The nuclear spin-lattice relaxation rate in the PuMGa$_5$ materials

A. V. Balatsky; M. J. Graf; N. J. Curro; Yunkyu Bang

REVIEW

The nuclear spin-lattice relaxation rate in the PuMGa$_5$ materials

Yunkyu Bang, M. J. Graf, N. J. Curro, A. V. Balatsky

Strongly Correlated Electrons · Submitted 2 Mar 2006 · 29 references

Pith's verdict: Plausible dirty d-wave and pseudogap picture for Pu-115, but the gap-ratio comparison leans on an unverified Tc shift.

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arXiv IDcond-mat/0603042 (version 2)

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CitationA. V. Balatsky, M. J. Graf, N. J. Curro, Yunkyu Bang. "The nuclear spin-lattice relaxation rate in the PuMGa$_5$ materials." arXiv:cond-mat/0603042 (2006). https://pith.science/paper/77TZ6BL4

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Claim paper

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Editor's note

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Colleague,

The headline result—PuRhGa5 and PuCoGa5 are both dirty d-wave superconductors with gap ratios 2Δ0/kBTc of about 5 and 8—is plausible but not proven; the entire comparison leans on an unverified shift of Tc from 8.5 K to 7.6 K for the PuRhGa5 fit. That is the thing to know before you cite the gap ratio.

What is genuinely new is the normal-state interpretation: the round-off in T1T of PuRhGa5 below ~20 K is treated as a pseudogap, and the two-component spin-fermion model with a spin gap and a fermion pseudogap does capture the temperature dependence. The T-matrix impurity formalism is not new—it comes from the authors' earlier work—but the self-consistent treatment and the fit to Sakai's data are competently done. The Knight-shift prediction in Fig. 3 is a useful falsifiable statement, and the near-total suppression of the coherence peak is a sensible diagnostic. I also credit the authors for stating plainly, in Section III, that there are no direct measurements of magnetic correlations in PuMGa5, and for flagging the Tc ambiguity in their final remark.

The soft spots are real but not fatal. The Tc normalization is the biggest: at 8.5 K the same gap gives 2Δ0/kBTc ≈ 4.5, and the "excess relaxation just below Tc" is exactly the kind of feature that could come from an additional channel rather than a shifted Tc. The gap ratio is an input, not a derived output, so the inference that pairing glue is stronger in PuCoGa5 is only as strong as the fit. In the normal state, Eq. (8) has several effectively free parameters; ΔFPG is not uniquely pinned—for PuCoGa5 they say it can be as large as 20 K and still fit. The pseudogap form is asserted phenomenologically, with the microscopic derivation deferred to a later paper. So the quantitative numbers (5 vs 8, ΔSG=20 K, ΔFPG=8 K) should be read as plausible estimates, not measurements.

Who it is for: people working on actinide superconductivity and heavy-fermion/cuprate connections. The qualitative picture—dirty d-wave plus a magnetic pseudogap—is a reasonable organizing hypothesis. The paper deserves a serious referee: the weakness is addressable, and the central argument is coherent even if the load-bearing Tc choice bothers me. If I were editing, I'd send it out with instructions to push the authors on the Tc normalization and to soften the quantitative claims accordingly.

Referee report

Significance

If the quantitative claims were robust, the paper would establish that both Pu-115 compounds are dirty d-wave superconductors in the unitary limit, that the lower Tc of PuRhGa5 is intrinsic rather than impurity-induced, and that the pairing interaction is significantly stronger in PuCoGa5 than in PuRhGa5. The normal-state analysis would also make PuMGa5 a bridge between heavy-fermion and cuprate phenomenology, with a testable Knight-shift prediction. The paper's strengths are its self-consistent impurity-scattering framework, the explicit comparison of two gap scenarios, and the honesty with which it flags unresolved questions, including the Tc normalization ambiguity and the absence of direct magnetic-correlation measurements. However, the central quantitative comparison depends on an unverified shift of the reported Tc, and the normal-state explanation rests on a phenomenological density-of-states formula with several adjustable parameters; these issues currently limit the certainty of the conclusions.

Major objections (3)

[§II, Figs. 1–2 and text near 'One final remark'] The central gap-ratio claim for PuRhGa5 rests on normalizing the experimental data with Tc = 7.6 K instead of the reported Tc = 8.5 K. The manuscript itself states that the origin of this ambiguity is 'not clear' and that using 8.5 K gives a poorer fit with an unexplained excess relaxation rate just below Tc. If the true Tc is 8.5 K, the same fitted gap corresponds to 2Δ0/kBTc ≈ 4.5 rather than 5; moreover, if an additional relaxation channel is present just below Tc, the extraction of both Γ and Δ0 from the data is underdetermined. Because the conclusions that the lower Tc of PuRhGa5 is intrinsic and that the gap ratios differ strongly between the two compounds depend directly on this normalization, the authors need either to justify the 7.6 K choice with independent evidence or to refit the data using the reported 8.5 K and show how the conclusions change.
[§II, after Eq. (1)] The gap ratio 2Δ0/kBTc is treated as an input parameter, not as a quantity derived from the calculation: the gap equation is the BCS equation, which would give the weak-coupling value 2.14, and the strong-coupling temperature dependence is inserted through the phenomenological form with chosen β and Δ0/kBTc. The comparison in Figs. 1 and 2 is therefore a comparison of two manually preselected ratios, not a systematic determination. Without a parameter scan over Δ0/Tc and Γ, or a quantitative goodness-of-fit criterion, the extracted values 2Δ0/kBTc ≈ 5 and ≈ 8 are not uniquely constrained by the data, and the statement that the comparison 'is a useful tool for determining the strong-coupling gap value' is stronger than the presented evidence.
[§III, Eq. (8) and Fig. 4(b)] The normal-state explanation relies on a phenomenological fermionic-pseudogap density of states with several free parameters (A, B, ΔSG, ΔFPG, and Γ), and the spin-gap scale is not independently measured. The paper notes both that there are 'yet no direct measurements of the magnetic correlations' in PuMGa5 and that for PuCoGa5 values of ΔFPG as large as 20 K produce fits of the same quality. As a result, the extracted energy scales ΔSG = 20 K and ΔFPG = 8 K, and the schematic phase diagram in Fig. 5, are not falsifiable on the basis of the 1/T1 data alone; additional magnetic measurements or a derivation of the FPG form from the model would be required to support the claimed crossover interpretation.

Minor objections (4)

[§II and §III] The symbol Γ is used for the impurity scattering rate in the superconducting-state fits (Γ/Δ0 ≈ 0.032 for PuRhGa5) and for a damping rate in Eq. (8) (Γ = 25 K); using the same symbol for two unrelated quantities is confusing and should be corrected.
[Eq. (3)] The typesetting of Eq. (3) is garbled: the angular averages with the square roots should be displayed with explicit factors so that the two contributions to 1/T1T are unambiguous.
[§II, text near Eq. (2)] The sensitivity statement about the parameter b is confusing: the text says b = 0.5 is not sensitive 'unless ξ is very large (b < 0.1), i.e., within the range of 0.1 < b < 1'; this wording should be clarified to specify the actual range over which results are insensitive.
[Reference 24 and text near 'Knight shift measurements'] The text cites 'Sakai et al. 24' for Knight-shift measurements, but Ref. 24 is listed as a private communication from H. Yasuoka; the citation should be corrected and, if possible, replaced with a published reference.

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The paper

The paper claims

The nuclear relaxation data of PuCoGa5 and PuRhGa5 are consistent with d-wave superconductivity in the unitary impurity-scattering limit, with gap ratios $2\Delta_0/k_BT_c \approx 8$ and $\approx 5$, and with PuRhGa5's lower $T_c$ being…

The reading

Using nuclear spin-lattice relaxation data, the paper argues that both plutonium superconductors PuCoGa5 and PuRhGa5 are dirty d-wave superconductors, meaning their pairing gap has line nodes and impurity scattering is strong enough to fill the low-energy gap with states. The fits give gap-over-$T_c$ ratios $2\Delta_0/k_BT_c \approx 8$ and $\approx 5$ for PuCoGa5 and PuRhGa5, which the authors take as evidence that the magnetic pairing interaction is stronger in PuCoGa5. They further conclude that PuRhGa5's lower $T_c$ is intrinsic: impurities suppress $T_c$ by only about 0.5 K. In the normal state, they attribute the observed suppression of $(T_1T)^{-1}$ in PuRhGa5 below about 25 K to a crossover from two-dimensional quantum-antiferromagnetic spin fluctuations to a fermion pseudogap, described by a two-component spin-fermion model. If correct, the PuMGa5 family sits between heavy-fermion and cuprate superconductors and should be tunable by Co/Rh substitution.

What carries the argument

The central machinery is a self-consistent T-matrix approximation for pointlike impurities in a d-wave superconductor: impurities renormalize the Matsubara frequency through $\Sigma_0 = \Gamma T_0$, and the gap equation is solved together with the impurity self-energy, so the relaxation rate $1/T_1T$ and spin susceptibility $\chi_S$ are computed from the renormalized quasiparticle spectrum. The paper works in the unitary scattering limit, where impurities generate a large residual density of states at zero energy. For the normal state, the key object is the two-component spin-fermion model: spin fluctuations come from local $5f$-electron spins and have a Lorentzian susceptibility with spin gap $\Delta_{SG}$, while the two-dimensional quantum-antiferromagnet correlation length follows a quantum-critical to quantum-disordered crossover, and a phenomenological fermionic pseudogap $\Delta_{FPG}$ suppresses the fermionic density of states entering the Landau-damping term. These pieces are combined in a single fitting formula for $1/T_1T$ and applied to both compounds.

Extended reading notes

Core claim

The central claim is that a single framework, d-wave pairing with unitary-limit impurity scattering for the superconducting state and a two-component spin-fermion model for the normal state, accounts for the relaxation rates of both compounds. In the superconducting state, the self-consistent T-matrix calculation reproduces the measured $1/T_1$ of PuRhGa5 with $2\Delta_0/k_BT_c \approx 5$ and $\Gamma/\Delta_0 \approx 0.032$, and of PuCoGa5 with the previously reported $2\Delta_0/k_BT_c \approx 8$ and $\Gamma/\Delta_0 \approx 0.01$; in absolute terms the two scattering rates are nearly equal ($\Gamma \approx 0.6$–$0.7$ K), consistent with self-irradiation damage. The paper therefore concludes PuRhGa5 is intrinsically lower in $T_c$, with weaker spin-fluctuation pairing glue. Normal-state fits use a spin gap $\Delta_{SG} \approx 20$ K plus a fermion pseudogap $\Delta_{FPG} \approx 8$ K for PuRhGa5, and $\Delta_{SG} \approx 0$ with $\Delta_{FPG} \approx 8$ K for PuCoGa5, locating PuCoGa5 near a two-dimensional quantum critical point and PuRhGa5 in a quantum-disordered regime.

Load-bearing premise

The model assumes the magnetic correlations are two-dimensional and originate from local $5f$-electron spins of plutonium, with a specific Lorentzian spin susceptibility, even though no direct measurements of these correlations in PuMGa5 exist.

Editorial extensions

If this is right

A residual density of states of order 25% of the normal-state value should survive to $T=0$ in both compounds, and the Knight shift should show a correspondingly large residual value; the paper calculates this explicitly.
Impurity damage cannot explain the $T_c$ difference between the two compounds: in the unitary-limit fit, the same absolute scattering rate suppresses $T_c$ of PuRhGa5 by only about 0.5 K, so the $T_c$ gap is intrinsic.
Because PuCoGa5 sits near a two-dimensional quantum critical point and PuRhGa5 on the quantum-disordered side, the alloy series Pu(Co,Rh)Ga5 should tune continuously through this crossover without changing carrier density.
The high-temperature normal-state data should obey a quantum-critical scaling relation involving the spin-spin relaxation time $T_{2G}$; measuring $T_{2G}$ would fix the dynamic exponent and test the two-dimensional antiferromagnetic assignment.
The pseudogap in PuRhGa5 should appear as two separate energy scales in other probes: a spin gap of about 20 K in magnetic measurements and a fermionic density-of-states suppression of about 8 K in thermodynamic measurements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the model's picture is correct, the fitted pseudogap scales should vary systematically with Rh/Co substitution; an NMR survey of the alloy series could map $\Delta_{SG}$ and $\Delta_{FPG}$ versus composition, something the paper sketches but does not compute.
The predicted residual Knight shift is a sharper test than the relaxation fit itself: a low-temperature Knight-shift measurement on PuRhGa5 would confirm or rule out the unitary-impurity scenario independently of the T-matrix details.
The normal-state explanation assumes the magnetic correlations are local-spin in origin, so an independent probe of the fermionic density of states, such as specific heat, should reveal the assumed $\Delta_{FPG}$ suppression in PuRhGa5; if it does not, the pseudogap would need a different source.

Original abstract

We examine the nuclear spin-lattice relaxation rates $1/T_1$ of PuRhGa$_5$ and PuCoGa$_5$, both in the superconducting and normal states, as well as their Knight shifts. Results for both compounds are consistent with a superconducting gap of d-wave symmetry in the presence of strong impurity scattering, though with quite different gap-over-$T_c$ ratios $2 \Delta_{0}/ k_B T_c$ (8 for PuCoGa$_5$ and 5 for PuRhGa$_5$). In the normal state, PuRhGa$_5$ exhibits a gradual suppression of $(T_1 T)^{-1}$ below 25 K, while measurements for PuCoGa$_5$ reveal a monotonic increase down to $T_c$. We propose that this behavior is consistently understood by the crossover from the two-dimensional quantum antiferromagnetic regime of the local $5f$-electron spins of Pu to the concomitant formation of the fermion pseudogap based on the two-component spin-fermion model.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,920 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.02 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.24 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We examine the nuclear spin-lattice relaxation rates $1/T_1$ of PuRhGa$_5$ and PuCoGa$_5$, both in the superconducting and normal states, as well as their Knight shifts. Results for both compounds are consistent with a superconducting gap of d-wave symmetry in the presence of strong impurity scattering, though with quite different gap-over-$T_c$ ratios $2 \Delta_{0}/ k_B T_c$ (8 for PuCoGa$_5$ and 5 for PuRhGa$_5$). In the normal state, PuRhGa$_5$ exhibits a gradual suppression of $(T_1 T)^{-1}$ below 25 K, while measurements for PuCoGa$_5$ reveal a monotonic increase down to $T_c$. We propose that this behavior is consistently understood by the crossover from the two-dimensional quantum antiferromagnetic regime of the local $5f$-electron spins of Pu to the concomitant formation of the fermion pseudogap based on the two-component spin-fermion model.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We examine the nuclear spin-lattice relaxation rates $1/T_1$ of PuRhGa$_5$ and PuCoGa$_5$, both in the superconducting and normal states, as well as their Knight shifts.	contribution 1.00	0.95	0.03	asserted 0.81
s1	Results for both compounds are consistent with a superconducting gap of d-wave symmetry in the presence of strong impurity scattering, though with quite different gap-over-$T_c$ ratios $2 \Delta_{0}/ k_B T_c$ (8 for PuCoGa$_5$ and 5 for PuRhGa$_5$).	result 0.85	0.89	0.47	hedged possibility 0.98
s2	In the normal state, PuRhGa$_5$ exhibits a gradual suppression of $(T_1 T)^{-1}$ below 25 K, while measurements for PuCoGa$_5$ reveal a monotonic increase down to $T_c$.	result 1.00	0.93	0.07	asserted 1.00
s3	We propose that this behavior is consistently understood by the crossover from the two-dimensional quantum antiferromagnetic regime of the local $5f$-electron spins of Pu to the concomitant formation of the fermion pseudogap based on the two-component spin-fermion model.	interpretation 1.00	0.48	0.78	hedged possibility 0.49

What kind of paper the classifier reports this is: empirical study 0.99 · evidence from observation or experiment 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: The NQR spin-lattice relaxation rate

**Figure 2.** FIG. 2: The NQR spin-lattice relaxation rate

**Figure 5.** FIG. 5: The schematic phase diagram for the alloy system

Showing 4 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603042 by the authors.

References

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We assume that the AFM correlation in PuMGa _5 is 2D like because the band calculations and the crystal structure of PuMGa _5 compounds, isostructural to CeMCo _5 compounds, indicate the 2D character of the electronic properties

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. L. Sarrao, L. A. Morales, J. D. Thompson, B. L. Scott, G. R. Stewart, F. Wastin, J. Rebizant, P. Boulet, E. Colineau, and G. H. Lander, Nature (London) 420 , 297 (2002)

View on Pith 2002

[2] [2]

F. Wastin, P. Boulet, J. Rebizant, E. Collineau, and G. H. Lander, J. Phys.: Condens. Matter 15 , S2279 (2003)

View on Pith 2003

[3] [3]

N. J. Curro, T. Caldwell, E. D. Bauer, L. A. Morales, M. J. Graf, Y. Bang, A. V. Balatsky, J. D. Thompson, and J. L. Sarrao, Nature (London) 434 , 622 (2005)

View on Pith 2005

[4] [4]

Y. Bang, A. V. Balatsky, F. Wastin, and J. D. Thompson Phys. Rev. B 70 , 104512 (2004)

View on Pith 2004

[5] [5]

E. D. Bauer, J. D. Thompson, J. L. Sarrao, L. A. Morales, F. Wastin, J. Rebizant, J. C. Griveau, P. Javorsky, P. Boulet, E. Colineau, G. H. Lander, and G. R. Stewart Phys. Rev. Lett. 93 , 147005 (2004)

View on Pith 2004

[6] [6]

H. Sakai, Y. Tokunaga, T. Fujimoto, S. Kambe, R. E. Walstedt, H. Yasuoka, D. Aoki, Y. Homma, E. Yamamoto, A. Nakamura, Y. Shiokawa, K. Nakajima, Y. Arai, T. D. Matsuda, Y. Haga, and Y. \= O nuki, to appear in J. Phys. Soc. Jpn

View on Pith

[7] [7]

K. Magishi, Y. Kitaoka, G.-q. Zheng, K. Asayama, T. Kondo, Y. Shimakawa, T. Manako, and Y. Kubo Phys. Rev. B 54 , 10131-10142 (1996)

View on Pith 1996

[8] [8]

Y. Bang, I. Martin, and A. V. Balatsky Phys. Rev. B 66 , 224501 (2002)

View on Pith 2002

[9] [9]

Y. Bang, M. J. Graf, A. V. Balatsky, and J. D. Thompson, Phys. Rev. B 69 , 014505 (2004); Y. Bang, M. J. Graf, and A. V. Balatsky, Phys. Rev. B 68 , 212504 (2003)

View on Pith 2004

[10] [10]

P. J. Hirschfeld, D. Vollhardt, and P. W\"olfle, Solid State Commun. 59 , 111 (1986); P. J. Hirschfeld, P. W\"olfle, and D. Einzel, Phys. Rev. B 37 , 83 (1988)

View on Pith 1986

[11] [11]

S. Schmitt-Rink, K. Miyake, and C. M. Varma, Phys. Rev. Lett. 57 , 2575 (1986)

View on Pith 1986

[12] [12]

H. Monien, K. Scharnberg, and D. Walker, Solid State Commun. 63 , 263 (1987)

View on Pith 1987

[13] [13]

V. A. Sidorov, M. Nicklas, P. G. Pagliuso, J. L. Sarrao, Y. Bang, A. V. Balatsky, and J. D. Thompson, Phys. Rev. Lett. 89 , 157004 (2002)

View on Pith 2002

[14] [14]

T. Mito, S. Kawasaki, Y. Kawasaki, G. -q. Zheng, Y. Kitaoka, D. Aoki, Y. . Haga, and Y. \= O nuki, Phys. Rev. Lett. 90 , 077004 (2003)

View on Pith 2003

[15] [15]

S. Kawasaki, T. Mito, G.-q. Zheng, C. Thessieu, Y. Kawasaki, K. Ishida, Y. Kitaoka, T. Muramatsu, T. C. Kobayashi, D. Aoki, S. Araki, Y. Haga, R. Settai, and Y. \= O nuki, Phys. Rev. B 65 , 020504 (2002)

View on Pith 2002

[16] [16]

See for example: P. Monthoux, A.V. Balatsky and D. Pines, Phys. Rev. B 46 , 14803, (1992)

View on Pith 1992

[17] [17]

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51 , 14874-14891 (1995); Y. L. Liu and Z.B. Su, Phys. Lett. A 200 , 393-398 (1995)

View on Pith 1995

[18] [18]

We assume that the AFM correlation in PuMGa _5 is 2D like because the band calculations and the crystal structure of PuMGa _5 compounds, isostructural to CeMCo _5 compounds, indicate the 2D character of the electronic properties

View on Pith

[19] [19]

A. Sokol and D. Pines, Phys. Rev. Lett. 71 , 2813 (1993)

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