COMPONENTE RURAL
ARTICULO 151 ZONIFICACION RURAL A PARTIR DEL USO POTENCIAL
now undergoes a periodic perturbation, so we can utilise Bloch’s theorem. The
perturbation Hamiltonian is given as
N
^ = A jim V |jo )(ja |,
(4.6)
N —>oo ^ '
w here A is a continuous p a ra m e te r t h a t d eterm ines tlie s tre n g th of th e p e rtu rb a tio n . T h is corresponds to a w eighted D irac com b p o te n tia l in real space.
If we cellularise th e system according to th e rule
x' = j a + X (4.7)
w here j labels cells an d x is an intra-cell index, th e n we can w rite th e p e rtu rb a tio n H am ilto n ian as N V ^ X l i m J 2 \ j , 0 ) { j , 0 \ , (4.8) N —>oo ^ ' j= 0 w here \ j , x ) \ j a + x). (4.9)
Here j is an intercell index, while x is an intracell index. We in tro d u c e th e m ixed basis \ k , x ) , w here
\ k , x ) = \k)\x) (4.10)
and
j = 0
H ere k is & w avenum ber; since we have (discrete) tra n sla tio n a l sy n n netry, we can ap p ly th e Bloch theorem . T h e q u a n tity x labels th e po in ts inside a single u n it cell.
W ith th is definition a little alg eb ra leads to
27T a l> = A lim
V|A:,0)(/c,0|,
(4.12) TV— * 0 0 k= 0 w here (4.13)system and th e p e rtu rb a tio n H am iltonian in th e form
0 = Y . d k , (4.14)
k
w here
( \ : ^ { k \ d \ k ' ) S ^ , , (4.15)
is a p a rtia l p ro jectio n of th e relevant o p e ra to r, we can w rite th e Lloyd form ula as
“ 9 k ( E ) V k ) ^ ) ^ ■ (4-16)
In th e presen t case, it is possible to shov.^ th a t b o th g an d V can be w ritte n in th e
form of Eq. (4.15), w here
Vk = X\ k,Xo){k,Xo\ (4,17)
and
9 k — / d x \ k , x ) { k , x \ g \ k , x ) {k, x\ . (4-18)
Jo
F u rth e rm o re , as has been discussed in c h a p te r 2 in th e case of a single im purity,
th e d e te rm in a n t collapses to th e sim ple scalar function 1 — A { k , Xo| g |A’, Xq) . S o th e
change in th e den sity of s ta te s is given as
-^ p ( ^ ) = ^ (log (1 - A { k , jjol g |A:, Xq)))^ . (4.19)
l b ev alu a te th e q u a n tity { k , X o \ g \ k , X o ) , we s ta r t w ith th e observation t h a t th e real space m a trix elem ent of th e G reen function of a free electron gas in one dim ension is given by th e form ula
{ x \ g \ x ' ) = {xo\ g\ xo)e^^’>^^^\^-^\ (4.20)
± \ / 2 E + 'iO+, Im (kp) > 0. By Eq. (4.11) and presumed orthonorm ality of the real space kets,
^ i j k a
( j , x \ k , x o ) ^ - ^ 6{ x - Xo) . (4.21)
W ith this result we can calculate { k , x o \ g \ k , X o ) :
p N a p N a
g = / dx'\x){x'\g^^^: Jo Jo
3,f
= V T d x £ d x ' \ j , x ) i f , (4.22)
This can be projected down to yield:
r a pa k p \ { j - f ) a + x - x ' \ ra pa { k , X o \ g \ k , x o ) = J
J
d x ' { k , X o \ j , x ) { f , x'\k, X o )g m e '^ '’ j j ' _ S ^ ’^^^tak{j'-j)^ikpa\{j-j')\ (4 23) i j 'which by shifting the index (in the limit th a t N —>■ oo) is the same as
OO
(k, Xol gl k, Xo) = goo ^ e“ '=^ei a k j i kpa\ j \ ] = -oo
(
- 1 oo ^ giafcjgtfcpaljl ^ ^ i a k j ^ ^ k , a \ i \+
1 j = - o o j = l j(
j= l OO j=l CXD \//
g i a ( k p - k ) ^ i a{ k p+k)I 2
f , i a { k p - k )gi a{ kp+k)
^
F ig u re 4.3: R elativ e energy change p er p e rtu rb a tio n for a one-dim ensional free elec tro n gas p e rtu rb e d by a series of equally spaced d e lta-fu n ctio n sc a tte re rs for (i) 2 sc a tte re rs (d ash ed line), (ii) 7 sc a tte re rs (continuous line), (iii) Infinite nu m b er of sc a tte re rs (d o tte d line). Ayv is a dim ensionless qu an tity , and is defined in Eq. (4.25). T h e a m p litu d e of th e sc a tte rin g p o ten tia l is A = —0.3 in all cases. T h e distan ce betw een successive m inim a (m axim a) is w here k f is th e Ferm i-w avem m iber of th e gas.