. The insensitivity of Lan-dau'ss result to the nature of the boundary has been (finally) proven by Angelescu et al [5]] for a broad class of boundary conditions, at least up to first order in the magnetic field. field.
Ass a side-mark it is worth noting that the concept of spin introduces another form of magnetismm in the free-electron gas: spin-paramagnetism. In fact the magnetisation due
1.2.1.2. Basic techniques 5 5
too Landau-diamagnetism in the free-electron gas given in (1.7) is exacdy one-third of thee magnetisation due to spin-paramagnerism, so the free-electron gas ultimately shows paramagneticc behaviour. In a real metal however the ratio of dia- and paramagnetism willl be different, because of the interactions of the electrons with the periodic potential off the lattice. This could lead to a net diamagnetism.
Evenn though the magnetisation is largely independent of the boundary conditions, it iss still interesting to investigate how the electron gas behaves in the neighbourhood of thee wall - behaviour that does depend strongly on the particular boundary conditions. Therefore,, a major part of this thesis is devoted to an investigation of the behaviour off the (charge) density and current density profiles close to the wall. In this investiga-tionn we will also incorporate the effects of quantum statistics. Landau's original deriva-tionn of the diamagnetic response does take into account the Fermi-Dirac statistics of thee electrons, but involves an approximation that breaks down for low temperatures and/orr strong magnetic fields. In fact, the final result would have been identical if he hadd started out with Maxwell-Boltzmann statistics. As we will see in this thesis, the behaviourr of a completely degenerate electron gas is considerably different from the Maxwell-Boltzmannn case.
1.22 Basic techniques
Throughoutt this thesis we will use some basic techniques to calculate density profiles andd correlation functions. Before we give an outline of the rest of this thesis, we will providee a short explanation of these techniques.
1.2.11 Green functions
Thee traditional approach to quantum mechanics has been to solve the Schrödinger equationn directly. And indeed in chapter 2 we will take this approach. Since the Schrö-dingerr equation is a partial differential equation a whole set of tools exist for solving it. Quitee a few of the techniques for constructing solutions for partial differential equa-tionss make use of Green functions.
Supposee that H is a differential operator, which in the context of the Schrödinger equa-tionn would be the Hamiltonian. Then the traditional definition of the Green function forr the eigenvalue equation Hi|>(r) = E\J>(r) is
withh appropriate boundary conditions. More specifically, if we want to describe a parti-clee that is restricted to a certain domain D, with boundary 3D, the appropriate bound-aryy condition for \J)(r) is
i M r ) = 0 ,, r e 3D (1.9) whichh translates into
Gu( r , r ' )) = 0 r e 3D and/or r ' e 3D (1.10)
forr the boundary condition on the Green function. In terms of the eigenfunctions i^nn (T) we can write
G u ( r , r ' )) = Y. — i H > u ( r N > ; ( r ' ) ( l . H )
nn u tn
forr complex u.
Inn this thesis we will encounter a few other functions that are derived from this basic Greenn function. The first is the discontinuity of Gu( r , r ' ) at u = E
GE( r , r ' )) = ^ [ Gu = E + io { r , r ' ) - Gu = E- i o ( r , r ' ) 3 = X >n( r N > ; ( r ' ) 6(En - E).
(1.12) ) Heree iO is an infinitesimal imaginary number. From this energy Green function it is but aa small step towards the temperature or thermodynamic Green function
(
'OO Od E e - PEG E ( r , r ' )) = 5 " e ^E" W r ) C ( r ' ) . (1.13)
00
n
Thiss G p ( r , r ' ) can be used to calculate physical quantities like charge and current densityy and correlation functions.
Noww suppose that G £ ( r , r ' ) is the Green function for an infinite domain. In order too construct the Green function for the boundary problem we must find a correction G u (r>r' )) t n a t satisfies
( H - u ) G < ( r , r ' ) = 00 (1.14) forr all r, r ' 6 D / 3 D (that is, inside D), with the boundary condition
G u ( r , r ' ) = - G j ( r , r/)) r e 3D and/or r ' e 3D. (1.15) Baliann and Bloch [8] give a way to construct such a correction for the wave equation
1.2.1.2. Basic techniques 7 7
formm of this multiple-reflection expansion depends on the nature of the boundary con-ditions.. John and Suttorp [31] looked a bit closer at this multiple-reflection expansion forr the Schrödinger equation and the Dirichlet boundary conditions (1.9) considered here.. Their version of the multiple-reflection expansion is given by
G£(r,T')) = - f d o - "wn " [ Vr" G £ ( r , r " ) ]r^r» w G i ( r "w, r ' ) JdD D
++ [ d < r "w[ d<j",wn"-[Vr»G°u(ry')]r^T„w
JdDD J3D
x n ' " . [ V1- G ; ( r "w, r lr,T i r, w G i ( r ' "v v(r ' )) + . . . (1.16)
wheree n " denotes the normal vector, directed outwards perpendicular to the surface elementt dcr"w, at the point r " on the boundary. The symbol W is used to stress the factt that a symbol stands for a coordinate at the boundary, and r'" XT T ' "W stands for thee average (half the sum) of the limit r "' —» r "/ W from the inside and from the outside off the domain. John and Suttorp then use this multiple-reflection expansion to derive severall physical quantities for a weakly magnetised electron gas in the neighbourhood off a hard wall for Maxwell-Boltzmann statistics.
1.2.22 Path integrals
Thee Feynman path integral provides an alternative to direcdy solving the Schrödinger equation.. In a way it is closer to physical intuition. Indeed, we will see that the fact that itt can be interpreted as a sum over trajectories, makes path integrals a suitable tool for investigatingg inhomogeneous quantum systems. And if one would wish to somehow extendd the analysis given here to a full quantum plasma, by taking into account the Coulombb forces between charged particles, the use of path integrals seems to be almost unavoidable.. At least they have been widely used in recent calculations on homoge-neouss quantum plasmas [13, 14].
TheThe Feynman-Kac formula
Considerr a particle in an external potential V(r), i.e. with the Hamiltonian
HH = ^ + V(r) (1.17) wheree we have chosen units in such a way that the particle mass m drops out. The
governedd by the temperature Green function G${r\ r ) , with 3 the inverse temperature. Itss path-integral representation is given by the Feynman-Kac formula
Gp( r ' , r )) = <r'|e-p H|r> =
.r',0 0
dnroP(cu)exp p
-I I
£ £
dxV(cu(T)) ) (1.18) )
wheree CU(T) describes the path and d u j f is the conditional Wiener measure. For sim-plicityy Tl has been set to 1.
Lett us take a closer look at this Wiener measure. The relevant underlying space is the sett CI of all paths cu with cu(0) = r and cu((3) = r ' . The most important property of thee Wiener measure is that for arbitrary (small) |r — r ' | and (3
»i(Q) )
L L
du{o>)) = (27t0)-3 / 2e Ï F (1.19) ) Too define the Wiener measure one has to assign a measure to certain subsets of CI, thee so-called cylinder sets. To construct these cylinder sets, we choose a sequence Ti,, T 2 , . . . , xn and for each Ti a "window" At (i.e. some Borel set) through which thepathss in the cylinder set should pass. Denoting the cylinder set with C1A, the measure
iss then given by
ee 2 ( Ti + 1 -Ti >
H(QA)) = f dki(cü) = f d n f d rn T7[27t(Tl+1 - Ti)]"
J nAA J A , JAn f=Q
(1.20) ) wheree ro = r, rn +i = r ' , To = 0 and Tn +i = 3- By varying n, and the sequences
Ttt and Ai, we obtain sufficiently many cylinder sets to define the conditional Wiener measure.. Since both the underlying space CI and the measure \i depend on the points rr and r ' and the "time" interval [0, |3], we usually write |i.^ jf instead of u.. The Wiener measuree allows us to construct a (Lebesgue) integral
- ƒ ƒ
1(f)) = du(a>)f(o>) (1.21) ) off a function f (a>) that depends on the path tu. Such integrals are usually referred to as
pathpath integrals.
Thee Feynman-Kac formula is often interpreted as a sum over all possible paths from r too r' where each path gets an appropriate weight that consists of a factor that depends onlyy on its shape (the Wiener measure) and a factor that depends on the potential V(r).
1.2.1.2. Basic techniques 9 9
Notee that while the paths U>(T) should be continuous, there is no need for them to be differentiable.. In feet the class of differentiable paths has measure zero.
Evenn though a number of techniques to actually calculate path integrals exist, only aa limited set of path integrals can be solved exacdy. In most cases, we will have to be satisfiedd with an approximation of some kind. Fortunately, several useful approximation techniquess for path integrals do exist.
AA standard way of actually calculating path integrals is by using a process known as "timee slicing" (in fact this is often used as a constructional definition of a path integral, andd reminiscent of the definition of the Wiener measure). In using this method, one startss by dividing the "time" interval [0, |3] into n + 1 subintervals [Tm,Tm +i] of equal
lengthh en, that is
rmm = m en m = 0, . . . , r i en =
P P
nn + 1 (1.22) ) Uponn doing this, the exponential that incorporates the effect of the (external) potential
exp p Jo o dTV(cu(T)) ) becomes s exp p -- Y_ €nV(cu(Tm) .. m=0 (1.23) ) (1.24) ) Fromm (1.19) we see that the Wiener measure is essentially Gaussian and on the interval [Tm,, Tm +i ] it gives a factor
(27t€, , \ " 3 / 2 , , i i (1.25) ) Noww if we define cu(Tm) = rm, we can write the "time sliced" path integral as
Ge > n( r > )) = J d3rn. - J d3T ! ( 2 7 t en) -3^+ 1) / Vi I^ rLe -£"v^ ) e lr2_-rirr e v, _ , rT- r | '
— 7 7 ^ —P- enV ( T i ) __ jT—
'n'n e
Thee full path integral can be found by taking the continuum limit: G p ( r ' , r ) == lim G ^ r » .
(1.26) )
Too actually calculate the path integrals using this technique, we will have to be able to doo the integrations over rm in (1.26). When these integrations are Gaussian, that is if
thee potential is quadratic, this is straightforward. But for more complicated potentials itt is hardly ever possible to find an exact result.
Inn many cases (including the problems explored in this thesis) the region in space that iss accessible to a particle is restricted. This means that one should only sum over those pathss that stay inside that region. This can be accomplished by setting the potential to infinityy outside this region, but in practice one will most often restrict the integrations overr rm in (1.26) to the region accessible to the particle. In most cases one is forced
too make (further) approximations to be able to do this, even if the unrestricted path integrall is solvable exacdy since the integration over the components of rm no longer
extendss from minus infinity to plus infinity.
TheThe Feynman-Kac-Ito formula
Thuss far we have only considered particles moving in ordinary potentials. But in order too describe the magnetised electron gas using path integrals, we also need to incorporate thee effects of an external magnetic field. In terms of the vector potential (which we assumee to be time-independent), the path integral representation of G p ( r ' , r ) is given byy the Feynman-Kac-Itó formula [45]:
G p ( r ' , r }} = J d ^ ^ ( u ) ) e x p - J C!TV(CÜ(T)) + i d r "" A ( r " ) (1.28) ) wheree in addition to h and m, the electric charge e and the velocity of light c have beenn set to 1 as well. It is not a priori clear what is meant by the integral J ^ d r A ( r ) . Inn principle, this is the integral of the vector potential along the path CU(T). But since thee majority of the paths consists of paths that are not differentiable, this is not simply aa line integral. In the literature [47, 45] one can find basically two approaches to the definitionn of "stochastic integrals" such as J ^ d r - A ( r ) . These approaches are known by thee names of their inventors: Stratonovitch and Ito. Both approaches define the integral ass the continuum limit of a sum over finite intervals
TV V
d r - A ( r ) == lim Y Am Am (1.29)
n—K»» *— m = 0 0
wheree Am = tu(Tm +i) — cu(xm). But they differ in the definition of Am.
Itoo uses Am = A(cü(Tm)). This is the convention used by mathematicians, and
1.2.1.2. Basic techniques 11 1
off the vector potential "in the future". Unfortunately it has one "unphysicaT aspect: it doess not transform in a correct way under a gauge transformation.
Considerr the gauge transformation A —> A + Vf. Under such a gauge transformation thee Green function Gp{r',r) should transform as G p ( r ' , r ) —> e, [ f { r ) _ f { r ) 1G p ( r '(r ) .
Inn other words A[iv) = ƒ ^ d r - A f r ) should transform as A{iv) —»A{tv)+1[r') — f{r). Itóss Lemma [47] states that for a function f (r) and a path cu with cu(0) = r, tu(3) = r ' wee have
[[ d r " (Vf )(r") = f (r') - f (r) - \ \ dT V Vf (cu(t)). (1.30)
Ja>> 2 Jo
Soo in order to restore gauge-independence when interpreting A{iv) as an Ito integral, onee needs to add the term ^ Jo dT V A(CU(T)).
Thee Stratonovitch approach uses Am = j[A(cü(Tm +i)) + A(cu(Tm))] and has
gauge-independencee built-in, since the difference between Itö and Stratonovitch is exacdy thee divergence term \ ƒ£ dT V A(CU(T)) mentioned above. For this reason it is more popularr among physicists. Fortunately, the difference between Ito and Stratonovitch disappearss when V A = 0. Therefore we are free to choose how to interpret the stochasticc integral in (1.28) as long as the vector potential is divergence-less. This may bee convenient since the mathematical framework for Itö integrals (Itó calculus) is much moree developed.
Notee that (1.29) provides the recipe for "slicing" the vector potential term in (1.28) that resembless the "time-slicing" described in (1.22). It is a convenient method to use when actuallyy calculating path integrals that involve the interaction with an electromagnetic field. field.
1.2.33 Degeneracy
Inn the previous sections we have treated some techniques that make it possible to cal-culatee properties of a free-electron gas for Maxwell-Boltzmann statistics. But by doing soo we would ignore the fermionic (or bosonic) character of the particles, and its con-sequencess for the statistics. This is fine for high temperatures and/or weak magnetic fields,fields, where the average occupancy of the states is small, but it does not provide a cor-rectt description for a cold and dense electron gas in a strong magnetic field. Some ad-hocc attempts at treating the (completely) degenerate electron gas have been made [43], butt a more elegant method exists: it turns out to be possible to derive the properties off a degenerate gas of non-interacting particles by starting from results for Maxwell-Boltzmannn statistics. In particular for T = 0 — the completely degenerate case — the two
statisticss are related by a Laplace transform, as was shown by Sondheimer and Wilson [48].. In their paper they make a more thorough re-examination of the free-electron gas inn a magnetic field (with Fermi-Dirac statistics), in order to put the older results on a firmerr theoretical basis. The essence of their approach is explained in this section. Considerr a single-particle operator ai(r), where i is the label of a particle. We are interestedd in the expectation value for the operator A(r) = 2^i=i ai (r) in a g35 °f* N noninteractingg particles. Since all particles are identical we can drop the particle label i and,, if we ignore the quantum statistics, we can write the expectation value of A(r) at inversee temperature (3 in the canonical ensemble as
<A(r))pp = ^ T r [ e -p Ha ( r ) ] . (1.31)
Heree V is the volume occupied by the gas and Z is the partition function per unit volume.. In other words
(A(r))pp = Np E nX e - PE" ( n | a ( r ) ) n ) (1.32)
2 _ nee n n
wheree En is the energy level corresponding to the single-particle state n .
Iff we would take into account the quantum statistics for spin- j fermions we would get
( A ( T ) ) » , , = 2 £e > [ EJ| 00 + 1( n | a ( r ) | n ) (1.33)
insteadd (in the grand canonical ensemble). Here we assumed that the hamiltonian is independentt of the spin, hence the overall factor 2. The chemical potential (x is deter-minedd by 2 ^ 1n[ e| 3 ( E n - > i' + 1]_T = N (or rather the other way around: the chemical
potentiall u determines the number of particles). In particular at T = 0, i.e. |3 = oo the factorr [ e ^E n _^ + I ]- 1 becomes a step-function which yields
{ A ( r ) ^ = 2 ^ 0 ( u . - En) < n | a ( r ) | n )) (1.34) n n
wheree n o w 2 ^n0 ( | x — En) = N.
Sincee the Laplace transform of the step-function is given by i i
d u e - ^ e ( ^ -- En) = d n e - < ^ = - e ~p E" (1.35)
1.3.1.3. Outline 13 3
wee see that
1
000 2 7d n e - ^ < A ( r ) ) ^^ = — <A(r))p (1.36)
wheree p = N / V is the particle density. In other words, (A(r))^ is the inverse Laplace transformm of |§{A(r)) p with respect to 0
ii fc+oo 2 7
<
A(r)>"-5dL
d|le"
l,'
l^<
Alr»»--
( u 7 )AA similar, though somewhat more complicated relation holds for T ^ 0:
(AfrJV.uu =
-r
a , i
'^(i^)Tï)
< A ( r ) >
^--
(1
'
38)
Inn this thesis however, we will restrict ourselves to the case T = 0.
1.33 Outline
Noww that we have explained these basic techniques, we can give an outline of the rest off this thesis. In chapters 2 and 3 we will calculate charge and current density profiles forr the completely degenerate magnetised free-electron gas near a wall. The expressions forr these density profiles are asymptotic expansions in terms of the distance from the walll and give information on how the profiles decay towards their bulk values.
Chapterr 2 uses the Green function approach outlined in section 1.2.1. In chapter 3 wee will use the path integral techniques from section 1.2.2. The final result, the afore-mentionedd asymptotic expressions for charge and current density, will be very similar, butt nevertheless, both methods are of interest. The Green function approach makes it straightforwardd to calculate more terms in the asymptotic expansions for the densities. Itt also yields an interesting integral relation for parabolic cylinder functions. The path integrall approach from chapter 3 on the other hand, makes it possible to give an intu-itivee derivation of the multiple-reflection expansion. We will see that the higher-order termss in this expansion correspond to paths reflecting from the wall a number of times. Usingg the multiple-reflection expansion, we will then derive the density profiles for Maxwell-Boltzmannn statistics, and via the inverse Laplace transform technique from sectionn 1.2.3, the profiles for the completely degenerate case.
Thee density profiles derived in chapter 2 and 3 have the form of a sum over Landau levels.. For weak magnetic fields the number of Landau levels will become very large,
andd the region where the asymptotic expressions for the profiles are valid, shirts further andd further away from the wall. In chapter 4 we will therefore investigate the weak field limit,, by looking first at the much simpler case where the magnetic field is perpendicular too the wall. After a comparison of our results with numerical data we will treat the originall geometry, where the magnetic field is parallel to the wall, in a similar way. Finally,, in the last chapter, we will derive correlation functions for the inhomogeneous electronn gas. Before we calculate those correlation functions in the magnetised case we willl derive the correlation functions for the field-free case in order to be able to com-paree the magnetised case with the field-free case. When the external magnetic field is present,, the full two-point correlation function, i.e. the correlation function where the componentss of both coordinates are all different, is difficult to evaluate. However, the correlationn function for points at the same distance from the wall, with the difference vectorr either parallel or perpendicular to the magnetic field, is much more manageable. Wee will derive asymptotic expansions for these correlation functions both for small and largee distances from the wall.
