electron cloud build-up: theory and data

Electron Cloud Build-Up: Theory and Data

Miguel Furman

LBNL

M. Furman - ECLOUD10 p. 1

LBNL

[email protected]

http://mafurman.lbl.gov

ECLOUD10 Workshop

Cornell, 8-12 Oct, 2010

Summary

• What is the electron-cloud effect (ECE)

• Brief history

• Primary and secondary electrons

• Simulations and data

• Mitigation


• Conclusions

Acknowledgments: I am grateful for collaboration and discussions over time with: A.

Adelmann, G. Arduini, V. Baglin, S. Berg, M. Blaskiewicz, O. Brüning, Y. H. Cai, J.

Calvey, F. Caspers, C. Celata, R. Cimino, R. Cohen, I. Collins, J. Crittenden, F.-J. Decker,

G. Dugan, N. Eddy, A. Friedman, O. Gröbner, K. Harkay, S. Heifets, N. Hilleret, U. Iriso, J.

M. Jiménez, R. Kirby, I. Kourbanis, G. Lambertson, R. Macek, A. Molvik, K. Ohmi, M.

Palmer, S. Peggs, G. Penn, M. Pivi, C. Prior, A. Rossi, F. Ruggiero, G. Rumolo, D. Sagan,

K. Sonnad, D. Schulte, P. Stoltz, J.-L. Vay, M. Venturini, L. Wang, S. Y. Zhang, X. Zhang,

A. Zholents, F. Zimmermann, R. Zwaska,…

My apologies to the experts – this is a very basic talk

What is the ECE(illustrated with the LHC cartoon by F. Ruggiero)

25 ns25 ns 25 ns25 ns


• Beam emits synchrotron radiation:

– provides source of photo-electrons

– other sources: beam-gas ionization, stray protons→wall

• Photo-electrons get rattled around the chamber from multibunch passages

—especially for intense positively-charged beams (e+, protons, heavy ions)

• Photoelectrons yield secondary electrons

– yield is determined by the secondary emission yield (SEY) function δ(E):

– characterized by peak value δmax

– e– reflectivity δ(0): determines survival time of e–

•Typical e– densities: ne=1010–1013 m–3 (~a few nC/m)

• Possible consequences:— single-bunch instability

— multibunch instability

— emittance blowup

— gas desorption from chamber walls

— excessive energy deposition on the chamber walls (important for superconducting machines, eg. LHC)

— particle losses, interference with diagnostics,…

• In summary: the ECE is a consequence of the interplay between the beam

Consequences


• In summary: the ECE is a consequence of the interplay between the beam and the vacuum chamber “rich physics”

— many possible ingredients: bunch intensity, bunch shape, beam loss rate, fill

pattern, photoelectric yield, photon reflectivity, SEY, vacuum pressure, vacuum

chamber size and geometry, …

• The ECE is closely related to the mechanism of photo-amplifiers

* IT IS ALWAYS UNDESIRABLE IN PARTICLE ACCELERATORS

* IT IS A USUALLY A PERFORMANCE-LIMITING PROBLEM

* IT IS CHALLENGING TO PROPERLY QUANTIFY, PREDICT AND EXTRAPOLATE

More...

• NOTE: if conditions are such that the bunch spacing

in time is equal to the traversal time of the electrons

across the chamber, you get a resonance condition

• “beam-induced multipacting” (BIM)

• First observed at ISR mid-70’s

—Usually dramatic consequences: gas desorption


—Usually dramatic consequences: gas desorption

from the vacuum chamber walls

—Beam is rapidly lost

—Or, trigger beam abort (e.g., at RHIC)

Our goals…

• Identify the relevant variables in each case

• Predict and measure

• If possible, minimize the effect in the

design stages of new machines

• Implement mitigation mechanisms

• Passive


• Passive

• low-emission coatings

• grooves

• weak B-fields to sweep electrons

• Active

• Adjust the chromaticity

• Feedback systems

• Tailoring bunch patterns

• Typically, both passive and active

• And wait with crossed fingers …

Brief history: BCE and CE

• BCE: effect first seen many years ago in proton storage rings:

— two-stream instabilities (in space-charge compensated coasting beams)

• BINP, mid 60’s: G. I. Budker, V. G. Dudnikov, …

• ISR, early 70’s: E. Keil, B. Zotter, H. G. Hereward,…

• Bevatron (LBL), early 70’s: H. Grunder, G. Lambertson…

— beam-induced multipacting (ISR, mid 70’s, bunched beams)

• O. Gröbner, ICHEA 1977

• multibunch effect; pressure rise instability

— High-intensity instability at PSR (LANL), since mid 80’s


— High-intensity instability at PSR (LANL), since mid 80’s

• single-long-bunch effect

• Fairly conclusively identified as an electron effect in 1991 (D. Neuffer, E. Colton, R.

Macek et al.)

• CE: started in early 90’s, KEK Photon Factory:

— M. Izawa, Y. Sato and T. Toyomasu, PRL 74, 5044 (1995)

• First observation of instability sensitivity to beam-charge sign in a lepton ring

• Electrons in the chamber were immediately suspected

• Quick decision to add an antechamber to the PEP-II e+ ring chamber

• Caveat: an electron-beam interaction had been previously observed at CESR (J. Rogers et al; “anomalous antidamping”)

ECE at KEK Photon FactoryIzawa, Sato & Toyomasu, PRL 74, 5044 (1995)

• Qualitative difference in coherent spectrum of e+ vs. e– multibunch beams under otherwise identical conditions:

electron beam spectrumpositron beam spectrum


Fast multibunch instability for e+ beam:

— insensitive to “clearing gap”

— sensitive to bunch spacing

— electrons in the chamber were immediately suspected

— first simulations: K. Ohmi, PRL 75, 1526 (1995); “photoelectron instability” (PEI)

— immediate concern for the B factories’ design

LHC

• 1995-96: concerns that electrons would spoil LHC vacuum (based on ISR experience, O. Gröbner)

• Early 1997: first simulations by F. Zimmermann that included photoelectrons showed a significant ECE

— first proton machine with significant synchrotron radiation:

critical energy of photon spectrum:

intensity: photons/proton/bend


— main concern: excessive power deposition

— initial estimates: ~a few W/m, vs. 0.5 W/m cryo capacity

— “LHC crash programme” started 1997 by F. Ruggiero

— big simulation effort, along with measurements

— conclusion: main sensitivity is SEY

— current consensus: peak SEY must be <~ 1.1–1.3 to avoid the problem

— we’ll know in a couple of years, when the LHC reaches nominal intensity

Importance of the EC

• ECE has been observed at many other machines:

— PEP-II, KEKB, BEPC, PS, SPS, APS, RHIC, Tevatron, MI, SNS, CESRTA …— diminished performance

and/or— dedicated experiments

• PEP-II and KEKB:

— controlling the EC was essential to achieve and exceed luminosity goals

—Antechamber: lets ~99% of photons escape

— TiN coating at PEP-II: suppresses SEY

—Solenoidal B-fields, B~20 G (at both machines) trap electrons near chamber surface


—Solenoidal B-fields, B~20 G (at both machines) trap electrons near chamber surface

—Complicated beam fill patterns were used for a while

• PSR: high-current instability, beam loss

− Decision to coat SNS vacuum chamber with TiN

• RHIC: fast vacuum pressure rise instability at high current forces beam dump (in some fill

patterns)

− Not any more (TiZrV coatings suppress SEY)

• Concern for future machines (LHC, ILC DR’s, MI upgrade,…)

• CESRTA is most significant, dedicated, systematic program to understand the ECE in e+e– rings

• Funding started ~3 yrs ago

• Great progress! ECLOUD10 workshop rightfully sited at Cornell

Simulations of the ECE

• Ideally, a single description of the combined beam+EC dynamics

• Such “self-consistent codes” are maturing, but not yet ready for regular, steady

use

• Complicated dynamics, many variables, some more relevant than other

• Slow

• So, there are 2 kinds of codes typically in use:

1. Build-up codes: simulate the development of the EC by the action of a given,

prescribed beam (ECLOUD, POSINST, PEI,...)


prescribed beam (ECLOUD, POSINST, PEI,...)

• This is the subject of this talk

2. Beam dynamics codes: simulate the dynamics of the beam by te action of a

given, prescribed EC (WARP, CLOUDLAND, PEHTS, HEADTAIL,...)

• Typically, both approaches are good approximations (“1st-order” approximations)

Code “POSINST” features(M. Furman and M. Pivi)

• Electrons are dynamical• represented by macroparticles

• Beam is not dynamical• represented by a prescribed function of time and space

• A simulated photoelectron is generated on the chamber surface• It is then “tracked” (F=ma) under the action of the beam• When it strikes the chamber wall, there is a probabilistic process:

• Absorbed


• Absorbed• Bounces elastically• Generate secondary electrons

• secondary electron emission: detailed model (M. Furman & M. Pivi,

PRSTAB/v5/i12/e124404 (2003))

• field-free region, dipole field, solenoidal field, others…

• round or elliptical vacuum chamber geometry (with a possible antechamber)

• perfect-conductor BCs (surface charges included)

•EC density reaches saturation, one way or the other

Secondary e– emission:two essential ingredients

E0

EnE2

E1

..

=SEY=no. of emitted electrons per

incident electron (incident energy, angle)

(1)

Note: δ=1 means one e– in,

one e– out


=emitted electron energy spectrum(2)

incident electron (incident energy, angle)

Secondary emission is an event-by-event simulation:– event=one electron-wall collision– instantaneous generation of n secondaries (or absorption)– detailed phenomenological model for δ(E0,θ0) and dδ/dE

• model parameters obtained from simultaneous fits to bench

measurements for δ and dδ/dE for Cu, St.St., Al and TiN• some parameters not well-known

Two sample measurements of SEY

2.0

1.5

1.0

measured data (R. Kirby) model fit (Furman-Pivi)

Stainless steel sample (data R. Kirby)

2.0

1.5

1.0

fit (Furman-Pivi) measured data

E0tspk=276.812

Copper sample (Hilleret data)

Cu St. steel


0.5

0.010009008007006005004003002001000

E0 [eV]

E0ts=0

E0tspk=310

dtspk=1.22

powts=1.813

P1epk=0.5

P1einf=0.07

E0epk=0

powe=0.9

E0w=100

P1rinf=0.74

Ecr=40

qr=1

0.5

0.010009008007006005004003002001000

E0 [eV]

E0tspk=276.812

dtspk=1.8848

powts=1.54033

E0ts=0

P1epk=0.496229

P1einf=0.02

E0epk=0

powe=1

E0w=60.8614

P1rinf=0.2

Ecr=0.0409225

qr=0.104045

• caveat: samples not fully conditioned!

(N. Hilleret; R. Kirby)

Sample spectrum: dδδδδ/dEThree main components: elastics, rediffused, true secondaries

St. St. sample, E0=300 eV, normal incidence, (Kirby-King,

NIMPR A469, 1 (2001))

0.08

0.06

Secondary energy spectrum

St. St., E0=300 eV, normal incidence

true secondaries

st. steel sample

δ = 2.04

δe = 6%

δr = 37%

δ =57%

Cu sample

δ = 2.05

δe = 1%

δr = 9%

δ =90%


0.04

0.02

0.00300250200150100500

Secondary electron energy [eV]

(area[0,50]=1.17)

backscattered

(area[295,305]=0.12)

rediffused

(area[50,295]=0.75)

r

δts =57%

δe+δr =43%

– Hilleret’s group CERN: Baglin et al, CERN-LHC-PR 472.

– Other measurements: Cimino and Collins, 2003

δts =90%

δe+δr =10%

Simulated movie, CESRTAfield-free region, 10 bunch passages


Simulation vs. experiment at CESRTA (G. Dugan)1.885 GeV tune shift data-central density 0.75 mA/bunch

POSINST simulation- Al chamber, peak SE energy 310 eV, SEY=1.8

Technique: measure “bunch tune shift”

roughly ∝ EC density

10-bunch train,


10-bunch train, followed by a “witness bunch”

Simulated movie, PSRfield-free region, 2 bunch passages


PSR: benchmark code POSINST

• Bunch length >> ∆t

— a portion the EC phase space is in resonance with the “bounce frequency”

— “trailing edge multipacting” (Macek; Blaskiewicz, Danilov, Alexandrov,…)

ED42Y electron detector signal

8µC/pulse beam

435 µA/cm2

electron signal


435 µA/cm2

measured (R. Macek) simulated (M. Pivi)

(δmax=2.05)

Simulated movie, LHCexternal dipole bending field


High-density regions form where Ew(x)=Emax

called “stripes” (F. Zimmermann)

Controlling the ECE

• Modify the vacuum chamber geometry (suppress both photoemission and SEY)

— add an antechamber (PEP-II: let photons escape)

— add transverse grooves (eg., LHC beam screen: suppress photoemission by ~x2)

— add longitudinal grooves (SLAC tests): suppress effective SEY (~x2)

• Modify the vacuum chamber electronic properties: low-SEY coatings

— TiN (PEP-II, SNS)

— TiZrV (RHIC and LHC RT regions – requires activation), …

— Amorphous carbon coating (under tests at CERN)


— Amorphous carbon coating (under tests at CERN)

— Note: most coatings require activation to become effective

— Clearing electrodes

• Use solenoidal B-fields (~20 G)— confines electrons near the chamber, away from the beam

• used extensively at KEKB and PEP-II

• significant improvement in performance

• Tailor the bunch fill pattern— add strategic gaps in the train

• Use feedback systems to actively counteract instabilities that arise

Conditioning effect of SEY

• The SEY usually dominates the EC build-up

• But, the SEY naturally decreases with electron bombardment

• “self-conditioning effect”

• Clearly seen in many cases

• Q: 1) is it fast enough? (Y)

Copper SEY (CERN)


• 2) does it go far enough? (N?)

• Copper sample:

• note δ(0)≈1

— consequences of “fish hook” not fully

explored

— But known to be unfavorable

because δ(0) controls the

dissipation rate of the EC

— Evidence from PSR that δmax➘, but

� δ(0) remains ∼ constant

(R. Cimino and I. Collins, proc.

ASTEC2003, Daresbury Jan. 03)

Conclusions

• The ECE is an ubiquitous phenomenon for intense beams

— spans broad range of charged-particle machines

• It is important inasmuch as it limits the machine performance

— Especially for high-intensity future machines

• It is interesting, as it involves in an essential way various areas of physics:

— Surface geometry and surface electronics

— Beam intensity and particle distribution

— Beam energy

— Residual vacuum pressure

— Certain magnetic features of the storage ring


— Certain magnetic features of the storage ring

• Simulation codes are getting better and better in their detailed modeling capabilities

• Enormous progress has been made since 1994

— With a disproportionate credit due to CESRTA over the past ~3 years

— Better and more refined e– detection mechanisms

— Simulation codes are getting better and better calibrated against measurements

— Phenomelogical “rules of thumb” are appearing that tell you when the ECE is serious• But not when it’s weak and safe

• But mysteries remain...

— Not a year has gone by without a couple of big surprises

— I encourage workshop speakers to emphasize the flies in the ointment

In closing...

Thanks to our Cornell colleagues, especially to Mark

Palmer, for organizing this workshop

I look forward to lively and productive discussions

THANK YOU FOR YOUR ATTENTION


THANK YOU FOR YOUR ATTENTION

Backup material


Secondary e– emission: effective SEY

if δeff>1: Ne~exp(t/τ)• EC density grows exponentially until space-charge limit

• close to beam neutralization level

if δeff<1: Ne~exp(–t/τ)

• walls are net absorber of electrons

• EC density saturates when no. of emitted primaries=no. of absorbed e–


• EC density saturates when no. of emitted primaries=no. of absorbed e

• exponential decay is seen upon beam extraction

What is δeff?

• δeff is a complicated function of Nb, bunch fill pattern, bunch shape, vacuum

chamber material, chamber geometry, …

• δeff is not known a priori

Conditioning effects: beam scrubbing

• PSR “prompt” e– signal (BIM) is subject to conditioning:

—signal is stronger for st.st. than for TiN

—sensitive to location and N

—signal does not saturate as N increases up to ~8x1013

—conditioning: down by factor ~5 in sector 4 after few weeks (low current)

• PSR “swept” e– signal is not:

—signal saturates beyond N~5x1013


—signal saturates beyond N~5x1013

— electron decay time τ ≈ 200 ns, independent of:

• N

• location

• conditioning state

• st. st. or TiN

• Tentative conclusion: beam scrubbing conditions δmax but leaves δ(0) unchanged

BIM in the APS: benchmark code POSINST

120

100

80

60

aver. electron-wall current [nA/cm2]

APS, positron beamDetector Current vs. Bunch Spacing

(10 bunches, 2 mA/bunch in all cases; measurements courtesy K. Harkay, ANL)

region of BIM

sB=d2/(reN), b<d<a


40

20

0

aver. electron-wall current [nA/cm

35302520151050

bunch spacing sB [RF buckets]

measured simulated

e+ beam, 10-bunch train, field-free region

Simulated

(code POSINST)

measured

(Furman, Pivi, Harkay,

Rosenberg, PAC01)

Lowering the SEY

• Low-SEY coatings

- TiN (used in PEP-II, SNS; tested at PSR)

- TiZrV: studied at CERN

• fully suppresses multipacting after activation (SPS tests)

• used in RHIC warm sections (“works better than solenoids”)

• will be used in LHC warm straights

• drawback: cannot be used in cold regions (needs activation ~160-200 C)


• SEY decreases with e– bombardment: “scrubbing”

– self-conditioning effect

• SPS ECE studies:

– ~5 years of dedicated EC studies with dedicated instrumentation

– scrubbing very efficient; favorable effects seen in:

• vacuum pressure

• in-situ SEY measurements

• electron flux at wall

Results for e– line density vs. t (one turn)


MI: sample time-averaged EC density

2

1

0

y [cm]

1.2x107

1.0

0.8

0.6

cm**-3

MI_1p3_6_spc1-K


-1

-2

y [cm]

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

x [cm]

0.4

0.2

0.0

Conclusions for FNAL MI

• There seems to be a critical value Nb~1.25x1011 at which the EC grows exponentially and reaches saturation (≈beam neutralization level) within ~110 ns

— this assumes a specific model for the SEY, and δmax=1.3

— also assumes a drift section of the MI

• What to do next:

— vary δmax; find Nb as a function of δmax


max b max

— look at different models of SEY

— look at magnetic sections (dipoles, quads)

— vary sb (?)

— study effects of EC on beam

• this is outside the scope of POSINST

• For a full 3D self-consistent simulation, see seminar by Jean-Luc Vay next week here (almost ready for quantitative predictions)

MI: preliminary results for a drift section

• Choose E=8 GeV (f=1.2% of beam lost during ∆tinj=0.4 s):

(assumes ηeff=100 e/p, from PSR experience)

σ


• Assume T=305 K, P=20 nTorr, σi=2 Mbarns:

• Assume δmax=1.3, model “K” (from fits to old St.St. SLAC

data; see PRSTAB/v5/i12/e124404 (2003))

Calculated azimuthal distribution of photons(from G. Dugan)

P=0

P=0.5

P=-0.5

P=±1

P=0.25

P=-0.75 P=-0.25

P=0.75


x-axis: P= scaled perimeter, from -1 to 1

P=-0.5P=-0.75 P=-0.25

vac. chamber cross section

EC formation: “seed” or “primary” electrons

Three main “primary electron” processes:

• photoelectrons

• residual gas ionization

• beam-particle losses

Instead of use = no. of e– generated per proton per meter of beam traversal (units m–1)


P = vac. pressure, T = temperature

ηeff = eff. e– yield per proton-wall collision

n’pl = beam particle loss rate per unit length per beam particle

Nb = bunch population

Yeff = eff. quantum efficiency (e– yield per γ)

σi = ioniz. cross-section per beam particle

LHC EC power deposition

(F. Zimmermann - ECLOUD’02)


Sensitive to model for secondary emission (peak SEY, spectrum, fraction of elastics/rediffused/true secondaries)

EC dissipation after beam extractionsimplest analysis

N

N’2b

• beam has been extracted, or gap between bunches• field-free region, or constant B field • assume monoenergetic blob of electrons• neglect space-charge forces


If not monoenergetic and not along a straight line, then

where K=f(angles)≈1.1–1.2

simulations show that this formula

works to within ~20%

and τ = dissipation time

EC dissipation in PSR after beam extraction

• “Sweeping e– detector”

—measures electrons in the

bulk

—τ ≈ 200 ns

—⇒ δeff ≈ 0.5 if E = 2–4 eV


—⇒ δeff ≈ 0.5 if E = 2–4 eV

—since δeff ≈ δ(0), you infer δ(0)

—well supported by

simulations (see next

slide)(measurements by Macek and Browman)

(PAC03, paper RPPB035)

EC dissipation after beam extraction:PSR simulation

10

100

1000

line density [nC/m]

EC line density beam line density

PSRdissip3

aver. neutralization level

PSR simulationfield-free section, N=5e13

p loss rate=4e-6/m, yield=100 e/pNB: primary e– rate

is 100 x nominal

input SEY:

δ = 1.7

EC line density vs. time (field-free region)


0.01

0.1

1

line density [nC/m]

2.0x10-6

1.81.61.41.21.00.80.60.40.20.0

time [s]

exponential decay(slope=2e-07 s)

δmax = 1.7

δ(0) = 0.4slope = 200 ns

MI: beam neutralization factor vs. Nb


Sensitivity to relative ratios of δδδδe, δδδδr and δδδδts: case of LHC

800

600

aver. power deposition [W/m]

LHC arc dipole simulation: electron-cloud power deposition

photoelectrons: outer edge only

n'e(γ)=6.3e-4 e/m, δmax=2.05

beam signal (arb. units)

Copper

Stainless steel

Copper, true sec. only

power deposition vs. time (LHC arc dipole)

800

600δe+δr = 43%


400

200

0

aver. power deposition [W/m]

1.4x10-6

1.21.00.80.60.40.20.0

time_sm [s]

Copper, true sec. only

Aver. power deposition in 0.5<t<1.2 µs

copper: 11 W/m

st. st.: 152 W/m

copper, TS only: 2.1 W/m.

δe+δr = 10%

400

200

01.060x10

-6

1.0501.0401.0301.020

time_sm [s]

δe+δr = 0

EC in the LHC (contd.)

• Later in 1997 it became apparent, both from CERN and LBNL simulations, that the main concern for the LHC is the energy deposition by the electrons on the vacuum chamber screen

• LHC is first storage ring ever in which this is a potential problem

• Initial estimates for heat load were ~several W/m

—Exceeds the available cooling capacity of the LHC cryogenic system.


—Exceeds the available cooling capacity of the LHC cryogenic system.

—Cryogenic system was designed before the effect was discovered

—At face value, would have to cut Nb or increase sb by factors of ~a few to accommodate heat load

⇒ operational limitation!

• This was the motivation of the “Electron-Cloud Crash Program” at CERN

• And of the LARP involvement in LHC EC research

More history: EC in the LHC

• 1995-96: concerns from the EC on LHC vacuum by O. Gröbner based on ISR experience

• Early 1997: first simulations by F. Zimmermann that included photoelectrons showed a significant ECE; concern about electron energy deposition

• LHC is the 1st proton machine in which synchrotron radiation is significant:

critical energy of photon spectrum: at 7 TeV


—The ECE in the LHC is dominated by secondary electron emission, not by the photoelectrons

critical energy of photon spectrum:

intensity: photons/proton/bend

at 7 TeV

⇒ lots of photoelectrons!

EC formation: beam-induced multipacting (BIM)

• train of short bunches, each of charge Q=NZe, separated by sb

• ∆t = e– chamber traversal time

e−

e−

e−

e−

+ + + + + +

γ or p


• ∆t = e chamber traversal time

• b = chamber radius (or half-height if rectangular)

The parameter defines 3 regimes:

If G = 1 and δeff > 1, EC can grow dramatically (O. Gröbner, ISR; 1977)

PSR Layout

PSR Layout

Skew Quad

Merging Dipole Stripper Foil

C Magnets

Bump Magnets

Matching SectionH- Beam

Final Bend

Extraction Line

H-/H0 Dump Line

ED02ED92

ROED1

Circumference = 90m

Beam energy = 798 MeV


11/17/00 RJM_ICANS-XV.ppt4

ED42

ED52

ED92Beam energy = 798 MeV

Revolution frequency =2.8 MHz

Bunch length ~ 250 ns (~63 m)

Accumulation time ~ 750 ms

~2000 turns

PSR instability

BPM ∆∆∆∆V signal

CM42 (4.2 µµµµC)(Circulating Beam

(R. Macek)


(200 µs/div)

Growth time ~ 75 µµµµs or ~200 turns

High frequency ~ 70 – 200 MHz

Controlled primarily by rf buncher

voltage

(Circulating BeamCurrent)

SPS spectrum(K. Cornelis, ECLOUD02)


electron cloud build-up: theory and data

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