LEVAL, Circuit Judge:
Appellants Rosie Baker and Vance Baker, who are mother and son, were convicted, following a two week jury trial in the United States District Court for the Eastern District of New York (Jacob Mishler, J.), of conspiracy to commit murder with intent to obstruct justice, 18 U.S.C. § 371 [126]*126(Count 1); murder with intent to obstruct justice, 18 U.S.C. § 1512(a)(1) (Count 2); conspiracy to use a firearm during and in relation to a crime of violence, 18 U.S.C. § 924(o) (Count 3); use of a firearm during and in relation to a crime of violence, 18 U.S.C. § 924(c) (Count 4); and, as to Vance Baker only, possession of a firearm with an obliterated serial number, 18 U.S.C. §§ 922(k) & 924(a)(1)(B) (Count 6). In addition, after the trial, appellants each pleaded guilty to one count of conspiracy to defraud the United States, 18 U.S.C. § 286. They were sentenced primarily to life imprisonment for murder, conspiracy to commit murder, and conspiracy to use a firearm; 46 months concurrent for conspiracy to defraud; 60 months consecutive for use of a firearm; 78 months concurrent for possession of a firearm with an obliterated serial number (Vance Baker only); and restitution to the New York City Human Resources Administration in the amount of $1,926,576.
On appeal, appellants contend: (1) the district court improperly invaded the secrecy of the jury’s deliberations and improperly dismissed one of the jurors after the deliberations had begun; and (2) the district court was precluded from sentencing the defendants to the term of life imprisonment authorized for murder to obstruct justice because on the murder count the district court failed to charge the jury on the essential elements of murder under section 1512(a)(1), charging only on the essential elements of the lesser included offense of using physical force under section 1512(b). We reject the first contention, but find merit in the second. We therefore vacate the judgment of conviction and remand.
BACKGROUND
A. The Evidence at Trial
The evidence at trial demonstrated that Vance Baker and Rosie Baker (hereinafter “Vance” and “Rosie”) killed Dr. Daniel Hodge in connection with a long-running Medicaid fraud scheme. Rosie was the director of Long Life Home Health Care, a nonprofit Medicaid services provider located in Brooklyn, New York, that provided home health care aides for Medicaid patients pursuant to a program administered by the New York City Human Resources Administration. Rosie, with the assistance of her son Vance, devised a number of ways to defraud the government by diverting funds from Long Life’s budget, including kick-backs for certain medical procedures, and self-dealing rental and services contracts between Long Life and several entities owned by Rosie. Between 1988 and 1996, Rosie appropriated almost $2 million from Long Life to her personal benefit.
The Bakers’ Medicaid fraud scheme began to encounter difficulties in September 1996, when a dispute arose between Rosie and a contractor named Andre Greene, who was renovating Long Life’s offices. The dispute culminated in a fight between Greene and Vance, in which Vance pointed a handgun with a removed serial number at Greene and fired two or three times, missing him. Greene managed to wrest the gun away from Vance and took it to a nearby police station. Vance then hid in the offices, but Dr. Daniel Hodge, who was associated with Long Life, led the police to Vance, and Vance was arrested.
Dr. Hodge had been involved in the Bakers’ Medicaid fraud scheme. Hodge provided medical services to Long Life employees and paid kick-backs to Rosie. Hodge and Rosie had become romantically involved, and Hodge lived with Rosie. After Hodge identified Vance for the police, however, the relationship between Hodge and Rosie deteriorated. Rosie began to exclude Hodge from Long Life, and [127]*127threatened to kill him. Hodge, meanwhile, demanded that Rosie pay him money allegedly due to him in connection with the Long Life Medicaid scheme. Hodge even filed suit in New York State court to collect these funds. Moreover, Hodge threatened to go to law enforcement authorities with evidence of the Long Life Medicaid fraud.
In May 1997, Rosie sought to enlist Michael Davis — a former boyfriend and sometime-participant in the Long Life scheme — to kill Hodge. Davis declined, at which point Rosie told him that she would get Vance to arrange for the killing.
One month later, on June 26, 1997, a witness saw two men in a late-model car in the parking lot adjacent to an apartment complex where Hodge was staying. A second witness heard gunshots and observed a man fleeing from Hodge’s car, carrying a briefcase. The man leapt over a fence and jumped into the passenger side of a late-model car, which then sped away. A police officer responding to a 911 call discovered Hodge slumped over the front seat of his car. Hodge had been killed by two gunshot wounds to his upper back fired from a revolver at point-blank range.
At trial, the government presented evidence that in a telephone conversation, Vance stated that a computer on which Hodge claimed to have saved incriminating evidence about Long Life had been destroyed. Vance was also overheard asking his father on the telephone whether he had gotten rid of the revolver and the shells. Furthermore, Davis testified that in late June or early July, Rosie told him that she had arranged that Vance would get someone to kill Dr. Hodge, but that Vance had done it himself, making it look like a robbery.
B. The Jury Charge
At the close of evidence, the district judge conferred with counsel on the instructions to be given to the jury. Counsel for the government advised that in the Court’s draft instructions the wrong statutory language was used for the elements of Count 2 of the indictment. While Count 2 charged appellants with murder with intent to obstruct justice, in violation of 18 U.S.C. § 1512(a)(1), the district court’s draft charge on Count 2 instructed the jury on the use of physical force with intent to obstruct justice, in violation of 18 U.S.C. § 1512(b). The district judge responded that the draft charge was in the language that the government itself had requested. The judge nonetheless encouraged the government’s attorney to request a modification. But on recognizing that the court’s draft version came from the government’s own request to charge, the government’s counsel withdrew the objection.
As a result, at the government’s request, the district court’s instructions on the murder count of the indictment (Count 2) set forth the essential elements of the lesser included offense under section 1512(b) (use of physical force to obstruct justice), rather than the essential elements of murder with intent to obstruct justice under section 1512(a)(1). The instructions given to the jury were:
In order to prove the defendant guilty of the charges in Count 2 of the indictment, the government must prove each of the following elements beyond a reasonable doubt.
First, that on or about the date charged, the defendant used physical force or attempted to do so.
Second, that the defendant acted knowingly and with intent to prevent Dr. Daniel Hodge from communicating to a law enforcement officer or judge of the United States information relating [128]*128to the commission or possible commission of a federal offense.
The district judge then defined the use of physical force as follows:
The first element the government must prove beyond a reasonable doubt is that the defendant used physical force or attempted to do so. Physical force is self-explanatory. In this case, the murder of Dr. Daniel Hodge constitutes physical force.
After the government withdrew its objection at the charge conference, neither the government nor the appellants raised any issue related to this charge until after the jury returned its verdict.
C. The Sentencing
Before sentencing, the district judge invited briefing on the question whether the defendants could be sentenced for the crime charged in Count 2 — murder with intent to obstruct justice, in violation of 18 U.S.C. § 1512(a)(1) — notwithstanding that the jury had been charged on the elements of the lesser included offense under 18 U.S.C. § 1512(b): using physical force to obstruct justice. The defendants submitted papers arguing that they could not be sentenced above the ten year maximum term authorized by section 1512(b), and therefore not to the life term authorized by section 1512(a).
The district judge concluded it was proper to sentence the defendants to life terms under section 1512(a)(1). In the judge’s view, it was “clear in the charge that the physical force that the jury was charged on was the murder of Dr. Hodge, putting a gun to his head and pulling [the] trigger twice_[T]he jury clearly understood that the defendants were charged with the crime of murder.” Accordingly, the court sentenced the defendants to a term of life imprisonment on Count 2, as authorized by section 1512(a)(1).
D. The Dismissal of Juror Number 12
Jury deliberations commenced at roughly 2:30 p.m., after the conclusion of the jury instructions. Less then two hours later, the district judge received a note from the jury foreperson, stating:
We have one juror who says that the evidence is not going to change her mind. She has a feeling and that’s it. She refuses to deliberate on any of the counts. She won’t even look at any of the evidence.
After consulting with counsel, the district judge decided to send the jury home for the evening, suspending deliberations until the morning. Before the dismissal of the jury that afternoon, the foreperson sent a second note, which stated:
Please advise us what to do. Now this person is saying that no evidence will change her mind. She also said that she feels we are against her. Now she refuses to even answer us. She said that she tried to get off this case from the beginning [and] she wasn’t allowed.
In a third note, the foreperson requested that the juror be replaced by one of the alternates.1 The third note stated that the juror was “not abiding by [her] oath”; that she “refuses to discuss anything with us”; that “she will not look at any of the evidence”; and that “she sits with her arms folded, refusing to communicate with any of us.” Furthermore, the note stated that “eleven of [the jurors] feel” that they “have invested much time in this case”; that they “have given up many things in [their] lives for more than a two week [129]*129period”; and that they “want to see this case through to the end.”
The district judge and counsel agreed to have the foreperson identify the juror in question and to interview the juror. The foreperson identified the juror as Juror Number 12, Nora Hunt. With counsel and the defendants present, the district judge proceeded to interview Juror Number 12. The juror stated at first that she had agreed to follow the law, that she had considered the evidence in the case to determine whether or not the government had proved guilt beyond a reasonable doubt, and that her conclusion was “based on the evidence.” The district judge asked her whether she felt that “the defendants were unfairly prosecuted,” to which she replied “[i]n some way.” The juror further stated that she had not come to her conclusion before the deliberations had begun, but then inconsistently added that during “the last day” of argument by counsel she had decided that she would not consider the evidence. She contended that she had discussed her views with her fellow jurors and that she had “listened very well,” but acknowledged that she stopped participating in the deliberations somewhere between one half hour and one hour after they had begun, and that there was no possibility of changing her mind.
The district judge made a finding that the juror “decided this case during the summation by counsel,” and refused to participate with the jurors in the exchange of views “a half-hour or an hour after deliberations [began].” The district judge excused the juror based on the findings that she (1) “made up her mind before deliberations started”; (2) “refused to participate in the exchange of views with the other jurors after the deliberations started”; and (3) “said that if [the district court] sent her back [to continue deliberations] she would refuse to participate.”
After the dismissal of Juror Number 12, the jury resumed its deliberations with eleven jurors. Approximately three hours later, the jury returned verdicts of guilty on all counts.
DISCUSSION
The defendants contend (1) that the district court’s dismissal of Juror Number 12 after the beginning of deliberations was improper under United States v. Thomas, 116 F.3d 606 (2d Cir.1997); and (2) that the district court improperly sentenced them under Count 2 for murder with intent to obstruct justice, because the jury had not been instructed on that offense but the lesser included offense of using physical force to obstruct justice.
A. The Dismissal of Juror Number 12
Federal Rule of Criminal Procedure 23(b) provides that “if the court finds it necessary to excuse a juror for just cause after the jury has retired to consider its verdict, in the discretion of the court a valid verdict may be returned by the remaining 11 jurors.” It is well-settled that the question whether “ ‘just cause’ exists to excuse a juror is a matter within the discretion of the district court.” United States v. Casamento, 887 F.2d 1141, 1187 (2d Cir.1989); see also United States v. Ruggiero, 928 F.2d 1289, 1300 (2d Cir.1991). Indeed, we have emphasized that the questions “[w]hether and to what extent a juror should be questioned regarding the circumstances of a need to be excused [are] also within the trial judge’s sound discretion.” United States v. Reese, 33 F.3d 166, 173 (2d Cir.1994); see also United States v. Register, 182 F.3d 820, 840 (11th Cir.1999) (“[T]he court also enjoys substantial discretion in choosing the investigative procedure to be used in checking for juror misconduct.” (internal quotation marks omitted)), cert, denied, [130]*130530 U.S. 1250, 120 S.Ct. 2703, 147 L.Ed.2d 973 (2000). As Judge Newman explained in United States v. Stratton, 779 F.2d 820, 832 (2d Cir.1985), the “just cause” standard of Rule 23(b) “encompass[es] a variety of ... problems that may arise during jury deliberations, confronting the trial judge with the need to exercise sound discretion as to the procedure to be followed at a particularly sensitive stage of the trial.”
In our view, the district judge acted within its discretion in carefully interviewing Juror Number 12; in excusing her on the grounds that she had improperly made up her mind prior to the beginning of deliberations and refused to engage in deliberations with the other jurors; and in allowing deliberations to proceed with eleven jurors.
It is well-settled that jurors have a duty to deliberate. Jurors “should examine the question submitted with candor, and with a proper regard and deference to the opinions of each other.” Allen v. United States, 164 U.S. 492, 501, 17 S.Ct. 154, 41 L.Ed. 528 (1896). “[I]t [is] their duty to decide the case if they c[an] conscientiously do so,” and thus “they should listen, with a disposition to be convinced, to each other’s arguments.” Id. As the Supreme Court explained over a century ago,
The very object of the jury system is to secure unanimity by a comparison of views, and by arguments among the jurors themselves. It certainly cannot be the law that each juror should not listen with deference to the arguments, and with a distrust of his own judgment, if he finds a large majority of the jury taking a different view of the case from what he does himself. It cannot be that each juror should go to the jury room with a blind determination that the verdict shall represent his opinion of the case at that moment, or that he should close his ears to the arguments of men who are equally honest and intelligent as himself.
Id. at 501-02, 17 S.Ct. 154. To be sure, we have held that “jurors have no burden or obligation” actually to “convince one another that one outcome is preferable to another.” Smalls v. Batista, 191 F.3d 272, 280 (2d Cir.1999). But the Supreme Court has made clear that it is appropriate to charge a jury that “when [it] enter[s] the jury room it is [the jurors’] duty to consult with one another to consider each other’s views and to discuss the evidence with the objective of reaching a just verdict if [the jurors] can do so.” Lowenfield v. Phelps, 484 U.S. 231, 235, 241, 108 S.Ct. 546, 98 L.Ed.2d 568 (1988); see also United States v. Corcione, 592 F.2d 111, 117 & n. 5 (2d Cir.1979) (approving instruction that told jurors “it is your duty as jurors to consult with one another and to deliberate with a view to reaching agreement”); United States v. Green, 523 F.2d 229, 234, 236 (2d Cir.1975) (approving jury charge that instructed jurors “to listen to each other’s arguments with an open mind to see if they could agree in ‘good conscience’ ”).
A district court’s finding on the question whether a juror has impermissibly refused to participate in the deliberation process is a finding of fact to which appropriate deference is due. See United States v. Barone, 114 F.3d 1284, 1307 (1st Cir.1997); Ruggiero, 928 F.2d at 1300 (noting reluctance to overturn the conclusion of the trial judge as to whether just cause existed to dismiss a juror under Rule 23(b) given that the trial judge’s conclusion is “based in large measure upon personal observations [of the juror’s behavior] that cannot be captured on a paper record”); see also United States v. Torres, 128 F.3d 38, 44 (2d Cir.1997) (observing in the context of voir dire of prospective jurors that an appellate court cannot “easily [131]*131second guess the conclusions of the deci-sionmaker who heard and observed the witnesses”)- Here, the district court properly instructed the jury prior to the beginning of deliberations that it was their “duty, as jurors, to consult with one another and to deliberate with a view towards reaching an agreement.” Soon after the deliberations began, the district court carefully interviewed the juror in question and determined on the basis of her own statements, as well as on the basis of the notes sent by the jury foreperson, that one half hour after the jury retired the juror refused to participate in deliberations. The record amply supports the district judge’s determination. As we have noted already, the juror herself told the district judge that she had made up her mind in advance of the beginning of deliberations, and within a half hour determined that she would not deliberate with her fellow jurors. The notes from the other eleven jurors strongly supported these statements. The jurors informed the court that Juror Number 12 “refuses to deliberate on any of the counts” and would not “even look at any of the evidence.” Moreover, the jurors informed the court that Juror Number 12 had made an issue of the court’s refusal to excuse her from jury service during jury selection. Accordingly, the district judge reasonably concluded, based on strong evidence, that Juror Number 12 had improperly decided not to participate in deliberations and had improperly made up her mind prior to those deliberations. Cf. Barone, 114 F.3d at 1307 (“[T]he district court reasonably concluded that [the juror] could not continue to deliberate as a fair and impartial juror.... ”).
Appellants urge us to reverse the judgment in the light of United States v. Thomas, 116 F.3d 606, 618-20 (2d Cir.1997). In Thomas, the trial judge removed a juror because the juror engaged in nullification: “purposefully disregarding the court’s instructions on the law — in effect ... intending] to acquit the defendants regardless of the evidence of their guilt.” Id. at 608. We ruled that, in such cases, where the issue of the juror’s removal arises from his refusal to vote in accordance with the evidence, the court must take pains to avoid the risk that the juror is in fact being removed “because he is unpersuaded by the Government’s case.” Id. at 621. We ruled that such efforts to remove a juror must be denied “if the record evidence discloses any possibility that a complaint [by the other jurors] ... stems from the juror’s view of the sufficiency of the government’s evidence.” Id. at 622 (internal quotation marks omitted).
The difference between this case and Thomas is subtle, but important. The stringent rule announced in Thomas applies to removal of a juror by reason of the juror’s determination to vote without regard to the evidence. It is based on the difficulty in detecting the difference between a juror’s illegal act of nullification, by deciding to vote against the weight of the evidence, and the juror’s failure to be convinced of the defendant’s guilt. In order to guarantee the defendant the right to a unanimous jury and to protect the defendant against efforts to achieve unanimity by removing jurors who disagree with the majority, Thomas ruled that a juror may not be removed for refusal to allow his or her decision to be governed by the evidence unless it is clear the motivation for the removal is not in fact the juror’s nonconforming view of the sufficiency of the evidence to convict.
Here, the juror was not removed for her nonconforming view of the evidence. She was removed for her admitted refusal to perform her duty as a juror by deliberating together with the other jurors. The complaints of the other jurors, although [132]*132including reference to her statement that “the evidence was not going to change her mind,” focused on her “refus[al] to deliberate on any of the counts.” The juror acknowledged in the court’s interview that within a half hour she stopped participating in the deliberations and would not change her mind. The district court’s reasons for dismissing were that, as she admitted, she had made up her mind before deliberations started, she refused to participate in the exchange of views with other jurors, and she said she would continue to refuse to participate. Thus, the basis of removal in this case, unlike Thomas, was not the juror’s refusal to base her vote on the evidence — a reason that is dangerously difficult to distinguish from a refusal to view the evidence as viewed by the other jurors — but rather her refusal to engage in the process of jury deliberation. The Thomas opinion acknowledged that its decision might well have been different, and a different rule would apply, if the juror’s disruptive behavior, rather than his determination to vote for acquittal, had been the reason for the removal. See Thomas, 116 F.3d at 624 (“[W]e do not suggest ... that a juror’s disruptive behavior ... could not serve as a ground for dismissal.... ”).
Because the district court’s reason for dismissing the juror was that the juror refused to participate in deliberations as required by her obligations as a juror, the rule of Thomas does not apply, and the trial court was within its discretion in removing the juror.
Many of the observations of the Thomas opinion are nonetheless pertinent. It is far better, when possible, for the court while inquiring into a juror’s possible misconduct to make efforts to avoid learning the jurors’ views. See Thomas, 116 F.3d at 620 (noting that “the very act of judicial investigation can at times be expected to foment discord among the jurors”). If the judge has not learned the views of the misbehaving juror, the court’s remedial action is less likely to be subject to challenge as an effort on the court’s part to shape the verdict. Furthermore, if the court has not been told the jurors’ views, or how the jury is divided, the jurors will not interpret the court’s intervention as seeking to influence the jury in any particular direction. The district judge recognized this and thus at first instructed Juror Number 12, “[Djon’t tell me what [your] decision was.” Nonetheless, in the next question, the district court asked, “Do you feel that the defendants were unfairly prosecuted here?” — a question that sent the inquiry unnecessarily in the direction of the juror’s view of the merits, which were not revealed by the foreperson’s notes.
We believe that question was an error of judgment. It would have been a safer and better course to concentrate on the issue of the juror’s willingness or unwillingness to deliberate, avoiding insofar as possible the issue of the various jurors’ views of the merits, and the proportions of any split among the jurors. Nonetheless, we recognize that it is often difficult to steer such interviews clear of revealing the jurors’ views. Although it would have been preferable to avoid such revelations, we find no abuse of discretion in the conduct of the juror interview or in the decision to remove the juror based on her well-established, and admitted, refusal, beginning only one half hour after the start of deliberations, to engage in the deliberative process.
B. The Sentencing
“[Tjhe Due Process Clause protects the accused against conviction except upon proof beyond a reasonable doubt of every fact necessary to constitute the crime with which he is charged.” In re [133]*133Winship, 397 U.S. 358, 364, 90 S.Ct. 1068, 25 L.Ed.2d 368 (1970). The Sixth Amendment guarantees the accused a trial by jury. See U.S. Const, amend. 6. “Taken together, these rights indisputably entitle a criminal defendant to ‘a jury determination that [he] is guilty of every element of the crime with which he is charged, beyond a reasonable doubt.’ ” Apprendi v. New Jersey, 530 U.S. 466, 477, 120 S.Ct. 2348, 147 L.Ed.2d 435 (2000) (quoting United States v. Gaudin, 515 U.S. 506, 510, 115 S.Ct. 2310, 132 L.Ed.2d 444 (1995)).
The jury deliberating on Count 2 did not consider the essential elements of the crime of murder with intent to obstruct justice. It had been instructed to convict the defendants if it found that they (1) had “used physical force or attempted to do so,” and (2) had “acted knowingly and with intent to prevent Dr. Daniel Hodge from communicating to a law enforcement officer or judge of the United States information relating to the commission or possible commission of a federal offense.” This instruction allowed the jury to convict the defendants on the murder count, without finding that the defendants had murdered, or even killed, Dr. Hodge. Such a verdict cannot sustain a conviction for the offense of murder to obstruct justice, in violation of 18 U.S.C. § 1512(a)(1).
The government contends that even though the district judge gave the wrong charge, the error was harmless because the jury “was required to find that the Bakers caused the death of Dr. Hodge.” In the alternative, the government argues that, taken as a whole, the court’s instructions to the jury were “permeated” by the murder of Dr. Hodge and thus “made it abundantly clear that to convict on Count 2, the jury had to find that the defendants participated in the murder of Dr. Hodge.” In particular, the government observes that the district court read to the jury the correct section of the statute, which provides that “ ‘whoever kills or attempts to kill another person’ ... shall be guilty of a crime” (quoting 18 U.S.C. § 1512(a)(1)), and defined the use of physical force for the jury by stating that “[pjhysical force is self-explanatory. In this case, the murder of Dr. Daniel Hodge constitutes physical force.”
In our view, taking the district court’s instructions as a whole, the jury was not required to find beyond reasonable doubt that the defendants killed Dr. Hodge in order to find them guilty on Count 2. It is unquestionably true that the court’s instructions frequently referred to the murder of Dr. Hodge. Nonetheless, with respect to Count 2, when the jury was told what the government must prove to convict, it was told that the government must prove that “the defendant used physical force,” not that the defendant murdered anyone. Nor does it help the government that the court went on to say that “physical force is self-explanatory. In this case, the murder of Dr. Daniel Hodge constitutes physical force.” No doubt it does. But the instruction does not require the jury to find a murder in order to find that the defendants used physical force. The “self-explanatory” meaning of physical force is not limited to murder. Furthermore, the instruction did not include a definition of murder.
Because the district court’s instructions permitted the jury to convict defendants on Count 2 of the indictment without finding that the government had proved beyond a reasonable doubt every element of the crime of murder, the appellants cannot be sentenced on that count for murder. The sentence on Count 2 is therefore vacated.
Because of the error in sentencing on Count 2, we vacate the sentences on all [134]*134counts, allowing the district judge maximum flexibility in refashioning the sentences. We note that the district court imposed life sentences not only on Count 2, but also on Counts 1 and 3. On resen-tencing, the district court should consider, with the advice of counsel, whether the statutory maxima under 18 U.S.C. §§ 371 & 924(o) authorize life sentences on Counts 1 and 3.
CONCLUSION
The judgment of conviction is vacated and the case remanded to the district court for resentencing.