numerical renormalization group computation of magnetic relaxation rates
DESCRIPTION
We report an essentially exact numerical renormalization-group (NRG) computation of the temperature-dependent NMR rate $1/T_1$ of a probe at a distance $R$ from a magnetic impurity in a metallic host. We split the metallic states into two subsets, A and B. The former comprises electrons $a_k$ in $s$-wave states about the magnetic-impurity site. The coupling between the $a_k$ band and the impurity is described by the Anderson Hamiltonian, diagonalizable by the NRG procedure. Each state $b_k$ in the B subset is a linear combination of an $s$-wave state about the probe site with the degenerate $a_k$, constructed to be orthogonal to all the $a_k$'s. The $b_k$ band hence decouples from the impurity and is analytically treatable. We show that the relaxation rate has three components: (i) a constant associated with the $b_k$'s; (ii) a $T$-dependent term associated with the $a_k$'s, which decays in proportion to $1/(k_FR)^2$, where $k_F$ is the Fermi momentum; and (iii) another $T$-dependent term due to the interference between the $a_k$'s and the $b_k$'s. The interference term shows Friedel oscillations whose amplitude, proportional to $1/k_FR$, can be mapped onto the universal function of $T/T_K$ describing the Kondo resistivity. We compare our findings with results in the literature.TRANSCRIPT
Numerical Renormalization-Group computationof magnetic relaxation rates
Krissia de Zawadzki, Luiz Nunes de Oliveira, Jose Wilson M. Pinto
Instituto de FΔ±sica de Sao Carlos - Universidade de Sao Paulo
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
π π
π πΎ β πβ1πΎ
General consensus
π πΎ = ~π£πΉ /ππ΅ππΎ
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: π from the impurity
NRG computation of the spin
lattice relaxation rate 1/(π1π ) as
function of π and π
Can we measure π πΎ via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses π πΎ
Phase of low-π Friedel oscillations
also changes
LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
π π
π πΎ β πβ1πΎ
General consensus
π πΎ = ~π£πΉ /ππ΅ππΎ
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: π from the impurity
NRG computation of the spin
lattice relaxation rate 1/(π1π ) as
function of π and π
Can we measure π πΎ via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses π πΎ
Phase of low-π Friedel oscillations
also changes
LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
π π
π πΎ β πβ1πΎ
General consensus
π πΎ = ~π£πΉ /ππ΅ππΎ
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: π from the impurity
NRG computation of the spin
lattice relaxation rate 1/(π1π ) as
function of π and π
Can we measure π πΎ via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses π πΎ
Phase of low-π Friedel oscillations
also changes
LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
π π
π πΎ β πβ1πΎ
General consensus
π πΎ = ~π£πΉ /ππ΅ππΎ
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: π from the impurity
NRG computation of the spin
lattice relaxation rate 1/(π1π ) as
function of π and π
Can we measure π πΎ via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses π πΎ
Phase of low-π Friedel oscillations
also changes
LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Radius of Kondo screening cloud
Radius of Kondo screening cloud
π π
π πΎ β πβ1πΎ
General consensus
π πΎ = ~π£πΉ /ππ΅ππΎ
Boyce &Slichter
NMR:
Experimental arrangement:
NMR probe: π from the impurity
NRG computation of the spin
lattice relaxation rate 1/(π1π ) as
function of π and π
Can we measure π πΎ via NMR?
Our findings:
Yes, we can!
T dependence changes as probe
crosses π πΎ
Phase of low-π Friedel oscillations
also changes
LASZLO, B. PRB, 75 (2007). BOYCE, J.B; SLICHTER, C.P. PRL, 32, 61 (1974).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
π» =
π»ππππβ β βk
πkπβ kπk
+
π»πβ β πππ
β πππ + πππβππβ +
π»πππ‘β β βΞ
π(πβ
0ππ +π».π.)
π = π£πΉπ·
(π β ππΉ )
+π·
βπ·
ππΉ
π»πππππ = βπ΄[Ξ¨β
β(οΏ½οΏ½)Ξ¨β(οΏ½οΏ½)πΌβ +π».π.]
Ξ¨π =βk
ππk.Rπk
1
π1=
4π
~
βπΌ,πΉ
πβπ½πΈπΌ |β¨πΌ|π»πππππ|πΉ β©|2πΏ(πΈπΌ β πΈπΉ )
π0 =1βπ
βk
πk
π
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
π» =
π»ππππβ β βk
πkπβ kπk +
π»πβ β πππ
β πππ + πππβππβ
+
π»πππ‘β β βΞ
π(πβ
0ππ +π».π.)
π = π£πΉπ·
(π β ππΉ )
+π·
βπ·
ππΉ
π»πππππ = βπ΄[Ξ¨β
β(οΏ½οΏ½)Ξ¨β(οΏ½οΏ½)πΌβ +π».π.]
Ξ¨π =βk
ππk.Rπk
1
π1=
4π
~
βπΌ,πΉ
πβπ½πΈπΌ |β¨πΌ|π»πππππ|πΉ β©|2πΏ(πΈπΌ β πΈπΉ )
π0 =1βπ
βk
πk
π
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
π» =
π»ππππβ β βk
πkπβ kπk +
π»πβ β πππ
β πππ + πππβππβ +
π»πππ‘β β βΞ
π(πβ
0ππ +π».π.)
π = π£πΉπ·
(π β ππΉ )
+π·
βπ·
ππΉ
π»πππππ = βπ΄[Ξ¨β
β(οΏ½οΏ½)Ξ¨β(οΏ½οΏ½)πΌβ +π».π.]
Ξ¨π =βk
ππk.Rπk
1
π1=
4π
~
βπΌ,πΉ
πβπ½πΈπΌ |β¨πΌ|π»πππππ|πΉ β©|2πΏ(πΈπΌ β πΈπΉ )
π0 =1βπ
βk
πk
π
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
The quantum system
NRG Probe
Single-impurity Anderson model
π» =
π»ππππβ β βk
πkπβ kπk +
π»πβ β πππ
β πππ + πππβππβ +
π»πππ‘β β βΞ
π(πβ
0ππ +π».π.)
π = π£πΉπ·
(π β ππΉ )
+π·
βπ·
ππΉ
π»πππππ = βπ΄[Ξ¨β
β(οΏ½οΏ½)Ξ¨β(οΏ½οΏ½)πΌβ +π».π.]
Ξ¨π =βk
ππk.Rπk
1
π1=
4π
~
βπΌ,πΉ
πβπ½πΈπΌ |β¨πΌ|π»πππππ|πΉ β©|2πΏ(πΈπΌ β πΈπΉ )
π0 =1βπ
βk
πk
π
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
ππ =βk
π k πΏ(πβ πk) (around impurity)
ππ =βk
π k ππk.RπΏ(πβ πk) (around probe)
π π
π π
NRG
analytical
{πβ π, ππβ²} = sin(ππ )ππ πΏ(πβ πβ²)
Gram-Schmidt construction
πππ = 1β1βπ 2
(πππ βππππ)
π = π (π,π ) = sin(ππ )ππ
ππ = ππΉπ (1 + π
π·
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
ππ =βk
π k πΏ(πβ πk) (around impurity)
ππ =βk
π k ππk.RπΏ(πβ πk) (around probe)
π π
π π
NRG
analytical
{πβ π, ππβ²} = sin(ππ )ππ πΏ(πβ πβ²)
Gram-Schmidt construction
πππ = 1β1βπ 2
(πππ βππππ)
π = π (π,π ) = sin(ππ )ππ
ππ = ππΉπ (1 + π
π·
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Two-center basis
Two-center basis
Spherically symmetric operators
ππ =βk
π k πΏ(πβ πk) (around impurity)
ππ =βk
π k ππk.RπΏ(πβ πk) (around probe)
π π
π π
NRG
analytical
{πβ π, ππβ²} = sin(ππ )ππ πΏ(πβ πβ²)
Gram-Schmidt construction
πππ = 1β1βπ 2
(πππ βππππ)
π = π (π,π ) = sin(ππ )ππ
ππ = ππΉπ (1 + π
π·
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
π»π =1
ππ
(πβ1βπ=0
π‘π(πβ πππ+1 +π».π.) +
βΞ
π(πβ ππ0 +π».π.) +π»π
)NRG[4]
π»πππππ = βπ΄ [ πβ βπβ +Ξ¦β
βΞ¦β + (πβ βΞ¦β +Ξ¦β
βπβ) ] Iβ +π».π.
ππ(π ) β‘β« π·
βπ·
ππβ
1 β π (π,π )πππ Ξ¦π(π ) β‘βπ
πΎπππ
analytically
numerically
1
π1=
(1
π1
)ππβ β
1βπ 2πΉ
+
(1
π1
)ΦΦβ β
π 2πΉ
+
(1
π1
)Ξ¦πβ β
(1βππΉ )ππΉ
cte
ππΉπ βͺ
1
ππΉπ
β«1
DULL
SMALL
ππΉ =sin(ππΉπ )
ππΉπ
WILSON, K. Rev Mod Phys, 47, 773 (1975).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
π»π =1
ππ
(πβ1βπ=0
π‘π(πβ πππ+1 +π».π.) +
βΞ
π(πβ ππ0 +π».π.) +π»π
)NRG[4]
π»πππππ = βπ΄ [ πβ βπβ +Ξ¦β
βΞ¦β + (πβ βΞ¦β +Ξ¦β
βπβ) ] Iβ +π».π.
ππ(π ) β‘β« π·
βπ·
ππβ
1 β π (π,π )πππ Ξ¦π(π ) β‘βπ
πΎπππ
analytically
numerically
1
π1=
(1
π1
)ππβ β
1βπ 2πΉ
+
(1
π1
)ΦΦβ β
π 2πΉ
+
(1
π1
)Ξ¦πβ β
(1βππΉ )ππΉ
cte
ππΉπ βͺ
1
ππΉπ
β«1
DULL
SMALL
ππΉ =sin(ππΉπ )
ππΉπ
WILSON, K. Rev Mod Phys, 47, 773 (1975).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
π»π =1
ππ
(πβ1βπ=0
π‘π(πβ πππ+1 +π».π.) +
βΞ
π(πβ ππ0 +π».π.) +π»π
)NRG[4]
π»πππππ = βπ΄ [ πβ βπβ +Ξ¦β
βΞ¦β + (πβ βΞ¦β +Ξ¦β
βπβ) ] Iβ +π».π.
ππ(π ) β‘β« π·
βπ·
ππβ
1 β π (π,π )πππ Ξ¦π(π ) β‘βπ
πΎπππ
analytically
numerically
1
π1=
(1
π1
)ππβ β
1βπ 2πΉ
+
(1
π1
)ΦΦβ β
π 2πΉ
+
(1
π1
)Ξ¦πβ β
(1βππΉ )ππΉ
cte
ππΉπ βͺ
1
ππΉπ
β«1
DULL
SMALL
ππΉ =sin(ππΉπ )
ππΉπ
WILSON, K. Rev Mod Phys, 47, 773 (1975).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
π»π =1
ππ
(πβ1βπ=0
π‘π(πβ πππ+1 +π».π.) +
βΞ
π(πβ ππ0 +π».π.) +π»π
)NRG[4]
π»πππππ = βπ΄ [ πβ βπβ +Ξ¦β
βΞ¦β + (πβ βΞ¦β +Ξ¦β
βπβ) ] Iβ +π».π.
ππ(π ) β‘β« π·
βπ·
ππβ
1 β π (π,π )πππ Ξ¦π(π ) β‘βπ
πΎπππ
analytically
numerically
1
π1=
(1
π1
)ππβ β
1βπ 2πΉ
+
(1
π1
)ΦΦβ β
π 2πΉ
+
(1
π1
)Ξ¦πβ β
(1βππΉ )ππΉ
cte
ππΉπ βͺ
1
ππΉπ
β«1
DULL
SMALL
ππΉ =sin(ππΉπ )
ππΉπ
WILSON, K. Rev Mod Phys, 47, 773 (1975).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
NRG and Lanczos basis
NRG and Lanczos basis
π»π =1
ππ
(πβ1βπ=0
π‘π(πβ πππ+1 +π».π.) +
βΞ
π(πβ ππ0 +π».π.) +π»π
)NRG[4]
π»πππππ = βπ΄ [ πβ βπβ +Ξ¦β
βΞ¦β + (πβ βΞ¦β +Ξ¦β
βπβ) ] Iβ +π».π.
ππ(π ) β‘β« π·
βπ·
ππβ
1 β π (π,π )πππ Ξ¦π(π ) β‘βπ
πΎπππ
analytically
numerically
1
π1=
(1
π1
)ππβ β
1βπ 2πΉ
+
(1
π1
)ΦΦβ β
π 2πΉ
+
(1
π1
)Ξ¦πβ β
(1βππΉ )ππΉ
cte
ππΉπ βͺ
1
ππΉπ
β«1
DULL
SMALL
ππΉ =sin(ππΉπ )
ππΉπ
WILSON, K. Rev Mod Phys, 47, 773 (1975).
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 5 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Friedel oscillations
Friedel oscillations
9.8 10.0 10.2 10.4 10.6 10.8 11.0 11.2kFRΟ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
10.5Ο
10Ο 10.25Ο
T=1.7569eβ4
T=9.8711eβ8
10.0 10.5 11.0
0.16
0.17
1T
1T(kFR
)2
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 6 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Relaxation rate - temperature dependence
ππ΅ππΎ = 1.25Γ 10β5
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
(101+0.25)Ο
(102+0.25)Ο
(103+0.25)Ο
(104+0.25)Ο
(105+0.25)Ο
(106+0.25)Ο
(107+0.25)Ο
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 7 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Interference relaxation rate
Interference relaxation rate
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-20
1
2
3
4
5
6
7
RβRK
outside
inside
(kFR
)2
kBT
( 1 T1
)
kB T
kFR=(n+14)Ο
Ξ»B =2ΟvFkB T
de Broglie
n=102
n=105
n=107
n=102
n=105
n=107
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 8 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Friedel oscillations
Friedel oscillations
101 102 103 104 105 106 107 108
kFRΟ
1.58
1.60
1.62
1.64
1.66
1.68
1.70
1.72
RK βTK =1.25eβ05
nΟ
(n+1/2)Ο
104 105 106
0.00010
0.00015
+1.6131
1T
1T(kFR
)2
Tβ5.62eβ11
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 9 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Conclusions
NMR to measure π πΎ : OK!
Inside cloud, π -dependent rate follows universal curve
Outside cloud, rate follows different curve
Phase of Friedel oscillations reverses around π = π πΎ
Future prospects:
Other geometries
P-h symmetric case differs from assymetric ?
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 10 / 11
Introduction NRG calculations Numerical results Conclusions Acknowledgment
Acknowledgment
Thank you!
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 11 / 11
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 1 / 4
Additional results
Relaxation rate - π’π πππ(π ) profile
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
(101+0.5)Ο
(102+0.5)Ο
(103+0.5)Ο
(104+0.5)Ο
(105+0.5)Ο
(106+0.5)Ο
(107+0.5)Ο
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 2 / 4
Additional results
Relaxation rate - π’ππΈπ (π ) profile
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
101 Ο
102 Ο
103 Ο
104 Ο
105 Ο
106 Ο
107 Ο
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 3 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/π1 as function of π
ππΉπ = ππ and ππΉπ = (π+ 12 )π, π = 10
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-9 10-8 10-71.4
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/π1 as function of π
ππΉπ = ππ and ππΉπ = (π+ 12 )π, π = 103
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-8 10-71.4
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/π1 as function of π
ππΉπ = ππ and ππΉπ = (π+ 12 )π, π = 105
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-7
1.6
1.8
2.0
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4
Additional results
Particle-hole symmetric case
Particle-hole symmetric case: 1/π1 as function of π
ππΉπ = ππ and ππΉπ = (π+ 12 )π, π = 107
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
kB T
0
1
2
3
4
5
6
7
10-91
2
3
4
(kFR
)2
kBT
( 1 T1
)
Zawadzki, K. de; Oliveira, L.N.; Pinto, J.W.M. NRG computation of nuclear magnetic relaxation rates 4 / 4