To the theory of interaction between electron and
nuclear systems.
V.L.Marchenko, A.M.Savchenko
Coupled electron-nuclear oscillations in antiferromagnetics with anisotropy of “light plane” type in the strong external
ε of these magnetic field are under consideration. The new mode
2k
coupled oscillations is obtained for antiferromagnetic systems
by using ?u-v?-Bogolubov’s unitary transformations. The dynamic shift of the frequency of nuclear magnetic resonance concerned with this mode is obtained.
PACS:74.70.H
Keywords: Antiferromagnetic, Magnetic field, Oscillations, Magnetostriction
It’s well known that an increase of the constant external magnetic field leads to the gradual collapse
of spins of magnetic sublattices in antiferromagnetic. The second order transition takes place in the strong enough magnetic field: magnetic moments line up along the field (paramagnetic phase). Moreover, in the vicinity of the critical collapse field
H long-wave
c
collective oscillations in the nuclear spin system are
in existence. These are nuclear spin waves initiated by electron spin waves of the quasiantiferromagnetic branch. The aim of this work is to investigate the interaction between nuclear spin oscillations and electron spin waves.
Let’s consider the Hamiltonian of our model in the form [1,2]:
()()()[]
()()
?
??
?????+?????????+??
???
?????++??????++=∑∑∑∑∑∑∑∑∑f g g g f f f g g f n g g f f f
g
g f g f g f g f fg S I S I A I I H S S H n S n
S K n S n S K S S r J H r r r r r r r r r
r r r r r r r r r r
r r 0221,2μμ (1)
where electron spins g f S S r
r , belong to different
sublattices “f” and “g”, g f I I r r , - nuclear spins, ()fg r J r
-
the integral of exchange interaction, g f fg r r r r
r r ?=, 21,K K -
anisotropy constants, n r
- normal to plane of light magnetization, 0A - the constant of electron-nuclear
interaction.
Now represent our Hamiltonian using the second
quantization operators. We can express operators [3] j S r
using operators of spin deflections +j j a ,a with the help of Holstein-Primakoff transformation. Further apply Fourier representation
1/2
j
k
1,a a ,2,j
ikr j f
N e j g
νν?=?==?=?∑r r r
and introduce new operators 1/2112a
2[a (1)a ]k k k νν??=+?r r r %. So the Hamiltonian of our model assume the form
{}"""1/2
1012,,11a a (a a a a )(a [(1)]..22z z n f g k k k k k k k k k f g k k H A B I I A S X X эсνννννννννννν
ω+++?????=++?+++?+????∑∑∑∑r r r r r r r r r r r
%%%%%%%
where
θμθθθsin 2)cos(222cos )0(20121H I A S K S J A A n k k +???++=+r r
S K S k J A A k k 2221sin )(2?=?θr
r r
S K B B k k 1212?=+r s
∑=+?=?f
r k i fg k k fg e r J k J S K S k J B B r r r
r r r )()(,cos )(222
21θ
])cos()sin()[('
''2
/11n y f n n x f n n z f f
r k i iI I I I e
N
X f
+??????=∑?θθθθr r
])cos()sin()([''''''2
/12n y g
n n x g
n n z g
g
r k i iI I I I
e
N
X g
+???????=∑?θθθθr r
Electron and nuclear spin systems of double-sublattice antiferromagnetic have four resonance frequencies which correspond to similar types of spin precession. With neglect of dynamic mode coupling of these systems nuclear resonance frequencies prove to be degenerate and are equal to n ω. If we take into account this dynamic coupling we obtain the shift of resonance frequencies, which is for nuclear frequency inversely proportional to electron resonance frequency squared.
Activation energy of spin waves of quasiantiferromagnetic branch in antiferromagnetics of the “light plane” type [4] (in magnetic fields E H H <<) is caused by magnetic anisotropy and crystal exchange interaction field,2/120)2(E A H H με?. While the activation energy of spin waves of quasiferromagnetic branch in these magnetic fields is already caused by spontaneous magnetostriction, exchange energy and hyperfine interaction, 2/1010)](2[N ms E H H H +?με. It follows that the shift of the nuclear resonance frequency coupled with the quasiantiferromagnetic mode in magnetic fields E H H << is sufficiently great (about ten percents).
1020εε<<, if H and c H are of the same order, therefore, in this case the dynamic shift of nuclear magnetic resonance frequency coupled with quasiantiferromagnetic mode is quite possible.
And exactly in this case long-wave collective oscillations (in the nuclear spin system), so called nuclear spin waves, exist. They interact with electron spin waves of the quasiantiferromagnetic branch and hereinafter just this phenomenon is under consideration.
Let’s determine the spectrum of coupled oscillations of electron and spin systems. Apply Holstein-Primakoff
transformation to nuclear spin operators ,,,g f j I j =r
using Bose-operators j j αα,+. Keeping in the Hamiltonian only terms quadratic in nuclear spin operators, we obtain
{}++++21k 1k 22k 2-k 2k 2-k 2221k 222k 221a a (a a a a )a a ..2n n k k k k k k k k k k k H A B C D эсωααααωα++
???=+++++++???
?∑∑r r r r r r r r r r r r r r r r r r r
%%%%%%% (2) где 1/21/2021221/2
0212(),[1cos()](),2
1
[1cos()]()2
n n N k k k k n n N k C D αα
αθθωωθθωω?=?=??=?+?r r r r r %%
We can diagonalize the quadratic form (2) by using ?u-v? Bogolubov’s unitary transformation [5]
2k 222222222a u v u v ,u v u
v ,
eek ek eek e k enk nk enk n k
k
nek
ek
nek e k
nnk
nk
nnk n k
C C C C C
C
C
C
α
++
??+
+
??=+++=+++r r r r r r r r r r r r r r r r r r %
where functions u и v are defined by
()()1/2
1/2
020********/2
1/2
00221/2
222222u
,
v ,u u , v v ,u
, v
22n n N k
N k nek
enk nk k nk k
n N n N nnk nek nnk enk n n nk nk k k k k eek
eek
k k A
B A A ωωωωεεωωωωωωεεεε????==????????????
??
=
=
??+???
?+?==??
??
????
?
?
r r
r r r r
r r r r r r r
r
r r
r r r r
r
r
()()1/2
1/2
1/2
0022,
u u , v v .
n N n N nek
eek nek
eek n n ek ek ωωωωωω????
=
=
??+?r r r r r
r
And so our Hamiltonian assumes the form
222222()ek ek ek nk nk nk k
H C C C C ++=?+?∑r r r r r r r
%,
where 2222,,,ek ek nk nk C C C C ++r r r r - creation and destruction operators of normal modes of quasielectron and
quasinuclear spin waves with frequencies 22 и , ek nk ??r r which can be obtained from the dispersion equation
()(
)()()
2
2222222222222222
22222222()40
n n k k k k k k n k k k k
k
C D A C D B C D C D εωωω?
???+???++++?=r r r r r r r r r r r
If we take into account the static influence of the nuclear subsystem we’ll obtain the energy gap in electron spectrum 2k ε (for 0k =):
()(){
}
{}
1/2
1/22211
2221/2
11
sin cos 2(0)cos 2[()(0)sin cos()2(0)[()(0)]N n k
k
k
A N n A
B H H J S J k J S
H H H J S J k J S
εμθθθμθμθθθμμ????=?=+?+??×
×++?+??r r r r
r
The result for 2k ? in the case of antiferromagnetic of “light plane” type in a long-wave approximation for magnetic fields c H H can be written in the form
{}
1/222222222
00222221(1)2()4(1)()82
j j n N n n N no n n n N k
k k k k A ωεωωωεωωωεωωω???
=++?±++?++??r r r r r
where 1,2j =.
It follows from the last formula that the dynamic coupling of electron and nuclear systems leads to the nullification of one of two frequencies of normal oscillations in the point of phase transformation to paramagnetic state. In this phase one of spin-wave branches is high activative and the other – low activative in the vicinity of phase transition point to paramagnetic state.
If {}22
2004max ,n n N εωωω , the spectrum of quasielectron (quasinuclear) oscillations for the “light plane” type antiferromagnetic can be presented in the form
21/22022221k k N k n k A εωωε????????????????????????
?r r
r r
In this case there are two oscillation branches. One –electron and one – quasinuclear.
References.
1. V.L. Marchenko, B.I. Sadovnikov, A.M. Savchenko, Bull. Moscow State University, No.6, 2003.
2. E.M. Pikalev, M.A. Savchenko, I. Shoiom , Zh.Eksp.Teor.Fiz, v.51, p.111, 1968.
3. T.Holstein, H.Primakoff//Phys.Rev., v.58, p.1098, 1940.
4. E.R. Alaberdin, M.B.Sadovnikova, A.M.Savchenko, Bull. Moscow State University, No. 6, 1999.
5. N.N. Bogolubov, V.V. Tolmachev, D.V. Shirkov, New Method in the Theory of Superconductivity, Moscow, 1958.